I don't know how you focus on and maintain such clarity over so many domains. It's like watching a miracle in real time. May the force ever be with you!
We need sociology of theoretical physics as well. We have these untestable theories like string theory for example. Unfortunately these fields suck up a lot of research funding, which goes at the cost of other less popular fields, which could be more promising in the end.
@Chris Django I agree completely. Theoretical physics, like pure mathematics, has been able to shield itself from the gaze of sociologists due to the inherent difficulties and other-worldliness of the subject. While there is pretty vigorous debate in certain physics circles re the pros and cons of string theories, it would be great to see a rigorous analysis of just how the power structures and social pressures have contributed to the elevation of this topic beyond what one might naively expect, given its disconnect from observations and predictions.
@Daniel G Whether or not 'the majority' of sociology is hand-waving these days, I have no idea myself. However, there *is* actual science happening under the label of 'sociology'. The closer to scientific method it follows, the more useful it will be, of course. Even if 90% (or more) of sociology today is crap, Sturgeon's Law applies: "90% of everything is crap." There are legit scientifically-minded sociologists working today, and if any of them focused some attention on pure mathematics, I'm sure they'd be able to find something interesting out about it.
@@ThePallidor Do you mean einstein ripping poincare, lorenz's and hilbert's works as whittaker has established? By the way minkowski's recasting of relativity is the superior theory.
@@ThePallidor I know, it's a bunch of different theories that kind of work in their own right. Althought chromodynamics show some promise as far as tentative theories go.
The main problem of mathematical physics is the paradigm of amputation of introspection from empirism, and as consequence applied math of physics invading and taking over pure math, leading to deep logical and empirical problems of inconsistency as well as fragmented academic sociology. A rather strong hypothesis is that many of the deep problems of mathematical physics (unified theory, mind-body problem, measurement problem) are artifacts of unsatisfactory underlying theory of mathematics. Intuitionist constructionism could offer more coherent and empirically sound approaches to get rid of the failing mathematical artifacts caused by Formalist constructions. Mother Nature resists obeying any and all arbitrary axiomatics implied by the current language game of mathematical physics, though they can serve heuristic function in the evolution.
Such a fantastic video - I am so interested to see where to you take this! Like others, I was going to note that this is also such a big issue for theoretical physics and, though difficult, is such an important discussion to have. I think it really helps to view this video in the context of the mathematics and issues presented in your Math Foundations Series - it felt very strange to me to have to go right back to the very beginning of the Math Foundation series and work through it all - but I must say its completely changed how I think about and undertake mathematics. Its also provided a really natural context for your discussion here. Thank you so much for all your amazing work and please keep it up!
I think Norman notices that the high priests, the unquestionable authorities, of this present age are telling lies. He wonders why the populace can't see or won't accept that modern math is built on lies and deception. It can be demonstrated that it is all lies. Norman can not see that Rabbinites have controlled all social inputs for more than 100 years. It's brainwashing by mass media and education. Without fake math space and evolution disappear. Without Lorentz transforms we live in a motionless world and "outer space" disappears. Modern society believes Rabbinite Hellywood and not reasonable conclusions.
The fundamental theorem of algebra is a statement on the existence of zeros; it makes no assertions on the difficulty of finding them, or even if a finite closed form representation can be found. Requiring an infinite sequence of digits to represent a zero doesn't change the fact that it exists (and can be approximated).
Yes, but I don't think this is the issue he is raising. The FTA is true in our current formulation of mathematics; this is undeniable. The issue, as I see it, is this: the development of nonconstructive theories may be leading the mathematical community astray in the sense of losing out on many interesting applied problems/settings.
@@AlekFrohlich There is no universal ‘interesting’ object that mathematicians are being lead astray from. What people find interesting is purely subjective and if people find constructive or non-constructive mathematics interesting (or both!) then they will study it. Neither are ‘wrong’ or any ‘better’ by any objective means.
Throughout the history of mathematics has mathematics slowly secured itself a very powerful and influential position in society. This power and influence is ever increasing with the rise of artificial intelligence as it is structured upon and simultaneously bounded by (discrete) mathematics. The immense success and influence of mathematics has resulted into vast parts of today’s society becoming structured on mathematics and is ever widely present. However, this presence and influential position of mathematics is rarely identified and acknowledged, especially by mathematicians as their main objective is often far from understanding its relation to society and reality. (This is of course not the case for philosophers or historians of mathematics as many of them are mathematicians themselves.) This lack of acknowledgement is however not that surprising since the image of mathematics is often distorted and there being a lack of understanding of what mathematics is really about and how it functions and develops. One can simply observe this production of this image distortion in high school class rooms where many students often manifest to their teacher or among each other their confusion about what mathematics is about (“why is 1+1=2?”) and what it is useful for (“like we would ever use this in real life”). When these students approach their teacher of mathematics about this issue, the teacher often does not really answer their questions, tries to answer it but fails to satisfy the philosophical demands of the students, or is awkward or obscure in its explanation as if it is embarrassed of not knowing the answer or that the teacher might belief that the role of being a mathematicians comes with understanding its ontology and is confronted with a failure to fulfil this role. Nevertheless, the majority of those that are confused about mathematics will force themselves through the mathematical literature because its sufficient completion is a requirement of achieving a high school diploma. For many high school students the sufficient completion of mathematics thus functions more as a status symbol rather than equipping oneself with a philosophical account of reality. Gaining a sufficient grade on a math test is rather a reflection of obedience towards the societal system rather than the acquisition of “objective” knowledge. It is the systematic imposition of a logic and knowledge that society or those in power desire its citizens to perceive as a dominant form of logic and knowledge. The goal of teaching mathematics in high school is not to question its “objectivity”, as the requirement of becoming a high school math teacher is not to be a philosopher of mathematics but to be an individual who is capable of transferring a particular form of logic and knowledge onto students that by law are obliged to follow the classes. Attempts to become absent in math classes will not only be directly punished by the school and government but also often by one’s social environment such as family and the future career market. This interaction between the confused high school students and math teachers in which the young student is confused about the ontology of mathematics and the teacher (regardless of believing to known or not know what mathematics is about) educating this philosophical account to the students reflects the underlying forces of society that unknowingly and unintentionally has guided or pressured these individuals into this exact interaction. This interaction can be observed over and over again each year throughout many high schools in various countries and cultures. This leads society to reproduce a system of teaching that causes this “dysfunctional” unintended pattern to repeat itself. Exactly these manifestations portray not only the power relations that are operating across all of society but also the unstable parts of society that many are oblivious to, this while it could be of vital importance to understand these interactions for the sake of the persistence of society, humankind, or even earth itself. What can be seen by many as a minor frustrating interaction for the student and a potentially embarrassing one for the teacher and something that would not seem to reflect the instability of society, can be a reflection of the failure or incomplete or in process exertion of power onto this young generation of students. The ultimate form of power would be to succeed in eliminating this above described social interaction such that these philosophical questions about mathematics would be eliminated from one’s mind. And this is done exactly within society by accentuating the societal usefulness of mathematics and concealing its fictionalism or social construction by highly institutionalizing itself throughout history. And when something becomes institutionalized it recedes from consciousness. Meaning, that if x becomes institutionalized, then x becomes routine and ordinary such that x becomes part of the background for those who are part of an institution.
I completely agree, math turned from being an applied tool of discovering how our world works, into being the goal in and of itself. The massive amount of notation and over specialization, and the laziness of mathematicians when they write a paper, instead of reducing problems so that everything is written in simple algebraic terms, they use the over bloated notation that could only be understood by a handful of people. The worst part is that this kind of thing is becoming the norm, even if someone comes up with some incredible discovery, it would be drowned in the sea of paper pushers in most of academia. Hearing a professor express these thoughts leaves me with some hope though.
It's interesting that intuitionistic logic was once controversial because it seemed overly restrictive. Then it turned out to play an important role in the Curry-Howard correspondence, which sheds a lot of light on the shared foundations of mathematics, logic, and computer science.
I think it's fair to say that it's still controversial. Barely any mathematician works under the framework, apart from people dealing with internal logic of topoi (and apart from logicians and computer scientists, when we count them to mathematicians). Same goes for modal logics. It's worth pointing out that by double-negation-translation, every classical theorem is equivalent to a constructively provable statement that is classically equivalent to it. But, of course, finding the constructive proves is harder than finding the classical ones. As for your second sentence, I'm not sure if shining a light on foundations of mathematics is of interest to many mathematicians.
@@NikolajKuntner The problem is that you're talking about a field that very few people are familiar with, let alone whether your interpretations match an interpretation I would give to the results you're talking about. And I've read a chunk of simpson's subsystems of second order arithmetic, so I'm not a complete ignoramus at it. Now as far as mathematicians go, intuition is the rule of the game, developing it is exactly what is called mathematical maturity, making someone get familiar enough with mathematical material as to intuitively understand it. But beyond that, the training of mathematicians reflects it, nobody goes with the bourbaki tomes or some other fully rigorous development. Even riemannian geometry, we do it in real and not banach spaces for precisely that reason, when it's no harder to do it in banach spaces.
Nikolaj-K spitballing there are 10^5 mathematicians and 10^7 cs in the world. So if only 1% are mathematicians the numbers are on par. I think this is a conservative estimate re cs. At a minimum we can say it has divided math community into leibnitzian Calculemus half and the other half
Thank you for this video professor Wildberger, really looking forward to this series. The social dynamics of our discipline, which I feel are totally neglected to this day, have many far-reaching consequences that affect us negatively. I'm confident that initiating this discussion will make visible many of the toxic attitudes, ideologies, and ways of interaction that occur within the culture of pure mathematics.
Even though I don't share your philosophical views, I do think that your channel is interesting and your discussions are worthwhile. That said, I'd like to discuss my interpretation of this video. -- There is no logical problem with modern (post zfc) mathematics in the sense of consistency; the problem is that theories built upon it may not be the best in terms of usefulness for investigating other scientific fields such as physics, computer science, and engineering. This combined with the fact that maths is THE playground upon which formal theories are built results in a slow down of scientific progress, which may be seen as a far-reaching consequence of the adoption of certain philosophical ideas in maths. I mean, maths, philosophy and physics were born together; it is kind of strange to separate them so fiercely as we do today. In summary, some philosophical ideas impregnated in maths by sociological reasons are leading us, as a community, astray. We could be better spending our research efforts elsewhere for the sake of scientific progress. -- What do you think of this?
Economics is not yet a science. . Any field of enquiry goes through four stages. . You have the observational stage, where certain regularities are noted, but not clearly understood. In Economics this is represented by The Wealth of Nations by Adam Smith. Then follows the religious stage, where the feelings are written down in 'holy books'. In Economics you have, for example, Das Kapital and The Communist Manifesto'. These are social religious books. Then follows the philosophical stage, whereby the context is constructed in the right way. In Economics this stage is represented by 'Human Action' of Ludwig von Mises. And, lastly, you have the scientific stage. One that I am working on. Only if a principle has been established to make a distinction between useful and useless statements, _totally belonging to the science itself,_ the field becomes a science. . I assert, that the science of economic has to deal with both utility and value. Two concepts that are in my development purely separate. That is, you can have something having tremendous utility, but no value at all. Like the air we breathe. And you can have something having tremendous value, but no utility at all. A certain amount of money in the form of a cryptocurrency is an example of that. . This means, that value _cannot be the result of utility + scarcity!_ . I hope to publish my first book on utility and value this month.
@@peterosudar1636 I will. . Thanks for your interest. . The content of the book is already finished. I only have to format it, so that it can be sold by Amazon.
Can we expect to see more episodes of this series soon? For me this story as in some ways as interesting as the actual mathematics the 20th century has produced.
Prof. Wildberger, someone in the comments mentioned there should be a sociology of theoretical physics as well. That comment reminded me of the theoretical physicist and fellow TH-camr, Sabine Hossenfelder, who has many interesting videos, several of which include critiques (of the sociology, without calling it as such) of theoretical physics. I would highly recommend at least checking out some of her videos (they tend to be less than 10 min each, but also tend to be very high signal-to-noise ratio, so the time-investment would probably be worth it), because, IMHO, she's kind of an intellectual 'bird of a feather' in many ways with your own thinking (but mainly on theoretical physics rather than mathematics). Just search for 'Sabine Hossenfelder', should be easy to find. Who knows? If you find her thoughts/critiques interesting, perhaps you two could collaborate on an interview or something. Just an idea! She has done a few interviews, including outside of strictly theoretical physics (for example, on the issue of 'the reproducibility crisis' in psychology and other science sub-fields), and they tend to be very enlightening, IMHO. So, she might also be interested in your perspective as an internal critic/skeptic within mathematics as well. After all, fundamentally, *there is* a huge overlap between theoretical physics and mathematics. Very interesting to see you delving into the broader role of mathematics within society, and as part of society! I applaud the exploration, and look forward to more! 😃 Cheers!
Everyone knows that the more modern theories of physics have not been established, you can even find it written on intro physics books. The problem is the researchers, not the field of physics.
@ ZetaOfS In mathematics, questions of "existence" are often red herrings. If we stick to things we can write down (the mathematical equivalent of the physicists requirement that physically meaningful objects/concepts are observable) then the difficulty disappears.
@@njwildberger This is a 'beating around the bush' answer. . I think there are two approaches to mathematics. The _analytic_ approach, and the _synthetic_ approach. The analytic approach is bottom-up, and the synthetic apoprocah is top-down. In the analytic approach, everything has to be constructible and constructed in order to exist. Also, you have to start with something you _define_ to exist. You cannot, for example, in the analytic approach, just state that you begin with a continuous number line. You have to define what you consider to exist. You can, for example, not just state that there is a point on the number line corresponding to the square root of two. . What you _can_ do is just introduce a mathematical object, having the property that the square of that object is equal to 2. And then you can say that there are two of them: Sqrt(2) and -Sqrt(2), and then extend the field of rational numbers, so that, from that point on, you can consider Sqrt(2) to exist, and derive all sorts of mathematical constructs from that. . In the _synthetic_ approach, you begin with a set of axioms. These are implicit definitions of all things you _exclude!_ . I always use the following example to clarify what I mean by this. . Suppose my grandfather makes a photo of an animal. But, alas, due to some defect the photo is hazy. However, you _can_ make a number of certain statements about what the photo shows. You can see that the animal, whatever it is, has four legs. You can see that it is about 0.5 meters. You can see that it has a color that becomes grey in a black and white picture. . You can then introduce postulates, which assert. Axiom 1: The animal has 4 legs. Axiom 2: The animal is 0.5 meters long. Axiom 3: The animal has a definite color. . From this, you can construct _several models!_ Model 1: the animal is a cat. Model 2: the animal is a dog. Model 3: the animal is an anteater . From the axioms alone you cannot say which of the three models is the correct one. The axioms do not provide a thorough enough exclusion to identify the one animal my grandfather has photographed. The same can be said about most mathematically axiomatic systems. They consist of certain symbols, certain statements, axioms, that tell what the mathematical objects have to satisfy, and then some manipulation rules that can be used to derive consequences from the axioms, which then apply to _all models_ that satisfy the axioms. . This I call _synthetic mathematics,_ because this approach _synthesizes many models_ into _one_ overarching system of insights, all of which are satisfied by all models satisfying the axioms. . _Almost every_ synthetic system of mathematics (except those that are categorical) suffers from this defect. And it has consequences for physics. . In physics, for example, you have quantum mechanics. Quantum mechanics, as it is taught now, begins with a number of postulates. This is a synthetic approach. And it least immediately to a huge problem. Because of this, we have _several models_ of quantum mechanics, also called interpretations. You have the Kopenhagen interpretation, the Many Worlds interpretation, the Ensemble interpretation, and several others. . Since _all_ of them satisfy the postulates of quantum mechanics, quantum mechanics as such _is not understood_ in the analytical sense. _That_ is, what Bohr meant when he said that when you think you understand quantum mechanics, you don't. And Feynman, likewise, meant when he said that _nobody_ understands quantum mechanics. . We _do understand_ quantum mechanics synthetically. But we _do not understand_ quantum mechanics analytically. Our _understanding_ of quantum mechanics, as a description of the world, _is incomplete! That is the problem with quantum mechanics!_ . The same, by the way, can be said about the special theory of relativity. This also is built upon two postulates. The Galilean Principle, and the constancy of the velocity of light statement. . Einstein himself constructed a model of the special theory of relativity. It is the one that is taught in all textbooks. . Yet, I myself have constructed a second model of the special theory of relativity, just to see whether it can be done. . So, questions of existence _are not red herrings!_ The very problem of existence points at two fundamentally different approaches to mathematics. Brouwer limited himself to only those things that are constructible. It is a bottom-up approach. But the other form of mathematics, promoted by Hilbert, is also a legitimate form of mathematics. It works with exclusions. It is a top-down approach. . As a mathematician, we do not have to make a choice between analytic or synthetic mathematics. We only have to recognize that there are two forms of mathematics. And sometimes, these two forms come together. This happens for example in propositional logic, where you first have the analytic approach, represented by truth tables. And then you have the synthetic approach, given by axioms defining the concept of tautology. And then you can prove that every tautology in the analytic sense is also a tautology in the synthetic sense and vice versa. Whenever that happens, the theory is called _complete_ or also _categorical!_ . Almost all theories of mathematics are either analytic or synthetic. Some are both. And from some we can prove that they can't be extended in such a way that they become both analytic and synthetic. We know, for example, that we cannot make a complete _synthetic_ theory of natural numbers. That is Gödel's theorem. But we _can_ make a complete _analytic_ theory of numbers, by using the Peano postulates.
@@konradswart4069 I found your comment very interesting and worth reading. I agree with much of what you say in it. However, I think perhaps you are being a little too quick to dismiss Wildberger's response to the original question, and what it is really pointing at. In both of your types of mathematics, there *still* remains the *practical* question of: Can you write this down? (And, following up an affirmative answer by *actually* writing it down.) Whether synthetic or analytic, there is the potential to face serious difficulties in actually writing down an actual answer to any given problem. In either case, there may be problems for which writing down an answer is not just 'difficult', but fundamentally impossible, given the constraints of the physical universe. I believe what Wildberger is suggesting (based on my personal understanding of his videos over the years -- so, I may be wrong on this, but hopefully not) is that if you shift your focus onto that 'writing it down' aspect, the question of the *existence* of various concepts becomes a moot point. For example, one could define a number sqrt2 into existence, with definition (sqrt2)^2 = 2, and then proceed to use this number (just focusing on the positive one, for sake of simplicity of discussion) in all sorts of perfectly logical algebra, all *written down* in terms of the regular mathematical symbols, plus the special symbol "sqrt2" to refer to this specially defined number. Wildberger would have absolutely no problem with that. And so, in that case, under those conditions, he'd be perfectly willing to accept that 'a square root of 2' *exists* within that framework. But he wouldn't care so much about the word 'exist', he'd just be focused on "Well, it can be *written down,* so there's no issue of 'existence' to even worry about." But in the case of developing a general mathematics of 'Real numbers' which includes 'the square roots' of all non-negative Reals, as well as all the transcendentals and all other numbers typically included in the Reals, *then* you run into the issue of being able to write them all down *and* perform general mathematical operations on them at the same time (as is essentially 'promised' by the definition in the first place). It's here that Wildberger would start poking at holes in the (various, logically conflicting or incomplete) definitions of the Reals and start asking, "Can you actually write down the positive square root of 2?" One might reply, "Yes, of course, as √2." To which he will poke further, "Okay, now how do I add Pi or e to that? (As is 'promised' by the def'n of Reals as continued decimals, for example.)" It's here that you run into the immediate *practical problem* with actually *writing down* the answer. You can never actually finish that, as a continued decimal, and so have to resort to just writing down the expression "√2 + Pi". And then Wildberger will ask about "Okay, so what is the z-th decimal digit of √2 + Pi?" (The number 'z' is defined by Wildberger in a previous video (see "Famous Math Problems 1" in his videos list). Needless to say, it is an incredibly huge number. Even using a clever formula that can isolate specific digits of Pi, for example, would be physically impossible to apply with an input parameter as big as z.) So, does that z-th digit 'exist'? Well, Wildberger doesn't really care, because you can't even get past the hurdle of determining what that digit actually is in the first place. It's impossible to *actually* write it down (in this physical universe; and as a result of a logically accurate calculation). Thus, my whole point is simply that Wildberger's original answer is not such a 'beating around the bush' answer as you have made it out to be. It's quite a bit more serious and deeper than that. Just thought I'd try my hand at elaborating on that point. Cheers!
@@robharwood3538 Thank you for your extensive response. . The very question: 'can you write it down?' already betrays an _analytic bias!_ 'To be able to write something down' is the same as 'to be able to construct it'. . My point in synthetic mathematics is, that the concept of _existence_ does not occur. You just sum up some criteria, which are the axioms. By writing the axioms you implicitly are saying that you _exclude everything that does not correspond to the axioms!_ So you are _excluding,_ and then you use the rules of the axioms + logic to see, what else is excluded. . Therefore writing down does not matter. . By using synthetic mathematics, you can deal with the infinite, which cannot be done with analytic mathematics. Infinite is simply _not finite!_ Whenever something is infinite, it is incomplete. If I say, that the set of positive natural numbers is infinite, I am saying that it is _not finite._ To be precise, what I am saying is that the set of positive integers _lacks_ a property. Namely that there is a _largest_ number. So an infinite set is not a definite object, but an _incomplete_ object. And therefore it is not an object. The idea of an infinite set contains many instances, all of which are finite, and just 'very large'. . Yet, you _can_ work with incomplete sets. What they define are not mental objects, but mental exclusions. . The only thing you can say about an infinite set is that it is _not_ the case that it does _not_ exist. Just like in Popper's theory that you cannot say that a statement is true. You can only say that a statement is not false. . My approach is not to see mathematics as one thing. I rather see it as a plurality of constructs. If you can capture some regularity into a set of tokens, a way to construct these tokens into formulas, then formulate axioms that exclude, and then add operations to this whole, you get a mathematics. That is, _only_ if it satisfies the rules of logic. . In my eyes, logic fulfils the same role in mathematics as the experiment fulfills in physics (and chemistry). We can make distinctions between useful and useless theories. Logic is the test of theories of mathematics. There is not just one mathematics. There are many mathematics. The experiment is the test of theories of physics. . Sqrt(2) does not have a finite representation in the decimal system. Which just means, that the decimal system is not powerful enough to construct finite representations of Sqrt(2). In fact, the decimal system cannot even make finite representations of 1/3, 1/7, 20/21, or, in general, a fraction with a numerator that does not consists of multiples of 2, of 5, or combinations of multiples of 2 and 5. . Just to demonstrate how powerful thinking in terms of exclusion is. The paradox of Russell can be eliminated in a simple way, by specifying what a definition must be. . A definition is not allowed to contain a negative. If it does, it is not a definition of something that can be constructed. So if somebody says he defines an infinite set, he is talking nonsense, because the definition of any infinite set contains a negation. (Not a largest member, not a fractional representation, etc..) . So, to return to the Paradox of Russell. He begins with a set that does _not_ contain itself as a member. That is _not_ a definition, because it contains a negative. Therefore the whole paradox of Russell is just confused thinking. . To give a simple example: if I say that some animal _is not_ an elephant, I am _not_ giving a definition of a non-elephantic animal. No, I am just talking nonsense. . Denials are allowed in mathematical _arguments,_ but _not_ in mathematical definitions.
@@konradswart4069 What is fascinating in QM is that it kinda disproves it's own postulates. Superpositions and entanglements have strong taste of continuities without LEM - and that is an ituitive a top down idea, as discussed e.g. by Bohmian philosophy. Wildberger's work is in spirit closely related to Whitehead's notion of "region" in his point-fee geometry, and this suggest abandoning Hilbert space as a postulate, as Hilbert's reductionistic axioms of geometry obfuscate empirical and logical phenomenology of QM. No-Communication theorem and Unitarity also don't seem absolutely necessary requirements.
I think you would be interested in Imre Lakatos who wrote "Proofs and Refutations". It dwelves into the social aspects of mathematics. I suspect you're already familiar with him.
Excellent content ! I think AI will shift the focus of human mathematicians from doing calculations, to making visualizations, and trying to interpret results from the machines. In a way it may make maths look more like experimental physics. Also, perhaps we will also have to learn to think more like machines to appreciate the "paths of greatest interest" which future AI's will uncover, as they automatically search for good results.
I greatly enjoyed your projective geometry series here on youtube Mr. Southwell. It was one of my first mathematical forays outside of the world of calculus. Cool to see you here.
Thank you very much Prof. Wildberger. Is there a historically oriented text or short bibliography that you can recommend on that 19th century nexus on the foundations of math? Thank you.
Professor, may I ask if you have any books that you'd recommend to a sociologist of science interested in the sociology of mathematics? I'm interested both in understanding how you're approaching the topic, and also thinking constructively about how the relatively small field of sociology of maths could be expanded. I think a lot of the questions you ask toward the end of this video have interesting answers in other contexts. For example, I'm thinking of Ludwik Fleck, "Genesis and Development of a Scientific Fact"; Paul Feyerabend, "Against Method"; Bruno Latour, "Science in Action"; David Bloor, "Knowledge and Social Imagery"; and perhaps most relevant to a discussion of sociology of maths, the work of Reviel Netz including "The Shaping of Deduction in Greek Mathematics" and "Ludic Proof: Greek Mathematics and the Alexandrian Aesthetic." Do you have any additional works you think should be added to the reading list?
Couldn't agree more with your caveat re AI and its impact on all areas of life, including the rarefied atmosphere of pure mathematics. I see mankind dividing into two groups----those that accept the dictates of AI and those who ferociously oppose its influence, to the point of violence even....
Solid excellent teachers like Dr Wildberger. Existing as a simulation is ridiculous. History is important. Math is important. Too much government red tape lowers educational standards. The frame of learning must keep the highest possible standards.
@@AlekFrohlich and sir: since you offered me a soap box. I will continue. Being an older person who never used a computer but later learned the use of yt. I experienced shock that being a simulation implied. I can only imagine the detrimental effect this would have on a young and impressionable mind. Almost as if it is an overwhelming exposure of why you have consciousness. Children need nurture and guidance in those areas. Not a simulation. The teachers need support that have higher values.
Greetings from California, USA! I've been following your videos since I first started my undergrad studies and just wanna say how much I really appreciate all of your videos as they've helped me along my maths journey. Currently I'm about to start a PhD program here in SoCal. Anyways, this may be unrelated, however, I wonder if you've ever given thought to the typical occupational route of the mathematician, i.e. either the usual professorship route or industry position. I know most would prefer the professorship route since the potential of working on their own research interests is highest, though I wonder if there could be an alternative route? I've recently come across the business model of the "cooperative enterprise" and would like to get your thoughts on whether or not you think such a cooperative organization, which would consist of mathematicians (pure & applied), scientists from various disciplines, etc., could become a fresh alternative? The idea is that these highly disciplined minds would be working on research and innovation for the benefit of humanity, whilst sharing the profits of their products and technologies amongst themselves and the organization. Thus it would be community-centric (primarily scientific) such as those found in academia but without depending on external (government) funding, as the end products/results would essentially provide the funds necessary for business operations, such as in industry but without the same amount of overhead of a regular corporation. I understand if this seems far-fetched and/or premature, but would appreciate your thoughts on this or if you think the traditional occupational route of the mathematician is fine as it is, as in "don't fix what's not broken".
Pure mathematics is IMO best considered a main branch of philosophy, both historically and in terms of deep philosophical questions of philosophy of mathematics. Formalist school (Hilbert, Dedekind etc) seems closely associated with the paradigm of Materialism and Reductionism, Intuitionism (Brouwer, Heyting, Weyl etc.) almost by definition belongs to the Idealist approach to metaphysics. I think it's fairly obvious that this ideological division is behind much of academic sociology. I wonder what is the presence of intuitionism in the contemporary academic sociology...?
Maybe, The 'Intuitionists and/or idealists' in sociology tend to be those who gravitate towards 'Qualitative-Participant-insider/Observation Methodologies' ... The 'get out of the Ivory towers dudes' ... or 'The Zeligs' ...
It's weird to see a mathematician call for the sociological study of mathematical academia. There are already sociologists who study science and mathematics sociology. But that's not the real point I wanted to make. It's weird that your reason for calling on sociologists is that you seem to think they'll show mainstream mathematicians are under some social curse, preventing them from seeing you're right. That's a very strange way to respond to people finding your arguments unconvincing. It's perfectly okay to hold minority/deviant opinions. But take a moment (at minimum) to consider that your arguments thus far have simply not been convincing. I myself don't find myself moved by your arguments.
In my eyes, pure mathematics is just synthetic mathematics. And synthetic mathematics is axiomatic mathematics. Applied mathematics is practical mathematics, and it is also analytical mathematics. So all analytical mathematics (like matrix theory, number theory, analytic geometry, differential equations) is applied mathematics. . And about the matter of science. You must first define when something is a science. . A body of knowledge is a science, when there exists _a method of testing anyone can use_ to make a distinction between useful and useless statements within the context defined by the potential body of knowledge. That method of testing must consist of terms belonging to the language used by the body of knowledge. . For example, physics is about space, time and dead matter. That is its context. (Chemistry is a part of physics.) The method of testing is the experiment. If a candidate of a theory of physics violates the experimental test within a certain domain of applicability, or cannot even be tested (string theory) this candidate is not physics, or at best _not yet_ physics. . Mathematics is about symbolic understanding. The method of testing consists of logic. That is why mathematics is a science. . Biology is about living beings. The method of testing is Neo-Darwinian evolution. If a theory violates that (like Lamarckism) it is not a theory of biology. . Astronomy is a science. Its method of testing is just observation with instruments. It can be enriched by physics. You then get astrophysics. . Astrology _is not a science,_ because, although it uses numbers, and thus mathematics, which _is_ a science, the numbers and mathematics used do not uniquely belong to astrology. Therefore astrology does not have a method of testing belonging uniquely to astrology, and which leads to usable results. . Psychology, sociology and cultural anthropology are not sciences. This is because there are no methods of testing _uniquely belonging_ to these areas of investigation. . Marxism is not a science, although many adherents believe it is, and Marx himself believed it also. Its logic is dialectic materialism, which is not logic at all. And its method of 'testing' is 'trying to shut up everybody who disagrees with through violence or threat of violence'. That is why Marxism _is a religion!_ . Economics is not a science. But it is close to it. I myself am working on a method of testing of economic theories that can turn it into a genuine science. . _Justice and lawmaking_ is also not a science. But the criterion of testing exists. Only it either it is not known, or not recognized universally. Therefore it is better to describe it as an _unaccepted science!_ . Because sociology is not a science, while mathematics is, connecting the two is just ridiculous. . It makes me think of the following: . When Einstein heard about: '100 scientists against Einstein's theory' he said? Why 100? One should be enough! . And, indeed, science is a matter of the individual. If some body of knowledge is a science, _anyone_ capable of using the test method belonging to that science, _can see for himself_ whether what is told to him is a correct scientific statement or rubbish. _Real science is independent of any groups!_ People can work together, but _every individual must be able, by himself, to make a distinction between useful content and rubbish!_ . If you have a barrel of sewage, and you put a drop of wine in it, you end up with a barrel of sewage. . If you have a barrel of wine, and you put a drop of sewage in it, you end up with a barrel of sewage.
The term *pure* when it's being incororated with Maths, turned out to be mere utterance of sanctity which securing itself from being held as captive by *application* , either in terms of supplying one(s) or contributing to solving certain well-posed prior problem(s). So-called truth in sociology, *can be socially-constructed* and be propagated through various social institutions and challenges pre-existing truths. We don't, yet, yield to such sophistication in Maths. Why? Because the rigidity involved in perceiving truthfulness. It may sound plausible for you to argue for such shady grounds to oppose FTA, but tht won't be recognized as being the truth in maths, let alone pure maths
Living in the US, I cannot help but position the tenets of this discussion against a general populace who cannot discern the difference between a contagion and a conspiracy.
It seems misleading to claim that other pure mathematicians couldn't give you a factorisation. That quintic could be factorised using Bring radicals or generalised hypergeometric functions in a way that would be acceptable to almost all mathematicians. I think you just object to roots being expressed using anything other than a rational number. You may as well state that you don't think anyone could provide a factorisation of x²-2=0, it's basically the same objection on your part.
@@njwildberger Define "exhibit". The fundamental theorem of algebra guarantees the existence of these roots. In a sense, you can't exhibit sqrt(2) either, you just define it as the root of X^2 - 2. In the same sense, you define these roots just this way: as the roots of this polynomial. Not being able to express these numbers in another meaningful way doesn't mean that they don't exist.
@@breizhkatolik defining things you do not understand is against what any thinking man should be doing... invoking the "thats the fundamental theorem" is exactly the type of problem Prof. Wildberger mentions. Saying "Not being able to express these numbers in another meaningful way doesn't mean that they don't exist" is like saying " If you don't see flying elephants, it does not mean that they dont exist". And that is true, if I don't see them it does not mean that they dont exist, but we are not the ones claiming that they do... you are.
@@העבד Except if said flying elephants are proven to exist, and their existence is relevant to pretty much all of math; if that's the case, who cares if you can't see them. That's even the case in physics, where some subatomic objects cannot be observed (I'll admit I lack the knowledge to be more precise, but I'm pretty sure that's the case). Just because we cannot define these roots in simple terms doesn't mean that they do not exist and should be discarded as useless or even worse, non-existent. For example, we can only describe one non-trivial element of the absolute Galois group of Q, yet it's one of the most studied group and has applications in a HUGE number of fields of research. Had mathematics only been concerned with what could be described, we would still be at the same level the mesopotamians were 3000 years ago.
@@ThePallidor Why should numbers be "finite"? Why restrict ourselves to such an arbitrary notion of "numbers"? Even integers and rational numbers (which, if I remember correctly, are accepted by NJ Wildberger) are usually defined as equivalence classes of infinite sets, and are therefore infinite sets themselves.
@Paul Wray, Actually the example of x^2=2 would not work so well, as there is a particular symbolics that has been invented to cover this special situation. That is, the pure mathematicians have invented a particular symbol sqrt(2) which supposedly captures the "infinite decimal" which begins 1.41421356237309504880168872420969807856967187537694807317667973 .. and carries on in some fashion or other. So the modern pure mathematician can get away with gaming the question, and "factoring" x^2-2 as (x-sqrt(2))(x+sqrt(2)). However with my example of a degree 5 polynomial, such sleight of hand is much harder to come by, and the computational unreality of the FTA becomes clearer.
Pure math does not care about sociology....... however my choice of your video may have something to do with it. By the way , the polynomial represented here would be more interesting if free coefficient = 360. (sum=400)
Can we have a real discussion on Sociology and Mathematics without first discussing the nature of consciousness? As Einstein said, "The most incomprehensible thing about the universe is that it is comprehensible."
@Ross Templeman I agree. "Concept formation" is key and should define the field of exploration. Ultimately, we need some axioms that establish the rules for the field. Euclid was able to determine a lot of truths within the simple field of lines and circles on a plane. But if we switch the plane for a sphere, the whole thing falls apart.
@Ross Templeman And on "finitism," the field of exploration is also the determining factor. We may not be able to determine the exact numeric value of the diagonal of a square, but geometrically, that diagonal is as "real" as the square itself.
Pure means how much pure it will be? To the what extent? I believe Mathematics was already pure. It is free from prejudices, biases etc. it's foundations are already strong. Simply because Artificial intelligence taking over, there is a need for making mathematics more pure? Numerical methods works very well where we don't need exact solutions. So mathematics and AI can combiningly work much better.
Sociology of pure maths is empirical science, so let's provide a case example of current state of sociology in the mathstackexchange. There's a sincere question about how can the statement "empty set is disjoint from itself" be consistent in the axioms of ZFC? Three downvotes, no answer, just three misleading and/or unhelpfull comments: math.stackexchange.com/questions/3781111/how-can-statement-empty-set-is-disjoint-from-itself-be-consistent-in-the-axiom
Now mathstackexchange has closed the question and answers are not allowed. I edited the question to try to make it more clear as requested by moderator message, dunno whether that changes anything in the sociological aspect. Here's the latest edition of the question: How exactly is empty set disjoint from itself in ZFC? Wiki on disjoint sets does not specify which set theory it is referring to, but states generally: "For families the notion of pairwise disjoint or mutually disjoint is sometimes defined in a subtly different manner, in that repeated identical members are allowed: the family is pairwise disjoint if A i ∩ A j = ∅ whenever A i ≠ A j (every two distinct sets in the family are disjoint)" I don't understand the meaning of this when applied to empty sets, and whether "sometimes" applies to ZFC. Meaning of "distinct" is not clear to me, how does it apply to empty sets if it does? ZFC is one-sorted theory with empty set as the only element, can {}i and {}j be distinct but same sort elements in the axioms of ZFC? According to axiom of extensionality {{}} and {{}} are same set, but can they be both same and distinct sets? Or both same and disjoint sets? Axiom of regularity forbids set of two empty sets. {{},{}} is hence not allowed in ZFC, but can this be taken as the definition of "distinct" empty sets? When disjointness is defined by intersection, "It follows from this definition that every set is disjoint from the empty set, and that the empty set is the only set that is disjoint from itself.", and that "empty family of sets is pairwise disjoint" (math.stackexchange.com/questions/1211584/is-the-empty-family-of-sets-pairwise-disjoint?noredirect=1&lq=1). Notion of "family of sets" is a metalanguage concept in relation to ZFC. Can ZFC formally contain the notion of "empty set disjoint from itself", and if so, how? Or is the notion inconsistent with axioms of ZFC? If the notion of "empty set disjoint from itself" is not consistent in ZFC, that would imply that the set {{}} is forbidden by axiom or regularity. Background motivation: this question arose when trying to understand relation of ZFC with metric theories such as Lebesque measure, and whether number theoretical mass is an optional interpretation or a logical consequence when assigning a metric theory to ZFC.
The meaning of "disjoint" and "pairwise disjoint" have meanings that depend upon the definition that is being used. The wikipedia article is saying that there are many definitions that one could use. But, the author should make clear which definition they are using: that determines what is meant. You seem to want to apply all the definitions as though they were equivalent. Here is my summary of the wikipedia article (or at least the beginning portion). I will use 0 for the empty set. I will use ^ for intersection. Def 1. Let A and B be sets. A and B are disjoint means A ^ B = 0. Def 2. Let F be a set. F is pairwise disjoint means x in F and y in F implies x = y or x ^ y = 0. Def 3. Let I be a set. Let W be a set. Let f: I -> W be a function from I onto W. f is pairwise disjoint means i in I and j in I implies f(i) = f(j) or f(i) ^ f(j) = 0. Def 4. Let I be a set. Let W be a set. Let f: I -> W be a function from I onto W. f is pairwise disjoint means i in I and j in I implies f(i) ^ f(j) = 0. Def 5. Let F be a multiset. etc. Def 5 is not common and I will ignore it. Using definition 1: 0 and 0 are disjoint . Using definition 2: 0 is pairwise disjoint. Using definition 2: {0, 0} = { 0 } is pairwise disjoint. Let I = {1, 2, 3}. Let f = {(1, 0), (2, 0), (3, {0})}. Using definition 3: f is pairwise disjoint. Using definition 4: f is not pairwise disjoint. Note: You say "Axiom of regularity forbids set of two empty sets. {{},{}} is hence not allowed in ZFC" But, this is not true. {{},{}} is allowed in ZFC. Note that {{},{}} is just equal to {{}}.
@@billh17 You are saying that ZFC contains multiple different definitions and semantics. That's news to me. "Using definition 3: f is pairwise disjoint. Using definition 4: f is not pairwise disjoint." You are saying that the different definitions are inconsistent with each other. Case closed, ZFC is inconsistent.
@@santerisatama5409 said "Case closed, ZFC is inconsistent." ZFC is inconsistent means that the axioms are inconsistent (that is, a false statement can be correctly derived from them). Definitions or (incorrectly proved) theorems don't make an axiom system inconsistent. You are just saying that one (or several) of the definitions is incorrect. That is one stance to take: you are essentially saying that each defined term can have only one meaning. This is one approach (and probably is the best approach). The metamath proof system takes this approach. However, the standard approach is to allow the same term to be used with multiple meaning. Of course, one would need to be aware of the context that the term is being employed. This is similar to the use of "scope" in a computer language (where the definition of one variable or one function can be 'shadowed' by a new definition in the new scope). Are you trying to prove ZFC is inconsistent or are you trying to understand what 'pairwise disjoint' means? It looks like you are doing the former.
@@billh17 I'm not trying to semantically interprete ZFC, you are. ZFC is a Formalist theory under bivalent logic, not semantic theory with various semantic interpretations according to intuitionist, paraconsistent or what ever logic. Bivalent logic is strict, it means that ZFC either consistent or inconsistent, there's no "included middle" of "supposing that ZFC is consistent" or "according to this semantic interpretation of the formalism ZFC can be this or that or anything..." Formalism aka axiomatic set theories concocted to justify "real numbers" are by definition post-modern, post-truth language games, which claim that mathematics is nothing but arbitrary language games of theorems begging the question by circular reasoning of arbitrary axioms. Formalism is post-truth philosophy of post-modernism be claiming that emprical methods of intuition and constructibility are not truth conditions of mathematics. It's very Orwellian. The most blatant dishonesty is the outright lie is that real numbers can do basic field arithmetics. Practically all real numbers are non-computable, non-demonstrable pseudonumbers which exist only in the same way that Emperor's New Clothes exist. No, real numbers don't compute, they can't do arithmetics. They don't exist.
Sociology is a mixture of standard cynical and classical thought, along with a bunch of ignorant politically oriented new theories. So just naming it sociology undermines its value.
@ebanfield Perhaps that is the case, but perhaps it is more a case of one approach to the discipline is correct, and the other is not. In any case, there is already dispute, as I claim strongly that the FTA is not exactly correct, and is only valid as an approximate statement of applied mathematics. The aim of science should not be to avoid controversy, but to resolve it.
I think that is already de facto the case. Essentially all undercover mathematicians employed in CS departments are operating in various flavors of intuitionism constructivism realizability and finitism - little controversy there. Most mathematicians employed in Math departments are platonists formalists or “don’t care-about-it-ists”. The “dispute” is whether the later departments should be associated with school of Theology or Natural Science. And FWIW IMO there is always need for dispute and disputation - it sharpens the mind, the argument and understanding. Two go in, one comes out.
@@njwildberger But pure mathematics is not science! It's related to science in that it is a good playground for ideas and strategies, which latter on (may or may not) serve to inspire solutions to applied problems.
@@DraganIGajic Why theology? Pure math is not about beliefs; it is about playing around with ideas. That's why most mathematicians don't care about foundations. It is important to note that ideas aren't necessarily tied up to logical frameworks; they may spawn out of physical intuition, for example. That's why ideas may be latter refurbished outside their place of birth. In conclusion, even though pure maths is not directly related to science, it has proven to be the driving force behind many of it's developments. So it is a worthwhile endeavour.
Fascinating inquire ... I can not even discern how to begin to think in that ... All ears for the series ... I wonder how dudes like Grothendieck (impossible), Connes, Perelman, or Mochizuki (possible) would expose their opinion about the topic ... If I were the sociologists, I would begin by questioning as most mathematicians as possible about the topic, try to find 'common individual but general schemes' ... and by their 'verbal report' incomes, elaborate another sequence of questions ... Well, at least, it seems that we are going to get the first insider perspective of the topic. Thanks.
Grothendieck, Ramanujan, Fermat and many more great minds of math had problems and struggled in academia (if they formal math education at all) and independently discovered many theorems.
@@העבד yep, it seems that there are the follow the leader and 'the artists' ... the thing is the capacity in the artists for generating a collective intersubjective school of thought that ends believing in the artist's abstract architecture as an objective universal idealization that allows more layers of 'objective' abstraction ... When/where the inner subjectivity ends as an outer objectivity ...
I get what you are doing, but take the "existence" of those roots: in terms of physical matter the curve is not even a dense dust, but even if so then the space between the matter dust is still continuously connected, albeit let's say fractal. So the mathematician is entitled (sociologically speaking) to idealize and think in terms of the ideal set of reals as a "space" complement of the dust of matter. Now move away from physics into abstraction and you have roots that cannot be finitely written as concrete numbers, but which nevertheless have a "life" in-between your "matter" rationals. But to be honest, if one is abstracting from physics then matter is not even comparable to the rationals, it has to be a finite set, unless like me you are prepared to abstract. At best what is continuous is the spacetime position of the matter. I write all this keeping in mind using physics as a basis for mathematics is not so simple, and I think flawed. Mathematics _proper_ is the domain where thought has freedom from physics, and physics is only a guide. You might counter that: "thought" is based in brains (matter), and I would just say you can never prove that is so, not until you can well-define the concept of "thought" and I say you can never do that and still capture the essence of thought (subjective qualia)--- rather than some trivial computational abstraction which is never going to be satisfactory (take that as a challenge). I will be afraid I cannot likely answer any challenge because you will want to "define away" the subjective "qualae" aspect of conscious thought which I consider to be essential, and then I am lost... if I accept your definition. So I won't, and we will be at stalemate, since no objective experiment can decide the issue. Upon that which one is forever ignorant one must be silent. Moral "victory" to me though, since we are silent on precisely that sort of realm of experience (idealizations, abstractions, the infinite & the mind) where I think we must for now remain silent. Having said all that, doing rational set based mathematics is good for the soul, good praxis, it is pressing upon weaknesses in transfinite mathematics theory that need to be pressed. The two points of view (intuitionists/constructivists say versus idealists/platonists) can only benefit from such a battle. What I would want to do is use both schools to crush the formalists who are, in sociological terms, mathematical psychopaths or at best sociopaths. (I am a physicist, so whatever mathematics I can make out of physics I see as concrete, real, and admitting abstractions that are platonic, all recovering platonists are welcome back to the fold at any time(!), so I say death to the formalists... [to their austerity, not their personage].)
This video makes me want to revisit some Foucault and imagine what he would do with the subject-spoiler alert: no answers but a lot of inspired problematization.
You need to invest in audio equipment and lights doc. Your making the viewers concentrate more on what you said due to low volume instead of focusing on what is being said.
After so many year, watching your videos keeps giving me the feeling of traveling back in time, as if the very concept of "language" has never been developed further, and in particular formal languages. Meanwhile there are projects such as Lean, fruit of people's efforts spent trying to teach computers how to understand a formal language suitable for usual mathematics. Its framework gives room even for axiom of choice. Yeah, we cannot give explicit closed formulas for the roots of that degree 5 polynomial, but that's missing the point entirely: we developed a language able to tell such roots do exist (with respect to our interpretation of said language) even if not able to give explicit evaluations (with respect to our interpreation of said language). Why should we do this? Because something is better than nothing, and existence information is enough to say interesting things. What do we get if we insist o avoid this possibility? Absolutely nothing. Long story short, at the time of writing there exist (!) computers accepting concepts you and a lot of your followers (they do behave as cultists) refuse.
@fmnq You can get a computer system to "work with" a lot of things, if you hard wire them appropriately. For example, you can get the computer to use the term "real number" and make statements about "arithmetic with real numbers". But with a bit of imagination, you can tweak those programs to also work with "leprechauns" and "loch ness monsters". You can then, if you are clever, run those computers to try to convince the uninitiated that such objects are legitimate, because see! our computers are "working with them". To reveal the vacuity of such systems, it is best to get down to specific computational questions. Don't ask about "cohomology of real-valued functions on normed sigma spaces". Rather ask, as I do, what is pi + e+ sqrt(2)?? Why don't you open up your favourite "formal language computer system" and get it to reveal the answer to this simple (?) arithmetical conundrum. I and my fellow cultists would be much enlightened.
@@njwildberger «You can get a computer system to "work with" a lot of things, if you hard wire them appropriately. For example, you can get the computer to use the term "real number" and make statements about "arithmetic with real numbers". But with a bit of imagination, you can tweak those programs to also work with "leprechauns" and "loch ness monsters". You can then, if you are clever, run those computers to try to convince the uninitiated that such objects are legitimate, because see! our computers are "working with them".» Yes, of course. Formal languages cannot see natural languages, since we use them as metalanguages, and terms are meaningless if not defined. You can be talking about leprechauns and loch ness monsters or whatever your favourite fairy tales might be, but you first must define them, and even if one would come up with a definition and claim their existence, it would just literally be taking a "particular string" for granted, not an ontological claim. Moreover, if a leprechaun was defined in the same way as a spectral sequence, they would be the same thing, similarly to how we cannot distinguish between two objects if whatever test we may use to get informations from them give us the same answer. And this should be obvious to someone with your background. «To reveal the vacuity of such systems, it is best to get down to specific computational questions. Don't ask about "cohomology of real-valued functions on normed sigma spaces". Rather ask, as I do, what is pi + e+ sqrt(2)?? Why don't you open up your favourite "formal language computer system" and get it to reveal the answer to this simple (?) arithmetical conundrum. I and my fellow cultists would be much enlightened.» This "don't do this, rather do that" is a kind of cherry picking and is based on a huge misunderstanding of proof assistants. Being able to verify proofs on generalized cohomology theories, perfectoid spaces or whatever your favourite topic might be is the precise reason these tools were developed. Requiring such a tool to provide an answer to your question is not why they were developed. Being sure results such as the classification of finite simple groups are correct is the motivation behind these projects. Also what do you mean by "what is"? Isn't the tautological "pi + e + sqrt(2)" answer satisfactory to you? We can give numerical approximations; having a deterministic algorithm to compute digits still is not satisfactory? Is the impossibility to list all of them at once an issue? It only involves evaluation (as an integers string), since computers can now avoid this problem by working directly on the formal language level. So, what are you asking exactly? A finite integers string compatible with usual arithmetic rules? What would be a satisfactory answer to you, and, more importantly, why?
I am not quite understanding the issue here with those 5th-(and larger) degree polynomials. If we can devise algorithms that produce increasingly better approximations of those roots (and NW seems to suggest that was possible, if only for the polynomial at hand) then, using the Dedekind-cut definitions of real numbers, we would actually get those numbers - because at the end of the day that's what all those real numbers really are anyway. I know NW does not like Dedekind-cuts, but if you move away from them and define real numbers differently then you get different theorems, doh. You might as well give those numbers then a different name, because they are simply an entirely different entity. On the other hand, I would not take it as a given (simply because I do not know either way) that it is always possible to construct Dedekind-cut-style solutions to polynomials, because there could be a possible gap between orthodox pure math and computational math, where pure math says "these things exist by classical logic and set theory", and computational maths says "I am happy with a never-ending process spitting out increasingly better intervals, but if you cannot give me that these solutions are a figment of your imagination". One does not have to go to real/complex numbers for issues of that kind, we even get such logical interferences with integers, e.g. cardinalities of finite non-recursive sets.
I have split my answer, in two parts, because TH-cam does not allow for such long explanations. I also want to add, that what I write here is the result of thinking for many years, maybe already a decennium, about the many things you yourself have said about mathematics in the many videos you have posted. In particular about the infinite. PART 1: You know, I think that you ask too much from the axiomatic approach. The axiomatic, synthetic approach is top-down, while the constructive, analytic approach is bottom-up. . As you know, the Logicism of Frege has failed. But that is because it began with a false premise. They thought that you could build mathematics from the ground up, by starting with pure logic, and then derive all mathematics from it. That is a false approach. . I see the relationship between logic and mathematics not as constructivist, but as epistemological. What I mean is that logic is only for _testing_ mathematical theories. If a mathematical theory does not satisfy logic, then it is not a mathematical theory, but rubbish. But you cannot _construct_ all of mathematics from logic. . It is comparable to physics. Every theory of physics must satisfy the experimental test in the sense of Popper. You _begin_ with just a conjecture, which is a fancy word for fantasy. Then you use _testing_ to see whether you can refute the theory. If you perform many tests and do not succeed to find a test that refutes the theory, then the theory is valid in that domain within which you test. And _thus_ the theory is useful within that domain. You can use it, within that domain you have tested it, to be certain that constructions you make will work. In this way fantasies are transformed into theories. That is, fantasies become constructions that are applicable in reality. That is how physics progresses. . But if you try the 'Logistic approach' to physics, you would try to use the idea of testing itself, to derive all of the laws of nature. And that cannot be done, because that is not how physics works. . In the same way, what the experimental test is for physics, is what _logic_ is for mathematics. Logic is a way _to test_ mathematical theories. . In general, you can _define science itself_ as an extension of philosophy. I even consider it the next step. . Philosophy is about _finding systems of definitions_ that transform your mind in such a way, that you are able _to become aware_ of certain parts of the world, whether it is the outside world or the inner world. Therefore there does not exist something like philosophy as such. Only _philosophizing_ as a mental activity exists. Every time you study the work of some philosopher, you should not be interested in 'truth', but only in 'what did the philosopher see, become aware of, and what tries his theory _make me see!_ . A philosophy must always be a tautology, because the distinction between truth and falsity is not important in philosophy. . But then, in the next step, science begins. Now that the philosopher has made it possible to see the world through the language the philosopher has created, which of the statements in his language _can be applied to reality,_ so that I can make a distinction between statements that are useful, and those that are useless. This is the epistemological step, which transforms a philosophy into a science. . So as soon as the philosopher has defined a field of inquiry, and within that field of inquiry, there is found some method with which I can make a distinction between useful things or garbage. Or, better, if I can make a distinction between statements that are true or false within the domain of applicability of the philosophy, the philosophy has become a science. In short, whenever there is a method of testing, the domain of inquiry changes into a science. . I have told you earlier, that so-called 'infinite sets' cannot be understood analytically. But they _can_ be understood synthetically. An infinite set, like those of the integers, do not _have_ the property of being infinite. No, they _lack_ the property of being finite. Infinite = not finite. So the infinite cannot be understood analytically. But it _can_ be understood synthetically. . Just see for yourself. The definition of an uncountable infinite set, as given by Cantor, consists of showing that an exhaustive list of real numbers _cannot be constructed._ That is, a list that, if you extend it enough, will reach_every_ number between 0 and 1, as long as you go far enough in that list. . So, by introducing the idea _of elimination,_ you can turn Cantor's theory of transfinite numbers into a sensible form of mathematics. And that _it is_ logical, follows from the fact, that Cantor's theory satisfies logic. That is enough to acknowledge it as sensible mathematics. . I think that this whole analysis shows that your approach to mathematics is both right and wrong.
If you don't want (very) smart people get involved in real worldly affairs, then let them play with complex ideas and theories that do not interfere with the real world. This kind of magic is applied to all universities and sciences unfortunately.
PART 2: Therefore, _logic_ is the thing that makes mathematics a science. The experimental test is the thing that makes a candidate for a theory of physics into a theory of physics or refutes it immediately, after which it is rubbish. Or there exists a domain of non-invalidity in the Popper sense. . In the same manner, an axiomatic system is something else than a model. Group theory, for example, _gives an exact definition of symmetry, wherever it can be._ If you then show something, you can _recognize_ whether some aspect of it satisfies the definition of a group, and thus you have found a symmetry in that thing. The thing itself becomes an example of symmetry. Or, if you wish, a model of a certain kind of symmetry. But if the thing does not satisfy the axioms of group theory, then it is _definitely not_ a form of symmetry. . In other words, when you have a form of mathematics, that is based on axioms, the axioms _cause that form of mathematics to become a science!_ This is, because the relationship between axioms and the construction that satisfy them is not a matter of derivation, but the relationship between axioms and the particular theory that the axioms define is _an epistemological_ one. The axioms are their _to test whether your mathematical constructions are done in the right way!_ . Therefore, you must not ask from the axioms themselves that you _construct_ models from them. No, you use the axioms _to focus your mind_ so that you make constructions _that satisfy the axioms._ And when you do, and you have made many derivations from the axioms, you also know that thát what you construct must also satisfy everything that follows from it. In particular, if you think of something within a certain field of mathematics, and that what you think of _violates_ one of the many things you derive from the axions, you _know_ that you have made a mistake. . To give a simple example. If you begin with Euclid's axioms, and you construct a triangle within which the sum of the interior angles is not equal to 2 right angles, _you know_ that you have made a mistake. _That_ is the function of axioms. . When I did not know all this, and I started to study Hilbert's extension of Euclid's axioms, I always had the feeling that something was wrong. And now I know it. Hilbert tried to construct a set of axions that really _define exactly_ what we are supposed to consider to be Euclidean geometry. Everything that was implicit in Euclid's work itself had to be made explicit, so that he had a theory that tells us _exactly_ what Euclidean geometry is. So he attempted to make _all_ assumptions of Euclid explicit. . I think now that this is futile. You can make so many constructions in the plane. Not only lines with definite lengths by defining a unit length, and constructions with straight edge and compass. You can have curves of third degree, fourth degree etc. You can have all kinds of shapes. So you will never know whether you make something that requires an extra axiom. Therefore, using axioms as tools _for construction_ is the wrong way to look at axioms. Axioms are not for construction, but to give you _certainty after you have constructed patterns!_ . But that is not the function of axioms! Axioms are just methods that prevent you from making mistakes! Just like the example I gave of the photo of the four-legged animal. If you want to prevent including dolphins as possible things your photograph might show, then the four-legged 'axiom' derived from the photo is a way to avoid that mistake. . My point is: the analytic approach to mathematics is about construction. It is bottom-up. The synthetic method of mathematics is 'destruction'. It is top-down, and its function _is not_ to find mathematical structures or patterns, but _to test_ if the mathematical structure or pattern you are investigating ar constructed without flaws. . I must add, that my vision of mathematics is based on my many years of meditation. . There exists a 'breakthrough' point in meditation, wherein a _separation_ between consciousness and experience occurs. It is known in the East as 'Dhyana' in India, 'Chan' in China, and Zen in Japan. It is a point of development of the human mind, wherein all thought and thinking stops. And when that happens, _experiencing itself_ stops, and there is _only consciousness._ . My vision of mathematics is based on this capacity of my mind. . I _reconginze_ consciousness as the ability of the human mind _to eliminate,_ or to _silence._ And I recognize _experiencing_ as the ability of the human mind _to generate_ thought and thinking. The function_ of consciousness is the ability to eliminate errors. If you make computer programs, you know that every computer programming system has a way to catch mistakes. Since we depend for our survival on our capacity to learn, that is, to program our brains, we have a similar function. And that is _exactly_ what consciousness is! . It is possible, to have thoughts without being conscious. That happens when you sleep and dream. The brains are disconnected from the body, and the brains are then able to allow _all_ thoughts you can possibly think,_ to take place. This function exists to support memory. . The opposite is also possible. You can be conscious, without there being any experience. It is _total elimination of all thought and thinking._ In that state, the ability to observe your mistakes is maximized. This is why psychotherapy can help, especially Gestalt psychology. Gestalt psychology is a way to make you aware of the automatic thoughts that are part of what 'you are', and control you, and which bring you into trouble. If you see a connection between an automatic thought that is part of your identity, an action you take, _why_ you do it, that is, what you try to achieve, and _become conscious_ of _why_ this does not work, the troubling thought is silenced, and the therapy is a success. But let me return to mathematics. . In the _synthetic, axiomatic_ approach mathematicians makes use of the power of consciousness. Of course, without knowing this. And in the analytical approach mathematicians make use of the power of experience. . If you choose for one approach, as you say you do, and declare that approach as _the only sensible mathematics,_ you _exclude yourself_ from using the full power of your mind, because both approaches complement each other. They complement each other even _perfectly!_ . Or, more concretely, if you _reject_ the axiomatic approach of mathematics, you will never feel completely satisfied with mathematics, because you will never feel full certainty about your mathematical constructs.
I posted above a question about disjoint empty set and consistency of that notion in the axioms of ZFC. Do you have a comment on that question, can you answer it so that we can have full certainty of the axiomatic consistency ZFC (if we forget about Gödel and other stuff that set theoreticians don't consider important)? The question concerns only definition of disjoint sets, axiom of extensionality and axiom of regularity, which is the kid stuff, so if ZFC is axiomatically consistent, proving that the question does not lead to paradox should be easy peasy. If you can't deny that axioms of ZFC lead to ridiculously blatant paradox, what does that mean for your more general philosophical argument about axioms providing full certainty in the sociological context? If people have believed 100 years that ZFC avoids blatant paradoxes, but it doesn't, have people believed in logical consistency of Cantor's paradise as such, or in Zermelo's sophistry and arguments from authority?
@@santerisatama5409 What a nice question! . Just for the record: \exists x\, \forall y\, \lnot (y \in x) E(x)A(y)~(y€x) . In my approach, predicate logic and set theory are equivalent. What I find interesting, is that predicate logic can be defined by a meta-axiomatic system. A meta-axiomatic system is a system that tells when something is an axiom. . There is a nice description of this in many logic books. I myself have studied, and have used the book: Logic for Mathematicians by Hamilton also to teach logic. The first chapter gives an _analytical_ description of propositional logic. In the first chapter the truth table is defined, and it shows, in the last theorem, that there is a connection between tautologies and logical proofs. Every logical proof can be written as a tautology. And, conversely (not in the book) every tautology can be reformulated as a set of logical proofs. In this first chapter an _analytical definition_ (in my sense of the meaning) of a tautology is given, as a formula whose truth table only results in 1, that is, true, whatever the truth values assigned to the propositions in the formula. . In the second chapter of Hamilton, a _synthetic_ definition of the propositional logic is given. It shows a meta-axiomatic definition of propositional logic, and a synthetic definition of tautology. It shows a system of meta-axioms from which exactly _all_ tautologies can be derived.. . In chapter 3 and 4 it does the same for an extension of the concept of tautology. It looks at predicate logic, and logical valid statements. It shows that you can have logical statements that are true for every possible interpretation, that is, in every possible mathematical theory. . In chapter 3 it gives an analytic definition, and in chapter 4 a synthetic definition of predicate logic. . Set theory is equivalent to predicate logic. Therefore set theory is a way to test mathematical theories., and is not a mathematical theory itself. Set theory is a _meta mathematical_ theory. It _tests_ whether mathematical theories are formulated correctly, whether analytical or synthetical. . Therefore, the axiom E(x)A(y)~(y€x) = -O- . is not an axiom that shows a mathematical theory. No, it describes a circumstance in which a construction of an analytical mathematical theory, _or an axiomatic system as such_ should be refuted. . In general, it says something every mathematician knows. You should _reject_ every mathematical theory that contains a contradiction. Or, a mathematical construction based on a logical derivation which contains a contradiction does not lead to a construction, but leads to something you must refute. . And that statement is not a mathematical statement itself, but is a meta-mathematical statement _about_ systems of mathematics, whether analytical or synthetic. . Therefore, my solution to your very interesting question is that it is a false question. You are looking for something that does not exist, because you (and many other mathematicians) look at set theory as a mathematical theory, which it is not. . My solution to this problem is very similar to Popper's solution to the induction problem. (The problem of how to move from particular facts to the genera statementsl.) Before Popper, people tried to construct a logical theory of induction, similar to a logical theory of deduction. Popper showed, that such a thing does not exist. The only thing that exists are conjectures (fantasies) and refutations. Therefore you cannot prove the truth of an induction by some logical process of induction. The only thing you can prove is that a theory is _not false_ as far as we know. Through testing you _construct_ a domain of non-invalidity, which you can _never_ assume to be infinite. (Absolute truths do not exist.) . In the same manner, _logic_ is meta-mathematics. Logic as such consists of methods _of testing_ mathematical theories. If you eliminate all content of a mathematical theory, you are left not with nothing, but with thought and thinking as such. Logic tries to define certainty about thought and thinking itself. And you can try to catch that in axioms. . The axiom E(x)A(y)~(y€x) = -0- tells in what circumstance you should _reject_ both an axiomatic system of mathematics, _and_ a synthetic system of mathematics. In itself it is neither a construction nor an elimination. It tells when we _should_ reject a construction, or when _we should_ trash an axiomatic system.
@@konradswart4069 ZFC is by definition a first order theory, so I can't consider your metamathematical sophistry about 2nd order logic etc. relevant to claim that the simple question is false question. Am I correct to interpret that you imply that ZFC does not need to be consistent because it is not a mathematical theory? Then why all the bother to try to avoid paradoxes of naive set theory by developing ZFC? Hasn't the primary motivation behind ZFC been to construct a logically sound theory of real numbers and number line continuum? Those motivations are now outside of mathematics? Perhaps to you, perphaps not, but the context of the discussion is also current sociology of mathematics, and the reactions to the simple question offer fascinating sociological data. It appears as if the question is taken as heretical violation of a religious dogma, which is ironical, as the question seems to refer to the same logical problem already pointed out by Bishop Berkeley, now hidden under a new disguise. Is there a yes or no answer to the question, "does disjoint empty set exist in ZFC", on the level of first order logic? Or is the question undecidable or something even more weird in relation to binary logic? And please let me repeat that the context of the question is only ZFC, not set theory as a whole, what ever that could mean. And to cover all sides, let's not exclude the possibility that there is some divine reason and purpose behind all this. Would space rockets etc. stop functioning and would we loss capability to prevent massive asteroid impacts if faith in ZFC was lost? I pose this question in all seriousness.
@@santerisatama5409 I am afraid, you still don't get it. . You argue from the paradigm of logicism. That is, you assume that logic and set theory is a foundational point of departure. In the idea of logicism you try to define the integers in a way you know: 0 = -O-, 1 = { -O-}, 2 = { -O-, { -O-}} ... n + 1 = {n, { -O-}} And then you build Z by a pair. Then Q by a pair. Then R by a Dedekind cut. And then _build_ mathematics from that. That is not how I see it. What I see are two things, corresponding to the two functions of the brains. You have experience, and you have consciousness. Experience is construction. Consciousness is elimination. Understanding consists of both. Logic is not a tool for construction, but it is a tool for testing. Therefore you begin with analytic mathematics, a bottom-up approach out of which deduction has developed. And then the synthetic method was discovered by Euclid, which begins by defining a context through axioms, and then show processes of elimination. If I say, for example: you can draw one and only one straight line through two points, I define the method of the straight edge without units. Because I do not specify the points, and also do not specify the straight line, I mean _any_ two points, and _any_ straight line, as long as it goes through those two points. The theorem then states that from all the straight lines I can draw in the plane, _if_ it goes through those two points, then there is no other straight line going through those two points. Therefore I am _eliminating_ the possibility that there can be another straight line going through those two points. This is an example of what I mean by synthetic thinking. Synthetic thinking is eliminating thinking. Our brains have two functions. In a fully developed human brain consciousness is in the left hemisphere. Experience is in the right hemisphere. If they work together, you have conscious experience, which most people confuse with consciousness itself, not realizing that with the common term 'consciousness' _two functions_ are meant. The right hemisphere is primary a generator. Thoughts are the manifestation of experience. The right hemisphere is the experiencing half. You can experience without being conscious. This is what happens when you sleep and dream. The other thing is much harder. Consciousness without experience. This happens, when the elimination function of the left hemisphere has developed so far, that it is capable of silencing all thought and thinking. The differentiation of the two halves of the brains has then become complete. The dominance of the left hemisphere over the right hemisphere has become complete, with the result that both hemispheres are then, for the first time, able to work together. And, contrary to what many believe, this type of 'enlightenment' is not pleasant at all. It is, especially at the beginning, very painful. Why do I tell you all of this? Because I have discovered that such a differentiation between experience and consciousness happens, exactly when you _realize, that there cannot be something like 'nothingness'!_ In other words, there cannot be something that is nothing. But that is _exactly,_ what this axiom asserts! The empty set is a thing that is nothing. If you 'accuse' me (not personally meant, we are having an exchange) of sophistry, well, _that_ is _to me, extreme sophistry!_ So there cannot be an object that is nothing. It was exactly when I saw this, after a tremendous mental struggle, that I understood what is meant with 'the empty set'. Namely, that there cannot be something like an empty set as a thought. Nothing is not nothingness. Nothingness is a confusion. Look at the following link: darksucker and the theory of dark. www.lightandsound.net.au/darksuckerpage.html _That_ is the confusion of the empty set! But although there is not a 'thing' that is an empty set. There _can be understanding of why something like the empty set appears in mathematics!_ You can understand, that the empty set is not a mathematical object, but a symbol standing for the outcome of a process of elimination that become _so extreme,_ that _nothing remains!_ And that is a form of _synthetic understanding!_ One of the most difficult, I may ask, because if you really understand _exactly_ what the empty set _is,_ you understand that there can be _no thought representing the empty set!_ Because if you understand what the word 'nothing' _really means,_ you understand that 'nothing _cannot be nothing-ness!_ And when that understanding emerges in you, _all thought, all thinking, and all experiencing stops!_ And that is _quite a realization!_ One that causes a tremendous transformation of the brains. One that gives a very strange understanding of 'what' your 'I' is. The 'I' itself consists of thoughts. Analytic and synthetic mathematics are two ways of understanding. You have understanding through thought and thinking. This is understanding through experiencing And you can have understanding through elimination. This is understanding through consciousness. So the two modes of understanding lead to two forms of mathematics. And that was the conflict between Brouwer and Hilbert. They were talking about two different things, and thought they were talking about the same thing. And about your question. Mathematics is the ultimate way to program your own two brains. It unleashes the full potential of the brains. Our brains have two halves. _Everybody_ who has _any real skill in mathematics, is able to both construct and eliminate!_ And the thing what makes mathematics so difficult for many people, is the eliminative part. Learning mathematics is not like history. In history you only accumulate. In mathematics you must be able to both accumulate _and_ eliminate! It is especially _elimination_ that takes the effort. Learning to face mistakes, and correct them. _That is what makes learning mathematics so hard!_ I do not know whether you program computers. But any programming language has a detector of errors in the code. Sometimes they are very sophisticated. In fact, programmers could only then make very sophisticated computer programs when the error detectors in programming environments became very good. The better they became, the better programs computer programmers could make. Our own consciousness has that role It is an error detector. That is why it is primarily an elimination function. And synthetic mathematics is what grew out of that function. We do not have to have faith in ZFC or any other theory to destroy comets. Faith as such is only needed, if you have no awareness of your own power to correct your mistakes. Faith as such stops that. What we must learn is _understanding,_ which leads neither to faith nor truth. What it leads to is _certainty!_ And that is a never-ending quest, because all understanding requires contexts. And that can always be improved. For example. First, we thought that the sun, moon, and earth revolve around the earth. Then we learned that only the moon revolves around the earth, and the earth + moon revolves around the sun. And finally, we learn from the General Theory of Relativity, that nothing 'moves', but all stands as still as possible in a warped space-time. 'Standing still' also gets a different meaning. As a human being we do not have to understand everything. The only thing we must do, is understanding enough. We already know that when we see a comet coming that threatens us, and we know this early enough, we know that we must not try to destroy it, but give it a nudge, so that it misses the earth. We know enough about mechanics and computers to calculate how large such a nudge must be to be successful. The greatest challenge is understanding understanding itself. I assert, that there are two forms of understanding: _experiencing_ and _consciousness_ which can work together. And that is how far I have gotten.
@@konradswart4069 Thank you for sharing, now I understand your position better. I also feel the pleasurable obligation to offer my complementary view. What you call experiencing and conscience, I would call phenomenal and metacognitive experiencing. Yes, both can shut down in various relations for various durations and do so regularly during deep sleep without dreams. However, during deep sleep the 3rd level which I call (transpersonal and transcendental) awareness, continues also without phenomenal and/or metacognitive content. The latter can be triggered on from deep sleep by alarm bell etc. to give localized awareness experiental content in the relations you analyze, etc. Awareness, where you practice and observe your most rigorous philosophical skepticism, is the the ontological irreducible primitive, which continues in absence of metacognitive thoughts whether wake or asleep. Awareness continues also when breathing stops. I'm sorry for your pain and hope you are feeling better now. As for my motivation, it is not logicism. I subscribe to intuitionist philosophy and in that respect what could be called empirical idealism, as phenomenal and intuitive mathematics has also transpersonal and transcendental level of existence in general awareness, meaning that mathematics is not limited only to subjective and intersubjective communication. Here I disagree with Brouwer and agree with later intuitionists who supported more generally shared and common idealist metaphysics beyond solipsist limitations. My motivation is ethical. When we construct linguistic expressions for our intuitions - as well as for our axiomatic heuristic games - our languages should be as clear and accessible as possible, and honest to and loyal to the heuristic game rules we are playing. ZFC or any other set theory or attempt to present completed infinity as consistent with LNC does not satisfy ethical requirement of honesty and clarity of language. ZFC is presented as a formal axiomatic theory. When confronted with potential formal inconsistency, no formal answer or conclusion is given by supporters of ZFC. Only semantic obfuscation, censorship and denial. The sociology of ZFC is not playing by it's own formal rules and it is behaving in dishonest and unethical way. My approach to language of ZFC is deconstructive with motivation of liberation and healing a sociologically unhealthy language game of mathematics. I'm not a logicist, I'm deeply radical revolutionary following the path of heart. I take seriously Badiou's view of set theoretical ontology, as well as his process philosophical truth theory. Set theoretical ontology and sociology in it's many aspects is suffering from post-truth syndrome, and we can't live without some kind of truth. A better truth that allows honesty. At least you are engaging in valuable and genuine philosophical dialogue from your own non-conformist point of view, which is very much appreciated and I don't question your honesty nor authenticity of your experience, though I may have slightly different but benevolent metaphysical interpretation of it. PS: in my language word for 'dark/black' is 'musta', and expression 'musta tuntuu' means both "It feels to me" and "dark is felt". I liked your link very much, because warmth of heart cannot be seen, only felt.
Given the tenor of so many of your very interesting talks, it seems to me that you need to be looking first at foundational - philosophical - issues; at Brouwer and Intuitionism, for example, and how a pure mathematics of that nature would impact upon physics. A sociology of mathematics would then be seen forv what it is - a second-order activity, the first-order activity being truly foundational. And you're very, very wrong in worrying about AI. AI has gone almost nowhere in fifty years. The howls of protest from its supporters should take a look at where they thought they were going to be by now with AI all those years ago. Philosophically, it's a busted flush. Robots making cars isn't what they had in mind: what they had in mind was an "intelligence" that would, for example, let a machine recognise a face. Billions of dollars later, they've almost (but not quite) got to the point of matching what every three-week old baby reaches with its mother.)
Hi, Norman , what's Norbert Wiener's equations to control human behavior all about ? Teach cybernetics math ,what a hot video series that would be . It's not that i want to control people - it's just so interesting and more to the point real [be it very abstract to ] . Norbert Wiener and his associates laid the foundations for this . where to from here ? this subject needs love in it or it kills by it's ability's to control human beings and limit there potentials . it's good stuff ,with Claude Shannon communication and game theory and um..machines and humans .. politic's ..the lot.. fascinating story but folks don't understand the math behind it. please teach cybernetics math Norman. Blaise pascal is more worthy than norbert Wiener ,for he was smart at theology ,very pleasing his Pensee's book of thoughts , this sentence is my attempt winning favor .
Why aren't you precise when you talk? Rather than saying punchlines like "I claim that it is not true" (which non-mathematicians might take as granted and true), why don't you say "I choose intuitionist constructivism as a more natural, more beautiful form of logic, I have a religious relationship to the logic I choose, as if this choice had anything to do with the ideas of truth, reality, God, because I'm not rationalist but rather I'm idealist; and in that choice of logic, such a claim is not true because transcendant number are not a thing". (Because I think you refer to transcendant numbers, because you wrote a polynomial, and I don't know whether the roots of this polynomial are transcendant or "just" irrational (sorry for incomplete knowledge of mathematics), but maybe it's simpler and you refuse even irrational numbers but then I can't see why you chose such a complicated argument rather than sqrt(2)).
you have to make a AI society to go with it , and AI relationships , and a AI leader of all Ai's. Moral ai's and imoral Ai's. seems a bit silly..but God made man , for relationship. He did not make us for a realtionship with Ai idol. What does God want..i think better relationships with humans and ..this makes us Fully human. the devil has a plan for that to.. unfortunately the devil is very bright.. which may be Mans greatest problem. this video has more relevance to us all. still waiting for an answer about Norbert wieners math is about ? ( - :. AI could make us whole lot less human . Ai could end up being more of a Satan than human.
Dear Prof. Wildberger, some do, but that doesn't mean they are not committing the same mistake. Similarly, just because some tendencies might explain some attitudes or behavior, this first has to be proven and for that to succeed, alternative explanations need to be excluded. I reacted to your statement that apparently concluded that tendencies are key. If I am wrong, I am more than happy to retract everything and beg for apologies. I understand the human brain to be an organ that receives a large amount of information and from this creates a subjective and more-or-less linear narrative that only contains the absolute minimum necessary from the information received by it, and most of it indirectly, in forms of relations. Because of this, there is a human tendency to try to look for a single narrative, a single cause, a single modulator, a single possible correct way to do pure mathematics, a single possible way to communicate. If you meant this tendency to be the root cause behind such theorems like the FTA, I actually agree with you, but that said, I am even more anxious to not just assume that we are right but actually go through the process of verifying it. That said, I would still approach this not by trying to match behavior with tendencies, but with circumstances with perceived options. People can act against their tendencies if there is motivation and clear options to do so. Without such options, without a belief that things can be different, it's rather useless to point out to us, that we are less than what we present ourselves.
@@ashnur Thanks for the nice comment. I completely agree that we should not take any assumptions into the investigation. Perhaps "innate human tendencies" are present, but ultimately not nearly as important as societal or cultural or perhaps economic forces. The relative importance of different factors in determining individual or group behaviors is an interesting question. So... you might have to take my comments in this sociological direction with a grain of salt -- I am hoping others more skilled will take up the challenge.
@@njwildberger A Biologist or psychologist probably works with innate tendencies of individuals. Sociology studies the patterns that emerge from interaction between individuals that can not be explained by biology or psychology. Some social patterns might be explained by genetics - i.e. are innate - An example might be parents that care for their offspring. However that is not a sociological finding, it is a biological "law" (we do not need a fancy new thing as sociology to explain this). Sociology starts were biology and psychology ends. Innate tendencies - i.e genetically based tendencies - are not part of the repertoire of sociology.
@Me Too Even on this planet, we have thought of numbers differently throughout history. Dr. Wildberger has a great series on Babylonian system, which may have given them different insights and abilities.
Awesome. I love how, amid the chaos of 2020, our man continues to attack the concept of an infinite set.
I don't know how you focus on and maintain such clarity over so many domains. It's like watching a miracle in real time. May the force ever be with you!
Thanks Robert!
We need sociology of theoretical physics as well. We have these untestable theories like string theory for example. Unfortunately these fields suck up a lot of research funding, which goes at the cost of other less popular fields, which could be more promising in the end.
@Chris Django I agree completely. Theoretical physics, like pure mathematics, has been able to shield itself from the gaze of sociologists due to the inherent difficulties and other-worldliness of the subject. While there is pretty vigorous debate in certain physics circles re the pros and cons of string theories, it would be great to see a rigorous analysis of just how the power structures and social pressures have contributed to the elevation of this topic beyond what one might naively expect, given its disconnect from observations and predictions.
@Daniel G Whether or not 'the majority' of sociology is hand-waving these days, I have no idea myself. However, there *is* actual science happening under the label of 'sociology'. The closer to scientific method it follows, the more useful it will be, of course.
Even if 90% (or more) of sociology today is crap, Sturgeon's Law applies: "90% of everything is crap." There are legit scientifically-minded sociologists working today, and if any of them focused some attention on pure mathematics, I'm sure they'd be able to find something interesting out about it.
@@ThePallidor Do you mean einstein ripping poincare, lorenz's and hilbert's works as whittaker has established? By the way minkowski's recasting of relativity is the superior theory.
@@ThePallidor I know, it's a bunch of different theories that kind of work in their own right. Althought chromodynamics show some promise as far as tentative theories go.
The main problem of mathematical physics is the paradigm of amputation of introspection from empirism, and as consequence applied math of physics invading and taking over pure math, leading to deep logical and empirical problems of inconsistency as well as fragmented academic sociology. A rather strong hypothesis is that many of the deep problems of mathematical physics (unified theory, mind-body problem, measurement problem) are artifacts of unsatisfactory underlying theory of mathematics.
Intuitionist constructionism could offer more coherent and empirically sound approaches to get rid of the failing mathematical artifacts caused by Formalist constructions. Mother Nature resists obeying any and all arbitrary axiomatics implied by the current language game of mathematical physics, though they can serve heuristic function in the evolution.
Such a fantastic video - I am so interested to see where to you take this! Like others, I was going to note that this is also such a big issue for theoretical physics and, though difficult, is such an important discussion to have.
I think it really helps to view this video in the context of the mathematics and issues presented in your Math Foundations Series - it felt very strange to me to have to go right back to the very beginning of the Math Foundation series and work through it all - but I must say its completely changed how I think about and undertake mathematics. Its also provided a really natural context for your discussion here.
Thank you so much for all your amazing work and please keep it up!
Thanks Jamie.
As an aspiring graduate student in pure math, I’m eagerly awaiting the rest of the lectures in this series.
Everything I have found on this channel so far has been fantastic. Appreciate this generous sharing knowledge. 👍👍
Very interesting introduction to an interesting idea. I just wish the audio were a little clearer. Many thanks.
I think Norman notices that the high priests, the unquestionable authorities, of this present age are telling lies. He wonders why the populace can't see or won't accept that modern math is built on lies and deception. It can be demonstrated that it is all lies. Norman can not see that Rabbinites have controlled all social inputs for more than 100 years. It's brainwashing by mass media and education. Without fake math space and evolution disappear. Without Lorentz transforms we live in a motionless world and "outer space" disappears. Modern society believes Rabbinite Hellywood and not reasonable conclusions.
Much respect to you sir i hope you carry on giving us your perspective,
The fundamental theorem of algebra is a statement on the existence of zeros; it makes no assertions on the difficulty of finding them, or even if a finite closed form representation can be found. Requiring an infinite sequence of digits to represent a zero doesn't change the fact that it exists (and can be approximated).
Yes, but I don't think this is the issue he is raising. The FTA is true in our current formulation of mathematics; this is undeniable.
The issue, as I see it, is this: the development of nonconstructive theories may be leading the mathematical community astray in the sense of losing out on many interesting applied problems/settings.
@@AlekFrohlich There is no universal ‘interesting’ object that mathematicians are being lead astray from. What people find interesting is purely subjective and if people find constructive or non-constructive mathematics interesting (or both!) then they will study it.
Neither are ‘wrong’ or any ‘better’ by any objective means.
This video is heading, I hope, to the true point that you are trying to make all of these years.I love your thinking professor.
Throughout the history of mathematics has mathematics slowly secured itself a very powerful and influential position in society. This power and influence is ever increasing with the rise of artificial intelligence as it is structured upon and simultaneously bounded by (discrete) mathematics. The immense success and influence of mathematics has resulted into vast parts of today’s society becoming structured on mathematics and is ever widely present. However, this presence and influential position of mathematics is rarely identified and acknowledged, especially by mathematicians as their main objective is often far from understanding its relation to society and reality. (This is of course not the case for philosophers or historians of mathematics as many of them are mathematicians themselves.) This lack of acknowledgement is however not that surprising since the image of mathematics is often distorted and there being a lack of understanding of what mathematics is really about and how it functions and develops. One can simply observe this production of this image distortion in high school class rooms where many students often manifest to their teacher or among each other their confusion about what mathematics is about (“why is 1+1=2?”) and what it is useful for (“like we would ever use this in real life”). When these students approach their teacher of mathematics about this issue, the teacher often does not really answer their questions, tries to answer it but fails to satisfy the philosophical demands of the students, or is awkward or obscure in its explanation as if it is embarrassed of not knowing the answer or that the teacher might belief that the role of being a mathematicians comes with understanding its ontology and is confronted with a failure to fulfil this role. Nevertheless, the majority of those that are confused about mathematics will force themselves through the mathematical literature because its sufficient completion is a requirement of achieving a high school diploma. For many high school students the sufficient completion of mathematics thus functions more as a status symbol rather than equipping oneself with a philosophical account of reality. Gaining a sufficient grade on a math test is rather a reflection of obedience towards the societal system rather than the acquisition of “objective” knowledge. It is the systematic imposition of a logic and knowledge that society or those in power desire its citizens to perceive as a dominant form of logic and knowledge. The goal of teaching mathematics in high school is not to question its “objectivity”, as the requirement of becoming a high school math teacher is not to be a philosopher of mathematics but to be an individual who is capable of transferring a particular form of logic and knowledge onto students that by law are obliged to follow the classes. Attempts to become absent in math classes will not only be directly punished by the school and government but also often by one’s social environment such as family and the future career market.
This interaction between the confused high school students and math teachers in which the young student is confused about the ontology of mathematics and the teacher (regardless of believing to known or not know what mathematics is about) educating this philosophical account to the students reflects the underlying forces of society that unknowingly and unintentionally has guided or pressured these individuals into this exact interaction. This interaction can be observed over and over again each year throughout many high schools in various countries and cultures. This leads society to reproduce a system of teaching that causes this “dysfunctional” unintended pattern to repeat itself. Exactly these manifestations portray not only the power relations that are operating across all of society but also the unstable parts of society that many are oblivious to, this while it could be of vital importance to understand these interactions for the sake of the persistence of society, humankind, or even earth itself. What can be seen by many as a minor frustrating interaction for the student and a potentially embarrassing one for the teacher and something that would not seem to reflect the instability of society, can be a reflection of the failure or incomplete or in process exertion of power onto this young generation of students. The ultimate form of power would be to succeed in eliminating this above described social interaction such that these philosophical questions about mathematics would be eliminated from one’s mind. And this is done exactly within society by accentuating the societal usefulness of mathematics and concealing its fictionalism or social construction by highly institutionalizing itself throughout history. And when something becomes institutionalized it recedes from consciousness. Meaning, that if x becomes institutionalized, then x becomes routine and ordinary such that x becomes part of the background for those who are part of an institution.
I completely agree, math turned from being an applied tool of discovering how our world works, into being the goal in and of itself. The massive amount of notation and over specialization, and the laziness of mathematicians when they write a paper, instead of reducing problems so that everything is written in simple algebraic terms, they use the over bloated notation that could only be understood by a handful of people. The worst part is that this kind of thing is becoming the norm, even if someone comes up with some incredible discovery, it would be drowned in the sea of paper pushers in most of academia. Hearing a professor express these thoughts leaves me with some hope though.
It's interesting that intuitionistic logic was once controversial because it seemed overly restrictive. Then it turned out to play an important role in the Curry-Howard correspondence, which sheds a lot of light on the shared foundations of mathematics, logic, and computer science.
I think it's fair to say that it's still controversial. Barely any mathematician works under the framework, apart from people dealing with internal logic of topoi (and apart from logicians and computer scientists, when we count them to mathematicians). Same goes for modal logics. It's worth pointing out that by double-negation-translation, every classical theorem is equivalent to a constructively provable statement that is classically equivalent to it. But, of course, finding the constructive proves is harder than finding the classical ones. As for your second sentence, I'm not sure if shining a light on foundations of mathematics is of interest to many mathematicians.
It's well recognized by intelligence researchers that what makes the genius is his intuition for a subject, not the formalism.
@@NikolajKuntner The problem is that you're talking about a field that very few people are familiar with, let alone whether your interpretations match an interpretation I would give to the results you're talking about. And I've read a chunk of simpson's subsystems of second order arithmetic, so I'm not a complete ignoramus at it.
Now as far as mathematicians go, intuition is the rule of the game, developing it is exactly what is called mathematical maturity, making someone get familiar enough with mathematical material as to intuitively understand it. But beyond that, the training of mathematicians reflects it, nobody goes with the bourbaki tomes or some other fully rigorous development. Even riemannian geometry, we do it in real and not banach spaces for precisely that reason, when it's no harder to do it in banach spaces.
Nikolaj-K spitballing there are 10^5 mathematicians and 10^7 cs in the world. So if only 1% are mathematicians the numbers are on par. I think this is a conservative estimate re cs. At a minimum we can say it has divided math community into leibnitzian Calculemus half and the other half
Thank you for this video professor Wildberger, really looking forward to this series. The social dynamics of our discipline, which I feel are totally neglected to this day, have many far-reaching consequences that affect us negatively. I'm confident that initiating this discussion will make visible many of the toxic attitudes, ideologies, and ways of interaction that occur within the culture of pure mathematics.
Can you name a few of these far-reaching consequences? Also, what would be some examples of toxic behavior in the community?
Even though I don't share your philosophical views, I do think that your channel is interesting and your discussions are worthwhile.
That said, I'd like to discuss my interpretation of this video.
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There is no logical problem with modern (post zfc) mathematics in the sense of consistency; the problem is that theories built upon it may not be the best in terms of usefulness for investigating other scientific fields such as physics, computer science, and engineering. This combined with the fact that maths is THE playground upon which formal theories are built results in a slow down of scientific progress, which may be seen as a far-reaching consequence of the adoption of certain philosophical ideas in maths. I mean, maths, philosophy and physics were born together; it is kind of strange to separate them so fiercely as we do today.
In summary, some philosophical ideas impregnated in maths by sociological reasons are leading us, as a community, astray. We could be better spending our research efforts elsewhere for the sake of scientific progress.
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What do you think of this?
Thank you very much for this intersting approach on mathematics and crossed look.. Ahmed
Im joining you!!
Thankyou.
The same can be said about economics. Check out the Cambridge controversies or the history of schools of thought based on Keynes.
Economics is not yet a science.
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Any field of enquiry goes through four stages.
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You have the observational stage, where certain regularities are noted, but not clearly understood. In Economics this is represented by The Wealth of Nations by Adam Smith.
Then follows the religious stage, where the feelings are written down in 'holy books'. In Economics you have, for example, Das Kapital and The Communist Manifesto'. These are social religious books.
Then follows the philosophical stage, whereby the context is constructed in the right way. In Economics this stage is represented by 'Human Action' of Ludwig von Mises.
And, lastly, you have the scientific stage. One that I am working on. Only if a principle has been established to make a distinction between useful and useless statements, _totally belonging to the science itself,_ the field becomes a science.
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I assert, that the science of economic has to deal with both utility and value. Two concepts that are in my development purely separate. That is, you can have something having tremendous utility, but no value at all. Like the air we breathe. And you can have something having tremendous value, but no utility at all. A certain amount of money in the form of a cryptocurrency is an example of that.
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This means, that value _cannot be the result of utility + scarcity!_
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I hope to publish my first book on utility and value this month.
@@konradswart4069 Please share info when complete!
@@peterosudar1636 I will.
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Thanks for your interest.
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The content of the book is already finished. I only have to format it, so that it can be sold by Amazon.
I also agree. It can not be philosophy or pure math when it is both. Like a double slit experiment it also is both.
Can we expect to see more episodes of this series soon? For me this story as in some ways as interesting as the actual mathematics the 20th century has produced.
Prof. Wildberger, someone in the comments mentioned there should be a sociology of theoretical physics as well. That comment reminded me of the theoretical physicist and fellow TH-camr, Sabine Hossenfelder, who has many interesting videos, several of which include critiques (of the sociology, without calling it as such) of theoretical physics. I would highly recommend at least checking out some of her videos (they tend to be less than 10 min each, but also tend to be very high signal-to-noise ratio, so the time-investment would probably be worth it), because, IMHO, she's kind of an intellectual 'bird of a feather' in many ways with your own thinking (but mainly on theoretical physics rather than mathematics). Just search for 'Sabine Hossenfelder', should be easy to find.
Who knows? If you find her thoughts/critiques interesting, perhaps you two could collaborate on an interview or something. Just an idea! She has done a few interviews, including outside of strictly theoretical physics (for example, on the issue of 'the reproducibility crisis' in psychology and other science sub-fields), and they tend to be very enlightening, IMHO. So, she might also be interested in your perspective as an internal critic/skeptic within mathematics as well. After all, fundamentally, *there is* a huge overlap between theoretical physics and mathematics.
Very interesting to see you delving into the broader role of mathematics within society, and as part of society! I applaud the exploration, and look forward to more! 😃
Cheers!
Everyone knows that the more modern theories of physics have not been established, you can even find it written on intro physics books. The problem is the researchers, not the field of physics.
PLEASE MORE OF THIS
A quick question, do you find arguments which ''prove'' existence of some ''X'' without a construction valid?
@ ZetaOfS In mathematics, questions of "existence" are often red herrings. If we stick to things we can write down (the mathematical equivalent of the physicists requirement that physically meaningful objects/concepts are observable) then the difficulty disappears.
@@njwildberger This is a 'beating around the bush' answer.
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I think there are two approaches to mathematics. The _analytic_ approach, and the _synthetic_ approach. The analytic approach is bottom-up, and the synthetic apoprocah is top-down. In the analytic approach, everything has to be constructible and constructed in order to exist. Also, you have to start with something you _define_ to exist. You cannot, for example, in the analytic approach, just state that you begin with a continuous number line. You have to define what you consider to exist. You can, for example, not just state that there is a point on the number line corresponding to the square root of two.
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What you _can_ do is just introduce a mathematical object, having the property that the square of that object is equal to 2. And then you can say that there are two of them:
Sqrt(2) and -Sqrt(2), and then extend the field of rational numbers, so that, from that point on, you can consider Sqrt(2) to exist, and derive all sorts of mathematical constructs from that.
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In the _synthetic_ approach, you begin with a set of axioms. These are implicit definitions of all things you _exclude!_
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I always use the following example to clarify what I mean by this.
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Suppose my grandfather makes a photo of an animal. But, alas, due to some defect the photo is hazy. However, you _can_ make a number of certain statements about what the photo shows. You can see that the animal, whatever it is, has four legs. You can see that it is about 0.5 meters. You can see that it has a color that becomes grey in a black and white picture.
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You can then introduce postulates, which assert.
Axiom 1: The animal has 4 legs.
Axiom 2: The animal is 0.5 meters long.
Axiom 3: The animal has a definite color.
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From this, you can construct _several models!_
Model 1: the animal is a cat.
Model 2: the animal is a dog.
Model 3: the animal is an anteater
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From the axioms alone you cannot say which of the three models is the correct one. The axioms do not provide a thorough enough exclusion to identify the one animal my grandfather has photographed. The same can be said about most mathematically axiomatic systems. They consist of certain symbols, certain statements, axioms, that tell what the mathematical objects have to satisfy, and then some manipulation rules that can be used to derive consequences from the axioms, which then apply to _all models_ that satisfy the axioms.
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This I call _synthetic mathematics,_ because this approach _synthesizes many models_ into _one_ overarching system of insights, all of which are satisfied by all models satisfying the axioms.
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_Almost every_ synthetic system of mathematics (except those that are categorical) suffers from this defect. And it has consequences for physics.
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In physics, for example, you have quantum mechanics. Quantum mechanics, as it is taught now, begins with a number of postulates. This is a synthetic approach. And it least immediately to a huge problem. Because of this, we have _several models_ of quantum mechanics, also called interpretations. You have the Kopenhagen interpretation, the Many Worlds interpretation, the Ensemble interpretation, and several others.
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Since _all_ of them satisfy the postulates of quantum mechanics, quantum mechanics as such _is not understood_ in the analytical sense. _That_ is, what Bohr meant when he said that when you think you understand quantum mechanics, you don't. And Feynman, likewise, meant when he said that _nobody_ understands quantum mechanics.
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We _do understand_ quantum mechanics synthetically. But we _do not understand_ quantum mechanics analytically. Our _understanding_ of quantum mechanics, as a description of the world, _is incomplete! That is the problem with quantum mechanics!_
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The same, by the way, can be said about the special theory of relativity. This also is built upon two postulates. The Galilean Principle, and the constancy of the velocity of light statement.
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Einstein himself constructed a model of the special theory of relativity. It is the one that is taught in all textbooks.
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Yet, I myself have constructed a second model of the special theory of relativity, just to see whether it can be done.
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So, questions of existence _are not red herrings!_ The very problem of existence points at two fundamentally different approaches to mathematics. Brouwer limited himself to only those things that are constructible. It is a bottom-up approach. But the other form of mathematics, promoted by Hilbert, is also a legitimate form of mathematics. It works with exclusions. It is a top-down approach.
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As a mathematician, we do not have to make a choice between analytic or synthetic mathematics. We only have to recognize that there are two forms of mathematics. And sometimes, these two forms come together. This happens for example in propositional logic, where you first have the analytic approach, represented by truth tables. And then you have the synthetic approach, given by axioms defining the concept of tautology. And then you can prove that every tautology in the analytic sense is also a tautology in the synthetic sense and vice versa. Whenever that happens, the theory is called _complete_ or also _categorical!_
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Almost all theories of mathematics are either analytic or synthetic. Some are both. And from some we can prove that they can't be extended in such a way that they become both analytic and synthetic. We know, for example, that we cannot make a complete _synthetic_ theory of natural numbers. That is Gödel's theorem. But we _can_ make a complete _analytic_ theory of numbers, by using the Peano postulates.
@@konradswart4069 I found your comment very interesting and worth reading. I agree with much of what you say in it. However, I think perhaps you are being a little too quick to dismiss Wildberger's response to the original question, and what it is really pointing at.
In both of your types of mathematics, there *still* remains the *practical* question of: Can you write this down? (And, following up an affirmative answer by *actually* writing it down.)
Whether synthetic or analytic, there is the potential to face serious difficulties in actually writing down an actual answer to any given problem. In either case, there may be problems for which writing down an answer is not just 'difficult', but fundamentally impossible, given the constraints of the physical universe.
I believe what Wildberger is suggesting (based on my personal understanding of his videos over the years -- so, I may be wrong on this, but hopefully not) is that if you shift your focus onto that 'writing it down' aspect, the question of the *existence* of various concepts becomes a moot point.
For example, one could define a number sqrt2 into existence, with definition (sqrt2)^2 = 2, and then proceed to use this number (just focusing on the positive one, for sake of simplicity of discussion) in all sorts of perfectly logical algebra, all *written down* in terms of the regular mathematical symbols, plus the special symbol "sqrt2" to refer to this specially defined number.
Wildberger would have absolutely no problem with that. And so, in that case, under those conditions, he'd be perfectly willing to accept that 'a square root of 2' *exists* within that framework. But he wouldn't care so much about the word 'exist', he'd just be focused on "Well, it can be *written down,* so there's no issue of 'existence' to even worry about."
But in the case of developing a general mathematics of 'Real numbers' which includes 'the square roots' of all non-negative Reals, as well as all the transcendentals and all other numbers typically included in the Reals, *then* you run into the issue of being able to write them all down *and* perform general mathematical operations on them at the same time (as is essentially 'promised' by the definition in the first place).
It's here that Wildberger would start poking at holes in the (various, logically conflicting or incomplete) definitions of the Reals and start asking, "Can you actually write down the positive square root of 2?" One might reply, "Yes, of course, as √2." To which he will poke further, "Okay, now how do I add Pi or e to that? (As is 'promised' by the def'n of Reals as continued decimals, for example.)" It's here that you run into the immediate *practical problem* with actually *writing down* the answer. You can never actually finish that, as a continued decimal, and so have to resort to just writing down the expression "√2 + Pi".
And then Wildberger will ask about "Okay, so what is the z-th decimal digit of √2 + Pi?" (The number 'z' is defined by Wildberger in a previous video (see "Famous Math Problems 1" in his videos list). Needless to say, it is an incredibly huge number. Even using a clever formula that can isolate specific digits of Pi, for example, would be physically impossible to apply with an input parameter as big as z.)
So, does that z-th digit 'exist'? Well, Wildberger doesn't really care, because you can't even get past the hurdle of determining what that digit actually is in the first place. It's impossible to *actually* write it down (in this physical universe; and as a result of a logically accurate calculation).
Thus, my whole point is simply that Wildberger's original answer is not such a 'beating around the bush' answer as you have made it out to be. It's quite a bit more serious and deeper than that. Just thought I'd try my hand at elaborating on that point. Cheers!
@@robharwood3538 Thank you for your extensive response.
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The very question: 'can you write it down?' already betrays an _analytic bias!_ 'To be able to write something down' is the same as 'to be able to construct it'.
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My point in synthetic mathematics is, that the concept of _existence_ does not occur. You just sum up some criteria, which are the axioms. By writing the axioms you implicitly are saying that you _exclude everything that does not correspond to the axioms!_ So you are _excluding,_ and then you use the rules of the axioms + logic to see, what else is excluded.
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Therefore writing down does not matter.
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By using synthetic mathematics, you can deal with the infinite, which cannot be done with analytic mathematics. Infinite is simply _not finite!_ Whenever something is infinite, it is incomplete. If I say, that the set of positive natural numbers is infinite, I am saying that it is _not finite._ To be precise, what I am saying is that the set of positive integers _lacks_ a property. Namely that there is a _largest_ number. So an infinite set is not a definite object, but an _incomplete_ object. And therefore it is not an object. The idea of an infinite set contains many instances, all of which are finite, and just 'very large'.
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Yet, you _can_ work with incomplete sets. What they define are not mental objects, but mental exclusions.
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The only thing you can say about an infinite set is that it is _not_ the case that it does _not_ exist. Just like in Popper's theory that you cannot say that a statement is true. You can only say that a statement is not false.
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My approach is not to see mathematics as one thing. I rather see it as a plurality of constructs. If you can capture some regularity into a set of tokens, a way to construct these tokens into formulas, then formulate axioms that exclude, and then add operations to this whole, you get a mathematics. That is, _only_ if it satisfies the rules of logic.
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In my eyes, logic fulfils the same role in mathematics as the experiment fulfills in physics (and chemistry). We can make distinctions between useful and useless theories. Logic is the test of theories of mathematics. There is not just one mathematics. There are many mathematics. The experiment is the test of theories of physics.
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Sqrt(2) does not have a finite representation in the decimal system. Which just means, that the decimal system is not powerful enough to construct finite representations of Sqrt(2). In fact, the decimal system cannot even make finite representations of 1/3, 1/7, 20/21, or, in general, a fraction with a numerator that does not consists of multiples of 2, of 5, or combinations of multiples of 2 and 5.
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Just to demonstrate how powerful thinking in terms of exclusion is. The paradox of Russell can be eliminated in a simple way, by specifying what a definition must be.
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A definition is not allowed to contain a negative. If it does, it is not a definition of something that can be constructed. So if somebody says he defines an infinite set, he is talking nonsense, because the definition of any infinite set contains a negation. (Not a largest member, not a fractional representation, etc..)
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So, to return to the Paradox of Russell. He begins with a set that does _not_ contain itself as a member. That is _not_ a definition, because it contains a negative. Therefore the whole paradox of Russell is just confused thinking.
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To give a simple example: if I say that some animal _is not_ an elephant, I am _not_ giving a definition of a non-elephantic animal. No, I am just talking nonsense.
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Denials are allowed in mathematical _arguments,_ but _not_ in mathematical definitions.
@@konradswart4069 What is fascinating in QM is that it kinda disproves it's own postulates. Superpositions and entanglements have strong taste of continuities without LEM - and that is an ituitive a top down idea, as discussed e.g. by Bohmian philosophy. Wildberger's work is in spirit closely related to Whitehead's notion of "region" in his point-fee geometry, and this suggest abandoning Hilbert space as a postulate, as Hilbert's reductionistic axioms of geometry obfuscate empirical and logical phenomenology of QM. No-Communication theorem and Unitarity also don't seem absolutely necessary requirements.
I think you would be interested in Imre Lakatos who wrote "Proofs and Refutations". It dwelves into the social aspects of mathematics. I suspect you're already familiar with him.
Excellent content ! I think AI will shift the focus of human mathematicians from doing calculations, to making visualizations, and trying to interpret results from the machines. In a way it may make maths look more like experimental physics. Also, perhaps we will also have to learn to think more like machines to appreciate the "paths of greatest interest" which future AI's will uncover, as they automatically search for good results.
@Richard Southwell I agree completely with what you are saying.
I greatly enjoyed your projective geometry series here on youtube Mr. Southwell. It was one of my first mathematical forays outside of the world of calculus. Cool to see you here.
@abstractRange thank you. Twas inspired by Wildberger's hyperbolic projective geometry series. You may enjoy that also
Thank you very much Prof. Wildberger. Is there a historically oriented text or short bibliography that you can recommend on that 19th century nexus on the foundations of math? Thank you.
Professor, may I ask if you have any books that you'd recommend to a sociologist of science interested in the sociology of mathematics? I'm interested both in understanding how you're approaching the topic, and also thinking constructively about how the relatively small field of sociology of maths could be expanded.
I think a lot of the questions you ask toward the end of this video have interesting answers in other contexts. For example, I'm thinking of Ludwik Fleck, "Genesis and Development of a Scientific Fact"; Paul Feyerabend, "Against Method"; Bruno Latour, "Science in Action"; David Bloor, "Knowledge and Social Imagery"; and perhaps most relevant to a discussion of sociology of maths, the work of Reviel Netz including "The Shaping of Deduction in Greek Mathematics" and "Ludic Proof: Greek Mathematics and the Alexandrian Aesthetic." Do you have any additional works you think should be added to the reading list?
Couldn't agree more with your caveat re AI and its impact on all areas of life, including the rarefied atmosphere of pure mathematics. I see mankind dividing into two groups----those that accept the dictates of AI and those who ferociously oppose its influence, to the point of violence even....
Times are changing and young people do need the extra attention that the changing of time requires
What changes are you talking about?
Solid excellent teachers like Dr Wildberger. Existing as a simulation is ridiculous. History is important. Math is important. Too much government red tape lowers educational standards. The frame of learning must keep the highest possible standards.
@@brendawilliams8062 "Existence as a simulation"? "Government red tape"?
@@AlekFrohlich and sir: since you offered me a soap box. I will continue. Being an older person who never used a computer but later learned the use of yt. I experienced shock that being a simulation implied. I can only imagine the detrimental effect this would have on a young and impressionable mind. Almost as if it is an overwhelming exposure of why you have consciousness. Children need nurture and guidance in those areas. Not a simulation. The teachers need support that have higher values.
@@AlekFrohlich little house on the prairie offered more than robotic brotherhood.
Future for pure maths, in terms of AI, will be super interesting!
Greetings from California, USA! I've been following your videos since I first started my undergrad studies and just wanna say how much I really appreciate all of your videos as they've helped me along my maths journey. Currently I'm about to start a PhD program here in SoCal. Anyways, this may be unrelated, however, I wonder if you've ever given thought to the typical occupational route of the mathematician, i.e. either the usual professorship route or industry position. I know most would prefer the professorship route since the potential of working on their own research interests is highest, though I wonder if there could be an alternative route? I've recently come across the business model of the "cooperative enterprise" and would like to get your thoughts on whether or not you think such a cooperative organization, which would consist of mathematicians (pure & applied), scientists from various disciplines, etc., could become a fresh alternative? The idea is that these highly disciplined minds would be working on research and innovation for the benefit of humanity, whilst sharing the profits of their products and technologies amongst themselves and the organization. Thus it would be community-centric (primarily scientific) such as those found in academia but without depending on external (government) funding, as the end products/results would essentially provide the funds necessary for business operations, such as in industry but without the same amount of overhead of a regular corporation. I understand if this seems far-fetched and/or premature, but would appreciate your thoughts on this or if you think the traditional occupational route of the mathematician is fine as it is, as in "don't fix what's not broken".
Are their equations of society which exists in sociology?
No, no equations. Some statistics, but that's always a snapshot, as society is changing constantly.
Pure mathematics is IMO best considered a main branch of philosophy, both historically and in terms of deep philosophical questions of philosophy of mathematics. Formalist school (Hilbert, Dedekind etc) seems closely associated with the paradigm of Materialism and Reductionism, Intuitionism (Brouwer, Heyting, Weyl etc.) almost by definition belongs to the Idealist approach to metaphysics. I think it's fairly obvious that this ideological division is behind much of academic sociology. I wonder what is the presence of intuitionism in the contemporary academic sociology...?
Maybe, The 'Intuitionists and/or idealists' in sociology tend to be those who gravitate towards 'Qualitative-Participant-insider/Observation Methodologies' ...
The 'get out of the Ivory towers dudes' ... or 'The Zeligs' ...
It's weird to see a mathematician call for the sociological study of mathematical academia. There are already sociologists who study science and mathematics sociology. But that's not the real point I wanted to make. It's weird that your reason for calling on sociologists is that you seem to think they'll show mainstream mathematicians are under some social curse, preventing them from seeing you're right. That's a very strange way to respond to people finding your arguments unconvincing. It's perfectly okay to hold minority/deviant opinions. But take a moment (at minimum) to consider that your arguments thus far have simply not been convincing. I myself don't find myself moved by your arguments.
In my eyes, pure mathematics is just synthetic mathematics. And synthetic mathematics is axiomatic mathematics. Applied mathematics is practical mathematics, and it is also analytical mathematics. So all analytical mathematics (like matrix theory, number theory, analytic geometry, differential equations) is applied mathematics.
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And about the matter of science. You must first define when something is a science.
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A body of knowledge is a science, when there exists _a method of testing anyone can use_ to make a distinction between useful and useless statements within the context defined by the potential body of knowledge. That method of testing must consist of terms belonging to the language used by the body of knowledge.
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For example, physics is about space, time and dead matter. That is its context. (Chemistry is a part of physics.) The method of testing is the experiment. If a candidate of a theory of physics violates the experimental test within a certain domain of applicability, or cannot even be tested (string theory) this candidate is not physics, or at best _not yet_ physics.
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Mathematics is about symbolic understanding. The method of testing consists of logic. That is why mathematics is a science.
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Biology is about living beings. The method of testing is Neo-Darwinian evolution. If a theory violates that (like Lamarckism) it is not a theory of biology.
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Astronomy is a science. Its method of testing is just observation with instruments. It can be enriched by physics. You then get astrophysics.
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Astrology _is not a science,_ because, although it uses numbers, and thus mathematics, which _is_ a science, the numbers and mathematics used do not uniquely belong to astrology. Therefore astrology does not have a method of testing belonging uniquely to astrology, and which leads to usable results.
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Psychology, sociology and cultural anthropology are not sciences. This is because there are no methods of testing _uniquely belonging_ to these areas of investigation.
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Marxism is not a science, although many adherents believe it is, and Marx himself believed it also. Its logic is dialectic materialism, which is not logic at all. And its method of 'testing' is 'trying to shut up everybody who disagrees with through violence or threat of violence'. That is why Marxism _is a religion!_
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Economics is not a science. But it is close to it. I myself am working on a method of testing of economic theories that can turn it into a genuine science.
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_Justice and lawmaking_ is also not a science. But the criterion of testing exists. Only it either it is not known, or not recognized universally. Therefore it is better to describe it as an _unaccepted science!_
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Because sociology is not a science, while mathematics is, connecting the two is just ridiculous.
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It makes me think of the following:
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When Einstein heard about: '100 scientists against Einstein's theory' he said? Why 100? One should be enough!
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And, indeed, science is a matter of the individual. If some body of knowledge is a science, _anyone_ capable of using the test method belonging to that science, _can see for himself_ whether what is told to him is a correct scientific statement or rubbish. _Real science is independent of any groups!_ People can work together, but _every individual must be able, by himself, to make a distinction between useful content and rubbish!_
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If you have a barrel of sewage, and you put a drop of wine in it, you end up with a barrel of sewage.
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If you have a barrel of wine, and you put a drop of sewage in it, you end up with a barrel of sewage.
The term *pure* when it's being incororated with Maths, turned out to be mere utterance of sanctity which securing itself from being held as captive by *application* , either in terms of supplying one(s) or contributing to solving certain well-posed prior problem(s).
So-called truth in sociology, *can be socially-constructed* and be propagated through various social institutions and challenges pre-existing truths.
We don't, yet, yield to such sophistication in Maths. Why? Because the rigidity involved in perceiving truthfulness.
It may sound plausible for you to argue for such shady grounds to oppose FTA, but tht won't be recognized as being the truth in maths, let alone pure maths
Logical limitarions, eg, that zero is an even number. That does not compute. 0/1=0/2. Why? n/0 cannot be done! Why?
You can define n/0 if you want to. Most mathematicians simply choose not to most of the time.
Living in the US, I cannot help but position the tenets of this discussion against a general populace who cannot discern the difference between a contagion and a conspiracy.
It seems misleading to claim that other pure mathematicians couldn't give you a factorisation. That quintic could be factorised using Bring radicals or generalised hypergeometric functions in a way that would be acceptable to almost all mathematicians. I think you just object to roots being expressed using anything other than a rational number. You may as well state that you don't think anyone could provide a factorisation of x²-2=0, it's basically the same objection on your part.
@Binary Agenda, Why not just exhibit such a factorization, and we can then decide whether it is correct or not?
@@njwildberger Define "exhibit". The fundamental theorem of algebra guarantees the existence of these roots. In a sense, you can't exhibit sqrt(2) either, you just define it as the root of X^2 - 2. In the same sense, you define these roots just this way: as the roots of this polynomial. Not being able to express these numbers in another meaningful way doesn't mean that they don't exist.
@@breizhkatolik defining things you do not understand is against what any thinking man should be doing... invoking the "thats the fundamental theorem" is exactly the type of problem Prof. Wildberger mentions. Saying "Not being able to express these numbers in another meaningful way doesn't mean that they don't exist" is like saying " If you don't see flying elephants, it does not mean that they dont exist". And that is true, if I don't see them it does not mean that they dont exist, but we are not the ones claiming that they do... you are.
@@העבד Except if said flying elephants are proven to exist, and their existence is relevant to pretty much all of math; if that's the case, who cares if you can't see them. That's even the case in physics, where some subatomic objects cannot be observed (I'll admit I lack the knowledge to be more precise, but I'm pretty sure that's the case). Just because we cannot define these roots in simple terms doesn't mean that they do not exist and should be discarded as useless or even worse, non-existent. For example, we can only describe one non-trivial element of the absolute Galois group of Q, yet it's one of the most studied group and has applications in a HUGE number of fields of research. Had mathematics only been concerned with what could be described, we would still be at the same level the mesopotamians were 3000 years ago.
@@ThePallidor Why should numbers be "finite"? Why restrict ourselves to such an arbitrary notion of "numbers"? Even integers and rational numbers (which, if I remember correctly, are accepted by NJ Wildberger) are usually defined as equivalence classes of infinite sets, and are therefore infinite sets themselves.
Your argument about roots doesn't require a fifth order equation, does it? The equation x^2 = 2 would do to illustrate your point, no?
@Paul Wray, Actually the example of x^2=2 would not work so well, as there is a particular symbolics that has been invented to cover this special situation. That is, the pure mathematicians have invented a particular symbol sqrt(2) which supposedly captures the "infinite decimal" which begins 1.41421356237309504880168872420969807856967187537694807317667973 .. and carries on in some fashion or other. So the modern pure mathematician can get away with gaming the question, and "factoring" x^2-2 as (x-sqrt(2))(x+sqrt(2)). However with my example of a degree 5 polynomial, such sleight of hand is much harder to come by, and the computational unreality of the FTA becomes clearer.
Pure math does not care about sociology....... however my choice of your video may have something to do with it. By the way , the polynomial represented here would be more interesting if free coefficient = 360. (sum=400)
Can we have a real discussion on Sociology and Mathematics without first discussing the nature of consciousness? As Einstein said, "The most incomprehensible thing about the universe is that it is comprehensible."
@Ross Templeman I agree. "Concept formation" is key and should define the field of exploration. Ultimately, we need some axioms that establish the rules for the field. Euclid was able to determine a lot of truths within the simple field of lines and circles on a plane. But if we switch the plane for a sphere, the whole thing falls apart.
@Ross Templeman And on "finitism," the field of exploration is also the determining factor. We may not be able to determine the exact numeric value of the diagonal of a square, but geometrically, that diagonal is as "real" as the square itself.
Pure means how much pure it will be?
To the what extent?
I believe Mathematics was already pure. It is free from prejudices, biases etc. it's foundations are already strong. Simply because Artificial intelligence taking over, there is a need for making mathematics more pure?
Numerical methods works very well where we don't need exact solutions. So mathematics and AI can combiningly work much better.
AI is useless in mathematics. Numerical methods are very limited and need quite a bit of mathematical background to understand their nuance.
Sociology of pure maths is empirical science, so let's provide a case example of current state of sociology in the mathstackexchange. There's a sincere question about how can the statement "empty set is disjoint from itself" be consistent in the axioms of ZFC? Three downvotes, no answer, just three misleading and/or unhelpfull comments:
math.stackexchange.com/questions/3781111/how-can-statement-empty-set-is-disjoint-from-itself-be-consistent-in-the-axiom
Now mathstackexchange has closed the question and answers are not allowed. I edited the question to try to make it more clear as requested by moderator message, dunno whether that changes anything in the sociological aspect. Here's the latest edition of the question:
How exactly is empty set disjoint from itself in ZFC?
Wiki on disjoint sets does not specify which set theory it is referring to, but states generally: "For families the notion of pairwise disjoint or mutually disjoint is sometimes defined in a subtly different manner, in that repeated identical members are allowed: the family is pairwise disjoint if A i ∩ A j = ∅ whenever A i ≠ A j (every two distinct sets in the family are disjoint)"
I don't understand the meaning of this when applied to empty sets, and whether "sometimes" applies to ZFC. Meaning of "distinct" is not clear to me, how does it apply to empty sets if it does? ZFC is one-sorted theory with empty set as the only element, can {}i and {}j be distinct but same sort elements in the axioms of ZFC? According to axiom of extensionality {{}} and {{}} are same set, but can they be both same and distinct sets? Or both same and disjoint sets?
Axiom of regularity forbids set of two empty sets. {{},{}} is hence not allowed in ZFC, but can this be taken as the definition of "distinct" empty sets?
When disjointness is defined by intersection, "It follows from this definition that every set is disjoint from the empty set, and that the empty set is the only set that is disjoint from itself.", and that "empty family of sets is pairwise disjoint"
(math.stackexchange.com/questions/1211584/is-the-empty-family-of-sets-pairwise-disjoint?noredirect=1&lq=1).
Notion of "family of sets" is a metalanguage concept in relation to ZFC. Can ZFC formally contain the notion of "empty set disjoint from itself", and if so, how? Or is the notion inconsistent with axioms of ZFC? If the notion of "empty set disjoint from itself" is not consistent in ZFC, that would imply that the set {{}} is forbidden by axiom or regularity.
Background motivation: this question arose when trying to understand relation of ZFC with metric theories such as Lebesque measure, and whether number theoretical mass is an optional interpretation or a logical consequence when assigning a metric theory to ZFC.
The meaning of "disjoint" and "pairwise disjoint" have meanings that depend upon the definition that is being used. The wikipedia article is saying that there are many definitions that one could use. But, the author should make clear which definition they are using: that determines what is meant. You seem to want to apply all the definitions as though they were equivalent.
Here is my summary of the wikipedia article (or at least the beginning portion).
I will use 0 for the empty set.
I will use ^ for intersection.
Def 1.
Let A and B be sets.
A and B are disjoint means A ^ B = 0.
Def 2.
Let F be a set.
F is pairwise disjoint means x in F and y in F implies x = y or x ^ y = 0.
Def 3.
Let I be a set.
Let W be a set.
Let f: I -> W be a function from I onto W.
f is pairwise disjoint means i in I and j in I implies f(i) = f(j) or f(i) ^ f(j) = 0.
Def 4.
Let I be a set.
Let W be a set.
Let f: I -> W be a function from I onto W.
f is pairwise disjoint means i in I and j in I implies f(i) ^ f(j) = 0.
Def 5.
Let F be a multiset.
etc.
Def 5 is not common and I will ignore it.
Using definition 1: 0 and 0 are disjoint .
Using definition 2: 0 is pairwise disjoint.
Using definition 2: {0, 0} = { 0 } is pairwise disjoint.
Let I = {1, 2, 3}.
Let f = {(1, 0), (2, 0), (3, {0})}.
Using definition 3: f is pairwise disjoint.
Using definition 4: f is not pairwise disjoint.
Note: You say "Axiom of regularity forbids set of two empty sets. {{},{}} is hence not allowed in ZFC"
But, this is not true. {{},{}} is allowed in ZFC. Note that {{},{}} is just equal to {{}}.
@@billh17 You are saying that ZFC contains multiple different definitions and semantics. That's news to me.
"Using definition 3: f is pairwise disjoint.
Using definition 4: f is not pairwise disjoint."
You are saying that the different definitions are inconsistent with each other. Case closed, ZFC is inconsistent.
@@santerisatama5409 said "Case closed, ZFC is inconsistent."
ZFC is inconsistent means that the axioms are inconsistent (that is, a false statement can be correctly derived from them). Definitions or (incorrectly proved) theorems don't make an axiom system inconsistent.
You are just saying that one (or several) of the definitions is incorrect. That is one stance to take: you are essentially saying that each defined term can have only one meaning. This is one approach (and probably is the best approach). The metamath proof system takes this approach.
However, the standard approach is to allow the same term to be used with multiple meaning. Of course, one would need to be aware of the context that the term is being employed.
This is similar to the use of "scope" in a computer language (where the definition of one variable or one function can be 'shadowed' by a new definition in the new scope).
Are you trying to prove ZFC is inconsistent or are you trying to understand what 'pairwise disjoint' means? It looks like you are doing the former.
@@billh17 I'm not trying to semantically interprete ZFC, you are. ZFC is a Formalist theory under bivalent logic, not semantic theory with various semantic interpretations according to intuitionist, paraconsistent or what ever logic.
Bivalent logic is strict, it means that ZFC either consistent or inconsistent, there's no "included middle" of "supposing that ZFC is consistent" or "according to this semantic interpretation of the formalism ZFC can be this or that or anything..."
Formalism aka axiomatic set theories concocted to justify "real numbers" are by definition post-modern, post-truth language games, which claim that mathematics is nothing but arbitrary language games of theorems begging the question by circular reasoning of arbitrary axioms. Formalism is post-truth philosophy of post-modernism be claiming that emprical methods of intuition and constructibility are not truth conditions of mathematics. It's very Orwellian.
The most blatant dishonesty is the outright lie is that real numbers can do basic field arithmetics. Practically all real numbers are non-computable, non-demonstrable pseudonumbers which exist only in the same way that Emperor's New Clothes exist.
No, real numbers don't compute, they can't do arithmetics. They don't exist.
Sociology is a mixture of standard cynical and classical thought, along with a bunch of ignorant politically oriented new theories. So just naming it sociology undermines its value.
What can they do to bring the value of the term 'sociology'? Because the discipline isn't going anywhere.
Why not say that there are two overlapping disciplines? No need for dispute or controversy.
@ebanfield Perhaps that is the case, but perhaps it is more a case of one approach to the discipline is correct, and the other is not. In any case, there is already dispute, as I claim strongly that the FTA is not exactly correct, and is only valid as an approximate statement of applied mathematics. The aim of science should not be to avoid controversy, but to resolve it.
I think that is already de facto the case. Essentially all undercover mathematicians employed in CS departments are operating in various flavors of intuitionism constructivism realizability and finitism - little controversy there. Most mathematicians employed in Math departments are platonists formalists or “don’t care-about-it-ists”. The “dispute” is whether the later departments should be associated with school of
Theology or Natural Science. And FWIW IMO there is always need for dispute and disputation - it sharpens the mind, the argument and understanding. Two go in, one comes out.
@@njwildberger But pure mathematics is not science! It's related to science in that it is a good playground for ideas and strategies, which latter on (may or may not) serve to inspire solutions to applied problems.
@@DraganIGajic Why theology? Pure math is not about beliefs; it is about playing around with ideas. That's why most mathematicians don't care about foundations.
It is important to note that ideas aren't necessarily tied up to logical frameworks; they may spawn out of physical intuition, for example. That's why ideas may be latter refurbished outside their place of birth.
In conclusion, even though pure maths is not directly related to science, it has proven to be the driving force behind many of it's developments. So it is a worthwhile endeavour.
Fascinating inquire ... I can not even discern how to begin to think in that ... All ears for the series ...
I wonder how dudes like Grothendieck (impossible), Connes, Perelman, or Mochizuki (possible) would expose their opinion about the topic ...
If I were the sociologists, I would begin by questioning as most mathematicians as possible about the topic, try to find 'common individual but general schemes' ... and by their 'verbal report' incomes, elaborate another sequence of questions ...
Well, at least, it seems that we are going to get the first insider perspective of the topic.
Thanks.
Grothendieck, Ramanujan, Fermat and many more great minds of math had problems and struggled in academia (if they formal math education at all) and independently discovered many theorems.
@@העבד yep, it seems that there are the follow the leader and 'the artists' ... the thing is the capacity in the artists for generating a collective intersubjective school of thought that ends believing in the artist's abstract architecture as an objective universal idealization that allows more layers of 'objective' abstraction ...
When/where the inner subjectivity ends as an outer objectivity ...
dr wildberger misses the fact that for some people mathematics is a thinking, not just a rational formalization
I get what you are doing, but take the "existence" of those roots: in terms of physical matter the curve is not even a dense dust, but even if so then the space between the matter dust is still continuously connected, albeit let's say fractal. So the mathematician is entitled (sociologically speaking) to idealize and think in terms of the ideal set of reals as a "space" complement of the dust of matter. Now move away from physics into abstraction and you have roots that cannot be finitely written as concrete numbers, but which nevertheless have a "life" in-between your "matter" rationals. But to be honest, if one is abstracting from physics then matter is not even comparable to the rationals, it has to be a finite set, unless like me you are prepared to abstract. At best what is continuous is the spacetime position of the matter.
I write all this keeping in mind using physics as a basis for mathematics is not so simple, and I think flawed. Mathematics _proper_ is the domain where thought has freedom from physics, and physics is only a guide. You might counter that: "thought" is based in brains (matter), and I would just say you can never prove that is so, not until you can well-define the concept of "thought" and I say you can never do that and still capture the essence of thought (subjective qualia)--- rather than some trivial computational abstraction which is never going to be satisfactory (take that as a challenge). I will be afraid I cannot likely answer any challenge because you will want to "define away" the subjective "qualae" aspect of conscious thought which I consider to be essential, and then I am lost... if I accept your definition. So I won't, and we will be at stalemate, since no objective experiment can decide the issue. Upon that which one is forever ignorant one must be silent. Moral "victory" to me though, since we are silent on precisely that sort of realm of experience (idealizations, abstractions, the infinite & the mind) where I think we must for now remain silent.
Having said all that, doing rational set based mathematics is good for the soul, good praxis, it is pressing upon weaknesses in transfinite mathematics theory that need to be pressed. The two points of view (intuitionists/constructivists say versus idealists/platonists) can only benefit from such a battle. What I would want to do is use both schools to crush the formalists who are, in sociological terms, mathematical psychopaths or at best sociopaths. (I am a physicist, so whatever mathematics I can make out of physics I see as concrete, real, and admitting abstractions that are platonic, all recovering platonists are welcome back to the fold at any time(!), so I say death to the formalists... [to their austerity, not their personage].)
This video makes me want to revisit some Foucault and imagine what he would do with the subject-spoiler alert: no answers but a lot of inspired problematization.
@Reckless Abandon you sure drank all your Kool-Aid, bud.
You need to invest in audio equipment and lights doc. Your making the viewers concentrate more on what you said due to low volume instead of focusing on what is being said.
I think, Pure mathematics is mostly intuitive, it's like mysticism. while applied mathematics is analytical ..
After so many year, watching your videos keeps giving me the feeling of traveling back in time, as if the very concept of "language" has never been developed further, and in particular formal languages. Meanwhile there are projects such as Lean, fruit of people's efforts spent trying to teach computers how to understand a formal language suitable for usual mathematics. Its framework gives room even for axiom of choice. Yeah, we cannot give explicit closed formulas for the roots of that degree 5 polynomial, but that's missing the point entirely: we developed a language able to tell such roots do exist (with respect to our interpretation of said language) even if not able to give explicit evaluations (with respect to our interpreation of said language). Why should we do this? Because something is better than nothing, and existence information is enough to say interesting things. What do we get if we insist o avoid this possibility? Absolutely nothing.
Long story short, at the time of writing there exist (!) computers accepting concepts you and a lot of your followers (they do behave as cultists) refuse.
@fmnq You can get a computer system to "work with" a lot of things, if you hard wire them appropriately. For example, you can get the computer to use the term "real number" and make statements about "arithmetic with real numbers". But with a bit of imagination, you can tweak those programs to also work with "leprechauns" and "loch ness monsters". You can then, if you are clever, run those computers to try to convince the uninitiated that such objects are legitimate, because see! our computers are "working with them".
To reveal the vacuity of such systems, it is best to get down to specific computational questions. Don't ask about "cohomology of real-valued functions on normed sigma spaces". Rather ask, as I do, what is pi + e+ sqrt(2)??
Why don't you open up your favourite "formal language computer system" and get it to reveal the answer to this simple (?) arithmetical conundrum. I and my fellow cultists would be much enlightened.
@@njwildberger «You can get a computer system to "work with" a lot of things, if you hard wire them appropriately. For example, you can get the computer to use the term "real number" and make statements about "arithmetic with real numbers". But with a bit of imagination, you can tweak those programs to also work with "leprechauns" and "loch ness monsters". You can then, if you are clever, run those computers to try to convince the uninitiated that such objects are legitimate, because see! our computers are "working with them".»
Yes, of course. Formal languages cannot see natural languages, since we use them as metalanguages, and terms are meaningless if not defined. You can be talking about leprechauns and loch ness monsters or whatever your favourite fairy tales might be, but you first must define them, and even if one would come up with a definition and claim their existence, it would just literally be taking a "particular string" for granted, not an ontological claim. Moreover, if a leprechaun was defined in the same way as a spectral sequence, they would be the same thing, similarly to how we cannot distinguish between two objects if whatever test we may use to get informations from them give us the same answer. And this should be obvious to someone with your background.
«To reveal the vacuity of such systems, it is best to get down to specific computational questions. Don't ask about "cohomology of real-valued functions on normed sigma spaces". Rather ask, as I do, what is pi + e+ sqrt(2)??
Why don't you open up your favourite "formal language computer system" and get it to reveal the answer to this simple (?) arithmetical conundrum. I and my fellow cultists would be much enlightened.»
This "don't do this, rather do that" is a kind of cherry picking and is based on a huge misunderstanding of proof assistants. Being able to verify proofs on generalized cohomology theories, perfectoid spaces or whatever your favourite topic might be is the precise reason these tools were developed. Requiring such a tool to provide an answer to your question is not why they were developed. Being sure results such as the classification of finite simple groups are correct is the motivation behind these projects.
Also what do you mean by "what is"? Isn't the tautological "pi + e + sqrt(2)" answer satisfactory to you? We can give numerical approximations; having a deterministic algorithm to compute digits still is not satisfactory? Is the impossibility to list all of them at once an issue? It only involves evaluation (as an integers string), since computers can now avoid this problem by working directly on the formal language level. So, what are you asking exactly? A finite integers string compatible with usual arithmetic rules? What would be a satisfactory answer to you, and, more importantly, why?
@@FractalMannequin Great response.
Cultists unite against the Fancifullists!
I am not quite understanding the issue here with those 5th-(and larger) degree polynomials. If we can devise algorithms that produce increasingly better approximations of those roots (and NW seems to suggest that was possible, if only for the polynomial at hand) then, using the Dedekind-cut definitions of real numbers, we would actually get those numbers - because at the end of the day that's what all those real numbers really are anyway. I know NW does not like Dedekind-cuts, but if you move away from them and define real numbers differently then you get different theorems, doh. You might as well give those numbers then a different name, because they are simply an entirely different entity. On the other hand, I would not take it as a given (simply because I do not know either way) that it is always possible to construct Dedekind-cut-style solutions to polynomials, because there could be a possible gap between orthodox pure math and computational math, where pure math says "these things exist by classical logic and set theory", and computational maths says "I am happy with a never-ending process spitting out increasingly better intervals, but if you cannot give me that these solutions are a figment of your imagination". One does not have to go to real/complex numbers for issues of that kind, we even get such logical interferences with integers, e.g. cardinalities of finite non-recursive sets.
I have split my answer, in two parts, because TH-cam does not allow for such long explanations.
I also want to add, that what I write here is the result of thinking for many years, maybe already a decennium, about the many things you yourself have said about mathematics in the many videos you have posted. In particular about the infinite.
PART 1:
You know, I think that you ask too much from the axiomatic approach. The axiomatic, synthetic approach is top-down, while the constructive, analytic approach is bottom-up.
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As you know, the Logicism of Frege has failed. But that is because it began with a false premise. They thought that you could build mathematics from the ground up, by starting with pure logic, and then derive all mathematics from it. That is a false approach.
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I see the relationship between logic and mathematics not as constructivist, but as epistemological. What I mean is that logic is only for _testing_ mathematical theories. If a mathematical theory does not satisfy logic, then it is not a mathematical theory, but rubbish. But you cannot _construct_ all of mathematics from logic.
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It is comparable to physics. Every theory of physics must satisfy the experimental test in the sense of Popper. You _begin_ with just a conjecture, which is a fancy word for fantasy. Then you use _testing_ to see whether you can refute the theory. If you perform many tests and do not succeed to find a test that refutes the theory, then the theory is valid in that domain within which you test. And _thus_ the theory is useful within that domain. You can use it, within that domain you have tested it, to be certain that constructions you make will work. In this way fantasies are transformed into theories. That is, fantasies become constructions that are applicable in reality. That is how physics progresses.
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But if you try the 'Logistic approach' to physics, you would try to use the idea of testing itself, to derive all of the laws of nature. And that cannot be done, because that is not how physics works.
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In the same way, what the experimental test is for physics, is what _logic_ is for mathematics. Logic is a way _to test_ mathematical theories.
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In general, you can _define science itself_ as an extension of philosophy. I even consider it the next step.
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Philosophy is about _finding systems of definitions_ that transform your mind in such a way, that you are able _to become aware_ of certain parts of the world, whether it is the outside world or the inner world. Therefore there does not exist something like philosophy as such. Only _philosophizing_ as a mental activity exists. Every time you study the work of some philosopher, you should not be interested in 'truth', but only in 'what did the philosopher see, become aware of, and what tries his theory _make me see!_
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A philosophy must always be a tautology, because the distinction between truth and falsity is not important in philosophy.
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But then, in the next step, science begins. Now that the philosopher has made it possible to see the world through the language the philosopher has created, which of the statements in his language _can be applied to reality,_ so that I can make a distinction between statements that are useful, and those that are useless. This is the epistemological step, which transforms a philosophy into a science.
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So as soon as the philosopher has defined a field of inquiry, and within that field of inquiry, there is found some method with which I can make a distinction between useful things or garbage. Or, better, if I can make a distinction between statements that are true or false within the domain of applicability of the philosophy, the philosophy has become a science. In short, whenever there is a method of testing, the domain of inquiry changes into a science.
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I have told you earlier, that so-called 'infinite sets' cannot be understood analytically. But they _can_ be understood synthetically. An infinite set, like those of the integers, do not _have_ the property of being infinite. No, they _lack_ the property of being finite. Infinite = not finite. So the infinite cannot be understood analytically. But it _can_ be understood synthetically.
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Just see for yourself. The definition of an uncountable infinite set, as given by Cantor, consists of showing that an exhaustive list of real numbers _cannot be constructed._ That is, a list that, if you extend it enough, will reach_every_ number between 0 and 1, as long as you go far enough in that list.
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So, by introducing the idea _of elimination,_ you can turn Cantor's theory of transfinite numbers into a sensible form of mathematics. And that _it is_ logical, follows from the fact, that Cantor's theory satisfies logic. That is enough to acknowledge it as sensible mathematics.
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I think that this whole analysis shows that your approach to mathematics is both right and wrong.
'Sir' don't you think our languages have also influenced mathematics from ancient times..?
If you don't want (very) smart people get involved in real worldly affairs, then let them play with complex ideas and theories that do not interfere with the real world. This kind of magic is applied to all universities and sciences unfortunately.
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PART 2:
Therefore, _logic_ is the thing that makes mathematics a science. The experimental test is the thing that makes a candidate for a theory of physics into a theory of physics or refutes it immediately, after which it is rubbish. Or there exists a domain of non-invalidity in the Popper sense.
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In the same manner, an axiomatic system is something else than a model. Group theory, for example, _gives an exact definition of symmetry, wherever it can be._ If you then show something, you can _recognize_ whether some aspect of it satisfies the definition of a group, and thus you have found a symmetry in that thing. The thing itself becomes an example of symmetry. Or, if you wish, a model of a certain kind of symmetry. But if the thing does not satisfy the axioms of group theory, then it is _definitely not_ a form of symmetry.
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In other words, when you have a form of mathematics, that is based on axioms, the axioms _cause that form of mathematics to become a science!_ This is, because the relationship between axioms and the construction that satisfy them is not a matter of derivation, but the relationship between axioms and the particular theory that the axioms define is _an epistemological_ one. The axioms are their _to test whether your mathematical constructions are done in the right way!_
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Therefore, you must not ask from the axioms themselves that you _construct_ models from them. No, you use the axioms _to focus your mind_ so that you make constructions _that satisfy the axioms._ And when you do, and you have made many derivations from the axioms, you also know that thát what you construct must also satisfy everything that follows from it. In particular, if you think of something within a certain field of mathematics, and that what you think of _violates_ one of the many things you derive from the axions, you _know_ that you have made a mistake.
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To give a simple example. If you begin with Euclid's axioms, and you construct a triangle within which the sum of the interior angles is not equal to 2 right angles, _you know_ that you have made a mistake. _That_ is the function of axioms.
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When I did not know all this, and I started to study Hilbert's extension of Euclid's axioms, I always had the feeling that something was wrong. And now I know it. Hilbert tried to construct a set of axions that really _define exactly_ what we are supposed to consider to be Euclidean geometry. Everything that was implicit in Euclid's work itself had to be made explicit, so that he had a theory that tells us _exactly_ what Euclidean geometry is. So he attempted to make _all_ assumptions of Euclid explicit.
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I think now that this is futile. You can make so many constructions in the plane. Not only lines with definite lengths by defining a unit length, and constructions with straight edge and compass. You can have curves of third degree, fourth degree etc. You can have all kinds of shapes. So you will never know whether you make something that requires an extra axiom. Therefore, using axioms as tools _for construction_ is the wrong way to look at axioms. Axioms are not for construction, but to give you _certainty after you have constructed patterns!_
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But that is not the function of axioms! Axioms are just methods that prevent you from making mistakes! Just like the example I gave of the photo of the four-legged animal. If you want to prevent including dolphins as possible things your photograph might show, then the four-legged 'axiom' derived from the photo is a way to avoid that mistake.
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My point is: the analytic approach to mathematics is about construction. It is bottom-up. The synthetic method of mathematics is 'destruction'. It is top-down, and its function _is not_ to find mathematical structures or patterns, but _to test_ if the mathematical structure or pattern you are investigating ar constructed without flaws.
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I must add, that my vision of mathematics is based on my many years of meditation.
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There exists a 'breakthrough' point in meditation, wherein a _separation_ between consciousness and experience occurs. It is known in the East as 'Dhyana' in India, 'Chan' in China, and Zen in Japan. It is a point of development of the human mind, wherein all thought and thinking stops. And when that happens, _experiencing itself_ stops, and there is _only consciousness._
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My vision of mathematics is based on this capacity of my mind.
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I _reconginze_ consciousness as the ability of the human mind _to eliminate,_ or to _silence._ And I recognize _experiencing_ as the ability of the human mind _to generate_ thought and thinking. The function_ of consciousness is the ability to eliminate errors. If you make computer programs, you know that every computer programming system has a way to catch mistakes. Since we depend for our survival on our capacity to learn, that is, to program our brains, we have a similar function. And that is _exactly_ what consciousness is!
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It is possible, to have thoughts without being conscious. That happens when you sleep and dream. The brains are disconnected from the body, and the brains are then able to allow _all_ thoughts you can possibly think,_ to take place. This function exists to support memory.
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The opposite is also possible. You can be conscious, without there being any experience. It is _total elimination of all thought and thinking._ In that state, the ability to observe your mistakes is maximized. This is why psychotherapy can help, especially Gestalt psychology. Gestalt psychology is a way to make you aware of the automatic thoughts that are part of what 'you are', and control you, and which bring you into trouble. If you see a connection between an automatic thought that is part of your identity, an action you take, _why_ you do it, that is, what you try to achieve, and _become conscious_ of _why_ this does not work, the troubling thought is silenced, and the therapy is a success. But let me return to mathematics.
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In the _synthetic, axiomatic_ approach mathematicians makes use of the power of consciousness. Of course, without knowing this. And in the analytical approach mathematicians make use of the power of experience.
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If you choose for one approach, as you say you do, and declare that approach as _the only sensible mathematics,_ you _exclude yourself_ from using the full power of your mind, because both approaches complement each other. They complement each other even _perfectly!_
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Or, more concretely, if you _reject_ the axiomatic approach of mathematics, you will never feel completely satisfied with mathematics, because you will never feel full certainty about your mathematical constructs.
I posted above a question about disjoint empty set and consistency of that notion in the axioms of ZFC. Do you have a comment on that question, can you answer it so that we can have full certainty of the axiomatic consistency ZFC (if we forget about Gödel and other stuff that set theoreticians don't consider important)? The question concerns only definition of disjoint sets, axiom of extensionality and axiom of regularity, which is the kid stuff, so if ZFC is axiomatically consistent, proving that the question does not lead to paradox should be easy peasy.
If you can't deny that axioms of ZFC lead to ridiculously blatant paradox, what does that mean for your more general philosophical argument about axioms providing full certainty in the sociological context? If people have believed 100 years that ZFC avoids blatant paradoxes, but it doesn't, have people believed in logical consistency of Cantor's paradise as such, or in Zermelo's sophistry and arguments from authority?
@@santerisatama5409 What a nice question!
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Just for the record:
\exists x\, \forall y\, \lnot (y \in x)
E(x)A(y)~(y€x)
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In my approach, predicate logic and set theory are equivalent. What I find interesting, is that predicate logic can be defined by a meta-axiomatic system. A meta-axiomatic system is a system that tells when something is an axiom.
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There is a nice description of this in many logic books. I myself have studied, and have used the book: Logic for Mathematicians by Hamilton also to teach logic. The first chapter gives an _analytical_ description of propositional logic. In the first chapter the truth table is defined, and it shows, in the last theorem, that there is a connection between tautologies and logical proofs. Every logical proof can be written as a tautology. And, conversely (not in the book) every tautology can be reformulated as a set of logical proofs. In this first chapter an _analytical definition_ (in my sense of the meaning) of a tautology is given, as a formula whose truth table only results in 1, that is, true, whatever the truth values assigned to the propositions in the formula.
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In the second chapter of Hamilton, a _synthetic_ definition of the propositional logic is given. It shows a meta-axiomatic definition of propositional logic, and a synthetic definition of tautology. It shows a system of meta-axioms from which exactly _all_ tautologies can be derived..
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In chapter 3 and 4 it does the same for an extension of the concept of tautology. It looks at predicate logic, and logical valid statements. It shows that you can have logical statements that are true for every possible interpretation, that is, in every possible mathematical theory.
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In chapter 3 it gives an analytic definition, and in chapter 4 a synthetic definition of predicate logic.
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Set theory is equivalent to predicate logic. Therefore set theory is a way to test mathematical theories., and is not a mathematical theory itself. Set theory is a _meta mathematical_ theory. It _tests_ whether mathematical theories are formulated correctly, whether analytical or synthetical.
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Therefore, the axiom E(x)A(y)~(y€x) = -O-
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is not an axiom that shows a mathematical theory. No, it describes a circumstance in which a construction of an analytical mathematical theory, _or an axiomatic system as such_ should be refuted.
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In general, it says something every mathematician knows. You should _reject_ every mathematical theory that contains a contradiction. Or, a mathematical construction based on a logical derivation which contains a contradiction does not lead to a construction, but leads to something you must refute.
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And that statement is not a mathematical statement itself, but is a meta-mathematical statement _about_ systems of mathematics, whether analytical or synthetic.
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Therefore, my solution to your very interesting question is that it is a false question. You are looking for something that does not exist, because you (and many other mathematicians) look at set theory as a mathematical theory, which it is not.
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My solution to this problem is very similar to Popper's solution to the induction problem. (The problem of how to move from particular facts to the genera statementsl.) Before Popper, people tried to construct a logical theory of induction, similar to a logical theory of deduction. Popper showed, that such a thing does not exist. The only thing that exists are conjectures (fantasies) and refutations. Therefore you cannot prove the truth of an induction by some logical process of induction. The only thing you can prove is that a theory is _not false_ as far as we know. Through testing you _construct_ a domain of non-invalidity, which you can _never_ assume to be infinite. (Absolute truths do not exist.)
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In the same manner, _logic_ is meta-mathematics. Logic as such consists of methods _of testing_ mathematical theories. If you eliminate all content of a mathematical theory, you are left not with nothing, but with thought and thinking as such. Logic tries to define certainty about thought and thinking itself. And you can try to catch that in axioms.
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The axiom E(x)A(y)~(y€x) = -0- tells in what circumstance you should _reject_ both an axiomatic system of mathematics, _and_ a synthetic system of mathematics. In itself it is neither a construction nor an elimination. It tells when we _should_ reject a construction, or when _we should_ trash an axiomatic system.
@@konradswart4069 ZFC is by definition a first order theory, so I can't consider your metamathematical sophistry about 2nd order logic etc. relevant to claim that the simple question is false question. Am I correct to interpret that you imply that ZFC does not need to be consistent because it is not a mathematical theory? Then why all the bother to try to avoid paradoxes of naive set theory by developing ZFC? Hasn't the primary motivation behind ZFC been to construct a logically sound theory of real numbers and number line continuum? Those motivations are now outside of mathematics? Perhaps to you, perphaps not, but the context of the discussion is also current sociology of mathematics, and the reactions to the simple question offer fascinating sociological data. It appears as if the question is taken as heretical violation of a religious dogma, which is ironical, as the question seems to refer to the same logical problem already pointed out by Bishop Berkeley, now hidden under a new disguise.
Is there a yes or no answer to the question, "does disjoint empty set exist in ZFC", on the level of first order logic? Or is the question undecidable or something even more weird in relation to binary logic? And please let me repeat that the context of the question is only ZFC, not set theory as a whole, what ever that could mean.
And to cover all sides, let's not exclude the possibility that there is some divine reason and purpose behind all this. Would space rockets etc. stop functioning and would we loss capability to prevent massive asteroid impacts if faith in ZFC was lost? I pose this question in all seriousness.
@@santerisatama5409 I am afraid, you still don't get it.
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You argue from the paradigm of logicism. That is, you assume that logic and set theory is a foundational point of departure. In the idea of logicism you try to define the integers in a way you know:
0 = -O-,
1 = { -O-},
2 = { -O-, { -O-}}
...
n + 1 = {n, { -O-}}
And then you build Z by a pair. Then Q by a pair. Then R by a Dedekind cut. And then _build_ mathematics from that.
That is not how I see it. What I see are two things, corresponding to the two functions of the brains. You have experience, and you have consciousness. Experience is construction. Consciousness is elimination. Understanding consists of both.
Logic is not a tool for construction, but it is a tool for testing. Therefore you begin with analytic mathematics, a bottom-up approach out of which deduction has developed. And then the synthetic method was discovered by Euclid, which begins by defining a context through axioms, and then show processes of elimination.
If I say, for example: you can draw one and only one straight line through two points, I define the method of the straight edge without units. Because I do not specify the points, and also do not specify the straight line, I mean _any_ two points, and _any_ straight line, as long as it goes through those two points. The theorem then states that from all the straight lines I can draw in the plane, _if_ it goes through those two points, then there is no other straight line going through those two points.
Therefore I am _eliminating_ the possibility that there can be another straight line going through those two points.
This is an example of what I mean by synthetic thinking. Synthetic thinking is eliminating thinking.
Our brains have two functions. In a fully developed human brain consciousness is in the left hemisphere. Experience is in the right hemisphere. If they work together, you have conscious experience, which most people confuse with consciousness itself, not realizing that with the common term 'consciousness' _two functions_ are meant.
The right hemisphere is primary a generator. Thoughts are the manifestation of experience. The right hemisphere is the experiencing half. You can experience without being conscious. This is what happens when you sleep and dream.
The other thing is much harder. Consciousness without experience. This happens, when the elimination function of the left hemisphere has developed so far, that it is capable of silencing all thought and thinking. The differentiation of the two halves of the brains has then become complete. The dominance of the left hemisphere over the right hemisphere has become complete, with the result that both hemispheres are then, for the first time, able to work together. And, contrary to what many believe, this type of 'enlightenment' is not pleasant at all. It is, especially at the beginning, very painful.
Why do I tell you all of this?
Because I have discovered that such a differentiation between experience and consciousness happens, exactly when you _realize, that there cannot be something like 'nothingness'!_ In other words, there cannot be something that is nothing.
But that is _exactly,_ what this axiom asserts!
The empty set is a thing that is nothing. If you 'accuse' me (not personally meant, we are having an exchange) of sophistry, well, _that_ is _to me, extreme sophistry!_
So there cannot be an object that is nothing. It was exactly when I saw this, after a tremendous mental struggle, that I understood what is meant with 'the empty set'. Namely, that there cannot be something like an empty set as a thought.
Nothing is not nothingness. Nothingness is a confusion.
Look at the following link: darksucker and the theory of dark.
www.lightandsound.net.au/darksuckerpage.html
_That_ is the confusion of the empty set!
But although there is not a 'thing' that is an empty set. There _can be understanding of why something like the empty set appears in mathematics!_ You can understand, that the empty set is not a mathematical object, but a symbol standing for the outcome of a process of elimination that become _so extreme,_ that _nothing remains!_ And that is a form of _synthetic understanding!_ One of the most difficult, I may ask, because if you really understand _exactly_ what the empty set _is,_ you understand that there can be _no thought representing the empty set!_ Because if you understand what the word 'nothing' _really means,_ you understand that 'nothing _cannot be nothing-ness!_ And when that understanding emerges in you, _all thought, all thinking, and all experiencing stops!_ And that is _quite a realization!_ One that causes a tremendous transformation of the brains. One that gives a very strange understanding of 'what' your 'I' is. The 'I' itself consists of thoughts.
Analytic and synthetic mathematics are two ways of understanding. You have understanding through thought and thinking. This is understanding through experiencing And you can have understanding through elimination. This is understanding through consciousness. So the two modes of understanding lead to two forms of mathematics. And that was the conflict between Brouwer and Hilbert. They were talking about two different things, and thought they were talking about the same thing.
And about your question. Mathematics is the ultimate way to program your own two brains. It unleashes the full potential of the brains. Our brains have two halves. _Everybody_ who has _any real skill in mathematics, is able to both construct and eliminate!_ And the thing what makes mathematics so difficult for many people, is the eliminative part. Learning mathematics is not like history. In history you only accumulate. In mathematics you must be able to both accumulate _and_ eliminate! It is especially _elimination_ that takes the effort. Learning to face mistakes, and correct them. _That is what makes learning mathematics so hard!_
I do not know whether you program computers. But any programming language has a detector of errors in the code. Sometimes they are very sophisticated. In fact, programmers could only then make very sophisticated computer programs when the error detectors in programming environments became very good. The better they became, the better programs computer programmers could make.
Our own consciousness has that role It is an error detector. That is why it is primarily an elimination function. And synthetic mathematics is what grew out of that function.
We do not have to have faith in ZFC or any other theory to destroy comets. Faith as such is only needed, if you have no awareness of your own power to correct your mistakes. Faith as such stops that. What we must learn is _understanding,_ which leads neither to faith nor truth. What it leads to is _certainty!_ And that is a never-ending quest, because all understanding requires contexts. And that can always be improved.
For example. First, we thought that the sun, moon, and earth revolve around the earth. Then we learned that only the moon revolves around the earth, and the earth + moon revolves around the sun. And finally, we learn from the General Theory of Relativity, that nothing 'moves', but all stands as still as possible in a warped space-time. 'Standing still' also gets a different meaning.
As a human being we do not have to understand everything. The only thing we must do, is understanding enough. We already know that when we see a comet coming that threatens us, and we know this early enough, we know that we must not try to destroy it, but give it a nudge, so that it misses the earth. We know enough about mechanics and computers to calculate how large such a nudge must be to be successful.
The greatest challenge is understanding understanding itself. I assert, that there are two forms of understanding: _experiencing_ and _consciousness_ which can work together. And that is how far I have gotten.
@@konradswart4069 Thank you for sharing, now I understand your position better. I also feel the pleasurable obligation to offer my complementary view.
What you call experiencing and conscience, I would call phenomenal and metacognitive experiencing. Yes, both can shut down in various relations for various durations and do so regularly during deep sleep without dreams. However, during deep sleep the 3rd level which I call (transpersonal and transcendental) awareness, continues also without phenomenal and/or metacognitive content. The latter can be triggered on from deep sleep by alarm bell etc. to give localized awareness experiental content in the relations you analyze, etc. Awareness, where you practice and observe your most rigorous philosophical skepticism, is the the ontological irreducible primitive, which continues in absence of metacognitive thoughts whether wake or asleep. Awareness continues also when breathing stops. I'm sorry for your pain and hope you are feeling better now.
As for my motivation, it is not logicism. I subscribe to intuitionist philosophy and in that respect what could be called empirical idealism, as phenomenal and intuitive mathematics has also transpersonal and transcendental level of existence in general awareness, meaning that mathematics is not limited only to subjective and intersubjective communication. Here I disagree with Brouwer and agree with later intuitionists who supported more generally shared and common idealist metaphysics beyond solipsist limitations.
My motivation is ethical. When we construct linguistic expressions for our intuitions - as well as for our axiomatic heuristic games - our languages should be as clear and accessible as possible, and honest to and loyal to the heuristic game rules we are playing. ZFC or any other set theory or attempt to present completed infinity as consistent with LNC does not satisfy ethical requirement of honesty and clarity of language.
ZFC is presented as a formal axiomatic theory. When confronted with potential formal inconsistency, no formal answer or conclusion is given by supporters of ZFC. Only semantic obfuscation, censorship and denial. The sociology of ZFC is not playing by it's own formal rules and it is behaving in dishonest and unethical way. My approach to language of ZFC is deconstructive with motivation of liberation and healing a sociologically unhealthy language game of mathematics. I'm not a logicist, I'm deeply radical revolutionary following the path of heart. I take seriously Badiou's view of set theoretical ontology, as well as his process philosophical truth theory. Set theoretical ontology and sociology in it's many aspects is suffering from post-truth syndrome, and we can't live without some kind of truth. A better truth that allows honesty.
At least you are engaging in valuable and genuine philosophical dialogue from your own non-conformist point of view, which is very much appreciated and I don't question your honesty nor authenticity of your experience, though I may have slightly different but benevolent metaphysical interpretation of it.
PS: in my language word for 'dark/black' is 'musta', and expression 'musta tuntuu' means both "It feels to me" and "dark is felt". I liked your link very much, because warmth of heart cannot be seen, only felt.
Given the tenor of so many of your very interesting talks, it seems to me that you need to be looking first at foundational - philosophical - issues; at Brouwer and Intuitionism, for example, and how a pure mathematics of that nature would impact upon physics. A sociology of mathematics would then be seen forv what it is - a second-order activity, the first-order activity being truly foundational. And you're very, very wrong in worrying about AI. AI has gone almost nowhere in fifty years. The howls of protest from its supporters should take a look at where they thought they were going to be by now with AI all those years ago. Philosophically, it's a busted flush. Robots making cars isn't what they had in mind: what they had in mind was an "intelligence" that would, for example, let a machine recognise a face. Billions of dollars later, they've almost (but not quite) got to the point of matching what every three-week old baby reaches with its mother.)
@A W I think your skepticism about AI might have been possible 5 or 10 years ago, but not any more. I will be explaining why in this series.
Hi, Norman , what's Norbert Wiener's equations to control human behavior all about ? Teach cybernetics math ,what a hot video series that would be . It's not that i want to control people - it's just so interesting and more to the point real [be it very abstract to ] . Norbert Wiener and his associates laid the foundations for this . where to from here ? this subject needs love in it or it kills by it's ability's to control human beings and limit there potentials . it's good stuff ,with Claude Shannon communication and game theory and um..machines and humans .. politic's ..the lot.. fascinating story but folks don't understand the math behind it. please teach cybernetics math Norman. Blaise pascal is more worthy than norbert Wiener ,for he was smart at theology ,very pleasing his Pensee's book of thoughts , this sentence is my attempt winning favor .
Further, the rot creeps.
The rot? 😂
Why aren't you precise when you talk? Rather than saying punchlines like "I claim that it is not true" (which non-mathematicians might take as granted and true), why don't you say "I choose intuitionist constructivism as a more natural, more beautiful form of logic, I have a religious relationship to the logic I choose, as if this choice had anything to do with the ideas of truth, reality, God, because I'm not rationalist but rather I'm idealist; and in that choice of logic, such a claim is not true because transcendant number are not a thing".
(Because I think you refer to transcendant numbers, because you wrote a polynomial, and I don't know whether the roots of this polynomial are transcendant or "just" irrational (sorry for incomplete knowledge of mathematics), but maybe it's simpler and you refuse even irrational numbers but then I can't see why you chose such a complicated argument rather than sqrt(2)).
Can an AI be more human? Can a person be less human?
you have to make a AI society to go with it , and AI relationships , and a AI leader of all Ai's. Moral ai's and imoral Ai's. seems a bit silly..but God made man , for relationship. He did not make us for a realtionship with Ai idol. What does God want..i think better relationships with humans and ..this makes us Fully human. the devil has a plan for that to.. unfortunately the devil is very bright.. which may be Mans greatest problem. this video has more relevance to us all. still waiting for an answer about Norbert wieners math is about ? ( - :. AI could make us whole lot less human . Ai could end up being more of a Satan than human.
"innate human tendencies' - ?!. Petitio principii in just under 2 minutes.
@Aron Gabor Do sociologists believe that there are "innate human tendencies"? I am pretty sure they do.
Dear Prof. Wildberger, some do, but that doesn't mean they are not committing the same mistake. Similarly, just because some tendencies might explain some attitudes or behavior, this first has to be proven and for that to succeed, alternative explanations need to be excluded. I reacted to your statement that apparently concluded that tendencies are key. If I am wrong, I am more than happy to retract everything and beg for apologies.
I understand the human brain to be an organ that receives a large amount of information and from this creates a subjective and more-or-less linear narrative that only contains the absolute minimum necessary from the information received by it, and most of it indirectly, in forms of relations.
Because of this, there is a human tendency to try to look for a single narrative, a single cause, a single modulator, a single possible correct way to do pure mathematics, a single possible way to communicate.
If you meant this tendency to be the root cause behind such theorems like the FTA, I actually agree with you, but that said, I am even more anxious to not just assume that we are right but actually go through the process of verifying it.
That said, I would still approach this not by trying to match behavior with tendencies, but with circumstances with perceived options. People can act against their tendencies if there is motivation and clear options to do so. Without such options, without a belief that things can be different, it's rather useless to point out to us, that we are less than what we present ourselves.
@@ashnur Thanks for the nice comment. I completely agree that we should not take any assumptions into the investigation. Perhaps "innate human tendencies" are present, but ultimately not nearly as important as societal or cultural or perhaps economic forces. The relative importance of different factors in determining individual or group behaviors is an interesting question. So... you might have to take my comments in this sociological direction with a grain of salt -- I am hoping others more skilled will take up the challenge.
@@njwildberger A Biologist or psychologist probably works with innate tendencies of individuals. Sociology studies the patterns that emerge from interaction between individuals that can not be explained by biology or psychology. Some social patterns might be explained by genetics - i.e. are innate - An example might be parents that care for their offspring. However that is not a sociological finding, it is a biological "law" (we do not need a fancy new thing as sociology to explain this). Sociology starts were biology and psychology ends. Innate tendencies - i.e genetically based tendencies - are not part of the repertoire of sociology.
@Me Too Even on this planet, we have thought of numbers differently throughout history. Dr. Wildberger has a great series on Babylonian system, which may have given them different insights and abilities.
I miss the old music 😢