Crisis in the Foundation of Mathematics | Infinite Series
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- เผยแพร่เมื่อ 14 มิ.ย. 2024
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What if the foundation that all of mathematics is built upon isn't as firm as we thought it was?
Note: The natural numbers sometimes include zero and sometimes don't -- it depends on how you define it. Within logic, zero is always included as a natural number.
Correction - The image shown at 8:15 is of Netwon's Principia and not Russell and Whitehead's Principia Mathematica as was intended.
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Mathematics is cumulative -- it builds on itself. That’s part of why you take math courses in a fairly prescribed order. To learn about matrices - big blocks of numbers - and the procedure for multiplying matrices, you need to know about numbers. Matrices are defined in terms of - in other words, constructed from - more fundamental objects: numbers.
References::
Probability website mentioned in comments: people.ischool.berkeley.edu/~n...
plato.stanford.edu/entries/ph...
plato.stanford.edu/entries/lo...
Ernst Snapper :: www.maa.org/sites/default/fil...
Philosophy of mathematics (Selected readings) edited by Paul Benacerraf and Hilary Putnam
Written and Hosted by Kelsey Houston-Edwards
Produced by Rusty Ward
Graphics by Ray Lux
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this is one of the best PBS's shows. They should hire her back!!
So true. When I need something explained about math, I wish Kelsey had a video.
She's a treasure. Should definitely be hosting a regular online program on PBS
@Persephone[Percy] she had to quit to finish her dissertation.
WHAAAA! This is my first video, I just subscribed because this was awesome (a lot of which often has to do with the presenter,) and this is the first comment?!? Noooo.
@@eleandrocustodio Great news, and I hope we see more of her. Shows like this are helping me through school. If she gets hired as a professor somewhere, I would 100% take a course. A good prof is essential to breaking down a text like those mentioned here - I'm not sure I'd have a chance on my without help for the books by Frege or Russell/Whitehead!!!
I feel very comfortable not having to write out a few hundred pages of justification whenever I need to add 1 and 1.
Mathematicians: "We prove the obvious, so that you don't have to!" =;o}
I love math for the fact I avoid papers lmao. But as a child, I always loved numbers and puzzles
And after proving that 1+1=2, he noted: "The above proposition is occasionally useful". (Of course, this means that it would be used to prove some other theorems.)
I count on a mental set of ten finger-numbers. This has saved me a lifetime's worth of fence-post errors, at least two of which were valuable to have missed.
One of Einstein's many advantages over the rest of us was apparently that his mental counting images included his toes.
@@therealpbristow Most of theorems of mathematics are nor obvious nor intuitive...
Astronaut #1: Wait, it's all just turtles?
Astronaut #2: Always has been.
Astronaut#3: I second that.
This is still one of my favorite episodes, dealing with one of my favorite areas of math / philosophy. Infinite Series, as well as other channels like 3blue1brown, go a long way towards making complex math a lot simpler for peeps like me to follow. This / the world completely fascinates me :-)
Same. I literally have dreams about the set theory. They have such an aura of mystery
That's actually Newton's Principia in the graphics.
hahaha! You are right Kushal! It's an example perhaps of the modern malaise of visuals over text. "Every webpage needs a picture" is the modern mantra. Yet it seems people have not learned how to as accurately proof read images as well as text. The result is TH-camr's and bloggers can just throw up images as eye candy without much thought.
I was scrolling the comments to see if anyone else noticed that lol. It says Newton right on the page!
Lol that's sad...
I thought Hey but that's Newton's work, when I read the publication titles. I didn't zoom in to check. Thanks for confirmation!
I didn't notice that but I did feel the cover was way to old looking for Russell's time
My crisis in math is that I've covered vectors in 4 different classes now, and I still don't know any more about them than did after the 1st class.
If you extend your arm and point at something, THAT IS A VECTOR!
A vector is a list of n numbers that define a point in n dimensional space. That is clear to me. But first you have to know what a space is.
@@simonmultiverse6349 so the thing you point at is a vector?
@@LucidStew NoooooooooooooooooooooooooOOOOOOOOOOOOoooooooOOOOOOOOOo!!!!
@@LucidStew I'm sorry but this statement explains why you don't understand vectors. You don't care enough to learn and that isn't really your fault.
@9:00 I like how we just cruise past the implications of Gödel.
Totally haha
It wasn't quite the foundational crisis like Russel's Paradox was. Besides, it came 3 decades after Russel's Paradox, and nearly a decade after the ZF axioms resolved that crisis. Sure, it was disappointing that, starting with a handful of axioms, you couldn't prove or disprove every possible mathematical statement. But many mathematicians already suspected that. After all, the first axiom alone cannot prove or disprove the second axiom; the first and second together cannot prove or disprove the third, and so on. You need at least 7 axioms to prove other important results; 9 is even better. But are that many axioms sufficient to prove _everything else?_ Clearly not.
@@imengaginginclown-to-clown9363 The 9 axioms of ZFC are mentioned in the video (starting at 8:30), which I hope you watched. Feel free to take it up with her if you disagree.
I'm guessing you're talking of the _axiom schemas_ used in the _first order_ theories of Presburger and Peano arithmetic, as well as ZFC.
The same theories can be formulated using second-order logic without using axiom schemas, that is, using a strictly finite number of axioms.
Regardless, Gödel's limitation applies to Peano and ZFC, but not Presburger. So I'm not sure what your point is.
@@imengaginginclown-to-clown9363 I feel we may be arguing semantics while basically agreeing on the underlying math. At the risk of repeating myself, and of telling you things you already know, let me explain once again:
First-order and second-order logic are _not_ theories by themselves. They are languages used to build theories. Presburger, Peano, and ZFC can be formulated in either first-order or second-order logic.
In first-order logic, you need axiom schemas which are, technically, an infinite number of axioms. But these are not arbitrary axioms; every axiom in a schema has exactly the same form.
This is important because, given any arbitrary mathematical statement, you must be able to tell, within a finite number of steps, whether that statement is an axiom or not. With a finite number of axioms, that's easy: You compare the statement with every axiom one-by-one. But what if you have an infinite number of axioms? If you start comparing the statement with the axioms one-by-one, it could take you forever-particularly if the statement is _not_ an axiom. This is unacceptable. So, even if your axioms are infinite, they cannot be arbitrary. They have to be comparable in a finite number of steps. See, that's where the requirement of some form of finiteness with regards to axioms comes from. In our case, we have a finite number of axiom schemas. So, given an arbitrary statement, we can quickly tell, from its form alone, whether it's an axiom in the schema or not.
This is completely different from having an infinite number of _arbitrary_ axioms. Some people (wrongly) suggest that you can overcome Gödel's limitation as follows: Every time you come across an undecidable statement, convert it (or it's converse) into an axiom, until all statements are either axioms, or decidable. That won't work because there are an infinite number of undecidable statements, and you cannot have an infinite number of _arbitrary_ axioms, as I explained earlier.
So go ahead and assume that I said "finite number of axiom _schemas"_ instead "finite number of axioms." My original summary of Gödel remains unchanged: "Starting with a finite number of axiom _schemas,_ you cannot prove or disprove every one of an infinite number of statements; some must remain undecidable."
Again: We do *not* use an infinite number of *_arbitrary_* axioms.
In second-order logic, you do _not_ need axiom schemas. Induction can be directly defined. You have a strictly finite number of axioms. But you still get the same results, including Gödel's undecidability.
*That's why I said that mathematical theories are based on a finite number of axioms.*
I was referring to either axioms in second-order logic, or axiom _schemas_ in first-order logic.
Let's not debate this any further; it's fruitless. I can't explain any better than I have.
Presburger is important to the theory of math, particularly as a comparison to Peano. It's not used in practice because it doesn't have multiplication. 🤣
Many people have wrong notions about Gödel. That's why Presburger is useful to clarify the concepts. For example:
- Is the undecidability of Peano and ZFC due to the axiom schemas (infinite axioms)? No, because Presburger also has them, but it is still decidable.
- Are _all_ mathematical theories subject to Gödel's limitations, that is, are they all undecidable, incomplete, and their consistency cannot be proved? No, Presburger is not subject to Gödel's limitations. It is proved to be decidable, consistent, and complete.
@@imengaginginclown-to-clown9363 Dude, I've already explained this before. Are you sure you're not just yanking my chain? Again: _I_ did not come up with the numbers 7 or 9. Please see the video again! The presenter mentioned that ZFC has 9 axioms, ZF without C has 8, and only 7 of the 9 ZFC axioms are proper axioms (or something like that). My point was that with a finite number of axioms / schemas, you cannot prove or disprove every possible statement. I could've used any finite number as an example. *I agree with you that the actual number is not important!* I used 7 and 9 because those were the specific numbers used *by the presenter.* Reusing the same numbers made my example more concrete. It let readers infer that ZFC (which, *according to the video,* has 9 axioms, of which 7 are "pure") is incomplete according to Gödel. The presenter didn't go into how a large number of axioms can be combined into a smaller number of axioms, and so I didn't either. These axioms are always taught as separate axioms; I learnt them as separate axioms; and they are used as separate axioms in proving theorems.
This is like the problem of how do you define a word when its definition must contain another word and as such eventually you must do the nonsensical thing of using a word to define a piece of what defines it. Despite this we can communicate clearly and that just seems magical to me, and similarly maths seems solid and logical and clear despite it being based on axioms themselves with limited foundation. I think this type of thinking is just a wonderful part of the human condition as weird as it is to understand something you can't define without using itself.
The phenomena you describe with words is quite fascinating. Yet there is no first axiom of words, perhaps it is the "I" which exists and allows all else to follow.
Wow you just blew my mind with that first sentence tbh lol. Super interesting comment
What underlies our modern languages now I think, is a set of words that a person associates, or can associate, directly with thoughts. If this wasn't the case, language wouldn't work. From a more general perspective, human language evolved continuously from, and in parallel to, life itself. From direct mechanical and chemical cues between cells, to more sophisticated sensory input, to non-verbal cues and facial expressions, to miming and gestures, to grunts/whistles/clicks/calls, to a super rudimentary "language" of a few basic nouns and verbs, and so on so forth. So the machinery of language, and thus the "axioms" of language, are quite literally in our DNA.
Yep, we are coded to understand language, but we don't understand it very precisely. We all have different beliefs about what every word represents, yet it is still useful. Math can be like that sometimes, but also it isn't. Every computer is coded to know that 1 and zero have some concrete, objective relationship to physical reality (they represent physical electrons).
You'd enjoy Derrida's work on language.
The title scared me. I was expecting a Gödel's incompleteness theorem type discovery but more deadly.
Jane Black yeah, you can prove that math is not actually false or inconsistent, so incomplete and undecidable is literally the worse it'll ever be. I don't think people appreciate how devastating that simple truth is.
wolfspam- can you explain a bit more so I can appreciate it?
Señor Gooba At least it wasn't a basilisk hack.
Free Diugh the best explanation is a little technical, relying on second order logic, but I'll give it a try.
In one sentence, the math we already know is beyond doubt, but Gödel's theorem cast literally everything else in math into doubt. In more detail...
No system can be proven false, or more accurately, no axiom can be proven false within a system, and we know that math "works" (aligns with our intuition and experience).
Consistency is the bare minimum measure of explanatory power of a formal system. In an inconsistent system, every single well formed sentence is true. Math neatly divides true and false statements, therefore it's consistent.
This leaves only completeness and decidability. Completeness means that any well formed sentence in a system can be proven true or false. In math that was the assumption that any conjecture or question you could correctly express in the mathematical language could, given enough work, lead to a theorem. It's the assumption any mathematician makes when they start working on a problem: that there is a solution.
What happened is that Gödel proved that some sentences simply could not be proven true or false, shattering the idea that we could neatly explore all of math from a finite set of axioms. Immediately after, he also proved undecidability, that is, that some problems were not only unsolvable, but it could not be known wether they were solvable.
This caused an actual crisis in mathematics, and deservedly so, because mathematicians were faced with proof that they might be working in vain to prove something that literally cannot be proven.
It also cooled down the formalists' efforts to eradicate "islands", or branches of mathematics that relied on their on some specific set of axioms, by expressing everything in terms of a minimal set, because we now knew that islands must exist, since no finite set of axioms could ever describe the whole of mathematics. That's what fuels the debates regarding the axiom of choice, the Continuum axiom etc.
That was it. Mathematics was quite literally incomplete. There was proof that a chunk of unknowable size was missing. Now we knew that there were things about math that we will never know and that we couldn't even know what those things were or how much of it there was.
wolfspam- that's very interesting and well explained. Why is "Consistency is the bare minimum measure of explanatory power of a formal system"? and what are some examples of known problems without completeness and decidability? Wouldn't discovering something that can't be proven true or false be the same as finding an axiom?
8:16 Wrong Principia... The editors must be out to lunch again.
Good catch!
I was just about to make the same observation, glad someone else spotted it too!
It even starts as Philosophiae naturalis and has Newton as the author. How do you mess that up?
I was going to say, I don't think Bertrand Russell's philosophical tome had a title all in Latin, or "Autore I.S. Newton" anywhere on the cover. And I'm sure it wasn't printed in a hand cranked printing press. It's a silly little error though and it made me laugh so I don't mind
Spotted this too. Busted!
Admittedly I keep videos like this to help me fall asleep. Have to wake up early because I got a Cs and Ds in high school.
Since subscribing to your Chanel, I find myself picuring you explaining thing to me when I study maths for an examn or had to proove soume engineering theory mathematicly. You restored my love for maths! Thank you for that!
Your pic of Russell's book is of Newton's Principia, which is 200 years older...
Lol. I was wondering why they choose to publish their book in the style of an ancient manuscript. I didn't catch that, thanks! :)
Newton is the superior product
Just a minor and natural mistake, the Dedekind cut for the square root of two is wrong. The sets should be {x in Q | x^2 < 2 or x < 0} and {x in Q | x^2 >= 2 and x >= 0}.
Jane Black What is used is not a Dedekind cut because it does not divide the rationals in a initial and a terminal sets, the set {x in Q | x^2 < 2} are the rationals in ]-sqrt(2), sqrt(2)[. You don't use Dedekind cuts to define the imaginary numbers, you use the already defined real numbers, the imaginary numbers are R^2 with the product (a, b) * (c, d) = (ac - bd, ad + bc)
Yeah. I also came down here to point out that there are negative solutions to x^2 < 2 that are missing from the diagram.
This is absolutely true but I'd regard it as entirely proper to not bring up the idea of x
It has nothing to do with imaginary numbers, just with negative numbers, which were already introduced. If x is -3, then it would not belong to the left part of the cut (and it should).
João Enes shut up egg head.
The video I've been waiting to watch my entire life.
Wait, was that really your entire life that I've been watching right now?
I swear when you said "What grounds mathematics?" I thought "It's turtles all the way down" and then you said it. I must be psychic
"All the things you can get by dividing integers"
*1/0*
no u
Ha, doesn't matter there's no rational basis for infinity or repeating decimals (if it's impossible for every digit to exist at the same time--it's not a number, in logic), so the whole system really stops at rational numbers anyway. Everything else is a totally different, and less rigorous logic.
@@onetwothree4148 No. There is no less rigorous logic in the definition of square roots or pi. You simply don't understand how these things are defined if you think that.
@@skepticmoderate5790 or you don't understand the logic of defining finite entities. You can't add infinity in the logic of arithmetic. Try programing a computer with an irrational number. No computation with irrational numbers is possible unless they cancel out.
@Paul Guaguin you're confusing mathematics with logic. Many branches of mathematics have nothing to do with logic or the world we live in
Halfway through Russel's Introduction to Mathematical Philosophy, a pleasant serendipity! :)
A video on foundations without mentioning Cantor, that’s something
we got the set theory out of this very exercise courtesy of Cantor.
Frege: finally I can put a foundation to math
Russell: think again bro
Frege: woah you log-blocked me bro
Wittgenstein has entered the chat.
I'd love if you could dig deeper into the ZF axioms and provide an example of how the axioms could be used to derive a theorem in classical mathematics.
It all depends on how hard you want to make the proof. Lets say we want to probe the Pythagorean Triangle Theorem with ZF.
The "easy" way is to first use ZF to work with sets and construct the Natural numbers. This implies proving everything you might need until then, so there is quite a lot of things to show. Then, you drop ZF and you can work with the Natural numbers like an ancient Greek.
The hard way would be to keep working with the integers as sets instead of numbers. Or sticking to the ZF axioms.
In general, you want to use the biggest blocks you can in proofs or problem. Once a block or theorem is proved, just use it. Otherwise things can get tedious fast. That is generally why we only see ZF used when working with sets.
Fnors also, if you look very closely, some blocks are not actually connected but assumed to be connected. Much of the work to unify all math cooled down after the crisis described in this video. But yeah, it's generally easier to prove a working framework in an axiom set instead of each individual already proven theorem.
Trevor Kafka ZF? I love Soviet Womble
If you're into overly verbose proofs, I can only recommend the book "mathematics made difficult" by Carl Linderholm. It's hilarious!
The investigation of the ZF axioms is quite removed from naïve set theory, so no mathematician would be interested in doing this. In other words, axiomatic set theory studies the axioms of ZF almost exclusively, whereas naïve set theory studies the consequences of these axioms. Hypothetically one could write such a proof involving both these fields of math, but it would be overly cumbersome to the point of being unreadable, even to mathematicians who were well versed in both subjects. It's similar to why mathematicians don't usually cite, e.g., the ZF axioms when writing an analysis proof. Sure the underlying logical structure is there, but really you get the entire picture from studying the two components and melding them together explicitly really just makes for a monstrous, overly complicated proof.
For anyone looking for the last link:
people.ischool.berkeley.edu/~nick/aaronson-oracle/index.html
It may be added to the description soon, but might save you typing it out.
THANK YOU!
If I concentrate and intentionally try to be random, I can keep it at 50-51%. If I make a relaxed run, it's around 58-60%.
You can train it on a certain pattern and then break it. For me it achieved 16% accuracy. Of course, I wasn't random, but it failed to predict my moves. Glorious victory for the fleshlings!
What pattern did you use?
Whatever the pattern he used, when I said 51%, I assumed 250, 300 key presses. I don't think it really counts if it's fewer than that.
This video gets better with time. Rewatching now, years later than the first time, I find deeper meaning to all the statements and facts presented here; really, it works on many levels and has something for everyone
I always come back to this channel. Really love those math videos 😋
No, please! Not another fascinating TH-cam Channel! There isn't enough time to watch them all! ... oh well... SUBSCRIBED.
The link to the webpage about guessing the key pressed at random is not in the description.
Im so happy these videos are getting 200K + views. Great work!
Hearing you talk about Principia Mathematica is beautiful!
6:53 Did you record that one afterwards? I'm interested how you pronounced it before overplaying it xD
Dedekind cuts are usually (e.g. in Landau's book) just defined for positive numbers, and then negative reals are defined on top of that. The problem is that the lower set in the video contained numbers like -7, and (-7)^2
Yes, the cut for x^2
> the condition defining the sets has to give you ideals
What exactly is meant by "ideal," here? It can't be algebraic since the only proper ideal of *Q* is trivial.
@@NateROCKS112 Ah. Failed to notice that the term has more than one usage in maths. I meant "ideal" as in mathematical order theory.
POWERFUL STUFF! THANKS FOR THIS AND ALL PBS DIGITAL STUDIOS SERIES!
Rudy Rucker's book "Infinity and the Mind" sent me down this and other lovely rabbit holes some twenty plus years ago. I'm encouraged that there are people out there that enjoy these things, though I've met precious few in person. Kudos.
After advanced calculus comes tooth decay.
What does that joke mean?
@@mpcc2022 "Calculus" is another word for tartar.
Dental joke
Biting humor.
I was REALLY hoping for Wittgenstein to show up after Russel, tearing down his work using their own "logicism methods". Mostly because he does it in such a badass way!
WOW, I started with set theory a week ago, ever googled anything, and now this pops up on my front page, thank you, youtube
I'm not an expert here, but If I remember well, what Godel states is that coherence can't be implied from ZFC, meaning that you will always find a model of ZFC which is coherent and another model which is not. So if you find a possible inconsistency in your model, you always have another model which will be consistent, because if there was none, then non coherence could be implied from ZFC. So even though we might find inconsistencies in models it isn't a big deal. And when we think about it, it's kind of what we always did in math, every time we found a true paradox, we just reshaped math so that it would not be affected by that paradox, like Russel's paradox.
4:26 One issue with the Dedekind cut for the square root of 2. It defines the lower set as “All rational numbers x such that x^2 < 2”. This would cause there to be two cuts as the lower set would have two boundaries with the upper set! It should be instead defined as “All rational numbers x such that x^2 < 2 OR x < 0”. This would create a lower set with only one cut as portrayed in the video by including all negative x values. The same logic applies to the upper set, being redefined as “All rational numbers y such that y^2 >= 2 AND y >= 0”. This creates an upper set with the same cut as portrayed in the video, by restricting it to positive x values and 0.
Feel free to tell me if I’ve interpreted this video wrong! If this cut was meant to define both the negative and positive square root of 2, then your definitions are valid but do not agree with your video graphic and are not mentioned. This could cause some confusion.
Of course you are absolutely right. Though the explanation of a Dedekind cut, using the √2 example, is so botched that the whole concept likely isn't supposed to be of importance for the video anyway.
I think many people would enjoy more fundamental expositions of category theory, topos theory or homotopic type theory at this point. Or maybe just a hint to wetten the appetite.
Can category theory serve as a foundation for mathematics?
Vincent Goossens as categories are generalizations of generalizations it's more likely to represent the top of the math pyramid
Vincent Goossens Yes, category theory can be used as another foundation of math by using toposes... which are themselves a kind of category. math.stackexchange.com/questions/1519330/is-it-possible-to-formulate-category-theory-without-set-theory
Leandro Suzano Also partially true. A lot of set-theoretic math can be derived from studying the category of sets as well. en.m.wikipedia.org/wiki/Category_of_sets?wprov=sfla1
I would personally like to see something more along the lines of analytic number theory or complex analysis... I mean those actually involve infinite series.
I have just discovered your channel and I think I love even more maths and women now
< sigh > I enjoyed these videos _so_ much when the series was ongoing, and I still enjoy them to this day.
*has an Gödel's incompleteness theorem feeling, lets see if this plays out*
Oh we breezed right by him. I love incompleteness cause deep in my heart I want to be a neo-Platonist, I want to buy into the idea that maths and logic have some kind of link and tell us something more about the world than just something in our heads...that maths and logic are part of the universe we are tapping into and not just a strange quirk of humans and our monkey brains.
Incompleteness kind of breaks this, but also rejoins it at the same time...its weird. In a way, I think most major philosophers of science, logic and maths still don't know what to do with incompleteness exactly...hell, even Gödel remained a neo-Platonist to his dying day! The great paper "the unreasonable effectiveness of mathematics in the natural sciences" is really the crux of this seeming problem, that math seems to have this other-worldy way to understand the universe, but as of yet, seems to be a complete fabrication of our minds. Most of the fabrications of our minds are completely nuts...dare I saw 99.99 repeating. But math, for some odd reason, seems to explain the universe in greater deal and precision than anything ever...and yet, it remains completely without foundation of a set of proved axioms...See, I can just talk myself into a fit with this stuff, and it just goes round and round like some evil barber paradox!
What compels you to think that mathematics is a fabrication of our minds? How does incompleteness lead to that in any way?
Yes, let me clear that up. Well, I am actually compelled the other way, like I say, my bias is a neo-Platonist one. Incompleteness doesn't strictly lead to a death of neo-Platonism either, more that most of my life has been an ebbing away of things that exist absolutely and external to myself to that which is completely derived by subjective experience. Or to say it another way, the only objective things are things of my own mind, and not the Kantian "Noumea" as I like to have pretended.
I would like to believe, however, that there is something special about mathematics and logic. Something that isn't strictly a human creation of minds, but some kind of "thing" about the universe which we understand. That math and logic ARE something real about the universe and not something like my experience of red. While my experience of red is the only kind of truth thing I can say exists for me, it is also trapped in that subjective solipsism I always wish wasn't the case.
While incompleteness more accurately says 'there are truth things that can't be proved' than anything about monkey brains, Noumea, and subjective reality....it seems to poke some holes in my ability to form a coherent bridge from subjective solipsism to Noumea about the universe itself. If there are truth things that can't be proved it is hard to suppose we can be able to truly say we can make a knowledgeable claim about some kind of objective, external reality.
That is the incompleteness part, the math part being a fabrication of our brains is a necessary understanding of math if one puts aside the idea that there is some kind of objective reality and things which exist independent of our ability to think about them. Many sciency people would suppose there IS an objective world, and we are experiencing it with our senses. I, on the other hand, with my own brand of epistemology submit it is a foolish thing to suppose anything about the world and we can only talk about the world of our senses. That subjective reality is the actual objective reality we experience and anything we suppose is outside ourselves is an unsupportable supposition. So the idea that math exists OUTSIDE human minds is a silly-nonsense thing in this way of viewing reality. Math and numbers HAS to come from our minds, because that is where ALL things about us come from. The idea that there is "math" out there in the universe has to be proven, and so far has not been so...not to my satisfaction anyway.
This is even though I desperately would like it to be the opposite. Math SEEEMS so different from all other human creations. It does seem special. It does seem to take on this thing I could imagine is objective and external. I just don't know that it is, there is still this gulf for me to be able to doubt it with good supporting reasons that are way to long and boring for a youtube post, I have gone into some of them but I know it is insufficient cause it is like 20 years of my own personal philosophy and has nooks and crannies I haven't gotten into.
TL DR: Because all reality is subjective reality, subjective reality is the only reality we can know, it is the objective reality so all things must come from human minds. If incompleteness tells us there are facts that are true that can't be proven, then there hope of understanding some kind of external reality absolutely is not certain to be the case. Human knowledge and understanding might come up against a wall for which is impossible to surpass.
I think this whole subject is much clearer in terms of computability. Alan Turing showed us that any (useful) axiomatic system can be encapsulated in a Universal Turing Machine. The important part is that it can be written down in a finite number of bits. Even something like pi which has an infinite number of digits can still be expressed by a finite computer program. A statement or object in mathematics is provable iff it is computable. The reason why something is not provable is because it would require an infinite amount of information to describe it. An example of this is Chaitin’s Constant, which has a specific value but it's not computable.
Saying math has no foundation is a bit confusing. I think it make more sense to recognize that the real issue is that we are bounded by the laws of computability i.e. we are bound to objects which only require a finite amount of information to express (a finite kolmogorov complexity).
Computability, in my view, has no bearing on "truth". And humans aren't really bound by computability, we deal with infinities all the time, well, as long as we establish certain rules for dealing with them. And there inlays the problem. There are no set of axioms we can use math to prove are the right axioms. Just as Gödel points out to us, there is no system that is complete that is also consistent. All complete systems must be inconsistent and all consistent systems must be incomplete. This is the crux of it for me. More pragmatic things like "useful" axiomatic systems are something I use for my more daily life affairs obviously, but for my big picture stuff, pragmatic things like computability are like logical positimism and beg more questions than I care for my epistemology!
But perhaps I misunderstood your meaning, please do go on if I miss the point, I have a habit of that on youtube for lofty conversations such as these
Ummm ... okay ... so are hyperreal numbers of "finite kolmogorov complexity"? And if so, what's the program to compute the value of any the non-real hyperreal number (i.e. a hyperreal number that isn't also a real number)? Just curious.
"Turtles all the way down" -really? XD
MrDivinity22 +
THAT reference!
I liked that one to be honest.
The phrase has been used in philosophy to refer to this kind of infinite regress for quite a while. Wiki says it may have been coined by Bertrand Russel so it probably would have turned up in the script anyway.
Tortoises actually.
Take my subscription, that's a thought I've been having everyday a lot of time before watching this video, and it inspires me to continue searching a solid answer, thanks 😎🇨🇴
The algorithm blessed me with another viewing of this video after several years. I miss Kelsey.
I LOVE PBS INFINITE SERIES
The question of whether math was discovered or invented has always blown my mind.
Shannon Wiggins definitely invented
it's an invention founded on discovered basic principles
Definitely discovered. All the tools and logic are invented but their properties are discovered, they'd come about regardless of how we invented things.
+Shannon Wiggins It's actually a bit of both. We invent the axioms of math, but then we discover what the consequences of those axioms are. But we can change the axioms on a whim and see what happens (that's how a lot of new math is discovered), and so they are most certainly invented by us. But it's not possible to see beforehand what all the consequences are of any set of axioms, and so that part we need to discover.
That's why math has a feel of both to it.
100% invented.
One of your better videos! Keep up the good work.
hi is there any way u can mesaure speed if we consider time as infinite and if we dont have variable time t
OMG! Your pronunciation of "Raphael" is perfect. Either you speak a romantic language fluently or your math skills make you linguistically smart.
Time link please!
I love ur typo, it so lovely.
@@amazinggrace5692 10:45
Smart people are just smart.
It's a Hebrew name, but her pronunciation is decent.
I first heard about "turtles all the way down" in Douglas Hofstadter "Gödel, Escher, Bach: An Eternal Golden Braid" 1979 book, and now here. I guess I've completed a closed path...
There's also a John green book called "turtles all the way down".
The anecdote is also mentioned in Stephen Hawking's 'Brief History of Time'.
Uve literally just said uve read one book (pretty hyped up, but dull). Read thousands.
These videos never get old. Wish Kelsey was still making them.
How do you get transcendental numbers with Dedekind cut? I only see it working for creating irrational numbers.
alt title: kid got mad about having to show his work when adding single digit numbers and took it personally
Wasn't that Newtons Principia at 8:22!?
For the website mentioned in the endnotes, the way I beat (got a 50% guess score for large set) it was by randomly selecting large even numbers and if it was divisible by four, D otherwise F: this worked really well in letting me select a number an outcome randomly by obfuscating the outcome from my brain during the selection process: until of course my brain picked up on things well enough to know whether my number would translate to f or d before the number was fully generated.
My anecdotal conclusion is that creating random outcomes is easier if you overload your brain's pattern recognition: the catch is that doing so will make the generation process slower.
combining pi and e - this sounds like a worthwhile saturday
Thank you! I never before understood the context of why it would take so long to prove 1 + 1 = 2 (what I thought was a defining axiom).
Also, Whitehead's and Russel's _Principia Mathematica_ is one of the few (possibly _only_ ) real-world example of a scholarly volume that broke the mind of the author (at least in Russel's case, according to his colleagues. He went a bit crazy afterwards). Unlike Lovecraft's other fictional works, I suspect it was a matter of the meticulosity and tedium necessary to follow it, not so much truths that are too terrible for mere mortals to behold. (Ia!)
Too often, people label things they don't understand as crazy. May be he was sane, but the people around him couldn't understand his _supposedly_ weird ideas.
*Question:* Why is mathematics in school taught so robotically, leaning so heavily on rote, when the other sciences build on observation and intuition? It makes math seem so impersonal that it's something to be avoided at all costs.
Because no child is left behind.
Because it's a whole lot easier to do it that way. Because we have to teach kids in elementary school.
And to be fair, we also have to memorize the periodic table.
I had to memorize the first thirty elements (up to zinc!) -- not because we were told that's what chemistry is all about -- but because it's more convenient than having to refer back to the table every time we saw a compound.
I have to agree with my teacher on that one. Some things are good to memorize.
+pixeldictator Fair enough. Plus, it's awfully hard to judge critical reasoning.
Who ever made you memorize the periodic table? What a waste of resources. Never memorize what can be so easily looked up!
Awesome video! Thanks for the enlightment
Great video.
I was waiting for you to bring up Kurt Godel's famous 1931 result proving that any non-trivial system of mathematical logic is either incomplete or inconsistent.
Dedekind was really dedicated, to establish the Foundations of mathematics
8:15 oops, that's Newton's Principea Mathematica not Russell's.
Russel and Newton both used the same title for their works.
@@javaguy418 I know but on that page it clearly states that it was written by Newton.
You are only about the 500th person to make this comment. Is it really too hard to check the comments before posting something so obvious?
was at the start and at the end of the channel. the feels.
I expected this video to be about univalent foundations, was surprised ;)
Hm, your √2 example is not a Dedekind cut because {x | x² < 2} isn't really a lower set because (-5)² ≮ 2. But I don't know how to fix that part of the video easily.
(As far as I understand Dedekind cuts. I might be wrong.)
Maybe do a ³√2 example instead?
Well, she defined that if there is no smallest value in the upper set (as in {x | x² < 2}), then this defines the gap between the two sets. Now with "gap" I assume she means the size of the gap between the positive and negative subset of x, or in other words the size of y.
But then she would've actually defined 2*sqrt(2) and not sqrt(2).
And btw we would need a proper definition for "size", which would go way beyond the point she was going to make.
My point is that the pair of sets is a cut for both √2 and -√2. Although the latter is not shown in the video.
Oh, the fix is easy, the lower set should be just:
{x >= 0 | x^2 < 2} U {x < 0}
(and the upper set the rest of Q)
Pretty sure that's a picture of Newton's Principia at 8:13 ...
My first year of community college, I decided to really apply myself. Math was giving me problems and I made a fuss the entire semester saying that I don't understand. I did the final exam feeling like I was in the dark and I came out of it completely deflated.
I nearly aced it to the surprise of the teacher and myself. Relying on rote was not a problem, but it never satisfied me, when I told my teacher I didn't understand, the explanations in this video is what I was craving.
This is why I'm an engineer and not a mathematician. I'm fine with the "foundation", as long as it does a good job of predicting what I'm about to build will actually work!
That's Principia Mathematica by Issac Newton, not Russell and Whitehead
Please do a video on the proof of why tree(3) isn't infinite
because some proven theorem says it. And that theorem is like hieroglyphics to anyone not really advanced in some specific area of mathematics.
Numberphile i assume.
yup
Ethanol 314 , you know iy
Parakmi I, I agreed, but PBS spacetime spends multiple episodes covering the more esoteric topics in physics, building to the answer over time. I suggest they take a similar approach here. It would be an interesting ride.
A great explanation. Thanks
6:53 - the name is on another audio channel, which is slightly more loud and distorted.
The black background is so cool, I hope it stays :)
keep up the good work!
Biology is Applied Chemistry
Chemistry is Applied Physics
Physics is Applied Mathematics
Mathematics is Applied Philosophy
Philosophy is Applied Pre-Frontal Cortex Neurons
Pre-Frontal Cortex Neurons is Applied Biology
Leonardo Celente okay I actually laughed out loud
Rather than considering it a joke, I would say that could be considered quite insightful.
*_Woooahhhhhhhhh_*
yeah..that's not accurate enough. not all chemistry has to do with biology, not all physics has to do with chemistry, not all math has to do with physics and not all philosophy has to do with math
Physics is Applied Philosophy too
*Yet one more crisis.* 2017 has truly been a sucky year.
It's almost beating 2016
Already has, John Dunsworth has passed :(
That crisis was like 100 years ago. Sure it isn't solved, but if you're that shocked right now you must've been living under a proverbial rock.
@0x: These issues tend to get swept under the rug. I'm not a mathematician, but I do have a Bachelor's of Science and have a layman's interest in math (including formal methods), and this is the first time I've heard of the problem with the Axiom of Infinity (it always did strike me as odd). I have heard about the controversy surrounding the Axiom of Choice. Math is in a weird place between formalism and intuitionism.
Bad things happen? Oh boy what a bad current year.
She mentions Principia Mathematica by Russell and Whitehead 1903, but the screen shows Philosophiæ Naturalis Principia Mathematica by Newton 1686
I am glad I'm not the only person that noticed that! Looks to me like the video editor doesn't know the difference, and pulled in the first image that came up on a Google search for "Principia Mathematica"...
I was terrified when I first read about this years ago, and in a way I still am.
Kudos for the "turtles, all the way down" reference.
It's a shame you simply glossed over Godel's incompleteness theorem- profound and incredibly important to the Hilbert program to establish foundations
Agreed! If anything, Godel's Theorem - which is logically PROVEN - has become (quite bizarrely) the unofficial 'bottom' of the the pyramid! So it's the logically proven impossibility of EVER having completeness that is THE fundamental truth beneath it all! Who'd have thunk it?!? "You said that the Set of All Sets - which was the Set He had written in the Book..." Exactly! tavi.
oh such a nice and nostalgic discussion, it reminds me the similar discusdion on logicism and math in my first Bachelor math class
Absolutely a great video
I am quite disturbed by the choice of the law of excluded middle as an example of an obviously true statement since it isn’t even true in some contexts, such as constructive type theory.
How do you get Transcendental Numbers from Dedikind Cuts? It seems it would only allow you to construct the Algebraic Numbers?
That's a good question. I'd like to know the answer to that one as well.
rngwrldngnr the cuts don't have to be definable, you just take them all. You must do so since there are more real numbers than formulae defining sets of rational numbers.
Dear rngwrldngnr: The fact that the example chosen of a Dedekind cut happens to involve the solution of an algebraic equation does not mean transcendentals fall outside the real numbers. As Larry Niven would have told you (he being a mathematician), transcendental numbers are real, and so indeed can be defined by Dedekind (not Dedikind) cuts. They form a subset of the irrational numbers which in turn form a subset of the real numbers.
Dedikind Cuts seem like they are pre-supposing the absolute location of the cut is definable. How do you pin-down the location of an infinite, non-repeating decimal string? How many irrationals are between the two nearest rationals? Do they alternate? Is there an infinite number of irrationals between any two rationals? You're talking about infinite length representations (decimal digits); What's an extra infinitesimal or two? In my opinion, it is a logical mistake to attribute any significance to infinitesimal differences that cannot in any practical way be measured or detected. You're opening a huge mathematical can of worms by allowing this. Just because a process tends towards a value (limit) does not automatically mean the value must exist or be definable on its own.
Up to 5:20 there is the explanation of bulding up Integers, Rationals, and Reals. Later she says that ZFC was supposed to be based solely in Logic, but the axioms of Infinity and Choice are not so, they talk of physical stuff. So the foundation of modern mathematics is not purely logic.
Desde algún lugar de éste planeta doy gracias por éste contenido, muchas gracias.
Judging from the title, I actually thought she was going to talk about issues with Common Core Math...
thishadowithin even I thought so
Yes, they should have clarified that the video was about the history of math not some recent crisis.
Apparently there's a crisis in reading comprehension
@@iii-ei5cv - who do you think lacked in reading comprehension and why do you think so?
@@iii-ei5cv
Oh, screw off. The clickbait title obviously claims that some error was *recently* found in the foundations of math. That's what a crisis is.
I think logic should be the first thing kids learn in school before numbers.... I think it would do wonders for our society lol
They will never give people the education they need to overthrow those who rule society.
@@rantan1618 and the ability to think critically I believe is number one on that list.
In fact that is how I was taught during the 1970s: in pre-school, we had a board game with Venn diagrams and little cards with faces: I had to put the laughing round face with curly hair in the intersection of the laughing AND curly AND round sets. At age six, I learned using parallel switches (OR) and serial switches (AND) to light a lamp. We also had classes of "logical thinking" throughout primary school, where we had to solve all kinds of puzzles.
It was called "moderne wiskunde" (modern maths), and was heavily influenced by the Bourbaki Programme. However, by the time I went to high school in the 1980s that approach had been mostly abandoned and was considered a failure, and it has a very bad reputation today - which I find rather unfortunate since it worked out great for me, but seemingly I was in the minority in that respect.
@@koenlefever it’s funny because I was just thinking of how to make logic into a kids game my gf is an Elementary school teacher and if I can come up with something she will probly implement it as game for the students after normal curriculum, she understands the importance of critical thinking skills and I think we can all agree logic and math is really the best tool to teach that, in my opinion that is why it is taught because most people will never need to actually do the computations in real life but the theory behind the computations is the real value.
So the inverse of the operations created new numbers (subtraction for addition created negative numbers, division for multiplication created fractions, nth root for exponential created complex numbers). Can we extend this progression? Does super root for tetration create any new numbers? Pentation?
Oh my God this just blew my mind
I made a quick program with python that put out pseudo random strings of "f"s and "d"s and I put it in that website, it could predict what I was going to input about 51% of the time
import random
while True:
input()
a=random.randint(0, 1)
if a==1: print('f')
else: print('d')
in the future I intend do this again but with the program putting inputs directly into the website so that it's faster.
And of course, 50% correct predictions over the long run, is indicative of failure to predict, in this case.
I always thought of the 'foundation' of maths being numbers themselves.
To take the numbers stuff down further, we define natural numbers with 0 and an increment function.
Start with zero and say that 1 is simply an increment function applied to zero. 2 is inc applied to inc applied to zero. 3 is inc applied to inc applied to inc applied to zero, and so on.
This is the idea of "Church Numerals," part of lambda calculus.
would be interesting to cover next the Wittgenstein's interpretation of this Principia Mathematica by Russell ...
In college I always wondered why "if-and only if" was adequate to be the proof of a mathematical theorem. I never fully resolved it in my own mind: Guess no one could come up with a better logical system.
If and only if means no other possibility of the implication being true. When we say that P implies Q, we also mean that another event such as X could also imply Q. So when we say that P if and only if Q, we are saying that only P could imply Q. That is, it is one-to-one relationship or map. Hope that helps explain the meaning, but I don’t mean to say that the logic is totally robust. For example, what if we could have a relationship which could behave both ways simultaneously? This may seem bizarre from logic point of view, but in a bizarre world like that of quantum mechanics, an object could physically be at two different places at the same time. Two objects the sum of which mathematically could exceed the speed of light could never in reality exceed the speed of light. The problem with logic is that we believe in it as result of our daily experiences. If our world changes, our logic might as well change too. The other problem is that: is our reality complete? Do we see the world in its all possibilities? If not, how could logic, which is based on our real encounters, could safely be used as the measuring stick of the rest of our reality, let alone being a foundation for mathematics? All we could say is that mathematical foundations makes sense; otherwise, sense is undefined!
@@dalisabe62 Thanks. I still have not resolved it in my mind. But cannot think of a better way to prove theorems. So I guess I'm stuck using it.
If you take logic it makes perfect sense. Think of it this way, if X is True, then Y is true and X is true only if Y is true. This means that X cannot be true when Y is false and vice versa.
X iff Y means that Y is a consequence of X, and X is a consequence of Y.
@@VicvicW But why is that adequate.? Do we just assume it is adequate?
The set of natural numbers doesn't include 0. Natural numbers start from 1. Whole numbers include 0 and natural numbers
In set theory the convention is to consider 0 to be a natural number. This allows us to define natural numbers like this: 0 is the empty set; 1 is {0}, or the set whose only element is the empty set; 2 is {0,1}, 3 is {0,1,2}, and so on; given any natural number n, n is the set of all numbers less than n, and n+1 is the union of sets n and {n}.
Great video!
Nice green screening work, the lighting is pretty good
why the pyramid of mathematics is upside down?
Ikr! it should have been a tree, and they are looking for the root from where it emerges
It's a Ponzi scheme.
Math is an upside down pyramid because it was built by academics.
Can we have a video about alternative set theories?
Turtles all the way down! :D Thank you, John Green for your wonderful new book. :)
Great video! Although it’s crazy how Dedekind gets all the lovin. Using Cauchy sequences of rational numbers (which are guaranteed to converge) to generate equivalence classes that converge to some limit point x (you will find that theses will fill out the whole Real Line) and having the representative for such a class be the real number (which can be Q or I) is math sexy.
Oh! And with a little Topology you can easily show the set has the Least Upper Bound property and boom, you got the Real Number Field.
Real numbers should be called Rubber numbers. They're not really numbers, but objects for investigating/gluing smooth topological spaces.
They only start to behave like numbers when you add a unit, or truncate to a finite sum assuming unit 1.