30:50 The distinction between the two sets of infinitesimals, and the existence of “inaccessible” numbers forming the majority of the number continuum, if this exists, are interesting and mind blowing . Wow!
Hi Norman. As a very junior mathematician, I find your videos extremely interesting and informative. I especially appreciate your novel take on hyperbolic geometry. That doesn't mean that I agree with all of your views. For example, I don't think our ability to write something down explicitly relates to its validity as a mathematical concept. Nevertheless I wanted to express my gratitude and admiration for your TH-cam channel. Fascinating stuff, keep it coming!
>For example, I don't think our ability to write something down explicitly relates to its validity as a mathematical concept. If you can't write it down, what concept is it? How is anyone supposed to know?
great, I was wondering what is your view on another modern orthodoxy both in math and physics ..randomness..is that fundamental to nature?..again we know that probability theories also use the infinity concept a lot?
My opinion is that the mathematical underpinnings of probability theory are generally rather weak. And the question of what is randomness is hard to entangle from issues about we know, and can know.
Thanks, another point, in my own research, which is an evolving field of applying quantum theory (the mathematical foundation) in decision making theories including financial decision making, quantum probability formulation is used. We are seeing that many long standing puzzles/ fallacies can be resolved if quantum probability formulation (which is fundamentally different from Kolgomorov foundation) is adapted as the basis of decision making...there are plenty of experimental data to show that..However some philosophers like David Wallace think that square of the amplitudes which are used as probabilities in quantum theory cant be called as probabilities at all...is this a fair remark?
sudip patra Do you happen to have any papers around your research, on arXiv or otherwise? I'm currently in grad school working on stochastic systems and optimization though I hadn't ever considered employing quantum probabilities for stochastic decision problems so I'm very interested to see what your findings are.
Maybe I am being too platonist, but the beauty of mathematics is that it is NOT limited by physics. Our minds are free to think beyond physical limitations. This is not mathematics, but it is darn interesting philosophy. It is one of the most profound treasures of mathematics, that by thinking meta-matheamtically we can see that our mathematics has indeed gone far beyond physics. I know Wildeberger thinks this is all fairies and unicroins, but I still think it is deep and profound, and maybe future generations will appreciate that it is telling us the human mind is something more than a lump of meat we call a brain. Our minds do seem to be able to penetrate into concepts that are just plainly "non-physical". I say that with all due humility, I think it is just an undeniable fact. It is the most beautiful things I have ever found in the life of the intellect, even more profound, dare I say, than many religions or other fantastical notions like parallel universes or multiverses. In mathematics we are already seeing vast worlds of ideas beyond physical reality. It is quite astounding. We try to bring these ideas not the physical world through books and lectures, but honestly, only a mind can grasp the ideas. The scribbling on paper has to be interpreted by some mind. But I guess you need to spend some time deeply exploring mathematics to appreciate this, I wish more people would do so.
Here is reply I gave to somebody on what learning math is (to me). I think math has a problem that Conrad Wolfram said "Math is taught today as the ability to rote memorize the procedures of calculation". Mathematics is not a series of notation symbols to be calculated. It is about finding ways to explore highly abstract regions of the imagination that are quite remote from reality. Einstein himself was quite certain that mathematics would further become the geometrization of abstraction(if my memory serves). From the preface of The Art of Science (2014) "As Leonardo’s pictorial style is related to the geometry of nature, so is Hilbert’s mathematical “style” related to his vision of geometric forms. The “ideal” style Hilbert conceived for mathematical knowledge results in a “general theory of forms”. In particular, if we look for evidence of our claim that “the faculty of vision is essential as it turns ‘imagination’ into visual and graspable forms”, we should pay attention to his essays on “intuitive geometry”. To fully appreciate the potentialities embedded in Hilbert’s picture of mathematical theories and their impact on the development of physical concepts, we should look at Hermann Weyl’s writings" Another close reference to Einstein "geometrization". Going back to teaching "real" mathematics, here is Dr. Alan Kay's (computer scientist) note on the function of computers for teaching math. "However - as a former mathematician (unfrocked now I guess) - my main questions would be about what parts of mathematics would be most suitable - and epistemologically transforming - for the children? For example, one could argue that starting with vectors as underlying representations for numbers these are very nice ways for children and everyone to think about magnitudes from an early age, etc. and bypassing most of the content of trig and analytic geometry - in favor of the kinds of discrete differential and integral calculus that the computer allows would provide a much stronger basis of mathematical thinking for both maths in general, and for maths as used in the sciences. I think of “computing” as processes that help one deal with ideas. And this covers a lot of ground (including various forms of mathematics, etc.) So: computers are things that manifest processes that help one deal with ideas. "
If you frame mathematics as "free to think beyond physical limitations", then everyone can just not accept it and be done with it, as there is no guarantee or physical structure involved.
I think Professor Wildberger has a great appreciation for mathematical abstractions as objects in themselves. What he objects to is mathematical objects which are built upon weak logical foundations and poor definitions such as infinite sets, infinite sums, infinite vector spaces, real number arithmetic, and so on.
even without physical limits, how does any idea that one "reaches" or "goes to" infinity mean anything. even in an unbounded universe the idea of arriving at infinity makes no sense semantically or operationally. the ideas that float around in terms of axiom of infinity or choice really have no relation to anything remotely intuitive, little own physically absurd. at best they seem to confuse metaphors.
Thanks, Dr. Wildberger, for a very clear account of 'infinitist' difficulties. I've been following your channel for a while, with pleasure, amusement and for instruction. My training (40 yrs ago!) is in physics and so 'applied' math, with sidelong looks at the 'pure'. (A distinction beloved of university administrators; Euler & co would probably have said, 'Huh?' to it.) But there are real differences in the way various 'factions' ground math. To physicists and anyone using math for essentially non-math purposes - bridge building to high-energy physics - the 'foundational' justification is empirical - does the bridge stand up, or not? No arguing with that! Pure mathematicians who worry about 'justifying' math to themselves, now use axioms. You can't argue with an axiom (by itself; you can say two or more are inconsistent). Cantor grounded his transfinite numbers, as between the finite and the 'Absolute Infinite' (God?) in the mind of his God. Which is such a relief - if you're that kind of theist. Those of us who are no kind of theist are denied this comfort. Which leaves pure math as being a product of the human imagination. Constrained or justified how? Maybe computationally, to a point. In your own terms, by my ability to communicate math ideas and relations between them to you. So the 'problem' of the existence of math objects is solved by saying, 'To be a mathematical object, is to be coherently communicable to another human imagination.' Which itself is limited ... and may agree to something it's merely misunderstood! I'm tempted to say there is no 'ultimate foundation' for math, merely a deep breath and an imaginative leap! Quite to where, no one knows! But that's the whole point of adventure.
I would posit a slightly different possibility: that there is a concrete, down-to-earth mathematics firmly rooted in agreed manipulations that we make with concrete expressions that we, or our machines, write down. And then there are the flights of imagination and visualizations that we bring to interpret these expressions and their relations, but these are secondary. The expressions themselves are primary. Or to put it another way: show me the formula!
I pretty well got booted from a site for noticing that Reals are assumed to achieve their limit, i.e., they are not becoming but have achieved their apparent infinite sum. So the rational numbers 1/3 or 1 are in fact 0.333... and 0.999.... I can think of the square root of 2 as a series and play with the idea that I would recognize its series expansion in a way similar to 0.333..., say, then the square root of 2 would also achieve its infinite series limit. So, we have a completed set of completed infinite elements. I note also, that it is necessary to define the word "fraction" for some rather simple Mathematical manipulations because of this, e.g., 0.999... is not a fraction at least in most instances or uses. I tried to think about the possibility of an infinitesimal, d, such that 0.999... = 1.000... - d but came to the conclusion that the concept did not seem to correspond to the way we use the Reals. I have a problem with just accepting completed infinities without any forewarning that that is what we are doing. An impractical practical instance of how it matters is that Zeno's Paradoxes have not been resolved by our theory of infinite series. It would be interesting to have better examples of the consequences.
I am a huge fan of your ideas...since a more platonic view of things will help us better understand things and eliminate mathematics of any errors....and I have watched your video real fish real jobs about a hundred times I just love to way you narrate ..its really very entertaining and enlightening
Do you know of other contemporary mathematicians who have similar thoughts or who are pursuing problems in this vein? If I were to do a PhD (and I'd like to) this seems like an important, useful, and interesting field of study that I'd like to pursue. Keep up the awesome work!
Thanks for the names! Maybe I wasn't really clear, but I was more interested in current mathematicians that may take on graduate students. I would love to work on this type of project (providing solid types and algorithms for mathematical foundations) and wonder who else is following similar lines of thought in modern mathematics circles
Dr. Wildberger, thanks for the video! Awhile back I commented on one of your other videos about your rejection of the idea of "computable real numbers", and I think after watching this video I better understand your perspective, since you say the axiom of choice requires the existence of these uncomputable reals. I was wondering, have you looked into Martin Lof Type Theory, where the "axiom of choice" is actually a theorem, and can be interpreted in a concrete, computational context? I believe, if I recall correctly, the assumption that all functions are computable is actually consistent with this system. Essentially, *everything* is either finite, or a finite description of a computational process, so I think it fits well with your "concrete" approach to mathematics. A type (under Marin Lof's interpretation) is not identified with the "completed infinity" of its elements, but with the rules for which we may computationally decide membership of terms in the type.
I think that this is an interesting direction, but I have not seen it mapped out clearly enough. Admittedly I have not looked at Martin Lof Type Theory. But I wan't to impress upon us the need for not assuming beforehand that we known what the word "function" means. That is an essential aspect of the problem. By shifting it to the idea of a "computational process" is well and good, but then we need to specify clearly what that means. Unfortunately I don't believe that computer scientist have a clear unambiguous agreement as to what this term refers to, although we all have a range of informal intuitions. For example, I am fascinated by the remarkable leaps that the Go playing AI machines such as AlphaGo/Master and DeepZenGo have made. Are these computational processes? They are neural nets of some complicated kind, but no-one really exactly knows what goes on inside them now. Except that they are very very good at what they do. There are very big challenges in trying to come up with a mathematically concise and consistent notion of "computational process" or "algorithm". Currently I reckon this is still in the too-hard basket.
I never thought I could be interested in mathematics before watching your videos. I believe your works and idea will shine out through the old heavy blanket and get popular more and more.
+njwildberger You mentioned geometric points and lines as though you have no issue with them. But how can a line have no breadth (i.e. be infinitely thin) and how can a point have no size at all (i.e. be infinitely small) and how can a line contain/traverse infinitely many points? Surely these are key fundamental concepts that need to be replaced by concepts without any implied infinities. My position is as follows: In any real or abstract system, there must be a smallest part of ‘space’ that cannot be sub-divided. The size of this smallest part does not need to be stated explicitly, it can remain as a variable parameter. Then a point could be said to refer to the position of one of these smallest parts of space. A line could extend indefinitely (not infinitely, but without an end point being specified), and a line segment could be defined as a path comprising of a number of these ‘smallest part of space’ points.
+Karma Peny Was thinking the exact thing, but it's essentially considering bounded subsets of the natural numbers. For clarification, in your definition is the distance from a point to itself 0?
In terms of measuring a length, it is a matter of expressing that length in terms of how many 'points' (= smallest parts of space) we need to get from one end to the other. So I would not say the "distance of a point to itself"; I would say anything that occupies the smallest possible part of space measures one point. My 'measurement' is ultimately expressed in terms of smallest points. We could get zero by asking how many points difference are there between length 1 and length 2, where both lengths are the same number of points.
+Karma Peny I see. Yes, we necessarily need to use bounded subsets of the integers (if we used rationals, then we would have the property that for any two rationals p < q, there is a rational inbetween them, (p+q)/2, which then gives the same problem) So let S be a bounded subset of positive integers. For this example, let's say S = {1,2,3,...10^10^10^10^10^10}. Then we can consider any line segment on S as [a,b]. where a and b are in S. We can define our length of the line segment as d([a,b]) = b - a. The integers are "far apart" with respect to the reals, but really if we want, we can in some sense consider them "close" if we define open sets to be balls of radius x, where x can be 10^10. Let's say that our space is so big that these fixed balls contain many points but there size is still very small with respect to how many integers we have. Now, we can define a function f: S -> S, to be continuous at point p if for every ball of radius n*x (n positive integer bounded greater than 1) centered around f(p), the ball of radius m*x (m positive integer bounded greater than or equal to 1) centered around p so that the image of the ball of radius s is contained in the ball centered around f(p). What do you think? Any suggestions?
For me, a measurement is a quantity of smallest spaces. For example, let's say we have a lot of ball bearings all of the same size; we can use these ball bearings to make shapes and lines. In this system we could define one ball bearing as the smallest part. So if p is a 'line' consisting of three ball bearings placed together in a row, and q is a 'line' consisting of four ball bearings placed together in a row, then we cannot construct a line between them of length (p+q)/2 because one ball bearing is the smallest unit of measure. Line segments must always be a whole number of smallest parts. My view of maths is that it is a tool we use to model the real world. At the simplest level it deals with the manipulation of quantities (expressed as natural numbers). The next level deals with the movement of objects through space. The plus and minus signs imply direction, and so signed numbers (like integers) are really vectors. If positive is East and negative is West, then +i is North and -i is South. Thus integers and so-called imaginary numbers can be treated exactly the same way, as they are all just vectors. We need to change the terminology, relate maths to reality and demystify concepts like complex numbers. The third level deals with the movement of objects through space and time. This should effectively be a finite approach to calculus. You said "We can define our length of the line segment as d([a,b]) = b - a" I don't like the idea of length possibly being negative. You could use the modulus |b-a| to guarantee an unsigned result. Also I personally would avoid talking about sets (scandalous I know) because the concept of a 'set' has never been clearly defined (as Prof Wildberger mentions in this video). You said "we can define a function f: S -> S, to be continuous at point p if ..." I don't accept the argument that a function or anything else can be continuous. I want a finite quantized approach to all of mathematics. Presumably your 'x' is a real number? I agree with Professor Wildberger that reals are not well defined an so cannot be used in any rigorous argument.
"Presumably your 'x' is a real number": It is a positive integer. The whole point was S was a bounded subset of positive integers (i.e. has a finite number points). By treating each "smallest point of space" as an integer (7 is not an amount of smallest points of space, each integer represents a unique smallest point of space), I defined the length of a line segment to be "the number of smallest points". We can stick with abs(b-a). Regarding continuity, this doesn't require you to accept the notion of continuity in the infinite sense. I defined a notion in the finite sense. The reason I defined a finite notion of continuity was to proceed to calculus in a finite approach (you need a notion of a derivative, which is why you need a notion of continuity). Think about it a bit more, I do not think we are in disagreement.
Excellent discussion. Thank you. It makes me think that much of modern mathematics is like fancy CGI animation. It looks okay at first glance but it’s not possible for physical objects. It’s not a simulation of reality, it’s a virtual reality. QUESTION: Is it correct to say “there is no continuum on the number line” or “to have a continuum on the number line requires an infinite amount of work, which is not possible”.
My 5 year old daughter had an interesting question concerning the nature of numbers. It was her first year at school. She said: "2 + 3 = 5 and 3 + 2 = 5. " i said that was true. But she then asked me: "Is this true for all numbers?" I had done University Pure Mathematics, and all I could say was everyone just assumes it to be true. On further thought, whole positive integers, or Natural Numbers (from which all our other numbers are constructed) are for COUNTING DISCRETE OBJECTS,, e.g., pebbles, cows, people etc. If I had shown her a group of 2 pebbles and a second group of 3 pebbles then she would have seen why the same answer is obtained no matter in what order they are combined. I believe that Numbers had been introduced as an abstract concept rather too quickly, but she saw the problem. However, Numbers came to be used to measure lengths of lines, which is a more complicated matter. Suppose we have a straight line of length one inch. We think of the line as consisting of points with no space between them (that is, it is continuous). However a point has zero width; then how many points does the line contain? With whole numbers, no matter how big the number is, one can always add one to it. Therefore there is no largest number. This concept or process is often called "infinity", but usually one excludes the existence of a Number which is "Infinity". If one allows it to exist, its properties are different from all other numbers. But the question about the number of points in a line will not go away. For example, think of the function defined as one at rational points between 0 and 1, and zero elsewhere. Then the Integral of this function is zero. Therefore, non-rational points must be allowed to exist. Surely, we have to allow sqrt(2) and other irrationals to exist. If not, the proof of a basic Euclid Theorem is wrong: if irrational numbers do not exist, then there are gaps in lines. In proving that an equilateral triangle can be constructed on any given line segment, Euclid assumes that two lines (circles) intersect at a point. The proof consists of drawing the intersecting circles, but gives no formal guarantee that the point of intersection does in fact exist: if there are gaps in a line, two lines might pass through one another but have no common point of intersection. When measuring a length in the real world, or in computations, we use only rational numbers. However, Irrational Numbers exist in Mathematics, and are required to make the Maths of the Continuum consistent. This was done by Dedekind and Cantor. I can do without "Infinity" as a Number (not really because I like to use Infinitesimals in Calculus, they are so intuitive. Robinson developed the Hypereals using Cantor's Transfinite Numbers). However Infinite Sets are required to make the Irrationals a logical part of Mathematics. One can consider the set of Natural Numbers, and it has to be a called an Infinite Set.
Mathematics is the language in which physics is written. But not only is mathematics is undefined as the number system cannot define infinity, but also QM rests on the uncertainty theorem of Heisenberg, yet Schrodinger's wave function guarantees unitary evolution of life from a single embryo. Same as unitary evolution of the entire universe.
It is a mistake to apply cardinality to infinite sets as it is to finite sets. Been working on it today, as a matter of fact, in order to resolve the Continuum Hypothesis. Not solve it, resolve it. Wash away the question. The cardinality of an infinite set should be described as a rate at which the elements approach the limit. Let's start simple by taking the positive integers, and compare then to the even positive integers. Our intuition is not wrong, one is half the size of the other, because... In both infinities, the elements which are the even positive integers appear in both sets. Yes, I used a reference to infinite sets. If the even positive integers are removed from the set of all positive integers, the set of positive integers has remaining elements. When the even positive integers are disjuncted from Z+, Z+ is not empty. So what then would be the cardinality of an infinite set? The rate at which the elements approach the limits, if the set is well ordered. So why do it differently for infinite sets and finite sets? Well, no reason... So let's apply the idea of rate of approaching he limit to finite sets. What is a limit if it CAN be reached? Because in a finite set, it is reached. This would be the major difference which has us invoke the idea of approaching a limit at a rate for infinite sets, and so seems not to apply, but... Take: {1, 2, 3} It does not infinitely approach anything, since it isn't infinite, but it does approach a finite limit which it never reaches. That would be "anything greater than 3". It will not ever reach a cardinality of anything greater than 3. What about the rate this finite set approaches the limit of "greater than 3"? It appears to have a change of 1 per 1 element. Taking: {2, 4, 6} This finite set approaches a limit of "greater than 6" at a rate of "2 per 1 element". Ok, so far today, that's all I have. I didn't even create a notation. So I will just stop there, leaving more questions than answers provided. We can tear apart Cantor when we get to that bridge.
Well done to both of you Prof. Wildberger. I appreciate your opposition to infinite sets from a computational point of view. Indeed, the practice of mathematics involves writing concepts down. Though I for one would not yet chuck out infinite sets as I feel that requires a strict constructivist position, I am not yet prepared to go, I do think that for the sake of reasoning about sets containing infinite elements it is still not a bad idea. For example it is quite useful in Model Theory.
Professor Wildberger, you are awesome. It is amazing how clearly you present complex ideas. I love all of your videos. Thank you for making them.
You are welcome, glad you enjoy them.
A very enjoyable interview that I will recommend to friends as an introduction to Professor Wildberger's ideas!
Thanks!
30:50 The distinction between the two sets of infinitesimals, and the existence of “inaccessible” numbers forming the majority of the number continuum, if this exists, are interesting and mind blowing . Wow!
Hi Norman. As a very junior mathematician, I find your videos extremely interesting and informative. I especially appreciate your novel take on hyperbolic geometry. That doesn't mean that I agree with all of your views. For example, I don't think our ability to write something down explicitly relates to its validity as a mathematical concept. Nevertheless I wanted to express my gratitude and admiration for your TH-cam channel. Fascinating stuff, keep it coming!
>For example, I don't think our ability to write something down explicitly relates to its validity as a mathematical concept.
If you can't write it down, what concept is it? How is anyone supposed to know?
This was a great interview. Steve patterson was an insightful interviewer. And N J Wildberger - brilliant as ever.
It is a great dialogue between a mathematician and a philosopher.
love the way you teach and talk about mathematics, your videos are awesome.
Thanks!
great, I was wondering what is your view on another modern orthodoxy both in math and physics ..randomness..is that fundamental to nature?..again we know that probability theories also use the infinity concept a lot?
My opinion is that the mathematical underpinnings of probability theory are generally rather weak. And the question of what is randomness is hard to entangle from issues about we know, and can know.
Thanks, another point, in my own research, which is an evolving field of applying quantum theory (the mathematical foundation) in decision making theories including financial decision making, quantum probability formulation is used. We are seeing that many long standing puzzles/ fallacies can be resolved if quantum probability formulation (which is fundamentally different from Kolgomorov foundation) is adapted as the basis of decision making...there are plenty of experimental data to show that..However some philosophers like David Wallace think that square of the amplitudes which are used as probabilities in quantum theory cant be called as probabilities at all...is this a fair remark?
sudip patra Do you happen to have any papers around your research, on arXiv or otherwise? I'm currently in grad school working on stochastic systems and optimization though I hadn't ever considered employing quantum probabilities for stochastic decision problems so I'm very interested to see what your findings are.
Maybe I am being too platonist, but the beauty of mathematics is that it is NOT limited by physics. Our minds are free to think beyond physical limitations. This is not mathematics, but it is darn interesting philosophy. It is one of the most profound treasures of mathematics, that by thinking meta-matheamtically we can see that our mathematics has indeed gone far beyond physics. I know Wildeberger thinks this is all fairies and unicroins, but I still think it is deep and profound, and maybe future generations will appreciate that it is telling us the human mind is something more than a lump of meat we call a brain. Our minds do seem to be able to penetrate into concepts that are just plainly "non-physical". I say that with all due humility, I think it is just an undeniable fact. It is the most beautiful things I have ever found in the life of the intellect, even more profound, dare I say, than many religions or other fantastical notions like parallel universes or multiverses. In mathematics we are already seeing vast worlds of ideas beyond physical reality. It is quite astounding. We try to bring these ideas not the physical world through books and lectures, but honestly, only a mind can grasp the ideas. The scribbling on paper has to be interpreted by some mind. But I guess you need to spend some time deeply exploring mathematics to appreciate this, I wish more people would do so.
Here is reply I gave to somebody on what learning math is (to me).
I think math has a problem that Conrad Wolfram said "Math is taught today as the ability to rote memorize the procedures of calculation". Mathematics is not a series of notation symbols to be calculated. It is about finding ways to explore highly abstract regions of the imagination that are quite remote from reality. Einstein himself was quite certain that mathematics would further become the geometrization of abstraction(if my memory serves).
From the preface of The Art of Science (2014) "As Leonardo’s pictorial style is related to the geometry of nature, so is Hilbert’s mathematical “style” related to his vision of geometric forms. The “ideal” style Hilbert conceived for mathematical knowledge results in a “general theory of forms”. In particular, if we look for evidence of our claim that “the faculty of vision is essential as it turns ‘imagination’ into visual and graspable forms”, we should pay attention to his essays on “intuitive geometry”. To fully appreciate the potentialities embedded in Hilbert’s picture of mathematical theories and their impact on the development of physical concepts, we should look at Hermann Weyl’s writings" Another close reference to Einstein "geometrization".
Going back to teaching "real" mathematics, here is Dr. Alan Kay's (computer scientist) note on the function of computers for teaching math.
"However - as a former mathematician (unfrocked now I guess) - my main questions would be about what
parts of mathematics would be most suitable - and epistemologically transforming - for the children?
For example, one could argue that starting with vectors as underlying representations for numbers
these are very nice ways for children and everyone to think about magnitudes from an early age, etc. and bypassing most of the content of trig and analytic geometry - in favor of the kinds of discrete differential and integral calculus that the computer allows would provide a much stronger basis of mathematical thinking for both maths in general, and for maths as used in the sciences.
I think of “computing” as processes that help one deal with ideas. And this covers a lot of ground (including various forms of mathematics, etc.) So: computers are things that manifest processes that help one deal with ideas.
"
If you frame mathematics as "free to think beyond physical limitations", then everyone can just not accept it and be done with it, as there is no guarantee or physical structure involved.
@ absolutely, I do take a very aristotelian view.
I think Professor Wildberger has a great appreciation for mathematical abstractions as objects in themselves. What he objects to is mathematical objects which are built upon weak logical foundations and poor definitions such as infinite sets, infinite sums, infinite vector spaces, real number arithmetic, and so on.
even without physical limits, how does any idea that one "reaches" or "goes to" infinity mean anything. even in an unbounded universe the idea of arriving at infinity makes no sense semantically or operationally. the ideas that float around in terms of axiom of infinity or choice really have no relation to anything remotely intuitive, little own physically absurd. at best they seem to confuse metaphors.
Thanks, Dr. Wildberger, for a very clear account of 'infinitist' difficulties. I've been following your channel for a while, with pleasure, amusement and for instruction. My training (40 yrs ago!) is in physics and so 'applied' math, with sidelong looks at the 'pure'. (A distinction beloved of university administrators; Euler & co would probably have said, 'Huh?' to it.)
But there are real differences in the way various 'factions' ground math. To physicists and anyone using math for essentially non-math purposes - bridge building to high-energy physics - the 'foundational' justification is empirical - does the bridge stand up, or not? No arguing with that!
Pure mathematicians who worry about 'justifying' math to themselves, now use axioms. You can't argue with an axiom (by itself; you can say two or more are inconsistent).
Cantor grounded his transfinite numbers, as between the finite and the 'Absolute Infinite' (God?) in the mind of his God. Which is such a relief - if you're that kind of theist. Those of us who are no kind of theist are denied this comfort.
Which leaves pure math as being a product of the human imagination. Constrained or justified how? Maybe computationally, to a point. In your own terms, by my ability to communicate math ideas and relations between them to you. So the 'problem' of the existence of math objects is solved by saying, 'To be a mathematical object, is to be coherently communicable to another human imagination.' Which itself is limited ... and may agree to something it's merely misunderstood!
I'm tempted to say there is no 'ultimate foundation' for math, merely a deep breath and an imaginative leap! Quite to where, no one knows! But that's the whole point of adventure.
I would posit a slightly different possibility: that there is a concrete, down-to-earth mathematics firmly rooted in agreed manipulations that we make with concrete expressions that we, or our machines, write down. And then there are the flights of imagination and visualizations that we bring to interpret these expressions and their relations, but these are secondary. The expressions themselves are primary. Or to put it another way: show me the formula!
WOW, extremely eye opening! Looking forward to read your book “Divine Proportions: Rational Trigonometry to Universal geometry”!
Great video! The more videos I see the more I understand the skepticism of infinity! Your work is very important!
Thankyou so much for your time. Generous.
This is great, and I'm very happy to learn about the definition of Dedekind cuts, which basically is the definition of void
I am interested in your comment. Could you elaborate a little?
Thank you very much, Prof Wildberger. You're one of my heroes.
I pretty well got booted from a site for noticing that Reals are assumed to achieve their limit, i.e., they are not becoming but have achieved their apparent infinite sum. So the rational numbers 1/3 or 1 are in fact 0.333... and 0.999.... I can think of the square root of 2 as a series and play with the idea that I would recognize its series expansion in a way similar to 0.333..., say, then the square root of 2 would also achieve its infinite series limit. So, we have a completed set of completed infinite elements. I note also, that it is necessary to define the word "fraction" for some rather simple Mathematical manipulations because of this, e.g., 0.999... is not a fraction at least in most instances or uses. I tried to think about the possibility of an infinitesimal, d, such that 0.999... = 1.000... - d but came to the conclusion that the concept did not seem to correspond to the way we use the Reals.
I have a problem with just accepting completed infinities without any forewarning that that is what we are doing. An impractical practical instance of how it matters is that Zeno's Paradoxes have not been resolved by our theory of infinite series. It would be interesting to have better examples of the consequences.
Great conversation, makes infinite sense!
I am a huge fan of your ideas...since a more platonic view of things will help us better understand things and eliminate mathematics of any errors....and I have watched your video real fish real jobs about a hundred times I just love to way you narrate ..its really very entertaining and enlightening
Thanks Alice!
Do you know of other contemporary mathematicians who have similar thoughts or who are pursuing problems in this vein? If I were to do a PhD (and I'd like to) this seems like an important, useful, and interesting field of study that I'd like to pursue.
Keep up the awesome work!
There are plenty: Gregory Chaitin, Per Martin-Löf, Edsger Dijkstra, Doron Zeilberger, and so on and so forth.
Thanks for the names! Maybe I wasn't really clear, but I was more interested in current mathematicians that may take on graduate students. I would love to work on this type of project (providing solid types and algorithms for mathematical foundations) and wonder who else is following similar lines of thought in modern mathematics circles
Very nice! I would love to see more interviews and debates.
Yes it is a nice format.
Dr. Wildberger, thanks for the video! Awhile back I commented on one of your other videos about your rejection of the idea of "computable real numbers", and I think after watching this video I better understand your perspective, since you say the axiom of choice requires the existence of these uncomputable reals.
I was wondering, have you looked into Martin Lof Type Theory, where the "axiom of choice" is actually a theorem, and can be interpreted in a concrete, computational context? I believe, if I recall correctly, the assumption that all functions are computable is actually consistent with this system. Essentially, *everything* is either finite, or a finite description of a computational process, so I think it fits well with your "concrete" approach to mathematics. A type (under Marin Lof's interpretation) is not identified with the "completed infinity" of its elements, but with the rules for which we may computationally decide membership of terms in the type.
I think that this is an interesting direction, but I have not seen it mapped out clearly enough. Admittedly I have not looked at Martin Lof Type Theory. But I wan't to impress upon us the need for not assuming beforehand that we known what the word "function" means. That is an essential aspect of the problem. By shifting it to the idea of a "computational process" is well and good, but then we need to specify clearly what that means. Unfortunately I don't believe that computer scientist have a clear unambiguous agreement as to what this term refers to, although we all have a range of informal intuitions.
For example, I am fascinated by the remarkable leaps that the Go playing AI machines such as AlphaGo/Master and DeepZenGo have made. Are these computational processes? They are neural nets of some complicated kind, but no-one really exactly knows what goes on inside them now. Except that they are very very good at what they do.
There are very big challenges in trying to come up with a mathematically concise and consistent notion of "computational process" or "algorithm". Currently I reckon this is still in the too-hard basket.
I never thought I could be interested in mathematics before watching your videos. I believe your works and idea will shine out through the old heavy blanket and get popular more and more.
Thank you!
I absolutely love this video
+njwildberger You mentioned geometric points and lines as though you have no issue with them. But how can a line have no breadth (i.e. be infinitely thin) and how can a point have no size at all (i.e. be infinitely small) and how can a line contain/traverse infinitely many points? Surely these are key fundamental concepts that need to be replaced by concepts without any implied infinities.
My position is as follows: In any real or abstract system, there must be a smallest part of ‘space’ that cannot be sub-divided. The size of this smallest part does not need to be stated explicitly, it can remain as a variable parameter. Then a point could be said to refer to the position of one of these smallest parts of space. A line could extend indefinitely (not infinitely, but without an end point being specified), and a line segment could be defined as a path comprising of a number of these ‘smallest part of space’ points.
+Karma Peny Was thinking the exact thing, but it's essentially considering bounded subsets of the natural numbers.
For clarification, in your definition is the distance from a point to itself 0?
In terms of measuring a length, it is a matter of expressing that length in terms of how many 'points' (= smallest parts of space) we need to get from one end to the other.
So I would not say the "distance of a point to itself"; I would say anything that occupies the smallest possible part of space measures one point. My 'measurement' is ultimately expressed in terms of smallest points.
We could get zero by asking how many points difference are there between length 1 and length 2, where both lengths are the same number of points.
+Karma Peny
I see. Yes, we necessarily need to use bounded subsets of the integers (if we used rationals, then we would have the property that for any two rationals p < q, there is a rational inbetween them, (p+q)/2, which then gives the same problem)
So let S be a bounded subset of positive integers. For this example, let's say S = {1,2,3,...10^10^10^10^10^10}.
Then we can consider any line segment on S as [a,b]. where a and b are in S.
We can define our length of the line segment as d([a,b]) = b - a. The integers are "far apart" with respect to the reals, but really if we want, we can in some sense consider them "close" if we define open sets to be balls of radius x, where x can be 10^10. Let's say that our space is so big that these fixed balls contain many points but there size is still very small with respect to how many integers we have.
Now, we can define a function f: S -> S, to be continuous at point p if for every ball of radius n*x (n positive integer bounded greater than 1) centered around f(p), the ball of radius m*x (m positive integer bounded greater than or equal to 1) centered around p so that the image of the ball of radius s is contained in the ball centered around f(p).
What do you think? Any suggestions?
For me, a measurement is a quantity of smallest spaces. For example, let's say we have a lot of ball bearings all of the same size; we can use these ball bearings to make shapes and lines. In this system we could define one ball bearing as the smallest part.
So if p is a 'line' consisting of three ball bearings placed together in a row, and q is a 'line' consisting of four ball bearings placed together in a row, then we cannot construct a line between them of length (p+q)/2 because one ball bearing is the smallest unit of measure. Line segments must always be a whole number of smallest parts.
My view of maths is that it is a tool we use to model the real world.
At the simplest level it deals with the manipulation of quantities (expressed as natural numbers).
The next level deals with the movement of objects through space. The plus and minus signs imply direction, and so signed numbers (like integers) are really vectors. If positive is East and negative is West, then +i is North and -i is South. Thus integers and so-called imaginary numbers can be treated exactly the same way, as they are all just vectors. We need to change the terminology, relate maths to reality and demystify concepts like complex numbers.
The third level deals with the movement of objects through space and time. This should effectively be a finite approach to calculus.
You said "We can define our length of the line segment as d([a,b]) = b - a"
I don't like the idea of length possibly being negative. You could use the modulus |b-a| to guarantee an unsigned result. Also I personally would avoid talking about sets (scandalous I know) because the concept of a 'set' has never been clearly defined (as Prof Wildberger mentions in this video).
You said "we can define a function f: S -> S, to be continuous at point p if ..."
I don't accept the argument that a function or anything else can be continuous. I want a finite quantized approach to all of mathematics. Presumably your 'x' is a real number? I agree with Professor Wildberger that reals are not well defined an so cannot be used in any rigorous argument.
"Presumably your 'x' is a real number": It is a positive integer. The whole point was S was a bounded subset of positive integers (i.e. has a finite number points).
By treating each "smallest point of space" as an integer (7 is not an amount of smallest points of space, each integer represents a unique smallest point of space), I defined the length of a line segment to be "the number of smallest points". We can stick with abs(b-a).
Regarding continuity, this doesn't require you to accept the notion of continuity in the infinite sense. I defined a notion in the finite sense. The reason I defined a finite notion of continuity was to proceed to calculus in a finite approach (you need a notion of a derivative, which is why you need a notion of continuity). Think about it a bit more, I do not think we are in disagreement.
Excellent discussion. Thank you.
It makes me think that much of modern mathematics is like fancy CGI animation. It looks okay at first glance but it’s not possible for physical objects. It’s not a simulation of reality, it’s a virtual reality.
QUESTION: Is it correct to say “there is no continuum on the number line” or “to have a continuum on the number line requires an infinite amount of work, which is not possible”.
My 5 year old daughter had an interesting question concerning the nature of numbers. It was her first year at school. She said: "2 + 3 = 5 and 3 + 2 = 5. " i said that was true. But she then asked me: "Is this true for all numbers?" I had done University Pure Mathematics, and all I could say was everyone just assumes it to be true.
On further thought, whole positive integers, or Natural Numbers (from which all our other numbers are constructed) are for COUNTING DISCRETE OBJECTS,, e.g., pebbles, cows, people etc. If I had shown her a group of 2 pebbles and a second group of 3 pebbles then she would have seen why the same answer is obtained no matter in what order they are combined. I believe that Numbers had been introduced as an abstract concept rather too quickly, but she saw the problem.
However, Numbers came to be used to measure lengths of lines, which is a more complicated matter.
Suppose we have a straight line of length one inch. We think of the line as consisting of points with no space between them (that is, it is continuous). However a point has zero width; then how many points does the line contain?
With whole numbers, no matter how big the number is, one can always add one to it. Therefore there is no largest number. This concept or process is often called "infinity", but usually one excludes the existence of a Number which is "Infinity". If one allows it to exist, its properties are different from all other numbers.
But the question about the number of points in a line will not go away. For example, think of the function defined as one at rational points between 0 and 1, and zero elsewhere. Then the Integral of this function is zero. Therefore, non-rational points must be allowed to exist.
Surely, we have to allow sqrt(2) and other irrationals to exist. If not, the proof of a basic Euclid Theorem is wrong: if irrational numbers do not exist, then there are gaps in lines. In proving that an equilateral triangle can be constructed on any given line segment, Euclid assumes that two lines (circles) intersect at a point. The proof consists of drawing the intersecting circles, but gives no formal guarantee that the point of intersection does in fact exist: if there are gaps in a line, two lines might pass through one another but have no common point of intersection.
When measuring a length in the real world, or in computations, we use only rational numbers. However, Irrational Numbers exist in Mathematics, and are required to make the Maths of the Continuum consistent. This was done by Dedekind and Cantor. I can do without "Infinity" as a Number (not really because I like to use Infinitesimals in Calculus, they are so intuitive. Robinson developed the Hypereals using Cantor's Transfinite Numbers). However Infinite Sets are required to make the Irrationals a logical part of Mathematics. One can consider the set of Natural Numbers, and it has to be a called an Infinite Set.
Mathematics is the language in which physics is written. But not only is mathematics is undefined as the number system cannot define infinity, but also QM rests on the uncertainty theorem of Heisenberg, yet Schrodinger's wave function guarantees unitary evolution of life from a single embryo. Same as unitary evolution of the entire universe.
It is a mistake to apply cardinality to infinite sets as it is to finite sets.
Been working on it today, as a matter of fact, in order to resolve the Continuum Hypothesis. Not solve it, resolve it. Wash away the question.
The cardinality of an infinite set should be described as a rate at which the elements approach the limit.
Let's start simple by taking the positive integers, and compare then to the even positive integers. Our intuition is not wrong, one is half the size of the other, because...
In both infinities, the elements which are the even positive integers appear in both sets. Yes, I used a reference to infinite sets. If the even positive integers are removed from the set of all positive integers, the set of positive integers has remaining elements.
When the even positive integers are disjuncted from Z+, Z+ is not empty.
So what then would be the cardinality of an infinite set? The rate at which the elements approach the limits, if the set is well ordered.
So why do it differently for infinite sets and finite sets? Well, no reason... So let's apply the idea of rate of approaching he limit to finite sets.
What is a limit if it CAN be reached? Because in a finite set, it is reached. This would be the major difference which has us invoke the idea of approaching a limit at a rate for infinite sets, and so seems not to apply, but...
Take: {1, 2, 3}
It does not infinitely approach anything, since it isn't infinite, but it does approach a finite limit which it never reaches. That would be "anything greater than 3". It will not ever reach a cardinality of anything greater than 3.
What about the rate this finite set approaches the limit of "greater than 3"? It appears to have a change of 1 per 1 element.
Taking: {2, 4, 6}
This finite set approaches a limit of "greater than 6" at a rate of "2 per 1 element".
Ok, so far today, that's all I have. I didn't even create a notation. So I will just stop there, leaving more questions than answers provided. We can tear apart Cantor when we get to that bridge.
Oh, stopped there because of finite sets that appear to not have a constant rate of change.
Well done to both of you Prof. Wildberger. I appreciate your opposition to infinite sets from a computational point of view. Indeed, the practice of mathematics involves writing concepts down. Though I for one would not yet chuck out infinite sets as I feel that requires a strict constructivist position, I am not yet prepared to go, I do think that for the sake of reasoning about sets containing infinite elements it is still not a bad idea. For example it is quite useful in Model Theory.
wow
NJW from just a couple of months ago
This dude is a genius
Thankyou.
when you match the even numbers to the natural numbers , all one is doing is changing the labels... it doesn't say much about numbers at all