A very enjoyable interview that I will recommend to friends as an introduction to Professor Wildberger's ideas!

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.

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.

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 KZfaq channel. Fascinating stuff, keep it coming!

This was a great interview. Steve patterson was an insightful interviewer. And N J Wildberger - brilliant as ever.

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!

love the way you teach and talk about mathematics, your videos are awesome.

Thank you very much, Prof Wildberger. You're one of my heroes.

It is a great dialogue between a mathematician and a philosopher.

Thankyou so much for your time. Generous.

Great conversation, makes infinite sense!

WOW, extremely eye opening! Looking forward to read your book “Divine Proportions: Rational Trigonometry to Universal geometry”!

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 absolutely love this video

Very nice! I would love to see more interviews and debates.

Great video! The more videos I see the more I understand the skepticism of infinity! Your work is very important!

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

This is great, and I'm very happy to learn about the definition of Dedekind cuts, which basically is the definition of void

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.

Thank you!

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.

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!

Thankyou.

wow NJW from just a couple of months ago This dude is a genius

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”.

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.

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.

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.

+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.

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.

8/11=7272... 9×8=72. 8+11=19. 1+9=10. 1+0=1. 7+2=9+1=10 1+0=1. So n+1 to every digit in the decimal. 7+1 2+2 7+3 2+4 7+5 2+6 7+7 2+8 7+9. 8 4 10 6 12 8 14 10 16. -> 8 4 1 6 3 8 5 1 7 8+4+1=13 6+3+8=17 5+1+7=13 -> (4 8 4 )/4=1,2,1,[or 9×12=108, 108+1=109, 10+9=19, 1+9=10 1+0=1. Anyway this is done there is n+1] 1+2+1=4. 4+9=13. 1+3=4. (.999...)/9=.111... .111/9=12345679. S=36 3+6=9. 11+8=19 1+9=10. 1+0=1 1+1+1+1+1+1+1+1=8. 8×9=72. 7+2=9. 19 9n+y=x. 9×1=9 9+9=18, 1+8=9. There are further expansions. Conclusion 8/11 is controlled as well as the whole number system, by the rule of 9. Maybe, not stop infinity, but surely infinity can be changed. Two rules of numbers: 1.) Numbers are infinite. 2.) Numbers can not be terminated only changed. 3.) I wonder what the third rule will be? All numbers are causality connected.