Norman Wildberger: The Problem with Infinity in Math

Prof. Wildberger is a huge discovery for me. Thank you very much for this interview!

I have come to believe infinity should stop at the point a particle pops into existence and starts back at one because the universe is probably fractal and maybe the speed of light is the top number

What I basically got from his argumentation can be simplified to “affine”, “elementary/digital”, and without openly saying it “integer/whole”. Even with these supposed complete personal views/perspectives/systems I feel like he is granting more power than such a notion could either allow or even handle. He seemingly means to recreate the power of the “real” numbers by means of a more clarified “rational” symbolic system… without openly calling it such. He essentially describes what math already is but one that would satisfy his perceived misgivings.

As as a theoretical physicist myself, I cannot thank Professor Wildberger enough for his contribution to teaching mathematics in a more pragmatic and intuitive way to students. I do not think mathematics education should be simpler, but clearer, with many computable examples.
As the Professor pointed out in this interview, we do not need brilliance, but clarity. Brilliance is a by-product of the latter, not the other way around. Nobody is required to show off or impress to be regarded as a genius, unless we suffer from some ultra-ego complex.
If you cannot explain any subject, being math, physics or wood-working, using day-to-day intuition, then you are fooling yourself and your audience in presuming you have understood anything at all.
Regardless of whether the real numbers do exist or not, it is mandatory for any educator out there to be able to convince their students that your theory is solid. From the very foundations. If you are a student, never assume what you are taught is correct, always ask questions and require precise answers. This is neither presumption nor arrogance, it is the true nature of what makes us humans, therefore curious about Mother Nature.

that's it

I came here because I am trying to implement infinite sequences in my own programming language, and while struggling, remembered the professor's view on the topic. The truth is, many programming languages do have constructs for dealing with possibly infinite things. In some languages, like Haskell, it's a fundamental concept. In others like JavaScript, it's a newer feature that was more recently tacked on because it has been shown to be often very useful.

I think Curt is trying really hard to find a "real world problem" that runs into trouble under Dr. Wildberger's objections. But I don't know that there is one - as Dr. Wildberger said, "you physicists are fine." His objections are not on the "practical front." They're leveled against the foundations of pure math. I think we can go our merry way and do practical calculations to our heart's content and he'd have no issue with it. It's just not the nature of his concerns.

Great discussion when is part 2 comming out

Has anyone on here heard of a lattice called Post's Lattice? Technically speaking, it's the "lattice of clones on a two element set". The lattice is a beautiful piece of mathematics that contains countably infinite parts but has a finite global structure (the lattice has a top and a bottom). Can we design an algorithm that generates this lattice?

Also, to get an idea of complexity of Clone Theory, "the lattice of clones on a three element set" is unknown and its accurate description would earn you at least a masters, if not a doctorate...

What's my point? To disregard "an infinite set of objects", countable or otherwise, particularly in the computational example above, would be completely erroneous.

P.S. Great convseration guys!

NJ Wildberger, would or could you say the Babylonian system is actually based on three modulus? "Base 60" could be further fractioned into base 10, base square-root of 2, base square-root of 3. A far less restrictive 'base/modulus'? -Matt

Regarding the bit at the end of the video where you mention that some commenters get upset that you would interview someone with "right-wing views" (whatever that means) my question to them would be "Why does it matter?". At what point in the history of humanity has politics ever helped anything scientific? I'd argue the opposite in fact. Nothing gets in the way of science and truth better than politics. Now with that being said, I LOVE your videos and can think of nothing less relevant to the interviews than the politics of the person you are interviewing. Anyone that gets bent out of shape over something as irrelevant as that should really ask themselves some fundamental questions about their own existence. Keep up the great work and ignore those dinguses! :)

Personally I am convinced the issues with math presented in this video are (at least in part) the fundamental reasons why we don't (and may never be able to) understand gravity on very large and very small scales.

This reminds me of how observation itself effects the state of things on a quantum level, because as soon as you’ve “grasped” infinity it escapes you and becomes something greater. So it’s an imaginary thing.

to my mind, this is the best of the best, Today. Thank You.

I have been pointing out a weakness or flaw in the convention of having both rational and irrational numbers on the same number line since it is clear that these type of numbers don’t have a common number line.

Some irrational numbers may be relative to two or more axes. These particular type of irrational numbers are derived from special formulas that have x and y or x, y, and z components.

Other type of irrational numbers are constant values that are also derived from special formulas, but have no dimensions - they are not relative to any axis.

All irrational numbers lack precision because they are nonterminating and nonrepeating in their decimal value representation.

On the other hand natural numbers, whole numbers, integers, and rational numbers can be represented by precise values on a single number line or axis.

What then is the point of having real number line when irrational numbers are off from the number line of rational numbers, and irrational numbers could not be directly added to rational numbers to produce precise numbers?

Would it not just make more sense to have another label of classification instead of real numbers?

I'm not sure I agree with the finitist view... it was interesting to hear professor WIldberger's thoughts, but I think they are unfounded. Not to be the devil's advocate here. Just because our current computers can't handle some process does not mean that they the concepts aren't useful.

Great iconoclast!

I used to hate putting on Duvet covers.

What is x/x when x = the least of the positive real numbers that can't be specified? Is it 1, undefined, or indeterminate?

Don't we define e^x as the limit of a summation, rather than the total of a summation?

Same thing said about infinity coukd be said about complex numbers: no counterpart in the real world. Maybe we should not take math so seri ously and just use it as a tool.

Professor Wilderberger's series of lectures on the history of mathematics is really great: check it out. I was delighted to see him being interviewed on this channel. Looking forward to the second interview.

Would help if Prof would say Irrational numbers when he says Real numbers. The points he’s making don’t apply to the rationals. He’s talking about irrationals!! The rationals are a subset of the reals.

Incidentally, infinity and computation is a subject I was thinking about lately as well. Pretty cool! here is some of my own gibberish about it in an off chance someone wants to actually read it:

When it comes to infinity and cardinality - every computation takes non-zero amount of time. That single fact tells us something very specific about computer limitations vs universe's 'limitations'.

I had an immediate reaction to 'it can't exist until you write it down': The null value is a well defined object in computers (some programmers have hard times with it and some use it to advantage of the program but that's a diffirent subject). In our minds there is a concept of non-existence too. It also, obviously, exist in mathematics. But I haven't heard of proof that 'nonexistence' is an object in reality.

All and all I think computation shares some properties with the universe (current state is a result of former state, complexity stems out of simplicity, self-similarity), but they might not overlap and the latter is just much, much more.

Yeah, what does it mean "to do"? NJW says you can't do an infinite number of things as if we are talking about some sort of time-dependent manual process.

The perspective holds base 10 for social convenience. Just for the math linguistics I'm not ready to burn my bridges to base 10 describing. Sometimes to see or understand we let go of our means to describe easily. Math and numbers are only an expanded language set, to me. What we have done with log and e we may also assess another vectorable perspective to assist in describing time flexibility in the mass apparency. Sort of a finitable set framework to take on some portion of areas that we now describe as undefined or infinite. Edit: Wow! Now I hear just this moment that your guest mentions the Babylonian base 60. This might be worth learning to use as a bilingual addition to base 10 which I have already learned. May be worth learning with also keeping base 10 for translation. His statement is a breakthrough, interest building! By coincidence the dodecahedron has 12 Pentagons & 5x12=60. I simply state the coincidence as a personal point of interest. Wonderful understanding and perspectives that stimulate my interest! Thanks!

Talks about his sponsors for first 3 minutes of video

The halting problem in computer science comes from infinities. It seems bizarre at first but once you understand it you realize that it can't be any other way.

Good content in mutual respect, love it, thank you!

Excuse my naive perspective but isn't "infinity" a way of describing an unbounded aspect of a process? That is, in principle, we treat an aspect of an object or concept as if it is unbounded (hence refer to infinite) as a matter of utility? e.g. we may choose to model space as continuous (in the formal real number sense) but we aren't necessarily asserting that it is indeed continuous, just that we do not a priori assume a bounds on precision. Assuming unboundedness is extremely useful as it prevents us from biasing calculations to some fixed precision / scale.

If you're interested in interviewing people on the UFO phenomenon, and you want a perspective that isn't in the mainstream, you should organize an interview Fr. Spyridon Bailey, an Orthodox Christian Priest who recently wrote a book called "The UFO Deception", which continues the work which Fr. Seraphim Rose started in "Orthodoxy and the Religion of the Future" which talked about UFOs, their place in new age cults, and their striking similarity to what is reported as demonic phenomena from every culture since before the modern era.

The universe is smooth and continuous. Epsilon nought and Mu nought describe the physical attributes of space. The smallest "drop" of space that naturally forms is a Planck. When energy pressie outs less the Planck size is larger. When energy density< is high Planck is smaller. Gravitational lensing is light mooching through more dense substance. Space with all that gravity vector energy. Simple refraction.
The limits are the cosmological constant at low end and Planck particle energy density at high end. When a neutron is crushed by gravity so that it touches another neutron it becomes the vacuum energy, a sheet of photons 90deg to is which create isospin , and then re emerge in deepest voids,lowest energy points of universe. A physical process to maintain fine tuning.
Neutron decay cosmology is inevitable.

For many years prof. Wildeberger is obscesed with real numbers. I like his lectures on alg. topolology but his attempts to exclude real numbers from math are childish and unsubstantiated. As far as I know he wrote some booklet claiming (I hope I'm not wrong ?) that he can put all the mathematics on the rational numbers foundation. I'm afraid that it is just a loss of time.
I also understand that "going against academic mainstream" is very attractive for laymen and journalists having very limited capability to understand subtle and complex core ideas.
All these problems were resolved years ago (in 20. century) in the fundamental works of Hilbert, Gödel, Zermelo, Cohen, Vopěnka, von Neumann and many others.

Real numbers are real

Real numbers are real but they are mathematical points representing separate distinct locations!!

If you had a line segment you would have infinities upon infinities of separate locations on the segment each location represented by a real number!!

If you imagine a small circle, it would have infinities of separate locations.

If you had a pin or pen with its point tapering to a mathematical point, you could repeatedly poke the circle anywhere and the chances of you poking the same location twice would be 1/infinities upon infinities, it would be practically impossible to land the pen on the same location twice!! This is pointed out by the Banach-Tarski paradox!!

Nonsense.

While i very much enjoyed the interview and it led me down the rabbit hole of hypergroups and helped me find a new algebraic structure based on my rho-calculus, i felt you were too star struck to ask probing questions. For example, a huge number of physical processes that are relevant to humans are iterative. Biological processes certainly are. But, even before we reach the complexity of biological phenomena processes like diffusion as NW studies in hypergroups are also iterative. Iterative processes are extremely susceptible to chaos. This means that even very tiny differences in input result in arbitrarily large differences in output. This means that arithmetic that depends on some finite precision can be very unreliable for a wide range of physical processes. It would have been interesting to get NW’s take on the relationship between chaos and his ultra finitist view of the reals. Likewise, Conway’s conception of numbers as games offers a very compelling and computationally grounded notion of quantity. It would have been interesting to get NW’s perspective on Conway’s construction.

Such lovely intro ❤

i've always felt that infinity was a direction, not a number

If I had a chance to ask "God" one question, my loyalty to society would oblige me to submit the P=NP hypothesis because the answer would have profound implications for many fields in applied computer science. But my soul would compel me to ask whether the real numbers exist, because it would change my spiritual view of the cosmos itself.

What's the reason for Babylonians to use base 60 and not base 30?

When talking about the 30th decimal place, they are neglecting to think about chaos theory.