If I understand correctly, he goes even farther and claims even really big but finite numbers don't "exist". And through this he claims to have "resolved" the Goldbach conjecture and other strange things.
He is not alone, this is a known branch of mathematics called finitism: https://en.wikipedia.org/wiki/Finitism
I am certainly happy to be corrected, but I don’t think we have found any event in nature that requires infinities to explain it.
Ultimately it comes down to a trade off. Which axiomatic system does one to work in ? Introducing infinity does have its fair share of warts (by warts I mean consequences that flies in the face of intuition).
You can truncate the series at an appropriate place of your choosing. But yes, \Pi as a number does not even exist unless you add the Reals, so that would need infinity. What I am saying is, one can work with the Rationals.
1 = 0.999... 1/2 = 0.49999...
etc.
In some ways it's a property of the representation, not something intrinsic to the number.
Or is it impossible to have contrarian views to basic tenets of a human-made axiomatic system such as math without being called a blasphemer?
>If I understand correctly, he goes even farther and claims even really big but finite numbers don't "exist". And through this he claims to have "resolved" the Goldbach conjecture and other strange things.
Does he just "claim" or does he give proof? Because if he has the proper system, and he proves his theorems based on his axioms, and does that correctly, that's not much different than e.g. non euclidean geometries.
>Does he just "claim" or does he give proof? Because if he has the proper system, and he proves his theorems based on his axioms...
I might be misremembering this, but I think his argument was just that "computers have tested all possible counterexamples up to some big number, therefore it's true.
All this talk of different axiom systems is giving ultrafinitism vastly more credit than it's due. It's not a different but coherent mathematical system. It's a "philosophy" based entirely on refusing to accept a result because it feels counterintuitive. And then coming to vastly more counterintuitive and absurd results as a consequence of trying to resolve it.
I mean they really believe that there exists a biggest number and there are no numbers bigger than that. Even regular finitists are kind of absurd, denying the existence of pi or the square root of 2.
A conceptual circle and a real circle differ markedly because of the lumpiness of matter, quantisation, uncertainty. Every real circle is an approximation (yay Plato!).
Pi doesn't appear to me to be real, as in writ in matter/energy -- it's a ratio of measurements of an imaginary article, circles don't exist in our experiential 4-space.
Irrational numbers like root-of-2 just fall out of basic maths though, there's not going to be a real line with that measure just as there's no line with a predefined unit measure (ie unless you define that line at that point in time as your unit, but a real line isn't even straight).
I'd be really interested in refutation of this -- it's one of those positions I've held since Uni, at least, but never explored as I've never seen it challenged/supported nor named.
When we say "a human hand has 5 fingers", we use the abstract concept of "finger" to make that description. It is abstract because each "finger" is actually unique. It's just a handy approximation we use (pun intended) for descriptive goals.
Basic maths is not reality. Therefore no, irrational numbers do not "fall out of it".
Edit: format
So why describe reality with an ideal and infinities and not based on fixed/arbitrary precision but limited math?
Circles appear in the shape of that same s orbital, though, so I don't see how to avoid pi.
The parent argument works here as much as for fingers though, those 2 electrons are identical but their spin differs. All electrons differ in the Standard Model by either spin or location.
Even without classification we can create arbitrary groupings of physical objects, those groups have cardinalities. (Unless your materialism denies the existence of more than one thing/substance; philosophy is fun, eh.)
I'm for pragmatism in such things however.
But even if we knew for sure it was, so what? Theorems about real numbers apply perfectly well to finite approximations. You don't gain anything by restricting yourself to finite math, and you lose quite a bit and make a lot of things much harder.
Time and space are effectively discretised, Planck scale and time. Even aside from that: for considerations of physical measure we always use something with an explicit size, a wavelength of light say, as our rod which limits our precision.
For example, a proton has three real quarks. But the mass of the three quarks is only the 1% of the mass of the proton https://en.wikipedia.org/wiki/Quark#Mass the rest is crap, or in more technical terms https://physics.stackexchange.com/a/81284 . When you measure with more energy, you see more and more virtual particles, infinite virtual particles.
At low energy the effect of the virtual particles is negligible (except in a very few weird cases like the https://en.wikipedia.org/wiki/Lamb_shift ). At high energy the probability of the virtual particles is bigger, so their effect is easier to measure.
I like this drawing of that shows the firs three steps of diagrams with virtual particles that are necessary to calculate the magnetic moment of the electron: http://www.strings.ph.qmul.ac.uk/~bigdraw/feynman/slide3.htm...
Infinity, axiom of choice, .. are all tools. We use them because their presence gives us other helpful results. None of it has anything to do with "lumpiness of matter", "matter/energy", "experiental 4-space", "real line".
No, but there's no requirement that the useful/only way to do math is with "platonic" / ideal math objects either.
Refutation as in "you overlooked this standard point that undermines your position", like maybe "polarised light describes a perfect circle", or "the gravitational field of a black-hole (when shaved) is a sphere".
AFAIK there's nothing truly infinite, nor perfectly geometric, that we can experience in our physical spacetime -- I'd like to hear from those who can show me that's wrong.
The world itself isn't axiomatic, if certain theories/systems that are unfalsified show parts of it are infinite/etc. -- despite that just being a model -- that's an interesting result.
He is unorthadox in his pushing of rational numbers, sure, but he gets through an extremely large amount of maths correctly. It is well within his job domain reflect on odd academic questions about what happens outside the world of human measurement.
He is not being called a blasphemer, he is being dismissed with reasons. There is a difference.
When everybody else believed that euclidean geometry was the only one possible, someone that insisted otherwise would also be labelled a crank (and he would be right in calling others wrong).
I am sure you understand that being labeled a crank is not evidence of actually having greater insight. Instead of arguing from a dubious analogy of uncertain applicability, you would be much more convincing if you first took the specific objections to Wildberger's position from the article, and refuted them.
Infinity is not a number. It's a limit.
Treating it as a number only confuses real understanding of it.
The quarrel Wildberger has is not with the proportion 1:0, but with the concept of a limit.