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Comment by coldtea | Hacker News Reader

The kind of crank with a PhD from Yale that has taught at Toronto uni, Stanford etc?

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.

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Smart people can be cranks. In fact they are far more likely to be so, perhaps because they overestimate their own ability. See e.g. http://lesswrong.com/lw/j8/the_crackpot_offer/ In any case I didn't call him a crank, but many actual mathematicians do.

>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.

Under some interpretations of physics there is a finite number of objects in the universe, hence a largest possible cardinality of things that is finite. Even sets like the number of possible states of the Universe appear to me to be finite, though so large as to be intractable to my imagination.

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.

What do you mean "refutation"? This is making you look as much of a crank as they are. There is no imaginary requirement to have a direct counterpart in the physical world, in "matter/energy".

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".

Perhaps I am a "crank".

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.

>There is no imaginary requirement to have a direct counterpart in the physical world, in "matter/energy".

No, but there's no requirement that the useful/only way to do math is with "platonic" / ideal math objects either.

sounds like you belong to the pythagoreans[0]

[0] https://en.wikipedia.org/wiki/Pythagoreanism

I'm certainly sympathetic to some Platonic ideas.

As far as we know the universe isn't discrete like a cellular automata. So it's state contains infinite information for starters. I'm not sure what the current theories of physics say, but I don't believe the universe is finite in time or space either. And who knows what new discoveries physics might make in the future and what weird math it might require. There's no a priori reason to expect the universe to be modelled by finitist math.

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.

Current theories AFAIAA are that the universe is finite in all dimensions. If it had finite particles, with finite (quantised) states, with finitely many discernible locations, to be found at finitely many discernible times, then the states overall are finite.

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.

There may be finite real particles, but there are infinite virtual particles (and some times, the difference between a real and a virtual particle is not too clear).

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.

But if the energy is finite, soesnt it mean that you can only produce so much virtual particles?

That's the nice part of virtual particles. If they are short lived enough they can be so many or so massive as you wish. Well, not as you wish because there are a lot of rules that relate the amount of particles, the mas and momentum of each one and the available energy.

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...

You seem to forget that maths and physics are meant to describe reality, not be reality. Any abstract concept (like numbers, forces, shapes, temperatures, energy levels, ...) we use to describe a real thing is different from that thing itself. If you think about it, even 2 isn't "more real" that Pi, because there are no 2 identical things in reality.

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

Two is the number of electrons in He, and the number of slots in the lowest orbital. Two appears in nature.

Circles appear in the shape of that same s orbital, though, so I don't see how to avoid pi.

There's no such thing as an S-orbital, there a probability distribution where a bounded probability produces an approximate sphere, in any universe with more than two charges that sphere is going to be 'irregular'.

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.

So you deny classification of similarity - fingers have structural similarities that allow them to form a class of object/item, this class and it's per-person sub-classes have a reality that to me is not entirely abstract.

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.

>You seem to forget that maths and physics are meant to describe reality, not be reality.

So why describe reality with an ideal and infinities and not based on fixed/arbitrary precision but limited math?

We've entered fuzzy-definition territory; but I'd say it is a stretch of the language to imply Wildberger is a crank. Which you have now done twice with no references.

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.

Post-euclidean geometers did not claim that Euclid was wrong, but if the article is to be believed, this guy is saying everyone else is wrong, apparently with arguments that are sketchy at best.

He is not being called a blasphemer, he is being dismissed with reasons. There is a difference.

>Post-euclidean geometers did not claim that Euclid was wrong, but if the article is to be believed, this guy is saying everyone else is wrong, apparently with arguments that are sketchy at best.

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).

You are making an invalid comparison here. In your possibly hypothetical example (were the founders of non-euclidean geometry dismissed as cranks by their peers?) your lone post-euclidean would be justified in calling the others wrong for dismissing his mathematics, and also justified in denying their claim that Euclidean geometry was the only geometry, but he would be wrong if he were to say that Euclidean geometry itself was wrong. Wildberger is doing the equivalent of the latter, apparently.

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.

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