Comment by WiltedSpinach | Hacker News Reader

Comment by WiltedSpinach | Hacker News Reader

Dr. Zeilberger writes a rather pleasant rant. He appears frustrated with the lack of imagination in mathematicians which occurs when they focus too strongly on the axiomatic nature of mathematics, and seems to wish that we take a step back from the powerful axiomatic tools we have developed, so that we may search for what is 'really true about the world' rather than 'playing a mathematical game/believing in a mathematical religion'.

Three specific complaints he has involve a belief in infinity (and limits), which he asserts do not exist in the real world; a delay in the publication of a pair of papers on which he worked, due to a variety of circumstances; and a "pernicious" influence of axiomatic mathematics which leads to "stupid" questions such as Hilbert's Second Problem.

The failed publication of one paper is particularly notable, as it claimed a counterexample of Fermat's Last Theorem (according to Dr. Zeilberger's "Opinion 123", on his Rutgers website), and (ibid) was recognized by Andrew Wiles himself as one of three possible counterexamples: the reason given for this oversight was in fact the acceptance of the decidedly non-rigourous statement "it is easily seen ..." (ibid). This particular instance seems to contradict the main thrust of Dr. Zeilberger's rant against a mathematics overburdened by rules.

Unfortunately, little example of what mathematics SHOULD look like is offered - beyond a statement that 'obvious things should be treated as such' and an assertion that infinities and continuities do not occur in real life/nature/the universe. While such a statement may be understandable coming from a respected (and clearly accomplished) combinatorist such as Dr. Zeilberger, physics has yet to demonstrate conclusively that space and time are discrete - undisproven interpretations of quantum mechanics exist which allow for continuities, and 'the size of the universe' is defined as that which we can see (ie, within ~13.8 billion light-years of Earth/Sol) so that actual infinities are ruled out only because of our inability to perceive them.

Perhaps Dr. Zeilberger needs to think outside his own discrete box.

Re: infinities and continuities, to my understanding, it has been proved than interaction with the world is inheriently discrete determined by Planks constant - I am not a physicist, so may be wrong.

If we cannot interact with arbitrarily precision and can only ever observe a finite universe, does it make sense to even ask if the it is any other way? Should it be figure out how to interact with the world using some new fundamental mechanic, other than a wave bound by C and h, it opens the question.

Even if the world is fundamentally discrete, that does not mean that continuous models are not valuable. They are.

We are used to doing discrete approximations to continuous problems, but the other way around works too. For example, evaluating the sum

1 + 1/4 + ... + 1/n^2

is quite hard. On the other hand, we may approximate by the integral over x^2, which remarkably has an easy formula.

Continuous and discrete models complements each other, they are not mutually exlusive.

I wasn't trying to say that continues models are not valuable, only that the question "is the world continues?" does not make sense if we can only interact with it in a discrete way.

We don't interact with complex numbers but "Are quantum 'probabilities' complexL" is still a meaningful question.

That is simply a rewording of the question: is it useful to use continuity to model the world.

This is getting quite philosophical now. It depends on the intrepetation of "world". If we mean the universe then no, it is to our current understanding of it, not useful to model it's entirety using continuity, if we can only messure it discretely. If by world we mean a subset of the universe, then it makes perfect sense, because the world might be on the macro level where this "discretness" is far smaller than we ever want to go.

I think you should concentrate on the world model in my sentence rather than world. I have something I want to analyse. How is it useful to think about it and what techniques do I want to use?

Planck's constant arises in continuous theories of quantum mechanics, such as Bohmian mechanics and its QFT extensions.

There are limitations to how much we can know, but not limits to what can be.

I haven't studied Bohmian mechanics to any reasonable degree, so I cannot comment on it. But preliminary reading has me sceptical because it relies on infinity. Time will tell I suppose.

Then read "The Cellular Automaton Interpretation of Quantum Mechanics" by Gerard 't Hooft (https://en.wikipedia.org/wiki/Gerard_%27t_Hooft), which tries to formulate a superdeterministic version of quantum mechanics based on cellular automata (i.e. much more discrete):

> https://arxiv.org/abs/1405.1548

EDIT: also cf. https://link.springer.com/book/10.1007%2F978-3-319-41285-6

There is some light bedtime reading for a decade or two... Thanks for the link, I actually look forward to dive into the topic.

A slight aside- due to the expansion of space, we can see objects that are currently further away than the simple amount of time it took the light to reach us. GN-z11 is one of the most distant objects imaged, has a light-travel distance of 13.4 billion light-years, but a proper distance of little more than double , 32 billion light-years.

https://en.wikipedia.org/wiki/GN-z11

I remember Dr. Zeillberger from my graduate days at Rutgers. He clearly had a lot of interesting stuff to say which he packages in sensational and emotional form. I found it off-putting then as well as now.

But to be charitable, I think it comes from a place of pushing with tremendous force a pendulum of thought back from the axiomatic nature towards more exploratory and relevant mathematics. It reminds me of Howard Zinn's intro to A People's History of the US in which he says that he purposefully did not water down his arguments to be more neutral because of the overwhelming nature of the opposition.

The infinity issue is quite interesting to ponder. Obviously with our current understanding of the universe and humans, the number of numbers ever encountered by all humans over all history would seem to be finite regardless of the fundamental nature of the universe itself. There is a very strong notion of our world being discrete in that way.

And it is very interesting to pursue these ideas. For example, the proof of infinitely many integers is to assume there is a largest and we add 1. But what if the largest is unknown as is the case much of the time? This is true, for example, with computers (it can be known for the native format, but that can obviously be extended in programming in an arbitrary way)

It is a practical question to ask about existence questions that require an axiom (axiom of choice often) which gives no constructive way of doing something. To what extent is that actually useful? And if we had an approximate construction of something without the axiom, but the thing itself may or may not exist depending on the axiom chosen, what would that mean?

It is also interesting to think about doing away with infinity and thinking about how annoying it would be to not be able to talk about pi or e or even sqrt(2). There would be no irrational numbers, presumably, in this framework.

One can then see the wonders that embracing infinity can lead to as a crutch for dealing with the horrendously messy world of the finite and discrete.

Discussing these issues when learning mathematics strikes me as elevating a great deal of bland rule absorption and gives students power over the tools of mathematics. So this debate is useful in my opinion though it would be absurd to try and draw a conclusion.

One last note about his discussion of the lack of rigor in QFT and that it was good. Recent work (at Rutgers[0]) suggests that at least some of the infinities and problems come from not appreciating the proper physical picture of what is actually happening. Nonetheless, the standard procedures worked to give us needed answers before such insight was discovered.

I think the basic answer should be that formal and exploratory mathematics are both useful and it is important to avoid dogma from either side.

[0]: https://arxiv.org/pdf/1703.04476.pdf

> It is a practical question to ask about existence questions that require an axiom (axiom of choice often) which gives no constructive way of doing something. To what extent is that actually useful?

I'm by no means a mathematician, but I do sympathize somewhat with this. To put it a bit controversially, the axiom of choice feels to me almost like the string theory of mathematics.

There's a lot of really cool stuff you can do with it, but we've left the realm of even potentially useful, and are off approaching a form of rigorous philosophy.

I was looking into Banach–Tarski recently. Is it actually that useful to discuss something which almost certainly can never physically exist in our universe?

Now of course you could play devil's advocate and say as some point people claimed you could never in the universe have a negative amount of something, so integers aren't useful, or a square root of a negative isn't useful so no imaginary numbers. Things are created all the type in math that we don't find practical applications for for decades or centuries.

Yes that's true, but it's not a question of absolute, it's a matter of degree. Has math swung too far into the world non-relevant "play" at the expense of other discoveries?

Recently we've just come across one of the most incredible tools of the past 1000 years. Computers! As a programmer aware of Hindley-Milner it boggles my mind the amount of programming people have been doing doing centuries without the benefit of computers to run them on. Now that we have them we can course correct. Previously we shooting in the dark, and were surprisingly accurate. Now that we have computers this utilize them to focus more and prioritize the amazing things that come from powerful computability.

Let's explore the systems that can be constructed using these new amazing tools. Infinity in itself is ok as a model common real world applications (pi, e, sqrt 2).

Getting out of my knowledge area, here, but what tools would we discover if we were focusing on a math without the Axiom of Choice? What proofs would be reformulated and what tools would we develop to do this work.

Again, everyone should be free to think about whatever they like, but as a field it should also be cautious of how it spends it's resources. Particularly it's most valuable resource of intellectual capital.

One thing which the author of this essay gets right is that contemporary mathematics is a bit dogmatic when it comes to the logical foundations. We should be looking at the Banach-Tarski theorem as evidence that a particular logical framework for reasoning about volumes/probability is more complicated than strictly necessary. Presumably no non-measurable sets exist in the "real world". At this point there are two possibilities:

- A theory with non-measurable sets is much simpler/more expressive than the alternatives and thus is still a useful tool.

- There is a simpler theory without this defect, which is at least as expressive when reasoning about real world phenomena.

In this case, the latter possibility turns out to be true. It is just unbelievably difficult to convince mathematicians to change the rules of the game, even if you can point to concrete gains.

> There is a simpler theory without this defect, which is at least as expressive when reasoning about real world phenomena.

This is really interesting to know, and was exactly the type of think that I was wondering about. And I assume the author felt similarly although didn't draw identical conclusions.

Out of curiosity could you describe or link me to this other theory?

I was thinking about categorical probability theory or measure theory based on locales. Unfortunately there are (to the best of my knowledge) no textbooks or good writeups available for either, merely long lines of research papers. As said, it's a bit of a niche area, since most mathematicians don't want to think about reworking foundations.

Are there any papers you would recommend as good starting points?

Without the axiom of choice you get similarly bizarre results. For instance wothout choice there exists a surjection from the reals to a set of greater cardinality.

That's because the reals don't exist, which is Doron's point. We need the Axiom of Choice because we invented the reals.

The existence of the reals is independent of the axiom of choice. The collection of mathematicians that don't believe infinite sets exist has maybe a few members. Constructive mathematicians believe the reals exists. They can be constructed.

>Constructive mathematicians believe the reals exists.

I think when people mention the "reals" they are usually referring to the uncountable reals, which you can't construct, compute, name, or know:

https://arxiv.org/abs/math/0404335

...(maybe skip to chapter 5 to get to the meat of it). So constructivists don't believe in those types of numbers, and so get "choice" as a theorem instead of an axiom.

Constructive mathematicians definitely do believe the reals exist and are uncountable. Not all reals are computable or definable. The set of computable reals is countable. There are different varieties of constructive mathematics. A very large majority of constructive mathematicians believe the reals are uncountable. In intuitionistic mathematics the reals are uncountable.

In some constructive versions of math you get weird things like there being an injection from R to N but there not being a bijection due to diagonalization.

I think you are confusing computable/nameable with constructive as that term is used by most mathematicians.

Note that Chaitan's Constant is not computable.

This link may be of interest to you:

https://mathoverflow.net/a/30694

What do you mean? If ZF without choice shows that a statement P holds then certainly ZFC also proves the same statement. Do you mean that there is a consistent extension of ZF with the statement "There exists a surjection from R to P(R)"?

There is a theorem that states that either there is a nonmeasurable subset of the reals or that there is a surjection of the reals to a larger set. The theorem is by Sierpinski. This is way out of my area but I'm guessing ZF is not strong enough to show either all sets are measurable or the existence of a non measurable set.

The common criticism of AC is Banach-Tarski. So if you don't agree with Banach-Tarski and want every subset of R to be measurable then you have to conclude an equally bizarre result.

As an algebraist I accepted AC. Instead of constantly saying, "let V be a vector space over k with a basis" it's easier to just assume all vector spaces have a basis. I think analysts need AC more than other branches, The Intermediate Value Theorem isn't provable without AC. I think.

What I found after Googling is that there is a theorem by Sierpinski which shows that if we make an additional assumption about the cardinality of [R]^\omega, which seems to be approximately the same as the boolean prime ideal axiom, then there is a non-measurable set. Thus the existence of non-measurable sets is implied by principles weaker than full choice, but that's not too surprising.

The case of measure theory is particularly interesting, though, since you don't even need to change your underlying logic to get a better model. For instance, if you base your "measure theory" on valuations on locales instead of measures on sigma algebras then the theory itself becomes simpler and the Banach-Tarski "paradox" goes away.

Briefly, in locale theory, your "measure" is defined on sublocales instead of subsets. While there are more sublocales than subsets, the condition for when two sublocales are disjoint is stronger. This is what breaks the Banach-Tarski construction. The orbit subsets used in Banach-Tarski still exist, but while they are disjoint as sets, they are not disjoint as sublocales and thus don't decompose the volume of the sphere.

I'm not familiar with locale theory. Thanks for the reference. I'm guessing there will be some non-intuitive results. My non-expert impression is that whatever one chooses in terms of logic and set theory there will be bizarre results when dealing with sets of cardinality of the continuum and not dealing with the continuum leaves out too much.

Here is a reference to a mathiverflow comment.

https://mathoverflow.net/a/22935

You should read about constructivism.

https://en.m.wikipedia.org/wiki/Constructivism_(mathematics)

Better yet, read the book Zeilberger mentions at the beginning of his essay:

"The Mathematical Experience" by Phillip J. Davis, and Reuben Hersh

http://isbn.nu/9780395929681

...which is a great book, composed of a bunch of different essays comparing and contrasting three philosophies of mathematics (Platonism, Formalism, Intuitionism).

There is a whole body of mathematics which takes these things seriously (constructive math, intuitionistic foundations) and it's, if anything, even more heavily formalized. I think the author has some technical points, but his desire for "make math more like science" is outmoded in many areas. Mathematicians went toward formalism for good reasons. They might not have made all the right decisions, but that doesn't mean that upending the table and starting fresh makes sense.

Zeilberger's Opinion 123 is an April Fool's Day joke.

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