http://math.ucr.edu/home/baez/rosetta.pdf

Graduate real analysis is the hardest thing I've ever had to do. This **** is killing me.

Friend was into ESO and wanted to code a bot but didn't know where to start, so I wrote this up for him yesterday:



Gonna let him figure out how to take the core movement engine and do some stuff with it..

the client has some convenient "features" too, like how moving the mouse 1 pixel corresponds to exactly 0.002 radians. doesn't work as neatly in many game clients

I think there's no problem really with knowing the distinction, but I think there's not much outside the collection "¬¬P~P for all predicates P" that's useful.

lol

The reason I linked to that paper is because the way you talk about intuitionist logic makes it sound as though you think that only people with some kind of ideological ax to grind could possibly be interested in structure that emerges when you introduce finer distinctions.

You may not have a use for it, but don't dismiss entire branches of computer science because you think it's a waste of time.

I'm pretty sure that a vibrant subfield of mathematics / computer science doesn't boil down to a misunderstanding exemplified by some confused undergraduate you tutored.

Not sure what you're talking about mate, other than you're seemingly taking something as a criticism that I didn't intend.

Good to know.

Can you remind me why you brought up my name in the context of your discussion with your confused friend? Because I am having trouble seeing the connection.

They weren't really related, I should have put them in separate posts. I really just mean that I've noticed more and more that programmers have a way of thinking about problems that is more finitist/constructivist, and I suppose that's related to the craft.

OK, that's fair.

I was thinking today about finitism, and it's kind of tricky to deal with. For instance, proving "there are infinitely many primes" becomes much more complicated if you only have potential infinities. If you don't believe actual infinities exist, then this statement:



Becomes really hard to show. Why? Because it can't be proved over all n, because a set of all such n doesn't exist. You can only show that, if a_1,...,a_n is prime, and the current largest number is big enough, that a_1 x ... x a_n + 1 exists and is coprime, implying the existence of another prime. Meaning the proof no longer can just assume the implication, it has to show that, for each number m, then for a number k large enough, we can construct the above proof to show m+1 many primes exist. So I believe if you're a finitist, you have to sit there and actually demonstrate the potentially infinite amount of proofs exist for the above.

Moreover, I'm not certainly you can actually show a k exists that guarantees the appropriate product exists without being far more patient than i am.

I don't think I am a finitist, but Doron Zeilberger would say that there are plenty of true statements that humans can't can't hope to prove, and that simply adopting classical logic doesn't really help much at all in coping with this terrible reality. Rather, something more drastic is required: either give up the goal of being able to understand every single step of the proof, and use brute force through computer, or pay less attention to proofs and more attention to the structure of the subject, and do as the physicists do, and treat proofs simply as very, very inefficient heuristics that guide understanding, and which typically aren't competitive with faster and looser sources of evidence for a conjecture being true. Zeilberger thinks that people who rely too much on classical logic simply haven't figured out how to go beyond human intuition because they haven't bothered to formalize their subject well enough to feed it to a computer.

http://sites.math.rutgers.edu/~zeilberg/Opinion69.html

Implicit in the title of that essay is that finitism doesn't even give a **** about intuitionist logic, since even it tries too hard to salvage the parts of classical mathematics that we already understand (the work by Paul Taylor tries to formalize heuristics used by classical mathematicians, using intuitionist logic, category theory, and type theory as tools toward that end). Zeilberger is saying that we can't hope to understand any of it once we realize that compared to computers, humans can only prove trivial statements, so we might as well jettison all pretension and treat the subject as combinatorics.

I mean, Zeilberger would say: this or that philosophical interpretation to mathematics could be "right", and it still wouldn't matter if your goal is simply to prove conjectures in combinatorics (and that all mathematics is secretly combinatorics).

Actually, I don't think finitism is even possible. Well, not in any pure sense. It's possible to be resistant to infinity in some cases, but I don't think it's possible to be resistant to infinity over all things.. and I'm starting to believe like it's more misleading a thing to think than not.

It's simply not possible in my view to extricate "completed infinities" - ever. Because it's never possible to make claims about arbitrary things without reference to one, if a potential infinity exists. Consider Peano arithmetic. We start with 0, and have a successor function. At any one time, we only have finitely many numbers. But the Peano axioms state e.g. "for all natural numbers x, x=x". This statement can't possibly make sense unless it's implicitly referring to the set of all natural numbers. Because a "potential infinity" only allows finitely many things to exist at any instant, never an infinite collection, so "for all natural numbers x, x=x" can only be defined on the natural numbers which exist up to how many times you've repeated the successor function. But then you can't know if it holds true past that - if it holds true past that, then what you're saying is you know for all natural numbers n, n=n, including the ones I haven't defined yet. If we allow for potential infinity, then we've actually just kinda made a completed infinity by how our axioms work.

In other words, the very definitions themselves imply the completed infinity - if you don't, then the definitions only make sense for finitely many things, and it's never possible to define the natural numbers, because we can't know if any of the statements hold true past some n.

In fact, this seems to be recognized - the "simplest" formalization of the natural numbers I can find is called primitive recursive arithmetic, which presumes the existence of, you guessed it:

- A countably infinite number of variables x, y, z,....

Even a statement such as "the non-logical axioms are: S(x)\neq 0" seems to refer implicitly to completed infinities in the same manner as I said above. If such a statement is always true, it's always true for all n, and that can't possible make sense in any way that makes potential infinity coherent..

It seems to me that finitism is more like "i don't like this certain type of infinity in this certain area" than anything.

Course I might just be speaking nonsense. IANALogician.

lol, You should send that to Doron.

I doubt he cares though.

no desire to speak to him

It probably wouldn't be a pleasant conversation.

[quote=Doron Zeilberger]
Continuous analysis and geometry are just degenerate approximations to the discrete world, made necessary by the very limited resources of the human intellect. While discrete analysis is conceptually simpler (and truer) than continuous analysis, technically it is (usually) much more diffcult.. Granted, real geometry and analysis were necessary simplifications to enable humans to make progress in science and mathematics, but now that the digital Messiah has arrived, we can start to study discrete math in greater depth, and do real , i.e. discrete, analysis.
[/quote]

.

You're right, his paper makes him sound annoying

https://www.youtube.com/results?search_query=doron+zeilberger

Look at the titles of some of those videos

"The Rise and Fall of Astrology and the Future Fall of the so-called Infinity Part 1 "

epic

My view of Doron goes back and forth between crackpot, troll, and genius. He is definitely a goofball. But his papers are quite inspiring if you are into combinatorics.

Eh, many people who do fine work have crackpot beliefs. I think his beliefs on infinity tend towards crackpot, but that's no indictment of his work in combinatorics. Newton was a crackpot, after all.

I think it's weird though for a person to step over the line from "I personally don't like infinity" to criticizing the whole field for using it. As far as I can tell, the cutoff point for where someone thinks the amount of infinity they're using is "acceptable" is arbitrary, and having read up enough on the kinds of ultrafinitism which actually extricate infinity enough to be "truly" not infinite, they're so barebones it's hard to even call it mathematics. I think I've concluded my meta-survey on the issue this way, and decided that whether you use infinity amounts to nothing more than a personal preference on how you like to see proofs done.

But I've not been convinced at all that the axiom of infinity is a bad assumption by any means, or that there's anything wrong with assuming it. I draw the line there, when a person criticizes other people for doing it, as though they're making a mistake for doing the assumption. I get pointing out the distinction, I get preferring it stylistically, I get pointing out the degree to which some assumptions about infinity are unnecessary, but I can't find any argument why it's necessarily bad, i.e. some kind of proof that infinity creates contradictions that's logically sound, something indeniable.

I think the historical problem with "completed infinity" was that people were often using it incorrectly, or without understanding what they meant by it.



The reason people like Doron are in the dark about modern notions of infinity is likely because, however thoroughly the idea has been clarified, they just don't get why they should care about it, if they never even need it. I think Doron would be perfectly happy doing math with the technology available to James Stirling.

Let me put it this way: is logical necessity a good enough reason for modern workers to care too much about something that is largely a technical set of results that happened to be fashionable in the past for philosophical reasons?

This is why I am sympathetic to those who don't give a rat's ass about fiddling about with Cantor's machinations:



https://www.math.uh.edu/~tomforde/Articles/DeathOfProof.pdf

If there's any point to the insane amount of focus on proofs and sets that the early 20th century witnessed, it's probably in teaching computers to prove things (since humans can't generally hope for proofs to be intuitive). But if you're going to do that, you might as well just cut to the chase and learn about type theory and intuitionist logic, because guess what, you aren't easily going to teach computers classical logic (correct me if I am wrong). And what the heck does that say about the coherence of the idea!

Cantor was in part motivated by philosophical notions of infinity and God when he invented set theory. And yet Bertrand Russell had to invent type theory just to fix the bugs!

Rather than clinging to a "god of the gaps", if we are going to talk about a universal language for mathematics, why not base it on something that is actually elegant and concise like category theory and type theory?

Really, I am pretty sure that set theory is just a domain specific language that does a good job of describing things that involve measure theory. But that doesn't mean that the language ought to be universal, when most mathematicians in other topics (like algebra) get by simply using category theory first and foremost.

But maybe it's not a question of whether or not this "axiom of infinity" you speak of is "right". Maybe it's one of those things where the only winning move is not to play, i.e., the mistake was thinking it was a good idea to assume any such axioms at all in the first place.

I mean, I would have thought that the cluster**** that was 20th century set theory and logic would have made for as good a negative result about the universality of simple ideas in logic and set theory.



If I can say the same thing about the logical consistency of the existence of an all-powerful creator, that doesn't change the fact that to some people this is unelegant word salad that is only "self-evident" to true believers.

Look, don't get me wrong. There are plenty of places in mathematics where this stuff is necessary. But as far as I'm concerned, a body of theory this ugly is best thought of as a tenuous collection of technical results.

Let me put it this way. If we need axioms to figure out wtf is going on with set theory, and we have to resort to books like Naive Set Theory which are desparate to get the hell out of dodge (Halmos said the book should be read once and thrown away), that's not exactly a ringing endorsement, is it?

Category theory and type theory open up problems that set theory works for. It doesn't solve things, it just moves problems from one area to another one. Set theory was the elegant solution, the difficulties of "nice theories" are never apparent beforehand, they become problems after working on it. So you can't know that category theory and type theory are going to be "more elegant" - in fact, for some proofs they may prove to be far less elegant. So the criticism I don't buy is that set theory needs replacing now, or that replacing it will solve all issues without creating new ones. Those seem like wildly bold claims to make and I do not think they're supported by evidence, just by the dreams of a few people. And it's 1) not possible and 2) not a good idea to require everyone to drop set theory off of this rather petty aesthetic concern.



No, it's not a ringing endorsement, but it doesn't mean set theory has to be dropped to work on a hypothetical other project which someone believes will fix everything and make math great againn. The history of "projects which fix all old issues and don't create new ones" is pretty slim.



It was a good idea, it lets us solve new problems that you otherwise couldn't. And it's so far completely noncontradictory. So I don't think there's any non-aesthetic ground to criticize it.

I don't think I ever contended that set theory shouldn't be used where it's natural to do so. My contention is that you don't gain much by shoehorning all of mathematics into a highly problematic language simply because it has achieved prefered status among mathematicians for frankly what turned out to be misguided philosophical reasons. There's a reason why people started using categories to describe the actual structure of mathematical ideas in topics like algebra and topology. Spending more than a trivial amount of time thinking about set theory anywhere outside of real analysis, dynamics, or the theory of computation is really just a waste of time as far as I can tell.

I'm not saying that set theory needs to "go away", any more than a battered religious person needs to prove that God doesn't exist simply in order to stop thinking about it.

I mean: FSM exists for all I know, but I try not to think about it.

The upshot is that the whole "you're a finitist!" canard is either religious dogma shouting down a heretic, or a strawman. I think that set theory aficionados care far more about these distinctions than those who just want to get on with the actual mathematics.

Construct the real numbers up from ZFC via Dedekind cuts, or define the real numbers as an ordered field with completeness axiom, it won’t make much difference to most people.