I'm sorry to say this, but if you really mean to argue that the complexity of a theory of finite sets is in any way comparable to the problems that arise from infinite sets, then I'm not sure what to say.

I mean, pretty much all of analysis is about looking for ways to reason about infinite sets using the same machinery that work for finite sets (this is what compactness is all about).

If you mean to argue that finite sets are just as complicated as the entire range of potentially non-compact sets, then please tell me why every finite set is compact?

Also, if we want to talk about this stuff, we should really first take a class in computer science, because people have already thought of it. Maybe the closest thing in computer science to infinite sets is recursive function theory.

Whose pipe dream is this?

This is like saying that arabic script for algebra can't possibly solve every problem that wasn't already potentially addressed by pre-algebraic attempts to solve equations by expressing them in natural language.

Of course algebra doesn't solve every issue. Nobody said it does! But... not using it puts you at an extreme disadvantage.

Literally what my contributions in this thread are about. :downs:

My only contention is that time will show these directions to be monotonically increasing improvements. That is of course more subjective, but even I confessed that the jury would still be out for decades if not centuries.

I am not saying that the study of "infinite" sets isn't going to be an interesting activity in its own right. Or that we're going to be able to translate even half of that activity into more explicit, computable language.

Just that refraining from invoking them before they are absolutely necessary is a worthy scientific endeavor, because we learn something that was hidden before by our overly powerful language. (This is benefit another similarity shared with computer science.)

Look, if you don't want to use constructive or intuitionistic formulations even when they work perfectly well, nobody will stop you if you find set theory more natural. I would use it myself probably.

A research program to designed to achieve the same results as before, while succeeding in the banishment of a known class of problematic objects--in order to enjoy the benefits of new properties to which the allowance of said problematic objects were counterexamples--is emphatically not a branch of "philosophical skepticism".

If you think this, then you should also think that pure functional programming languages arise from "philosophical skepticism" of side effects.

Of course people who work on pure functional programming may harbor lingering skepticism of side effects. You may even call it paranoia leading to ideology. But to cast aspersions on their results, which are merely artifacts of technology, because you find their philosophical motivations distasteful? Haha.

You're sorry to say you're not sure what to say?

Maybe you're confused, I wasn't speaking then about theorem checkers, I was speaking about finitism.

Theorem checkers are entirely whatever to me. I'm not dispassionate, not passionate, the only refrains I have there are that I doubt they will be the revolution implied earlier in the thread.

Basically, this thread is about me trying to convince you of the relative merits of some cool technical results in a branch of mathematics which you think shouldn't be considered as important as I do because you find the philosophical motivations of many of its progenitors distasteful or unnecessary.

If I hadn't been the one to begin this discussion with over the top criticisms of some of the problems in set theory that motivated this research, or if I hadn't overly enthusiastically tried to get you interested in something which you clearly don't care too much about, I would tell you to go pound sand by now.

Well, yeah, constructivist proofs are more insightful than non-constructivist ones. I've never doubted or denied that. You're probably misreading my statements to find more hostility than actually exists.

I acknowledged what I think its merits are, and put limitations on what I find to be the more ridiculous statements you've made on the meta-implications of it. For some reason you really don't like that I oppose your perception that computer science threatens a coup against mathematics. Since then you seem to be spinning your tires on the issue.

Look Reid, my patience has run out. You clearly want to make this a discussion that is far more philosophical than I ever cared to. I really don't care about your war with finitists, since I never professed to be one. I couldn't care less if you think the work that I've shared is "finitist". I also don't really care about purging mathematics of infinite sets, or telling people not to use them. Really, I don't. Maybe if you would get this through your head, you could move on and stop harassing me about this. I apologize if I got you so interested in slaying some kind of finitist dragon with my over the top recoiling over infinite sets, but really, it was only ever a technical matter and not a philosophical one like you're trying to make it.

Bottom line: I don't care to argue with you. I couldn't care less what you think. About the only thing you could do to interest me in this discussion is explain to me why Paul Taylor's research is a waste of time. I've spent a great deal of effort trying to explain its merits to you in its own terms, but nothing you've written since has given me any indication that you've even read any of it for any other reason than to find something philosophical to nitpick about.

Finally, I couldn't care less if you're interested in it at this point. You've shown over and over again that you're not interested.

Give me one reason why I should continue this conversation other than to be "educated" by you on how to better avoid provoking self-proclaimed professors on philosophical matters that I couldn't care less about with my over-the-top language?

tl;dr: name one thing that either of us care about on this topic which we both have in common.

(Hint: it's a compact set.)

Sorry if that came across as rude. lol

I'm not about to read about your interests to explain it to you. If you want to talk about Paul Taylor, I'd be interested to read content of substance about his work.

Part of the trouble is I find it difficult to know what you actually want to discuss when there's pages of your meta-discussion on the topic. Unless you want to get into the substance of Paul Taylor's work, there isn't much else to say, so I don't know how to parse your many posts of meta-commentary on the subject. So I gave the only meta-commentary I saw interesting to give related to what you posted. I'm trying to engage meaningfully here, but I find it challenging.

Also, if we want to study Paul Taylor's work, I believe I'm particularly lacking in knowledge to read it. I would need to, at minimum, read an intro text on type theory. Until then I'm not sure I can properly understand at all the contents of his work. And currently, I have enough projects that I don't think this will be a likely venture any time soon, so if you want to discuss his work, the effort will need to come from you.

I genuinely apologize for the confusion. It's actually my fault entirely. (Not being sarcastic here.)

I don't know much about the work myself. I'm not suggesting that you should study it. But I wanted to share it because I will be working through tangentially related topics in the present.

Most of the confusion between the two of us arises from divergent interpretations of language resembling value judgements. In my interpretation, I use strong language to imply that something is "good" or "bad" in the sense that it's either worth looking at or worth avoiding: in other words, they are aesthetic judgements. In your interpretation, you seem to have taken such aesthetic judgements as categorical conclusions which ought to withstand academic scrutiny. But by then I've moved on. I recognize that this is a genuine source of confusion, and I really do apologize. When I share things or say this or that sucks, it's just to try to spark interest. I do think that this confusion wouldn't happen in real life.

If you think I am trying to back out of having made some philosophical remark which I really believed, but can't bring myself to acknowledge now having backpeddled: this is certainly possible. You probably taught me to think over some of the things I took for granted, and made my own understanding more clear. But at the same time, this is immensely frustrating (again, my fault), since this was really never what I intended to talk about (again, my fault).

Let me put it this way: in my heuristic judgement, there are clear aesthetic reasons for preferring the kind of work that Paul Taylor's represents over, say, something that is philosophically opposed to his motivations for producing it. But I am not particularly interested in having a discussion about those philosophical aspects, because I think the work stands for itself. I never meant to tout it as a panacea, and I feel that by careening this discussion into a philosophical and academic one, I was cornered into trying to make it one. Really, it's not that significant a body of work. It's just that I harbor very hard to explicate aesthetic reasons for wanting to care a lot of it, which (intuitively, at least) seemed reason enough to want to share it with somebody who might well share enough of my aesthetic judgements to want to share that interest (rather than using it as an opportunity to annihilate some construction of a philosophical justification for that interest).

Anyway, this is all my fault. I thought about it this morning, and came to the following conclusion (addressed to myself, not you): If you have a perspective on something, whatever you do, do NOT use that perspective to criticize things that fit into conflicting perspectives. And especially don't think you can then be entertained in trying to explain the contents of that perspective without such an explanation getting subsumed into fodder for a counterattack to your original incendiary volley.

Now, to invert the thrust of this discussion: say I was a dogmatic finitist. Then you could expect me, say, to simply cite Paul Taylor's work, and then go on to any number of other similar works, perhaps without even having read them, in some kind of attempt to build up a case for categorical statements I sought to prove in some kind of debate.

Of course, that would presume I actually cared to win a debate with you, rather than fish for expertise by floating my interests stated in unrepentantly strong language, in the hopes of learning something new.

Most of the B.S. I post on this forum is meant to help me learn something new by sharing my unfiltered beliefs about things I am interested in. I couldn't care less about "winning" a debate. In fact I would rather be wrong, since this is really my whole point for posting in the first place.

Perhaps I should have better communicated what my interests were, but I feel that at some point I really did do a pretty good job of that. I think on some level we have different assumptions about what kinds of statements are more interesting than others. Although part of it is simply the result of me relying on assumptions about the relative merits about the things I am sharing which I haven't bothered to justify at all. Of course I could do that, but then I'd have to share less, because this takes quite a bit of effort to do explicitly in language. I usually only make it half way, which is good enough to attract a subject matter expert to correct me, but unfortunately it's also bad enough to attract a non-subject matter expert who disagrees with the thrust of it, and wants to endlessly attack me for having brought it up.

Anyway, that leaves me with two options: either tone down the language, or ignore people who it provokes who nevertheless don't have the expertise to set me straight in any meaningful way except to get angry. Of course I still want to provoke the subject matter experts (and such language is necessary to fully express my true sentiments in their most unabashed form), so it's not an option to tone down the language. Of course, as a courtesy to the non-subject matter experts, I should make explicit my particular reasons for doing this, so they don't waste their time trying to disabuse me of something I haven't been kind enough to let them know I don't care at all about.

As a rule: when I bring something up in the course of a discussion, it's usually not a citation to demonstrate a truth, because generally I don't care to believe that things are true or not, and I don't care what people who mostly disagree with me already want to think. Rather, it's a "side effect" of explicit interest in it, and a predictor that it will be loosely related to something I will work on in the future, or have aesthetic reasons for believing it's related to something I've already worked on. We have a word for this: "cool". If I share something that I think is cool, I'll typically want to learn more about it, and I might share strong reasons for doing so. Those reasons might be wrong, but I will probably still think the thing is cool, unless my reasons are completely misguided. But I am very unlikely to respond at all to posts from people who attempt to tell me why my stated reasons are misguided, unless they ALSO can teach me something new about the cool thing.

I think this discussion could have resulted in me learning something from Reid, but unfortunately my aesthetic reasons for caring about the "cool" things are far too faint and strong to be much affected by anyone who isn't already interested in category theory or type theory. The points he made about intuitionism were somewhat infomative I suppose since they pointed me to an exact page in Paul Taylor's book where he discussed very similar things, which I hadn't bothered to read before.

Anyway, I wish I could teach Reid something about what he is interested in regarding this topic, but unfortunately I am too far up my own ass on this stuff to be anything but headstrong in pushing ahead in the particular direction I've set out on. I actually do think that classical set theory is interesting, I really do. It's just that it's a completely different topic to talk about it for its own sake, and I am working on this other stuff right now. (Of course, this last sentence is oddly symmetric with what Reid said about being open to Paul Taylor's work, but not having time right now. So I suppose we are two mathematicians each with disparate bottom lines, talking past one another and getting mired in philosophical concerns that neither one really cares for.)

Regarding C++ alternatives:

The core of the problem is that different developers write C++ in very different ways. I’m not talking about company-level or even product-level differences here, I’m talking about translation unit-level differences. Some systemy-perfy guy can write up a generic shared memory cache in C-ish and a mess of syscalls, a template wizard library guy can use it as a backing store for some magic data structure that nobody else understands, a data guy can use the template in a pipeline, and an apps guy can hook up the pipeline to a standard OOP GUI. C++ is broadly good enough for all of us despite wildly varying conventions, design patterns, and non-functional requirements.

Every proposed alternative language (or runtime) has been variously better for some people and variously much worse for others. That’s a problem. I don’t want to rag on specific languages and I’m sure it won’t be hard for you to find places where these alternatives fall down, especially considering the scenario I gave you above, so I’m not going to provide examples. I’ll just say that the cure has to be better than the disease, and today it isn’t. The alternatives don’t cover the use cases, C++ isn’t bad enough to bear the complexity overhead of gluing multiples of these alternative languages together, and even if you did, you can say goodbye to templates forever because no other language has anything quite like them, and you’ll never be able to meaningfully use templated code or feasibly marshall template instances across any FFI.

Pragmatically speaking, I don’t think a real C++ alternative is going to happen for a long time. As far as I’m concerned nobody is even really working on the problem, at least not the right one. The best hope (and what I’d do if I were interested enough) is to fork Clang and comment out all of the stupid ****. And that’s a perfect C++ answer to the problem: one that’s just good enough.

That's pretty interesting, because if it's true (and from what I gather it is), then it means that C++:

1. is not one language, but a family of languages (this I had heard before), and
2. is forever

For amusement, futurists might occupy themselves with the question: will neither, either, or both of Java and C++ survive as long as modern human civilization?

That said, it's sort of an easy question, seeing we've still got legacy stuff from the 50's running.

I think I know now why Alan Kay once extolled the internet for having not one single line of code in it from the original ARPANET.

By the way, regarding the whole "finitist" thing: I probably am a finitist, but I don't particularly care that I am. And this is an interesting place to be, for the following reason.

I think the resistance I showed to efforts to try to convince me why I should care that I am might as well apply to similar crusades against other "-isms". In particular, I think the same difficulty arises in trying to convince people why they shouldn't be racist. It's just not going to work, unless they wish to be humbled. But you're up against a lifetime of experience that has led them to a particular point of view that they themselves don't quite understand where it has come from (since they scarcely acknowledge that they even might be racist, but don't care to try too hard to refute it either).

In other words, it allows you to be pointlessly adversarial and draw attention onto yourself.

And just like harboring racist beliefs, finitists like Zeilberger (or even myself, I suppose) are prone to offensive outbursts of strong language, which derail the conversation.

It does allow that. But sometimes it also just slips out, which is also true of racism. And then a racist person can just throw up his hands and say he doesn't care, and wants to move on to more important things without acknowledging it.

But the root of the problem is that the perception of people harboring one kind of -ism or another is typically a greenlight to try to disabuse the person of holding a philosophy which s/he may not even care to become aware of, let alone explicate.

In this vein, my links to Paul Taylor's research would be akin to eugenics motivated IQ studies, or Holocaust denial research: something that nobody really asked for, but which usually comes out to justify having made an outburst.

Of course, maybe someone is just really interested in the IQ studies, and thinks that it may help better illustrate the limitations of certain mainstream beliefs, perhaps even in the interest of helping the very groups it criticizes. But by then, the discussion is already toxic, because it rings far too much like something that most people find reprehensible, so the subject itself becomes taboo.

And at the same time, academics have a field day accusing people of racism, when the real issue is something else, like economics.

That all said, the fact that something can be so taboo that just raising the issue provokes responses similar to those reserved for racists... well, that just sort of shows what an orthodoxy academia really can be. Of course there are crackpots who never shut up, who are certainly a problem. Although I don't particularly think I am one of them (also, I think crackpots are more analogous to conscious racists, since they don't simply have uncontrolled outbursts, but actually fully embrace their beliefs... so they are more like the alt. right), since in the end I don't actually care too much, nor do I think there is some kind of conspiracy against people who "question" axiomatic set theory (despite my invocation of ecclesiastic analogy).

Now that C++ is getting regular standard updates, it’s probably a better candidate for a forever-language than anything else out there right now. It’s not perfect but you shouldn’t underestimate the utility of effortlessly switching between half a dozen “just okay” languages versus one really great one.

He says, in English.

And how many academics who dismiss IQ studies cited by eugenicists have really read them, and followed through on all the possibilities the studies might have some merit? I imagine not many, because: does one really need to know the findings of such a study to still reject eugenics on alone on ethical and moral grounds?

On the other hand, the consequences of eugenics make for far more a zero sum game than those of constructivist mathematics, which is hardly a zero sum game! The whole idea of "pluralism" I advocated for was all about coexisting with orthodox foundations while not being particularly satisfied by them.

Well this is honestly a helpful piece of information for me. I think I am going to make a point of it to learn / use... um, one of the C++'s.

True statement when the outburst happens when the necessary parties for Cunningham's Law to apply are not present, false when they are. But I agree when I talk to you about this stuff I am mostly tilting against windmills, and of course it was rude.

In other words, I allow myself to have racist outbursts, because I hope that you will step up to the plate and help teach me why racism is actually a good thing. But in order to do that, you have to listen long enough to my philosophical justifications for being racist before chiding me for holding them in the first place.