This reminds of of that one time when I was on a date with a girl from the history department who somehow bemusedly sat through my entire mini-lecture on comparing infinite sets. Twenty years and three kids later, she'll still occasionally look me straight in the eye and declare "my infinity is bigger than your infinity."
Wow, I did a very similar thing on the first date with my now wife. I explained the halting problem, and Godel's incompleteness theorems. We also talked about her (biomedical) research, so it wasn't a one sided conversation.
I think dominating on a first date is a risk (which I was mindful of) but just being yourself, and talking about something you're truly passionate about is the key.
"So you see if the chance of pregnancy is constant per..uh..encounter, and given that the condom just broke, we're on a spectrum from the chance of a second round roughly doubling the odds but the overall chance is still small, or it doesn't make much difference anyway. Either way, the numbers say we should go again."
Way back then, calculus was a culture war battleground. Bishop Berkeley famously argued the foundations of calculus weren't any better that those of theology. This sort of thing motivated much work into shoring them up, getting rid of infinitesimals and the like (or, later, making infinitesimals rigorous in nonstandard analysis).
There’s something called “not giving a fuck” that works in those situations. The crux of it though is you need to “know thyself” or you’ll be forever your worst critic and enemy.
Also you're being open and readable to the other person. You're not being deceptive or putting on a show, which is usually what rustles people's jimmies.
Get this incel trolling out of here. No, all that would happen if your date weren’t interested in math/whatever you’re into is a polite message “I had a great time but I don’t think we have much in common” and leave it at that.
His oddly specific fear of being "plastered on a bunch of facebook new-york-dating-experience groups" sounds like women have had to warn each other about him before, and it probably wasn't about his interest in math, but something much worse.
I'm curious. Did either of you ever notice the implicit philosophical assumptions that you have to make to come to the conclusion that one infinity can be larger than another?
Despite the fact that this was actively debated for decades, modern math courses seldom acknowledge the fact that they are making unprovable intellectual leaps along the way.
> Despite the fact that this was actively debated for decades, modern math courses seldom acknowledge the fact that they are making unprovable intellectual leaps along the way.
That’s not at all true at the level where you are dealing with different infinities, usually, which tends to come after the (usually, fairly early) part dealing with proofs and the fact that all mathematics is dealing with “unprovable intellectual leaps” which are encoded into axioms, and everything in math which is provable is only provable based on a particular chosen set of axioms.
It may be true that math beyond that basic level doesn’t make a point of going back and explicitly reviewing that point, but it is just kind of implicit in everything later.
I guarantee that a naive presentation doesn't actually include the axioms, and doesn't address the philosophical questions dividing formalism from constructivism.
Uncountable need not mean more. It can mean that there are things that you can't figure out whether to count, because they are undecidable.
> I guarantee that a naive presentation doesn't actually include the axioms
But you said "modern math courses". Are you now talking about a casual conversation? I mean the OP's story is that his wife just liked listening to him talk about his passions.
> Uncountable need not mean more.
Sure. But that doesn't mean that there aren't differing categories. However you slice it, we can operate on these things in different ways. Real or not the logic isn't consistent between these things but they do fall out into differing categories.
If you're trying to find mistakes in the logic does it not make sense to push it at its bounds? Look at the Banach-Tarski Paradox. Sure, normal people hear about it and go "oh wow, cool." But when it was presented in my math course it was used as a discussion of why we might want to question the Axiom of Choice, but that removing it creates new concerns. Really the "paradox" was explored to push the bounds of the axiom of choice in the first place. They asked "can this axiom be abused?" And the answer is yes. Now the question is "does this matter, since infinity is non-physical? Or does it despite infinity being non-physics?"
You seem to think mathematicians, physicists, and scientists in general believe infinities are physical. As one of those people, I'm not sure why you think that. We don't. I mean math is a language. A language used because it is pedantic and precise. Much the same way we use programming languages. I'm not so sure why you're upset that people are trying to push the bounds of the language and find out what works and doesn't work. Or are you upset that non-professionals misunderstand the nuances of a field? Well... that's a whole other conversation, isn't it...
Vitali's counter-example for the existence of a non-trivial measure requires the Axiom of Choice. But not assuming the Axiom of Choice then prohibits choosing and actually using that measure. There is more to it than just Banach-Tarski.
Your guesses at what I seem to think are completely off base and insulting.
When I say "modern math courses", I mean like the standard courses that most future mathematicians take on their way to various degrees. For all that we mumble ZFC, it is darned easy to get a PhD in mathematics without actually learning the axioms of ZFC. And without learning anything about the historical debates in the foundations of mathematics.
Honestly it's difficult to understand exactly what you're arguing. Because I understand laymen not understanding your argument about infinities not being real (and even many HN users don't understand code is math bit a CS degree doesn't take you far in math. Some calc and maybe lin alg) but are we concerned about laymen? I too am frustrated by nonexperts having strong opinions and having difficulties updating them, but that's not a culture problem. We're on HN and we know the CS stereotypes, right?
If instead you're talking about experts then I learned about what you're talking about in my Linear 2 course in a physics undergrad and have seen the topic appear many times since even outside my own reading of set theory. The axiom of choice seems to have even entered more main stream nerd knowledge. It's very hard to learn why AoC is a problem without learning about how infinities can be abused. But honestly I don't know any person that's even an amateur mathematician that thinks infinities are physical
The fact that you think I'm talking about the axiom of choice, demonstrates that you didn't understand what I'm talking about. I would also be willing to bet a reasonable sum of money that this topic did not come up in your Linear 2 course in physics undergrad.
The arguments between the different schools of philosophy in math are something that most professional mathematicians are unaware of. Those who know about them, generally learned them while learning about either the history of math, or the philosophy of math. I personally only became aware of them while reading https://www.amazon.com/Mathematical-Experience-Phillip-J-Dav.... I didn't learn more about the topic until I was in grad school, and that was from personal conversations. It was never covered in any course that I took on, either in undergraduate or graduate schools.
Now I'm curious. Was there anything that I said that should have been said more clearly? Or was it hard to understand because you were trying to fit what I said into what you know about an entirely unrelated debate about the axiom of choice?
> The fact that you think I'm talking about the axiom of choice, demonstrates that you didn't understand what I'm talking about.
Dude... just a minute ago you were complaining about ZFC... Sure, I brought up AoC but your time to protest was then.
The reason I brought up AoC is because it is a common way to learn about the abuse of infinity and where axioms need be discussed. Both things you brought up. I think you are reading further into this than I intended.
> Now I'm curious. Was there anything that I said that should have been said more clearly?
Is this a joke?
When someone says
>> Honestly it's difficult to understand exactly what you're arguing.
That's your chance to explain. It is someone explicitly saying... I'm trying to understand but you are not communicating efficiently.
This is even more frustrating as you keep pointing out that this is not common knowledge. So why are you also communicating like it is?! If it is something so few know about then be fucking clear. Don't make anyone guess. Don't link a book, use your own words and link a book if you want to suggest further reading, but not "this is the entire concept I'm talking about". Otherwise we just have to guess and you getting pissed off that we guess wrong is just down right your own fault.
So stop shooting yourself in the foot and blaming others. If people aren't understanding you, try assuming they can't read your mind and don't have the exact same knowledge you do. Talk about fundamental principles...
That point being that what we mean by "exists" is fundamentally a philosophical question. And our conclusions about what mathematical things exist will depend on how we answer that question. And very specifically, there are well-studied mathematical philosophies in which uncountable sets do not have larger cardinalities than countable ones.
If none of those explanations wind up being clear for you, then I'm going to need feedback from you to have a chance to explain this to you. Because you haven't told me enough for me to make any reasonable guess what the sticking point is between you and understanding. And without that, I have no chance of guessing what would clarify this for you.
The "philosophical questions" dividing formalism from constructivism are greatly overstated. The point of having those degrees of undecidability or uncountability is precisely to be able to say things like "even if you happen to be operating under strong additional assumptions that let you decide/count X, that still doesn't let you decide/count Y in general." That's what formalism is: a handy way of making statements about what you can't do constructively in the general case.
To be fair, constructivists tend to prefer talk about different "universes" as opposed to different "sizes" of sets, but that's all it is: little more than a mere difference in terminology! You can show equiconsistency statements across these different points of view.
Yes, you can show such equiconsistency statements. As Gödel proved, for any set of classical axioms, there is a corresponding set of intuitionistic axioms. And if the classical axioms are inconsistent, then so is the intuitionistic equivalent. (Given that intuitionistic reasoning is classically valid, an inconsistency in the intuitionistic axioms trivially gives you one in the classical axioms.)
So the care that intuitionists take does not lead to any improvement in consistency.
However the two approaches lead to very different notions of what it means for something to mathematically exist. Despite the formal correspondences, they lead to very different concepts of mathematics.
I'm firmly of the belief that constructivism leads to concepts of existence that better fit the lay public than formalism does.
Probably not. But this one time we had an argument and I made a statement along the lines of "I'm right, naturally." She went irrational. I lost the argument.
Here's a hint. When someone makes a reference to something that was actively debated for decades, and you're not familiar with said debates, you should probably assume that you're missing some piece of relevant knowledge.
What leaps are "unprovable"? I'm curious, that doesn't sound right.
For sure there are valid arguments on whether or not to use certain axioms which allow or disallow some set theoretical constructions, but given ZFC, is there anything that follows that is unprovable?
When you say "given ZFC", you're assuming a lot. Including a notion of mathematical existence which bears little relation to any concept that most lay people have of what mathematical existence might mean.
In particular, you have made sufficient assumptions to prove that almost all real numbers that exist can never be specified in any possible finite description. In what sense do they exist? You also wind up with weirder things. Such as well-specified finite problems that provably have a polynomial time algorithm to solve...but for which it is impossible to find or verify that algorithm, or put an upper bound on the constants in the algorithm. In what sense does that algorithm exist, and is finite?
Does that sound impossible? An example of an open problem whose algorithm may have those characteristics is an algorithm to decide which graphs can be drawn on a torus without any self-crossings.
If our notion of "exists" is "constructable", all possible mathematical things can fit inside of a countable universe. No set can have more than that.
> When you say "given ZFC", you're assuming a lot.
Errr, I'm just assuming the axioms of ZFC. That's literally all I'm doing.
> In what sense do [numbers that can't be finitely specified] exist?
In the sense that we can describe rules that lead to them, and describe how to work with them.
I understand that you're trying to tie the notion of "existence" to constructability, and that's fine. That's one way to play the game. Another is to use ZFC and be fine with "weird, unintuitive to laypeople" outcomes. Both are interesting and valid things to do IMO. I'm just not sure why one is obviously "better" or "more real" or something. At the end, it's all just coming up with rules and figuring out what comes out of them.
My point is that going from a lay understanding of mathematics to "just accept ZFC" means jumping past a variety of debatable philosophical points, and accepting a standard collection of answers to them. Mathematicians gloss over that.
On the other hand, I think it's really cool to teach laypeople about things like "sizes of infinities", etc. They are deep math concepts that can be taught with relatively simple analogies that most people understand, and they're interesting things to know. I know that I personally loved learning about them as a kid, before I had almost any knowledge of math - it's one of the reasons that while I initially didn't connect with other areas of math, I found set theory delightful as a kid.
I just feel like if you need to first walk people through a bunch of philosophical back and forth on constructionism, you'll never get to the fun stuff.
We each find different things delightful. What I like, you may not. And vice versa.
But it is easy to present deep ideas from constructivism, without mentioning the word constructivism. Or even acknowledging that the philosophy exists.
For example the second half of https://math.stackexchange.com/questions/5074503/can-pa-prov... is an important constructivist thing. It shows why everything that a constructivist could ever be interested in mathematically, can be embedded in the natural numbers. With all of the constructions needing nothing more than the Peano Axioms. (Proving the results may need stronger axioms though...)
From my point of view, https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach does something similar. That book got a lot of people interested in basic concepts around recursion, computation, and what it means to think. Absolutely everything in it works constructively. And yet that philosophy is not mentioned. Not even once.
The only point where a constructivist need discuss all of the philosophical back and forth on constructivism, is in explaining why a constructivist need not accept various claims coming out of classical mathematics. And even that discussion would not be so painful if people who have learned classical mathematics were more aware of the philosophical assumptions that they are making.
> We each find different things delightful. What I like, you may not. And vice versa.
To be honest, I don't feel like I know enough about the constructivist philosophy. What would be a good place to start if I want to learn more about it?
I haven't yet read your PA proving Goodstein sequences article, though I have skimmed it and it is, indeed, super interesting.
And for the record, Godel, Escher, Bach was probably the single most important influence on me even starting to get interested in computation, etc.
You are being very cryptic. Are you trying to say that the existence of uncountable sets requires the axiom of choice? If you are, that's false. If you aren't, I'm not sure what you are trying to say.
I'm definitely not trying to say that the existence of uncountable sets requires the axiom of choice. Cantor's diagonalization argument for the reals demonstrates otherwise.
I'm saying that to go from the uncountability of the reals to the idea that this implies that the infinity of the reals is larger, requires making some important philosophical assumptions. Constructivism demonstrates that uncountable need not mean more.
On the algorithm example, you could have asked what I was referring to.
The result that I was referencing follows from the https://en.wikipedia.org/wiki/Robertson%E2%80%93Seymour_theo.... The theorem says that any class of finite graphs which is closed under graph minors, must be completely characterized by a finite set of forbidden minors. Given that set of forbidden minors, we can construct a polynomial time test for membership in the class - just test each forbidden minor in turn.
The problem is that the theorem is nonconstructive. While it classically proves that the set exists, it provides no way to find it. Worse yet, it can be proven that in general there is no way to find or verify the minimal solution. Or even to provide an upper bound on the number of forbidden minors that will be required.
This need not hold in special cases. For example planar graphs are characterized by 2 forbidden minors.
For the toroidal graphs, as https://en.wikipedia.org/wiki/Toroidal_graph will verify, the list of known forbidden minors currently has 17,523 graphs. We have no idea how many more there will be. Nor do we have any reason to believe that it is possible to verify the complete list in ZFC. Therefore the polynomial time algorithm that Robinson-Seymour says must exist, does not seem to exist in any meaningful and useful way. Such as, for example, being findable or provably correct from ZFC.
> you're assuming a lot. Including a notion of mathematical existence which bears little relation to any concept that most lay people have of what mathematical existence might mean.
John Horton Conway:
> It's a funny thing that happens with mathematicians. What's the ontology of mathematical things? How do they exist? In what sense do they exist? There's no doubt that they do exist but you can't poke and prod them except by thinking about them. It's quite astonishing and I still don't understand it, having been a mathematician all my life. How can things be there without actually being there? There's no doubt that 2 is there or 3 or the square root of omega. They're very real things. I still don't know the sense in which mathematical objects exist, but they do. Of course, it's hard to say in what sense a cat is out there, too, but we know it is, very definitely. Cats have a stubborn reality but maybe numbers are stubborner still. You can't push a cat in a direction it doesn't want to go. You can't do it with a number either.
In the sense that all statements of non-constructive "existence" are made, viz. "you can't prove that they don't exist in the general case", so you are allowed to work under the stronger assumption that they also exist constructively, without any contradiction resulting. That can certainly be useful in some applications.
Sure, we can choose to work in a set of axioms that says that there exists an oracle that can solve the Halting problem.
But the fact that such systems don't create contradictions emphatically *DOES NOT* demonstrate the constructive existence of such an oracle. Doubly not given that in various usual constructivist systems, it is easily provable that nothing that exists can serve as such an oracle.
If such a system proved that the answer to some decidable question was x, when the actual answer was y, then the system would prove a contradiction. If the system doesn’t prove a contradiction, then that situation doesn’t happen, so you can trust its answers to decidable questions.
If the only questions you accept as meaningful are the decidable ones, then you can trust its answers for all the questions you accept as meaningful and for which it has answers.
Also, “provable that nothing that exists can serve as such an oracle” seems pretty presumptive about what things can exist? Shouldn’t that be more like, “nothing which can be given in such-and-such way (essentially, no computable procedure) can be such an oracle”?
Why treat it as axiomatic that nothing that isn’t Turing-computable can exist? It seems unlikely that any finite physical object can compute any deterministic non-Turing-computable function (because it seems like state spaces for bounded regions of space have bounded dimension), but that’s not something that should be a priori, I think.
I guess it wouldn’t really be verifiable if such a machine did exist, because we would have no way to confirm that it never errs? Ah, wait, no, maybe using the MIP* = RE result, maybe we could in principle use that to test it?
You're literally talking about how I should regard the hypothetical answers that might be produced by something that I think doesn't exist. There's a pretty clear case of putting the cart before the horse here.
On being presumptive about what things can exist, that's the whole point of constructivism. Things only exist when you can construct them.
We start with things that everyone accepts, like the natural numbers. We add to that all of the mathematical entities that can be constructed from those things. This provides us with a closed and countable universe of possible mathematical entities. We have a pretty clear notion of what it means for something in this universe to exist. We cannot be convinced of the existence of anything that is outside of the universe without making extra philosophical assumptions. Philosophical assumptions of exactly the kind that constructivists do not like.
This constructible universe includes a model of computation that fits Turing machines. But it does not contain the ability to describe or run any procedure that can't fit onto a Turing machine.
Therefore an oracle to decide the Halting problem does not exist within the constructible universe. And so your ability to imagine such an oracle, won't convince a constructivist to accept its existence.
You can think that something doesn't exist in the general case, while still allowing that it might exist in unspecified narrow cases where additional constraints could apply. For example, there might be algorithms that can decide the halting problem for some non-Turing complete class of programs. Being able to talk in full generality about how such special cases might work is the whole point of non-constructive reasoning. It's "non-constructive" in that it states "I'm not going to construct this just yet".
Well yes. We can certainly make a function that acts something like that oracle in some special cases. But my point was to give an example of something that cannot be constructively created. The oracle that I described cannot exist within the universe of constructable things.
> Therefore an oracle to decide the Halting problem does not exist within the constructible universe.
I might be confused here, but isn't an Oracle to decide the halting problem something that everyone agrees doesn't exist?
The whole idea is for this to be a thought experiment. "If we magically had a way to decide the halting problem, how would that affect things" seems like a normal hypothetical question.
You literally cannot doubt the existence of this oracle, without doubting what existence means in classical mathematics.
Here is why a classical mathematician would say that this oracle exists.
Let f(program, input, n) be 1 or 0 depending on whether the program program, given input input, is still running at step n. This is a perfectly well-behaved mathematical function. In fact it is a computable one - we can compute it by merely running a simulation of a computer for a fixed number of steps.
Let oracle(program, input) be the limit, as n goes to infinity, of f(program, input, n). Classically this limit always exists, and always gives us 0 or 1. The fact that we happen to be unable to compute it, doesn't change the fact that this is a perfectly well-defined function according to classical mathematics.
If you give up the existence of this oracle, you might as well give up the existence of any real numbers that do not have a finite description. Which is to say, almost all of them. Why? Because the set of finite descriptions is countable, and therefore the set of real numbers that admit a finite description is also only countable. But there are an uncountable number of real numbers, so almost all real numbers do not admit a finite description.
The real question isn't whether this oracle exists. It is what you want the word "exists" to mean.
If I'm following you, then most "mathematical" CS is based on constructivist foundations? E.g. while a halting problem Oracle might "exist" in the mathematical sense, it's not considered to "exist" for most purposes of deciding complexity classes, etc.
> The real question isn't whether this oracle exists. It is what you want the word "exists" to mean.
I was going to say the same thing. I'm not sure what "exists" means in some of these discussions.
It would be more accurate to say that most mathematical CS fits inside of constructivist foundations. Of course it also fits inside of classical foundations. So someone with constructivist inclinations may be drawn to that field. But participation in that field doesn't make you a constructivist.
As for what exists means, here are the three basic philosophies of mathematics.
The oldest is Platonism. It is the belief that mathematics is real, and we are trying to discover the right way to do it. Ours is not to understand how it is to exist, it is to try to figure out what actually exists. Kurt Gödel is a good example of someone who argued for this. See https://journals.openedition.org/philosophiascientiae/661 for a more detailed exploration of his views, and how they changed over time. (His Platonism does seem to have softened over time.)
Historically this philosophy is rooted in Plato's theory of Forms. Where our real world reflects an ideal world created by a divine Demiurge. With the rise of Christianity, that divine being is obviously God. This fit well with the common idea during the Scientific Revolution that the study of science and mathematics was an exploration of the mind of God.
Formalism dates back to David Hilbert. In Hilbert's own description, it reduces mathematics to formal symbol manipulation according to formal rules. It's a game to figure out what the consequences are of the axioms that were chosen. As for existence, "If the arbitrarily posited axioms together with all their consequences do not contradict each other, then they are true and the things defined by these axioms exist. For me, this is the criterion of truth and existence." See page 39 of https://philsci-archive.pitt.edu/17600/1/bde.pdf for a reference.
In other words if we make up any set of axioms and they don't contradict each other, the things that those axioms define have mathematical existence. Whether or not we can individually describe those things, or learn about them.
Over on the constructivist side of the fence, there are a wide range of possible views. But they share the idea that mathematical things can only exist when there is a way to construct them. But that begs the question.
Finitism only accepts the existence of finite things. In an extreme form, even the set of natural numbers doesn't exist. Only individual natural numbers. Goodstein of the Goodstein sequence is a good example of a finitist.
Intuitionism has the view that mathematics only exists in the minds of men. Anything not accessible to the minds of men, doesn't exist. The best known adherent of this philosophy is Brouwer.
My sympathies generally lie with the Russian school, founded by Markov. (Yes, the Markov that Markov chains are named after.) It roots mathematics in computability.
Erret Bishop is an example of a more pragmatic version of constructivism. Rather than focus on the philosophical claims, he pragmatically focuses on what can be demonstrated constructively. https://www.amazon.com/Foundations-Constructive-Analysis-Err... is his best known work.
Everyone agrees that you can't write an algorithm for a Turing machine (or computational equivalent) that decides the Halting problem for Turing machines in every case. Since this is explicitly worded as "you can't write an algorithm for..." it's in fact talking about a kind of constructive existence and saying that it doesn't apply. The oracle concept is normally phrased about "If we magically had a way to decide this undecidable problem in every case" but it's real utility from a constructive POV is talking about special cases that you haven't bothered to narrow down just yet.
This is exactly what I’m saying is presumptive! If constructivism is to earn the merit of being less presumptive by virtue of not assuming the existence of various things, it should also not assume the non-existence of those things.
Which, I think many visions of constructivism do earn this merit, but not your description of it.
The underlying problem is that constructivism and non-constructive reasoning are using the word "exists" (and, relatedly, the logical disjunction) to mean very different things. The constructive meaning for "exists" is certainly more intuitive, so it makes sense that constructivists would want it by 'default'; but the non-constructive operator (which a constructivist would preferably understand as "is merely allowed to exist"), while somewhat more subtle, has a usefulness of its own.
So having a different philosophy from you makes me presumptive?
What makes you presume that you have any business telling someone with different beliefs from you, what is OK to believe? You may believe in the existence of whatever you like. Whether that be numbers that cannot be specified, or invisible pink unicorns.
I'll be over in the corner saying that your belief does not compel me to agree with you on the question of what exists. Not when your belief follows from formalism, which explicitly abandons any pretense of meaningfulness to its abstract symbol manipulation.
No, that’s not what I said. Thinking you can determine a-priori that something that is logically self-consistent, cannot exist, if there is no reason that such a thing being physically instantiated would imply a logical contradiction, is the thing I think is presumptive.
Merely believing that such a thing (a halting oracle) doesn’t exist, isn’t something I meant to call presumptive, only believing that you can know a-priori (with certainty) that such things cannot exist.
I don’t claim that you are obligated to agree with me that they do exist. Someone who believes they don’t, but doesn’t believe they can know this as certain a-priori knowledge, would be no more presumptive than I am, and someone who is agnostic on the question of whether they exist would be less presumptive than I am.
Also, I disagree with your notion of “meaningfulness”. At a minimum, all statements in the arithmetic hierarchy are meaningful. The continuum hypothesis might in a certain sense not be meaningful.
> Merely believing that such a thing (a halting oracle) doesn’t exist, isn’t something I meant to call presumptive, only believing that you can know a-priori (with certainty) that such things cannot exist.
If you think that I was making that case, then you have misunderstood something important.
Constructivism is a statement about what kinds of arguments will convince me that things exist.
Could things exist that I don't believe in? Absolutely! There could well be a bank account with my name on it that I don't know about. Its existence is possible, and my lack of belief in it is no skin off of its back. But I still don't believe that it exists.
Similarly, the Platonists could be correct. There could be an omniscient God whose perfect mind gives existence to a perfect system of mathematics, beyond human comprehension. I have no way to prove that there isn't such a God, and therefore that there isn't such a perfect mathematics.
However the potential for such things to exist is a point of theology. I do not believe in their existence. Just as I do not believe in the existence of Santa. In neither case can I prove that they don't exist. And if you choose to believe in them, that's your business. Not mine.
There is nothing presumptive in my laying out the rules of reason that I will accept as convincing to me. There is a lot of presumption if anyone else comes along and tells me that I should think differently about unprovable propositions.
Now it happens to be the case that from the rules of reason that I use, I provably can't be convinced of the existence of certain things. That's a mathematical theorem. But the fact that I can't be convinced, doesn't prove that you shouldn't be convinced. You are free to be convinced of all of the unprovable assertions that you wish. And it is also true that on something like this, I have no way to convince you that it doesn't exist.
On meaningfulness, meaning is in the eye of the beholder. For example there are people who are willing to pay a million dollars for a century old stamp which was misprinted with the airplane upside-down. (See https://en.wikipedia.org/wiki/Inverted_Jenny to verify that.) They clearly find great meaning in that stamp. But I don't.
So again, you're free to find meaning in whatever you want. But you're in the wrong to object that I don't find meaning in what you consider important.
Ah, that I wouldn't call presumptuous. I just interpreted a previous statement you made as saying that one can prove that certain such things can't exist. So, I guess there was just a miscommunication (I guess I misunderstood.)
> emphatically DOES NOT demonstrate the constructive existence of such an oracle
Of course, but it shows that you can assume that such an oracle exists whenever you are working under additional conditions where the existence of such a "special case" oracle makes sense to you, even though you can't show its existence in the general case. This outlook generalizes to all non-constructive existence statements (and disjunctive statements, as appropriate). It's emphatically not the same as constructive existence, but it can nonetheless be useful.
That argument ought to convince you that there's a mere "possible world" where that bank account turns out to exist. Sometimes we are implicitly interested in these special-cased "possible worlds", even though they'll involve conditions that we aren't quite sure about. Non-constructive existence is nothing more than a handy way of talking about such things, compared to the constructively correct "it's not the case that the existence of X is always falsified".
It would be weird for a constructivist to be interested in a possible world that they don't believe exists.
Theoretically possible? Sure. But the kinds of questions that lead you there are generally in opposition to the kinds of principles that lead someone to prefer constructivism.
