Well, that certainly was not the common perception in the philosophy of mathematics in the first half of the 20th century. Sure, they believed mathematics is an exploration of the axioms, but also that the axioms must be true in some deep philosophical sense. Brouwer wrote: [A]n incorrect theory, even if it cannot be inhibited by any contradiction that would refute it, is none the less incorrect,[1] and Russell believed that all of mathematics could be deduced from laws of logic that are true in the most absolute sense.
It is also doubtful that this is the perception today. If we want to use mathematics to derive any result about the physical world -- and we most certainly do -- our axioms must be consistent with it. How do you ensure that that is the case?
But such questions are not part of mathematics itself, just as the scientific method is not part of physics, but rather belong in a more fundamental discipline, called foundations of mathematics, or the philosophy of mathematics, which is usually studied by philosophers/logicians.[2]
[1]: On the Significance of the Principle of Excluded Middle in Mathematics, 1923
> they believed mathematics is an exploration of the axioms, but also that the axioms must be true in some deep philosophical sense
That's a contradiction, and it's easy enough to show it: a mathematician can research the consequences of the axiom of choice, and can research the consequences of its negation. It would be silly to deny that such research is legitimate mathematics.
Brouwer's statement strikes me as circular. Beyond that, it seems to me that the law of excluded middle, is patently false. I already gave a counterexample: the axiom of choice. Neither the axiom, nor its negation, has a derivable truth value to settle upon.
Ah, I see I'm behind the times [0]
As to Russell, he of course turned out to be wrong. I refer of course to Gödel. I'm not sure I see your point in mentioning him, or for that matter Brouwer - do explain.
> our axioms must be consistent with it
They don't. We hope that we have the mathematics to advance physics, but we needn't hope that all mathematics is applicable to physics.
It's perfectly legitimate for a mathematician to explore the consequences of denying the axiom of choice. (I already linked to such.) Such work would presumably never have application in physics, but it's still legitimate mathematics.
> How do you ensure that that is the case?
As you later allude to, that isn't a question of mathematics, it's a question of how and why mathematical discoveries map onto observable realities like physics and statistics. This is a completely legitimate line of question. The canonical article about how remarkable this is: Wigner's "The Unreasonable Effectiveness of Mathematics in the Natural Sciences".
> such questions are not part of mathematics itself
It's not a matter of legitimacy (which is a social construct) but of foundation. Russell, Brouwer and Hilbert believed that mathematics must be based on a solid philosophical foundation so that it leads to some "Truth." I am not saying this is the only way to think about the philosophy of mathematics -- in fact, my original comment said just the opposite -- but it was very much at the center of the mathematical world in the early decades of the 20th century, and is still a matter of debate among logicians. The philosophy of mathematics is a deep and complex subject. It does not seek to mathematically derive theorems from mathematical axioms, but to derive mathematical axioms from philosophical underpinnings -- whether they are physical reality, some Platonic reality, or even common sense. Logicians don't think the subject is silly at all, and if you're interested in learning more about it, the SEP link I provided above is a great place to start.
> it was very much at the center of the mathematical world in the early decades of the 20th century
Ok, but those pre-Gödel mathematicians were profoundly mistaken about the nature of mathematics. To put it bluntly: they were wrong, so why should I care what they thought?
> It does not seek to mathematically derive theorems from mathematical axioms, but to derive mathematical axioms from philosophical underpinnings
I don't agree. If you don't add new axioms, then you are 'merely' deriving theorems. If you do add new axioms, fine, you're just adopting a new set of axioms to explore through derivation.
If some set of axioms gives rise to interesting mathematical consequences, does that mean these axioms must have philosophical underpinnings? No. Does the 'validity' of the mathematics depend on the philosophical underpinnings of the axioms? No: it counts as mathematics either way.
It would be pure silliness to dismiss non-Euclidean geometry on the grounds that Hey, you just made that up!
Perhaps there's a gap somewhere in my account of things, but I'm not seeing one so far.
> to derive mathematical axioms from philosophical underpinnings -- whether they are physical reality, some Platonic reality, or even common sense
Which of these underpins non-Euclidean geometry? How about fields with no practical applications? How about, as I've mentioned several times, research into what happens when you deny the axiom of choice?
They're all still valid fields of mathematics. The 'underpinnings' of the chosen axioms are of no consequence: it's valid mathematics either way.
The miracle is that we're able to be so successful with fields like physics and statistics. Picking the right special sets of axioms, we've been able to derive huge amounts.
> the SEP link I provided above is a great place to start
> Ok, but those pre-Gödel mathematicians were profoundly mistaken about the nature of mathematics. To put it bluntly: they were wrong, so why should I care what they thought?
How do you know they were wrong? Has there been some breakthrough discovery in philosophy? And if they were, Gödel was wrongest of them all. He was the most extreme Platonist of them all.
> If some set of axioms gives rise to interesting mathematical consequences, does that mean these axioms must have philosophical underpinnings? No. Does the 'validity' of the mathematics depend on the philosophical underpinnings of the axioms? No: it counts as mathematics either way.
Again, the points you are raising have been discussed by logicians as part of the philosophy of mathematics. As usual, things are not as simple as you may think.
> Perhaps there's a gap somewhere in my account of things, but I'm not seeing one so far.
Perhaps that's because you've been thinking about this issue for a little bit, while logicians have been grappling with it for over a century. I may have confused you with the reference to physics. It is not the reason for the problem of foundation, but I used it as an example for a situation where something external to mathematics determines our choice of axioms.
BTW, if you want just a teaser to what it is that you may be missing, here's one:
That mathematics explores the theorems derivable from axioms tells us little. Is mathematics about the formal manipulation of the symbolic expression of those axioms, some (nondeterministic) Turing machine, chewing up and spitting out strings of symbols? Nearly all mathematicians would say not (including Hilbert's Formalism, which some gravely mistake for saying mathematics is just what I described). But if not, this means that the linguistic symbols mean something, and in analytical philosophy, meaning is a sort of a mapping from linguistic terms to something outside the language. But if mathematical formulas mean something -- i.e. refer to something outside them -- what is the nature of that thing, and, more importantly, how do we learn about it?
> How do you know they were wrong? Has there been some breakthrough discovery in philosophy?
Of course there has! Gödel! As you said, Russell believed that all of mathematics could be deduced from laws of logic that are true in the most absolute sense. Gödel showed that this is impossible, and his discovery ended Russell's project overnight. Russell was fundamentally mistaken about the nature of mathematics. No two ways about it.
I already linked to the Wikipedia discussion of why Brouwer's idea about the law of the excluded middle, seems no longer to be taken seriously.
> Gödel was wrongest of them all. He was the most extreme Platonist of them all.
Maybe so, I'm afraid I really don't know his philosophy.
> if mathematical formulas mean something -- i.e. refer to something outside them -- what is the nature of that thing, and, more importantly, how do we learn about it?
But the mathematics derives the same with or without associating real-world meaning. Seems to me that this is enough to demonstrate that the mapping to something beyond the mathematics, is something that should be treated as quite separate from the mathematics itself (which is 'merely' axioms + consequences).
Indeed, we see this in action with fields of mathematics that find real-world applications only years after the discovery of the mathematics itself.
Suppose a field of mathematics is found to have several applications. Again, the mathematics is unchanged: the discovered applications are a different beast entirely.
Doubtless I'm not the first to take this line of thought.
> including Hilbert's Formalism, which some gravely mistake for saying mathematics is just what I described
I'm afraid you're ahead of me again here -- that's still on my reading list.
Also, perhaps just a nitpick: nondeterministic Turing machines might differ from deterministic ones in terms of space/time complexity (that's an open problem), but they're just as impotent against non-computable problems. Were you thinking of some kind of probabilistic machine?
You seem to think that philosophy is about settling issues. It's more about understanding the question better.
I also think you're confusing Russell with Hilbert, and it is not accurate that Gödel destroyed Hilbert's project, just changed its nature (or, rather, destroyed the original understanding but not subsequent ones). But Hilbert's idea was actually much more in line with yours -- he also believed that mathematics is about the process rather than some absolute truth, but in order to show that the process can lead anywhere it was important for him to prove consistency. After all, if our mathematics can prove any proposition (including, say, 1=0) then it's rather useless, whether or not it's "valid" in some sense.
> I already linked to the Wikipedia discussion of why Brouwer's idea about the law of the excluded middle, seems no longer to be taken seriously.
I think that neither Brouwer nor that Wikipedia article says what you think they do. :) There's a whole branch of logic, intuitionistic logic, and mathematics, constructive mathematics, that are largely based on Brouwer's ideas and rejection of LEM. They are particularly popular among some kind of computer scientists, BTW.
> But the mathematics derives the same with or without associating real-world meaning. Seems to me that this is enough to demonstrate that the mapping to something beyond the mathematics, is something that should be treated as quite separate from the mathematics itself
I'm not talking about real world meaning, nor "beyond mathematics", but beyond the formula (as a string). If the strings refer to anything, what is the nature of the things they refer to?
> Also, perhaps just a nitpick: nondeterministic Turing machines might differ from deterministic ones in terms of space/time complexity (that's an open problem), but they're just as impotent against non-computable problems. Were you thinking of some kind of probabilistic machine?
No, I simply meant that deduction in a formal system (like first-order logic) is a nondeterministic computation, and to emphasize the perspective that math is just a manipulation of symbols without meaning I referred to Turing machines. That they cannot settle non-computable problems is irrelevant, because neither can mathematicians, and whatever mathematicians can do, so can a TM.
> You seem to think that philosophy is about settling issues. It's more about understanding the question better.
I'm not sure that's the whole story. The decline of logical positivism was a process of philosophers discovering fatal flaws with it and discarding it, no?
> I also think you're confusing Russell with Hilbert
Oops! Absolutely right!
> it is not accurate that Gödel destroyed Hilbert's project, just changed its nature (or, rather, destroyed the original understanding but not subsequent ones)
Perhaps you're right - I'm not qualified to discuss things like proof theory I'm afraid.
> if our mathematics can prove any proposition (including, say, 1=0) then it's rather useless, whether or not it's "valid" in some sense.
Sure, provided we can maintain the distinction between a situation where the 'principle of explosion' applies, and situations where we're merely adopting exotic axioms like in non-Euclidean geometry. I imagine this distinction is fairly straightforward in practice: either there's contradiction and inconsistency, or there's not. (Ignoring of course the question of how we could know.)
> I think that neither Brouwer nor that Wikipedia article says what you think they do.
Perhaps so, but doesn't the existence of something like the axiom of choice, disprove the law of excluded middle? (At least when it's applied universally.)
> beyond the formula (as a string). If the strings refer to anything, what is the nature of the things they refer to?
Ah, right. Well I'm not arguing against 'notions not notations'. The maths derives the same whether we use conventional symbolic representations, or not.
So sure, there's a correspondence to something beyond the string itself (and not merely in the sense that you can substitute each symbol for an alternative, a la alpha equivalence). Presumably the correspondence is between the formula, and some space of facts.
I see a 'problem of interpretation' here - some amount of 'you have to know what it means', such as with clumsy ambiguous mathematical notations [0]. I don't know if that's a serious problem though.
> deduction in a formal system (like first-order logic) is a nondeterministic computation
I'm afraid I don't follow. You can perform a derivation as many times as you like, you'll always get the same result, no?
> That they cannot settle non-computable problems is irrelevant, because neither can mathematicians, and whatever mathematicians can do, so can a TM.
Agree -- there's no good reason to assume brains cannot be simulated by computers (which is what that question boils down to, roughly speaking), at least in principle.
A surprising number of people disagree on that point though. Wishful thinking, I suspect. The same patterns arises in discussions of free will.
> The decline of logical positivism was a process of philosophers discovering fatal flaws with it and discarding it, no?
Few things in philosophy are definitively discarded (and if they are, they can come back in some revised form); they may fall out of fashion.
> Ignoring of course the question of how we could know.
But that was precisely Hilbert's point. The problem was this: either you took Brouwer's intuitionism, which has no inconsistencies but is inconvenient, or you take classical mathematics because it's useful. But if you do, you need to know that it's consistent, but you can't.
> but doesn't the existence of something like the axiom of choice, disprove the law of excluded middle?
No. LEM is an axiom that you either include (in classical mathematics) or not (in constructive mathematics).
> I don't know if that's a serious problem though.
The problem is not exactly that of interpretation, but a more basic one: if the formulas refer to something, what kind of thing is that thing, and if that thing exists independently of the formulas, how do we know that the formulas tell us the truth about those things? Again, there is no definitive answer to this question, just different philosophies.
> You can perform a derivation as many times as you like, you'll always get the same result, no?
If you start with a set of axioms, you can apply any inference rule in any order to any subset of them. There's an infinity of possible derivations, that yield all the theorems in the theory. We can look at that as a form of nondeterministic computation (that nondeterministically chooses an inference) .
> A surprising number of people disagree on that point though.
> Few things in philosophy are definitively discarded (and if they are, they can come back in some revised form); they may fall out of fashion.
Disagree.
Some philosophical ideas seem increasingly unsustainable in the face of mounting scientific knowledge. Vitalism, for instance.
In a similar vein, I'd say we're seeing an ongoing erosion of dualism in philosophy of mind, which will be unlikely to recover. Like with vitalism, the more we learn from science, the less magic we need to explain ourselves, be it 'life', or consciousness.
In the case of logical positivism, it was a case of showing the position to be unsustainable on its own terms. That's analogous to disproving a scientific hypothesis - I don't see it coming back.
> either you took Brouwer's intuitionism, which has no inconsistencies but is inconvenient, or you take classical mathematics because it's useful. But if you do, you need to know that it's consistent, but you can't.
Neat. I see I'll have to do my homework.
> LEM is an axiom that you either include (in classical mathematics) or not (in constructive mathematics).
But if you do include it, don't you have to commit arbitrarily to either AC or ¬AC?
> if the formulas refer to something, what kind of thing is that thing, and if that thing exists independently of the formulas, how do we know that the formulas tell us the truth about those things?
Isn't this 'just' a question of knowing that the correspondence is valid?
> There's an infinity of possible derivations, that yield all the theorems in the theory. We can look at that as a form of nondeterministic computation (that nondeterministically chooses an inference) .
Right, I'm with you. Infinite graph traversal.
> Gödel did.
Roger Penrose too. Honestly I find his position to be almost risibly weak. It's plain old quantum mysticism and wishful thinking, as far as I can tell. Microtubules are meant to be essentially magical? Seriously? Here he is explaining his position (takes about 10 minutes) https://youtu.be/GEw0ePZUMHA?t=660 and here's the Wikipedia article on his book about it: https://en.wikipedia.org/wiki/The_Emperor%27s_New_Mind
> Some philosophical ideas seem increasingly unsustainable in the face of mounting scientific knowledge. Vitalism, for instance.
OK, but the philosophy of mathematics is a more slippery beast.
> But if you do include it, don't you have to commit arbitrarily to either AC or ¬AC?
No. You can have a theory that's consistent with either.
> Isn't this 'just' a question of knowing that the correspondence is valid?
Correspondence to what? Are those objects platonic ideals? Are they abstractions of physical reality? You can answer this question in many ways. But whichever way, this question of soundness (formal provability corresponds to semantic truth) rests on us knowing what is true.
Some philosophies of mathematics say that the question is unimportant: it doesn't matter if we get the "truth" part right, all that matters is that mathematics is a useful tool for whatever we choose to apply it to (this doesn't even mean applied mathematics; one valid "use" is intellectual amusement).
> No. You can have a theory that's consistent with either.
I think I see your point. If your theory has no way to express the question Is the AC true? then the LEM is no issue, as there's no 'proposition' to worry about at all.
> Are those objects platonic ideals? Are they abstractions of physical reality?
My intuition is to favour the former, as the latter seems a much stronger claim.
It seems reasonable to say that any mathematical system exists as a platonic ideal, but it's plain that not all mathematical systems abstract physical systems, and we shouldn't be quick to say that we know that any mathematical system does so. That's an empirical question that - true to Popper - can never be positively proved.
> If your theory has no way to express the question Is the AC true? then the LEM is no issue, as there's no 'proposition' to worry about at all.
That's not what I meant. I meant that choice can be neither proven nor disproven. The philosophical fact that, due to LEM, it must either be true or not is irrelevant, because you cannot rely on it being true or on it being false.
> My intuition is to favour the former, as the latter seems a much stronger claim.
Well, Platonism is a popular philosophy of mathematics among mathematicians, but many strongly reject it.
> can be neither proven nor disproven. The philosophical fact that, due to LEM, it must either be true or not is irrelevant, because you cannot rely on it being true or on it being false
So it's similar to, but not the same thing as, modifying the LEM to permit Unknowable as a third category (beyond the proposition is true and its negation is true). Instead we maintain that it has a truth value one way or the other, but that it happens to be both unknowable and of no consequence.
> modifying the LEM to permit Unknowable as a third category
LEM doesn't need to be modified, and it's not a third category. In classical logic (with LEM), unlike in constructive/intuitionistic logic (no LEM), truth and provability are simply not the same -- in fact, it is LEM that creates that difference.
> Instead we maintain that it has a truth value one way or the other, but that it happens to be both unknowable and of no consequence.
Precisely! That is the difference between classical and constructive logic. In constructive logic truth and provability are the same, and something is neither true nor false until it has been proven or refuted. But before you conclude that this kind of logic, without LEM, is obviously better (philosophically), you should know that any logic has weirdnesses. What happens without LEM, for example, is that it is not true that every subset of a finite set is necessarily finite. ¯\_(ツ)_/¯
It is also doubtful that this is the perception today. If we want to use mathematics to derive any result about the physical world -- and we most certainly do -- our axioms must be consistent with it. How do you ensure that that is the case?
But such questions are not part of mathematics itself, just as the scientific method is not part of physics, but rather belong in a more fundamental discipline, called foundations of mathematics, or the philosophy of mathematics, which is usually studied by philosophers/logicians.[2]
[1]: On the Significance of the Principle of Excluded Middle in Mathematics, 1923
[2]: https://plato.stanford.edu/entries/philosophy-mathematics/