Logical difficulties in modern mathematics (2012)

> 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.

The Wikipedia entry is also a good overview, but as usual, contains a few (sometimes serious) inaccuracies: https://en.wikipedia.org/wiki/Philosophy_of_mathematics

> 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'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.

...and we're back to philosophy :P

[0] This particular bit of indefensibly brain-dead notation never ceases to bother me. https://math.stackexchange.com/a/932916/

> 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

> 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.