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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 Hilbe

• MaxBarraclough 8y ago
  > 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/ https://math.stackexchange.com/a/932916/
  • pron 8y ago
    > 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. Gödel did.
    • MaxBarraclough 8y ago
      > 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 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 https://en.wikipedia.org/wiki/The_Emperor%27s_New_Mind