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Post by blowfly on Dec 4, 2009 0:50:32 GMT
This problem bothers me. I think of mathematics as inseparable from mathematicians, occurring entirely within on brains, and many layers of abstraction and separation away from the external world.
But then, the fact that mathematics is so friggin'ly darn good at describing the world, comes along and demands some explanation. I don't think the assumptions about mathematics above should be discarded, but there's certainly a rather awkward elephant in the room.
I don't have an answer, but I'd like to try and reduce the problem to something more answerable via a thought experiment.
Suppose we have two boxes. In one of them is a simple physics experiment - say, the double slit experiment. In the other, is the simplest possible computer circuitry required to mathematically simulate the double slit experiment. Both boxes have a screen, on which the outputs of the experiment and computation are shown.
We press the big red button and let them both run, and find that the pictures on the screens are essentially identical. For fun, we tweak them a little - change the size and location of slits in both boxes, and we find the pictures on the screens still match exactly. So the match isn't merely incidental, there is genuinely some relationship between the contents of the two boxes.
Which allows us to rewrite our original question as:
What is the isomorphic relationship between the physical contents of the two boxes?
I suspect this question is far more answerable than the vague "Why is mathematics so good?". I really do think it's answerable, though I'm not yet sure how.
Cheers,
-blowfly
P.S. I highly, highly recommend David Deutsch's "The Fabric of Reality". It's not on this exact question, but it's on related matters, including questions like - what qualifies as a good explanation? Do the laws of physics permit their own self description? Is life itself a form of virtual reality? Brilliant book.
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Post by ydoaPs on Dec 4, 2009 1:45:33 GMT
But then, the fact that mathematics is so friggin'ly darn good at describing the world, comes along and demands some explanation Why? It is the expected result as mathematics is an abstraction of the reality it describes. It's sort of like asking why logic is so logical. |
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Post by blowfly on Dec 4, 2009 4:25:26 GMT
But then, the fact that mathematics is so friggin'ly darn good at describing the world, comes along and demands some explanation Why? It is the expected result as mathematics is an abstraction of the reality it describes. Ahh, but it's exactly that which I don't believe. I think mathematics is nothing more than the smashing together of symbols inside a mathematician's brain (or computer) - nothing more. It doesn't have any direct relationship with elements of the external world, and the external world most certainly is not inherently mathematical. |
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hrg
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Post by hrg on Dec 4, 2009 10:40:34 GMT
This problem bothers me. I think of mathematics as inseparable from mathematicians, occurring entirely within on brains, and many layers of abstraction and separation away from the external world. But then, the fact that mathematics is so friggin'ly darn good at describing the world, comes along and demands some explanation. I don't think the assumptions about mathematics above should be discarded, but there's certainly a rather awkward elephant in the room. I think that you are putting the cart before the horse. We have preferably developed those areas of mathematrcs which contain good models of the real world (e.g. plane geometry, differential equations etc.). OTOH, there are modern areas of mathematics which don't contain such models. |
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Post by vazscep on Dec 4, 2009 11:30:08 GMT
I think that you are putting the cart before the horse. We have preferably developed those areas of mathematrcs which contain good models of the real world (e.g. plane geometry, differential equations etc.). OTOH, there are modern areas of mathematics which don't contain such models. And on the other other hand (the foot?) most of the real world isn't profitably described using mathematics. The entirety of social sciences, for just one set of examples, are not generally regarded as successful mathematical sciences, despite earnest attempts to make them so. I suggest that when we talk about how amazingly useful mathematics is for modelling the real world, we're counting hits and ignoring misses. |
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Post by vazscep on Dec 4, 2009 11:41:47 GMT
This problem bothers me. I think of mathematics as inseparable from mathematicians, occurring entirely within on brains, and many layers of abstraction and separation away from the external world. But then, the fact that mathematics is so friggin'ly darn good at describing the world, comes along and demands some explanation. I don't think the assumptions about mathematics above should be discarded, but there's certainly a rather awkward elephant in the room. You're running up against the same criticism the formalists faced from Quine. The formalists generally regarded mathematics as an entirely uninterpreted symbol crunching calculus that was merely useful for formulating theories to make predictions in physics. But so it goes, they have no reason to explain this utility. But the realists, presumably do, because they believe that all mathematical theorems express truths on equal footing with results in physics, so we can expect mathematical truths to be a guide to truths in physics. This is probably related to the second question I asked in the other thread. |
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Post by dahduh on Dec 4, 2009 19:46:38 GMT
I think that you are putting the cart before the horse. We have preferably developed those areas of mathematrcs which contain good models of the real world (e.g. plane geometry, differential equations etc.). OTOH, there are modern areas of mathematics which don't contain such models. And on the other other hand (the foot?) most of the real world isn't profitably described using mathematics. The entirety of social sciences, for just one set of examples, are not generally regarded as successful mathematical sciences, despite earnest attempts to make them so. I suggest that when we talk about how amazingly useful mathematics is for modelling the real world, we're counting hits and ignoring misses. I would not say that failure to accurately model the social sciences using mathematics is a failure of mathematics; rather it is a reflection of a lack of good modelling, or that the systems are inherently chaotic and sensitive to the smallest perturbations. Even the latter we can understand perfectly using mathematics, even as it imposes a limitation on what we can do with mathematics. |
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Post by dahduh on Dec 4, 2009 19:52:48 GMT
But then, the fact that mathematics is so friggin'ly darn good at describing the world, comes along and demands some explanation. How about you turn it around. If you have a deterministic system - that is a system that has a time coordinate and evolves along that coordinate without losing or gaining information from an external source - is there any way it could not be accurately modeled by mathematics? Or is your question "why is continuum mathematics so darn good at describing the world", because naively I would have expected the world to evolve more like the bits in a computer. And all the really nasty awkwardness in physics comes from the continuum - renormalization, singularities at the centers of black holes, and so on. |
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Post by blowfly on Dec 9, 2009 22:38:29 GMT
Hi all, sorry I've been a bit absent. This problem bothers me. I think of mathematics as inseparable from mathematicians, occurring entirely within on brains, and many layers of abstraction and separation away from the external world. But then, the fact that mathematics is so friggin'ly darn good at describing the world, comes along and demands some explanation. I don't think the assumptions about mathematics above should be discarded, but there's certainly a rather awkward elephant in the room. I think that you are putting the cart before the horse. We have preferably developed those areas of mathematrcs which contain good models of the real world (e.g. plane geometry, differential equations etc.). OTOH, there are modern areas of mathematics which don't contain such models. Sure, and for every mathematical formula which is good at describing the natural world, there are an infinite number which are useless. At a glance, it seems we've uncovered some Dawkins-style lottery fallacy in the original proposition - we focus on the formulae which seem to work and ignore the rest. But actually, I'm still unconvinced. The ones which are good at describing the world, are exceptionally, and consistently good at it. The equation "e=mc2" isn't the incidental output of detached empirical sifting through all possible equations, actually, it defines the exact operation of the calculator in the second box required to exactly replicate the output of the first box. And so I'm still left with the strong sensation that there is some noteworthy relationship between the physical contents of the two boxes, and I'm fascinated to know what it is. |
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Post by blowfly on Dec 9, 2009 22:40:15 GMT
Hi all, sorry I've been a bit absent. I think that you are putting the cart before the horse. We have preferably developed those areas of mathematrcs which contain good models of the real world (e.g. plane geometry, differential equations etc.). OTOH, there are modern areas of mathematics which don't contain such models. Sure, and for every mathematical formula which is good at describing the natural world, there are an infinite number which are useless. At a glance, it seems we've uncovered some Dawkins-style lottery fallacy in the original proposition - we focus on the formulae which seem to work and ignore the rest. But actually, I'm still unconvinced. The ones which are good at describing the world, are exceptionally, and consistently good at it. The equation "e=mc2" isn't the incidental output of detached empirical sifting through all possible equations, actually, it defines the exact operation of the calculator in the second box required to exactly replicate the output of the first box. And so I'm still left with the strong sensation that there is some noteworthy relationship between the physical contents of the two boxes, and I'm fascinated to know what it is. P.S. If it helps the discussion, we could talk about why are "these specific mathematical equations so good at describing stuff". In fact, arguable all areas of mathematics are good at describing stuff, namely the activities of mathematicians. |
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Post by blowfly on Dec 9, 2009 22:58:01 GMT
I think that you are putting the cart before the horse. We have preferably developed those areas of mathematrcs which contain good models of the real world (e.g. plane geometry, differential equations etc.). OTOH, there are modern areas of mathematics which don't contain such models. And on the other other hand (the foot?) most of the real world isn't profitably described using mathematics. The entirety of social sciences, for just one set of examples, are not generally regarded as successful mathematical sciences, despite earnest attempts to make them so. I suggest that when we talk about how amazingly useful mathematics is for modelling the real world, we're counting hits and ignoring misses. I'm comfortable writing this off as the practical limits of computation when modelling emergent phenomena. Hypothetically, with a sufficiently powerful computer, able to model (say) every atom in the earth, we could develop extremely accurate meteorological simulations, or social simulations, etc. And the fact remains that at the lowest reductive level we know about (quantum mechanics), our mathematical equations are exceptionally accurate - exact, for all practical purposes.
As I said above to HRG, yes we're counting hits, but I think the sheer quality of the hits we get deserves greater attention. And more generally - how these bits of plastic and silicon with electrons flowing through them can produce exactly the same outputs as entirely different physical constructions.
As a silly example - if I put a toaster in one box, and an elephant in another, and told you they will produce exactly the same output, you'd laugh. But the contents of the two boxes in my hypothetical are just as physically distinct as a toaster and an elephant, and yet produce exactly the same output. It interests me.
Gotta run, I'll hopefully get around to the other comments later today. |
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Post by dahduh on Dec 12, 2009 19:23:25 GMT
As I said above to HRG, yes we're counting hits, but I think the sheer quality of the hits we get deserves greater attention. I wonder if you are not just advancing a misapplied argument from incredulity here. The fact that two very different things have the same behavior is only grounds for supposing they are just different representations of the same underlying logic - not very astonishing, as many things can be represented in very different ways. To take one example from mathematics itself, the geometric rotations of a cube that leave the cube invariant is identical to a permutation group. The one is geometry, the other is group theory - a priori there's no reason to suspect a connection. But if you look closely its just that they both contain the same set of rules. Likewise for the universe and some mathematical construction. Why, given that the universe does follow rules (which is amazing!), is it noteworthy that these rules can be mimicked by mathematics? |
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Post by blowfly on Dec 13, 2009 23:19:23 GMT
As I said above to HRG, yes we're counting hits, but I think the sheer quality of the hits we get deserves greater attention. I wonder if you are not just advancing a misapplied argument from incredulity here. The fact that two very different things have the same behavior is only grounds for supposing they are just different representations of the same underlying logic - not very astonishing, as many things can be represented in very different ways. To take one example from mathematics itself, the geometric rotations of a cube that leave the cube invariant is identical to a permutation group. The one is geometry, the other is group theory - a priori there's no reason to suspect a connection. But if you look closely its just that they both contain the same set of rules. Sure. A bit like how "Hello world" can be implemented via completely different programming languages, which support the same underlying concepts. I believe it is noteworthy, for a very good reason - the universe is not performing computation in order to work out what to be next, which is why the analogy above doesn't fit. The two boxes experiment isn't a case of two computational systems with the same underlying concepts, it's a case of a computational system and a non-computational system producing exactly the same output. I see your point - there's nothing especially noteworthy about computational equivalence, but given that one of the boxes involves no computation, that can't be the relationship between them. Or to put it another way - the universe doesn't "follow rules". Rather, we make formalized descriptions of the universe. Really - it's near impossible to make a coherent philosophy of the universe actually "following rules" - it ultimately implies that electrons are whipping out calculators and running QED formulas in order to determine where to pop up next. Cheers, -blowfly |
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Post by vazscep on Dec 14, 2009 18:51:42 GMT
I would not say that failure to accurately model the social sciences using mathematics is a failure of mathematics; rather it is a reflection of a lack of good modelling, or that the systems are inherently chaotic and sensitive to the smallest perturbations. Even the latter we can understand perfectly using mathematics, even as it imposes a limitation on what we can do with mathematics. I look at it this way: had it turned out that the basic systems of physics were chaotic and sensitive to small perturbations, I don't think we'd be asking why mathematics is so wonderful at describing stuff, because we would not have got much use from it. Indeed, we might even wonder: "why isn't the world more usefully described with mathematics?" My point is that we are in this position with many things we want to talk about. I agree, in principle, that the world could be simulated to arbitrary precision and therefore described mathematically. But no-one's got a chance of doing that, so I wouldn't say this very dismal "in principle" prompts us to ask "why is mathematics so good at describing stuff." Besides, most of the time I spend understanding the world is time spent not talking in maths. |
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Post by dahduh on Dec 14, 2009 21:34:09 GMT
I believe it is noteworthy, for a very good reason - the universe is not performing computation in order to work out what to be next, which is why the analogy above doesn't fit. The two boxes experiment isn't a case of two computational systems with the same underlying concepts, it's a case of a computational system and a non-computational system producing exactly the same output. Now this is something I wish we could really be certain about. You say "the universe is not performing computation"; very well, let's be precise what is meant by 'computation'. Computation as I understand it requires a) causality, b) a state space, and c) some systematic means of transforming your state space. If we look at the world, we do indeed observe we have all of these things, and more: it appears that our transformations are lossless, so no information is ever gained or lost. So as far as I can see the universe is performing computation, but I have no idea how. The real dilemma as I see it, is that the universe is supposed to be, by definition, 'everything'; so whatever it is that is doing the computation, including the state space and the transformation rules, must be part of that computation. It is a bit like demanding that a Conway game of life, which we know can run an arbitrarily complex computer that can simulate anything, must simulate itself with perfect fidelity in real-time. Now maybe there exists some incredibly tricky configuration with very special redundancies in it that allows it to do just this, but frankly I just find the whole idea confusing. But pardon the discursion: I disagree, the universe definitely does perform computation, we just don't know how. I don't imagine it's quite so complex as this. For a long time there has been a suspicion that the quantum fields of the standard model are simply an effective theory; underneath it all there may be some very simple rules, with the apparent complexity an emergent phenomenon. For example, waves and whirlpools and turbulence are all very interesting phenomena described by a nice continuous differential equation, but this emerges from very simple particle-particle interactions. Maybe at the very simplest level there really are bits being manipulated by simple rules, but again what does the computation I have no idea. |
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