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If math was somehow "deep" in the manner Wigner suggests, you'd expect it would have a lot fewer rough edges. In particular you'd expect we wouldn't have to thr

If math was somehow "deep" in the manner Wigner suggests, you'd expect it would have a lot fewer rough edges. In particular you'd expect we wouldn't have to throw away solutions to our differential equations that are perfectly good mathematically, but unfortunately unphysical: http://www.tjradcliffe.com/?p=381 http://www.tjradcliffe.com/?p=381 The Dirac equation is notable as one of the very, very few mathematical descriptions of reality that describe only what is, and nothing of what is not. Pretty much any wave other equation has non-physical (time reversed) solutions, and the Navier-Stokes equation has mathematically OK solutions (if you consider the odd singularity "OK") that are unphysical. Even the Dirac equation has issues with pre-acceleration that are at best physically ambiguous. My own take is that math is a language we use to describe reality in a way that allows us to do precise book-keeping. Like any language, we are able to express "mimsy were the borogoves" as easily as perfectly meaningful things like "colourless green ideas sleep furiously" (as in: "Beneath the snows of winter that blanket the land, colourless green ideas sleep furiously"... there's actually a minor literature using this phrase in meaningful sentences, just to prove Chomsky was no poet.)

yequalsx 12y ago
I don't see why your first paragraph is true. Can you elaborate why we should expect mathematics to have 'fewer rough edges' in your parlance if mathematics was deep in the manner of Wigner? I don't see why this has to follow.
deleted 12y ago
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mej10 12y ago
This to me just implies that we are assuming things about the universe that aren't true. Like perhaps the "fact" that it is all continuous? The tools of continuous physics/math are so ingrained into the culture at this point that people seem to have forgotten the distinction between the map and the territory.
dTal 12y ago
I am not entirely sure I agree with you that reality is the master and mathematics, the servant - or that it's basically a human construction like language. For a start, you don't invent new mathematics - you discover it. You can't discover language. There's a strong sense when you look at things like the Mandelbrot set that mathematics is in some sense already "out there". What constrains maths is consistency - it is the study of inevitable consequences. Assuming that the laws of reality are indeed consistent (see hackinthebochs's comment for an argument to why they must be), this makes it a branch of mathematics. At this point considering the branch of mathematics that we personally can observe somehow more "real", in some vaguely-defined cosmic sense, is multiplying entities unnecessarily. I don't think it has "rough edges" either. If your equations don't predict the world well, it doesn't mean that they are bad mathematics, or that the world is "wrong". You're just misapplying them. As an aside, that Chomsky poem is the most beautiful thing I've read all week.
- dllthomas 12y ago
  "As an aside, that Chomsky poem is the most beautiful thing I've read all week." To clarify (unsure whether or not you understand this): "Colorless green ideas sleep furiously" was picked by Chomsky as a nonsense phrase that he asserted had no meaning despite being syntactically correct. That was an aside from his actual use of the phrase, which was to demonstrate that there is in fact something that we see as "grammatical" in even novel sentences that play by the rules - the point of the sentence was that it had not been uttered before. This was in the space of an argument at the time of whether we deem things grammatical because they are sensical, or deem things grammatical because we have encountered them before. I don't know offhand whether this was used directly as evidence or as pedagogy (certainly, I first encountered it in pedagogical context) but it makes it pretty clear that it cannot be the latter and it probably isn't the former, and we're really recognizing some deeper structure. A lot of this is more obvious to we who've been playing with compilers for ages - though I am only moderately confident that this "obvious" result is actually correct. Coming back... the first half of the quoted sentence is someone else's work, trying to place Chomsky's "nonsense" sentence in a situation where it has a meaning. I'm not convinced it was successful - I think it plays with mood rather than meaning. Regardless, that's the context...

yequalsx 12y ago

I don't see why your first paragraph is true. Can you elaborate why we should expect mathematics to have 'fewer rough edges' in your parlance if mathematics was deep in the manner of Wigner? I don't see why this has to follow.

deleted 12y ago

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mej10 12y ago

This to me just implies that we are assuming things about the universe that aren't true. Like perhaps the "fact" that it is all continuous? The tools of continuous physics/math are so ingrained into the culture at this point that people seem to have forgotten the distinction between the map and the territory.

dTal 12y ago

I am not entirely sure I agree with you that reality is the master and mathematics, the servant - or that it's basically a human construction like language. For a start, you don't invent new mathematics - you discover it. You can't discover language. There's a strong sense when you look at things like the Mandelbrot set that mathematics is in some sense already "out there". What constrains maths is consistency - it is the study of inevitable consequences. Assuming that the laws of reality are indeed consistent (see hackinthebochs's comment for an argument to why they must be), this makes it a branch of mathematics. At this point considering the branch of mathematics that we personally can observe somehow more "real", in some vaguely-defined cosmic sense, is multiplying entities unnecessarily. I don't think it has "rough edges" either. If your equations don't predict the world well, it doesn't mean that they are bad mathematics, or that the world is "wrong". You're just misapplying them. As an aside, that Chomsky poem is the most beautiful thing I've read all week.

dllthomas 12y ago
"As an aside, that Chomsky poem is the most beautiful thing I've read all week." To clarify (unsure whether or not you understand this): "Colorless green ideas sleep furiously" was picked by Chomsky as a nonsense phrase that he asserted had no meaning despite being syntactically correct. That was an aside from his actual use of the phrase, which was to demonstrate that there is in fact something that we see as "grammatical" in even novel sentences that play by the rules - the point of the sentence was that it had not been uttered before. This was in the space of an argument at the time of whether we deem things grammatical because they are sensical, or deem things grammatical because we have encountered them before. I don't know offhand whether this was used directly as evidence or as pedagogy (certainly, I first encountered it in pedagogical context) but it makes it pretty clear that it cannot be the latter and it probably isn't the former, and we're really recognizing some deeper structure. A lot of this is more obvious to we who've been playing with compilers for ages - though I am only moderately confident that this "obvious" result is actually correct. Coming back... the first half of the quoted sentence is someone else's work, trying to place Chomsky's "nonsense" sentence in a situation where it has a meaning. I'm not convinced it was successful - I think it plays with mood rather than meaning. Regardless, that's the context...