komponisto comments on Which fields of learning have clarified your thinking? How and why? - Less Wrong

12 [deleted] 11 November 2011 01:04AM

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Comment author: komponisto 11 November 2011 06:02:15PM 3 points [-]

Like I suspected, this is rife with confusion-of-levels.

There are a lot of phenomena -- in mathematics, in the cosmos, and in everyday experience -- that you cannot understand without knowing something about differential equations. There are hardly any phenomena that you can't understand without knowing the difference between a cardinal and an ordinal number. That's all I mean by "fundamental."

That's like saying that you can get through life without knowing about atoms more easily than you can without knowing about animals, and so biology must be more fundamental than physics. Completely the wrong sense of the word "fundamental".

Differential equations govern most of our everyday experiences, including the experience of writing out the axioms for set theory and deducing theorems from them.

This is a classic confusion of levels. It's the same mistake Eliezer makes when he allows himself to talk about "seeing" cardinal numbers, and when people say that special relativity disproves Euclidean geometry, or that quantum mechanics disproves classical logic.

And we can model differential equations in a first order theory of real numbers, which requires no set theory

Your conception of "differential equations" is probably too narrow for this to be true. Consider where set theory came from: Cantor was studying Fourier series, which are important in differential equations.

But likewise, the standard reduction to set theory does not illuminate differential equations

...and nor does the reduction of biology to physics "illuminate" human behavior. That just isn't the point!

Comment author: [deleted] 11 November 2011 06:13:04PM 1 point [-]

And we can model differential equations in a first order theory of real numbers, which requires no set theory

Your conception of "differential equations" is probably too narrow for this to be true.

Nope. It is literally possible to reduce the theory of Turing machines to real analytic ODEs. These can be modeled without set theory.

Comment author: komponisto 11 November 2011 07:31:49PM 0 points [-]

It is literally possible to reduce the theory of Turing machines to real analytic ODEs.

Okay, that sounds interesting (reference?), but what about the rest of my comment?

Comment author: [deleted] 13 November 2011 04:05:34PM *  0 points [-]

Here is Pour-El and Richards. Here is a more recent reference that makes my claim more explicitly. Both are gated.

What about the rest of my comment?

I'm not sure what to say. You've accused me of "confusing levels," but I'm exactly disputing the idea that sets are at a lower level than real numbers. Maybe I know how to address this:

But likewise, the standard reduction to set theory does not illuminate differential equations

...and nor does the reduction of biology to physics "illuminate" human behavior. That just isn't the point!

I don't know about human behavior, which isn't much illuminated by any subject at all. But the reduction of biology to physics absolutely does illuminate biology. Here's Feynman in six easy pieces:

Everything is made of atoms. That is the key hypothesis. The most important hypothesis in all of biology, for example, is that everything that animals do, atoms do. In other words, there is nothing that living things do that cannot be understood from the point of view that they are made of atoms acting according to the laws of physics. This was not known from the beginning: it took some experimenting and theorizing to suggest this hypothesis, but now it is accepted, and it is the most useful theory for producing new ideas in the field of biology.

You simply can't say the same thing -- even hyperbolically -- about the set-theoretic idea that everything in math is a set, made up of other sets.

Comment author: JoshuaZ 11 November 2011 08:20:18PM *  0 points [-]

Matiyasevich's book "Hilbert's 10th Problem" sketches out one way to do this.

Comment author: vi21maobk9vp 12 November 2011 11:25:46AM -1 points [-]

Hilbert's 10th problem is about polynomial equations in integer numbers. This is a vastly different thing.

Comment author: JoshuaZ 12 November 2011 01:51:51PM *  2 points [-]

Yes, Hilbert's 10th Problem was whether there was an algorithm for solving whether a given Diophantine equation has solutions over the integers. The answer turned out to be "no" and the proof (which took many years) in some sense amounted to showing that one could for any Turing machine and starting tape make a Diophantine equation that has a solution iff the Turing machine halts in an accepting state. Some of the results and techniques for doing that can be used to show that other classes of problems can model Turing machines, and that's the context that Matiyasevich discusses it.

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