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.”
But here is a joke answer that I think illustrates something. Differential equations govern most of our everyday experiences, including the experience of writing out the axioms for set theory and deducing theorems from them. And we can model differential equations in a first order theory of real numbers, which requires no set theory. A somewhat more serious point along these lines is made in some famous papers by Pour-El and Richards.
Is this a good way to think about set theory? Of course not. But likewise, the standard reduction to set theory does not illuminate differential equations. Boo set theory!
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!
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.
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.
I’m not sure, presumably to “+*=01>” one adds a bunch of special functions. The “o-minimal approach” to differential geometry requires no sequences. Functions are encodable as definable graphs, so are vector fields.
As you note, for completeness reasons an actual o-minimal theory is not as strong as set theory; one has to smuggle in e.g. natural numbers somehow, maybe with sin(x). I could have made a less tendentious point with Godel numbering.
Again, this is meant as a kind of joke, not as a natural way of looking at sets. The point is that I don’t regard von Neumann’s {{},{{}},{{},{{}}}} as a natural way of looking at the number three, either.
Once you say that functions are definable graphs, you are on a slippery slope. If you want to prove something about “all functions”, you have to be able to quantify over all formulas. This means you have already smuggled natural numbers into the model without defining their properties well...
When you consider a usual theory, you are only interested in the formulas as long as you can write—not so here, if you want to say something about all expressible functions.
And studying (among other things) effects of smuggling natural numbers used to count symbols in formulas into the theory is one of the easy-to-reach interesting things in set theory.
About natural numbers—direct set representation is quite unnatural; underlying idea of well-ordered set is just an expression of the idea that natural numbers are the numbers we can use for counting.
The true all-mathematical value of set theory is, of course to be a universal measure of weirdness: if your theory can be modelled inside ZFC, you can stop explaining why it has no contradictions.
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.”
But here is a joke answer that I think illustrates something. Differential equations govern most of our everyday experiences, including the experience of writing out the axioms for set theory and deducing theorems from them. And we can model differential equations in a first order theory of real numbers, which requires no set theory. A somewhat more serious point along these lines is made in some famous papers by Pour-El and Richards.
Is this a good way to think about set theory? Of course not. But likewise, the standard reduction to set theory does not illuminate differential equations. Boo set theory!
Like I suspected, this is rife with confusion-of-levels.
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”.
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.
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.
...and nor does the reduction of biology to physics “illuminate” human behavior. That just isn’t the point!
Nope. It is literally possible to reduce the theory of Turing machines to real analytic ODEs. These can be modeled without set theory.
Okay, that sounds interesting (reference?), but what about the rest of my comment?
Here is Pour-El and Richards. Here is a more recent reference that makes my claim more explicitly. Both are gated.
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:
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:
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.
Matiyasevich’s book “Hilbert’s 10th Problem” sketches out one way to do this.
Hilbert’s 10th problem is about polynomial equations in integer numbers. This is a vastly different thing.
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.
What signature do we need for it? Because in the first-order theory of real numbers without sets you cannot express functions or sequences.
For example, full theory of everything expressible about real numbers using “+, *, =, 0, 1, >” can be reolved algorithmically.
I’m not sure, presumably to “+*=01>” one adds a bunch of special functions. The “o-minimal approach” to differential geometry requires no sequences. Functions are encodable as definable graphs, so are vector fields.
As you note, for completeness reasons an actual o-minimal theory is not as strong as set theory; one has to smuggle in e.g. natural numbers somehow, maybe with sin(x). I could have made a less tendentious point with Godel numbering.
Again, this is meant as a kind of joke, not as a natural way of looking at sets. The point is that I don’t regard von Neumann’s {{},{{}},{{},{{}}}} as a natural way of looking at the number three, either.
Once you say that functions are definable graphs, you are on a slippery slope. If you want to prove something about “all functions”, you have to be able to quantify over all formulas. This means you have already smuggled natural numbers into the model without defining their properties well...
When you consider a usual theory, you are only interested in the formulas as long as you can write—not so here, if you want to say something about all expressible functions.
And studying (among other things) effects of smuggling natural numbers used to count symbols in formulas into the theory is one of the easy-to-reach interesting things in set theory.
About natural numbers—direct set representation is quite unnatural; underlying idea of well-ordered set is just an expression of the idea that natural numbers are the numbers we can use for counting.
The true all-mathematical value of set theory is, of course to be a universal measure of weirdness: if your theory can be modelled inside ZFC, you can stop explaining why it has no contradictions.