I’m not sure one is more or less fundamental than the other.
Really? You don’t think the demonstrable reducibility of other branches of mathematics to set theory means anything?
It does seem fair to say that as far as differential equations are concerned a completely different foundational setting wouldn’t make any difference.
This is actually a vacuous statement, because if it did make a difference, you wouldn’t call it “a completely different foundational setting” of the same subject. Similarly, it wouldn’t “make any difference” if, hypothetically, a 747 (as we understand it in terms of high-level properties) turned out to be made of something other than atoms; because by assumption the high-level properties of the thing we’re reducing are fixed.
The important point is whether something can be reduced, not whether it must be.
I really don’t know enough about programming to make a properly impressive analogy, but set theory is like a lower-level language or operating system on top of which other branches can be made to run.
I think the right analogy is not to building 747s out of parts, but to telling stories in different languages. The plot of “3 little pigs” has nothing to do with the English language, and the plot of Wiles’s proof of Fermat’s last theorem has nothing to do with set theory.
Really? You don’t think the demonstrable reducibility of other branches of mathematics to set theory means anything?
Not in any strong sense, no. I reduce things to other fundamental frameworks also. One could for example choose categories to be one’s fundamental objects and do pretty well. To extend your 747 analogy, this is closer to if we had two different 747s, one made of atoms and the other made from the four classical elements, and somehow for any 747 you could once it was assembled to decide to disassemble it into either atoms or earth, air, fire and water.
Epicycles are only a rough approximation that doesn’t work very well, and it doesn’t in any way give you Kepler’s third law (that there’s a relationship between the orbits). I’m also confused in that even if that were the case it wouldn’t make or Kepler or Newton’s mechanics worthless. What point are you trying to make?
Obviously Kepler’s astronomical model is superior and that line might have been rhetorical flourish but the Ptolemaic, Copernican and Tychonic models were by no means “rough approximations” that don’t “work very well”. Epicycles worked very well which is part of why it took so long to get rid of them- the deviations of theory from actual planetary paths were so small that they were only detectable over long periods of time or unprecedented observational accuracy (before Brahe).
(I don’t understand the grandparent’s point either and agree that mathematical reduction to set theory is a different sort of thing from physical reduction to quantum field theory—just pointing this one thing out.)
Really? You don’t think the demonstrable reducibility of other branches of mathematics to set theory means anything?
This is actually a vacuous statement, because if it did make a difference, you wouldn’t call it “a completely different foundational setting” of the same subject. Similarly, it wouldn’t “make any difference” if, hypothetically, a 747 (as we understand it in terms of high-level properties) turned out to be made of something other than atoms; because by assumption the high-level properties of the thing we’re reducing are fixed.
The important point is whether something can be reduced, not whether it must be.
I really don’t know enough about programming to make a properly impressive analogy, but set theory is like a lower-level language or operating system on top of which other branches can be made to run.
I think the right analogy is not to building 747s out of parts, but to telling stories in different languages. The plot of “3 little pigs” has nothing to do with the English language, and the plot of Wiles’s proof of Fermat’s last theorem has nothing to do with set theory.
Not in any strong sense, no. I reduce things to other fundamental frameworks also. One could for example choose categories to be one’s fundamental objects and do pretty well. To extend your 747 analogy, this is closer to if we had two different 747s, one made of atoms and the other made from the four classical elements, and somehow for any 747 you could once it was assembled to decide to disassemble it into either atoms or earth, air, fire and water.
Well, we can disassemble every planet orbit to epicycles. Does that mean that our astronomical knowledge based on Newton’s mechanics is worthless?
Epicycles are only a rough approximation that doesn’t work very well, and it doesn’t in any way give you Kepler’s third law (that there’s a relationship between the orbits). I’m also confused in that even if that were the case it wouldn’t make or Kepler or Newton’s mechanics worthless. What point are you trying to make?
Obviously Kepler’s astronomical model is superior and that line might have been rhetorical flourish but the Ptolemaic, Copernican and Tychonic models were by no means “rough approximations” that don’t “work very well”. Epicycles worked very well which is part of why it took so long to get rid of them- the deviations of theory from actual planetary paths were so small that they were only detectable over long periods of time or unprecedented observational accuracy (before Brahe).
(I don’t understand the grandparent’s point either and agree that mathematical reduction to set theory is a different sort of thing from physical reduction to quantum field theory—just pointing this one thing out.)
Yes, by doesn’t work very well, I mean more “doesn’t work very well when you have really good data.” I should have been more clear.
Epicycles can give you arbitrary precision if you use enough of them… It is quite similar to Fourier transform.
My point is that in most cases you can disassemble a 767 into various colections of parts.