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Gurkenglas (Jun 20 2022 at 16:34):Just as adjoining a solution for x²+1=0 to the reals gets the complex numbers, I'd like to adjoin a solution for f(x)=x, f'=0 to the functions. What is the result called? It should of course be closed under composition, satisfy the chain rule and product rule and so on. I am hoping to use it to distinguish "Prefer (if A, then B)." from "If A, then prefer B.".
Morgan Rogers (he/him) (Jun 21 2022 at 08:53):You're trying to add this to the differential ring of differentiable functions from the real to the reals, right? I'll call the ring R.
The first assumption is going to be a problem; when you formally add an element to a ring, you take a free extension (generate all formal expressions involving the new element f and elements of R), and then quotient by the axioms you want the element to satisfy. The first axiom amounts to f-x =0, which identifies f with x and returns us to the original ring. You need a weaker condition to avoid that collapse.
Gurkenglas (Jun 21 2022 at 12:56):f-x would have the same values as 0, but not the same derivative. f=g would no longer be equivalent to f(x)=g(x) for all x.
Morgan Rogers (he/him) (Jun 21 2022 at 13:06):I understand the property that you want, I'm just telling you that the naive interpretation of the context you want to perform this extension in (rings with a differential), it isn't possible to obtain a meaningful extension with this property starting from the ordinary ring of differentiable functions on the reals. So you need to decide upon a synthetic setting where what you're saying (namely, that an element of the ring is not determined by its values) makes sense. I say "synthetic" because that kind of property reminds me of the basic objects of synthetic differential geometry.
Gurkenglas (Jun 21 2022 at 13:17):I don't see what's wrong with taking the free extension R[f]. By not meaningful, do you mean that I won't find an interpretation of those formal expressions that doesn't feel detached from reality? Or are you claiming something formal?
Gurkenglas (Jun 21 2022 at 13:23):detached being a pun on Pytorch's detach which behaves like f :) (Pytorch's automatic differentiation apparatus skips over f.)
Morgan Rogers (he/him) (Jun 21 2022 at 13:23):I'm saying that as soon as you quotient by an ideal containing f-id, f gets identified with id (the unit of the ring of functions) and hence you end up with a trivial extension
Gurkenglas (Jun 21 2022 at 13:25):I'm not trying to quotient by an ideal containing f-id, because the first axiom is f(x)=id(x), not f=id.
Morgan Rogers (he/him) (Jun 21 2022 at 13:27):For the usual ring of functions on the reals, those are the same thing!!
Gurkenglas (Jun 21 2022 at 13:28):That's why I'm extending it. Extensions are allowed to break such statements, yes? For example, there are two real numbers with absolute value 1, but more than two such complex numbers.
Gurkenglas (Jun 21 2022 at 13:31):(I think we should also quotient out the identities arising from the chain rule.)
Morgan Rogers (he/him) (Jun 21 2022 at 13:43):In analogy with your original example, if you adjoin to the reals an element satisfying x-1 = 0, then x gets identified with 1 and you get the reals back out again.
Morgan Rogers (he/him) (Jun 21 2022 at 13:44):So I would suggest that you need to choose a bigger starting ring than the ring of functions from the reals to the reals.
Reid Barton (Jun 21 2022 at 13:50):I think you should start by specifying exactly what kind of algebraic structure you're talking about.
Jules Hedges (Jun 21 2022 at 14:15):I vaguely thought that the ring of dual numbers was , but that would fail for the same reason, was I mistaken all along?
Morgan Rogers (he/him) (Jun 21 2022 at 14:19):No, that works because you aren't insisting that the extension is a field.
Gurkenglas (Feb 01 2023 at 22:51):I suppose it would be a ring equipped with the additional structures of differentiation and composition, and the properties of chain rule and product rule. Would there be anything else to possibly add?
John Baez (Feb 01 2023 at 23:20):The ring consists of real polynomials in one variable. It has the structures of differentiation and composition, obeying the chain rule and product rule. It contains a solution to , whatever that means... namely it has an element , which we can call if we prefer. But this element has , of course.
John Baez (Feb 01 2023 at 23:21):I don't get why you want something that acts like the function yet has derivative zero. That's like saying you want .
Gurkenglas (Jun 03 2023 at 08:18):@John Baez the reason is to model detach, which is treated like this by pytorch's automatic differentiation. It lets you do gradient descent while ignoring the relationship between two variables. A theory of optimization processes that ignore a relationship would be useful to build an AI that doesn't prevent its operator from changing its instructions.
Notification Bot (Jun 05 2023 at 08:20):This topic was moved here from #learning: questions > f(x)=x, f'=0 by Matteo Capucci (he/him).
Morgan Rogers (he/him) (Jun 05 2023 at 08:53):Detach looks like it copies the value of a function and stops modifying it. That is, is a constant, which gets you the differential behaviour you were after.
To put it another way, there is an "evaluate now" operator which takes a function and produces the function which is constant at .