>>7708352

If by "analysist" you mean a guy who's studying classical analysis, then the question has probably no answer.

Dropping non-constructive axioms of your theories becomes relevant, for example, whenever you got to implement stuff. As you can't even list the element of [math]\mathbb R[/math], theorems in classical analysis will be vacuous for such realization. The good news is that people have spun the spiderwebs of theories like analysis in a constructive fashion long ago, see for example all the constrictive theorems that aggregate around the unprovable intermediate value theorem

https://en.wikipedia.org/wiki/Constructive_analysis#Examples

Syntetic analysis as in the pic from the book by Kock is relevant because the requirements for the theory are such that certain topoi give you internal analysis as a gift. That is you consider some topos, check if it has this and that property and then it might be implied that that topos really is about calculus/contains a theory of calculus and you can import all the theorems you already know - I mean that's the nice feat of category theory in general.

The category of sets [math]A, B, C,...[/math] (=objects) and functions [math]f, g ,h,...[/math] (=arrows) is a topos and thus has cartesian product [math]A\times B[/math] (=the categorical product) and function spaces [math]B^A[/math] (=internal, setty realizations of the arrow class from [math]A[/math] to [math]B[/math], which may be written [math]A \to B[/math])

A theorem of sets is that the function space

[math] (A\times B) \to C [/math]

is isomorphic to

[math] (B\times A) \to C [/math]

as well as

[math] A \to C^B [/math]

and

[math] B \to C^A [/math]

E.g. for A=B=C the reals the first space contains

[math] \langle a,b \rangle \mapsto \sin(a)+3b [/math]

which you can systematically map to the function

[math] \langle b,a \rangle \mapsto \sin(a)+3b [/math]