The essential part of the answer (page references) is contained in the comment of @Conifold. However his general conclusion is plain wrong, and I would like to put the things straight.
A scientist makes ASSUMPTIONS. Then develops a theory. And then compares with observations/experiments. If this comparison works, this CONFIRMS his assumptions.
For example, Newton (and others) assume the inverse square law. Then Newton (and others) derived a lot of consequences from it, which can be tested by observations. And the agreement with observations proves the initial conjecture. This is how science works.
Returning to Fourier. Of course, he could not prove mathematically the statement
that "arbitrary periodic function has a Fourier expansion", for the simple reason
that the modern notion of "arbitrary function" did not exist at that time.
It was first stated by Dirichlet, whose purpose was to give a mathematical justification of Fourier discoveries.
(Further attempts in this direction led to further evolution of the notion of function: "generalized functions" or "distributions" were also introduced ith the purpose to justify Fourier analysis.
Fourier himself was a scientist, first of all. And he lived at the time when science
was not separated from mathematics (it is still not separated completely). So his approach is that of a scientist: he makes assumptions, develops a theory and then tries to test it.