How do we explain the use of a software on a math paper?

Jaxon: 2 days ago

Suppose one has written a math/computer science paper that is more focused in the math part of it. I had a very complicated function and needed to find its maximum, so I used Mathematica (Wolfram) to do it. How do I explain that? "Using wolfram we find the maximum of $f$ to be $1.0328...$ therefore...". It looks very sloppy.

Everett: 2 days ago

Welcome to MO! I believe the answer to your question depends on what Mathematica command(s) you used to find the maximum.

If you used the command Maximize[], then its output is exact and, in my view, can in general be trusted no less than the work of about any human. A reason for this belief is that Maximize[] has probably been tested at least hundreds or thousands as many times as an average proof by an average mathematician.

Still, others may disagree with this comparison between human work and the outputs of commands such as Maximize[], and so, then you may have to try to obtain the maximum in a way that can be verified by hand. Also, of course, Maximize[] can only solve comparatively easy maximization problems.

On the other hand, if you just used the command NMaximize[] -- which tries to find the maximum numerically, then its output can only be considered a suggestion -- certainly not a proof.