Is there a term which refers strictly to the foundational assumption(s) that a proof is based off of? I don't mean this in the sense of an axiom, but rather, for example, stating that $m \in \mathbb{R}$ and $n \in \mathbb{Z}$ at the start of the proof. The term could then be used to refer to the collective of these two assertions unambiguously at any point in the proof.
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1One way to answer your own question is to follow Frank Adams' advice and learn to write proofs by reading the proofs of others. You will quickly learn to identify well-written proofs (they'll be the ones you either understand or want to understand). You will find that good proofs don't need the technical term you are asking for because in simple cases it will be obvious what assumptions are being talked about and in complex cases the author will have taken care to introduce labels or notations or terminology to make it clear what is going on (as in Arthur's answer). – Rob Arthan Oct 04 '18 at 20:11
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There isn't really. So you have to make your own. One common way is
Assume $$m\in\Bbb R,\quad n\in\Bbb Z\tag{*}$$
and you can then refer to it as $(\text *)$ at any point.
Arthur
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The usual term for this is "hypothesis" (or often "hypotheses", when there is more than one assumption). The hypotheses of a theorem are the assumptions made in the statement of the theorem which must hold in order for the conclusion to follow. More generally, a "hypothesis" of an argument is an assumption you make at the start of the argument which will prove that some consequences follow from the assumptions.
That said, the term "hypothesis" is not as strict and unambiguous in its meaning as you seem to be asking for. Since it can broadly refer to any assumption made in an argument, the exact meaning varies with context and it is not a totally precise term on its own. Still, it is probably the term that mathematicians most frequently use with the meaning you ask for.
Eric Wofsey
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