(1) Proofs in the manner of Euclid establish relationships between
assertions, no? In an axiomatic development, the theorems and lemmas, are,
as it were, assertions emphasized in one long development which constitutes
a kind of tree or tree-like structure of linked assertions, no? An
axiomatic development is one way to communicate a body of results in such a
way that the results are linked together in a single pattern, no? A nice
side effect is that if some assertion (theorem, or statement in a proof)
appears to be untrue or surprising, one can banish any doubts or lower any
raised eyebrows one has by working from positions one accepts as true or
finds unsurprising, or alternatively decide that one knows where a link has
failed, no? This is the case, I should think, provided one accepts the
axioms as a starting point, certain rules of inference, and the like. It's
nice, of course, if the axioms turn out to be mutually consistent.
(2) Explanations which aren't proofs in some strict axiomatic sense, or
aren't proofs in some sense approved by suitably trustworthy logicians, may
or may not establish relationships of this kind, no? One can explain how
an algorithm works without embedding it in an axiomatic system, no? Such
explanations are called "proofs" by some people, no?
(3) When trying to show somebody some relationship between mathematical
objects or concepts or the like, one uses what the traffic will bear, no?
For example, when a (pure) mathematician explains something about group
theory of interest to a physicist, isn't it often inappropriate to present
a proof based on a set of axioms for group theory? Hadn't one better try
to present an explanation of some sort which will reveal to the physicist
what he wants to know, by exploring what the physicist already knows, and
trying to link into that? Chances are that what the physicist knows about
group theory isn't lodged in his or her brain the way it's lodged in the
brain of a (pure) mathematician, right?
Gordon Fisher gfisher@shentel.net