The first question I take to be empirical. I am not an anthropologist, and
shall not engage with it, except to notice that unless it is an empirical
question, I have no idea of how to answer it, or even if there is an answer.
If the suggestion is that relativism goes all the way down, then I shall just
have to live with the fact that I sometimes won't understand what other people
say -as I already do. It does of course invite one to treat such utterances
politely and I would prefer to invite discussion that will enable me to
enlarge my understanding, rather than bullying the speaker I do not
understand. Philosophers have a rather bad reputation in this respect.
It is the other question that interests me. It seems clear that the
development of mathematics by noticing that one might provide proofs with some
generality is historically, but not logically, linked to a belief in the
existence of fundamental truths. In fact, Euclid does not speak of axioms at
all: what Aristotle calls 'axioms', Euclid calls simply 'common notions'. The
postulates are described by Proclus as being those assumptions that the
student needs to accept without demonstration for the study, while a number of
the definitions are never used in the actual derivations. The modern notion of
an axiomatic system is an idealized notion of what is to be found in Euclid.
It is the philosophers who think of this system as relying upon fundamental
insights. We know nothing about Euclid's own view of his axioms or
assumptions. The development of a 'deductive style' could have been conducted
piecemeal, without any search for axioms as such, and I cannot see that the
same mathematical results might not have been achieved. Thus approach might
not have produced the same ideas about insight and intuition, but this is
surely a matter for philosophy or psychology, not mathematics. I'd welcome
feedback here, I am a philosopher. I find it difficult to think that the
system we see in Euclid was not the end result of a much more piecemeal
development, permitted by the presence of a number of mathematicians in a
society that encouraged free inquiry.
One question that I have been considering is whether the production of
elements was not originally the result of an educational need in Athenian
society. We do not know in what form the first 'Elements' of Hippocrates
appeared. But the term 'elements' was also (perhaps originally) applied to the
elementary parts of words when these were learned by children. Plato takes the
contribution of letters to syllables as an example of thinking in terms of
elements in the Theaetetus. The notion of elementary parts is not directly
connected to the idea of assumptions, and any connection seems then to be
historically contingent.
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Janet D. Sisson
Department of Philosophy
University of Calgary
Home page: http://www.acs.ucalgary.ca/~jdsisson/