Much effort has gone into devising rigorous foundations for mathematics. Many variants exist, and long may the variation continue – for it has kicked up plenty of interesting material along the way. However, it suffices for my purposes, whatever variety of foundation you may chose to use, to introduce a single straightforward notion – the relation – from which arise tools familiar to any mathematician, out of which may be built all the mathematics a physicist ever needs. This being my aim, I attend to foundations only sufficiently to assert what little I assume of them, so as to render the rest of what I discuss compatible with a broad variety of foundations.
A relation, put simply, is characterized by: for any relation, r, any
values, x and y, one may legitimately ask
does r relate x to y ? –
that is, it's a valid and meaningful question, though not necessarily within our
power to answer in all cases. A relation is then understood simply in terms of
which values it relates to which values. Given an expression containing two
names not bound by context to any value, one can construct a relation by
choosing one of these names (choosing the other at this point gives the reverse
relation) and saying that one value relates to another precisely when,
substituting the former for the chosen name and the latter for the other, the
expression reads as a true statement.
From this beginning, one can define
pairs, mappings, collections, the natural numbers and lists, topology, binary
operators, scalars and linearity, measure and probability, smoothness, geometry
and curvature. I want these tools at my disposal when I come to discuss
physics. Alongside my introduction to relations and these consequences, I
define plaintext denotations for the
various kinds of value which arise in these discussions (and for the various
ways they interact), and introduce a
primitive entities. I have taken some care to ensure
that the denotations thus implied for the mathematical entities I'll be using in
physics can be clear and concise, yet specify a rich structure of well-defined
behaviour. Work out for yourself whether I've succeeded.