# Foundation

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 bestiary of 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.