Foundation's Shadow

I've managed to tame my ambition enough to focus on getting together the mathematical tools I need for the discussion of physics. On one hand, this involves holding back from the immense (and fascinating) topic of how reason can attempt to found itself on how little a foundation. On another, it has encouraged me to focus on what tools the foundations provide: in particular, the tools any foundation must provide, to suffice as a foundation for the discussion of physics.

It turns out that there is an interwoven little bundle of ideas, from any one of which it is possible to build the others, which suffice to define everything physics seems to need. I think of this little bundle as the shadow of the foundations: there are many objects which, suitably aligned with the light, will produce a given shadow; just so, there are many foundations which will produce this bundle, and any useful foundation will cast this shadow, if you put the screen in the right place to catch it.

If I start with mappings, I can obtain: identities, which I can interpret as collections, and; singleton mappings which accept just one input and produce just one output, thus providing a pair notion.

If I start with pairs, I can construct lists, which I can use to define collections – though only finite ones. Lisp programmers can assure you that pairs suffice to construct anything that matters, at least to a Lisp programmer – and Lisp is computationally complete, so pairs should serve my purposes entirely to my satisfaction. However, working from relations or mappings flows more naturally, especially around obstacles produced by implicit infinities.

If I start with the notion of collection – a primitive notion of which the stereotype is set – I can construct pairs: a collection of form {{x}, {x,y}} can be read as a pair, distinguishing x as the lone part and y as its paired part. Given collections and any pair notion, I can express a relation as a collection of pairs: a pair [x,y] is in the collection if the relation relates x to y.

If I start with relations, sensibly characterized, they instantly imply pairs, collections and mappings in delightfully natural ways. There is a natural composition of relations: s∘r relates x to z precisely if there is some y for which s relates x to y and r relates y to z. When the relations are mappings, so is the composite; which makes sense as a composition on mappings.

Wherever I start in this little bundle of interwoven notions, I can get the whole bundle. From the bundle I can obtain the successor relation, the natural numbers and lists. The bundle suffices for discussion of binary operators (as mappings whose inputs are pairs, or as mappings whose outputs are mappings), which suffice to build linearity, along with notions such as integration and probability, topology and continuity, differentiation and smoothness – in short, the tools I need to do modern physics.

Starting from the other end, I am certain that any discussion of theoretical physics needs, at least, the notion of linearity. I describe this in terms of mappings and do not see how it could be described without involving mappings, albeit perhaps under some disguise. So I require mappings from any foundation before I can hope to build the mathematical tools needed for theoretical physics. Conveniently, any mathematical tool-set which does support mappings inescapably also supports the bundle, hence the tools needed for physics.

I spent some time trying to work out which corner to start from: which to treat as fundamental. My final opinion was relations: but by the time I reached that conclusion, I was less concerned. Each gives all, so take any, without fear or favour, and use it to induce the others – or, at least, any of mapping, collection and relation, if you want tools for discussing implicit infinities, even if only for the sake of understanding how to make them redundant.

I chose to conduct my discourse from the foundation of relations, but with mappings taking centre stage. All four players in the bundle take their parts: I find them (and lists) to be helpful descriptive idioms. From these little acorns, the mighty oaks grow, which form the very heart of the forest of knowledge.

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