Preparing the ground for mathematics

My purpose in building mathematical tools is to ensure that I can do physics, notably quantum mechanics and general relativity, without ad hoc notational differences adding themselves to the conceptual difficulties standing in the way of mutual intelligibility between discourses. I also want to write my mathematics in HTML as plain text - by which I mean something in which an HTML 1.0-compliant browser without images will display the information content unambiguously to the patient and thoughtful reader. As much as anything else, this is the easiest form for me to use in typing the material - and these pages exist as my notes on a subject, tentatively placed where other folk can read them not out of expecting other folk to do so, but to allow the possibility (and to let me read them from anywhere).

Physics requires functions: I see no way of building anything resembling our modern physical theories without them. Questions like what values does this function accept ? (or produce) lead naturally to discussion of sets. In practice, mathematics also ends up involving relations (and they're a tool I want at my disposal in physics, even if I can do without them) and, as I'll show in due course, one can construe certain kinds of relation as mappings and collections which have substantially similar form to functions and sets.

Mathematics formalises the application of reason to real systems. Reason is something subtler than logic but closely related: logic, roughly speaking, formalises the application of reason to the process of reason. Mathematics is, accordingly, founded on logic. Various logical systems are possible, with various advantages and disadvantages: I chose to work upwards from the layer just above the foundations, if only because otherwise I'd be so busy vanishing up my own navel that I'd never get round to doing any physics. That layer is the discussion of relations, mappings and collections: during which I'll introduce most of the denotational forms I'll be using for mathematics and physics.

Because the ground-work I'll be doing is notionally built on any foundation that supports relations, I try hard to avoid presuming too much about the logical form of the foundations - merely that it supports enough structure that it will allow me to define relations. However, to conduct proofs I necessarily use reason in forms which I trust the foundations will also support; and all my definitions depend on the premise that they are sufficiently reasonable that the foundations will allow them. So my discourse aims to provide a reasoned account of things, rather than a formally logical one. The further we get above the foundations, the less this will be relevant - the preliminary tools of the ground-work embody the foundation's formalisation of certain reasonable truths.

I break the ground-work into the following pieces:

preliminary meta-discussion

covering the truth that any text is read in some context, and the ways that a context evolves during the text to which it gives meaning.

reason, or naïve logic

with the quiet acceptance that Gödel has taught us not to expect not(not(A)) to imply A.


and other basic notions, providing enough detail to make some denotations meaningful.

binary operators

their primitive properties and some more notational material.


which generalise equality.

These run in tandem with my introduction of denotational definitions and a bestiary which may be thought of as parts of a glossary. The denotations page tries to set out enough information for a parser-writer to have a good chance of working out how to parse my plaintext denotations for mathematical entities; the bestiary introduces various significant relations. The style in which I use patterns in the denotational definitions should be intelligible to anyone with experience of parser-writing.

I try to assume as little as possible about the context in which all those tools are to be used. Elsewhere, I'll introduce some contexts and apply the tools of the ground-work in those. One such context will be the context of pure finite collections, in which I can build the natural numbers and lists, which will call for more denotational forms and provide the tools with which to discuss richer structure flowing from the parts above. After that, we'll be cooking on gas.

Further Burbles

See also:

Zero, One and Many

or why I use 2-relations.

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