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:
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.
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.
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.
See also:
or why I use
2-relations
.
Written by Eddy.