FOM: Ferguson's points

[Neil Tennant]

Stephen Ferguson's email makes several very sensible points that I
would like to echo in welcoming him to the list. It may be useful
to summarize them, and seek reactions from the mathematicians.

1. There is a contrast between justificatory foundationalism (i.e.
determining what axioms are needed to prove what we want) and
conceptual foundationalism (i.e. determining what concepts are
'basic', serving for definitions of all the rest). Obviously the two
are closely linked; but how, exactly?

2. Special interest attaches to the phenomenology of 'studying a
particular structure' (such as the natural numbers) independently of
conceiving them as making up some fragment of the universe of sets.
Intellectually, pursuing truths about some unique intended structure
somehow 'feels different' from studying the commonalities and
interrelationships of structures of a 'laxer' kind, such as groups.
Stephen makes the interesting point that there is still a
justificatory step called for in the latter venture, namely arguing
for the appropriateness of the choice of, say, group axioms.  Is the
only justification offered by mathematicians that, once groups are
defined this way, one finds them all over mathematics (e.g. groups of
transformations in geometry); and that the group concept leads to
various interesting theorems about groups?

3. Perhaps one of the most important foundational problems is the
problem of justifying a particular logic (and choice of a language
with certain expressive powers) for different areas of mathematics;
and, possibly, tailoring the logic to the demands of the particular
area of mathematics in which one is working.

I would like to see some more discussion of these issues on the list.

Neil Tennant