Chris and Pat, (01)
I agree that Chris correctly diagnosed the distinction between
the words 'dismiss' and 'ignore'. (02)
But I said that I had "no preference" for either word.
Either one is adequate to make my point that foundational
work is "irrelevant" for the working mathematician -- in
any or every sense of the word 'irrelevant'. (03)
In any case there are many other related issues that were hot
topics at the time -- including the point that arithmetic is
much more solid than set theory, and it is misguided to think
that arithmetic could be made more secure by defining it in
terms of symbolic logic and set theory. (04)
In fact, one of the motivations for Goedel numbering is that
many mathematicians were highly suspicious of all that newfangled
logic. Any proof by contradiction was likely to be considered
a proof that all that logic stuff was foolishness. (05)
Therefore, Goedel defined a mapping from logic to the more
"solid" arithmetic because mathematicians had more confidence
in arithmetic than in any version of logic or set theory. (06)
PH> The term 'foundations' is well chosen. When building a house... (07)
The analogy with houses is misleading because gravity creates
a preferred direction of building (although there are many cases
where builders jack up a house to construct a new foundation
under it). For mathematics, there is no inherent directionality. (08)
PH> Similarly with mathematics and formal logic. (09)
Not really. Goedel's approach was much more reasonable than
Frege's or Russell's: adopt the more solid arithmetic as the
basis for modeling the newfangled logic. (010)
In any case, there is no reason to adopt any single system
as a "foundation" for any other. Mathematicians generally show
relative consistency by using the structures defined by one set
of axioms as a model for a different set of axioms. (011)
With that approach, no single system is the "foundation'.
Instead, you have an open-ended collection of systems, and
you can show relative consistency of one system in terms
of another. That is very different from the idea of a
single, privileged, universal foundation for everything. (012)
John (013)
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