FOM: difference, structure, congruence
Leo Harrington
leo at math.berkeley.edu
Tue Mar 17 15:08:52 EST 1998
Someone on fom last fall commented that among basic
foundational terms one might consider (I hope my memory serves):
difference structure congruence
I'd like to suggest that some of the conversations that are
now occurring on fom might benefit from an awareness of an intrinsic unity
among these terms, namely that everything will exhibit difference,
structure, and congruence.
For example, in the ongoing conversations involving Reuben Hersh:
a while ago (again if memory serves) Hersh used the imagery of a
daffodil - the daffodil grows and perishes and is replaced by other
daffodils, but there is a common stem. So we have
this daffodil (difference)
the stem (structure)
the daffodils (congruence)
Hersh then can be heard as expounding difference (our math is just
this daffodil) and denying congruence (math is math); whether this
gives Hersh a fair hearing is unclear (at least to me) - at the very
least Hersh is saying:
"the existence, the reality (and they are real) of mathematical
objects is neither physical, subjective or transcendental, but in the shared
thinking and communication of human beings." (Hersh on fom, 13 March 22:29)
For me, the main difficulty I have with this statement is that
it says "the existence of mathematical objects is in ..." and thus
decides to locate existence in a particular way. If this is read to mean
that that particular way is the only way, then, yes, we have something
that is claiming difference and denying congruence. BUT
if this is read as a particular way to encounter the desired
objects as existing, and so by its very nature this way is a particular
way and thus necessarily a way involving difference, then there is no
denial of congruence here.
(Under this last reading, what is going on here would be
similar to what went on with Dedekind (as we are learning from a
current conversation on fom); it would be ungrammatical to identify
real numbers with sets, and it would be unstructural to identify
mathematical objects with the shared thinking and communication
of human beings; but we can build the real numbers as sets, and so
maybe we build our mathematical objects through our thoughts and
communications).
The current conversation about boolean algebras and boolean
rings:
This clearly would seem to involve again a clash between
difference and congruence. To see what's at stake here, let's go
to Dedekind again. One way to view what we've been told through
the current conversation on Dedekind is: to bring something into
existence involves difference - the desired object must appear as
a real thing; but the treatment of the thing will involve congruence -
to refer to the scaffolding is unstructural.
A rather clear instance of this occurs in the usual axioms
of set theory: sets are intentionally described (difference), but
they are identified if they have the same extension (congruence).
In the conversation between Simpson and Pratt, at one point
Pratt remarks that it is usual these days for sets of axioms to be
identified if they have the same consequences. And various people
similarly remarked that languages modulo theories are identified if
they have the same expressive power. Etc.
I am not at all sure if I correctly understand the importance
for Simpson of this conversation, but he does say that it touches on
important foundational issues, so to avoid the risk of misrepresenting
his position I'll just say that for me the issue is that once again
difference, structure, and congruence simply go together, BUT
to get a foundation for the existence of things one might be
led to emphasize differences over congruences since differences are
what one ultimately points to when invoking existence.
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