FOM: What label to apply to this philosphy?
Dean H. Judson
djudson at unr.edu
Mon Jan 5 16:27:27 EST 1998
I've read Reuben Hersh's comments (and follow-ups) with much interest.
Yet, I find a position that sounds good to me but that doesn't fit well
with the existing typologies for what "math" is.
How about this (by way of citing sources, I'm drawing pretty heavily from
John Casti's ideas, with some from Hofstadter):
1) Mathematical (i.e. Number) systems are alphabets, with axioms and rules
of construction. In the usual way, we construct new strings from old ones
and call the new strings "theorems." (If we stopped here, we'd just call
ourselves formalists. But we don't.)
2) Occasionally, somebody makes an "operational" correspondence between
the strings in our system and something in the real world. I use the word
"operational" intentionally--to imply that somebody makes an "operational
definition" of a string. (e.g., "1" means 1 _finger_, "a+b" means put "a"
fingers together with "b" fingers, etc.)
3) Using this "operational" definition, our "theorems," which used to be
only abstract, now become _predictions_ about what we should see in the
world, if our model is true. With surprising regularity, these
predictions work. (Not always, however, 1+1(mod 2) does not make an
accurate prediction about what one does when one uses the above
operational definitions on ones fingers.)
Now, this isn't intuitionism; it isn't formalism; it surely isn't
platonism. Yet, it doesn't really seem like the word "humanism" (or
"social constructivism") fits, either. The "reality" (and usefulness) of
mathematics is in the encoding step. The logical power is in the
formalization and construction of new theorems.
It strikes me that, historically speaking, I've got the order backward: I
went from model to operational encoding to prediction and then
(presumably) back to model. Historically, it looks like humans started
with operational encoding, went to model and then back to encoding.
However, I like my ordering better, because it acknowledges the important
role of the model builder.
So, the questions: What "label" would one put on this philosophy? In
what ways do you find this point of view unsatisfying? Are there any
supporters for this notion in this community?
THANKS for your thoughts--my signature line follows:
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Dean H. Judson, Ph.D., Nevada State Demographer
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Statistics and Research Methods Laboratory and
Social Psychology Ph.D. Program
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