FOM: Objectivity of logical/mathematical truth?
barwise at phil.indiana.edu
Sun Dec 21 13:16:12 EST 1997
>if one believes (as I do) that
> mathematics is "socially constructed", how can logical/mathematical truth
> be objective in character? What we have to accept is that more or less
> objective communication is possible on a whole spectrum of socially
> constructed concepts from nationality, marriage, money, English
> grammar, the calendar, position in the university, to chess and
> mathematics. ...
> Sol Feferman
> Physicists don't try to figure out why anything exists. The main
> thing, the starting point for everything else, is this: there is a
> world, there is existence.
> Reuben Hersh
0. Some years ago, Lynn Arthur Steen described mathematics as
the science of pattern.
(I don't know if this originated with Steen, but I will call it Steen's
idea since that is where I first read it.) Devlin has recently published a
popular (and lovely) book on mathematics titled something like "The Science
of Pattern." To me Steen's idea is an attractive one.
1. Both numbers and geometrical objects can been seen as patterns among
physical things out there in the world. These particular patterns are
patterns among physical things, but are not the things themselves.
Patterns are thus abstract in accord with the way we think of mathematical
objects as abstract. And there are objective facts about these patterns,
these abstract objects.
2. One of the supposed mysteries about mathematics is its applicability.
How can matheamtics be so useful when it is so abstract? Steen's idea
makes the applicablity of mathematics less mysterious. For the
mathematical objects are patterns, some of which are patterns about things
in the physical world. So of course a science of patterns is going to shed
light on the world.
3. Steen' s idea also make room for people, in several ways. First, while
patterns are inherent in the things they are patterns among, they also
depend in SOME sense on pattern recognizers. People bring to the endeavor
certain abilities, abilities that let them recognize uniformities in the
world, these uniformities being patterns. Dogs recognize different
patterns that people do. Both sorts of patterns are all there in some
sense, independent of people and dogs, but in another sense patterns depend
on the pattern recognition abilities of some sort of agent.
4. Now there are many patterns that exist in things other than the world of
brute physics. Sol's list is a good one. There is, for example, language,
a particularly human artifact. The mathematics of language has to do with
patterns in syntax, semantics, etc., of human sound making.
5. The things in Sol's list all depend on people and in fact on society.
To the extent that we study patterns among such things, we are studying
abstract objects whose existence depends on the existence of society. But
it does not seem to me that that means that the patterns themselves are
socially constructed, nor are the truths about them necessarily socially
6. It does seem to be that there is another sense, though, in which social
forces are at work. For one thing, society exerts forces that causes us to
look in this direction rather than that, to discover these patterns rather
than those. Some of us choose to look at patterns among patterns among
patterns. Higher mathematics, metamathematics, ... But the computer age
is causing a lot of people to shift attention to patterns in computer
systems. More importantly, though, whether it is pure or applied, we
discover and explore patterns together. Science is a collaborative
enterprise, not a private one, and the science of patterns is not unlike
other science in this regard. To me this is the sense in which, in Sol's
phrase, `mathematics is "socially constructed"'.
7. While recogniize this aspect of mathematics, I would never choose to
claim that `mathematics is "socially constructed"' myself because it
suggests the black plague of postmodernism, which is far from what I think
Sol had in mind.
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