[OPE] A dialectical view of mathematics...

From: Jurriaan Bendien <adsl675281@tiscali.nl>
Date: Thu Feb 19 2009 - 13:11:01 EST

In Popper this determination to insulate science from the human foibles of
scientists produced the absurd notion of 'knowledge without a knowing
subject'. Dialectical philosophy, in contrast, typically recognises that
human minds, however fallible, are the only available vehicles for the
greater rationality of science. All that the dialectical philosopher need
insist on is that the general direction of a historical development is best
explained by an analysis of the concepts governing that development. This
assumption may be implausible in military, political or economic history,
but it is in just this respect that
science and mathematics differ from most other human activities.

Notice that the object of study is mathematical development rather than
truth or validity. Human rationality reveals itself in speech and action. It
is not a tenseless state that can be inspected without reference to time and
change. Similarly, the dialectical philosopher of mathematics seeks
rationality and integrity in the development of mathematics. Does the
mathematical community reach the right choices for the right reasons?

Remember we noticed that even on a deductivist picture of mathematical
argument it is necessary to choose the axioms. Real mathematicians have to
choose problems, techniques and proof-strategies. The
mathematical community has to decide when to treat a result as proven and
worthy of celebration. These choices (to reiterate) are neither arbitrary
nor 'simply subjective'. They are, however, time-bound. I choose a problem
now in the belief that it will be fruitful in the future. The community
declares a theorem proven, confident that it will not have to withdraw that
status later. When a result is judged to be significant, it is deemed to
have staying power.

Clearly the dialectical philosophy of mathematics sketched here owes
something to Hegel. This debt is compatible with the spirit of Lakatos, who
named Hegel as one of the three 'ideological sources' of the Ph.D. thesis
that eventually became "Proofs and Refutations".

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Received on Thu Feb 19 13:12:51 2009

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