From: Jurriaan Bendien <adsl675281@tiscali.nl>

Date: Thu Feb 19 2009 - 13:11:01 EST

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".

https://uhra.herts.ac.uk/dspace/bitstream/2299/456/1/100570.pdf

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Received on Thu Feb 19 13:12:51 2009

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