George Lakoff & Rafael Nunez on the Philosophy of Mathematics - more stuff for Les Schaffer
Jurriaan Bendien
bendien at tomaatnet.nl
Sun Jul 13 12:36:19 MDT 2003
Les,
In regard to what we were discussing, the following conclusions by Lakoff &
Nunez are interesting and pertinent:
"From a scientific perspective, there is no way to know whether there are
objectively existing, external mathematical entities or mathematical truths.
Human mathematics is embodied; it is grounded in bodily experience in the
world. Human mathematics is not about objectively existing, external
mathematical entities or mathematical truths. Human mathematics is primarily
a matter of mathematical ideas, which are significantly metaphorical in
nature. Mathematics is not purely literal; it is an imaginative, profoundly
metaphorical enterprise. There is no mathematics out there in the physical
world that mathematical scientific theories describe. (...) Mathematics,
being embodied, uses general mechanisms of embodied cognition and is
grounded in experiences in the world. Therefore, it is not arbitrary ! That
means: mathematics is not purely subjective; mathematics is not a matter of
mere social agreement; [and] mathematics is not purely historically and
culturally contingent. This is not to say [however] that historical and
cultural factors don't enter into mathematics. They do, in a very important
way... [...] Mathematics is creative and open-ended. By virtue of the use of
conceptual metaphors and conceptual blends, present mathematics can be
extended to create new forms by importing structures from one branch to
another, and by fusing mathematical ideas from different branches. Human
conceptual systems are not monolithic. They allow alternative versions of
concepts and multiple metaphorical perspectives of many (though by no means
all !) important aspects of our lives. Mathematics is every bit as
conceptually rich as any other part of the human conceptual system.
Moreover, mathematics allows for alternative visions and versions of
concepts. There is not one notion of infinity but many, and not one formal
logic but tens of thousands, not one set theory or geometry or statistics
but a wide range of them - all mathematics ! [...] The ortrait of
mathematics has a human face." (George Lakoff and Rafeal Nunez, Where
Mathematics Comes From; How the Embodied Mind brings Mathematics into Being,
NY: Basic Books, 2000, p. 365, 379).
I think this view is perfectly compatible with what Marx and Engels had to
say about it in the 19th century, and indeed confirms the validity of their
approach in the area of philosophy and mathematical theory, whatever be
their specific errors or by now outmoded utterances. I have already
mentioned my example of the person who sees 2 people where another person
sees 3 people, even although they observe the same thing, the difference
being that one notices that one of the observed people is a pregnant woman,
and the other doesn't, or takes a different view of what a "person" is. This
just shows in my opinion that mathematics is inseparable from
conceptualisation and categorisation processes, which reach beyond number
theory and point beyond mathematics.
All this is lost on Melvin P., who prefers a monolithic Stalinist metaphor
asserted on the point of a bayonet.
Regards
Jurriaan
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