A rchive Date
[ 22-05-2000 ]
Category
[ Philosophy ]
sub-Categoy
[ Greek ]
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[http://www.edge.org/discourse/index.cgi?OPTION=VIEW&THREAD=reuben-hersh/2-10-97/hersh
Plato and Pi
I have to say that I disagree with almost everything that Reuben Hersh says. I can start with what I agree with, which is that the platonist and formalist schools of philosophy of mathematics do not capture what mathematics is.
It must be said, however, that they are not stupid, or obviously wrong, I think that my disagreement with Platonism comes from two things: first from my philosophical commitment to the idea that the world we see is all there is, and that everything we see must be explained in terms of a network of relationships among real things. This leaves no place for a realm of real, eternal forms that transcends the particulars of the world, as well as no place for a view of the universe as if from outside of it. It also leaves no place for a platonic realm of mathematical form.
The second reason I disagree with platonism is that I think it is insufficient to make sense of the mathematical structures that arise in biology.
It is one thing to speak of every possible platonic solid, but should we think that every possible biological species, or every possible niche, or every possible ecology exists eternally in some realm of ideal platonic biology? What about every possible way of earning a living in human society? Stuart Kauffman has been arguing that it may not be possible to list these kinds of things in advance, and I am tempted to think he may be right. There are lots of things that apparently cannot be classified in mathematics, like algebras or knots or four dimensional manifolds. For this reason, I suspect that Platonism will eventually come to be seen as insufficient to encompass the variety of possible mathematical structures. The point is that I believe that novelty is both possible and important, there are novel structures being discovered all the time, both by natural selection and human intelligence, and some of these are mathematical.
Formalism is easier to put away; it was basically killed by Godel's theorem. So what then is mathematics?
I believe I understand the reasons why Hersh makes the move he does: that it is a shared construction of human beings, for that is some of it. There are an infinite number of possible mathematical structures, why some have been intensively thought about, while others were either thought uninteresting and most have not even been conceived of is a historical question. So historical and social questions may plausibly play some role in understanding why mathematics is as it is now. But this is not the same thing as to ask, what is a number, or what is mathematics.
I do not have an answer to this question that satisfies me, although I have thought a lot about it. In my opinion it is one of the really hard questions, like consciousness, or whether time might have begun, or might end. There are questions that I believe we cannot even conceive of satisfactory answers to given what we know presently. This does not mean they may not someday be solved-I think they may. There is a list of questions we are on the verge of solving, like the origin of life or the nature of space, that require twentieth century physics and mathematics, and that a nineteenth century person could not even have gotten started on.
Having said this, there are two thoughts that I find interesting when I try to think about what mathematics is.
The first is the observation that time may play an essential role because a mathematical paradox can become a feedback loop when time is introduced. Something cannot be both true and not true eternally, but it can be alternatively in time.
The second is the possibility that category theory may have profound implications for the question of what mathematics is, because it puts the emphasis exactly on relationships between different things. One might have looked down on category theory some years ago, but given the profound insights it has introduced into the relationships between different mathematical structures such as algebra and topology it seems very worth thinking about.
- Lee Smolin]
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