Mazur positions himself securely outside both the platonist and non-platonist camps, but is clear that he thinks that any adequate philosophy of mathematics must take into account the deeply perceived experience of working mathematicians. Regarding platonism, Mazur views it as a kind of irrationalism, akin to blind faith:
If we adopt the Platonic view that mathematics is dis-Later on, Mazur advices platonists to learn from poets and prophets how to spread the faith in transcendent mathematical forms. This way of equating platonism with irrationalist religion is, I believe, a caricature of platonistic philosophy, and an oversimplification. Many writers have presented several different rigorous rational arguments for platonism, so it is not an area where we must completely abandon rationality. Have these rational arguments proved platonism? Certainly not, but neither have the opposites of platonism been proved. Such is the nature of philosophical questions, but just because they are not easily answered once and for all doesn't mean that we have to give up our own thinking and resort to prophets. Almost all philosophical positions are in the end a matter of faith, but it is still possible to argue rationally about philosophical questions, and thus base one's faith on a rational ground.
covered, we are suddenly in surprising territory, for this
is a full-fledged theistic position. Not that it necessarily
posits a god, but rather that its stance is such that the only
way one can adequately express one’s faith in it, the only
way one can hope to persuade others of its truth, is by
abandoning the arsenal of rationality, and relying on the
resources of the prophets.
An interesting part of Mazur's essay consists of a list of advice for people who want to write about platonism versus non-platonism. As I am arguing for platonism, I will focus on what Mazur has to say to platonist writers. In essence, Mazur rightly observes that platonism implies a sort of disregard for rigorous proof. This is because platonists believe that mathematical facts are true irrespective of whether we can prove them or not, and that these truths are part of a platonist 'landscape' which it is our task to map. The challenge for the platonist, although Mazur does not say it explicitly, is thus to account for the demand for rigorous proof in mathematics while still maintaining that this demand is not strictly necessary. I would like to sketch an answer to this:
The method of rigorous proof is a tradition and social construction if there ever was one. The greatest minds of mathematics, from Euclid (who promoted the rigorous method, but nevertheless started his book with a proposition with a non-rigorous proof), through to Euler, Leibniz, and Newton, to modern day (not contemporary) Italian algebraic geometers and mathematical physicists (think Dirac's delta function and Feynmann's path integral) have worked on mathematics in a non-rigorous way. Clearly mathematics can be done in a not necessarily rigorous way, and fruitfully so! The insistence on rigour was, I guess, born some time around the time of Gauss and Abel, and developed in the hands of Weierstrass and others. Present day mathematics works in this tradition, even though one can observe cultural variations, e.g. in Russian mathematical exposition. Rigour has an 'hygienic' advantage in minimizing the number of incorrect statements in the literature, but it also no doubt slows down mathematical discovery. Where would Euler and Leibniz have been today if they had insisted on working rigorously?
Mathematics was discovered or created long before the idea of rigour, and it is not impossible that we may one day see the resurgence of not-necessarily rigorous mathematics. This has indeed been suggested, for example in the famous debate initiated by Jaffe and Quinn.
This is however not the end of the story. The platonist can argue that rigour, while not strictly necessary in the discovery of truth, is nevertheless our best tool for pinning down the exact nature of the mathematical forms, and so is valuable in that it gives us more exact knowledge.
Mazur demands that those in opposition to platonism must thoroughly account for the mathematician's perception of the transcendence and independence ("autonomy even") of mathematical concepts. I have written exactly the same thing in earlier posts on this blog. Whether non-platonists can ever convince us that we are in a deep illusion induced by spending much time thinking about abstract objects, and to lay bare some psychological mechanisms through which this illusion comes about, only time will tell. To me, it seems like a long shot. Until then those agreeing with Mazur have much reason to believe in mathematical platonism.
4 comments:
Well, higly interesting paper when Mazur finishes off by stating that non-platonists must "thoroughly account for the mathematician's perception of the transcendence and independence ("autonomy even") of mathematical concepts".
Good point, but not unanswered by neurologists. More and more studies pinpoint the nature of the heureka moment and neurologist get more and more understanding of how these complex thinking processes work. There is a lot of uncounscious thinking that lead up to the insight phenomenon and that is part of why it is hard to gauge. So just using your own experience of transcendence as an argument is to me futlie and not very insightful.
But do not think that Platonism and psychologism disagree on how mathematical sentences should be interpreted—they both state that mathematical sentences should be interpreted as being statements about abstract objects. but they disagree on the metaphysical question of whether abstract objects exist. it can be argued that the concept of an abstract object is so unclear that there is no objective, agreed-upon condition that would need to be satisfied in order for it to be true that there are abstract objects. I could easliy imagine a future where this debate is unsolved, but significatly put in perspective, however, it does in no way imply that platonism (as Plato described it) can be utilized in any other form discussing the rest of our experiences of life. For nothing we have seen thus far supports such a view.
Research on the insight moment of human beings:
- An Eye Movement Study of Insight Problem Solving. Guenther Knoblich
et al. in Memory & Cognition, Vol. 29, No. 7, pages 1000–1009;
October 1, 2001.
- Sleep Inspires Insight. Ullrich Wagner et al. in Nature, Vol. 427,
pages 352–355; January 22, 2004.
- New Approaches to Demystifying Insight. Edward M. Bowden et al. in
Trends in Cognitive Sciences, Vol. 9, No. 7, pages 322–328; July 2005.
Thanks for the comment! It is difficult to prove the existence of colours to someone who was born blind, so unfortunately the mere experience of mathematicians is never going to suffice as an argument on its own. However, there are other arguments such as the unreasonable effectiveness of mathematics in explaining natural phenomena, and the ubiquity of mathematical structures in biological organisms. Psychologism cannot explain this, unless one also assumes that all minds are connected and that physical reality itself is a product of the mind.
Regarding the meaning of existence of abstract objects, I think that Plato, Aristotle, and those classical guys already gave a good answer. First there is relative existence: A exists relatively to B if A cannot exist without the existence of B. We know that something exists (for example our own consciousness). Let us also for simplicity assume the common sense view that things exist in the universe at least roughly the way science describes them. Then the claim of platonism is that these things in our universe exist relatively to some forms, and that these forms are themselves not existing relative to physical objects or human minds.
Aristotle would not agree with the last bit, but that's his problem.
I'm not sure I understand your two last sentences. I have tried to illustrate how platonism pervades many parts of human life. I have mentioned music, and in sciences such as theoretical physics, platonism is *the* underlying, often unspoken, paradigm.
Yes empirical sciences can in some senses be said to utilize mathematics (Quine), but there are arguments against this view as well, that most or all empirical sciences can just drop mathematics and still work.
It is hard to make any smart claims about consciousness since no one has got any working definitions of it right now, but before I completely adopt mathematical Platonism I would like read more studies of human pattern recognition or developments of the so called epistemological argument. Where in the body lies the purported connection with transcendent, abstact objects?
Yes, I've heard of these guys who tried to show that maths is dispensable in science. What I read didn't convince me for two reasons. One of them replaces numerical maths by "qualitative" geometric arguments, but betweenness and such notions are also mathematical, part of topology or some other geometry. Not all maths is quantitative. The other one says that science doesn't need the existence of maths objects, only the *assumptions* that so-and-so exists. I think that most people will agree that this doesn't change anything.
Let me know if you find anything interesting about pattern recognition or any other cognitive aspects of maths.
The human connection with the transcendent objects, I would say, is 'dislocated'. See my post on the problem of interaction. However, I think it's fair to say that we grasp various aspects of the forms via all parts of ourselves, that is, both via the senses and through the mind's capability for abstraction.
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