I’m not sure what led me to David Mumford’s Why I am a Platonist, which appeared in a 2008 issue of the European Mathematical Society (EMS) Newsletter, but I’m happy I found it. David Mumford is currently Professor Emeritus at Brown and Harvard Universities. The EMS piece is a clear and straightforward exposition of Mumford’s Platonism, which he defines in this way:
The belief that there is a body of mathematical objects, relations and facts about them that is independent of and unaffected by human endeavors to discover them.
Mumford steers clear of the impulse to place these objects, relations, and facts somewhere, like outside time and space. He relies, instead, on the observation that the history of mathematics seems to tell us that mathematics is “universal and unchanging, invariant across time and space.” He illustrates this with some examples, and uses the opportunity to also argue for a more multicultural perspective on the history of mathematics than is generally taught.
But Mumford’s piece is more than a piece on what many now think is an irrelevant debate about whether mathematics is created or discovered. As the essay unwinds, he begins to talk about the kind of thing that, in many ways, steers the direction of my blog.
So if we believe that mathematical truth is universal and independent of culture, shouldn’t we ask whether this is uniquely the property of mathematical truth or whether it is true of more general aspects of cognition? In fact, “Platonism” comes from Plato’s Republic, Book VII and there you find that he proposes “an intellectual world”, a “world of knowledge” where all things pertaining to reason and truth and beauty and justice are to be found in their full glory (cf. http://classics.mit.edu/Plato/republic.8.vii.html).
Mumford briefly discusses the Platonic view of ethical principles and of language, making the observation that all human languages can be translated into each other (“with only occasional difficulties”). This, he argues, suggests a common conceptual substrate. He notes that people have used graphs of one kind or another to organize concepts and proceeds to argue that studies in cognition have extended this same idea into things like semantic nets or Bayesian networks, with the understanding that knowledge is the structure given to the world of concepts. Mathematics, then, is characterized as pure structure, the way we hold it all together.
And here’s the key to this discussion for me. Mumford proposes that the Platonic view can be understood by looking at the body.
Brian Davies argues that we should study fMRI’s of our brains when think about 5, about Gregory’s formula or about Archimedes’ proof and that these scans will provide a scientific test of Platonism. But the startling thing about the cortex of the human brain is how uniform its structure is and how it does not seem to have changed in any fundamental way within the whole class of mammals. This suggests that mental skills are all developments of much simpler skills possessed, e.g. by mice. What is this basic skill? I would suggest that it is the ability to convert the analog world of continuous valued signals into a discrete representation using concepts and to allow these activated concepts to interact via their graphical links. The ability of humans to think about math is one result of the huge expansion of memory in homo sapiens, which allows huge graphs of concepts and their relations to be stored and activated and understood at one and the same time in our brains. (emphasis added)
Mumford ends this piece with what is likely the beginning of another discussion:
How do I personally make peace with what Hersh calls “the fatal flaw” of dualism? I like to describe this as there being two orthogonal sides of reality. One is blood flow, neural spike trains, etc.; the other is the word ‘loyal’, the number 5, etc. But I think the latter is just as real, is not just an epiphenomenon and that mathematics provides its anchor. (emphasis added)
David Mumford now studies the mathematics of vision. He has a blog where he discusses his work on vision, his earlier work in Algebraic Geometry, and other things. In his introductory comments to the section on vision he says the following:
What is “vision”? It is not usually considered as a standard field of applied mathematics but in the last few decades it has assumed an identity of its own as a multi-disciplinary area drawing in engineers, computer scientists, statisticians, psychologists and biologists as well as mathematicians. For me, its importance is that it is a point of entry into the larger problem of the scientific modeling of thought and the brain. Vision is a cognitive skill that, on the one hand, is mastered by many lower animals while, on the other hand, has proved very hard to duplicate on a computer. This level of difficulty makes it an ideal test bed for theorizing on the subtler talents manifested by humans.
The section on vision provides a narrative that describes some of the hows and whys for particular mathematical efforts that are being used. And each one of these disciplines has its own section with links to references.
A 2011 post of mine was inspired, in part by the idea that what Plato was actually saying is consistent with even the most brain-based thoughts on how we come to know anything. I took note of a few statements from what has been called the simile of the sun.
And the power which the eye possesses is a sort of effluence which is dispensed from the sun
Then the sun is not sight, but the author of sight who is recognized by sight
And the soul is like the eye: when resting upon that on which truth and being shine, the soul perceives and understands, and is radiant with intelligence; but when turned toward the twilight of becoming and perishing, then she has opinion only, and goes blinking about, and is first of one opinion and then of another, and seems to have no intelligence.