December 28, 2002

Reuben Hersh

Department of Mathematics and Statistics

University of New Mexico

Albuquerque, New Mexico 87131-1141

Re: What Is Mathematics, Really?

Dear Professor Hersh:

Thank you for writing this book! I got a B.A. in math from U.C. Berkeley and an M.A. from Harvard, but then I got disillusioned with math and got a Ph.D. in Psychology. I wanted to "save the world", and didn't think math could help me do that. I was interested in math for its beauty, and teaching, and had no interest in research. Apparently, there is no place for such people (teaching) in our universities.

Your book gave me a good overview of the philosophy of mathematics. However, before I read it, I wondered if the word "Really?" was a bit presumptuous. My suspicion was borne out. Since you were very critical of many past philosophers, I hope that you enjoy debate, and that you won't be too upset if I tell you my honest feelings.

I fully appreciate all the obvious work and thought that went into your book, but I believe that your conclusions are clearly wrong. The "Blind Men and the Elephant" parable is very apropos. Many people have taken a narrow viewpoint of math, as you so well described. But just as your "humanistic" view takes a wider view than that of your predecessors, it is still anthropocentric (don't feel bad; you share that view with about 6 billion other people!).

On page 139 you say "Mathematics is a human product." Again (p.206): "Mathematics is an activity of the community. It doesn't exist apart from people." On page 223: "There's no thinking without a brain." (For a different view, see Donald Griffin, Animal Thinking.) On page 225: "communication with other humans is a precondition to mathematical knowledge". On page 236: "I [identify] mathematics as the study of certain social-historic-cultural objects." And on page 248: "Dealing with mathematics … can only be done in social-cultural-historic terms. This isn't controversial. It's a fact of life."

It is very easy to prove that you are wrong. Take a hungry chimpanzee and hold two bananas in front of him. Then turn your back on him, hide one banana in your shirt, and hand him the other banana. I predict that he will look for the missing banana -- because he has counted them! Try the experiment with larger numbers of bananas. As the numbers increase, his ability to detect the absence of one will decrease. But the same would be true of a human subject!

It's quite obvious that we aren't the only species that can do mathematics. Orb-weaving spiders are quite good at geometry. So are nest-building birds and squirrels. Recently I observed some of their nests withstanding a windstorm. Many animals seem to be able to approximate straight lines and other efficient travel paths. It is easy to find other examples.

Your "humanistic" view of mathematics completely fails to explain how animals (or very young children) can do mathematics. It also fails to explain mathematics's most salient feature: the unusual consensus among its users. Most aspects of culture display wide variations among people. We can't agree on why 1 + 1 = 2 (your book catalogued numerous different philosophies of mathematics), but we all (including animals) agree that it's true. This is a pretty strong indication that math is based on something external, not only to our culture, but even to our species!

In The Spirit in the Gene -- Humanity's Proud Illusion and the Laws of Nature, Reg Morrison argues that religion, because it is universal among cultures, must be genetically based. I would like to suggest that humans are, above all, biological organisms, and cannot be understood without taking that into account. Our brains come with a built-in ability to recognize certain patterns (e.g. faces). Most likely, that includes some understanding of counting and numbers. I don't see how we can explain our understanding of mathematics without enlisting knowledge of biological chemistry, physics, physiology, and genetics. I feel a new "ism" coming on. J

Sincerely yours,