Mathematics, Substance and Surmise: Views on the Meaning and Ontology of Mathematics


Ernest Davis & Philip J. Davis


Springer. In preparation, to appear fall 2015.


A collection of 17 essays, addressed to a general readership, discussing questions of mathematical ontology --- What is a mathematical object? What do mathematical statements assert? What is the nature of mathematical truth? --- from a variety of disciplinary perspectives, including mathematics, philosophy, computer science, history, sociology, semiotics, and psychology.



Mathematics discusses an enormous menagerie of mathematical objects: the number 28, the regular icosahedron, the finite field of size 169, the Gaussian distribution, and so on. It makes statements about them: 28 is a perfect number, the Gaussian distribution is symmetric about the mean. Yet it is not at all clear what kind of entity these objects are. Mathematical objects do not seem to be exactly like physical entities, like the Eiffel Tower; nor like fictional entities, like Hamlet; nor like socially constructed entities, like the English language or the U.S. Senate; nor like structures arbitrarily imposed on the world, like the constellation Orion. Do mathematicians invent mathematical objects; or posit them; or discover them? Perhaps objects emerge of themselves, from the sea of mathematical thinking, or perhaps mathematicians prod them into existence, as Michael Dummettt suggested.

Concomitant with these general questions are many more specific ones. Are the integer 28, the real number 28.0, the complex number 28.0 + 0i, the 1 x 1 matrix [28]; and the constant function f(x)=28 the same entity or different entities? Different programming languages have different answers. Is ``the integer 28'' a single entity or a collection of similar entities; the signed integer, the whole number, the ordinal, the cardinal, and so on? Did Euler mean the same thing that we do when he wrote an integral sign? For that matter, do a contemporary measure theorist, a PDE expert, a numerical analyst, and a statistician all mean the same thing when they use an integral sign?

Such questions have been debated among philosophers and mathematicians for at least two and a half millennia. But, though the questions are eternal, the answers may not be. The standpoint from which we view these issues is significantly different from Hilbert and Poincaré, to say nothing of Newton and Leibnitz, Plato and Pythagoras, reflecting the many changes the last century has brought. Mathematics itself has changed tremendously: vast new areas, new techniques, new modes of thought have opened up, while other areas have been largely abandoned. The applications and misapplications of mathematics to the sciences, engineering, the arts and humanities, and society have exploded. The electronic computer has arrived and has transformed the landscape. Computer technology offers a whole collection of new opportunities, new techniques, and new challenges for mathematical resesarch; it also brings along its own distinctive viewpoint on mathematics.

The past century has also seen enormous growth in our understanding of mathematics and mathematical concepts as a cognitive and cultural phenomenon. A great deal is now known about the psychology and even the neuroscience of basic mathematical ability; about mathematical concepts in other cultures; about mathematical reasoning in children, in pre-verbal infants, and in animals.

Moreover the larger intellectual environment has altered, and with it, our views of truth and knowledge generally. Works such as Kuhn's analysis of scientific progress and Foucault's analysis of the social aspects of knowledge have become part of the general intellectual currency. One can decide to reject them, but one cannot ignore them.