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


Ernest Davis & Philip J. Davis


Springer. In preparation, to appear fall 2015.     Book web page on

Hard-cover ISBN: 978-3-319-21472-6
eBook ISBN: 978-3-319-21473-3


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.


From the Introduction

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 they ``come into being as we probe'', as Michael Dummett suggested.

Most of us who have done mathematics have at least the strong impression that the truth of mathematical statements is independent both of human choices, unlike truths about Hamlet, and of the state of the external world, unlike truths about the planet Venus. Though it has sometimes been argued that mathematical facts are just statements that hold by definition, that certainly doesn't seem to be the case; the fact that the number of primes less than N is approximately N/ \log(N) is certainly not in any way an obvious restatement of the definition of a prime number. Is mathematical knowledge fundamentally different from other kinds of knowledge or is it simply on one end of a spectrum of certainty?

Similarly, the truth of mathematics --- like science in general, but even more strongly --- is traditionally viewed as independent of the quirks and flaws of human society and politics. We know, however, that math has often been used for political purposes, often beneficent ones, but all too often to justify and enable oppression and cruelty.* Most scientists would view such applications of mathematics as scientifically unwarranted; avoidable, at least in principle; and in any case irrelevant to the validity of the mathematics in its own terms. Others would argue that ``the validity of the mathematics in its own terms'' is an illusion and the phrase is propaganda; and that the study of mathematics, and the placing of mathematics on a pedestal, carry inherent political baggage. ``Freedom is the freedom to say that two plus two makes four'' wrote George Orwell, in a fiercely political book whose title is one of the most famous numbers in literature; was he right, or is the statement that two plus two makes four a subtle endorsement of power and subjection?

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 research; 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.

* The forthcoming book Weapons of Math Destruction by Cathy O'Neil studies how modern methods of data collection and analysis can feed this kind of abuse.

Note on the order of the chapters