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

eBook ISBN: 978-3-319-21473-3

- Introduction: Ernest Davis.
- Chapter 1: Ursula Martin & Alison Pease, ``Hardy, Littlewood, and polymath.''
- Chapter 2: David H. Bailey & Jonathan Borwein, ``Experimental computation as an ontological game changer: The impact of modern mathematical computation tools on the ontology of mathematics.''
- Chapter 3: Philip Davis, ``Mathematical Products.''
- Chapter 4: Ernest Davis, ``How Should Robots Think about Space?''
- Chapter 5: David Berlinski ``Mathematics and its Applications.''
- Chapter 6: Jody Azzouni "Nominalism: The nonexistence of mathematical objects"
- Chapter 7: Donald Gillies, ``An Aristotelian Approach to Mathematical Ontology.''
- Chapter 8:
Jesper Lützen,
``Let
*G*be a group.'' - Chapter 9: John Stillwell, ``From the Continuum to Large Cardinals.''
- Chapter 10: Jeremy Gray, ``Mathematics at Infinity''
- Chapter 11: Jeremy Avigad, ``Mathematics and Language.''
- Chapter 12: Micah Ross. ``Mathematics as Language.''
- Chapter 13: Kay O'Halloran ``Mathematics as Multimodal Semiotics.''
- Chapter 14: Steven Piantadosi, ``Problems in Philosophy of Mathematics: A View from Cognitive Science.''
- Chapter 15: Lance Rips. ``Beliefs about the Nature of Numbers''.
- Chapter 16: Nathalie Sinclair ``What kind of thing might number become?''
- Chapter 17: Helen Verran. ``Enumerated Entities in Public Policy and Governance.''

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 Leibniz, 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.