Mathematics, Substance and Surmise: Views on the Meaning and Ontology
Ernest Davis & Philip J. Davis
Springer. In preparation, to appear fall 2015.
A collection of 17 essays, addressed to a general readership,
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 and Language.''
"Nominalism: The nonexistence of mathematical objects"
David H. Bailey
``Experimental computation as an ontological
game changer: The impact of modern mathematical computation tools on the
ontology of mathematics.''
``Mathematics and its Applications.''
``How Should Robots Think about Space?''
``An Aristotelian Approach to Mathematical Ontology.''
``Mathematics at Infinity''
``Let G be a group.''
``Hardy, Littlewood, and polymath.''
``Problems in Philosophy of Mathematics: A View from Cognitive Science.''
``Beliefs about the Nature of Numbers''.
``Mathematics as Language.''
``What kind of thing might number become?''
``From the Continuum to Large Cardinals.''
``Enumerated Entities in Public Policy and Governance.''
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
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
of themselves, from the sea of mathematical thinking, or perhaps
mathematicians prod them into existence, as Michael Dummettt
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 ; 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.