Jessica asked:

What is mathematics? Is it our minds quantifying phenomena? Is it something within the world itself?
Where could I start reading to learn more on the subject?

While The Oxford English Reference Dictionarydefines with certainty mathematics as the "abstract
deductive science of number, quantity, arrangement, and space" a comprehensive philosophical
definition of mathematics is not really possible. The common quadripartition of the views into
Platonism, Intuitionism, Logicism and Formalism is not always clear, therefore I try to strike the
problem at the roots.

Mathematics appears to be different from other types of investigation because of it's apparently high
degree of certainty. Theorems like '2 + 2 = 4' or 'There are infinitely many prime numbers', are often
taken as necessary truths and a priori knowledge.

The most basic entities in mathematics are probably numbers. What exactly are they? Are they
synthetic, i.e. man-made, or objective— existing in reality? If they are real, then where or what are
they?

In the view of realism,all mathematical objects (like numbers, triangles, theorems, etc.) have an
objective existence independent of human consciousness. They don't exist in our physical universe
(Have you ever seen an actual number?) but in a separate realm of mathematical entities, thus
raising mathematics to the level of metaphysics. It follows that mathematics is a process of discovery
not invention.

This position was reiterated by Kant, particularly with respect to Euclidean geometry. Unfortunately,
the almost immediate discovery of non-Euclidean geometry put paid to any such idea, because it
showed that the axioms on which Euclidean geometry was based are not all necessarily true.

Alternatives to realism fall into two groups:

1. Those who agree that mathematics has a subject-matter, but hold that mathematical objects are
   not independent of the mind, conventions, or language of the mathematician. The most common
   views in this camp take mathematical objects to be mental constructions,and so are varieties of
   idealism.Within this group, there are two possibilities.

*mathematics is subjective, so each person has his own mathematics. A problem with this subjective
idealism, then, is to account for the intersubjectivity of mathematical assertions and the apparent
objectivity of mathematics.

*mathematics is both mind-dependent and objective, following Kant in asserting that mathematics
deals with structures common to human minds. This variation accounts for the necessity and apriority
of our subject by holding that mathematics represents ways we mustthink, perceive, and apprehend.

2. The other alternative to realism is to deny the subject-matter of mathematical objects. To avoid
   general scepticism, the burden is to give an account of mathematics, and its role in the intellectual
   enterprise that does not presuppose an ontology. One common manoeuvre in this direction is to
   reconstruct mathematical assertions in modal terms. For example, instead of asserting that there is a
   natural number with a given property, one asserts that there might be a system that exemplifies the
   natural-number structure in which there would be a number with the property, or one asserts that it is
   possible to construct a certain item with a certain property.

Another alternative is to construe mathematics as fiction, much like what we read in novels. At least
at first sight, fictional discourse does not invoke any ontological commitments. But this theorist might
then try to give an account of the role of fictional mathematics in presumably non-fictional discourses,
like science!

Yet another alternative in this area is to construe mathematical truths as analytic, true in virtue of the
meanings of their terms. Again, such a view may not involve an ontology, and it does account for the
necessity and apriority of mathematics. The necessity of mathematics is semantic, or linguistic, and
mathematical knowledge therefore is knowledge of meaning. The problem, however, is to square this
view with mathematics as practised. One needs to give an explication of the meanings of
mathematical terminology according to which every mathematical truth is analytic!

Finally, Wittgenstein's attempt is to accommodate mathematics in terms of the normative social
practice inherent in a linguistic community. This denies that mathematics is necessary and a priori,
but it does account for the perceived necessity of mathematics. We simply have to accept the basic
mathematic principles because we cannot imagine living any other way.

Here's a small list of books, which hopefully will help you getting more familiar with this fascinating
subject:

*Philip J. Davis, Reuben Hersh: The mathematical experienceNew ed. Publ. by: Mariner Book,
Boston, 1998

*Douglas R. Hofstadter: Gödel, Escher, Bach. an eternal golden braid20th-anniversary ed. Publ. By
Penguin, London, 2000

*Ludwig Wittgenstein (1956) Remarks on the Foundations of MathematicsOxford: Blackwell

*Bertrand Russell Introduction to Mathematical PhilosophyLondon: George Allen & Unwin; New York:
The Macmillan Company New Ed. By Dover Publications

*Paul Benacerraf and Hilary Putnam (eds.), Philosophy of Mathematics2nd edn. (Cambridge, 1983).

Simone Klein