Mathematical Platonism
David Terr, PhD
1/9/14
1 PLATONIC FORMS [1]
Plato believed that the objects we perceive are but approximations of “ideal” or “pure” forms, which
exist somewhere outside of this world. He believed this due to the fact that real-world objects are
changing, whereas he believed that ideal forms are eternal, i.e., they exist outside time and are
unaffected by the passage of time. Thus, for instance, a table is not a perfect form since it can be
moved, rotated, collapsed, or destroyed, whereas the ideal table, of which any other table is an
approximation, is perfect and unchanging.
2 MATHEMATICAL FORMS
Plato’s ideal forms apply to everything, including abstract things, such as numbers and other
mathematical concepts like shapes. For instance, the number 6 is a Platonic form which represents, say,
six cups. Similarly, a sphere is a platonic form which represents the shape of a circular object, say a ball.
Clearly, mathematical Platonic forms, such as the number 6 or a perfect sphere, do not exist as realworld objects. Nevertheless, they are quite useful concepts.
3 MATHEMATICAL PLATONISM
Although I am by no means a strict Platonist, I do consider myself a mathematical Platonist. There are
several reasons for this. First, I am awed by the fact that mathematical truths are universal, i.e.,
independent of the many cultures who have discovered them independently of one-another. Second,
mathematics seems to have at its heart some very deep truths. For instance, in the 19th century, Peano
discovered 10 very simple axioms from which all of arithmetic may be built up. [2] In addition, Euclidean
geometry is built up from just 5 axioms. Third, a wide variety of pure mathematical results, which at first
had no practical applications whatsoever, later found very important applications in a wide variety of
fields, including physics, engineering, and computer science. For instance, a variety of orthogonal
polynomials, such as Legendre, Hermite, and Laguerre polynomials, were discovered in the late 19th
century with no applications in mind, but later turned out to be indispensable tools in quantum
mechanics. In addition, modular arithmetic, a branch of number theory, which is usually considered to
be the purest area of mathematics, have recently been applied to cryptography, in particular the RSA
cryptosystem, which keeps our bank accounts and personal information safe.
3.1 MATHEMATICAL UNIVERSE HYPOTHESIS (MUH) [3]
Also known as the Ultimate Ensemble, this is a speculative Theory of Everything (TOE) which sees our
external physical reality as a mathematical structure.
4 MATHEMATICAL REALISM
Very much in contrast with mathematical Platonism is the mathematical realism. This philosophy asserts
that mathematical concepts are pure mental constructions. Although this captures the essence of how
we learn math, I think it fails to grasp the beauty and depth of the subject.
5 REFERENCES
[1] http://en.wikipedia.org/wiki/Theory_of_Forms
[2] http://en.wikipedia.org/wiki/Peano_axioms
[3] http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis