The title presents a fascinating question. I am a mathematics
teacher. I should be able to readily respond to the question. After
all, I do teach the subject. OK, my somewhat overly-simplified answer
is that mathematics is the language of our universe. Be you
evolutionist, creationist, or holder of some other belief, empirical
evidence overwhelmingly suggests that our universe is mathematically
designed. Still, if you are an intelligent reader (and I assume you
are), my response to the question is clearly unsatisfactory. Among
other things, it does not address questions such as "why does
mathematics work?" and "where does it come from?"
Despite my love and appreciation for the magnificent academic
discipline of mathematics, I do not have a good answer to the
question "what is mathematics?" Since the time of Plato,
mathematicians and philosophers have addressed this question.
Answers, many of them contradictory, have been produced for over two
thousand years, but none have survived the test of time and critical
analysis.

I believe it is possible that
students can gain a tremendous appreciation for mathematics if they
understand that the question "what is mathematics?" has been analyzed
and debated since the time of the Pythagoreans, a mystical cult that
surfaced around 550 B.C. I also believe that humans love and
appreciate a good mystery: The
very nature of mathematics is a mystery!

The purpose of this writing is to present a brief summary of types
of mathematical thought that have surfaced since the ancient Greeks
realized that humans have the mental capacity to reason. Prior to
doing this, it is important to note three discoveries that shocked
the mathematical world. These, and a few other discoveries, shattered
the historical beliefs of many intellectuals who thought that their
version of mathematics had a firm foundation. While a perfect analogy
is not possible, imagine yourself having purchased, and living in, a
luxurious tenth floor condominium in a very desirable location... and
then finding out the foundation of the building was unstable. The
shockers:

**Shocker #1:**

The discovery (invention?) of non-Euclidean geometries in the
19th century. Prior to this time, many schools of mathematical
thought had accepted the laws of Euclidean geometry (studied in
modern schools) as indubitable truths about the universe. Meaningful
geometries that did not conform to the laws of Euclid's famous
historical work, the *Elements, *were discovered by Hungarian
**Janos Bolyai** (1802-1860) and Russian **Nikolai
Lobachevsky** (1792-1856). For example, in some geometries,
Euclid's famous *parallel postulate* ("Through a point external
to a line, there exists, in a plane, exactly one line parallel to the
given line") is false.

**Shocker #2.**

*Godel's Incompleteness Theorem: *In the 20th century,
German **Kurt Godel** (1907-1978) proved that consistency can
never be established by methods of mathematical proof. Every logical
system must contain statements that can't be proved. In other words,
a formal mathematical system could never prove its own consistency.
Something must be accepted on pure faith. (In modern day geometry,
the terms point, line, and plane are never formally defined. And, a
postulate such as "In a plane, two non-parallel lines intersect at
unique point" is accepted as true without proof. However, there are
simple non-Euclidean geometry models where this postulate is not
true.)

These discoveries, and some others that have not been mentioned,
are important in understanding why many theories about the nature of
mathematics have failed to pass the test of time. What follows is a
very brief summary of some historical schools of thought which made
attempts to answer the question "what is mathematics?"

==============================
**Platonism:**

Greatly influenced by the earlier Pythagoreans, **Plato
**(c.427-c.347) asserted that mathematics represents a separate
universe of abstract objects existing outside of what we know as time
and space. Mathematical objects (such as numbers) aren't created by
humans. They always existed. (An analogy might be represented by a
great piece of sculpture. The end result was already there. The
sculptor simply removed the excess marble.) Platonism asserts that a
mathematician is an empirical scientist who can only discover what is
already there. He or she can't invent new mathematics. Mathematical
truth possesses absolute certainty.

[PROBLEMS WITH THIS VIEW: Platonists never really explain how
flesh-and-blood mathematicians come to interact with the external
universe of mathematics. The discovery of non-Euclidean geometries
contradicted the "absolute truth" view of the
Platonists. Other historical interpretations reject the mysticism
surrounding an external world of mathematics.]

===================================
**Formalism:**

German mathematician **David Hilbert** (1862-1943) headed this
group. Formalists assert that mathematics must be developed through
axiomatic systems. Formalist and Platonists agree on the principles
of mathematical proof, but Hilbert's followers don't recognize an
external world of mathematics. Formalists argue that are no
mathematical objects until we create them. Humans create the real
number system by establishing axioms to describe it. All mathematics
needs is inference rules to progress from one step to the next. The
Formalists tried to prove that within the framework of established
axioms, theorems, and definitions, a mathematical system is
consistent. In the mid-twentieth century, formalism became the
predominant philosophical attitude in math textbooks.

[PROBLEMS WITH THIS VIEW: Godel's Incompleteness Theorem
contradicts the consistency philosophy of formalism. It has been
pointed out that accepted results from theorems were used before
axioms were created to establish the theorems. The modern emphasis on
the concrete and the applicable is not consistent with the formalist
philosophy that you don't really do mathematics until you state a
hypothesis and begin a proof.]

==============================
**Logicism:**

English mathematicians and philosophers **Bertrand Russell**
(1872-1970) and **Alfred North Whitehead** (1861-1947) cofounded
this school of thought. This school claims that mathematics is a vast
tautology. All of mathematics is derivable from principles of logic.
Many of the logistic ideas are similar to those of the formalists,
but the latter group does not believe that mathematics can be deduced
from logic alone. Among other things, the Logicists attempted a
logical construction of the real number system, whereas the
Formalists constructed it axiomatically. Logicism also uses
mathematical sets in its logical development.

[PROBLEMS WITH THIS VIEW: Logicism, despite many attempts, could
not successfully resolve paradoxes that arose in set theory. Godel's
Incompleteness Theorem was a death blow to the "math is a tautology"
philosophy expounded in *Principia Mathematica*, a monumental
work constructed by Russell and Whitehead.]

=========================
**Intuitionism** (sometimes called
**Constructivism**):

Building on the philosophies of **Immanuel Kant **(1724-1804),
Dutch mathematician **Luitzen Brouwer** (1881-1966) emerged as the
leader of this school of thought, which differs considerably from
those previously discussed. Intuitionists claim that mathematics
originates and thrives within the mind. Human minds intuitively
possess the forms of space and time. The natural numbers are given
intuitively, and they represent the fundamental datum of mathematics
from which springs all meaningful mathematics. Mathematical laws are
not discovered by studying nature; rather, they are found in the
recesses of the human mind.

[PROBLEMS WITH THIS VIEW: The intuitionist view doesn't give any
insight as to why mathematics works. We don't know how intuitive
knowledge is held in the brain. Mental representations of concepts
such as love, hate, etc. differ considerably from human to human. Is
it realistic to assume humans share the same intuitive view of
mathematics? Why do we teach mathematics if it is all intuitive?]

=========================
In conclusion, despite the fact that I have taught mathematics for
many years, I really cannot explain what it is, where it comes from,
why it works, or how we can make such amazing use of things that
could be classified as fictional
Mathematics is, to a great extent,
a mystery. As a thinker, I can only say that my personal philosophy
of mathematics takes bits and pieces from each of the historical
schools of mathematical thought. As previously mentioned, I see
mathematics as a language. As we become more and more proficient with
this language, we will better understand the universe that we
inhabit. I believe the Creator (and you may define Creator however
you wish) put mathematics out there for us to discover, but I don't
believe humans will ever discover all of the mathematics that exists.
The mystery of our existence in a mathematically designed universe is
what makes living interesting and exciting.