The minds of those who choose
to explore the philosophy of mathematics are introduced to
an elaborate menu of theories, that each has their own fundamental
approach towards answering the epistemological and ontological
issue of mathematics. There are a number of traditional approaches
of gaining insight to the status and truth of mathematics,
such as responding to mathematics as a formal game, or postulating
that mathematical knowledge is acquired through intuition.
This essay will seek to break down the core claims of a Platonist's
approach to mathematics, one which reacts to the basic issues
of mathematics by treating mathematical objects as real objects,
as well as point out some of the fundamental problems that
underlie its methodology, justifying a clear understanding
of the Platonistic tactic.
Traditionally, a Platonist is
one who is seeded by Plato's "Theory of Forms,"
a theory that takes a metaphysical and epistemological approach
to deducing the universe. The metaphysical angle of the Theory
of Forms divides reality into two levels, the "world
of forms" and the "world of physical objects."
Within the two categories are juxtaposed elements of existence,
the aspect that exists in the world of forms, and its translation
in the world of physical objects. For example, the world of
forms is described as eternal, existing outside space and
time, while the world of physical objects is temporal and
dependant on spacetime. The world of forms is also intangible,
and unchangeable, and defines true reality while the world
of physical objects is both sensible and changeable and is
dependant on the world of forms. In this context, the Platonist
relates the two worlds of reality by claiming that physical
objects are "imperfect copies" of perfect forms;
that physical objects "participate" in the physical
representation of the forms.
The epistemological aspect of the theory of forms states that
true knowledge is based on knowledge of the general principles
and universal properties of the forms. And that knowledge
of the forms is obtained through reason alone. Plato indicates
that knowledge of the world of forms is innate, but forgotten
at birth, and we spend our maturing lives relearning what
we have forgotten of the world of forms. Plato breaks down
the Forms into four basic categories: Moral and aesthetic
ideals (such as love or justice), Mathematical objects (such
as circles or triangles), Natural Kinds (such as animals,
or vegetables), and Natural "stuff" (such as water,
or air.) Contemporary Platonists in the philosophy of mathematics,
only adopt the second principle concerning mathematical objects.
With that background it is easier
to understand the foundation of Platonism, as it relates to
mathematics. There are about seven core claims of the Platonists
approach to mathematics, as well as two fundamental problems
with the seven core claims.
The first and foremost of the
seven claims in Platonism seeks to answer the ontological
issue concerning mathematics by claiming that Mathematical
objects are real, and exist independently of the world of
physical objects. This claim answers the epistemological issue
of math by indicating that mathematicians "discover"
things, as opposed to a constructivist's approach who would
claim that mathematicians "make" things. Adopting
this approach gives Platonists an advantage over the other
theories in the philosophy of mathematics, who construct lengthy
proofs to account for mathematical truth. The Platonist accepts
mathematical claims as either true or false, depending on
whether or not they refer to "correct" mathematical
objects. It argues that mathematical truth, like other physical
truths, can be understood through standard semantics; that
a mathematical statement is true if the properties of the
statement exist (are true.) For example, the semantics that
makes the statement "Joe is wearing a blue shirt"
true are the same semantics that make the statement "3x3=9"
true, in the Platonist's approach.
The next two claims of the Platonist
argument is that mathematical objects are partially "abstract",
and exist outside of space and time. However, Platonists state
that mathematical entities are not necessarily universal.
What this means is that mathematical entities are not necessarily
related to the process of abstraction, which shows the relation
between a particular and its universal quality. An example
of the abstraction process would be the account of a "round
rock" and its participation in the universal property
of "roundness." Platonists argue that mathematics
need not abide by the universal v. particular abstraction
process; that numbers can be particular but that does not
imply they are subject to the same abstraction process. That
in fact, mathematical objects exist beyond the bounds of space
and time in abstract form.
This leads us to the next of the
core claims of Platonism, which is the claim that mathematics
is "a priori", not empirical. "A priori"
means independent of the senses; that mathematical truths
and mathematical objects are intuited through the "mind's
eye" and cannot be seen but grasped, while empirical
knowledge is obtained through the senses. This concept of
the "mind's eye" will give Platonists another edge
over other conventional theories, as it explains the psychological
impulse to believe such statements as "5+10=15"
just as one is compelled to believe the sky is blue, which
is far from the psychological response one has to a conventionalists
game rule such as "rooks move on horizontally or vertically."
This does in no way mean that "a priori" knowledge
is certain knowledge. A reasonable Platonist is aware that
the "mind's eye" is fallible and susceptible to
folly.
The last of the core claims in
Platonism is that Platonism, more so than any other branch
of Mathematical philosophy is open to nonstandard investigative
techniques, and gives credit to mathematical truth represented
in the form of proofs, diagrams, computations, pictures, etc.
It states that there are many ways in which we can access
the mathematical realm, and attune our mathematical perception.
But with this last claim, the first problem with the Platonist
theory arises, which is the problem of access.
The "problem of access"
is one of the counterarguments against the plausibility of
the Platonist approach. This argument challenges the legitimacy
of the "minds eye" and questions its reliability
in identifying what most theorists refer to as the "causal
theory of knowledge". The causal theory of knowledge
states that to know anything, there must be a causal relation
between the knower and the thing known. The Platonist rebuttal
is to reject the causal theory of knowledge because perception
is relative to the individual, and can be dependant on many
other factors like cultural norms or social standards. So
identifying the causal relation becomes interpretive, and
describing how exactly we come to form mathematical knowledge
from the perception of mathematical objects remains a mystery.
The Platonist is prone to accept that we intuit 5 + 5 = 10
in the same manner we could claim to see a ball on the floor,
by simply if there is a ball on the floor or not. There is
also a drawback of the causal theory of knowledge for the
Platonist to grab hold of; that if one can show that knowledge
can be obtained without having a causal relation with the
instance that beset knowledge, there is reason to believe
the causal theory of knowledge is not an efficient way of
disproving Platonist theory.
The second problem with Platonist
theory is the problem of certainty. Platonists claim that
mathematical knowledge is 'a priori', a property that many
debate the certainty of. If one is to assume that a priori
knowledge is certain knowledge, the Platonist argument can
fall apart. Some theorems need lengthy proofs to explain its
accounts, and therefore cannot be a priori knowledge. But
a Platonist will argue that mathematics is descriptive and
not stipulative. That although the mind's eye is fallible,
just as normal sense experience is, mathematical intuition
is not a sense experience. That while the statement "my
foot is cold" or "that dog is brown" are stipulative,
in the sense that what makes them true is articulation of
the fact, the statement "the square root of 2 is an irrational
number" is true not because we say it is, but because
it is descriptive of the a priori instance where the square
root of 2 is irrational. In this sense a Platonist can argue
the problem of certainty by stating that a priori knowledge
does not have to be certain. That lengthy proofs provide concepts
of formation, as opposed to certainty.
The Platonists methodology for
making accounts of mathematical knowledge can be summarized
through its seven fundamental claims concerning the status
and truth of mathematical knowledge. Platonists seek to explain
the acquisition of mathematical knowledge by reiterating that
we obtain this knowledge through reason alone, independently
of our empirical senses. The Platonist argument that mathematical
objects are real and exist independently of the physical world,
can weakly account for the problems with access to the mathematical
realm and the argument that a priori knowledge cannot be certain.
The upshots to Platonist theory are that under many circumstances
where other theories may have to make long accounts for mathematical
claims, the Platonist reduces and accepts mathematical knowledge
by stating that mathematical objects are in fact real while
its competitor theories, such as fomalism or empiricisim use
different methodology to account for mathematical knowledge
