If you are a Formalist at heart,
in reference to my last paper, the first Platonist approach
you would argue against is referring to mathematical objects
as real objects, and you'd do away with the existence of abstract
entities and the odd realm of objects that exist outside of
space and time. Indeed there are numerals in the world, but
what of numbers? A formalist will argue that mathematical
objects do not really exist, but serve the purpose of constructing
a formal game of rules and symbol manipulation by which you
can describe the physical world. To a formalist, mathematical
truth lies in our ability to form proofs to validate truth
by manipulating the symbols of a system, much like a game
is played. But the formalist platform is not without shortcomings,
some of which are best illustrated by Gödel's incompleteness
theorems. But before we can discuss the inadequacies of a
formalist approach, we should explore the core claims of formalism,
and its underlying motivations.
The father of modern Formalism,
David Hilbert reformed formalism theory motivated to make
the claim that one cannot assert mathematical truth without
proof of it. Hilbert had a Kantian perspective to mathematics,
and made many great contributions to the world of geometry,
theoretical physics and others. According to Kant, space and
time are essentially of human creation. That space and time
serve the purpose of description, relative to our forms of
perception.
From the perspective that space
and time do not objectively exist, Kant's view of the measurable
universe is fundamentally finitistic. Intuitively we know
that we cannot travel infinite distances or count to infinity.
"Of course, there are no upper bounds on what we can
do: no matter how far we move, we can always move a step further,
and no matter how many events we experience, we can always
experience one more. But at any point we will have acquired
only a finite amount of experience and have taken only a finite
number of steps." (Brown 65) It is under these conventions
that formalists claim the only legitimate infinity is a potential
infinity, and not an actual infinity. And consequentially,
if space and time are human creation, as humans we would know
their properties a priori, which would enable us to deduce
mathematical truths bound by our own perception, with certainty.
This Kantian background leads
us to Hilbert's second fundamental claim to formalism. Hilbert
also wanted to assert that in order for proofs to be made
to validate mathematical truth, a system must be consistent.
In order for a system to be consistent, there must not be
conflicting conditions within the system (of symbols and rules)
that would allow you to prove false statements or form contradictory
proofs. Hilbert saw that insisting a system remain consistent
would prevent theoretical consequences from arising, many
of which were previously posed by the incorporation of the
actually infinite to finitistic mathematics, and allow the
realm of applicable and truth bearing mathematics to be declared
without objection.
With this in mind, the formalist
identifies two different parts of math; the finite and meaningful,
and the infinite and meaningless. What is meant by meaningless
is that there is no substantially provable truth value within
the statement, and what is meant by meaningful is that it
is applicable to making predictions in the physical world.
So then how does one go about constructing meaningful proofs
to assert mathematical truth? Under Hilbert's reconstruction
of formalism, he insists that all existing theories be formalized.
Being that classical mathematics is mainly a mix of different
systems, symbols, and linguistics, formalism calls for mathematics
to be reformed into a rigorous analytical form of comprehensive
symbolic notation. Although it is now widely accepted that
Gödel's Incompleteness theorems have extinguished all
hope from ever carrying out Hilbert's reformation of mathematics
in its entirety.
Gödel's incompleteness theorem
has been by far the most devastating theoretical concepts
to formalism. For Hilbert and formalist's alike, in very simple
terms "completeness" of a system will assert the
truth of a proof with certitude. For Gödel, it is impossible
to prove the completeness of a formal system within itself.
Consider a formal system named B: The formal system B is complete,
just when for any B statement A, A is provable in B, if and
only if A is true in B. This illustrates how it is possible
to always prove the consistency of a system within itself,
and is the underlying motivation for Gödel's first incompleteness
theorem. Through an analytical proof, Gödel shows that
there is a statement A in formal arithmetic that is true and
not provable. Inspired by the liar paradox, which states "What
I am now saying is false;" a paradox because if it is
true it is false, and if false then it is true. Gödel
first incompleteness theorem roughly states that a function
F(x) is a formula that says "x is the least number not
named by any formula containing fewer than 10k symbols."
But we know there is a number n that has the property of being
fewer than 10k symbols, and analytical proof will show that
F(x) contains less than 10k symbols. So the formula F(x) does
not name n, making g unprovable. But we know g is true since
it says that n has the property described by F(x). In simpler
terms, Gödel "formulates a sentence of arithmetic
which says something like 'I am not provable' and sure enough
it is unprovable, which would show that it is a true yet unprovable
sentence of arithmetic, making the system of axioms incomplete"
(Brown 71)
Gödel's second incompleteness
theorem shatters Hilbert's consistency principle; it states
that there can be no finitistic proof of the consistency of
formal arithmetic. And if one cannot prove formal arithmetic
to be consistent, it would be impossible to prove a system
of greater complexity to be consistent in its own right. Gödel's
second incompleteness roughly states that if g is true if
and only if it is not provable, through analytic proof we
can't show that PA is consistent, since we cannot prove 0
= s0. Which shows that there is no proof in PA that it's consistent,
since you cannot prove that you cannot prove 0 = s0. Now since
Gödel's theorems illustrate the implausibility of the
consistency of formal arithmetic as well as the provability
of true statements, formalist hope for identifying truth with
provability, or consistency with certainty is obliterated.
Although Hilbert's program of
formalism is seriously injured by Gödel's incompleteness
theorems, it does in no way mean there is not good theoretical
value in the formalist standpoint. The most credible thing
formalists are known for is identifying how important notation
and proof can be, and how useful symbolic notation and formal
proofs can be in justifying mathematical truth. Formalists
can assert mathematical claims by rejecting abstract objects,
and reducing classical mathematics to finitistic analytical
systems. PreHilbert formalists approached mathematics as
a formal game of symbol manipulation, that make predictions
about the physical world around us. Formalists cleaned up
much of classical mathematics by emphasizing the need for
a more concrete and consistent proof of a claim, rather than
in models or mixed systems of symbols and linguistics. Formalists
may not have the ability to declare their fundamental claims
of correlating consistency with certitude, or truth with completeness,
but they justified in pointing out the necessity of formal
notation.
