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. Pre-Hilbert 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.