Mathematical Realism (PRO) vs. Mathematical Anti-Realism (CON)

I would like to begin by making two arguments against Platonism.The Epistemological ArgumentThe Epistemological Argument asserts that Platonism advances a view of reality which would entail an inability to understand mathematics at all. The argument can be formulated as follows:Premise One: Humans exist entirely within spacetime.Premise Two: If there exist any abstract mathematical objects, they exist outside of spacetime.Premise Three: Humans can only obtain knowledge of items that exist within spacetime.Premise Four: If there exist any abstract mathematical objects, humans could not obtain any knowledge of them.ThereforePremise Five: If Mathematical Platonism is correct, then human beings could not have mathematical knowledge.Premise Six: Human beings have mathematical knowledge.ThereforeConclusion: Mathematical Platonism is incorrect.I do not believe that there is any objection to Premises One, Two, or Six, so I will not waste time on them. Rather, I will focus on the justification of Premises Three and Four as a whole, as these will likely be the most contested.Human beings obtain knowledge through two means: emprical and rational. The empirical means require a usage of the senses, which are entirely based on physical phenomena. Thus, the human senses cannot be used to determine the nature of the non-spatiotemporal. When arguing on the ability of humans to perceive ideas such as those in mathematics, the discussion inevitably involves the rational faculties, so we must ask how they are involved.The answer is that by itself, pure logic cannot deduce anything outside of pure logic. Logical axioms such as DeMorgan's Law only apply when there is something non-abstract to apply it to. Otherwise, much like formulae in physics cannot deduce anything without values being substituted in for the variables, logical axioms cannot make deductions about real-world non-abstract phenomena unless they are being applied to known facts outside of logic. Hence, logic cannot be used to derive knowledge of items beyond known spacetime unless already given knowledge about said items.The Non-Uniqueness ArgumentThe Non-Uniqueness argument demonstrates that while Mathematical Platonism Premise One: If there are any sequences of abstract objects that satisfy the axioms of Peano Arithmetic, then there are infinitely many such sequences.Premise Two: There is nothing "metaphysically special" about any of these sequences that makes it stand out as the sequence of natural numbers.Premise Three: There is no unique sequence of abstract objects that is the natural numbers.Premise Four: Mathematical Platonism entails that there is a unique sequence of abstract objects that is the natural numbers.ThererforeConclusion: Mathematical Platonism is incorrect.The structure for the first and second premises is best summed up by Benacerraf, who argues for these ideas in terms of set theory. If we imagine infinite sequences of sets that obey a consistent change in structure concordant with the changes advanced by Peano Arithemtic's description of the natural numbers, then we can concieve of a number of sets with that sort of structure, e.g.Ø, {Ø}, {{Ø}}, {{{Ø}}},...orØ, {Ø {Ø}}, {Ø {Ø {Ø}}}, {Ø {Ø {Ø {Ø}}}},...All of these sets and many more like them have similar structural features that obey the structure of the set of the natural numbers as outlined in Peano Arithmetic. Ultimately, when seeking to represent the numbers of a sequences, the contents of our sequence of sets is not important, but the structure is. As long as the structure stays true, then the sequence can be thought of as the sequence of natural numbers. Knowing this, we can see that none of these infinite sequences have any redeeming characteristic to make them be seen as the sequence of natural numbers.I will now move on to rebutting my opponent's arguments.Rebuttal of Frege's Argument:This argument relies largely on a conflation between the logical sentences describing mathematics and the sentences used outside of it. While it is true that Classical Semantics does imply that all objects in statements must refer to real objects, there is no principle that excludes the referral to a conception of something. If we did not accept this, then Classical Semantics would result in absurd scenarios and deductions.For example, consider the statement: "Santa Claus lives at the North Pole." This statement is technically true due to how we have defined Santa Claus, in that he is a fat bearded man with elves and reindeer who lives at the North Pole. Within this definition, the statement is true. Clearly, if we did not allow ourselves to refer to conceptions, then Classical Semantics would entail that this statement refers to an actual Santa Claus. Clearly it isn't, and the reason why is that we are not referring to a real Santa Claus, but merely to our conception of Santa Claus.How does this apply to mathematics? Simply put, my stance of Fictionalism merely advocates that the objects of mathematics themselves are not refered to, however this says nothing about our conceptions of mathematical ideas. Mathematics is a system of rules that we have invented, and when we describe the properties of mathematical objects within those rules, we are describing the behavior of our conceptions. To give another example, consider Chess:Chess, like mathematics, has logical rules and consistency to it. We can describe the behavior of objects within Chess just as in mathematics and use the same language. We can say that "There exists a piece that can move both diagonally and horizontally." However, no one is willing to imply that there exist actual abstract Pawns and Kings, and so similarly we have no reason to believe in abstract number objects either.Rebuttal of Quine's Indispensability Argument:The main problem with this argument is that it is a misrepresentation of what mathematics really is. As I said before, mathematics is a system of logical rules that intertwine with each other. This system happens to be the best system that humans have invented to describe the way that reality behaves, but even so, it is not really a "theory" of reality. It is more of a tool that allows us to provide an input (the observations that we have about reality) and allows us to describe other properties of reality. In the same way, computer programs that sort lists have real-word counterparts, but we have to understand how to translate them. For example:Suppose a programmer writes a program designed to sort rubber balls by their color. The input for his code might be a list of the colors of the balls, or it might be a picture of them, or list of HEX color-codes. However, he cannot input the balls themselves. He must translate the balls into some form of information, feed that into his code, and then converts the information he recieves into a batch of sorted balls. But he can never simply give the balls to the program. He must translate them into information first.Similarly, our mathemtical formulae that describe reality do not do so directly. We cannot simply give two stars into Newton's formula for gravitation and recieve the amount of force between them. We must first translate the stars into two numbers representing masses and a number representing distance, and then we can determine a number representing force, which we can translate back into our conclusions. But mathematics by itself cannot do this.Rebuttal of Putnam's Succes Argument:This argument mostly falls under the same rebuttals as the previous one, but I would also like to point out that this is also an Argumentum ad Ignoratum. Simply because we haven't proven an explanation false does not automatically entail that said explanation is the correct one. This argument thusly fails.SOURCES:[1]http://www.calstatela.edu...[2]http://plato.stanford.edu...;