Exploring the Platonist Argument

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        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 space-time. 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 non-standard 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 counter-arguments 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

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