From the book “Philosophy of Mathematics”, the author James Brown addresses many problems involving Platonism’s view in mathematics; such as how other mathematicians and philosophers debunk Platonism’s view. Brown, however, saves Platonism by attacking the intruders first. Before reaching the stage where Brown fights the intruders, it is necessary to understand the following: (1) the Platonism’s claim in mathematics, which is described in the second paragraph; (2) its two main fundamentals problems, which appears in paragraph three; (3) how Conventionalism, Empiricism, and Nominalism rejects Platonism, as described in the fourth paragraph. In the fifth paragraph, Brown will defend Platonism by attacking the intruders. Finally in the sixth paragraph, I will give my opinion on how well Brown defends Platonism and whether he’s objective.
First, Platonists claim that “mathematical objects are perfectly real and exist independently of us” (Brown, p11). They are perfectly real in the sense that they do have a unique form like any ordinary object. Since they have a unique form, we should be able to describe them. Thus, Platonists came up with the idea that using standard semantics, which is the theory of how truth is assigned to sentences, we are able to describe these mathematical entities. For instance, if the statement ‘Jerry has one hat’ is assumed to be true, then a person named Jerry must exist. Similarly, ‘1 x 2 = 2’ and ‘1 > 0’ are true statements if and only if the number 1 exists. By showing that objects exist in these true statements, Platonists claim that mathematical objects exist as well. Second, they claim that all mathematical objects are abstract. Abstract doesn’t mean universal, but it means “outside of space and time, and they are neither concrete nor physical” (Brown, p12). For example, when we describe a red apple, the redness of the apple is universal, but the ‘red-apple’ itself is a particular because the apple has an instance color of red. Consequently, a mathematical object such as 7 is a particular, but also abstract—it exists outside of space and time. Third, they claim that men are born with limited mathematical perceptions, which is similar to the fact that men have limited perceptual access to the physical realm. In the mathematical realm, all men are intuited of simple addition such as ‘1 + 1 = 2’, but not the Fundamental Theorem of Calculus. Similarly, within in the physical realm, men can only see ordinary objects like trees, but not the exotic entities of science—electrons. Fourth, they claim that “mathematic is a priori, not empirical” (Brown, p13). Unlike in empirical knowledge, which is dependent upon the five physical senses, priori is independent from them. Priori, according to Platonist, is a physical-sense-independent mechanism that allows a person to experience mathematical objects through the “mind’s eye” (Brown, p13). Fifth, they claim that although mathematics is a priori, it’s not necessary certain. The mind’s eye, according to Platonists, is just as fallible as the physical senses. Finally, they claim that the different ways to access the mathematical realm includes proofs, conjectures, pictures and diagrams.
The first problem is called the problem of access, which derives from two circumstances. The first situation involves the Platonists claim that mathematical realms can be accessed through the mind’s eye. The mind’s eye, according to Platonists, is like the sixth sense that allows men to grasped mathematical entities. Platonists, however, cannot explain how the mind’s eye functions. Since they cannot explain the workings of the mind’s eye, their claim that mathematical knowledge is known a priori seem nonsensical. The second situation is derived from the epistemologist’s Causal Theory of Knowledge, which states that “to know anything at all, there must be some sort of causal connection between the object known and the knower” (Brown, 16). Using this theory, naturalists disagree to the Platonists claim that we can access abstract objects. Since abstract objects exist outside of space and time, they are “causally inert” (Brown, 16), according to naturalists. Thus, it infers that men cannot possibly interact with abstract objects. The second problem is known as the problem of certainty. From the previous paragraph, it is learned that a priori knowledge, which according to Platonists, is open for errors. Kitcher, however, disagrees and insists that priori means certain. If mathematics knowledge is known a priori and priori means certain, but mathematical proofs are not certain, then it implies that mathematical knowledge is not a priori.
Various objections to the Platonist’s view of how mathematical perception relates to the physical world are being held by the Conventionalists, Empiricists, and Nominalists. In terms of how mathematics relates to science, Platonists and conventionalists have opposing views. Platonists claim that “since mathematics is essential for science, it must be true; and since it’s true, there must exist such objects as sets, functions, numbers, etc” (Brown, 53). Using standard semantics, Platonists claim that abstract objects do exist. Nominalists, however, denies the existence of abstract objects. According to Field, a Nominalist, “mathematics is not essential, but only provides an extremely useful short cut” (Brown, 53). He shows the inessential of mathematics by stating that mathematical objects, such as numbers, can actually be substituted with symbols. Consequently, these numbers doesn’t refer to real objects. Thus, abstract mathematical entities are just a convenient way to represent the physical world, but they do not exist. Second, Conventionalists disagrees with Platonist’s claim that mathematics is descriptive because it is known a priori. That is, mathematics is not based on empirical knowledge, but through the experience of the mind’s eye. Conventionalists, however, claim that “the source of a priori knowledge is to be found in language or in convention” (Brown, p18). That means the human language, such as the statement ‘fawns are young deers’, is true by the language stipulation. There are also stipulations in games; in a game of chess, for example, ‘Rooks can move vertically or horizontally’. These stipulations create truths and these truths are known a priori. Thus, mathematics is stipulative, not descriptive. Finally, Empiricists reject Platonist’s claim that mathematical knowledge is experienced through the mind’s eye. According to Empiricists, all knowledge including mathematics, originates in empirical experience. A more important distinction view between Platonists and Empiricists is how mathematics relates to the physical world. Platonists believe that mathematics realm is independent of the physical realm, and the former represents the latter. In contrast, Empiricists claim that mathematics does not represent the physical world, but describes the world. For instance, the statement ‘2 + 3 = 5’ is not truth about a mathematical realm, but it is “general truth about the physical world” (Brown, 55). Consequently, Empiricists claim that mathematics is part of the physical world.
Although Brown appears to be very objective most of the time, he does show greater support for Platonism over other philosophical views. Brown is so loyal to Platonism such that he backs them up whenever their argument seems weak. Here are a few examples to show his loyalty toward Platonism. First, let’s recall that one of Platonist’s claims is that men have a thorough knowledge of the mechanism of physical perception but not how the mind’s eye functions. Surprisingly, Brown rejects this claim to show that he’s being fair-and-square. However, Brown later conclude that “the fact Platonists have nothing to offer in the way of an explicit mechanism for seeing with the mind’s eye does not in the least undermine the cogency of the Platonist claim” (Brown, 15). This is an obvious statement made by Brown to show his loyal toward Platonism despite the fact that they cannot offer an explanation how their claim is valid. Second, when Epistemologists use the Causal Theory of Knowledge to reject Platonists claim that men have access to abstract entities, Brown fights back by disproving the causal theory of knowledge with his example of an EPR-type set-up. With this experiment, Brown shows that there need not be any causal connection between entities in the physical realm. Thus, according to Brown, since the Causal Theory of Knowledge is refuted, “we can have knowledge even without a causal connection” (Brown, 17). As a third example, when Kitcher refute Platonists claim that mathematical knowledge is known a priori, Brown defends Platonists by stating that Kitcher’s argument will only works under the assumption that priori means certain, but “there is no reason in the world to make that assumption—even though it has often been made in the past” (Brown, 18).
By understanding Platonists core claims as well as its fundamental problems in mathematics, one can evaluate whether Brown does a thorough task in defending against other philosophers and mathematics who tries to debunk Platonist’s claims. In the case where Brown uses the ‘EPR-type set-up’ to show that Epistemologist’s Causal Theory of Knowledge is perhaps flawless, I believe he does a fine job. Nevertheless, I prefer the better argument presented by Tait in trying to prove the weakness of the Causal Theory of Knowledge; but that is not the subject of this paper. The next case Kitcher tries to debunk Platonist’s claim that mathematical knowledge is a priori; Brown does another excellent job here to defend Platonism. According to Brown, Kitcher’s argument is only valid under the assumption that priori means certain, but “there is no reason in the world to make that assumption” (Brown, 18). Although Brown sounds a little sarcastic, he’s right about it—who in the world will allow that assumption? Besides, Platonists already made the claim that priori means uncertain. Kitcher cannot just reverse Platonist’s claim to fit his argument. Nevertheless, I think Brown is too Platonism-oriented. Even when Platonists can’t explain how the mind’s eye mechanism works, Brown still tries to defend them by stating that “the fact Platonists have nothing to offer in the way of an explicit mechanism for seeing with the mind’s eye does not in the least undermine the cogency of the Platonist claim” (Brown, 15). This, I do not think Brown makes a good case for it.

SOURCES

Brown, J. (1999), Philosophy of Mathematics, Routledge.

Registered Members, login
Join now, it's free

Property of EssaySwap.com