Formalism
In this essay I will show that whilst formalism is an attractive view it does not provide us with an adequate account of mathematics. I will begin with a brief outline of the basic position before going on to discuss it. Finally, I will discuss Hilbert’s programme.
In brief, formalism is the view that mathematics is the study of formal systems. This however does not tell the whole story and formalism can be divided into term formalism and game formalism (Shapiro, 2000: pp. 141-148). Term formalism is the view that mathematics is about characters or symbols. That is, the number 2 is just the character ‘2’. Whereas, game formalism is the view that mathematics is a game in the same way that chess is a game. There are characters, or pieces, that can only be manipulated according to specific rules. Consequently, mathematical practice is just like a game of chess and similarly meaningless.
On first glance, these views seem attractive for two reasons. First, it seems perfectly natural to agree that maths is just about symbol manipulation, what else could it be about? Second, formalism causes issues about the existence of numbers to fall away. Term formalism identifies numbers with characters and game formalism holds that mathematical symbols just are symbols.
There are, however problems with both these views. First, term formalism. If numbers are to be identified with characters then we encounter a problem. Consider the character ‘0’ and the character ‘0’. If the term formalist identifies numbers then since we have two separate characters we also have two separate numbers. A view I’m sure the term formalist does not want to defend. In which case the term formalist has to draw a distinction between token and type. Fo...
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...cond part of Gödel’s theorem shows that if T is consistent then the consistency of T cannot be proven within T. This leaves Hilbert’s programme in tatters. Hilbert had hoped to give a proof of the consistency of mathematics using finite methods but Gödel’s theorem shows that such a proof cannot be found. Hilbert’s programme cannot establish the certainty of mathematics. (Brown, 2008: pp. 76-82)
In conclusion, I have shown that the basic positions of formalism are unconvincing. I have also shown that whilst Hilbert’s programme failed for technical reasons there are also philosophical problems with it.
Works Cited
Brown, J., 2008. Philosophy of mathematics: a contemporary introduction to the world of proofs and pictures. 2nd Edition. London: Routledge.
Shapiro, S., 2000. Thinking about mathematics: the philosophy of mathematics. Oxford: Oxford University Press.
The issue at stake on page 12 of Fred D’Agostino’s, “The Ethos of Games” is simply whether or not formalism, as interpreted through the dichotomization thesis, provides a satisfactory account of games. In this context, formalism means that a game can be defined solely by the formal rules of that specific game (D’Agostino, p. 7). At the same time, according to the dichotomization thesis, the rules of any game can be definitively separated into two categories, but never both (p. 11). One of those categories being regulative rules, which can be defined as any rule that invokes a penalty (p. 11). The other category, constitutive rules, are simply the set of rules that define a game (p. 11). Given these definitions, D’Agostino argues that through the dichotomization thesis, formalism does not provide a proper account of games (p. 12).
The Philosophical Writings of Descartes, Cottingham, John, and Robert Stoothoff, and Dugald Murdoch. (eds.) 1984. Cambridge: Cambridge University Press.
Edmund Husserl’s conception of mathematics was a unique blend of Platonist and formalist ideas. He believed that mathematics had reached a mixed state combining Platonic and formal elements and that both were important for the pursuit of the sciences, as well as for each other. However, he seemed to believe that only the Platonic aspects had significance for his science of phenomenology. Because of the significance of the distinction between these two types of mathematics, I will always use one of the adjectives “material” or “formal” when discussing any branch of mathematics, unless I specifically mean to include both.
Game Theory was said to have been introduced by Emile Borel in 1921. Borel was a French mathematician who published papers on the theory of games. From this standpoint and according to the article “Game Theory”, Borel could have been named the “first mathematician to envision an organized system for playing games” however; evidence has shown that Borel did not develop his ideas any further. This is the reason why most historians have given credit to John Von Neumann.
In conclusion, Wittgenstein’s Remarks on the Foundations of Mathematics offers a successful criticism of Cantor’s ideas. With particular reference to the Diagonal Argument, Wittgenstein has established that the difficulty in understanding or forming concepts of the sets which Cantor talks of makes it more difficult to impose theorems and proofs upon them. Although Wittgenstein can see some value in Cantor’s conclusion that the set of real numbers is non-denumerable, the steps and assumptions to reach such an idea are uncertain. As a result Cantor’s Diagonal Argument cannot be considered successful in its entirety and therefore one cannot consider there to exist more real numbers than natural numbers. It is unfathomable that an infinite number of infinities exist when we cannot form a thorough notion of infinity in general.
In his book, The Language of Thought, Jerry Fodor claims that i) Wittgenstein’s private language argument is not in fact against Fodor’s theory, and ii) Wittgenstein’s private language argument “isn’t really any good” (70). In this paper I hope to show that Fodor’s second claim is patently false. In aid of this I will consider Wittgenstein's Philosophical Investigations (243-363), Jerry Fodor's The Language of Thought (55-97), as well as Anthony Kenny’s Wittgenstein (178-202). First I shall summarize Wittgenstein’s argument; then I will examine Fodor’s response and explain why it is fallacious. In my view, Fodor is wrong because he takes Wittgenstein to be a verificationist, and also because he makes a false analogy between people and computers.
Wentzel, J and Vrede Van Huyssl-en. Mathematics as a Human Endevavor. New York: Macmillian Reference USA, 2003. Print.
This essay is written to introduce the Russell’s Theory on Definite Description. The main content of this essay including: the definition of definite description, the puzzles concerning definite description, Russell’s Theory on Definite Description, how this theory solves the puzzles, Strawson’s objection to this theory, my evaluation on the convincingness of Strawson’s objection and my evaluation on the convincingness of Russell’s Theory of Definite Description.
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To better attempt to understand Aristotle’s view on mathematical truths, further inquiry will be made in regards to a fictionalist versus a literalist view point of mathematical objects. Both literalism and fictionalism have been attributed to Aristotle
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