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.
I shall also expound Ayer's theory of knowledge, as related in his book. I will show this theory to contain logical errors, making his modified version of the principle flawed from a second angle.
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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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Ludwig Wittgenstein (1889-1951) produced two commonly recognised stages of thought in 20th century analytic philosophy, both of which are taken to be central and fundamental in their respective periods. His early philosophy in the Tractatus Logico-Philosophicus, first published in 1921, provided new insights into relationships between the world, thought, language and the nature of philosophy by showing the application of modern logic to metaphysics via language. His later philosophy, mostly found in Philosophical Investigations, published posthumously in 1953, controversially critiqued all traditional philosophy, including his own previous work. In this essay I will explain, contrast and evaluate both stages of his philosophy, highlighting strengths and weaknesses and concluding that Wittgenstein’s late philosophy has provided an interesting explanation for the meaning of language.
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Burton, D. (2011). The History of Mathematics: An Introduction. (Seventh Ed.) New York, NY. McGraw-Hill Companies, Inc.