Absolutism. The absolutist believes mathematics is:

Transcription

1 Fallibilism

2 Absolutism The absolutist believes mathematics is: universal objective certain discovered by mathematicians through intuition established by proof after discovery. Most mathematicians share this view. Platonists are absolutists.

3 Absolutism The absolutist believes mathematical knowledge is necessary, perfect and atemporal. Absolutists see the unreasonable effectiveness of mathematics in describing the world and scientific phenomena as supporting their view: Since mathematics is a pure endeavor separated from experiment or experience, the only way it could be successful in describing the world is if it were the way the world expresses itself to us.

4 Fallibilism The fallibilist view sees mathematics as: Incomplete A work in progress Corrigible Subject to revision Changing Invented, rather than discovered.

5 Who are the fallibilists? Ludwig Wittgenstein Early, Middle, and Late versions Remarks on the Foundations of Mathematics (late period, as in the Late Ludwig Wittgenstein ) Anti Platonist

6 Wittgenstein Wittgenstein believed that mathematics was nothing but calculation (sort of like a formalist), and the rules for doing that calculation are arbitrary (not as much like a formalist). Rules of counting and adding are only custom and habit, the way we do it. Rules are not self enforcing. They are enforced only by agreement of the people concerned. (Hersh, What Is Mathematics, Really?)

7 Wittgenstein Wittgenstein argues that we follow rules in mathematical reasoning because of well tried custom, not because of logical necessity. Wittgenstein s contribution is to point out that it is what mathematicians do in practice, and not what logical theories tell us, that is the engine driving the development of mathematical knowledge. (Paul Ernest)

8 Imre Laktos Imre Lakatos is another fallibilist. His major work is Proofs and Refutations and is based on his doctoral dissertation.

9 Proofs and Refutations In Proofs and Refutations Lakotos uses the metaphor of the mathematics classroom to recreate the historical development of the Euler Formula, F+V=E+2, which relates the number of faces (F), edges (E) and vertices (V) of a class of geometric solids. Lakatos point is that proving this fact took over a hundred years and proceeded in fits and starts.

10 Proofs and Refutations Among other things, the mathematical definitions of solids, faces, edges and vertices had to be modified and refined, as different proofs and corresponding counterexamples (refutations) were discovered. Proofs were developed and published, only to be shown to have weaknesses, and then modified and strengthened for the next round of refutations.

11 Lakatos Lakatos s general argument is that, just as in his example of the Euler Formula, no definitions or proofs in mathematics are ever absolutely final and beyond revision. He argues that mathematics grows by successive criticism and refinement of theories and the introduction of new, competing theories not the deductive pattern of formalized mathematics. (Hersh, What Is Mathematics, Really?)

12 Philip Kitcher, a British philosopher of science now living in the U.S., has his own take on fallibilism. In his book The Nature of Mathematical Knowledge, Kitcher argues that much mathematical knowledge is accepted based on the authority of mathematicians, and not because of proofs. Philip Kitcher

13 Philip Kitcher Kitcher s analysis of proof suggests the importance of social context. In proof, he suggests, much of the argument is tacit and draws on unspoken mathematical knowledge learned through practice ( To read his proofs, one must be privy to a whole subculture of motivations, standard arguments and examples, habits of thought and agreed upon modes of reasoning ). Since the informal, tacit, cultural knowledge of mathematics varies over time, mathematical proof cannot be described as absolute it is temporal.

14 Arguments from Absolutists Mathematicians argue that if mathematics has no absolute necessity and essential characteristics to it, then it must be arbitrary. Thus, they argue, anarchy prevails and anything goes in mathematics. (Ernest)

15 Arguments from Absolutists But Richard Rorty argues that the opposite of necessity is not arbitrariness but contingency. Arbitrariness means to be determined by chance or whim. The opposite of this notion is that of being determined based on judgment or reason.

16 Arguments from Absolutists Paul Ernest argues that Both contingencies and choices are at work in mathematics, so it cannot be claimed that the overall development is either necessary or arbitrary. Much of mathematics follows by logical necessity from its assumptions and adopted rules of reasoning, just as moves do in the game of chess.

17 Arguments from Absolutists This does not contradict fallibilism for none of the rules of reasoning and logic in mathematics are themselves absolute. Mathematics consists of language games with deeply entrenched rules and patterns that are very stable and enduring, but which always remain open to the possibility of change, and in the long term, do change. (Paul Ernest)

18 Arguments from Absolutists John Barrow asks, If mathematics is invented how can it account for the amazing utility and effectiveness of pure mathematics as the language of science? But if mathematics is invented in response to external forces and problems, as well as to internal ones, its utility is to be expected. Since mathematics studies pure structures which originate in practical problems, it is not surprising that its concepts help to organize our understanding of the world and the patterns within it.

19 Finishing up Fallibilism Once humans have invented something by laying down the rules for its existence, like chess, the theory of numbers, or the Mandelbrot set, the implications and patterns that emerge from the underlying constellation of rules may continue to surprise us. But this does not change the fact that we invented the game in the first place. It just shows what a rich invention it was. Paul Ernest