Abstract
‘What is progress in mathematics?’ The answer, at least a partial answer, seems easy: ‘I make mathematical progress when I solve a problem I wanted to solve or prove a theorem I wanted to prove.’ Now there’s surely something right about this, but the issue that brings us here isn’t my mathematical development or lack thereof; what we want to know is how mathematics as a discipline progresses. Does mathematics as a discipline progress when someone solves a problem she wanted to solve or proves a theorem she wanted to prove? Well, not if the problem had already been solved or the theorem had already been proved by someone else, at least as long as that previous solution or proof is well known to the mathematical community. But the shortcomings of the easy answer go deeper than this.
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References
Bulloff, J. et al. (Eds.), Foundations of Mathematics. Berlin: Springer.
Gale, D. and Stewart, F. (1953). “Infinite games with perfect information.” Contributions to the Theory of Games. Annals of Mathematical Studies. Vol. 28: 254–66.
Gödel, K. (1938). “The consistency of the axiom of choice and of the generalized continuum hypothesis.” Proceedings of the National Academy of Sciences. Vol. 24: 556–7. Reprinted in his (1990, 26–7).
Gödel, K. (1940). The Consistency of the Continuum Hypothesis. Princeton: Princeton University Press. Reprinted in his (1990, 33–101).
Gödel, K. (1947). “What is Cantor’s Continuum Problem?” 2nd ed. Reprinted in his (1990, 176–87).
Gödel, K. (1964). “What is Cantor’s Continuum Problem?” 2nd ed. Reprinted in his (1990, 254–70).
Gödel, K. (1990). Collected Works. Vol. II. S. Feffermann et al. (eds.), New York: Oxford University Press.
Grattan-Guiness, I. (1970). “An unpublished paper by Georg Cantor.” Acta Mathematica. Vol. 124: 65–107.
Hallet, M. (1984). Cantorian Set Theory and Limitation of Size. Oxford: Oxford University Press.
Kunen, K. (1970). “Some applications of iterated ultrapowers in set theory.” Annals of Mathematical Logic. Vol. 1: 179–227.
Kunen, K. (1971). “Elementary embeddings and infmitary combinatorics.” Journal of Symbolic Logic. Vol. 36: 407–13.
Kunen, K. (1980). Set Theory. Amsterdam: North Holland.
Lusin, N. (1917). “Sur la classification de M. Baire.” Comptes Rendus de l’Académie des Sciences. Vol. 164: 91–4.
Maddy, P. (1993). “Does V equal L?” Journal of Symbolic Logic. Vol. 58: 15–41.
Maddy, P. (1997). Naturalism in Mathematics. Oxford: Clarendon Press.
Martin, D. A. (1970). “Measurable cardinals and analytic games.” Fundamenta Mathematica. Vol. 66: 287 – 91.
Martin, D. A. (1975). “Bore! determinacy.” Annals of Mathematics. Vol. 102: 363–71.
Martin, D. A. (1980). “Infinite games.” Proceedings of the International Congress of Mathematicians, Helsinki 1978. Helsinki: Academia Scientiarum Fennica.
Martin, D. A. and Steel, J. (1988). “Projective determinacy.” Proceedings of the National Academy of Science. Vol 85, 6582–86.
Martin, D. A. and Steel, J. (1989). “A proof of projective determinacy.” Journal of the American Mathematical Society. Vol.2: 71–125.
Moschovakis, Y. (1980). Descriptive Set Theory. Amsterdam: North Holland.
Mycielski, J. and Steinhaus, H. (1962). “A mathematical axiom contradicting the axiom of choice.” Bulletin de l’Académie Polonaise des Sciences. Vol.10: 1 – 3.
Nagel, E., Suppes, P. Tarski, A. (Eds.). Logic, Methodology, and Philosophy of Science. Stanford: Stanford University Press.
Rowbottom, R. (1964). “Some strong axioms of infinity incompatible with the axiom of constructibility.” Ph.D. Dissertation, University of Wisconsin. Repr. in Annals of Mathematical Logic. Vol. 3: 1 – 44.
Scott, D. (1961). “Measurable cardinals and constructible sets.” Bulletin de l’Académie Polonaise des Sciences. Vol. 9: 521–4.
Scott, D. (Ed.). (1971). Axiomatic Set Theory. Providence: American Mathematical Society.
Silver, J. (1966). “Some applications of model theory in set theory.” Ph.D. Dissertation, University of California, Berkeley. Revised version in Annals of Mathematical Logic. Vol. 3: 45–110.
Silver, J. (1971). “The consistency of the GCH with the existence of a measurable cardinal,” in (Scott 1971, 391–6).
Silver, J. (1971a). “Measurable cardinals and Δ1 3 well-orderings.” Annals of Mathematics. Vol. 94: 414 – 46.
Solovay, R. M. (1969). “On the cardinality of Σ1 2 sets of reals,” in (Bulloff 1969, 58–73).
Tarski, A. (1962). “Some problems and results relevant to the foundations of set theory, “ in (Nagel et. al. 1962, 125–35).
Ulam, S. (1930). “Zur Masstheorie in der allgemeinen Mengenlehre.” Fundamenta Mathematica. Vol. 16: 140–50.
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Maddy, P. (2000). Mathematical Progress. In: Grosholz, E., Breger, H. (eds) The Growth of Mathematical Knowledge. Synthese Library, vol 289. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-9558-2_23
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DOI: https://doi.org/10.1007/978-94-015-9558-2_23
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