## Penelope J. Maddy

Professor, Logic & Philosophy of Science

School of Social Sciences

School of Social Sciences

PH.D., Princeton University

University of California, Irvine

759 SST

Irvine, CA 92697

759 SST

Irvine, CA 92697

**Research Interests**

philosophy of mathematics and logic, philosophy of science

**Academic Distinctions**

American Academy of Arts and Sciences

**Research Abstract**

My work centers in the philosophy of mathematics, especially the philosophy of set theory, with connections to related issues in logic, philosophy of science, epistemology, meta-philosophy, and even Wittgenstein (early and late).

Set theory bears a special relationship to the rest of mathematics because of the remarkable fact that all the objects of classical mathematics (numbers, functions, spaces, groups, whatever) can be thought of as sets and all the theorems of classical mathematics can be proved from the standard axioms of set theory (the theory ZFC). The catch is that there are natural-arising questions in many branches of mathematics -- from analysis and algebra to set theory itself -- that cannot be answered on the basis of these same axioms. One reaction is to conclude that these so-called 'independent' questions, despite their seeming naturalness, have no unambiguous answers, that they are true in some models of ZFC and false in others, that all these models are equally good, and that further inquiry into their absolute truth value is misguided.

This glib answer is not acceptable to many mathematicians who continue to pursue these questions, often by searching for new hypotheses to add to the list of accepted axioms. The philosophical question arises: on what grounds, by what standards, are these axiom candidates to be judged? This question is the central one that drives my research.

One answer is: an acceptable axiom must be true in the objectively existing world of sets. I defended an answer of this sort in my book REALISM IN MATHEMATICS (Oxford UP, 1990). Since then, I've become less satisfied with the realist answer, partly due to philosophical doubts about the indispensability arguments for mathematical realism and partly due to methodological doubts that proper set theoretic practice, in the case of particular axiom candidates, is what it would be were realism correct. I outline an alternative approach to these problems in NATURALISM IN MATHEMATICS (Oxford UP, 1997).

For more recent work in philosophy of logic and philosophy of science, see my home page, above.

Set theory bears a special relationship to the rest of mathematics because of the remarkable fact that all the objects of classical mathematics (numbers, functions, spaces, groups, whatever) can be thought of as sets and all the theorems of classical mathematics can be proved from the standard axioms of set theory (the theory ZFC). The catch is that there are natural-arising questions in many branches of mathematics -- from analysis and algebra to set theory itself -- that cannot be answered on the basis of these same axioms. One reaction is to conclude that these so-called 'independent' questions, despite their seeming naturalness, have no unambiguous answers, that they are true in some models of ZFC and false in others, that all these models are equally good, and that further inquiry into their absolute truth value is misguided.

This glib answer is not acceptable to many mathematicians who continue to pursue these questions, often by searching for new hypotheses to add to the list of accepted axioms. The philosophical question arises: on what grounds, by what standards, are these axiom candidates to be judged? This question is the central one that drives my research.

One answer is: an acceptable axiom must be true in the objectively existing world of sets. I defended an answer of this sort in my book REALISM IN MATHEMATICS (Oxford UP, 1990). Since then, I've become less satisfied with the realist answer, partly due to philosophical doubts about the indispensability arguments for mathematical realism and partly due to methodological doubts that proper set theoretic practice, in the case of particular axiom candidates, is what it would be were realism correct. I outline an alternative approach to these problems in NATURALISM IN MATHEMATICS (Oxford UP, 1997).

For more recent work in philosophy of logic and philosophy of science, see my home page, above.

**Publications**

'Believing the axioms' (Journal of Symbolic Logic, 1988)

REALISM IN MATHEMATICS (Oxford University Press, 1990)

NATURALISM IN MATHEMATICS (Oxford University Press, 1997)

'Naturalism and the a priori', forthcoming in A PRIORI KNOWLEDGE. (Downloadable from my home page.)

'Logic and the discursive intellect', forthcoming in PHILOSOPHIA MATHEMATICA. (Downloadable from my home page.)

**Professional Societies**

Association for Symbolic Logic

American Philosophical Association

**Research Centers**

History and Philosophy of Science

**Last updated**

01/10/2001