Penelope Maddy | |
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| Born | July 4, 1950 |
| Education | Princeton University (PhD, 1979) |
| Era | Contemporary philosophy |
| Region | Western philosophy |
| School | Analytic philosophy |
| Institutions | University of California, Irvine |
| Thesis | Set Theoretical Realism (1979) |
| Doctoral advisor | John P. Burgess |
Main interests | Philosophy of mathematics |
Notable ideas | Set-theoretic realism (also known as naturalized Platonism), [1] mathematical naturalism |
Penelope Maddy (born 4 July 1950) is an American philosopher. Maddy is Emerita UCI Distinguished Professor of Logic and Philosophy of Science and of Mathematics at the University of California, Irvine. She is well known for her influential work in the philosophy of mathematics, where she has worked on mathematical realism (especially set-theoretic realism) and mathematical naturalism.
Maddy received her Ph.D. from Princeton University in 1979. Her dissertation, Set Theoretical Realism, was supervised by John P. Burgess. [2] She taught at the University of Notre Dame and University of Illinois, Chicago before joining Irvine in 1987. [3]
She was elected a Fellow of the American Academy of Arts and Sciences in 1998. [4] The German Mathematical Society awarded her a Gauss Lectureship in 2006.
Maddy's early work, culminating in Realism in Mathematics, defended Kurt Gödel's position that mathematics is a true description of a mind-independent realm that we can access through our intuition. However, she suggested that some mathematical entities are in fact concrete, unlike, notably, Gödel, who assumed all mathematical objects are abstract. She suggested that sets can be causally efficacious, and in fact share all the causal and spatiotemporal properties of their elements. Thus, when one sees three cups on a table, one also sees the set. She used contemporary work in cognitive science and psychology to support this position, pointing out that just as at a certain age we begin to see objects rather than mere sense perceptions, there is also a certain age at which we begin to see sets rather than just objects.
In the 1990s, she moved away from this position, towards a position described in Naturalism in Mathematics. Her "naturalist" position, like Quine's, suggests that since science is our most successful project so far for knowing about the world, philosophers should adopt the methods of science in their own discipline, and especially when discussing science. As Maddy stated in an interview, "If you're a 'naturalist', you think that science shouldn't be held to extra-scientific standards, that it doesn't require extra-scientific ratification." [5] However, rather than a unified picture of the sciences like Quine's, her picture has mathematics as separate. That is, mathematics is neither supported nor undermined by the needs and goals of science but is allowed to obey its own criteria. This means that traditional metaphysical and epistemological concerns of the philosophy of mathematics are misplaced. Like Wittgenstein, she suggests that many of these puzzles arise merely because of the application of language outside its proper domain of significance.
She has been dedicated to understanding and explaining the methods that set theorists use in agreeing on axioms, especially those that go beyond ZFC.
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word ἀξίωμα (axíōma), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'.
In analytic philosophy, anti-realism is the position that the truth of a statement rests on its demonstrability through internal logic mechanisms, such as the context principle or intuitionistic logic, in direct opposition to the realist notion that the truth of a statement rests on its correspondence to an external, independent reality. In anti-realism, this external reality is hypothetical and is not assumed.
In metaphysics, nominalism is the view that universals and abstract objects do not actually exist other than being merely names or labels. There are at least two main versions of nominalism. One version denies the existence of universals – things that can be instantiated or exemplified by many particular things. The other version specifically denies the existence of abstract objects – objects that do not exist in space and time.
Willard Van Orman Quine was an American philosopher and logician in the analytic tradition, recognized as "one of the most influential philosophers of the twentieth century". He served as the Edgar Pierce Chair of Philosophy at Harvard University from 1956 to 1978.
In mathematical logic, Russell's paradox is a set-theoretic paradox published by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains an unrestricted comprehension principle leads to contradictions. The paradox had already been discovered independently in 1899 by the German mathematician Ernst Zermelo. However, Zermelo did not publish the idea, which remained known only to David Hilbert, Edmund Husserl, and other academics at the University of Göttingen. At the end of the 1890s, Georg Cantor – considered the founder of modern set theory – had already realized that his theory would lead to a contradiction, as he told Hilbert and Richard Dedekind by letter.
Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship with other human activities.
Friedrich Ludwig Gottlob Frege was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic philosophy, concentrating on the philosophy of language, logic, and mathematics. Though he was largely ignored during his lifetime, Giuseppe Peano (1858–1932), Bertrand Russell (1872–1970), and, to some extent, Ludwig Wittgenstein (1889–1951) introduced his work to later generations of philosophers. Frege is widely considered to be the greatest logician since Aristotle, and one of the most profound philosophers of mathematics ever.
Reductionism is any of several related philosophical ideas regarding the associations between phenomena which can be described in terms of simpler or more fundamental phenomena. It is also described as an intellectual and philosophical position that interprets a complex system as the sum of its parts.
Hilary Whitehall Putnam was an American philosopher, mathematician, computer scientist, and figure in analytic philosophy in the second half of the 20th century. He contributed to the studies of philosophy of mind, philosophy of language, philosophy of mathematics, and philosophy of science. Outside philosophy, Putnam contributed to mathematics and computer science. Together with Martin Davis he developed the Davis–Putnam algorithm for the Boolean satisfiability problem and he helped demonstrate the unsolvability of Hilbert's tenth problem.
Analytic philosophy is a broad, contemporary movement or tradition within Western philosophy, especially anglophone philosophy, focused on analysis. Analytic philosophy is characterized by a clarity of prose; rigor in arguments; and making use of formal logic and mathematics, and, to a lesser degree, the natural sciences. It is further characterized by an interest in language and meaning known as the linguistic turn. It has developed several new branches of philosophy and logic, notably philosophy of language, philosophy of mathematics, philosophy of science, modern predicate logic and mathematical logic.
Quasi-empiricism in mathematics is the attempt in the philosophy of mathematics to direct philosophers' attention to mathematical practice, in particular, relations with physics, social sciences, and computational mathematics, rather than solely to issues in the foundations of mathematics. Of concern to this discussion are several topics: the relationship of empiricism with mathematics, issues related to realism, the importance of culture, necessity of application, etc.
In set theory, a universal set is a set which contains all objects, including itself. In set theory as usually formulated, it can be proven in multiple ways that a universal set does not exist. However, some non-standard variants of set theory include a universal set.
In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large". The proposition that such cardinals exist cannot be proved in the most common axiomatization of set theory, namely ZFC, and such propositions can be viewed as ways of measuring how "much", beyond ZFC, one needs to assume to be able to prove certain desired results. In other words, they can be seen, in Dana Scott's phrase, as quantifying the fact "that if you want more you have to assume more".
In metaphysics, the distinction between abstract and concrete refers to a divide between two types of entities. Many philosophers hold that this difference has fundamental metaphysical significance. Examples of concrete objects include plants, human beings and planets while things like numbers, sets and propositions are abstract objects. There is no general consensus as to what the characteristic marks of concreteness and abstractness are. Popular suggestions include defining the distinction in terms of the difference between (1) existence inside or outside space-time, (2) having causes and effects or not, (3) having contingent or necessary existence, (4) being particular or universal and (5) belonging to either the physical or the mental realm or to neither. Despite this diversity of views, there is broad agreement concerning most objects as to whether they are abstract or concrete. So under most interpretations, all these views would agree that, for example, plants are concrete objects while numbers are abstract objects.
The Lakatos Award is given annually for an outstanding contribution to the philosophy of science, widely interpreted. The contribution must be in the form of a monograph, co-authored or single-authored, and published in English during the previous six years. The award is in memory of the influential Hungarian philosopher of science and mathematics Imre Lakatos, whose tenure as Professor of Logic at the London School of Economics and Political Science (LSE) was cut short by his early and unexpected death. While administered by an international management committee organised from the LSE, it is independent of the LSE Department of Philosophy, Logic, and Scientific Method, with many of the committee's members being academics from other institutions. The value of the award, which has been endowed by the Latsis Foundation, is £10,000, and to take it up a successful candidate must visit the LSE and deliver a public lecture.
Charles Dacre Parsons was an American philosopher best known for his work in the philosophy of mathematics and the study of the philosophy of Immanuel Kant. He was professor emeritus at Harvard University.
Originally, fallibilism is the philosophical principle that propositions can be accepted even though they cannot be conclusively proven or justified, or that neither knowledge nor belief is certain. The term was coined in the late nineteenth century by the American philosopher Charles Sanders Peirce, as a response to foundationalism. Theorists, following Austrian-British philosopher Karl Popper, may also refer to fallibilism as the notion that knowledge might turn out to be false. Furthermore, fallibilism is said to imply corrigibilism, the principle that propositions are open to revision. Fallibilism is often juxtaposed with infallibilism.
An Introduction to the Philosophy of Mathematics is a 2012 textbook on the philosophy of mathematics by Mark Colyvan. It has a focus on issues in contemporary philosophy, such as the mathematical realism–anti-realism debate and the philosophical significance of mathematical practice, and largely skips over historical debates. It covers a range of topics in contemporary philosophy of mathematics including various forms of mathematical realism, the Quine–Putnam indispensability argument, mathematical fictionalism, mathematical explanation, the "unreasonable effectiveness of mathematics", paraconsistent mathematics, and the role of mathematical notation in the progress of mathematics. The book was praised as accessible and well-written and the reaction to its contemporary focus was largely positive, although some academic reviewers felt that it should have covered the historical debates over logicism, formalism and intuitionism in more detail. Other aspects of the book that received praise were its coverage of mathematical explanation, its appeal to mathematicians and other non-philosophers, and its discussion questions and further readings, whilst its epilogue and short length received a more mixed reception.
The Quine–Putnam indispensability argument is an argument in the philosophy of mathematics for the existence of abstract mathematical objects such as numbers and sets, a position known as mathematical platonism. It was named after the philosophers Willard Quine and Hilary Putnam, and is one of the most important arguments in the philosophy of mathematics.
In the philosophy of mathematics, Aristotelian realism holds that mathematics studies properties such as symmetry, continuity and order that can be immanently realized in the physical world. It contrasts with Platonism in holding that the objects of mathematics, such as numbers, do not exist in an "abstract" world but can be physically realized. It contrasts with nominalism, fictionalism, and logicism in holding that mathematics is not about mere names or methods of inference or calculation but about certain real aspects of the world.
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