The Unreasonable Effectiveness of Mathematics in the Natural Sciences"The Unreasonable Effectiveness of Mathematics in the Natural Sciences" is a 1960 article by the physicist Eugene Wigner. In the paper, Wigner observes that a physical theory's mathematical structure often points the way to further advances in that theory and even to empirical predictions. Wigner begins his paper with the belief, common among those familiar with mathematics, that mathematical concepts have applicability far beyond the context in which they were originally developed.
PsychologismPsychologism is a family of philosophical positions, according to which certain psychological facts, laws, or entities play a central role in grounding or explaining certain non-psychological facts, laws, or entities. The word was coined by Johann Eduard Erdmann as Psychologismus, being translated into English as psychologism. The Oxford English Dictionary defines psychologism as: "The view or doctrine that a theory of psychology or ideas forms the basis of an account of metaphysics, epistemology, or meaning; (sometimes) spec.
Structuralism (philosophy of mathematics)Structuralism is a theory in the philosophy of mathematics that holds that mathematical theories describe structures of mathematical objects. Mathematical objects are exhaustively defined by their place in such structures. Consequently, structuralism maintains that mathematical objects do not possess any intrinsic properties but are defined by their external relations in a system. For instance, structuralism holds that the number 1 is exhaustively defined by being the successor of 0 in the structure of the theory of natural numbers.
Abstract object theoryAbstract object theory (AOT) is a branch of metaphysics regarding abstract objects. Originally devised by metaphysician Edward Zalta in 1981, the theory was an expansion of mathematical Platonism. Abstract Objects: An Introduction to Axiomatic Metaphysics (1983) is the title of a publication by Edward Zalta that outlines abstract object theory. AOT is a dual predication approach (also known as "dual copula strategy") to abstract objects influenced by the contributions of Alexius Meinong and his student Ernst Mally.
Quasi-empiricism in mathematicsQuasi-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 (see Penelope Maddy) with mathematics, issues related to realism, the importance of culture, necessity of application, etc.
Mathematical universe hypothesisIn physics and cosmology, the mathematical universe hypothesis (MUH), also known as the ultimate ensemble theory, is a speculative "theory of everything" (TOE) proposed by cosmologist Max Tegmark. Tegmark's MUH is the hypothesis that our external physical reality is a mathematical structure. That is, the physical universe is not merely described by mathematics, but is mathematics — specifically, a mathematical structure. Mathematical existence equals physical existence, and all structures that exist mathematically exist physically as well.
Benacerraf's identification problemIn the philosophy of mathematics, Benacerraf's identification problem is a philosophical argument developed by Paul Benacerraf against set-theoretic Platonism and published in 1965 in an article entitled "What Numbers Could Not Be". Historically, the work became a significant catalyst in motivating the development of mathematical structuralism. The identification problem argues that there exists a fundamental problem in reducing natural numbers to pure sets.
ConventionalismConventionalism is the philosophical attitude that fundamental principles of a certain kind are grounded on (explicit or implicit) agreements in society, rather than on external reality. Unspoken rules play a key role in the philosophy's structure. Although this attitude is commonly held with respect to the rules of grammar, its application to the propositions of ethics, law, science, biology, mathematics, and logic is more controversial. The debate on linguistic conventionalism goes back to Plato's Cratylus and the philosophy of Kumārila Bhaṭṭa.
Informal mathematicsInformal mathematics, also called naïve mathematics, has historically been the predominant form of mathematics at most times and in most cultures, and is the subject of modern ethno-cultural studies of mathematics. The philosopher Imre Lakatos in his Proofs and Refutations aimed to sharpen the formulation of informal mathematics, by reconstructing its role in nineteenth century mathematical debates and concept formation, opposing the predominant assumptions of mathematical formalism.
On Formally Undecidable Propositions of Principia Mathematica and Related Systems"Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I" ("On Formally Undecidable Propositions of Principia Mathematica and Related Systems I") is a paper in mathematical logic by Kurt Gödel. Submitted November 17, 1930, it was originally published in German in the 1931 volume of Monatshefte für Mathematik. Several English translations have appeared in print, and the paper has been included in two collections of classic mathematical logic papers.
