Philosophy of Mathematics
Philosophy of Mathematics is an active branch of Philosophy addressing questions about the character of Mathematics, the conduct of mathematical inquiry, and the role of mathematical objects in describing empirical phenomena. As a form of philosophical inquiry, it examines the record of mathematical inquiry and poses questions regarding its aims, its conduct, and its results. Although the questions are diverse and never-ending, a number of recurrent themes can be recognized:
- What are the sources of mathematical subject matter?
- What does it mean to refer to a mathematical object?
- What is the character of a mathematical proposition?
- What kinds of inquiry play a role in mathematics?
- What are the objectives of mathematical inquiry?
- What gives mathematics its grip on experience?
- What is the bearing of beauty on mathematics?
The terms "philosophy of mathematics'" and '"mathematical philosophy'" are not synonyms. The former refers to the study of the role and meaning of numbers and mathematical concepts, while the latter refers to the "doing" of actual theorems and various proofs in Mathematics and Meta-Logic.
Our concept of physical space is the result of a desire to order our experiences of the external world. This ordering process is accompanied by successive approximations and abstractions which lead to our concept of mathematical space. For the physicist the correspondence between the data of experience and his concept of physical space is all important. As the abstraction process continues, this correspondence becomes less significant, so that the mathematician feels free to concentrate upon the logical relations involved. (G. de B. Robinson, 5).
This is a process of abstraction that produces empirically bound concepts and formally free concepts in tandem, and that brings about a threefold relation among contingent experiences, concepts of physical space, and concepts of mathematical space. Achieving a more thorough understanding of this process, by which mathematical patterns are abstracted from concrete experience, developed as quasi-autonomous forms, and then applied back to experience in far-reaching and surprising ways, is one of the essential services that philosophical examination can perform for the benefit of mathematical thought.
Mathematical propositions, at least at first sight, appear to differ from other sorts of propositions, but in ways that have, historically speaking, been difficult to define precisely. One distinctive feature of mathematical propositions is, as Hilary Putnam sketched a common view of it, "the very wide variety of equivalent formulations that they possess", by which he does not mean the sheer number of ways of saying the same thing but "rather that in mathematics the number of ways of expressing what is in some sense the same fact (if the proposition is true) while apparently not talking about the same objects is especially striking" (Putnam, 170).
RealismMathematical Realism, like Realism in general, holds that mathematical entities exist independently of the human Mind. Thus humans do not invent Mathematics, but rather discover it, and any other intelligent beings in the Universe would presumably do the same. In this point of view, there is really one one sort of Mathematics that can be discovered: Triangles, for example, are real entities, not our creations.
PlatonismPlatonism is the form of Realism suggesting mathematical entities are abstract, have no spatiotemporal or causal properties, and are eternal and unchanging. This is often claimed to be the naive view most people have of numbers. The term Platonism is used because such a view is seen to parallel Plato's belief in a "World of Ideas", an unchanging ultimate reality that the everyday world can only imperfectly approximate. Plato's view partly came from Pythagoras, and his followers the Pythagoreans, who believed that the world was, quite literally, built up by the numbers.
LogicismLogicism is the provable thesis that Mathematics is reducible to Logic, and hence a branch of Logic (Carnap 1931/1883, 41). Logicists hold that Mathematics can be known a priori, but suggest that our knowledge of Mathematics is just part of our knowledge of Logic in general, and is thus analytic, not requiring any special faculty of mathematical intuition. In this view, Logic is the proper foundation of Mathematics, and all mathematical statements are necessary logical truths.
EmpiricismEmpiricism is a form of Realism that denies Mathematics can be known a priori at all. It says we discover mathematical facts by empirical research, just like facts in any of the other sciences. It is not one of the classical three positions advocated in the early 20th Century, but primarily arose in the middle of the century. This argues statements like "2 + 2 = 4" come out as uncertain, contingent truths, which we can only learn by observing instances of two pairs coming together and forming a quartet.
FormalismFormalism holds that mathematical statements may be thought of as statements about the consequences of certain string manipulation rules. For example, in the "game" of Euclidean Geometry (which is seen as consisting of some strings called "axioms", and some "rules of inference" to generate new strings from given ones), one can prove that the Pythagorean Theorem holds (that is, you can generate the string corresponding to the Pythagorean Theorem). Mathematical truths are not about numbers and sets and triangles and the like ? in fact, they aren't "about" anything at all. Thus, Formalism need not mean that mathematics is nothing more than a meaningless symbolic game. It is usually hoped that there exists some interpretation in which the rules of the game hold.
IntuitionismIntuitionism is a program of methodological reform whose motto is that "there are no non-experienced mathematical truths" (L.E.J. Brouwer). From this springboard, intuitionists seek to reconstruct what they consider to be the "corrigible" portion of Mathematics in accordance with concepts of constructibility, provability, intuition, and knowledge. Brouwer, the founder of the movement, held that mathematical objects arise from the a priori forms of the volitions that inform the perception of empirical objects. (CDP, 542)
ConstructivismLike Intuitionism, Constructivism involves the regulative principle that only mathematical entities which can be explicitly constructed in a certain sense should be admitted to mathematical discourse. In this view, Mathematics is an exercise of the human intuition, not a game played with meaningless symbols. Instead, it is about entities that we can create directly through mental activity. In addition, some adherents of these schools reject non-constructive proofs, such as a proof by contradiction.
FictionalismFictionalism was introduced in 1980 when Hartry Field rejected and in fact reversed Quine's indispensability argument, that Mathematics was indispensable for our best scientific theories, and therefore should be accepted as true. Field suggested that Mathematics was dispensable, and therefore should be rejected as false. He did this by giving a complete axiomatization of Newtonian Mechanics that didn't reference numbers or functions at all. He started with the "betweenness" axioms of Hilbert's Geometry to characterize Space without coordinatizing it, and then added extra relations between points to do the work formerly done with Vector Fields.
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- Philosophy of Language
- Philosophy of Science
- Scientific Method
- Set Theory
- Truth Theory
- History of Philosophy
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