The Foundations of Mathematics To understand this text, it might be helpful to keep track of some names; these are philosophers and mathematicians who have worked on the problems of logic and mathematics which are discussed below. In some cases, the boundaries are blurry between philosophers, mathematicians, and logicians. Charles Parsons – professor at Harvard; here cited as a historian of logical and mathematical developments. Gottlob Frege – professor in Germany; first to create a formal symbolic system, and the first to articulate the logicist point of view; he developed both propositional logic and quantified logic. Bertrand Russell – professor in England; refined propositional and quantified logics; refined logicist thought; generally a successor to Frege in logicism. W.V.O. Quine – professor at Harvard; worked within a logicist framework, rejecting some of Russell’s work and refining logicism; also worked within a set-theoretic framework. Immanuel Kant – professor in Germany; viewed arithmetic as the result of synthetic a priori reasoning, and based upon the preconditions of space and time, which are, in turn, structures created by human reason; laid the foundations for the development of intuitionism. Henri Poincaré – professor in France; wrote that mathematics was in no way based upon logic, but rather upon Kantian notions of space and time. L.E.J. Brouwer – professor in the Netherlands; carried forward Poincaré’s Kantian intuitionism, and opposed Hilbert’s formalism. David Hilbert – professor in Germany; axiomized mathematics and geometry; argued for formalism. Georg Cantor – professor in Germany; essentially founded modern set theory. Richard Dedekind – professor in Germany; refined set theory and number theory Ernst Zermelo – professor in Germany; axiomized set theory; refined the concept of well-ordered. Abraham Fraenkel – professor in Germany and Israel; refined Zermelo’s axiomization of set theory. Kurt Gödel – professor in Germany and Princeton; showed that the axiom of choice does not impair the ZF system. If you are reading this text and are not familiar with formal symbolic systems, please also consult the appendix at the end of this paper, where a brief overview of a propositional calculus is given. Of the many topics which are of mutual interest to both mathematicians and philosophers, one is the foundation of mathematics: as Charles Parsons wrote, “foundational research has always been concerned with the problem of justifying mathematical statements and principles, with understanding why certain evident propositions are evident, with providing justification of accepted principles which seem not quite evident, and with finding and casting off principles which are The Foundations of Mathematics – page 1 unjustified.” Philosophers and mathematicians who work in this field are concerned to show the reliability of mathematics, that it indeed has a solid foundation, and that its results are therefore trustworthy. While this is regarded in ordinary life as an obvious truth – who ever really doubts that 7 + 5 = 12 ? – it becomes reasonable to investigate when mathematics produces results that are either counter-intuitive or beyond the possibility of confirmation or verification by ordinary sensory experience, or by clear a priori rational thought. Such attention-demanding results are common in both contemporary physics, but also in ancient riddles. Thus philosophers like Bertrand Russell hoped to prove, at least, the consistency of ordinary arithmetic; but universally-acknowledged, or even nearly-universally-acknowledged, results have eluded the philosophical and mathematical communities. At stake is logical certainty and the degree to which we can rely on the results of reason: if we can’t take 7 + 5 = 12 as certain, what then could we take as certain? Charles Parsons notes that “two of the main qualities for which mathematics has always attracted the attention of philosophers are the great degree of systematization and the rigorous development of mathematical theories. The problem of systematization seems to be the initial problem in the foundations of mathematics, both because it has been a powerful force in the history of mathematics itself and because it sets the form of further investigations by picking out the fundamental concepts and principles. Also, the systematic integration of mathematics is an important basis of another philosophically prominent feature, its high degree of clarity and certainty. In mathematics systematization has taken a characteristic and highly developed form – the axiomatic method – which has from time to time been taken as a model for systematization in general.” We can take a look at various groupings of philosophers and mathematicians: notably, the logicists, the intuitionists, the formalists, and the set-theoreticians. These are not the only four schools of thought regarding the foundations of mathematics, but they are four good examples with which to begin. Logicism. - All mathematics, according to the logicist, are derived from logic. This belief is espoused by philosophers and mathematicians like Bertrand Russell and Gottlob Frege. A major text associated with the movement is Russell’s Principia Mathematica. Frege’s system was one of the first major attempts at a logicist foundation for mathematics. In addition to the usual axioms of logic, three other The Foundations of Mathematics – page 2 axioms are added, in logicist foundations of mathematics, to make most of mathematical arithmetic possible; numerous other lesser axioms are added for creating various other parts of mathematics. The as the number of axioms increases, the elegance and plausibility of logicist program decreases. The three main axioms added are: the axiom of reducibility, the axiom of infinity, and the axiom of choice. The axiom of reducibility allows for a function of one type with variables of a different types. These three axioms, added to the usual axioms of propositional and quantified logic, and a few lesser axioms, then make the rest of mathematics derivable. - Thus math is founded on logic and proceeds forth from out of logic. But the question arises, where is then the border between mathematics and logic? If math is founded upon logic, then all axioms belong to logic, but the three central axioms need for mathematics seem to be mathematical in nature: disguised mathematical postulates. So was math really drawn out of logic, or was math carefully hidden in logic by means of these axioms in order that it might seem that math would come forth form logic? So we ask, what is a mathematical axiom, what is a logical axiom? What is the border between math and logic? We must examine pure logic prior to the development of mathematical arithmetic, so that we can know truly what logic is, and how mathematics emerges from it. We must be sure that we have not used disguised mathematical axioms as logical axioms. Are the three axioms really logic, or merely disguised mathematics? What you count as logic determines whether or not logic can create mathematics. Frege, Russell, and others derived arithmetic from logic, but the question remains whether the non- arithmetical parts of mathematics can be additionally derived from logic. Can geometry and topology and probability be derived from the same logical basis which allegedly is the basis for arithmetic? Quine, the next generation of logicist, says yes, but many disagree with him. - Another question for the logicist, or another objection to logicism, asks why mathematics relates to the real world and the natural sciences, i.e., if logic generates mathematics, and logic is not a physical reality but rather a set of laws, prescriptive or descriptive, of thought, then why should mathematics bear any relation to physical reality? Why should math, if it is derived from thought, i.e., from logic, apply to anything but thought? Another way to ask this question is to ask whether math is self-referential or does it touch nature? The logicist must build a bridge from logic to nature to show how mathematics can be applied. Logicism also floundered on certain paradoxes, and the theory of types was introduced to remove those difficulties. The Foundations of Mathematics – page 3 Intuitionism. - This view arose as an opposition to logicism. The intuitionists rely on direct intuition rather than on logic. This view is therefore somewhat Kantian, because Kant wrote that time and space are intuitions and he deduced arithmetic and geometry from intuitions – but the intuitionist school of mathematics goes beyond any questions posed by Kant – Poincaré was the main founder of this school – he said that axioms are not needed, rather we begin with our clear intuitions, e.g., we have clear intuitions of 1 and 0 and therefore we can build arithmetic. So we don’t need logical definitions. After Poincaré, Brower developed intuitionism and wrote that mathematics is a mental activity and the mind knows its own intuitions better than anything else. Numbers and arithmetic are built up from intuitions of plurality and time. Mathematics is thus synthetic, composing the truth out of parts. Logic belongs to language, not to mathematics, the and introduction of logic into mathematics creates errors. Mathematics belongs to the mind and as such is inexpressible or at best only partially expressible. Logic belongs to expression and hence is posterior to mathematics. Logic is founded upon mathematics; logic is the rules of language, language used to express mathematics – or language attempting to express the inexpressible mathematics. Neither proofs nor paradoxes are important, and there is no obligation to be logical. And consistency is not important. A less ambitious, minimalist logic, an intuitionist logic, is allowable, but systems like Russell’s are discarded. For example, the intuitionists allow the principle of the excluded middle to be used only in certain circumstances, in a certain narrow scope, i.e., finite cases. But intuitionists do not allow the use of the principle of the excluded middle in dealing with infinite sets, indeed intuitionists avoid the use of infinite sets as unintuitive. They also avoid certain types of numbers as unintuitive, e.g., transcendental numbers and π .