I carry out in this paper a philosophical analysis of the principle of excluded middle (or, as it is often called in the version I favor here, principle of bivalence: any meaningful assertion is either true or false). This principle has been criticized, and sometimes rejected, on the charge that its validity depends on presuppositions that are not, some believe, universally obtainable; in particular, that any well-posed problem is solvable. My goal here is to show that, although excluded middle does (...) indeed rest on certain presuppositions, they do not have the character of hypotheses that may or may not be true, or matters of fact that may or may not be the case. These presuppositions have, I claim, a transcendental character. Hence, the acceptance of excluded middle does not necessarily require, as some have claimed, an allegiance to ontological realism or some sort of cognitive optimism, construed as factual theses concerning the ontological status of domains of objects and our capability of accessing them cognitively. DOI:10.5007/1808-1711.2011v15n2p333. (shrink)

It has been accepted since the early part of the Century that there is no problem formalizing mathematics in standard formal systems of axiomatic set theory. Most people feel that they know as much as they ever want to know about how one can reduce natural numbers, integers, rationals, reals, and complex numbers to sets, and prove all of their basic properties. Furthermore, that this can continue through more and more complicated material, and that there is never a real problem.

In mathematics we have demonstrably insoluble problems, one example being that of trisecting an arbitrary angle in elementary geometry. Every now and then, we encounter engineers and others who offer solutions of the insoluble and make some stir. To those who feel convinced of the demonstration of insolubility, these claimed solutions do not seem to deserve any serious considerations. In fact, such solutions have long since ceased to attract attention from mathematicians.The situation in philosophy seems different. Here we do not (...) possess demonstrations as clear and distinct as in mathematics. For example, it has been a fashion among philosophers for two centuries or so to answer Hume's scepticism concerning induction. Although, presumably, there never has been any answer or solution which is as widely accepted among philosophers as Hume's sceptical argument, yet philosophers continue to enjoy elaborating arguments for and against Hume's scepticism. And people who feel convinced by Hume's resaoning often find it difficult to understand why philosophers persist in trying to contradict Hume flatly instead of seeking for some more devious way out. (shrink)

I propose to sketch my views on several aspects of the philosophy of mathematics that I take to be especially relevant to philosophy as a whole. The relevance of my discussion would, I think, become more evident, if the reader keeps in mind the function of (the philosophy of) mathematics in philosophy in providing us with more transparent aspects of general issues. I shall consider: (1) three familiar examples; (2) logic and our conceptual frame; (3) communal agreement and objective certainty; (...) (4) the transcommunal universality of mathematics; (5) the big jump to the potential infinite; (6) the reconciliation of local creation with the hypothesis of discovery; (7) Platonism as realism plus conceptualism; (8) foundational studies and mathematical practice; and (9) the decomposition of philosophical disagreements. The views of Gödel and Wittgenstein are emphasized in order to add specificity to the discussions. (shrink)

I construe mathematical philosophy not in the narrow sense of philosophy of mathematics but in a broad indefinite sense of different manners of giving mathematics a privileged place in the study of philosophy. For example, in one way or another, mathematics plays an important part in the philosophy of Plato, Descartes, Spinoza, Leibniz, and Kant. In contrast, history plays a central role in the philosophy of Vico, Hegel, and Marx. In more recent times, Frege, Husserl, Russell, Ramsey, and Gödel all (...) began as mathematicians. One way of viewing Kant’s system of philosophy might be to stress that he was struck by the synthetic a priori character of mathematical propositions. He went on to offer a remarkable account of this fact and also look for and propose synthetic a priori foundations of physics, morality, and esthetics. (shrink)