Andrew D. Irvine

Physicalism in Mathematics

1. Epistemology & Nominalism.- 2. What Is Abstraction & What Is It Good For?.- 3. Beliefs About Mathematical Objects.- 5. Field & Fregean Platonism.- 5. ? in The Sky.- 6. Nominalism.- 7. The Logic of Physical Theory.- 8. Knowledge of Mathematical Objects.- 9. Physicalism, Reductionism & Hilbert.- 10. Physicalistic Platonism.- 11. Sets are Universals.- 12. Modal-Structural Mathematics.- 13. Logical & Philosophical Foundations for Arithmetical Logic.- 14. Criticisms of the Usual Rationale for Validity in Mathematics.- Contributors.- Index of Proper Names.

How Braess’ paradox solves Newcomb's problem*

Abstract Newcombs problem is regularly described as a problem arising from equally defensible yet contradictory models of rationality. Braess’ paradox is regularly described as nothing more than the existence of non‐intuitive (but ultimately non‐contradictory) equilibrium points within physical networks of various kinds. Yet it can be shown that Newcombs problem is structurally identical to Braess’ paradox. Both are instances of a well‐known result in game theory, namely that equilibria of non‐cooperative games are generally Pareto‐inefficient. Newcombs problem is simply a limiting case in which the number of players equals one. Braess’ paradox is another limiting case in which the ‘players’ need not be assumed to be discrete individuals. The result is that Newcombs problem is no more difficult to solve than (the easy to solve) Braess’ paradox.

Aristotle'S Early Logic

Aristotle is generally recognized as the founder of systematic logic or of what he called “analytics.” He has been the first to systematize universally valid logical laws. Aristotle has been also responsible for the remarkable accomplishment of developing logic in at least two distinct ways, including his almost complete theory of the syllogism and his complex and sophisticated theory of modal logic. In addition, he is noted for his work in axiomatic, and there is some evidence that he also began investigating what is now called “propositional logic,” although he did not develop these investigations systematically. It is widely accepted that Aristotles main contributions to logic begin—and some would say end—with the prior analytics. Aristotle holds what modern logicians call a classical conception of validity and, correspondingly, a classical conception of logical implication or entailment. This proposal is made in the face of the fact that Aristotle gives no account of these things anywhere in his writings.

Jack and Jill and Employment Equity

Jack and Jill have both applied for the same entry-level position at a local university. After interviewing the leading candidates, the members of the hiring committee agree that both Jack and Jill have all the necessary qualifications for appointment to the position. Both have the required education and training. Both have strong letters of recommendation from their Ph.D. supervisors and from their current employers. Both are similarly experienced and both are potentially capable of making important future contributions to their chosen discipline. The members of the hiring committee also agree that Jack and Jill are superior to all other applicants for the position. In short, in the judgment of the hiring committee, they are the two best qualified candidates and both meet their potential employers expectations concerning a successful applicant. Yet neither Jack nor Jill is clearly superior to the other.

Bertrand Russell's Logic

Bertrand Russell is generally recognized as one of the most important Englishspeaking philosophers, logicians and essayists of the twentieth century. Often cited along with G.E. Moore as one of the founders of modern analytic philosophy and along with Kurt Gödel as one of the most influential logicians of his time, Russell is also widely recognized for his sustained public contributions to many of the most controversial social, political and educational issues of his day. Even so, more than anything else, it is Russell’s work in logic and the foundations of mathematics that serves as his core contribution to intellectual history and that makes Russell the seminal thinker he is. His most significant achievements include

Frege on Number Properties

In the Grundlagen, Frege offers eight main arguments, together with a series of more minor supporting arguments, against Mill’s view that numbers are “properties of external things”. This paper reviews all eight of these arguments, arguing that none are conclusive.

Russell's Unknown Logicism: A Study in the History and Philosophy of Mathematics

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Aristotle on the Rule of Law

In Politics III.10 and IV.4, Aristotle discusses the difference between governments that are regulated by the rule of law and those that are not. Although he concludes that the rule of law helps guard against arbitrary and injudicious government action, Aristotle is also sensitive to the fact that in a democracy it is essential for the people to remain sovereign over the law. His discussion is helpful for understanding, not only the tension between the ‘rule of law’ and the ‘rule of men’, but also the complex role the rule of law plays in any modern democracy.