Does Mathematics Need New Axioms?

Abstract

Does mathematics need new axioms? was the second of three plenary panel discussions held at the ASL annual meeting, ASL 2000, in Urbana-Champaign, in June, 2000. Each panelist in turn presented brief opening remarks, followed by a second round for responding to what the others had said; the session concluded with a lively discussion from the floor. The four articles collected here represent reworked and expanded versions of the first two parts of those proceedings, presented in the same order as the speakers appeared at the original panel discussion: Solomon Feferman (pp. 401–413), Penelope Maddy (pp. 413–422), John Steel (pp. 422–433), and Harvey Friedman (pp. 434–446). The work of each author is printed separately, with separate references, but the portions consisting of comments on and replies to others are clearly marked.

Axioms as Definitions: Revisiting Poincaré and Hilbert

“…Our choice among all possible conventions is guided by experimental facts; but it remains free and is only limited by the necessity of avoiding every contradiction [...] In other words, the axioms of geometry (I do not speak of those of arithmetic) are only definitions in disguise. 2 Viewed as definitions the axioms are neither true, nor false.…”

The Axiom of Infinity and Transformations j: V → V

“…One of Cantor's perspectives about the mathematical universe is that the transition from finite to transfinite is reasonably smooth and consequently, the universe is fairly homogenous 2 See [12] for a survey of a variety of contemporary views on these issues. 3 A striking example of such an effort that bore great fruits was Silver's early efforts to prove the inconsistency of measurable cardinals; his efforts resulted in many of the foundational results on 0 # .…”

Concept analysis in programming language research: done well it is all right

“…Feferman [23] called a certain axioms "fundamental": these are "axioms for such fundamental concepts as number, set and function that underlie all mathematical concepts" (p. 403, emphasis in the original). Easwaran [21] argues, convincingly to me, that such axioms are a way for mathematicians to insulate mathematics from philosophy: mathematicians can decide individually on what philosophical basis they adopt them, but once they have each done that, they can work together to explore the mathematical reality they establish.…”

“…The modern mathematician tends to view axioms simplyin the words of Feferman [23]-as "definitions of kinds of structures which have been recognized to recur in various mathematical situations" (p. 403). But when trying to settle the concept of "type" I am not dealing with the abstraction of a recurring structure from particular instances into a general theory; or at least, it is precisely the question of what abstraction is best chosen that is at issue.…”