Professor Grünbaum's much-discussed refutation of Zeno's metrical paradox turns out to be ad hoc upon close examination of the relevant portion of measure theory. Although the modern theory of measure is able to defuse Zeno's reasoning, it is not capable of refuting Zeno in the sense of showing his error. I explain why the paradox is not refutable and argue that it is consequently more than a mere sophism.

In his monograph On Numbers and Games, J. H. Conway introduced a real-closed field containing the reals and the ordinals as well as a great many less familiar numbers including $-\omega, \,\omega/2, \,1/\omega, \sqrt{\omega}$ and $\omega-\pi$ to name only a few. Indeed, this particular real-closed field, which Conway calls No, is so remarkably inclusive that, subject to the proviso that numbers—construed here as members of ordered fields—be individually definable in terms of sets of NBG, it may be said to contain (...) “All Numbers Great and Small.” In this respect, No bears much the same relation to ordered fields that the system ℝ of real numbers bears to Archimedean ordered fields. In Part I of the present paper, we suggest that whereas $\mathbb{R}$should merely be regarded as constituting an arithmetic continuum, No may be regarded as a sort of absolute arithmetic continuum, and in Part II we draw attention to the unifying framework No provides not only for the reals and the ordinals but also for an array of non-Archimedean ordered number systems that have arisen in connection with the theories of non-Archimedean ordered algebraic and geometric systems, the theory of the rate of growth of real functions and nonstandard analysis. In addition to its inclusive structure as an ordered field, the system No of surreal numbers has a rich algebraico-tree-theoretic structure—a simplicity hierarchical structure—that emerges from the recursive clauses in terms of which it is defined. In the development of No outlined in the present paper, in which the surreals emerge vis-à-vis a generalization of the von Neumann ordinal construction, the simplicity hierarchical features of No are brought to the fore and play central roles in the aforementioned unification of systems of numbers great and small and in some of the more revealing characterizations of No as an absolute continuum. (shrink)

Much of Grünbaum's work may be regarded as a careful development and systematic elaboration of the Riemann-Poincaré thesis of the conventionality of congruence, the thesis that the continuous manifolds of space, time, and space-time are intrinsically metrically amorphous, i.e. are devoid of intrinsic metrics. Therefore, to appreciate Grünbaum's philosophical contributions, one must have a clear understanding of what he means by an intrinsic metric. The second and fourth sections of this paper are exegetical; in them we try to piece together, (...) from his sundry remarks about intrinsic metrics, what Grünbaum means by the term 'intrinsic metric.' We shall argue that, the customary carefulness and precision of Grünbaum's writings notwithstanding, there are residual unclarities and difficulties which beset his conception of an intrinsic metric. In the fifth section we shall propose an explication of Grünbaum's notion of an intrinsic metric which seems on the whole faithful to Grünbaum's intuitions and insights and which also seems capable of performing the philosophical services which his work demands of that notion. The third section is a digression on Zeno's metrical paradox of extension. (shrink)