MATHEMATICS AND THE SCIENCES — LASLEY 187 



Zeno saw the conflict between these opposites, and used what he saw 

 to deny the possibility of motion, to discourage placing bets on 

 Achilles in his historic race with the tortoise, and for other strange 

 and bewildering purposes. Even today one doesn't just rush in to 

 show where Zeno was wrong. In his antinomies are found the baffling 

 ideas of the infinite, the deceiving implications of continual divisi- 

 bility. Down the years we trace these difficulties like a colored skein 

 in the pattern of scientific thought. They face the scientist in his 

 effort to understand the constitution of matter. Is this paper from 

 which I read smooth and unbroken, or is it made of discrete particles 

 bounding about hither and yon — a veritable beehive ? The physicist 

 leans to the latter view. (This opinion may help explain the nature 

 of what is being read from these pages.) What, then, about action 

 at a distance? How are light, radiation, energy, gravitation con- 

 veyed from here to there? What? No ether? Can we have ether 

 without continuity? If the ether is a jellylike mass, is it not com- 

 posed of particles? If it is composed of particles, will not the 

 quantum behavior of matter nullify the continuity of the action ? If 

 we have a continuous exciting cause, is it not strange that energy 

 should emerge in units (quanta), or not at all? Are there no frac- 

 tions? The physicist says, "No, no fractions." De Vries claimed 

 that evolution proceeds by "explosions." But Darwin, Newton, 

 Kant, Leibniz all believed in continuity. Plank's quantum theory 

 replaces a continuum of states in an isolated system by a finite number 

 of discrete states. 



The mathematician has been through — I should say, is in — this 

 same turmoil. He has never fully recovered from the Pythagorean 

 shock of the irrational. For a time it was thought that Weierstrass, 

 Dedekind, and Cantor had laid the spectre, but the contrary views of 

 Knonecker, Brouwer, and Weyl on the calculus of Leibniz and New- 

 ton, the feeling that this calculus is making "bricks without straw," 

 must at least have a hearing. The critics of continuity claim that 

 nothing which cannot actually be constructed by a finite number of 

 steps can hope to lead to a discipline free of paradox. They maintain 

 that all analysis must eventually subject itself to the domination of 

 the positive integer. Karl Pearson, one of the nonmathematical 

 scientists who shares this view, states the position thus, "No scientist 

 has the right to use things unless their existence can be demonstrated." 



Some may meet these difficulties by what has been called "a con- 

 tinuous but discreet silence." 



Certain it is that a Thomas Wolfe may write "Of Time and the 

 River" with a much more glib assurance than may an Einstein. 



Simple things these — in time such a perfect continuity ; in number 

 such discreteness (I came near saying "discretion"), and in the 

 shadows an infinity trying to bridge the gap. 



