A Budget of Paradoxes, Volume I | Page 3

Zirkels, Leipsic, 1892; Thomas Muir, "Circle," in the
eleventh edition of the Encyclopædia Britannica; the various histories
of mathematics; and to his own article on "The Incommensurability of
[pi]" in Prof. J. W. A. Young's Monographs on Topics of Modern
Mathematics, New York, 1911.
The editor wishes to express his appreciation and thanks to Dr. Paul
Carus, editor of The Monist and The Open Court for the opportunity of
undertaking this work; to James Earl Russell, LL.D., Dean of Teachers
College, Columbia University, for his encouragement in its prosecution;
to Miss Caroline Eustis Seely for her intelligent and painstaking
assistance in securing material for the notes; and to Miss Lydia G.
Robinson and Miss Anna A. Kugler for their aid and helpful
suggestions in connection with the proof-sheets. Without the generous

help of all five this work would have been impossible.
DAVID EUGENE SMITH.
TEACHERS COLLEGE, COLUMBIA UNIVERSITY.
* * * * *
A BUDGET OF PARADOXES
{1}
INTRODUCTORY.
If I had before me a fly and an elephant, having never seen more than
one such magnitude of either kind; and if the fly were to endeavor to
persuade me that he was larger than the elephant, I might by possibility
be placed in a difficulty. The apparently little creature might use such
arguments about the effect of distance, and might appeal to such laws
of sight and hearing as I, if unlearned in those things, might be unable
wholly to reject. But if there were a thousand flies, all buzzing, to
appearance, about the great creature; and, to a fly, declaring, each one
for himself, that he was bigger than the quadruped; and all giving
different and frequently contradictory reasons; and each one despising
and opposing the reasons of the others--I should feel quite at my ease. I
should certainly say, My little friends, the case of each one of you is
destroyed by the rest. I intend to show flies in the swarm, with a few
larger animals, for reasons to be given.
In every age of the world there has been an established system, which
has been opposed from time to time by isolated and dissentient
reformers. The established system has sometimes fallen, slowly and
gradually: it has either been upset by the rising influence of some one
man, or it has been sapped by gradual change of opinion in the many.
I have insisted on the isolated character of the dissentients, as an
element of the a priori probabilities of the case. Show me a schism,
especially a growing schism, and it is another thing. The homeopathists,

for instance, shall be, if any one so think, as wrong as St. John Long;
but an {2} organized opposition, supported by the efforts of many
acting in concert, appealing to common arguments and experience, with
perpetual succession and a common seal, as the Queen says in the
charter, is, be the merit of the schism what it may, a thing wholly
different from the case of the isolated opponent in the mode of
opposition to it which reason points out.
During the last two centuries and a half, physical knowledge has been
gradually made to rest upon a basis which it had not before. It has
become mathematical. The question now is, not whether this or that
hypothesis is better or worse to the pure thought, but whether it accords
with observed phenomena in those consequences which can be shown
necessarily to follow from it, if it be true. Even in those sciences which
are not yet under the dominion of mathematics, and perhaps never will
be, a working copy of the mathematical process has been made. This is
not known to the followers of those sciences who are not themselves
mathematicians and who very often exalt their horns against the
mathematics in consequence. They might as well be squaring the circle,
for any sense they show in this particular.
A great many individuals, ever since the rise of the mathematical
method, have, each for himself, attacked its direct and indirect
consequences. I shall not here stop to point out how the very accuracy
of exact science gives better aim than the preceding state of things
could give. I shall call each of these persons a paradoxer, and his
system a paradox. I use the word in the old sense: a paradox is
something which is apart from general opinion, either in subject-matter,
method, or conclusion.
Many of the things brought forward would now be called crotchets,
which is the nearest word we have to old paradox. But there is this
difference, that by calling a thing a crotchet we mean to speak lightly
of it; which was not the necessary sense of paradox. Thus in the
sixteenth century many spoke of the earth's motion as the paradox of {3}