THE PARADOXES OF LOGIC
We have already[98] referred to the contempt shown by some
mathematicians for exact thought, which they condemn under the name
of "scholasticism." An example of this is given by Schoenflies in
the second part of his publication usually known as the _Bericht
ueber Mengenlehre_.[99] Here[100] a battle-cry in italics--
"_Against all resignation, but also against all scholasticism!_"--
found utterance. Later on, Schoenflies[101] became bolder and adopted a
more personal battle-cry, also in italics, and with a whole line to
itself:
"_For Cantorism but against Russellism!_"
"Cantorism" means the theory of transfinite aggregates and numbers
erected for the most part by Georg Cantor. Shortly speaking, the great
sin of "Russellism" is to have gone too far in the chain of logical
deduction for many mathematicians, who were perhaps, like
Schoenflies,[102] blinded by their rather uncritical love of
mathematics. Thus it comes about that Schoenflies[103] denounces
Russellism as "scholastic and unhealthy." This queer blend of qualities
would surely arouse the curiosity of the most _blase_ as to what strange
thing Russellism must be.[104]
Schoenflies[105] said that some mathematicians attributed to the logical
paradoxes which have given Russell so much trouble to clear up,
"especially to those that are artificially constructed, a significance
that they do not have." Yet no grounds were given for this assertion,
from which it might be concluded that the rigid examination of any
concept was unimportant. The paradoxes are simply the necessary results
of certain logical views which are currently held, which views do not,
except when they are examined rather closely, appear to contain any
difficulty. The contradiction is not felt, as it happens, by people who
confine their attention to the first few number-classes of Cantor, and
this seems to have given rise to the opinion, which it is a little
surprising to find that some still hold, that cases not usually met
with, though falling under the same concept as those usually met with,
are of little importance. One might just as well maintain that
continuous but not differentiable functions are unimportant because they
are artificially constructed--a term which I suppose means that they do
not present themselves when unasked for. Rather should we say that it is
by the discovery and investigation of such cases that the concept in
question can alone be judged, and the validity of certain theorems--if
they are valid--conclusively proved. That this has been done, chiefly by
the work of Russell, is simply a fact; that this work has been and is
misunderstood by many[106] is regrettable for this reason, among others,
that it proves that, at the present time, as in the days in which
_Gulliver's Travels_ were written, some mathematicians are bad
reasoners.[107]
Nearly all mathematicians agreed that the way to solve these paradoxes
was simply not to mention them; but there was some divergence of opinion
as to how they were to be unmentioned. It was clearly unsatisfactory
merely not to mention them. Thus Poincare was apparently of opinion that
the best way of avoiding such awkward subjects was to mention that they
were not to be mentioned. But[108] "one might as well, in talking to a
man with a long nose, say: 'When I speak of noses, I except such as are
inordinately long,' which would not be a very successful effort to avoid
a painful topic."
Schoenflies, in his paper of 1911 mentioned above, adopted the
convenient plan of referring these logical difficulties at the root of
mathematics to a department of knowledge which he called "philosophy."
He said[109] of the theory of aggregates that though "born of the
acuteness of the mathematical spirit, it has gradually fallen into
philosophical ways, and has lost to some extent the compelling force
which dwells in the mathematical process of conclusion."
The majority of mathematicians have followed Schoenflies rather than
Poincare, and have thus adopted tactics rather like those of the March
Hare and the Gryphon,[110] who promptly changed the subject when Alice
raised awkward questions. Indeed, the process of the first of these
creatures of a child's dream is rather preferable to that of
Schoenflies. The March Hare refused to discuss the subject because he
was bored when difficulties arose. Schoenflies would not say that he was
bored--he professed interest in philosophical matters, but simply called
the logical continuation of a subject by another name when he did not
wish to discuss the continuation, and thus implied that he had discussed
the whole subject. Further, Schoenflies would not apparently admit that
the one method of logic could be applied to the solution of both
mathematical and philosophical problems, in so far as these problems are
soluble at all; but the March Hare, shortly before the remark we have
just quoted, rightly showed great astonishment that butter did not help
to cure both hunger and watches that would not go.[111] The judgment of
Schoenflies by which certain apparently mathematical questions were
condemned as "philosophical," rested on grounds as flimsy as those in
the Dreyfus Case, or the Trial in _Wonderland_.[112]
FOOTNOTES:
[98] Chapters VII and XXXVI.
[99] _Die Entwickelung der Lehre von den Punktmannigfaltigkeiten._
Bericht, erstattet der deutschen Mathematiker-Vereinigung, Leipzig,
1908.
[100] _Ibid._, p. 7. The battle-cry is: "_Gegen jede Resignation, aber
auch gegen jede Scholastik!_"
[101] "Ueber die Stellung der Definition in der Axiomatik," _Jahresber,
der deutsch. Math.-Ver._, vol. xx., 1911, pp. 222-5. The battle-cry is
on p. 256 and is: "Fuer den Cantorismus aber gegen den Russellismus!"
[102] _Ibid._, p. 251. "Es ist also," he exclaims with the eloquence of
emotion and the emotion of eloquence, "nicht die Geringschaetzung der
Philosophie, die mich dabei treibt, sondern die Liebe zur
Mathematik;..."
[103] "Ueber die Stellung der Definition in der Axiomatik," _Jahresber,
der deutsch. Math.-Ver._, vol. xx., 1911, p. 251.
[104] [Cf. for this, _M._, vol. xxii., January 1912, pp. 149-58.--ED.]
[105] _Bericht_, 1908, p. 76, note; cf. p. 72.
[106] E.g. in F. Hausdorff's review of Russell's _Principles_ of 1903 in
the _Vierteljahrsschr. fuer wiss. Philos. und Soziologie_.
[107] [Cf. _M._, vol. xxv., 1915, pp. 333-8.--ED.]
[108] Russell, _A. J. M._, vol. xxx., 1908, p. 226.
[109] _Loc. cit._, p. 222.
[110] See Appendix Q.
[111] See Appendix R.
[112] See Appendix S.