Mathematical philosophy, a study of fate and freedom; lectures for educated laymen, by Cassius J. Keyser.

184 MATHEMATICAL PHILOSOPHY
doubled; neither is the product of integers preserved, for,
if xi and x2 be any two numbers in I, and their transforms
be yi and y2, then the product X1X2 is transformed into
yiy2, which is not xlx2 but is 4x1x2; sums and differences
are also variant, the transforms of xi+X2 and xi-x2
being respectively 2(x +x2) and 2(xi-x2); but ratios
yi 2j1 xi
are invariant for =2 =. Now suppose the law of
y2 2X2 X2
transformation to be: (2) y =x; then neither the property of being an integer, nor value, nor product nor sum
nor difference is invariant, but ratio is; so, too, is the
property of being a number, as was also the case under (I),
though not there mentioned. This latter property is
again invariant under the transformation: (3) y=x+ I;
but ratio is not. Is the property of being a number invariant under every transformation of I? No, it is not
invariant under the transformation converting the integers of I into an endless succession of days, di, d2 d,...
in accord with the law: (4) n -> d,; but even under
(4) we have an interesting invariant,-namely, the property of nextness,-for the transform of an integer next
after a given one is itself next after the given one's transform. Note that under none of the transformations
(I), (2), (3) is the class I invariant as a whole, being
converted by (i) and (3) into a part of itself and by (2)
into a class including it and another class besides-a class
of fractions. But I as a whole is invariant under many
transformations-for example, under the reciprocal oneto-one transformation: (5) I * — 2, 3 — > 4,....
For a different sort of example, consider the following.
Let D denote something very strong and solid, say a
diamond at a certain time and place; suppose it removed
gently to another place, and now denote it by D'; D and D'