THE HARDSHIPS OF A MAN WITH AN UNLIMITED INCOME
I once heard a man refer to his income as limited, in order to
illustrate the hardship of a class of men, of which he of course was
one, in having to pay a somewhat high income-tax. It is obvious that
this man spoke enviously, and consequently admitted the existence of
more fortunately placed individuals who had unlimited incomes. A little
reflection would have shown the man that he was not taking up a
paradoxical attitude. A "paradoxical attitude" is of course the
assertion of one or more propositions of which the truth cannot be
perceived by a philosopher--and particularly an idealist--and can be
perceived by a logician and occasionally, but not always, by a man of
common-sense. Such propositions are: "The cat is hungry," "Columbus
discovered America," and "A thing which is always at rest may move from
the position A to the different position B."
Now, if a man had an unlimited income, it is an immediate inference
that, however low income-tax might be, he would have to pay annually to
the Exchequer of his nation a sum equal in value to his whole income.
Further, if his income was derived from a capital invested at a finite
rate of interest (as is usual), the annual payments of income-tax would
each be equal in value to the man's whole capital. If, then, the man
with an unlimited income chose to be discontented, he would be sure of a
sympathetic audience among philosophers and business acquaintances; but
discontent could not last long, for the thought of the difficulties he
was putting in the way of the Chancellor of the Exchequer, who would
find the drawing up of his budget most puzzling, would be amusing.
Again, the discovery that, after paying an infinite income-tax, the
income would be quite undiminished, would obviously afford satisfaction,
though perhaps the satisfaction might be mixed with a slight uneasiness
as to any action the Commissioners of Income-Tax might take in view of
this fact.
A problem of a wholly different nature is connected with the possible
purchase by the man with an unlimited income of an enumerable infinity
of pairs of boots. If he wished to prove that he had an even number of
boots, it would be easy if right boots were distinguishable from left
ones, but if the man were a faddist of such a kind that he insisted that
his left boots should not be made in any way differently from his right
ones, it would not be possible for him to prove the theorem mentioned
unless he assumed what is known as "the multiplicative axiom." In fact
this axiom shows that it is legitimate to pick out an infinite
succession of members of an infinite class in an arbitrary way. In the
case of the pairs of boots, each pair contains two members, and if there
is no means of distinguishing between them, when we wish to pick out one
of them for each of the infinity of pairs, we cannot say which ones we
mean to pick out unless we assume, by means of the above axiom, that a
particularized member can always be found even with things of each of
which it can be said that, like Private James in the _Bab Ballads_,
No characteristic trait had he
Of any distinctive kind.
However, a solution of the puzzle was given by Dr. Denes Koenig of
Budapest. You first prove that there are points in space such that, if P
is one of them, not more than a finite number of pairs of boots are such
that each centre of mass of the two members of a pair is equidistant
from P. Taking a point P of this sort, select from each pair the boot
whose centre of mass is nearest P. (There may be a finite number of
pairs left over, but they can be dealt with arbitrarily.)
Another form of the problem is as follows. Every time the man bought a
pair of boots he also bought a pair of socks to go with it; he had an
enumerable infinity of pairs of each, and the problem is to prove that
he had as many boots as he had socks. In this case the boots, we will
suppose, can be divided into right and left, but the socks cannot. Thus
there are an enumerable infinity of boots, but the number of the socks
cannot be determined without admitting the axiom mentioned above. A
further difficulty might arise if the owner of the boots and socks lost
one leg in some accident, and told his butler to give away half his
socks. Naturally the butler would find great logical difficulties in so
doing, and it would seem to be an interesting ethical problem whether he
should be dismissed from his situation for failing to prove the
multiplicative axiom. Again, if the butler stole a pair of boots, the
millionaire would have as many pairs as before, but might have fewer
boots. There is as yet no evidence that the number of his boots is equal
to or greater than the number of pairs.