"""Once we allow some principle of recursive reasoning, we are landed with the infinite. There are an infinite number of natural numbers: for 0 is a natural number, and if anything is a natural number, its successor is a natural number. So there are natural numbers, and for any finite number, n, it is demonstrably false that there are exactly n natural numbers. We feel impelled to allow the question `How many natural numbers are there?', and the only possible answer seems to be `an infinite number'. But we have qualms.
Infinity seems out of this earth. It smacks of Platonism, mysticism and theology. The word `infinite' is a negative concept, contrasting not only with `(in other words, Cantor resides in Hades. Yay)
finite', in a strict mathematical sense, but with
`definite' and with `comprehensible'. Often, especially in theology and ancient philosophy, the Infinite is the Whole....the Universe, the Absolute, whose logic is difficult and fraught with inconsistencies. We are wise to be wary.
Although Parmenides, Plato, Augustine, William of Alnwick, Leibniz, Cantor, Dedekind and most modern mathematicians are fairly happy with infinity, other philosophers have had doubts. Aristotle allowed the existence of the potential infinite but denied
the actual infinite, and was followed by Aquinas and most of the Schoolmen. It is fair to place Weyl and the modern Intuitionists in that tradition. Locke had considerable difficulty in articulating a coherent and satisfactory account of infinity. Berkeley was deeply
critical of infinitesimals and mathematical infinity generally, and modern Finitists likewise reject every sort of infinity, and try to confine themsleves to finite numbers alone. But infinitistic arguments
keep seeping in, and strict Finitists distinguish themselves from their laxer brethren, and still are not stringent enough in their scepticism to escape the strictures of the ultra strict."""
It is easy to stifle doubts, and accept the infinite as part of the
mathematical exercise, justified by the general success of mathematics.
Geometry runs more smoothly if we allow points at infinity:
then we can say that two straight lines always intersect, parallel
lines meeting at a point at ifinity. If we add the symbol 1 to our
other numerical symbols, we are able to discuss series, sums and
integrals much more incisively. Once we allow the question `How
many natural numbers are there?' it is illuminating to answer @0.
It seems silly to let philosophical scruples deter us from entering
But credulity conduces to inconsistency. Easy acceptance is not
on. It is important, though di cult, to articulate the objections
to in nity that we are inclined to feel. What tends to happen is
that the objections are only half-formulated, and never adequately
considered; instead, the mathematician gets used to operating with
in nity, and, as it were, represses the doubts he once felt without
ever thinking them through properly and either seeing that they are
valid and accepting them or seeing that they are not valid and why.
As in psychoanalysis, repression causes trouble later on. Many Intuitionists
show symptoms of not having come to terms with in nity
in their youth, and su ering in consequence from psychological
lesions in middle age. The only prophylactic is to give full rein to
doubts at the first encounter. Doubt now, doubt bravely: dubita
fortiter to parody Luther.