Free Mathematics
Mathematics is in its development entirely free and is only bound
in the selfevident respect that its concepts must both be consistent
with each other and also stand in exact relationships, established
by definitions, to those concepts which have previously
been introduced and are already at hand and established.
In particular,
in the introduction of new numbers it is only obligated
to give definitions of them which will bestow such a determinacy
and, in certain circumstances, such a relationship to the older
numbers that they can in any given instance be precisely distinguished.
As soon as a number satisfies all these conditions it can
and must be regarded in mathematics as existent and real.
Foundations of a General Theory of Manifolds Section §8.

Immanent and Transcendent Reality
First, we may regard the whole numbers as real in so far as, on
the basis of definitions, they occupy an entirely determinate place
in our understanding, are well distinguished from all other parts
of our thought and stand to them in determinate relationships,
and thus modify the substance of our minds in a determinate way.
But then, reality can also be ascribed to numbers to the extent
that they must be taken as an expression or copy of the events and
relationships in the external world which confronts the intellect,
or to the extent that, for instance, the various number classes
are representatives of powers that actually occur in physical or
mental nature.


Fear of Danger
It is not necessary, I believe, to fear, as many do, that these principles
present any danger to science. For in
the first place the designated conditions, under which alone the
freedom to form numbers can be practiced, are of such a kind
as to allow only the narrowest scope for discretion (WillkÂ¨ur).
Moreover, every mathematical concept carries within itself the
necessary corrective: if it is fruitless or unsuited to its purpose,
then that appears very soon through its uselessness and it will be
abandoned for lack of success.

Definition of Set
By a manifold or set I understand any multiplicity which can
be thought of as one, i.e. any aggregate of determinate
elements which can be united into a whole by some law.

Inductive Definition of Number
X is a subset of Ω = S(X) ∈ Ω
[Ω is the class of all numbers, S(X) is the smallest number larger than any number in the set of numbers X]

