These were lifted from Tait's paper on Cantor's "Grundlagen" (Foundations of a General Theory of Manifolds).

Mathematics is in its development entirely free and is only bound in the self-evident 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.

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.

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 (Willk¨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.

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.

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]