[1] Of course, if we comprehend in our view only elementary
geometrical and algebraical science, it is easy to
show that they _do_ demand both axioms and intuitions.
Take, _e.g._ Euclid I. I., where in the construction
it is necessary to employ intuition for the assertion
that the arcs really cut one another. There is no
logical certainty that they do; in fact, in some other
conditions, _e.g._ in those of other space dimensions,
they might not.
[2] This is, of course, not the space of experience. Logic
and mathematics deal with implications of thought. See
B. Russell (_Hibbert Journal_, 1904, pp. 809-12), who
has shown that in all pure mathematics it is only the
implications that are asserted, not the premiss or the
consequence, as mathematicians used formerly to assume.
[3] De Morgan, Peirce, Schröder, and B. Russell have worked
out the logic of relations as well as the syllogism.
[4] See Taylor, “Elements of Metaphysics,” p. 13.
[5] See Dr. Caird, “Evolution of Theology in the Greek
Philosophers.”
[6] So Galileo, Newton, Huygens were philosophers in
science. Descartes, Pascal, Leibniz were mathematicians
as well as philosophers.
[7] See S. Augustine, _De Civitate Dei_, Book XII. ch.
xix.: “Ita vero suis quisque numerus proprietatibus
terminatur, ut nullus eorum par esse cuicumque alteri
possit. Ergo et dispares inter se atque diversi sunt,
et singuli quique finiti sunt, et omnes infiniti sunt.”
[8] See R. Dedekind, _Was sind und was sollen die Zahlen?_
1893.
[9] Two transfinite aggregates can have an ordinal
correspondence with one another.
[10] See G. Cantor, _Zur Lehre vom Transfiniten_. 1890.
[11] _e.g._ Mr. P. Jourdain, _Philosophical Magazine_.
1904.
[12] The same result is hinted at by Mr. Taylor.
Taylor, “Elements of Metaphysics,” p. 22.
[13] Linear order, 1, 2, 3, &c. Circular. A CD B,
A CD B.… The latter, it is true, involves the idea
of separation. But this idea can be developed from
those of inclusion and exclusion, which belong to the
fundamental laws of thought.
[14] The Absolute, according to a recent metaphysical
thinker, is “a conscious life which embraces the
totality of existence, all at once, and in a
perfect systematic unity, as the content of its
experience.”--Taylor, “Elements of Metaphysics,” p. 60.