philosopher, add: "And you can see that that exists." If, being a philosopher, you do add this,
you are uttering nonsense. To say "lions exist" means "there are lions," i.e. "'x is a lion' is true
for a suitable x." But we cannot say of the suitable x that it "exists"; we can only apply this verb
to a description, complete or incomplete. "Lion" is an incomplete description, because it applies
to many objects: "The largest lion in the Zoo" is complete, because it applies to only one object.
Now suppose that I am looking at a bright red patch. I may say "this is my present percept"; I
may also say "my present percept exists"; but I must not say "this exists," because the word
"exists" is only significant when applied to a description as opposed to a name.* This disposes
of existence as one of the things that the mind is aware of in objects.
I come now to understanding of numbers. Here there are two very different things to be
considered: on the one hand, the propositions of arithmetic, and on the other hand, empirical
propositions of enumeration. "2 + 2 = 4" is of the former kind; "I have ten fingers" is of the
latter.
I should agree with Plato that arithmetic, and pure mathematics generally, is not derived from
perception. Pure mathematics consists of tautologies, analogous to "men are men," but usually
more complicated. To know that a mathematical proposition is correct, we do not have to study
the world, but only the meanings of the symbols; and the symbols, when we dispense with
definitions (of which the purpose is merely abbreviation), are found to be such words as "or"
and "not," and "all" and "some," which do not, like "Socrates," denote anything in the actual
world. A mathematical equation asserts that two groups of symbols have the same meaning; and
so long as we confine ourselves to pure mathematics, this meaning must be one that can be
understood without knowing anything about what can be perceived. Mathematical truth,
therefore, is, as Plato contends, independent of perception; but it is truth of a very peculiar sort,
and is concerned only with symbols.
Propositions of enumeration, such as "I have ten fingers," are in quite a different category, and
are obviously, at least in part, dependent
* On this subject see the last chapter of the present work.