f making itself go are unchanging, we can calculate what
positions of imagined arrest it will exhibit hereafter under given
conditions. We can compute, for instance, at what point Achilles
will be, and where the tortoise will be, at the end of the twentieth
minute. Achilles may then be at a point far ahead; but the full detail
of how he will have managed practically to get there our logic never
gives us--we have seen, indeed, that it finds that its results
contradict the facts of nature. The computations which the other
sciences make differ in no respect from those of mathematics. The
concepts used are all of them dots through which, by interpolation or
extrapolation, curves are drawn, while along the curves other dots are
found as consequences. The latest refinements of logic dispense
with the curves altogether, and deal solely with the dots and their
correspondences each to each in various series. The authors of these
recent improvements tell us expressly that their aim is to abolish the
last vestiges of intuition, _videlicet_ of concrete reality, from the
field of reasoning, which then will operate literally on mental dots
or bare abstract units of discourse, and on the ways in which they may
be strung in naked series.

This is all very esoteric, and my own understanding of it is most
likely misunderstanding. So I speak here only by way of brief reminder
to those who know. For the rest of us it is enough to recognize this
fact, that altho by means of concepts cut out from the sensible flux
of the past, we can re-descend upon the future flux and, making
another cut, say what particular thing is likely to be found there;
and that altho in this sense concepts give us knowledge, and may be
said to have some theoretic value (especially when the particular
thing foretold is one in which we take no present practical interest);
yet in the deeper sense of giving _insight_ they have no theoretic
value, for they quite fail to connect us with the inner life of the
flux, or with the causes that govern its direction. Instead of being
interpreters of reality, concepts negate the inwardness of reality
altogether. They make the whole notion of a causal influence between
finite things incomprehensible. No real activities and indeed no real
connexions of any kind can obtain if we follow the conceptual logic;
for to be distinguishable, according to what I call intellectualism,
is to be incapable of connexion. The work begun by Zeno, and