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Like Spencer, we may begin by distinguishing in the sciences two groups
-- the _Abstract_ and the _Concrete_. The former group deals with the
conceptual equivalents of the modes under which the perceptive faculty
discriminates objects, the latter with the concepts by aid of which we
describe the contents of perception. We have the, to start with, the
following division:
_Perceptions_ (Sense-Impressions and Stored Impresses)
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_Modes of Perception_ _Contents of Perception_
Abstract Science Concrete Science
_Abstract Science_ may deal with the general relations of discrimination
... The general relations of discrimination may be either qualitative or
quantitative. The former branch is termed _Logic_, and discusses the
general laws by which we identify and discriminate things, or what are
frequently termed the laws of thought....
Both Time and Space lead us at once too the conception of quantity or
number, and we thus have a large and important branch of _Abstract
Science_ which deals with the laws of quantity. Now quantity may be
either _discrete_ and definite like the numbers of arithmetic 8, 100,
1/13, 17/4, etc...or it may be _continuous_ and changing with other
quantities...
Among the sciences which deal especially (if not entirely) with discrete
quantity, the best known are probably _Arithmetic_ and _Algebra_; but
there are a number of others we ought to briefly note. We want to know
how to measure quantity and what errors are likely to arise in its
measurement. Closely allied to this is the discussion of probable and
average quantities, dealing with cases where we cannot measure individual
quantity, but only approximate and average results. Hence arise the
_Theory of Measurement_, _Theory of Errors_, _Theory of Probability_,
_Theory of Statistics_, etc.
Passing to change in quantity,...to understand the mutual relationship
of quantities which are functions of each other is the scope of sciences
like the _Theory of Functions_ which teaches us how functions can be
represented and handled....Special branches are the _Differential
Calculus_ or _Calculus of Fluxions_...and the _Integral Calculus_ or
_Calculus of Sums_....
With this we complete our review of _Abstract Science_. We see that
it embraces all that is usually grouped as _Logic_ and _Pure Mathematics_.
In these branches we deal with conceptual modes of discrimination; and
since the concepts formed are in general narrowly defined and free from
the infinite comlexity of the contents of perception, we are able to
reason with great preciseness, so that the results of these sciences are
absolutely valid for all that falls under their definitions and axioms.
On the account the branches of _Abstract Science_ are frequently spoken
of as the _Exact Sciences._
-- James A Landau