![]() |
![]() |
4 |
![]() |
The Crisis in Psychology
249
16 is not only the addition of 16 unities, it is also the square
of
4 and the biquadrate
of
2
. . ..
Only even numbers can be divided by two
. . ..
For division by 3 we have
the rule
of
the sum of the
figutes.
. ..
For 7 there is a
special
law..
..
Zero
destroys
any other number by which it is multiplied; when it is made divisor or dividend with
regard to some other number, this
number
will in the first case become infinitely large,
in the
second case
infinitely small [Engels, 1925/1978, pp. 522/524].
About both concepts of mathematics one might say what Engels, in the words of
Hegel, says about zero: “The non-existence of something is a
specific
non-existence”
[ibid., p.
525],
i.e., in the end it is a real non-existence. [7] But maybe these qualities,
properties, the
specificity
of concepts as such, have no relation whatsoever to reality?
Engels [ibid., p. 530] clearly rejects the view that in mathematics we are dealing
with purely free creations and imaginations of the human mind to which nothing
in the objective world corresponds. Just the opposite is the case. We meet the pro-
totypes of each of these imaginary quantities in nature. The molecule possesses
exactly the same properties in relation to its corresponding mass as the mathemati-
cal differential in relation to its variable.
Nature operates with these differentials, the molecules, in exactly the same way and
according to the same laws as mathematics with
its
abstract differentials [ibid., p. 531].
In mathematics we forget all these analogies and that is why its abstractions turn
into something enigmatic. We can always find
the real relations from which the mathematical relation
. . .
was taken
. . .
and
even
the natural analogues of the mathematical way to make these relations manifest [ibid.,
p.
534].
The prototypes of mathematical infinity and other concepts lie in the real world.
The mathematical infinite is taken, albeit unconsciously, from
reality,
and that is why
it
can
only be explained on the basis of reality, and not on the basis of itself, the
mathematical abstraction (ibid., p. 534).
If this is
true
with respect to the highest possible, i.e., mathematical abstraction,
then bow much more obvious it is for the abstractions of the real natural sciences.
They must, of course, be explained only on the basis of the reality from which they
stem and not on the basis of themselves, the abstraction.
2. The second point that we need to make in order to present a fundamental
analysis of the problem of the general science is the opposite of the first. Whereas
the first claimed that the highest scientific abstraction contains an element of reality,
the second is the opposite theorem: even the most immediate, empirical, raw, sin-
gular natural scientific fact already contains a first abstraction. The real and the
scientific fact are distinct in that the scientific fact is a real fact included into a
certain system of knowledge, i.e., an abstraction of several features from the inex-
haustible sum of features of the natural fact. The material of science is not raw,
but logically elaborated, natural material which has been selected according to a
certain feature. Physical body, movement, matter—these are all abstractions. The
fact itself of naming a fact by a word means to frame this fact in a concept, to
single out one of its aspects; it is an act toward understanding this fact by including
it into a category of phenomena which have been empirically studied before. Each
word already is a theory, as linguists have noted for quite some time and as Po-
tebnya [1913/1993] has brilliantly demonstrated. [8]
Everything described as a fact is already a theory. These are the words of
Goethe to which MUnsterberg refers in arguing the need for a methodology. [9]
When we meet what is called a cow and say: “This is a cow,” we add the act of
thinking to the act of perception, bringing the given perception under a general
|
![]() |
![]() |
4 |
![]() |