The Competitiveness of Nations in a Global Knowledge-Based Economy

Kenneth E. Boulding

**The Limitations of
Mathematics: An Epistemological Critique **

Seminar in the Application of Mathematics to the Social Sciences

http://csf.colorado.edu/boulding/limits_of_math.html

It is a commonplace that mathematics is
a tool or a language. This is particularly true of applied mathematics where we
are interested in it not merely as an end in itself but as a means to some other
end. The “other end” in so far as mathematics is a tool is the increase of
knowledge: so far as mathematics is a language, it is the transmission of
knowledge.

Means, however, always in part determine
the ends for which they are ostensibly designed, and neither tools nor
languages are neutral in regard to the ends which they
serve. Tools help to determine the tasks we will undertake, and languages
influence what we will say. These notes are directed towards examining the
limitations of mathematics both as a tool and as a language, especially in
regard to possible distortions of the growth of knowledge which might result
from too exclusive a reliance on mathematical tools. I do not intend there
remarks to be in any way derogatory; I assume I can take as given the
extraordinary beauty and power of mathematical tools. The rounded growth of
knowledge however requires both mathematics and something that is not
mathematics. By inquiring into the limitations of mathematics we may become
clearer as to what is “not-mathematics.”

I will consider first three limitations
of mathematics as a tool.

(i) The
delicacy or coarseness of a tool has an important effect on the task which can
be done with it; we do not cut out cataracts with a buzz-saw or cut down trees
with a scalpel. Mathematics clearly has a bias on the side of delicacy and
exactness. Where the task requires delicacy, this is all to the good. If
however the empirical universe which we are trying to know is not delicate, too
great a reliance on mathematics may be misleading, if it is not checked by good
judgment about the nature of the empirical universe itself. This is a problem
of considerable importance for the social sciences, where the empirical
universe itself is frequently “coarse” in texture. A good example of this
difficulty is the theory of “rational behavior” in economics. The calculus is
too fine, relationships in the empirical world are not continuous, and the
theory of uncertainty is largely an attempt to discuss vagueness by means of
clear concepts! I do not imply, of course, that mathematics is incapable of
modifying itself in the direction of the buzz-saw for there are some signs of
this. The bias, however, is at present all towards the scalpel.

(ii) Another bias of mathematics is
towards rigidity. Any given mathematical process is a sausage-machine -- what
comes out is determined entirely by what goes in. A more exact analogy is that
of a calculating machine with a set program. We must distinguish here between
mathematical operations as such and the programming of these operations.
Mathematical operations can be done by a machine or a graduate student,
whichever is cheaper. Programming, however, is the real art of the
mathematician, and paradoxically enough, programming itself cannot be done wholly
by mathematical operations. The creative act in mathematics itself involves a
great deal of “not-mathematics.” An important bias here arises from the
difficulties of adjusting the program while the operations are proceeding.

(iii) The third bias is somewhat
connected with the second, and arises because mathematical tools are frequently
so expensive to acquire that there is a compulsion to use them, almost like the
compulsion to read. The mathematician who cannot think unless he has chalk and
a big blackboard is in a bad way.

I now want to consider certain
limitations of mathematics as a language. This is perhaps only a special aspect
Of its use as a tool, in this case as a tool of
communication. As a language mathematics exhibits all three biases mentioned
above. it must also be emphasized that it is not a
complete language-that is, it is a device for talking about some things, not
about all things. It is also further abstracted from the empirical world than
ordinary language. All language involves abstraction (orderly loss of
information), we can never talk about as much as we
know. Mathematics is an abstraction from ordinary language, an abstraction
which again involves loss of “richness” of information. There would be nothing
wrong with this if it did not tend to set up a self-perpetuating linguistic
barrier. The more mathematicians talk to each other the less skilled they
become, very often, in communicating with the outside world. Knowledge
monopolies (“mysteries”) also tend to lead to high status in any culture, and
there is a constant temptation of any priesthood towards a cult of
unintelligibility. Mathematicians are peculiarly exposed to this temptation;
they come to value style and elegance according to internal aesthetic patterns
which have little to do with communicability. The mathematician who explains
himself loses status with the other mathematicians, as he is obviously not
talking for them! We approach the point where only the mathematician can
criticize the mathematician, but his very identification with the inner circle
prevents him from criticism, from the point of view of the surrounding culture.

The biases of mathematics can, of
course, be foreseen and avoided. To do this, however, requires a theory of
knowledge. I suggest that any adequate theory of knowledge must take account of
the following elements:

(i) knowledge is an organic
structure which follows certain laws of organic growth.

(ii)
knowledge is not wholly intellectual, not wholly
conscious, nor wholly clear.

(iii)
knowledge grows by a “feedback” process of messages which are perceived as
related to previous acts, and filtered through a value system which is itself
part of the developing knowledge structure. It grows also by symbolic
communications of an authoritative nature. “Public,” that is, shared knowledge
grows through a complicated system of symbolic communication and feedback.

Summarized by Harold Slater.