Kenneth E. Boulding
The Limitations of Mathematics: An Epistemological Critique
Seminar in the Application of Mathematics to the Social Sciences
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.