20th WCP: Algebra As Thought Experiment

In the sixteenth century, physics was a part of the general subject known as philosophy. When Galileo wrote the Dialogues Concerning the Two Chief World Systems, he was commenting on some aspects of Aristotle's Philosophy. He was more favorable to the use of mathematics in various problems of physics than was current in his day. He may be described as trying to revive an Archimedean conception of motion in terms of geometry, which may be called "kinematics."

Galileo also introduced the experimental method into physics. The experimental method had been very successfully utilized in Alchemy, which was a low or a clandestine science at the time. Its success in physics brought the issue of experimentation into the spot light. Since there are few claims original to Galileo that we may still wish to defend, we may want to describe his success as the introduction of these techniques which he invented or adapted for subsequent physicists to employ. Experimental techniques have improved considerably since Galileo's day. But my concern today is with mathematical technique. The mathematical technique of modern physics has a great deal to do with the use of algebra, which was not known to Galileo, in fact. It was Descartes' great contribution to the technique of modern physics.

Compared with philosophers of before, what is striking about modern physics is the method employed: Typically, a physicist works with experiments, which are interpreted in terms of some algebraic formulae. These formulae are then manipulated, and a new result is obtained, which is then sought to be interpreted experimentally. Sometimes, the experimental result is not obvious. It needs to be investigated, which leads to new experimental discoveries. These new experimental results have to be interpreted in turn in formulaic terms, which can fail to be adequate, leading in its turn to the need to make mathematical innovations.

These are the respects in which physics is different from philosophy. In all other respects, I would suggest, physics is still much like philosophy of old. In a naive way, we can speak of this dual method of algebra and experiment that I have just sketched as the method of physics. Of course, in doing so we have to neglect the portion or aspect of physics which is like philosophy and concentrate our attention on where they differ. I call this a naive conception of method because it is only a prima facie observation. It does not solve the many problems concerning method which we may want to solve. It serves merely to introduce them.

The astonishing thing about this method, from the perspective of philosophers, is that it is so successful. This little branch of philosophy, which experimented and took up algebra came to be much bigger and more successful than the body out of which it grew. It is, even more than the lobster's claw, very much bigger than the lobster. Whether we give a philosophical or a social account of the success of physics there is little question that it made a very big mark in our conception of the world. Husserl thought, and I rather agree with him, that modern philosophy is merely an attempt to cope with the problems that are left unsolved by the extraordinary, but very partial, if growing, success of modern classical physics.

We need to understand why physics has been so successful over the last four hundred years. A simple answer is to interpret the success of physics as nothing less than the discovery of a fundamental truth about the world, which would of course explain its success very well. But if we mean by "the discovery of the truth" that what was discovered was once and for all to be declared as the truth, then in more recent times, Einstein, Bohr and others have shown us otherwise.

The philosopher who took this question up most seriously of all was Kant, who found that Hume's critique of induction and of necessity had left the predictive success of physics, and particularly its unexpected predictive success, inexplicable. Kant tried to resolve the difficulty by dividing our knowledge into two: There is, he said an experimental method, which gives us detailed knowledge of phenomena, and there is a mathematical method, which tells us something else. Mathematics does not give us detailed knowledge of particular phenomena, but rather the general conditions governing all phenomena. He suggested that the way to understand why mathematics successfully describes the world is that it is the only way to perceive or to conceive it.

Few of us today would want to defend all of Kant's views on mathematics, or on physics. But it is an astonishing fact that even when all of us are keen to disagree with him about this or that about mathematics or physics, we agree that the investigations of the philosophy of mathematics and of the philosophy of physics are different. Although Quine's account of holism has done much to blur the difference in any account of them, if we compare the professions which constitute the philosophy of mathematics with that of the philosophy of physics (or with that of the philosopher of the empirical sciences, in general) we find that these subjects still have little in common, and a great gulf separating them.

Like Quine, I would like to question the separation of physics from mathematics. Unlike Quine, I would like to separate the role of algebra from other parts of mathematics. My main theses will be these:

1. Mathematical physics is a single empirical subject, and not, as Kant thought, philosophically two.

Kant separated mathematics for special philosophical study, the study of how synthetic a priori judgment is possible, from the study of experimental physics, to be based on induction.

Both of these attempts have given us theoretical trouble. Concerning mathematics, we philosophers cannot give a realistic account of what it is about, while at the same time explaining how we can know the objects of mathematics. If we do the one, we cannot adequately do the other. In the philosophy of the empirical sciences, we can neither state the principles of induction which would explain the predictive success of physics, nor explain its success by avoiding reliance on induction. Since Kant we have had twin subjects each of which seems to fail precisely because they are separated, as far as I can see.

Kant's solution to the difficulty of understanding the success of mathematical physics is this: he describes mathematics as a piece of a synthetic knowledge which is nevertheless known a priori (or prior to the inductive generalization of any particular experiences) because geometry and arithmetic are presupposed by the very possibility of experience. Nothing can possibly be experienced which is not in space (analyzed in geometry) or in time (yielding arithmetic).

The difficulty in Kant's account of mathematical physics is that it can explain only one round of success in physics. If we say that Newton's success was to be understood as the discovery of the true mathematical laws presupposed by the very possibility of experience, then Einstein's revolutionary account of motion, or Bohr's non classical account of the stability of the atom cannot be explained. If instead we explain Einstein's success as the discovery of the true underlying nature of the presuppositions of experience, then we cannot also understand the success of Galileo, Descartes, Huygens, Newton, Maxwell or Bohr. The reason is that physics is not just a cumulation of mathematical results, but revolutionary (as Kore, Popper, Kuhn and Feyerabend have argued). Kant's explanation is not suited to multiple revolutions in physics, though it was a brilliant explanation of the one that he knew about.

Very few philosophers of science or of mathematics, perhaps none, accept Kant's account in its entirety of the success of mathematical physics. But we seem to have become accustomed to seeking the foundations of mathematics and of physics independently, as he set them out. In fact, we note that in this Twentieth World Congress in Philosophy, there is a section on the philosophy of science and one on the philosophy of mathematics, outlining clearly how Kant's distinction is institutionally structured in the very possibility of this congress.

2. The apparently unempirical part of mathematics has to do with the use of algebra.

By calling mathematics a separate study, I believe that we do ourselves a disservice. In classical times there were a number of mathematical subjects, such as astronomy, mechanics, optics, geometry, arithmetic and music. They were considered to be mathematical and empirical at once. It was Descartes who first proposed that the mathematical sciences are mathematical because they are all basically algebraic. I suggest that we give this up, because by treating all of mathematics as algebra, we project upon all of mathematics certain properties which are only those pertaining to algebra.

3. The success of modern mathematical physics (in Galileo's sense) owes a great deal to the introduction of algebra by Descartes to the empirical subject known as "mathematical physics."

The most obvious way in which Descartes influenced modern physics is by discovering space. Space as a unitary entity was not conceivable until Descartes had invented the method of algebraic geometry. This extended the study of geometry to the study of not just this or that object, which geometry did, but to the class of all shapes under a certain description, for instance. By describing space as a single individual substance all at once he became the first unified field theorist. Of course, the recently more coined expression "unified field theory" seems applicable only because the single substance of Descartes got disunified in the strange admixture of Maxwell's mechanical equation for waves and Newton's for particles.

4. Algebra is useful in physical investigations for the following reason: Algebras possess the property of closure.

Closure is the property that when you apply an operation to an object that is found in a system of algebraic objects. The result is another object within the system. Descartes first showed that geometry can be studied in an algebraic system that, in our later way of describing this feature, is closed under the four operations of arithmetic.

5. Investigating an operation by inventing a system of objects under which it is closed allows us to discover some very precise things about the operations, which we would be otherwise unable to know.

6. The system of objects which is closed under an operation is a system of invented objects, which will be real only if by a highly unlikely coincidence the fiction we invent is like the truth.

In algebra, the basic technique, as Descartes discovered, is to find a few operations and an assembly of objects which are closed under those operations. A system is "closed" under those operations if, whenever we use them on members of the system, we get objects that are also within the system. In order to ensure this, we must of course define these algebraic objects artificially. Numbers thus get redefined as natural, rational, real, complex, etc. Spaces are qualified as vector spaces, Banach spaces, Hausdorf space, Reimannian manifolds, and so on. In algebra we can thus succeed in making very precise formal discoveries about its objects precisely because they are artificial. Algebra is therefore a form of fiction. For instance, when Boole proposed the algebra of classes, he proposed that the classes in his system of ("Boolean") algebra are closed under the two operations of union and intersection. Any two classes can, of course, join together form a larger class. But what does it mean for two classes with no members in common to have a common "intersection?" Boole proposed that every class has the null class as a subclass. If we accept this artifice, then Boolean algebra works. Without it, there is no closure, nor Boolean algebra. Is the null class a piece of fiction, or is it real? Clearly, it is fictional, except, perhaps, by accident, if the world just happens to be as described in his laws of thought. It would seem that an algebraic system, like a novel, or like a shaggy dog story, is a fiction. But it is a fiction only in respect of the objects of the system, not in every respect.

7. Algebras bear on reality because of the similarities between the operations investigated in the algebras and the operations of the experimental physicist in the laboratory.

8. The way in which an algebraic system bears on reality is not by virtue of the reality of the fictional objects invented to allow algebraic closure but because of the similarities between the operations of the algebra and experimentally notable operations with which we can work. To take an example, Boolean algebra invents individuals and classes with the bare properties that they need in order to satisfy the operations of Boole's Laws of Thought. Under these operations the system is closed. But in order to bring about closure, we have to show how the operations of class union and class intersection are always defined, for any two classes. Intersection fails when classes have no part in common. So we make the rule that whenever two classes are disjoint they have in common the null class. Clearly, the null class is a Boolean invention which allows us to study the operations of classes with great vigor.

Suppose I now ask you whether there is really such a thing as the null class. Clearly, it is unlikely, since it was invented as a piece of theoretical fiction, just as Sherlocke Holmes was invented by Sir Arthur Conan Doyle. Of course, by accident there may have been someone who is just as Holmes is described by Doyle, unknown to us. But we would not waste our time on this as a serious speculation. We would bet against it. Why then should we worry about the null class?

Boolean algebra bears upon reality in the understanding of the operations of union and intersection. They seem to be capable of extension to something far more interestingly, as Georg Cantor showed, making some very fundamental mysteries of infinitely large and infinitely small quantities amenable to study. The lesson that we should draw from this is not that therefore the object called "individuals" and "classes" described by Boole or Cantor are real. It is, rather, that the operations employed in their algebraic treatment of classes are close analogues of operations on classes that we can actually perform, for instance in arithmetic.

8. Algebraic objects are not therefore real, or even necessarily realistic, but fictional, even though they are studied in the pursuit of nonfictional reality. They are therefore described best as thought experiments.

The use of operations in experiments - in physics, operations used for measurement, for instance - leads to the applicability of algebras to interpret physical systems. When an experiment is interpreted by a physicist, the experimental operations must meet the stringent criteria set by the algebra used. I note that this is not the same thing as testing a theory by experiment, which also happens in physics, quite copiously.

9. The success of modern mathematical physics has been due to two additional kinds of tests which theories have had to pass: They had to pass the skeptical scrutiny of the ingenious experimenter. This was not easy. Besides that they had to pass the scrutiny of the algebraist as a thought experimenter, who thus put theoretical constraints on an adequate theory of physics.

Insofar as we look at the role of algebra in the process of testing theories, they provide us with an additional means for testing them. A theory must not only pass the experimental tests, but it must also be interpretable algebraically. This makes physics more difficult to pursue than old fashioned medieval philosophy, of which it is an offshoot.

10. The success of modern physics lies in the extraordinary knowledge of esoteric phenomena winkled out by experimenters and in the extraordinary knowledge of mathematical facts which have not only grown, but grown exponentially since the introduction of algebra, in the seventeenth century, to the study of geometry, mechanics, numbers, inequalities, probability, statistics, and much more.

If I were to summarize my thoughts in a sentence, I would describe them as a realistic interpretation of mathematics and physics, and an instrumental interpretation of algebra. I propose the suggestion for your consideration that we do not divide our subjects into empirical physics and mathematics, whatever they are. Nor need we interpret it as only as a whole, as Quine will have it. But we divide it into an empirical mathematical physics, and a fictional or thought experimental algebraic adjunct to it. My suggestion is that we can thus make much better headway in solving the main problems of philosophy concerned with the antinomies, paralogisms and ideals that Kant struggled with so valiantly in his mature writings.