In A Philosophical Disconnect I discussed the disconnect between philosophy of law and political philosophy; in Another Disconnect I discussed the disconnect between economics and accounting; and in A Metaphysical Disconnect I discussed the disconnect between the philosophy of time and the philosophy of history. I have come to see these disconnects as important features that define the structure of our civilization fully as much as the connections that are, in fact, made, and which define our civilization. It was, then, with some interest that I realized that there is a profound disconnect between philosophy of mathematics and philosophy of science.
Mathematics has always had a problematic relationship to the empirical sciences. There has been a pervasive feeling that there is some deep connection between mathematics and science, but feeling the connection is entirely different from being able to spell it out explicitly. We understand that both science and mathematics are expressions of rational thought, unique attainments of intelligence, but also distinct (though overlapping) implementations of intelligence.
We tend to think of mathematics as part of the sciences, as one among the many sciences, and mathematics holds an honorary place as the “M” in STEM, but mostly we avoid trying to be explicit about the relationship between mathematics and empirical science because it is so difficult to formulate adequately. Only philosophers, who are pretty marginal to the contemporary intellectual enterprise, ask questions like this, but for the most part these questions are swept under the rug.
The philosophies of the special sciences —philosophies of biology, of physics, of archaeology, and so on — all easily fall within contemporary philosophy of science, and we can readily employ philosophical analyses of observation, method, and theory derived from any one of these to any of the others. But this is not true of mathematics. If we attempt to apply contemporary philosophy of science to mathematics we find that we come up short, and there just isn’t a straight-forward way to adapt the analytical methods employed for empirical scientific knowledge to mathematics. And mathematics has its own tradition of philosophical reflection that is quite separate from the philosophy of science. Philosophers of mathematics may pass between interests in logic and mathematics, but rarely cross over into philosophy of science.
Peter Godfrey-Smith in his textbook Theory and Reality: An Introduction to the Philosophy of Science acknowledged the difficulty of formulating a comprehensive account of the philosophy of the empirical sciences based on the central ideas of our time that are used in the exposition of science:
“When I wrote the proposal for this book, publishers sent it out for comments. One anonymous reviewer reacted against the idea that at the end we would have a happy three-way marriage of empiricism, naturalism, and scientific realism. The reviewer saw these as three ideas that could each be defended fairly well individually but which do not go well together. There are conflicts between them, or at least between some of the pairs.” (Peter Godfrey-Smith, Theory and Reality: An Introduction to Philosophy of Science, University of Chicago Press, 2003, p. 219)
Whether or not we can weave together empiricism, naturalism, and scientific realism in a single coherent philosophy of science, if we take these as the serious contenders for organizing themes in the philosophy of science, we can immediately see how little they have to do with the philosophy of mathematics.
Right off the mark, empiricism and naturalism seem to have nothing to do with mathematics. Of course, there are naturalistic philosophies of mathematics (for example, Naturalism in Mathematics by Penelope Maddy) and even empirical philosophies of mathematics, of which John Stuart Mill was a great representative, and Frege was a great critic.
Perhaps the trendiest approach to the philosophy of mathematics (sometimes called “neo-empiricism”) is to focus on mathematical practices, but it remains to be seen where this interest in the actual practices of actual mathematicians will take us. Here the old joke applies that mathematicians are Platonists while doing mathematics during the week, and formalists on the weekend when talking about what they do as mathematicians, so even here exactly what mathematicians are doing when engaged in mathematical practice is open to interpretation. Are mathematicians manipulating abstract objects as real as protons, neutrons, and electrons, or are they playing a game with symbols?
More importantly, and more to the point in the present context, is that mathematical practices aren’t much like the practices of those engaged in the empirical sciences, though we could certainly pursue this analogy. This doesn’t have much to do with empiricism, naturalism, or scientific realism, but Imre Lakatos’ approach to the philosophy of science through scientific research programs would be a profitable way to go about assimilating mathematical practice to scientific practice, but even the most successful pursuit of this analogy would yield limited returns.
In regard to scientific realism, this seems to have more applicability to the philosophy of mathematics, but the resemblance may be merely superficial. The question of the nature and the existence of mathematical objects has been a central concern of the philosophy of mathematics, but this seems to be rather a different question, pursued in a different way, from the question of the existence of the objects of empirical science. In traditional philosophical language, mathematics is concerned with the a priori, and, if we learn about the existence of mathematical objects, we learn about them a priori, while the empirical sciences are, by definition, a posteriori, and we learn about the existence of the objects of empirical science a posteriori.
Our ways of conceptualizing mathematics (philosophy of mathematics) and our ways of conceptualizing science (philosophy of science) are so profoundly different that philosophers of mathematics and philosophers of science scarcely talk to each other, and when they do talk to each other, they usually talk at cross purposes and each fails to understand the other. This is a disconnect that goes to the very root of the rational mind, and is perhaps more fundamental than the metaphysical disconnect between time and history that I examined in an earlier post. In so far as this kind of rationality is central to the project of western civilization, this is a disconnect at the foundation of western civilization.
Our inability to put science and mathematics together within the same conceptual framework is like our inability to put general relativity and quantum theory together in the same conceptual framework, only more consequential. In the latter case, we know that eventual unification of relativity and quantum theory will be within a scientific conceptual framework much as we know it today, even if this unification involves conceptual breakthroughs that we cannot guess today. The disconnect between science and mathematics is even more profound than the disconnect between general relativity and quantum theory. In the case of the science/mathematics disconnect, we cannot even say what a common conceptual framework would look like.
If we could overcome the disconnect between mathematics and empirical science we might attain a deeper insight into both, but, if the disconnect between the two is constitutive of either discipline or both disciplines (as is implied by the distinction between the a priori and the a posteriori), then overcoming the disconnect would mean overcoming science and mathematics as we have known them to date, and establishing some new and more comprehensive discipline that transcends contemporary science and mathematics — something more than science and more than mathematics, but distinct from both.
How might this overcoming, this transcendence of the distinction between science and mathematics, come about? The scope of mathematics could be expanded until it included the whole of the sciences by mathematicizing the sciences (making science a part of an extended mathematics), and this has already happened to a certain extent (and this may explain the presence of mathematics in STEM). Coming from the other side of this, contrariwise, the scope of science could be expanded to include mathematics (making mathematics part of an extended science), and one way to do this would be through a theory of scientific abstraction that recognized various degrees of abstraction involved in the many sciences, with mathematics occupying the ultimate degree, anchoring one end of a continuum of abstraction.
A great deal has been written about the scientific method, but much less about scientific abstraction, that is to say, the particular mode of abstraction employed in the sciences. One might even say that there is a disconnect here as well, between the explicit understanding of science in terms of scientific method and the absence of any similarly explicit formulation of scientific abstraction. Moreover, I would argue that scientific abstraction is equally as important as scientific method, but that scientific abstraction is relatively neglected both in science itself and in contemporary analytical philosophy of science, largely because it is assumed to be so simple as matter as to be unworthy of exposition, while it is, in fact, a much more difficult philosophical problem than that of scientific method.
Scientific method can be approached in an entirely pragmatic spirit. If a method works, by which I mean if it yields accurate predictions, then it can be adopted, and refinements to the scientific method can be adopted as they prove themselves in the production of scientific knowledge. Method is, by definition, a matter of practice, and practice has practical consequence and can be learned through observation and imitation.
Scientific abstraction is a different matter; in so far as it is a matter of practice, it is a matter of conceptual practice, and it cannot be learned by imitation, example, or observation. At best, scientific abstraction can only be inferred from scientific practice, and different observers will make different inferences.
Because scientific abstraction hasn’t been given a systematic and explicit exposition, scientists by-and-large follow their intuitions — indeed, it might be said that what distinguishes a great scientist from a mediocre scientist is the quality of their intuition, which means knowing what problems to work on and how to work on them — and these intuitions go largely unexamined if they are successful. Unsuccessful researchers might cast about for a more effective intuition on which to proceed, but science today is so large that there is no shortage of “normal science” problem solving for those who lack the imagination to frame a new intuitive research program.
Even though scientific abstraction is not well understand, and is far from being explicitly delineated, its importance is recognized. For example:
“…power in science springs from abstraction. Thus, although a feature of nature may be established by close observation of a concrete system, the scope of its application is extended enormously by expressing the observation in abstract terms.” (Peter Atkins, The Laws of Thermodynamics, p. 38)
The expression of scientific truths derived from empirical observation in abstract terms is a problem of formalization: finding the right abstract terms in which to express a theory is the same as finding the right formalization of an empirical theory, and this is much more difficult that designing an experiment, making observations, and recording measurements. Finding a new and more effective formalization is what Einstein did in proposing the theory of relativity, rather than working within the established paradigm of physics.
Here we glimpse one of the underlying difficulties of understanding scientific abstraction, and why it is such a difficult philosophical problem: it demands not merely an extrapolation of formal methods, but self-reflection on formal methods, and the latter is much more difficult, and far more rare, than the former. Logic and mathematics have almost exploded as disciplines over the past hundred years. (Keith Devlin says that mathematics today is in a “new golden age.”) While the quantity and sophistication of formal thought has grown by leaps and bounds, critical self-reflection on the methods of formalization — almost always taking place at a later stage of intellectual development than the stage of discovery — has not grown, and in some respects has remained notably stagnant over the same time period.
We should not be surprised at the stagnation of critical inquiry into formalization: discovery demands, or at least flourishes best, when it takes place against a uniform and stable background of assumptions, and one of these uniform and stable background conditions is scientific abstraction. The formal background of thought is likely to remain untouched as long as it continues to yield scientific knowledge (i.e., as long as scientific discovery continues). An explicit and thorough critical study of scientific abstraction will probably come later in the development of science. As long as science still remains largely within the paradigm of discovery that grew out of the scientific revolution, there will be little need to tamper with its fundamental assumptions, and indeed doing so may prove to be counterproductive from the standpoint of discovery alone.