Progress in Mathematics and Logic — II

[for a brief explanation of this ongoing series, as well as a full table of contents, go here]

History of mathematics: the philosophical approach

It is interesting to note that mathematicians and historians of mathematics have often taken what can be characterized as a decidedly philosophical approach to the understanding of the development of their field. One example is provided by Mehrtens’ (1976) influential paper about the applicability of the (then relatively recently articulated) ideas by Kuhn to the field of mathematics.

Mehrtens begins by suggesting that mathematics can be thought of as being about “something” that offers resistance (without having to go so far as to invoke a special ontology of mathematical problems or objects — as in the case of mathematical Platonism: Maddy, 1990; Balaguer 1998; Bigelow 1998), and while the problems of mathematics are more markedly internally generated when compared to those of the natural sciences, he thinks the analogy holds in the sense that “the relation between mathematicians and their subject is very much like that of the natural sciences” (Mehrtens 1976, 300). Does this “resistance” yield Kuhn-like periods of revolution in mathematics? Crowe (1975) had denied that possibility (more on his paper below), but Mehrtens advances the example of the shift to differential notation catalyzed by the work of Robert Woodhouse at the beginning of the 19th century as a possible instance of Kuhnian revolution in mathematics. Nonetheless, even Mehrtens agrees that there are few (if any) examples of mathematical theories that have been overthrown, with gradual change or increasing obsolescence accounting for most instances of change instead.

Even though it is debatable whether there have been paradigm shifts in the Kuhnian sense in mathematics, there certainly have been Lakatos-type (1970) research programs. As we have seen in the previous chapter, Lakatos was attempting to improve on both Popper’s (1963) prescriptive and Kuhn’s (1963) descriptive approaches in philosophy of science, proposing that at any given moment there may be more than one active research program pursued by scientists (or mathematicians). Recall that these programs have a hard, non-negotiable theoretical core, figuratively surrounded by a soften “protective belt” made of ancillary hypotheses and methods that can be negotiated, revised or abandoned during the development of the program (while retaining the core). A successful research program, then, is progressive in the sense that it keeps generating fruitful scholarship. But research programs may stall and eventually degenerate when they fail to lead to further insights or applications. In mathematics, an instance of shifting from a potentially degenerating to a progressive research program may have occurred at the turn of the 20th century, when there was a change in emphasis away from applying mathematics to logic and toward using instead symbolic logic to explore foundational questions in mathematics itself.

Setting aside revolutions, though, the Kuhnian approach to historiography is rich of other concepts that may still apply, at least partially, to mathematics, and help us understand in what sense and how it makes progress. For instance, there have been episodes in the history of the field that do resemble Kuhn’s description of scientific crises, except that the resolution of such crises did not cause the kind of paradigm shift that Kuhn famously described as taking place in, for instance, physics. The reason for the difference is that mathematicians, when faced with a crisis, have focused on the fruitfulness and applicability of their theories, and they have benefited from the interactions between mathematics and other fields, all of which — as Lakatos (1963/64) pointed out, essentially diffuses a Kuhnian crisis.

What about another element of the Kuhnian view, the treatment of anomalies within a given field, the accumulation of which in physics eventually leads to the onset of a crisis and the occurrence of a paradigm shift? The history of mathematics certainly presents a number of cases of anomalies, such as Euclid’s Fifth Postulate [4]. The 5th was an anomaly from the beginning, because unlike Euclid’s other four postulates, it is not self-evident. Moreover, without it, Euclid could not prove his theorems, which is why people sought a proof of it for two millennia. It was the realization, by the 19th century, that the search was going to be fruitless that led people to explore what today we call non-Euclidean geometries, as well as to abandon the “metaphysical” belief in a single unifying geometry. This sort of historical pattern, Mehrtens suggests, is rather general in the history of mathematics.

Historical patterns notwithstanding, the emotional response of the mathematical community to an anomaly is a question for sociology and psychology of the discipline and its practitioners (just as in the case of science under analogous circumstances), but it has sometimes made an impact, as in the famous case of the extremely negative reaction of the Pythagoreans to the idea of incommensurability, i.e., the existence of irrational numbers like π. Incommensurability was apparently discovered by the Pythagorean Hippasus of Metapontum. When Pythagoras — who allegedly was out of town at the time — got back and understood what Hippasus had done he was so upset that he had his pupil thrown overboard and drowned!

There are two other similarities between the way things work in mathematics and Kuhn’s description of the scientific process (and progress): the idea of normal science and the characteristics of the scholarly community itself. As we have seen, for Kuhn the history of a field is characterized largely by long periods of “normal science,” in between the relatively brief instances of crisis and paradigm shifting. Mehrtens readily agrees that much of what is done in everyday mathematical scholarship similarly falls under “normal mathematics,” and that it is this process that eventually leads to the sort of textbook-type streamlined and elegant formulations of a given theorem. Kuhn also thought of science as being characterized by a relatively well defined community of practitioners who share the same values (epistemic as well as others, including aesthetic ones) and procedures (theoretical as much as empirical). Again, this is certainly the case also for modern mathematics, although when one goes further back in time in the history of either science or mathematics, what counts as the relevant “community” is far more murky.

One of Kuhn’s more mature concepts — which actually replaced the initial idea of a paradigm in his later writings — is that of a disciplinary matrix (Chapter 4). This, too, unquestionably applies to mathematics. Mathematicians share concepts, theories, methods, terminology, values, and aesthetic preferences, though the importance of different values changes over time. For instance, throughout the history of imaginary numbers fruitfulness dominated over rigor, but the latter gained more currency as a value throughout the 19th century. Also important to mathematics’ disciplinary matrix are so-called exemplars, which include Euclid’s Elements, Gauss’ Disquisitiones Arithmeticae, and other standard works that characterize the field, its methods and problems. Exemplars include procedures, such as the geometric representation of complex numbers. Mehrtens mentions a number of standard problems in mathematics that are part of the disciplinary matrix, such as factorization procedures. These have wide application to a variety of specific mathematical problems, and yet do not require the availability of complete solutions. In fact, a good number of the so-called “open problems” presented in textbooks fall within this category. Concepts also play a Kuhnian-style role, that of symbolic generalizations, within mathematics’ disciplinary matrix. Consider, for instance, the fundamental role of the concept of function: according to Mehrtens, if a mathematician doesn’t care much about ontology (i.e., she is not metaphysically inclined) then concepts pretty much determine what she thinks exists (or doesn’t exist) in the realm of mathematics. All of these components influence the very way in which mathematicians think about their subject matter, just like standard works (Newton’s Principia, Darwin’s Origin, and so forth) and exemplars (Galileo’s thought experiments, the study of natural selection in Galapagos finches) play an analogous role in the natural sciences.

What does all of this tell us about progress in mathematics? Mehrtens points out that sometimes changes in the disciplinary matrix occur despite the conscious efforts of the originators of the changes themselves. In astronomy, Kepler struggled mightily before giving up the (metaphysical) assumption that the orbits of the planets had to be circular. In mathematics, Hamilton invented quaternions [5] after trying hard for a long time not to abandon the principle of commutativity (the idea that in a given operation changing the order of the operands does not change the result) that was characteristic of the then current disciplinary matrix. So change sometimes occurs despite the resistance of some of the very practitioners who are later seen as the agents of that change. And, as is the case in science, mathematical discoveries often appear to be “in the air,” meaning that several mathematicians converge on a particular solution to a given problem, a phenomenon that is likely explained by the social bounds within the community made possible by the field’s disciplinary matrix.

There are of course other ways of reflecting on the nature of mathematics, in some aspects diverging from the one sketched out by Mehrtens. Without pretending to be either exhaustive or in a position to adjudicate disagreements among historians of mathematics, I will devote the rest of this section to two classic papers by Michael J. Crowe (1975, 1988), because I think they provide the non mathematician with useful and insightful views on how mathematics works, and especially — as far as my purposes here are concerned — in which respects it is similar to or different from the natural sciences [6].

Crowe (1975) presented what he thought are ten “laws” of mathematical history, in an attempt to differentiate the history of mathematics from that of the natural sciences on the basis of the diverging conceptual structures of the two fields. A rapid glance at Crowe’s list, however, shows a rather complex picture, with more similarities between mathematics and science than he perhaps realized or was willing to grant. For instance, he begins with “New mathematical concepts frequently come forth not at the bidding, but against the efforts, at times strenuous efforts, of the mathematicians who create them,” which is something that more than occasionally happens in science as well, for example in the just mentioned case of Kepler’s initial (and prolonged) refusal to move away from the assumption of circularity of planetary orbits, or of Einstein’s famous regret at the introduction of his cosmological constant, which turned out to be an inspired and fruitful move after all.

“Many new mathematical concepts, even though logically acceptable, meet forceful resistance after their appearance and achieve acceptance only after an extended period of time,”

an example of which is the invectives deployed as a common response to the idea of square roots of negative numbers between the mid-16th century and the early 19th century. Then again, in science Alfred Wegener’s idea of continental drift and Lynn Margulis’ contention that several sub-cellular organelles originated by symbiosis between initially independent organisms were also greeted with scorn, just to mention a couple of instances from the history of science. Crowe continues: “Although the demands of logic, consistency, and rigor have at times urged the rejection of some concepts now accepted, the usefulness of these concepts has repeatedly forced mathematicians to accept and to tolerate them, even in the face of strong feelings of discomfort.” For instance, mathematicians accepted the idea of imaginary numbers for more than a century despite the lack of formal justification, because they turned out to be useful, both internally to mathematics and externally, as in the cases of applications to quantum physics, engineering, and computer science. Again, it’s not difficult to find analogous episodes in the history of science, though usually across significantly shorter time scales — as with the gradual acceptance, after a period of significant unease, of the idea of light quanta at the beginning of the 20th century (Baggott 2013).

“The rigor that permeates the textbook presentations of many areas of mathematics was frequently a late acquisition in the historical development of those areas and was frequently forced upon, rather than actively sought by, the pioneers in those fields.” We have encountered this above, during our discussion of the difference between the history of mathematics and the presentation of mathematical heritage. Again, examples are not difficult to find, and Crowe mentions increasing standards for the acceptability of proof, with those characteristics of mathematical practice before the 19th century having been superseded by new, more rigorous ones by the end of the 19th century, standards that in turn would not be acceptable in contemporary practice. Analogously, both observational and experimental standards have definitely been ratcheted up during the history of individual natural sciences, in part — obviously — as a result of technological improvement and the consequent amelioration of observational and experimental tools, but also because of theoretical-conceptual refinements, for instance the introduction of Bayesian thinking in disciplines as varied as medical and ecological research (Ogle 2009; Kadane 2011).

For Crowe “the ‘knowledge’ possessed by mathematicians concerning mathematics at any point in time is multilayered. A ‘metaphysics’ of mathematics, frequently invisible to the mathematician yet expressed in his writings and teaching in ways more subtle than simple declarative sentences, has existed and can be uncovered in historical research or becomes apparent in mathematical controversy,” e.g., in the case of Eugen Dühring, who in 1887 accused some of his colleagues of engaging in mysticism because they accepted the concept of imaginary numbers (apparently, something persistently hard to swallow for some members of the mathematical community, from Pythagoras on!). Accusations of pseudoscience — some founded, others not — also fly around scientific circles, as for instance in a notorious case where geneticist Michael Lynch (2007) labelled colleagues who take a different approach to certain conceptual issues in evolutionary theory as no better than Intelligent Design creationists [7]. Further, “the fame of the creator of a new mathematical concept has a powerful, almost a controlling, role in the acceptance of that mathematical concept, at least if the new concept breaks with tradition.” And so it goes in science. The already cited Lee Smolin (2007), for instance, has produced a fascinating philosophical, historical and even sociological analysis of the development of string theory in fundamental physics throughout the latter part of the 20th century. From it, Smolin concludes that the impact of a small number of highly influential people, and not just the inherent merits of the theory, has swayed (at least temporarily) an entire discipline into placing most of its conceptual eggs into one approach to the next fundamental theory, with the result that a whole generation of physicists has passed without a new empirically driven breakthrough [8], the first time such a thing has happened in at least a century.

Crowe also maintained that “multiple independent discoveries of mathematical concepts are the rule, not the exception,” recalling that complex numbers, for instance, were discovered (or where they invented?) independently by eight mathematicians, using two different methods. This type of convergent intellectual evolution is certainly not alien to the history of the natural sciences as well, just think of the spectacular case of the simultaneous independent discovery of the theory of natural selection by Charles Darwin and Alfred Russell Wallace (Wilson 2013). And finally: “Mathematicians have always possessed a vast repertoire of techniques for dissolving or avoiding the problems produced by apparent logical contradictions and thereby preventing crises in mathematics … Revolutions never occur in mathematics.” This is the already discussed point about the fact that a straightforward Kuhnian historiography of mathematics doesn’t work very well (it arguably doesn’t work all that well for much science outside of physics either). More specifically, mathematicians from Fourier to Moritz have remarked that mathematics makes progress slowly, and does so by continuously building on previous knowledge, not by replacing it. The standard example is Euclidean geometry which, contra popular perception, has not been replaced, but rather enlarged and complemented, by non-Euclidean approaches.

A few years after the original article, Crowe (1988) — who must have a penchant for decalogues — commented on what he considers ten misconceptions concerning mathematical practice. Again, several of his points are worth examining briefly, for the insight they provide into matters related to the idea of progress in mathematics, and hence our general quest to understand progress in disciplines that I consider allied to philosophy. Crowe begins by rejecting the common understanding that “the methodology of mathematics is deduction,” contra Hempel (1945/1953), who argued in the positive. Interestingly, later in his career Hempel himself (1966) published a simple proof that shows that deduction cannot be the sole method of mathematical reasoning, because deduction can only test the validity of a claim, it cannot, unaided, provide a method of discovery. It is also not the case that mathematics provides certain knowledge, according to Crowe, who again cites Hempel (1945/1953) and his demonstration that Euclidean geometry lacks a number of postulates that are actually necessary to prove several of its own propositions, an inconvenient fact that was not discovered for a couple of millennia.

That mathematics is cumulative is another generalization to which Crowe finds plenty of exceptions. While largely true (as is the case for science), it is not difficult to find counterexamples, such as the sidelining of the quaternion system (see above). To complicate matters, however, Quaternions have not in fact been shown to be an incorrect approach, and accordingly they are still used alongside a number of other techniques — for instance in calculations pertinent to 3D rotations, with applications in computer graphics (Goldman 2011). It is also not true that mathematical statements are invariably correct. A good example here is the work by Imre Lakatos (1963/1964) in his Proofs and Refutations, where he shows that one of Euler’s claims for polyhedra [9] has been falsified a number of times and that several proofs of the claim have been shown to have flaws. (For additional discussion of this point see also Philip Kitcher’s (1985) The Nature of Mathematical Knowledge.)

We have already encountered another misconception about mathematics, that its structure accurately reflects its history. This is clearly not true also in the case of science, and it once again relates to the difference between the actual development of a field and the way it is presented in textbooks (i.e., cleaned up and somewhat “mythified”). Crowe remarks, for instance, that most current presentations of mathematical problems begin with axiomatizations, which in reality tend to be achieved late in the development of our understanding of a particular problem. Let’s remember that Whitehead and Russell (1910) took 362 pages to prove that 1+1=2, a mathematical fact that was known for a long time before their Principia Mathematica saw the light. Relatedly, it is not the case that mathematical proof in unproblematic either. Here Crowe quotes none other than Hume (1739/40): “There is no … mathematician so expert … as to place entire confidence in any truth immediately upon his discovery of it, or regard it as any thing, but a mere probability. Every time he runs over his proofs, his confidence encreases; but still more by the approbation of his friends; and is rais’d to its utmost perfection by the universal assent and applauses of the learned world.” A well known case is Bell’s (1945) point that Euclid’s original proofs of his theorems have over time been demolished and completely replaced by better ones. Lakatos (1978) defined proofs as “a thought-experiment — or ‘quasi-experiment’ — which suggests a decomposition of the original conjecture into sub-conjectures or lemmas, thus embedding it in a possibly quite distant body of knowledge.” Accordingly, Lakatos encouraged mathematicians to search for counterexamples to accepted theorems, as well as not to abandon apparently refuted theorems too soon. This is all very much reminiscent of the sort of objections to naive falsificationism in natural science stemming from the Duhem-Quine thesis (Ariew 1984), and which we have seen in the last chapter were directly addressed by Lakatos himself.

A misconception cited by Crowe that is crucial for our discussion here (and that goes somewhat against his broad position as stated in the 1975 paper) is that the methodology of mathematics is radically different from the methodology of science. Take Christiaan Huygens’ argument that mathematicians use the same hypothetico-deductive method that is attributed to science, contending that axiom systems are accepted (at least initially) because they are helpful or interesting, not as deductive justifications of theorems. In a very important sense, then, things like non-Euclidean geometries and complex numbers should be thought of as “hypotheses” that mathematicians provisionally embraced and later tested, in the logical equivalent of what physicists do with empirically testable hypotheses. Perhaps Kitcher (1980) put it most clearly: “Although we can sometimes present parts of mathematics in axiomatic form … the statements taken as axioms usually lack the epistemological features which [deductivists] attribute to first principles. Our knowledge of the axioms is frequently less certain than our knowledge of the statements we derive from them … In fact, our knowledge of the axioms is sometimes obtained by nondeductive inference from knowledge of the theorems they are used to systematize.” A related questionable notion is that — unlike scientific hypotheses — mathematical claims admit of decisive falsification. There are examples (for instance concerning complex numbers) of apparently falsified claims that were successfully rescued by modifying other aspects of the web of mathematical knowledge, in direct analogy with Quine’s concept of a web of belief for the empirical sciences, which we have discussed in Chapters 2 and 3.

The final misconception tackled by Crowe is that the choice of methodologies used in mathematics is limited to empiricism, formalism, intuitionism, and Platonism. Here he makes a distinction between the (descriptive) history of mathematics and the (perhaps more prescriptive) philosophy of mathematics. In analogy with historically-minded philosophers of science (a la Kuhn) he suggests that the actual practice of mathematics as it has unfolded over the millennia is more messy and complex than it can be accounted for by any single one of the above mentioned categories.

Our brief analysis of how mathematics works is far from comprehensive, and the interested reader is directed to some of the much more in-depth treatments mentioned in the references. But a few general lessons can, I think, be safely drawn. First and foremost, there is no accounting for how mathematics makes progress without paying attention to the history of the field, as reconstructed by actual historians — as opposed to the sanitized, almost mythologized, version one often finds in textbooks. Second, mathematics is neither completely different in nature from, nor is it quite the same as, natural science. For instance, as we have seen argued on several occasions, simple falsification fails in both areas, and for similar, Duhem-Quine related, reasons. Moreover, it is also not true — contra popular perception — that mathematics proceeds by deduction only, leaving the more messy inductive processes to the empirical sciences. But it is certainly the case that deduction plays a much more significant role in mathematics than in the sciences, which do not have anything analogous to the idea of proving a theorem. Also, while we have seen that it is naive to think that once a mathematical truth has been proven it is done with, contributing to a monotonic, relentless increase in mathematical knowledge, it is the case that the paths taken by the natural sciences are much more untidy and prone to reversal — precisely because they depend substantially on empirical evidence and cannot rely on deductive proof [10].

The overall picture that emerges, then, is one in which there are both significant similarities and marked differences between the natural sciences and mathematics, which have consequences for our understanding of how the two make progress. I turn now for the rest of the chapter to the field of logic, where we will see fewer similarities with the natural sciences and more with mathematics, and that will help us further bridging the gap toward a reliable concept of progress in philosophy, the focal topic of the next chapter.

Notes

[4] Which says: if a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.

[5] Complex numbers of the form w + xi + yj + zk, where w, x, y, z are real numbers and i, j, k are imaginary units that satisfy certain conditions.

[6] For more recent entries on the history of mathematics, see, among others: Krantz, 2010; Cooke 2011; Radford 2014. On the related (as far as concepts of progress go) field of the philosophy of mathematics, see: Irvine, 2009; Colyvan 2012.

[7] Full disclosure: I fall squarely in the camp of those targeted by Lynch.

[8] The discoveries of the Higgs boson in 2013 and of gravitational waves in 2016 do not count, since they (spectacularly) confirmed the already established Standard Model and General Theory of Relativity, respectively.

[9] If you really wish to know, the claim is that V-E + F = 2, where V is the number of vertices, E the number of edges, and F the number of faces.

[10] It may have occurred to some readers that a number of scientists seem to think that mathematics actually is a science, because it historically got started with empirical observations about the geometric-mathematical properties of the world. I refer to this (misguided) view as radical empiricism. Here are my thoughts on the matter.

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