Progress in Mathematics and Logic — I

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

“Contrariwise, continued Tweedledee, if it was so, it might be;
and if it were so, it would be;
but as it isn’t, it ain’t. That’s logic.”
(Lewis Carroll)

Despite all the complications we have examined in the previous chapter — historical as well as epistemological — involved in the deceitfully simple task of making sense of the idea of progress in science, I think it is undeniable that science does, in fact, make progress, and indeed constitutes one of the clearest examples of a discipline (or, rather, a set of disciplines) that does so. Here I turn to the closely connected fields of mathematics and logic, for a number of reasons.

First, mathematics in particular is often thought of as not only having made clear progress throughout its history, but as having done so in an even less equivocal sense than that of science. After all — at first sight at least — scientific theories are often overturned (think of the replacement of Newtonian mechanics by Einstein’s relativity), while a mathematical theorem, once proven, stays firmly put into the set of things we (think we) know for sure. Second, the history of logic is more uneven, with clear hallmarks of progress, but also extremely long periods of stasis, and therefore presents us with a dynamics of “progress” that is definitely distinct from that of science as well as sufficiently different from the one that applies to mathematics to provide an additional useful contrast class. Third, I think the comparison among these three areas of human endeavor (science, mathematics, and logic) will help us pinpoint more exactly in what sense philosophy itself makes progress, as the latter discipline shares aspects of the other three without really being the same animal as any of them.

As a preliminary, however, it will help to briefly discuss the general relationship between mathematics and logic, as seen by practitioners in these fields, something I have become fascinated with ever since reading my first book on the philosophy of mathematics (Brown 2008). According to Cameron (2010) [1] there are two roles that logic plays in mathematics. The first deals with providing the foundations on which the mathematical enterprise is built. As he puts it: “No mathematician ever writes out a long complicated argument by going back to the notation and formalism of logic; but every mathematician must have the confidence that she could do so if it were demanded.” The second role is played by logic as a branch of mathematics, on the same level as, say, number theory. Here, according to Cameron, logic “develops by using the common culture of mathematics, and makes its own rather important contributions to this culture.” For him, therefore, the relationship between logic and mathematics is not along the lines of one being a branch of the other, exactly. Rather, certain logical systems can be deployed inside mathematics, while others are in an interesting sense outside of it, meaning that they provide (logical) justification for mathematics itself.

Berry (2010) approaches the issue of the relationship between logic and mathematics in a more comprehensive and systematic manner. She puts forth that the answer to the question depends (not surprisingly, perhaps) on what exactly one means by “logic.” In particular, she provides a useful classification of five meanings one might have in mind when using the word:

1. First order logic.

2. Fully general principles of good reasoning.

3. A collection of fully general principles which a person could learn and apply.

4. Principles of good reasoning that are not ontologically committal.

5. Principles of good reasoning that no sane person could doubt. [2]

Berry’s first point is that we know that it is not possible to program a computer to produce all and only the truths of number theory, but it is possible to program such computer to produce all the truths of first order logic. Which means that mathematics is not the same as logic if one understands the latter to be the first order stuff (option #1 above). Berry adds that if we make the further assumption that human reasoning can be modeled in a computer program, then logic doesn’t capture all of mathematics also in cases #3 and #5 above. What about #2, then? To quote Berry: “If by ‘logic’ you just mean … fully general principles of reasoning that would be generally valid (whether or not one could pack all of these principles into some finite human brain) — then we have no reason to think that math isn’t logic.” I am very sympathetic to this broader reading of logic, but we are still left with option #4. Apropos of that Berry reminds her readers that standard mathematics is reducible to set theory, and that the latter in turn has been shown to be reducible to second-order logic, thus implying that mathematics is, after all, a branch of logic.

Berry’s conclusion is that “it is fully possible to say … that math is the study of ‘logic’ in the sense of generally valid patterns of reasoning. However, if you say this, you must then admit that ‘logic’ is not finitely axiomatizable [because of Gödel’s theorems: Raatikainen 2015], and there are logical truths which are not provable from the obvious via obvious steps (indeed, plausibly ones which we can never know about). … What Incompleteness shows [then] is that not all logical truths can be gotten from the ones that we know about.”

This contrast between Berry’s and Cameron’s analyses helps me make the point that mathematics and logic are indeed deeply related, which further justifies treating them in a single chapter in this book, even though the exact nature of that relationship is still very much up for discussion among practitioners of these fields. My objective here, of course, regardless of whether you think mathematics is a branch of logic or vice versa (or, perhaps, that both disciplines are instantiations of a broader way of reasoning about abstract objects — as I am inclined to believe), is to show that both have clearly made progress over the past two millennia or more — albeit in an interestingly different fashion from each other and from science. Let us begin with mathematics first.

Progress in mathematics: some historical considerations

Somewhat surprisingly, mathematicians may experience a similar problem to that encountered by philosophers in precisely defining what it is that they do. Weil (1978) paraphrases Houseman on poetry, saying that “[the mathematician] may not be able to define what is a mathematical idea, but he likes to think that when he smells one he knows it.” He goes on to acknowledge the existence of transitional stages between folk ideas about the world and distinctly mathematical ones: the concept of a icosahedron, for instance, is definitely an example of a mathematical idea, while more mundane geometrical figures, like circles, rectangles and cubes are clearly part of the thinking tools of laypeople. Philosophers also often incorporate “folk concepts” (say, free will) in their parlance, but end up having a tougher time than either mathematicians or scientists in convincing outsiders that there is more technical content to what they do than it may appear from a cursory examination of their debates. While it is hard to imagine a non-mathematician commenting seriously on technical issues in geometry just because he knows what a square is, it seems like everyone feels free to blunder into discussions of metaphysics because they think they know what the pros are talking about (or, worse, that they certifiably know better than said pros).

In this section, and pretty much in the remainder of the chapter, I will take an historical approach to the subject matter under discussion, as I think it is by far the most informative and easy to follow. However, I need to acknowledge that there are a number of ways of conceiving of progress in mathematics that I will not entertain, or that I will mention only in passing. It is my belief that even if I did incorporate them (into what would then become a much longer, and significantly more unwieldy, chapter) they would not alter my main conclusion: that mathematics  does make progress, and that it does so in ways that are significantly different from how science proceeds.

To begin with, then, there is the thorny issue of mathematical Platonism, which I have tackled in part in the Introduction. Platonism, which comes in a variety of flavors (Balaguer 1998; Linnebo 2011), is a popular position among both mathematicians and philosophers of mathematics — though of course there are many who reject it, resulting into a large and fascinating literature. If one is a Platonist in mathematics, then one may deploy the mathematical equivalent of a correspondence theory of truth, which we have seen is the one assumed by most scientists (Chapter 4). As a result one might think that it is legitimate to conclude that mathematics makes progress in a very similar way to how science does.

There are two issues that arise in connection with this and are germane to my project. First, all the non trivial problems associated with the deployment of a correspondence theory of truth that we have seen in the previous chapter would still hold for its deployment with respect to the field of mathematics. Second, and more importantly, the sense of “correspondence” here must be different, since Platonic forms are neither to be found “out there” nor are they accessible by empirical means of any sort (indeed, one of the most powerful objections to Platonism is the lack of a reasonable account of how, precisely, mathematicians do gain access to the alleged Platonic realm). These are not insuperable problems, but I maintain that in order to deal with them one has to shift to talk of progress in conceptual (as opposed to empirical) space in mathematics, the very same talk that I think holds also in the cases of logic and philosophy. This in turn means that mathematics cannot be conceived in the same teleonomic fashion that most views of science allow, which means that my main argument here will stand. In fact, I am tempted to say that mathematics and logic, very much unlike what is usually argued for science, not only are not aiming at any “final theory,” but in fact generate their problems internally and expand their concerns in an autonomous fashion throughout the course of their history, without aiming at any unified goal. In that sense, they work perhaps in a way similar — though not identical — to how the sciences can be understood to work if one buys into a more radical notion of the disunity of science itself, similar to those presented by Fodor, Cartwright, Hacking and Dupré and discussed at the end of the last chapter.

A related worry may arise if a theorist sees mathematics as a set of formal systems that are interpretable in ways that aid natural scientific inquiry, similar to the Quineian “naturalistic” view that we have discussed in Chapter 3. From that standpoint, progress in mathematics would then consist in adding formal systems that can be deployed in new scientific ventures. I submit, however, that this would amount to a very impoverished view of mathematics, essentially relegating the enterprise to a subordinate role as handmaiden to science. I think there are obvious reasons to reject such a role in both the cases of mathematics and logic, reasons that are rooted in the historical record and the actual contemporary practices of mathematicians and logicians: many problems in both fields are internally generated, and are neither derived, nor do they get their validation, from science. This, of course, does not deny the obvious fact that both mathematics and logic are — more than occasionally — extremely useful to scientific practice.

Yet another conceivable view would be to sidestep the issue of mathematical truth altogether and approach mathematical progress directly in terms of the deepening of our mathematical understanding. Standard examples of progress in mathematics would then make sense accordingly: for example, Descartes’ development of analytic geometry enabled us to find a general method of solving locus problems; the tradition that runs from Lagrange to Galois successively deepens our understanding of methods for solving polynomial equations of different degrees; Dedekind’s construction of the real numbers enables us to understand the principles of real analysis [3], and so on. Mathematical progress, following this view, is a type of explanatory progress, a thesis that is ontologically neutral and needs not take sides on the issue of Platonism. This view of (internally generated) progress as deepening of our understanding of a given domain of knowledge can be extended to logic (and, in part, to philosophy), and is — I think — perfectly compatible with the thesis that I develop in this book.

The above qualifications having been made, there likely is no more obvious place to start for a brief historical overview of progress than by arching back to the ancient Greeks (Bergrren 1984). Broadly speaking, the Greeks deployed three approaches to mathematics: the axiomatic method, the method of analysis, and geometric algebra  — all of which embodied a more sophisticated modus operandi than that typical of Babylonian mathematics, by which the Greeks were clearly inspired.

The standard example of axiomatic method is Euclidean arithmetics (Szabó 1968), exemplified for instance by the theory of proportions. I am interested here in the connections that this approach had with philosophy, particularly the Eleatic school to which Zeno (he of the famous paradoxes) belonged, as well as disciplines that today we consider more remotely connected to either mathematics or philosophy. For instance, one of the problems that was initially addressed by means of Euclidean arithmetics had to do with musical theory, and in particular the issue of how to divide the octave into two equal intervals. This, in turn, fed back into theoretical mathematics, since it generated an interest in incommensurable quantities. The second approach, geometric analysis (Mahoney 1968), led to both a preoccupation with general problems in geometry and to the solution of specific issues, such as trisecting the angle. Again, there was a tight connection with philosophy at the time, since geometric analysis — while not actually invented by Plato, as it has sometimes been suggested — certainly inspired him to elaborate his ideas on dialectic philosophy. The third approach used by the Greeks to address mathematical-philosophical problems was geometric algebra, classically embodied by one of the most influential books ever written in the history of mathematics, Euclid’s Elements. Geometric algebra is interpreted by historians of mathematics as a translation of the Babylonian approach to algebra into geometric language, and thus provides us with an example of progress insofar such “translation” allowed the Greeks to address a much wider range of problems than the Babylonians had been capable of (or interested in).

What did the ancient Greeks accomplish with this array of new tools at their disposal? Quite a bit, as it turns out. The list is long and well known to anyone familiar with the basics of mathematics, but it is worth remembering that it includes the foundations of a whole class of geometry (Euclid), the theory of proportions (Pythagoras, Eudoxos), and the theory of incommensurables (early Pythagoreans, Theodoros, Theaetetos), to mention but a few. Archimedes’ work alone was of crucial importance. Setting aside his output on the theory of levers, Bashmakova (1956) has suggested that Archimedes’ had developed a method for finding extrema (i.e., the maximum or minimum, local or global, of a given function) that was rediscovered only in the 16th and 17th centuries by Ricci and Torricelli. And then there is Ptolemy, whose work was pivotal for the history of mathematical methodology and of trigonometry, and likely instrumental for the origin of the very idea of mathematical function. In fact, Greek geometry provided the foundations for a more modern treatment of continuous phenomena in general, a marked advance over the Babylonian focus on discrete quantities.

Beyond the Greeks the history of mathematics gets much more complex, and I will comment on a few more examples of what I think is obvious progress below. However, it is interesting to note what mathematicians and historians of mathematics say about the dynamics of the field. For instance, take the classic comments by André Weil (1978) on the unfolding of history of mathematics. He begins by highlighting the convergence of mathematical discoveries across time and cultures, as in the case of the discovery of certain classes of power series expansions, which took place independently in Europe, India and Japan. Or the solution to Pell’s equation, achieved first in India in the 12th century, then again in Europe — following work by Fermat — in 1657.

Weil echoes Leibnitz’s approach to getting acquainted with the nature of mathematical practice. Leibnitz thought that we should look at “illustrious examples” to learn about what he called “the art of discovery.” Doing that, according to Weil, clearly shows that mathematicians often directly concern themselves with long-range objectives, as opposed to the “puzzle solving” (to use Kuhn’s phrase) of much empirical science, which in turn means that practitioners benefit from an awareness of broad trends and the evolution of mathematical ideas over long periods of time. The reason I find this interesting is because it is not very different from the way philosophers regard the history of ideas in their own field, and very much in contrast to the general neglect of the history of science that scientists often display. Underlying this contrast may be that philosophy and mathematics (and, we shall soon see, logic) make progress across relatively long time spans, while scientific discovery is often fast paced by comparison, so that history rapidly becomes less relevant in the latter case.

When considering the history of ideas (and therefore the progress made) in mathematics, science, and philosophy, an important issue that is often overlooked is the distinction between history proper and what Grattan-Guiness (2004) labelled “heritage.” History, in this context, refers to the development of a given idea or result, while heritage refers to the impact that idea or result had on further work, when seen from a modern vantage point. Approaching things from a heritage perspective tends to focus on the results obtained, while a properly historical approach takes a comprehensive look also at the causes and motivations of certain developments. It should be obvious that it is important to keep the two separate, on penalty of incurring in anachronistic readings of the history of any given field, a recurring problem especially in the sciences, where practitioners learn a highly sanitized (one would almost want to say romanticized), introductory textbook version of the history of science.

A well documented example of the difference, discussed by Grattan-Guiness, is that of Book 2, Proposition 4 in the already mentioned Elements by Euclid, which concerns a theorem for the completion of a square, nowadays a method to solve quadratic equations. Beginning in the late 19th century there has been a tendency to interpret this as evidence that Euclid was a geometric algebraist, i.e., essentially focused on algebra. But historians of mathematics have pointed out that a more sound (and not anachronistic) reading of the actual history shows Euclid’s theorem to be firmly within the bounds of geometry. The point is that, although historians have set the record straight, a number of mathematicians still cling to the earlier interpretation, presumably because they are interested in the knowledge content, not the actual development, of Euclid’s ideas — thus confusing history and heritage.

Another example is provided by Lagrange’s work in the 18th century. He understood that there was a connection between the algebraic solvability of polynomial equations and the properties of some functions of the roots of those polynomials. Lagrange’s ideas played a crucial role in the subsequent development of what is now known as group theory (i.e., a theory of a broad type of algebraic structures), but the original insight should not be described in terms of group theory, because that would confuse things and distort the sequence of historical developments. As Grattan-Guiness (2004, 171) aptly puts it: “The inheritor [his term for people who disregard the history of ideas] may read something by, say, Lagrange and exclaims: ‘My word, Lagrange here is very modern!’; but the historian should reply: ‘No, we are very Lagrangian.’” This should strike a chord with both scientists and especially philosophers, who are at times guilty of precisely the same kind of historical reading-by-hindsight. Grattan-Guiness (2004, 169, 171) summarizes the difference between heritage and history in this way: “Heritage is likely to focus only upon positive influence, whereas history needs to take note also of negative influences, especially of a general kind, such as reaction against some notion or the practice of it or importance accorded some context … Heritage resembles Whig history, the seemingly inevitable success of the actual victors, with predecessors assessed primarily in terms of similarities with the dominant position.”

So, what do we learn from the actual history of mathematics, as opposed to a glorified consideration of its heritage? One of the fundamental notions that emerge is that the development of a number of mathematical theories has been slow, or even characterized by long periods of stasis. It is therefore an interesting exercise for the contemporary practicing mathematician to meditate on which current theories may be in a stage of arrested (but perhaps eventually to be resumed) development, since looking at mathematics from the standpoint of heritage may provide a false view of the field as inevitably progressing by accumulation of more refined theories, while the real story also includes a number of abandoned or faded away notions.

Following Grattan-Guiness, the history of mathematics — again like the history of science — has also demonstrably been affected by social and even political forces completely external to the field. For instance, in the 19th century French mathematicians worked largely on mathematical analysis, while their English colleagues focused on new algebras, just like, say, during the middle part of the 20th century Soviet geneticists (before the Lysenko disaster: Joravsky 1970) adopted a distinctly more developmental approach to the study of organisms than their Western European counterparts. The message is that it is worthwhile to escape from the all too easy trap of thinking that whatever question or approach is currently being pursued by one’s own intellectual community is the inherently best or most interesting one. These considerations also carry consequences in terms of education: mathematics — like much of science — even when taught with a historical perspective, is actually presented in heritage fashion, smoothing over the irregularities and the setbacks. Consequently, students get a misleading view of a series of finished products with no sense of why the questions were raised to begin with, or of what struggle ensued in the attempt to answer them. This in turn yields a false image of constant linear progress that artificially inflates the differences between disciplines that appear to make clear progress (mathematics, science) and those that appear more stagnant (philosophy, other parts of the humanities).

Notes

[1] Yes, I realize that this and the next citation refer to blog posts, not peer reviewed papers. We live in a brave new world, and sometimes interesting ideas are put out there in the blogosphere. More seriously, sometimes comments blogged by professionals in a field are more informative and insightful than what they write in the primary literature, for the simple fact that they can afford to lower their guard somewhat, expressing themselves more freely and creatively. I know because I do a lot of blogging myself…

[2] We will not get into a discussion of what constitutes “sanity” insofar as possibility #5 is concerned. Another time, perhaps.

[3] I am indebted to an anonymous reviewer for bringing up these examples in her/his critique of a previous draft of this chapter.

References

Balaguer, M. (1998) Platonism and Anti-Platonism in Mathematics. Oxford University Press.

Bashmakova, I.G. (1956) Differential methods in Archimedes’ works. In: Actes du VIII Congres Internationale d’Histoire des Sciences. Vinci: Gruppo Italiano di Storia delle Scienze, pp. 120-122.

Bergrren, J.L. (1984) History of Greek mathematics: a survey of recent research. Historia Mathematica 11:394-410.

Berry, S. (2010) Is math logic? (accessed on 16 April 2016)

Brown, J.R. (2008) Philosophy of Mathematics: A Contemporary Introduction to the World of Proofs and Pictures. Routledge.

Cameron, P. (2010) Mathematics and logic (accessed on 10 August 2015)

Grattan-Guiness, I. (2004) The mathematics of the past: distinguishing its history from our heritage. Historia Mathematica 31:163-185.

Joravsky, D. (1970) The Lysenko Affair. University of Chicago Press.

Linnebo, Ø. (2011) Platonism in the philosophy of mathematics. Stanford Encyclopedia of Philosophy (accessed on 11 October 2012).

Mahoney, M.S. (1968) Another look at Greek geometric analysis. Archive for History of Exact Sciences 5:318-348.

Raatikainen, P. (2015) Gödel’s Incompleteness Theorems. Stanford Encyclopedia of Philosophy (accessed on 10 August 2015)

Szabó, A. (1968) The Beginnings of Greek Mathematics. Reidel.

Weil, A. (1978) History of mathematics: why and how. In: Proceedings of the International Congress of Mathematicians. O. Lehto (ed.), American Mathematical Society, pp. 227-236.