Progress in Mathematics and Logic — II

mathematics[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.

References

Ariew, R. (1984) The Duhem thesis. British Journal for the Philosophy of Science 35:313-325.

Baggott, J. (2013) Farewell to Reality: How Modern Physics Has Betrayed the Search for Scientific Truth. Pegasus.

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

Bell, E.T. (1945) The Development of Mathematics. McGraw-Hill.

Bigelow, J. (1988) The Reality of Numbers: A Physicalist’s Philosophy of Mathematics. Clarendon.

Colyvan, M. (2012) An Introduction to the Philosophy of Mathematics. Cambridge University Press.

Cooke, R.L. (2011) The History of Mathematics: A Brief Course. John Wiley & Sons.

Crowe, M.J. (1975) Ten ‘laws’ concerning patterns of change in the history of mathematics. Historia Mathematica 2:161-166.

Crowe, M.J. (1988) Ten misconceptions about mathematics and its history. In: W. Asprey and P. Kiteher (eds.), History and Philosophy of Modern Mathematics, Minnesota Studies in the Philosophy of Science, Vol. XI, University of Minnesota Press, pp. 260-277.

Goldman, R. (2011) Understanding quaternions. Graphical Models 73:21-49.

Hempel, C.G. (1945/1953) Geometry and Empirical Science. In: P.P. Weiner (ed.), Readings in the Philosophy of Science, Appleton-Century-Crofts.

Hempel, C.G. (1966) Philosophy of Natural Science. Prentice- Hall.

Hume, D. (1739-40) A Treaty of Human Nature (accessed on 24 August 2012).

Kadane, J.B. (2011) Bayesian Methods and Ethics in a Clinical Trial Design. John Wiley & Sons.

Kitcher, P. (1980) Mathematical Rigor—Who Needs It? Nous 15: 490.

Kitcher, P. (1985) The Nature of Mathematical Knowledge. Oxford University Press.

Krantz, S.G. (2010) An Episodic History of Mathematics: Mathematical Culture Through Problem Solving. MAA.

Kuhn, T. (1963) The Structure of Scientific Revolutions. University of Chicago Press.

Irvine, A.D. (2009) Philosophy of Mathematics. North Holland.

Lakatos, I. (1963/64) Proofs and refutations. British Journal for the Philosophy of Science 14:1-25, 120-139, 221-243, 296-342.

Lakatos, I. (1970) Falsification and the methodology of scientific research programs. In: Criticism and the Growth of Knowledge, I. Lakatos and A. Musgrave (eds.), Cambridge University Press, pp. 170-196.

Lakatos, I. (1978) Infinite Regress and the Foundations of Mathematics, A Renaissance of Empiricism in Recent Philosophy of Mathematics, and Cauchy and the Continuum: The Significance of the Non-Standard Analysis for the History and Philosophy of Mathematics. In: J. Worrall and G. Currie (eds.) Lakatos’s Mathematics, Science and Epistemology. Philosophical Papers, Cambridge University Press.

Maddy, P. (1990) Realism in Mathematics. Clarendon.

Mehrtens, H. (1976) T.S. Kuhn’s theories and mathematics: a discussion paper on the ‘new historiography’ of mathematics. Historia Mathematica 3:297-320.

Ogle, K. (2009) Hierarchical Bayesian statistics: merging experimental and modeling approaches in ecology. Ecological Applications 19:577-581.

Popper, K. (1963) Conjectures and Refutations: The Growth of Scientific Knowledge. Routledge.

Radford, L. (2014) Reflections on history of mathematics. In: M. Fried and T. Dreyfus (eds.), Mathematics & Mathematics Education: Searching for Common Ground, Springer.

Smolin, L. (2007) The Trouble With Physics: The Rise of String Theory, The Fall of a Science, and What Comes Next. Mariner Books.

Whitehead, A.N. and Russell, B. (1910) Principia Mathematica. Cambridge University Press.

Wilson, J.G. (2013) Alfred Russel Wallace and Charles Darwin: perspectives on natural selection. Transactions of the Royal Society of South Australia 137:90-95.

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26 thoughts on “Progress in Mathematics and Logic — II

  1. 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.

    Although the Ancient Greeks, even before Euclid knew that there was no single unifying geometry, for example Aristotle says in “Physics”

    Necessity in mathematics is in a way similar to necessity in things which come to be through the operation of nature. Since a straight line is what it is, it is necessary that the angles of a triangle should equal two right angles. But not conversely; though if the angles are not equal to two right angles, then the straight line is not what it is either.

    This is obviously not an insight by Aristotle, as he only alludes to it by way of illustrating some other point, so it was likely general knowledge at the time. One of these things that get rediscovered, although without analytical geometry the ancient Greeks could hardly explore these ideas.

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  2. Hi Massimo,

    I really like this article (and learned quite a bit from it)! Repeatedly it emphasizes the close similarities between activity in maths and activity in science. I’d like to highlight some sentences that only make sense under my view that maths is, in essence, a real-world model just like physics.

    “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.”

    What do we mean by a “fact” that was “known” unless we mean it was adopted as being a real-world model and was known to be real-world true?

    [I guess we could mean that it was a “fact” that was “known” about some “Platonic realm”, but come on, that’s just silly, especially as, as you’ve said, we have no way of “knowing” about any such “realm”.]

    Quoting Kitcher: “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.”

    What does “knowledge of the statements/theorems” mean if not that we know that they are good real-world models?

    “Mehrtens begins by suggesting that mathematics can be thought of as being about “something” that offers resistance …”

    So if that “something” is not Platonic, then how about it being real-world correspondence? Are there other alternatives?

    “The 5th [postulate] was an anomaly from the beginning, because unlike Euclid’s other four postulates, it is not self-evident.”

    By “self-evident” we presumably mean obviously true as a real-world model? What else can we mean by it?

    Quoting Crowe: “Many new mathematical concepts, even though logically acceptable, meet forceful resistance after their appearance and achieve acceptance only after an extended period of time, […] the usefulness of these concepts has repeatedly forced mathematicians to accept and to tolerate them, …”

    So presumably the “forceful resistance” comes from initial intuitive rejection of them as being real-world sensible, and the acceptance then comes from realising that they are indeed powerful real-world models — this process being pretty much the same whether we’re talking about complex numbers or quantum mechanics.

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  3. A couple of other points:

    … 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).

    … nor inside physics really, I’ve never been that impressed by the Kuhnian account.

    Smolin concludes that the impact of a small number of highly influential people, […] with the result that a whole generation of physicists has passed without a new empirically driven breakthrough …

    I think this way over-states the role of strong theory. I suspect that the lack of empirically driven breakthroughs in fundamental physics results more from the lack of empirical results that are inconsistent with current models.

    Liked by 3 people

  4. That Pesky Word ‘Explain’

    I find the use of the word ‘explain’ by Weinberg in Dreams and others confusing. It explains why people are talking past each other…or should I say causes people to talk past each other.
    ex·plain

    verb: explain;
    3rd person present: explains; past tense: explained; past participle: explained; gerund or present participle: explaining

    -make (an idea, situation, or problem) clear to someone by describing it in more detail or revealing relevant facts or ideas.
    “they explained that their lives centered on the religious rituals”
    synonyms: describe, give an explanation of, make clear, make intelligible, spell out, put into words; More
    elucidate, expound, explicate, clarify, throw/shed light on; loss, interpret
    “a technician explained the procedure”

    -account for (an action or event) by giving a reason as excuse or justification.
    “Callie found it necessary to explain her blackened eye”
    synonyms: account for, give an explanation for, give a reason for; More
    justify, give a justification for, give an excuse for, vindicate, legitimize
    “nothing could explain his newfound wealth”

    –>_be the cause of_ or motivating factor for.
    “her father’s violence explains her pacifisms”

    You see ‘explain’ can mean cause; it can also mean to give an explanation.

    ‘’Atoms explain chemistry ‘can mean ‘Atoms cause chemistry.’

    ‘Atoms explain chemistry’ can also mean ‘That atoms cause chemistry is, at least part, of the explanation of chemistry’

    Weinberg seems to mix and mismatch these meanings leading people to think that particle physicist claim to explain why ‘anything exist.’

    We can’t and mostly we don’t!

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  5. Steven Weinberg:

    [Alvin Weinberg proposed a] criterion of scientific merit by proposing that, other things being equal, that field has the most scientific merit which contributes most heavily to and illuminates most brightly its neighboring scientific disciplines” (his italics). After reading an article of mine on these issues, Alvin wrote to me to remind me of his proposal. I had not forgotten it, but I had not agreed with it either. As I wrote in reply to Alvin, this sort of reasoning could be used to justify spending billions of dollars in classifying the butterflies of Texas, on the grounds that this would illuminate the classification of the butterflies of Oklahoma, and indeed of butterflies in general.

    Weinberg, Steven (2011-04-20). Dreams of a Final Theory: The Scientist’s Search for the Ultimate Laws of Nature (p. 60). Knopf Doubleday Publishing Group. Kindle Edition.

    –another straw man!

    –In fact, classification, however, tedious was an important contributor to Darwin figuring out how evolution works. In did illuminate neighboring disciplines. W/o Linnaeus there’s no Darwin.

    –who knows what a detail classification with DNA samples of Texas butter flies might reveal about evolution and ecology!

    -W miss directs the word neighbor from its metaphorical meaning to its literal geographic meaning. I doubt Alvin was impressed with the repost.

    Sorry Massimo. Weinberg is being illogical, but that’s not the point of your current post. I’ve got to stop reading this basically terrible book.

    Previous post was almost on topic.

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  6. Coel,

    > “maths is, in essence, a real-world model just like physics”

    I remember you made the same claim on an older blog of Massimo, but I also remember it was refuted. Maths can be used to model the real world, and it has been developed often with that aim in mind, but to say dat maths “is” a real-world model is simply wrong.

    I sympathize with your viewpoint in a certain sense. If renormalisation gives me accurate descriptions of reality (with certain caveats) and mathematics tells me that renormalisation makes no sense, I personally assume that mathematicians and theoretical physicists will sort out the mathematical problems one day – I believe, in other words, that mathematics is able to describe the real world.

    But maths is richer than that. The axioms of set theory contain an axiom of infinity and as far as I know infinity has never been observed in the real world. Then there’s the axiom of choice. One can use it to prove that every vector space has a basis – a desirable property if you want to model the real world – but one can also use it to prove that there exist sets that are not Lebesgue-measurable – and such a set has never been observed as far as I know. One can construct functions that are continuous everywhere but nowhere differentiable, one can construct shapes with a finite volume and a smooth but nevertheless infinite surface area etc. etc. Mathematics (set theory) starts with an axiom that doesn’t describe anything in the real world and it gives you more than you can find in the real world.

    In physics you can always say: “Oh, I have good physical reasons to neglect that term” or “my theory breaks down in that energy range”, but that trick doesn’t work in mathematics. Once you have set theory, you’ll have to swallow all those weird things you didn’t ask for. Mathematics is not a real world model “just like physics”.

    Liked by 1 person

  7. The axioms of set theory contain an axiom of infinity and as far as I know infinity has never been observed in the real world.

    Infinity never will be seen in the real world because it is a NaN, i.e., not-a-number. It is a concept which can be informally treated as a kind of number, but to handle it right you have to deal with limits and all that.

    An infinite sum that converges never quite reaches it’s limit, but you can get as close as you like by continuing to sum. That’s what a limit means, that’s how we use the concept of infinity.

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  8. Coel,

    thanks for the kind words.

    # I’d like to highlight some sentences that only make sense under my view that maths is, in essence, a real-world model just like physics #

    No, they don’t, and it ought to be obvious from the fact that I wrote, them…

    couvent, synred,

    those points have been made to Coel, a number of times. To no avail. I have no idea why he insists that mathematics is a model of the world while it very plainly isn’t. Since he’s a smart guy, I must conclude that ideological blinders (scientism) are at play. I honestly don’t mean it as an offense, just an anthropological observation.

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  9. Hi Coel,
    “… nor inside physics really, I’ve never been that impressed by the Kuhnian account.” Indeed, a fact well known by many ever since his book was published. A very well argued review of Kuhn’s book by Dudley Shapere (1964) shows some of the inadeqacies (both historical and “methodological”) which apply to Kuhn’s later attempt at elaboration as well.

    Hi Massimo,
    I realize that book suggestions are perhaps not really desirable in a sense, but surely they don’t hurt and sometimes one finds something of interest. Ladislav Kvasz’s award winning “Patterns of change” (http://fernando-gil.org.pt/en/nominees/2010/winner/) is a book on history and philosophy of mathematics which builds on some of the ideas taken from Kuhn and Lakatos. Might be worth at least a quick look if you are further interested in the development of math.

    Liked by 1 person

  10. To the issue of math being a real world model, how much is simply useful fiction?

    For instance, a dimensionless point is no more real than a dimensionless apple, because any multiple of zero is zero, but giving it some incremental spatial and temporal dimension complicates the desire for an ideal of location.

    So, for reasons of efficiency, it is far easier to overlook the fact it is self negating, than to have to deal with the fuzziness of an infinitesimal quantity of location, getting around the problem of “how many angels can dance on the head of a pin.”.

    Just as all of our other thought processes are a selecting of useful information, from its more chaotic context. Yet projecting from that useful bit of information, without taking the larger context into account, can be quite problematic. Such as issuing excessive monetary notes, but ignoring the inconvenient factor that every asset has to eventually be backed by an obligation.

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  11. Massimo,

    I already got that impression about Coel, but we can always try, can’t we?

    Now, if you don’t mind … I’m feeling more and more uncomfortable about the direction this project is taking.
    As I already stated, I don’t care if philosophy makes “progress” or not. For me philosophy is a valid and often interesting approach to certain problems, progress or not.
    I personally feel sometimes that, just like every discipline, phillosophy has a rather grand view of its own capabilities. It’s not entirely untrue that scientism – which I abhor – has a counterpoint in “philosophism”.
    But I also feel that your approach upto this point has been rather negative if I may say so.
    Everything I’ve read until now points to two possible conclusions.
    1) “Science” doesn’t make progress in any meaningful sense, so why should “philosophy” make progress?, or 2) if scientists are allowed to claim progess in these circumstances, then philosophers are allowed to claim progress too.
    I find this rather weak. I personally prefer a more positive approach. It’s far more convincing to confirm that science makes progress, philosophically speaking, to define that progess, and to prove that philosophy makes progress too.

    Another problem is that you may end up with more disciplines that make progress than you wish for.

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  12. A friend of mine, a math grad student called “Freaky Tom”, tried to make ‘progress’ in Astrology. He added the precession of the ecliptic as a correction to Astrological charts he tried to sell to sobriety girls at University of Illinois using the ‘improved’ calculation as a marketing ploy.

    I didn’t work. I’m not sure he sold even one improved chart.

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  13. Hi synred,

    I daresay Freaky Tom should have involved a grad student in marketing and PR, he would have sold a bundle. But then again, what would the marketing student need a maths student for?

    Liked by 1 person

  14. I also disagree with the “maths is a real world model” proposition. For a start it seems to be a category mistake. Maths is what you use to build models.

    Also, a mathematical model’s usefulness lies in the fact that it behaves somewhat like some physical system, or part thereof. But the same model could behave identically in a world which contained none of the things it models.

    Liked by 1 person

  15. Yeah, Tom was good at math, but not at marketing. A couple pages of impressive looking diagrams and correct calculations would not could not compete with a little old fashioned BS.

    He likely just liked doing the math and it was also something along the lines of making fun of Astrology, since they weren’t even calculating there own theory correctly. Maybe the potential customers could sense that he was teasing them.

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  16. I tend to think the information flows the other way. Math doesn’t intrinsically explain the world, but the world influences what math we come up. E.g., counting->number theory, but number theory is not proven by counting. Counting gave us the idea for number theory, but the fact that we count isn’t going to prove number theory is consistent. It seems likely it is consistent, but as long as it works well enough in the domain where we use it, we’ll use it.

    My buggy inconsistent computer programs work until it encounters the conditions that trigger the bug and it divides by 0 or get stuck in a loop. Inconsistences can lurk in a program for years while in merrily calculates track parameters from ‘hits’ in a detector, until one day some odd condition causes it to crash.

    Liked by 3 people

  17. Synred wrote:

    counting->number theory, but number theory is not proven by counting. Counting gave us the idea for number theory, but the fact that we count isn’t going to prove number theory is consistent

    ———————————————

    This is exactly right.

    Contingent experience is causally, but not epistemically prior to mathematical inquiry.

    Liked by 2 people

  18. couvent,

    # Everything I’ve read until now points to two possible conclusions.
    1) “Science” doesn’t make progress in any meaningful sense, so why should “philosophy” make progress?, or 2) if scientists are allowed to claim progess in these circumstances, then philosophers are allowed to claim progress too. #

    I’m not sure how you got either conclusion. I never claim that science doesn’t make progress, only that to provide a coherent account of it is much more difficult than scientists think. As for #2, we haven’t gotten to my positive account of progress in philosophy yet, though I have sketched enough of it that it should be clear that it is a positive one, and distinct from the way science does it.

    As for Coel, good luck my friend.

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  19. Hi couvent,

    I remember you made the same claim on an older blog of Massimo, but I also remember it was refuted.

    Disagreed with, rather than refuted! 😉

    Maths can be used to model the real world, and it has been developed often with that aim in mind, but to say dat maths “is” a real-world model is simply wrong.

    OK, what’s the difference? If I have something that “can be used as” a map of a country, and which works very well in that regard, allowing me to navigate accurately, in what way is it wrong to say that it “is” a map of that country?

    The axioms of set theory contain an axiom of infinity and as far as I know infinity has never been observed in the real world.

    True, but models are always abstractions and idealisations of reality. We haven’t observed infinity, but it’s a useful concept in modelling the world (physicists often use the concept of infinity in the sense of large enough that any boundary can be disregarded).

    Further, we’ve never observed an “inverse-square law”, and yet the concept is used in physical models because it is a useful model of reality.

    I’ve never seen contour lines in the real world. Hills don’t come with contour lines drawn on! And yet, a map containing contour lines is a good and useful representation of and model of reality.

    Then there’s the axiom of choice.

    Which is the one that everyone points to in such contexts. It is adopted because it is clearly real-world true for *finite* sets, though it is likely not “true”, or not a good real-world model applied to infinite sets.

    Mathematics (set theory) starts with an axiom that doesn’t describe anything in the real world and it gives you more than you can find in the real world.

    Sure, and the same is common in physics! You can start with the Friedmann equation of cosmology, and then find the solution that corresponds to our universe; but you can also find lots of solutions that do not correspond to our world or to anything else that is instantiated. Whether or not they correspond to “universes” that lie far beyond our observable horizon is something one can argue about.

    I really think that a lot of these attempts to draw distinctions between science and other things come from an insufficient appreciation of how science is. There are people here who think that science doesn’t do concepts or implications of concepts, and that conceptual analysis is outside science. Yet it isn’t. Maths-like conceptual analysis is fully present in physics.

    Hi Robin,

    For a start it seems to be a category mistake. Maths is what you use to build models.

    Yes, agreed, maths is both a model and a toolkit out of which one can build models. Similarly in physics, concepts such as “mass” or “electricity” are not in themselves models, but are components from which one builds models.

    But the same model could behave identically in a world which contained none of the things it models.

    So what? A “model of star formation” would behave the same in a universe with no stars.

    Liked by 2 people

  20. Dan,

    “This is exactly right.

    Contingent experience is causally, but not epistemically prior to mathematical inquiry.”

    Events, as well as our experiencing of them, have to occur in order for them to be ontologically, but mostly epistemically determined. The effective flow of time for events is future to past. They are in the present, before being in the past. Even though our mental experience is the flow from past to future, as it emerges from this physical state of being, i.e.. the present.

    Coel,

    “I’ve never seen contour lines in the real world. Hills don’t come with contour lines drawn on! And yet, a map containing contour lines is a good and useful representation of and model of reality.”

    And so a four dimensional representation of the flow of time and spatial coordinates doesn’t actually exist, as the “fabric of spacetime.” It only represents the changing configuration of what is occurring.

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  21. PS I note that none of those disagreeing with me gave any alternative interpretation of the sentences that I highlighted.

    One alternative is that maths is “about” a “Platonic realm”, but it beats me why anyone would want to invoke such a realm when there is a real and known realm staring us all in the face that does the job of being what maths is “about” just as well.

    Liked by 3 people

  22. Brodix wrote:

    Events, as well as our experiencing of them, have to occur in order for them to be ontologically, but mostly epistemically determined. The effective flow of time for events is future to past. They are in the present, before being in the past. Even though our mental experience is the flow from past to future, as it emerges from this physical state of being, i.e.. the present.

    ———————————————————-

    Wrong. Causal and epistemic priority are entirely separate and can come apart. This is a pretty basic idea in philosophy.

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  23. Kazalzs,

    I’m rather skeptical of the idea that the effectiveness of mathematics is an easily solved issue. As for Duhem-Quine being a straw man, I don’t think so. Why do you?

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