Does mathematics “progress”?

American Progress, 1872 painting by John Gast; this image is available from the United States Library of Congress‘s Prints and Photographs division under the digital ID ppmsca.09855.

An updated and expanded version of this post has been published at Silicon Reckoner. Comments on the new version are welcome below.

Mechanization of mathematics, at least in certain aspects, has been welcomed as “progress,” notably in a comment on this blog. Most readers of this blog, I suspect, will embrace the “progressive” label, if the alternative is what is being promoted on the Killing Obama’s Radical Progressive Agenda Facebook page. Nevertheless, as the above image reminds us, the notion of “progress” in its current usage is so thoroughly entwined with technological determinism, European colonialism, genocide, and environmental devastation, that it is a struggle to find an interpretation of the word, applicable to mathematics, whose connotations are unequivocally positive.

The OED traces the first use of the word, in the (originally metaphorical) sense of

Advancement to a further or higher stage, or to further or higher stages successively; growth; development, usually to a better state or condition; improvement…

to 1457 in the Acts of Parliament of Scotland, in the sentence

Sen Gode..hes send oure souerane lorde sik progres and prosperite, that [etc.].

The word’s current use evolved from the mid-18th through 19th centuries, from the Enlightenment through the Industrial Revolution. The OED quotes Benjamin Franklin using the word in 1780. In 1794 the Marquis de Condorcet wrote his Sketch for a Historical Picture of the Progress of the Human Mind, “perhaps the most influential formulation of the idea of progress ever written” according to Wikipedia, while hiding on the “rue des Fossoyeurs” (the present-day rue Servandoni) in Paris from the warrant for his arrest issued by the Convention. (By dying in prison, Condorcet escaped the guillotine, “a symbol of the penal, technological and humanitarian progress inspired by the Enlightenment” according to the Open University website.) In the 19th century “progress” was a watchword for thinkers as diverse as Hegel, Comte, Marx and Engels (“entire sections of the ruling class are, by the advance of industry, precipitated into the proletariat [… and] also supply the proletariat with fresh elements of enlightenment and progress”), Darwin, and Herbert Spencer (“the civilized man departs more widely from the general type of the placental mammalia than do the lower human races“).

Gast’s painting is one illustration of the principle, that was generally accepted by European colonizers, including the American settlers, and given clear expression by the German Rudolf Cronau in 1896:

The current inequality of the races is an indubitable fact. Under equally favorable climatic and land conditions the higher race always displaces the lower, i.e., contact with the culture of the higher race is a fatal poison for the lower race and kills them…. [American Indians] naturally succumb in the struggle, its race vanishes and civilization strides across their corpses…. Therein lies once again the great doctrine, that the evolution of humanity and of the individual nations progresses, not through moral principles, but rather by dint of the right of the stronger.

Rudolf Cronau, in Friedrich Hellwald, Kulturgeschichte in ihrer natürlichen Entwickelung, 4th ed., 4 vols. (Leipzig: Friesenhahn, 1896 ), IV: 615-16

Other colonial powers used the notion of progress in similar ways. Gast’s painting is well-known but I only became aware of it last month, when I watched Raoul Peck’s indispensable four-part documentary Exterminate all the Brutes, whose title is a quotation from the character Kurtz in Conrad’s Heart of Darkness, characterized as “an emissary of pity and science and progress, and devil knows what else.” “By the simple exercise of our will we can exert a power for good practically unbounded,” Kurtz said, a few lines before arriving at the words that served as Peck’s title. This was the intellectual matrix in which Hitler formed his world view. The word “progress,” in expressions like “human progress” or “progress of mankind,” appears dozens of times in Mein Kampf. A typical example:

“Not through [the Jew] does any progress of mankind occur.”

Mein Kampf, Chapter XI.

Now that Godwin’s law has been reconfirmed, possibly for the first but certainly not for the last time, in connection with mechanization of mathematics, I can quickly come to the point of this post, which is to draw attention to the questions that are not being asked when the desirability or feasibility of mechanization of mathematics is under debate. Arundhati Roy asked one such question in her first non-fiction book, about the Narmada Dam project:

How can you measure progress if you don’t know what it costs and who has paid for it?

Arundhati Roy, The Cost of Living, Random House of Canada, 1999

Twenty years after the book’s publication, when the dam has submerged at least 178 villages in Madhya Pradesh, Tina Stevens and Stuart Newman defended the precautionary principle as protection from the “hidden agendas of BioTechnical science”:

Precaution does not derail progress; rather, it affords us the time we need to ensure we progress in socially, economically, and environmentally just ways.

Tina Stevens and Stuart Newman, Biotech Juggernaut, Routledge, 2019

Most of the mathematicians and philosophers who promote mechanization are perfectly candid about their agendas, and cannot be suspected of genocidal tendencies. But the potential implications of the widespread adoption of technological solutions to perceived mathematical problems — “what it [will] cost… and who [will pay] for it,” not to mention the question of who stands to benefit — are simply not being acknowledged.

This post is meant to be the first of a series of texts exploring these questions and the reasons for the absence of any sustained discussion of these issues on the part of mathematicians, in contrast to the very visible public debate about the promises and dangers of AI. Much of MWA was devoted to a critique of the notion of “usefulness” in mathematics when, as is nearly always the case, it is not accompanied by a close examination of the perspectives in which an application of mathematics may or may not be seen as “useful.” The similarities with the intended critique of the uncritical use of the word “progress” are evident, but now I want to keep focused on the ideology surrounding mechanization — mechanical proof verification and automated theorem proving, in particular. So the plan is to continue this discussion in a different venue, and gradually to phase out this blog, as I have already tried and failed to do once before.

40 thoughts on “Does mathematics “progress”?

  1. Jon Awbrey

    ❝It is understandable that an engineer should be completely absorbed in his speciality, instead of pouring himself out into the freedom and vastness of the world of thought, even though his machines are being sent off to the ends of the earth; for he no more needs to be capable of applying to his own personal soul what is daring and new in the soul of his subject than a machine is in fact capable of applying to itself the differential calculus on which it is based. The same thing cannot, however, be said about mathematics; for here we have the new method of thought, pure intellect, the very well-spring of the times, the fons et origo of an unfathomable transformation.❞

    Robert Musil • The Man Without Qualities

    Like

    Reply
    1. ThS

      The extremely interesting work of Gowers and Ganesalingam described in this post https://gowers.wordpress.com/2013/04/14/answers-results-of-polls-and-a-brief-description-of-the-program/ and that was later posted to the arxiv https://gowers.wordpress.com/2013/09/19/preprint-about-theorem-proving-program-up-on-arxiv had motivations coming from linguistics and from a will to better understand mathematicians’ thought processes. There are no sinister motivations, it is at least progress for linguistics and cognitive science. It would also be progress for mathematics if more evolved versions of the program lead to profound theorems being proved, which to experts seemed totally untractable. For now, it is not the case.

      Like

      Reply
  2. Graham White

    One of the ways in which mathematics has been said to progress has been in rise of what is called abstract algebra, and the phasing out of the central role in mathematics of such things as automorphic forms (though, of course, automorphic forms have an after-life in the form of the Langlands programme and such like). Still, though, reading nineteenth century mathematics gives me more joy than reading most twentieth century mathematics.

    Like

    Reply
    1. mathematicswithoutapologies Post author

      “Mathematics has been said to progress” is a historical claim that can be justified by a study of texts. “Mathematics has progressed” is a value judgment that may or may not be consensual and may evolve in time. Automorphic forms are a good example of a field that at one time ceased to be fashionable. When I was a graduate student the Stone-Cech compactification was yesterday’s news. Now it’s back, like go-go boots.

      Like

      Reply
  3. Jon Awbrey

    Just off-hand, by way of getting grounded and oriented, there are a few questions I’d have to ask first.

    • Is mathematics a science, or a form of scientific inquiry?
    • If so, what is the place of mathematics within the sciences?
    • Does science progress?
    •• (I mean progress in the sense of progress toward a goal, not necessarily progressive jazz or progressions in music generally.)
    • If so, what is the goal of science?
    • If science has a goal, does mathematics serve it?
    • If so, how?

    I know, that’s a lot of •

    Like

    Reply
    1. mathematicswithoutapologies Post author

      Your parenthetical reference to “a goal” retraces the passage from the initial meaning of “progress,” from Latin progressus, as in

      “In oure progresse to outward werkis aftir þese now afore taken preisingis and preiyngis..we ouȝte to dispose vs wiþ holding to gidere of al þe bifore seid inward endewing” (R. Pecock, 1443)

      the first definition in the OED, to the second definition, which corresponds to the value judgment. The latter may or may not be consensual, depending on the goal.

      But I think the same question can be asked without reference to science. You can replace “science” by “art” or by “handicraft” or by “religious ritual” or “operation of power” in each of those questions.

      Like

      Reply
      1. Jon Awbrey

        By science or scientific inquiry I meant to focus on species of goal-directed activity with the specific goal of “knowledge” and to ask whether mathematical arts and crafts and rites fall within that ballpark. We come back to the antic Socratic question of whether we put another drachma in the machine for the sake of an external gain or simply to continue the play for its own sake.

        Like

      2. mathematicswithoutapologies Post author

        In this post I was less concerned with philosophy than with philology — which, by the way, is another example of a term that was once seen as hopelessly antiquated and musty but that has been revived recently.

        But when you introduce “knowledge” you have to grapple with “truth” and “objective reality.” Philosophy has been so unsuccessful at pinning those down that some would prefer to give up on them altogether. We can choose to call what mathematics generates “knowledge” but that just leads to the (philological) question of what this has to do with what other practices call “knowledge.”

        And by the way, I hope you don’t mind that I slightly edited your last comment.

        Like

    1. mathematicswithoutapologies Post author

      That Delphi paper, as you know, was a strong influence on parts of my book.

      But I have to confess — and this should be clearer if and when this blog migrates to another platform — that I’m trying to understand how promoting mechanization as “progress” is or is not similar to the promotion of smart toasters — to mention an item mentioned in passing in Godzilla vs. Kong (for me that was the most interesting moment in the movie, and it went by too quickly). If the authors of that film assumed their mass audience suspects smart technology to have a hidden agenda, we are entitled to question the inevitability of mechanization, not to mention the “progress” it is supposed to embody.

      But to return to philosophy: if philosophers can’t pin down what progress means (and I know there is an entry in the Stanford Encyclopedia, and a book by Nesbit), does that mean (metaphilosophically) that the word should be scrapped, or used in scare quotes, or just used sparingly (like nutmeg)? Or are my qualms about the word just a misplaced reaction as a mathematician to a term that lacks precise definition? And here I have to thank you again for providing an ostensive strategy for dealing with mathematical realism, on the model of “real ale.”

      Like

      Reply
  4. David Corfield

    The concept of ‘progress’ in mathematics is a strangely neglected subject. Where the physicist can point to evident progress, e.g, Weinberg–Salam electroweak theory, and pace some anti-realist scruples, we might understand this progress in terms of having found out something new about the world, I think it’s harder to say just what makes for mathematical progress. On the other hand, surely we can agree that, e.g., through Langlands’ work number theory progressed. My own proposed solution, as you know, was via the narrative structure of accounts of mathematics.

    Leaving aside mechanization, do you feel uncomfortable applying the idea of progress in your field because of its historical connotations? You surely need some language to say that a field has moved along, such that a practitioner today can address what was unknown or uncertain or only vaguely formulated in the past.

    Like

    Reply
    1. mathematicswithoutapologies Post author

      This is a helpful reminder, in particular because you proposed this in part as an alternative to Lakatos, and the post should have referred to his distinction between “progressive” and “degenerating” research programs. The names indicate which of the two kinds of research programs is to be preferred. An empirically-minded sociologist of mathematics could use a metric (h-index?) to demonstrate the objectivity of Lakatos’s distinction. Mathematicians would be skeptical about such a metric.

      I think neither Lakatos’s methodology of research programs nor your narrative approach can account for the potential gulf between mechanized and pre-mechanized mathematics, which could easily evolve into two distinct disciplines.

      I’m uneasy with the word “progress” because it can be used to short-circuit critical thought. My purpose in mentioning its historical associations with domination, eugenics, greed, genocide, white supremacy, and so on is to remind those who use it to do so in a critical spirit, rather than to suppose that labelling something as “progress” is an adequate substitute for rational analysis. In future texts I intend to explore the context in which proposals for mechanization arise, namely how the ideology of progress is used to advance powerful industrial interests. In the back of my mind is always the concern that these interests will persuade funding agencies that their mechanical practitioners represent progress over human mathematicians.

      Like

      Reply
  5. David Corfield

    I look forward to hearing your views on this. Since I am sympathetic to Thurston on progress in terms of increased understanding, I share your suspicions on mechanized mathematics as progress.

    As we’ve discussed before, however, even with the latter there is the difference between merely establishing correctness in a formal system and the human rethinking of concepts to adapt to a prover. If the latter, like Lean, is tied in with category-theoretic logic, then experience with it boosts one’s ‘internal logic’ skills. I know you’re rather sceptical about whether these skills really contribute to mathematical understanding, but, as I observed to you before, they do allow, say, Mike Shulman and Peter Scholze to have an exchange: https://golem.ph.utexas.edu/category/2020/03/pyknoticity_versus_cohesivenes.html#c057780

    Like

    Reply
    1. mathematicswithoutapologies Post author

      Maybe I wouldn’t be so skeptical if the expression “internal logic” didn’t sound like something invented by a marketing executive. The first impression it gives is of THE key to understanding, as in the sentence “When we figure out the internal logic of the brain we will understand consciousness.” Someone finally explained to me that it’s just the name for a certain functor. I find that perfectly respectable, even if (or especially if) it’s a functor that has little or no relevance to my own work. But the name does promise more than it delivers.

      But now, since your previous comment mentioned finding out something new about the world, I am reminded that I was brought up during the Cold War to believe that the arts also exhibit “progress.” So the modernist novel represents progress over the realist novel, cubism represents progress over impressionism, and so on. If I expand on this post in a new venue I will track down some quotes to that effect that were taken seriously at the time. Critics don’t write that way any more, though I guess it’s said that there has been some recent progress in the recognition and visibility of certain underrepresented groups, and I might even say something like that.

      Does this constitute proof that mathematics is not an art? Would the explanation be that mathematics generates knowledge and music, for example, does not? Who would defend such a claim?

      Like

      Reply
      1. David Corfield

        Bring alive heroes from the past. Don’t you have a different kind of conversation with Gauss or Riemann, than you do with Beethoven or Manet? They will want to know how things have gone in their field, and in the arts there are things to say about innovative techniques being introduced, but this can’t compare with ‘Have you demonstrated my conjecture yet? Was my geometry put to any scientific use? Is it clear why complex surfaces resemble arithmetic field extensions?’

        Like

      2. mathematicswithoutapologies Post author

        That’s helpful in understanding how mathematicians often do use the word “progress,” in relation to defined research programs.
        It leaves open the (quintessentially philosophical) question of what makes these people heroes, when other people, with whom few of us are now having conversations, were also asking questions.

        Like

  6. Jon Awbrey

    ♩ On A Related Note ♪ The Music Of The Primes ♫ Riffs & Rotes

    Now there’s a progression of progressions I could enjoy, musically speaking, ad infinitum, and yet this pilgrim would consider it progress, mathematical speaking, if he could understand why the sequentiae should be sequenced as they are.  Would that understanding add to my enjoyment?  On jugera …

    Like

    Reply
      1. mathematicswithoutapologies Post author

        And in the French context it would be the “Mission civilisatrice.” And this is a great opportunity to invite French speakers to watch Michèle Audin remind French-speaking listeners just what that kind of civilizing entailed in French-occupied Algeria, while celebrating the 150th anniversary of the Paris Commune, to which she has already dedicated several books of non-fiction, two novels, and an exceptionally detailed blog. You can start with https://www.facebook.com/editionssociales/videos/la-commune-de-paris-hier-et-aujourdhui/895189917939571/

        And while you’re at it, you can also read her account of her parents’ activism during the Algerian revolution and the tragic consequences.

        Like

  7. Jon Awbrey

    Re: In this post I was less concerned with philosophy than with philology

    Not that I don’t love wisdom and words, however often their stars may cross, but I invoked Musil rather for the way he syzygied the way of the engineer, the way of the mathematician, and the recursive point where their ways diverge.  It’s in this frame I think of the word entelechy, which I got from readings in Aristotle, Goethe, and Peirce and promptly gave a personal gloss as end in itself, partly on the influence of Conway’s game theory.

    And that’s where I remember all those two-bit pieces I gave up to pinball machines in the early 70s and the critical point in my own trajectory when I realized I would never beat those machines, not that way, not ever, and I turned to the more collaboratory ends of teaching machines how to learn and reason.

    Like

    Reply
  8. Pingback: Differential Logic and Dynamic Systems • Discussion 1 | Inquiry Into Inquiry

  9. mathematicswithoutapologies Post author

    I want to return to David Corfield’s question and suggest that the natural comparison is not between my conversation with Gauss and Riemann and my conversation with the artists, but rather between my conversation with the culture heroes of mathematics and the painter’s conversation with Manet and Co. Writers and painters, and I suppose musicians as well, often use the language of truth to describe their goals. It can be argued that Virginia Woolf’s representation of interior monologue is in some sense more accurate than Jane Austen’s — which counts as progress if that sort of thing is the goal of your “research program.”

    And I just rediscovered a sentence about progress in connection with the mechanization program, in John Harrison’s 2008 article in the AMS Notices:

    We welcome the prospect of formalizing mathematics. In our view, the traditional social process is an anachronism to be swept away by formalization, just as empiricism replaced a similar ‘social process.’

    A definition of “progress”: “the sweeping away of anachronisms,” like indigenous people and buffalo. I’ll be returning to Harrison’s article soon, in another venue, because upon rereading it carefully I am struck by the strangeness of its worldview.

    Like

    Reply
  10. Jon Awbrey

    In several places I can’t find right now I described formalization as an arrow.  A like idea occurs in a paper by Susan Awbrey and myself where we discussed the “dimension of increasing formalization in our mental models of the world” as one of the obstacles to integrating knowledge across various styles of inquiry.  An excerpt follows.

    Conceptual Barriers to Creating Integrative Universities

    The Trivializing of Integration

    From reviewing its philosophical sources, we can see that the trivialization of integration hypothesis presents barriers to creating an integrated learning environment. Below we focus on three closely interrelated problematics and the bearing that the triviality of integration hypothesis has on them:

    Problematic 1 is the tension that arises along a dimension of increasing formalization in our mental models of the world, between what we may call the ‘informal context’ of real-world practice and the ‘formal context’ of specialized study.

    Problematic 2 is the difficulty in communication that is created by differing mental models of the world, in other words, by the tendency among groups of specialists to form internally coherent but externally disparate systems of mental images.

    Problematic 3 is a special type of communication difficulty that commonly arises between the ‘Two Cultures’ of the scientific and the humanistic disciplines. A significant part of the problem derives from the differential emphasis that each group places on its use of symbolic and conceptual systems, limiting itself to either the denotative or the connotative planes of variation, but seldom integrating the two.

    Like

    Reply
  11. Pingback: Differential Logic and Dynamic Systems • Discussion 2 | Inquiry Into Inquiry

  12. A.B.

    I am very curious to read what you would say about the recent so called Liquid Tensor Experiment, in light of your past comments on the interaction between mathematicians and computers.

    Like

    Reply
  13. mathematicswithoutapologies Post author

    In that case you should write to the editors at Quanta. I sent them an article on the topic after Scholze’s first post on the Xena project blog last December, but they decided not to publish it. I have reworked it, with a reference to Scholze’s second post, and it may appear elsewhere. In the long term I am thinking of moving this blog to Substack and if I do I expect to devote one post there entirely to the Liquid Tensor Experiment, with attention to its mathematical as well as sociological implications.

    Like

    Reply
  14. Łukasz

    What does the phrase “mathematics progresses” mean, in the context of the result placed in the paper: T. J. Stępień, Ł. T. Stępień, „On the Consistency of the Arithmetic System”, Journal of Mathematics and System Science, vol. 7, 43 (2017), arXiv:1803.11072 ?
    There in this paper, was published a proof of the consistency of the Arithmetic System. This proof had been done within this Arithmetic System (the abstract related to this paper: T. J. Stepien and L. T. Stepien, “On the consistency of Peano’s Arithmetic System” , The Bulletin of Symbolic Logic, vol. 16, No. 1, 132 (2010)).
    This is well-known fact that Gödel’s Second Incompleteness Theorem tells that consistency of Arithmetic System is unprovable within this System, unless this System is inconsistent.

    Like

    Reply
      1. Łukasz

        The title of the issue raised in your post, is: “Does mathematics ‘progress’ ?”
        As I have it understood, your intention has been to wonder about making progress by mathematics.
        So, first of all I have written about the proof of consistency of Arithmetic, done within Arithmetic, contrary to (at least at face value) Gödel’s Second Incompleteness Theorem. Next, as I have recalled: Gödel’s Second Incompleteness Theorem tells that consistency of Arithmetic System is unprovable within this System, unless this System is inconsistent. So, my reflection is as follows: how about progress of mathematics in this context ?

        Like

      2. mathematicswithoutapologies Post author

        The point of my post is not to answer the question one way or another but to examine what “progress” may mean in mathematics The word is typically used without explanation and there’s no reason to think different people use it to mean the same thing. In particular, when those who promote mechanization as “progress” they presume the word is unambiguous, and moreover that it invariably has a positive connotation. I don’t believe that the word is unambiguous, and I included reminders of the history of the word in order to suggest that the connotation is not necessarily positive.

        So to answer your question, if you have read my post carefully you’ll know that I don’t claim to be able to answer your question, nor even to be able to make sense of your question unambiguously.

        Like

  15. Pingback: Announcing a new newsletter on mechanizing mathematics | Mathematics without Apologies, by Michael Harris

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s