Radical Rigor

18 September 2011


The Legacy of Wittgenstein and the

last section of his Philosophical Investigations

Not long ago in Beyond Anti-Philosophy I introduced the idea of conceptual naturalism:

“…science is a philosophical research program, and it is based upon a small set of philosophical principles that have proved themselves remarkably fruitful in the investigation of the natural world. Scientific concepts are amenable to exposition by methodological naturalism. We might call this conceptual naturalism. Failures of conceptual naturalism — like investigating past lives as past lives, rather than as reports and descriptions of lives — result in conceptual confusion, and no amount of observation or experiment will clarify conceptual confusion.”

There is, as I see it, a reciprocity between methodological naturalism and conceptual naturalism: each is necessary to the exposition of the other; each is clarified by the clarity of the other. Methodological naturalism converges on conceptual naturalism; conceptual naturalism enlarges the sphere of phenomenon to which we can bring the resources of methodological naturalism. Both stop short of naturalism simpliciter, and a fortiori of ontological naturalism. This is the province of philosophy rather than of science, and it is perhaps one of the sources of anti-philosophy in science that science is ultimately bounded by philosophy.

In any case, since I explicitly mentioned conceptual confusion in this passage it was my intention to cite Wittgenstein in this connection. There very last section of Wittgenstein’s Philosophical Investigations includes this:

“The confusion and barrenness of psychology is not to be explained by its being a ‘young science’; its state is not comparable with that of physics, for instance, in its beginnings. (Rather, with that of certain branches of mathematics. Set theory.) For in psychology, there are experimental methods and conceptual confusion. (As in the other case, conceptual confusion and methods of proof.)”

“The existence of the experimental method makes us think that we have the means of getting rid of the problems which trouble us; but problem and method pass one another by.”

“An investigation is possible in connection with mathematics that is entirely analogous to our investigation of psychology. It is just as little a mathematical investigation as the other is a psychological one. It will not contain calculations, so it is not for example logistic. It might deserve the name of the ‘foundations of mathematics’.”

Ludwig Wittgenstein, Philosophical Investigations, Blackwell, 2003, p. 197e (translation modified)

It is interesting to note in this section how Wittgenstein approaches the philosophy of mathematics almost gingerly in this passage. While the Philosophical Investigations touches upon the philosophy of mathematics in places, elsewhere in Wittgenstein’s oeuvre the philosophy of mathematics is central, the fons et origo of Wittgenstein’s thought. Here Wittgenstein seems to be coming at it again, although from a new angle: as though after the experience of formulating ordinary language linguistic philosophy he had passed through to the other side of thought and was prepared to return to the source of his thought, older and now wiser.

Wittgenstein’s Tractatus Logico-Philosophicus — the only book-length work published during this lifetime — is through and through concerned with the philosophy of mathematics. The only contemporary philosophers Wittgenstein cited in this work were Frege and Russell, who had pioneered the doctrine of logicism, which is the position that mathematics is simply a highly developed form of logic, which amounts to the claim that there are no uniquely mathematical ideas, only logical ideas.

So this early period of Wittgenstein’s thought was brought into being and sustained by philosophical reflection on mathematics. We know that this was true of the later period of Wittgenstein’s thought also. After revolutionizing contemporary philosophy with his Tractatus, Wittgenstein returned to Austria and hid out in the Alps as a village schoolmaster, where a few Anglo-American philosophers made the pilgrimage to seek him out and question him about the Tractatus. One of them managed to persuade him to travel to Vienna to attend a lecture by L. E. J. Brouwer, the father of intuitionism, then the most influential form of constructivist philosophy of mathematics. After this lecture, Wittgenstein began his slow, incremental return to philosophy. But it was a different philosophy.

The works that Wittgenstein wrote in this period, which have been published posthumously, are sometimes called his Middle Period, to mark them off from the better known Early Wittgenstein (the Tractatus) and Late Wittgenstein (the Philosophical Investigations). These middle period works, too, are pervasively concerned with the philosophy of mathematics. Last February in Nothing contrasts with the form of the world I commented on one of these middle period works, the Philosophical Remarks. Though not nearly as well known as the Tractatus or the Philosophical Investigations, the middle period works are intriguing and fruitful in their own way. They have been an influence on my own thought.

While Wittgenstein was writing the works of his later period he delved deeply into philosophical psychology. Several works of this nature have been published posthumously. The Philosophical Investigations is in a sense both the culmination of these efforts in philosophical psychology and a response to them. The response comes in the final section quoted above. Wittgenstein, in delving deeply into psychology, found psychology to be infected with conceptual confusions that would not, he thought, be ameliorated by workman-like progress based on the experimental method. Something more was needed, something different was needed, to deliver psychology from its conceptual confusions.

Wittgenstein put much of contemporary mathematics in the same basket by comparing the conceptual confusion of set theory to the conceptual confusion of psychology. Here I decisively part ways with Wittgenstein, since I agree about the conceptual confusion of psychology, but I am hesitant over the conceptual confusions of set theory. It is not that I deny these latter confusions, but rather than I am hopeful about them (and, I guess, I’m not that hopeful about the former). Of course, many people are and were hopeful about what might be called the set theorization of mathematics. In many of Gödel’s later posthumously published essays (those that make up the contents of Volume III of his collected papers) we can see Gödel consciously groping toward a better conceptual formulation of the foundations of set theory. He saw the need and attempted to fill it, but the conceptual infrastructure needed for the decisive breakthrough (the kind of conceptual breakthrough that make it possible for Cantor to formulate set theory in the first place) wasn’t there yet. But Gödel was headed in the right direction.

Although Gödel wasn’t influenced by the thought of the later Wittgenstein, the direction he was headed in was the direction that Wittgenstein outlined in the last section of his Philosophical Investigations, quoted above. That is to say, Gödel was doing conceptual work in the foundations of mathematics. This has been the exception rather than the rule. Since the time of Gödel and Wittgenstein the field has been dominated by technical work, work of the highest formal rigor, and also work of conceptual rigor, but not, it must be said, radical conceptual work.

It is very difficult to characterize radical philosophy. Husserl spent a career trying to do so, and in his last years took pride in being able to call himself a genuine beginner in philosophy. But Husserl’s legacy (very much like the legacy of Wittgenstein and Gödel) has been dominated by philosophers who have done work of technical and conceptual rigor, but not radical work. Another problem stems from the political connotations of “radical,” which are connected to the Marxist tradition, which retains a vital connection to contemporary philosophy. So if you talk about radical philosophical thought, many people will assume that you’re talking about Marxism or some species of far left anarcho-syndicalism, and that is not at all what I have in mind.

I made a first attempt to get at my conception of radical philosophical thought — which I see as following in the tradition of the later Husserl, the later Wittgenstein, and the later Gödel — in my post Jacob Bronowski and Radical Relflection. I haven’t returned to thus much, partly because of other work on which I have been engaged, and partly due to the intrinsic difficulty to radical philosophical thinking. But I want to note it in connection with the last section of the Philosophical Investigations quoted above.

Radical thought, as I conceive of it, would not only be philosophically radical, but also scientifically radical. That is why I wrote the above post on Jacob Bronowski, who most philosophers would not recognize as having made any contribution to philosophy. But Bronowski, as I attempted to describe, did engage in radical scientific thought (and even attempted to popularize it) and this in itself constitutes a contribution to radical philosophical thought. We must learn from this radicalism wherever and whenever we find it.

For a time it seemed that philosophical thought had been overtaken by science, and much of twentieth century philosophical thought seems like a self-parody as philosophers try to mimic the success of the physical sciences. This is what twentieth century logical empiricism and logical positivism is all about. But these philosophers learned the wrong lesson. Contemporary philosophers are starting to learn the right lessons. I have written several posts about the emerging school of philosophical thought called Object Oriented Philosophy (or object oriented ontology – “OOO”). One of the best things about this movement is the attempt to take science seriously as a source of insight for philosophical thought. A lot of analytical philosophers wouldn’t recognize this even to be the case, since OOO is largely formulated in the language of continental philosophy, though a close reading will make this obvious.

Radical philosophy, however, cannot rest with accepting the insights of science or even accepting scientific knowledge or the scientific method as its point of departure. This is an important point of departure, but it is only the beginning. As I have attempted to point out in several posts (most recently in An Aristotelian Definition of Science), science is part of philosophy, and philosophy must then take responsibility for science.

And for mathematics as well. You see, if philosophy must take science seriously, and science take mathematics seriously, then philosophy also must take mathematics seriously. Science, philosophy, and mathematics are all caught up in the same dilemma of needing radical conceptual clarification, even while each as it progresses adds more and more to the accumulated total based on a confused conceptual foundation.

Of course, Wittgenstein took mathematics seriously, which is one reason he devoted the better part of his philosophical career to the philosophy of mathematics. But while Wittgenstein mentions the conceptual confusions of psychology in the same section that he mentions the possibility of a foundational inquiry into mathematics parallel to his foundational inquiry into psychology, he doesn’t seem to have quite seen the full relationship between the two. But, then again, science and and especially psychology of that time was not mathematicized to the extent that it is today. All of the rigorous technical work that I mentioned above has had the consequence of accelerating the mathematization of the sciences (think of economics today, or even branches of biology like theoretical ecology).

Mathematics provides the framework whereby other bodies of knowledge are rendered scientific, but is mathematics itself scientific, or is it rather part of the structure of science itself, and therefore neither scientific nor non-scientific?

If mathematics is an assumption of and part of the structure of science, then it is to be put on a par with parsimony, induction, uniformitarianism, and methodological naturalism. If, on the other hand, mathematics is science, is a part of science, then it is not on the same level of the philosophical principles of science that I have just mentioned, but is subject to them just as is the rest of science.

It could be argued that the principles of mathematics make themselves manifest in science through the medium of mathematics, so that mathematical principles are ultimately also scientific principles, and they are to be understood as being on a level with the other principles of science (such as those I mentioned above). This is an interesting idea, and it is, in fact, my first reaction to this as I begin to think about it. There is even a sense in which this is parallel to logicism, in which logic and mathematics ultimately share the same principles. However, I want to immediately point out that I do not regard this as anything even approaching a definitive formulation. It is only a first, instinctive, intuitive response to the question I am attempting to pose to myself.

I have my conceptual work cut out for me: I need to systematically think through the relation of mathematics to the sciences from the perspective of the philosophical principles of science. In other words, I know that I need to think through the relation of mathematics to parsimony, uniformitarianism, induction, and methodological naturalism. This will be an unfamiliar and therefore difficult exercise of thought, because these philosophical principles of science are usually formulated in empirical terms, so they must be re-formulated in a priori terms in order to understand their consequences for mathematics (either that, or re-formulate mathematics in terms of the a posteriori, which some philosophers prefer to do). This is a tall order, and I won’t be finishing it any time soon. In fact, I have yet to begin. In any case, I leave you with this reflection and exhortation:

We need radical philosophical thought, but it is difficult to do, requiring a real conceptual effort above and beyond the norm — the “norm” of which might be called the norm of normal philosophy, conceived in parallel with what Kuhn called normal science — and so it is rare. We need technical and conceptual rigor as well. These are also difficult, but slightly less rare. Ultimately what we need is both: we need radical rigor.

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Grand Strategy Annex

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4 Responses to “Radical Rigor”

  1. As a mathematician with an interest in crises and their avoidance or amelioration the seriousness with which you take these vital issues seems necessarily radical.

    Much of our current affairs have been hugely influenced by science, and good advice about science is essential to their conduct. You regard parsimony, induction, uniformitarianism, and methodological naturalism as the key principles of science. But they often seem more honoured in the breech. Maybe this explains some the troubles we have in making decisions that ought to be informed about science?

    As an example, up until 2007 politicians and their economic advisers seem to have been adopting scientific principles as if economies were constrained by our theories – but I wish they hadn’t.

    Mathematics, of course, makes no assumptions that the world is simple, regular or lacking in innovation: as mathematics it makes no assumptions at all. It might be helpful to distinguish two faces of mathematics. One is a means to improved computation, as in financial mathematics. The other is a means to critique the assumptions of other theories, identifying implicit assumptions, as Keynes did for probability theory and economics. Both types of mathematcis are important, but we seem most in need of the latter, including a critique of scientific methods. I discuss on my blog. Regards.

    • geopolicraticus said

      Dear Dave,

      I was wondering if anyone would read this rather long post, so thanks very much for letting me know that my efforts were not in vain. It could have been longer if I had included more quotes from the thinkers I have mentioned. In fact, I may do a longer version in the future because I find this so fascinating.

      No doubt the weakest part of my recent writings on science has been my “laundry list” of scientific principles — parsimony, induction, uniformitarianism, and methodological naturalism — which I more or less came up with off the top of my head. This list is highly imperfect. I pass over the principles implicit in the experimental method, and by including induction and not mentioning falsifiability, it sounds like I am siding with Carnap again Popper.

      I really need to think more carefully and systematically about the philosophical principles of science and come up with a more thorough list. But you seem to have gotten the general drift of what I am getting at, despite the imperfections of my exposition. And it it probably true that science needs to be defined in terms of a Wittgensteinian family resemblance in terms of the principles that any particular piece of work in science exemplifies.

      Economics, to which you allude as a mathematicized science having profound practical implications, is a perfect example of a science that rapidly became mathematicized but this mathematization involved little reflection on the foundational principles of the formalization of economics. I am working on this problem in other manuscripts, but it is a long term project. I may need to incorporate the distinction you make between two faces of mathematics, which we might call a constructive face and a critical face of the foundations of mathematics. This distinction bears some resemblance to the distinction that Russell employs in the opening pages of his Introduction to Mathematical Philosophy.

      Thanks again for your stimulating response.

      I will continue to work on these problems.

      Very Respectfully Yours,


  2. Nick, I sometimes distinguish between science and scientism. Scientism takes some principles or methods which it thinks are characteristic of science, and applies them without checking for appropriateness, as in economics. It seems to me that anything that regarded its principles and methods as a priori universally applicable would be dogmatic and ideological. So good science must always be safeguarded by ‘good thinking’.

    It seems to me that we need some word for that quality that real-world problems have which mean that your laundry list (which many seem to implicitly assume) is not appropriate. Complex?

    I make an attempt at http://djmarsay.wordpress.com/notes/about-these-posts/work-in-progress/complexity/complexity-causality-sense-making-and-strategy/ to distinguish complex from simple, and to relate to theories of activity. Comments appreciated.

    • geopolicraticus said

      Dear Dave,

      Science deals in abstractions. Mathematics is perhaps the most abstract science of all, but all the sciences are abstract to some degree. We have discovered that these abstractions can be preternaturally powerful in describing the world and predicting outcomes.

      Scientific models can in fact be so powerful and persuasive that people forget that they are dealing with abstractions. This is when, as you say, principles and methods are treated as dogmatic and ideological. And this is also the point at which “good thinking” ought to step in. I will be so bold as to say that the “good thinking” that is needed is philosophical thinking.

      “Complex” is fine, but may I also suggest “concrete,” since concrete realities are complex in virtue of their concreteness? I might also suggest “transcendental” in the specific sense of “transcending the principles of science.”

      Perhaps only literature — and indeed, within literature, perhaps only poetry — can capture the concrete world in its full concreteness. This is not the task of science.

      Very Best Wishes,


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