Prior to the advent of civilization, the human condition was defined by nature. The biosphere of the Earth, with all its diverse flora and fauna, was the predominant fact of human experience. Very little that human beings did could have an effect on the human condition beyond the most immediate effects an individual might cause in the environment, such as gathering or hunting for food. Nothing was changed by the passage of human beings through an environment that was, for them, their home. Human beings had to conform themselves to this world or die.

The life of early human communities was defined by nature, not by human activity.

The life of early human communities was defined by nature, not by human activity.

Since the advent of civilization, it has been civilization and not nature that determines the human condition. As one civilization has succeeded another, and, more importantly, as one kind of civilization has succeeded another kind of civilization — which latter happens far less frequently, since like kinds of civilization tend to succeed each other except when this process of civilizational succession is preempted by the emergence of an historical anomaly on the order of the initial emergence of civilization itself — the overwhelming fact of human experience has been shaped by civilization and the products of civilization, rather than by nature. This transformation from being shaped by nature to being shaped by civilization is what makes the passage from hunter-gatherer nomadism to settled agrarian civilization such a radical discontinuity in human experience.

This transformation has been gradual. In the earliest period of human civilizations, an entire civilization might grow up from nothing, spread regionally, assimilating local peoples not previously included in the project of civilization, and then die out, all without coming into contact with another civilization. The growth of human civilization has meant a gradual and steady increase in the density of human populations. It has already been thousands of years since a civilization could flourish and fail without encountering another civilization. It has been, moreover, hundreds of years since all human communities were bound together through networks of trade and communication.

Civilization is now continuous across the surface of the planet. The world-city — Doxiadis’ ecumenopolis, which I discussed in Civilization and the Technium — is already an accomplished fact. We retain our green spaces and our nature reserves, but all human communities ultimately are contiguous with each other, and there is no direction that you can go on the surface of the Earth without encountering another human community.

The civilization of the present, which I call industrial-technological civilization, is as distinct from the agricultural civilization that preceded it as agricultural civilization was distinct from the nomadic hunter-gatherer paradigm that preceded it in turn. In other words, the emergence of industrialization interpolated a discontinuity in the human condition on the order of the emergence of civilization itself. One of the aspects of industrial-technological civilization that distinguishes it from earlier agricultural civilization is the effective regimentation and indeed rigorization of the human condition.

The emergence of organized human activity, which corresponds to the emergence of the species itself, and which is therefore to be found in hunter-gatherer nomadism as much as in agrarian or industrial civilization, meant the emergence of institutions. At first, these institutions were as unsystematic and implicit as everything else in human experience. When civilizations began to abut each other in the agrarian era, it became necessary to make these institutions explicit and to formulate them in codes of law and regulation. At first, this codification itself was unsystematic. It was the emergence of industrialization that forced human civilizations to make its institutions not only explicit, but also systematic.

This process of systematization and rigorization is most clearly seen in the most abstract realms of thought. In the nineteenth century, when industrialization was beginning to transform the world, we see at the same time a revolution in mathematics that went beyond all the earlier history of mathematics. While Euclid famously systematized geometry in classical antiquity, it was not until the nineteenth century that mathematical thought grew to a point of sophistication that outstripped and exceeded Euclid.

From classical antiquity up to industrialization, it was frequently thought, and frequently asserted, that Euclid was the perfection of human reason in mathematics and that Aristotle was the perfection of human reason in logic, and there was simply nothing more to be done in the these fields beyond learning to repeat the lessons of the masters of antiquity. In the nineteenth century, during the period of rapid industrialization, people began to think about mathematics and logic in a way that was more sophisticated and subtle than even the great achievements of Euclid and Aristotle. Separately, yet almost simultaneously, three different mathematicians (Bolyai, Lobachevski, and Riemann) formulated systems of non-Euclidean geometry. Similarly revolutionary work transformed logic from its Aristotelian syllogistic origins into what is now called mathematical logic, the result of the work of George Boole, Frege, Peano, Russell, Whitehead, and many others.

At the same time that geometry and logic were being transformed, the rest of mathematics was also being profoundly transformed. Many of these transformational forces have roots that go back hundreds of years in history. This is also true of the industrial revolution itself. The growth of European society as a result of state competition within the European peninsula, the explicit formulation of legal codes and the gradual departure from a strictly peasant subsistence economy, the similarly gradual yet steady spread of technology in the form of windmills and watermills, ready to be powered by steam when the steam engine was invented, are all developments that anticipate and point to the industrial revolution. But the point here is that the anticipations did not come to fruition until the nineteenth century.

And so with mathematics. Newton and Leibniz independently invented the calculus, but it was left on unsure foundations for centuries, and Descartes had made the calculus possible by the earlier innovation of analytical geometry. These developments anticipated and pointed to the rigorization of mathematics, but the development did not come to fruition until the nineteenth century. The fruition is sometimes called the arithmetization of analysis, and involved the substitution of the limit method for fluxions in Newton and infinitesimals in Leibniz. This rigorous formulation of the calculus made possible engineering in its contemporary form, and rigorous engineering made it possible to bring the most advanced science of the day to the practical problems of industry.

Historians of mathematics and industrialization would probably cringe at my potted sketch of history, but here it is in sententious outline:

● Rigorization of mathematics also called the arithmetization of analysis

● Mathematization of science

● Scientific systematization of technology

● Technological rationalization of industry

I have discussed part of this cycle in my writings on industrial-technological civilization and the disruption of the industrial-technological cycle. The origins of this cycle involve the additional steps that made the cycle possible, and much of the additional steps are those that made logic, mathematics, and science rigorous in the nineteenth century.

The reader should also keep in mind the parallel rigorization of social institutions that occurred, including the transformation of the social sciences after the model of the hard sciences. Economics, which is particularly central to the considerations of industrial-technological civilization, has been completely transformed into a technical, mathematicized science.

With the rigorization of social institutions, and especially the economic institutions that shape human life from cradle to grave, it has been inevitable that the human condition itself should be made rigorous.

I am not suggesting this this has been a desirable, pleasant, or welcome development. On the contrary, industrial-technological civilization is beset in its most advanced quarters by a persistent apocalypticism and declensionism as industrialized populations fantasize about the end of the social regime that has come to control almost every aspect of life.

I wrote about the social dissatisfaction that issues in apocalypticism in Fear of the Future. I’ve been thinking more about this recently, and I hope to return to this theme when I can formulate my thoughts with the appropriate degree of rigor. I am seeking a definitive formulation of apocalypticism and how it is related to industrialization.

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Grand Strategy Annex

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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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