27 December 2014
The human mind is a strange and complex entity, and while the mind possesses unappreciated subtlety (of the kind I attempted to describe in The Human Overview), rigorous thinking does not come naturally to it. Rigor is a hard-won achievement, not a gift. If we want to achieve some measure of conceptual clarity we must make a particular effort to think rigorously. This is not easy. If you let the mind do what comes naturally and easily to it, you will probably not be thinking rigorously, and you will probably not attain conceptual clarity.
But what is rigor? To ask this question puts us in a position not unlike Saint Augustine who asked, “What, then, is time?” If no one asks me, I know what rigor is. If I wish to explain it to him who asks, I do not know. What distinguishes rigorous thinking from ordinary thinking? And what distinguishes a rigorous life from an ordinary life? Is there any relation between the formal and existential senses of rigor?
As a first and rough approximation, we could say that rigor is the implementation of a precise idea of precision. Whether or not a precise idea of precision can be applied to the human condition, a question that I have addressed in The Human Condition Made Rigorous, is a question of whether the formal sense of rigor is basic, and existential rigor is an implementation of formal rigor in life.
Kierkegaard concerned himself with what I am here calling existential rigor, i.e., the idea of living a rigorous life. One of the central themes that runs through Kierkegaard’s substantial corpus is the question of how one becomes an authentic Christian in an inauthentic Christian society (though this is not how Kierkegaard himself expressed the problem that preoccupied him). Kierkegaard expresses himself in the traditional Christian idiom of suffering for the truth, but Kierkegaard’s suffering is not pointless or meaningless: it is conducive to existential rigor:
“My purpose is to make it difficult to become a Christian, yet not more difficult than it is, nor to make it difficult for stupid people, and easy for clever pates, but qualitatively difficult, and essentially difficult for every man equally, for essentially it is equally difficult for every man to relinquish his understanding and his thinking, and to keep his soul fixed upon the absurd; it is comparatively more difficult for a man if he has much understanding — if one will keep in mind that not everyone who has lost his understanding over Christianity thereby proves that he has any.”
KIERKEGAARD’S CONCLUDING UNSCIENTIFIC POSTSCRIPT, Translated from the Danish by DAVID F. SWENSON, PROFESSOR OF PHILOSOPHY AT THE UNIVERSITY OF MINNESOTA, Completed after his death and provided with Introduction and Notes by WALTER LOWRIE, PRINCETON: PRINCETON UNIVERSITY PRESS, p. 495
The whole of Kierkegaard’s book Attack Upon Christendom is an explicit attack upon “official” Christianity, which he saw as too safe, too comfortable, too well-connected to the machinery of the state. In Kierkegaard’s Denmark, no one was suffering in order to bear witness to the truth of Christianity:
“…hundreds of men are introduced who instead of following Christ are snugly and comfortably settled, with family and steady promotion, under the guise that their activity is the Christianity of the New Testament, and who live off the fact that others have had to suffer for the truth (which precisely is Christianity), so that the relationship is completely inverted, and Christianity, which came into the world as the truth men die for, has now become the truth upon which they live, with family and steady promotion — ‘Rejoice then in life while thy springtime lasts’.”
Søren Kierkegaard, Attack Upon Christendom, Princeton: Princeton University Press, 1946, p. 42
And from Kierkegaard’s journals…
“Could you not discover some way in which you too could help the age? Then I thought, what if I sat down and made everything difficult? For one must try to be useful in every possible way. Even if the age does not need ballast I must be loved by all those who make everything easy; for if no one is prepared it difficult it becomes all too easy — to make things easy.”
Søren Kierkegaard, The Soul of Kierkegaard: Selections from His Journals, 1845, p. 93
Kierkegaard is full of such passages, and if you read him through you will probably find more compelling instances of this idea than the quotes I have plucked out above.
Kierkegaard called into question the easy habits of belief that we follow mostly without questioning them; Russell called into question the intuitions that come naturally to us, to the human mind, and which we mostly do not question. Both Kierkegaard and Russell thought there was value in doing things the hard way, not in order to court difficulty for its own sake, but rather for the different perspective it affords us by not simply doing what comes naturally, but having to think things through for ourselves.
Russell’s approach to rigor is superficially antithetical to that of Kierkegaard. While Kierkegaard was interested in the individual and his individual existence, Russell was interested in universal logical principles that had nothing to do with individual existence. William James once wrote to Russell, “My dying words to you are ‘Say good-by to mathematical logic if you wish to preserve your relations with concrete realities!'” Russell’s response was perfect deadpan: “As for the advice to say goodbye to mathematical logic if I wish to preserve my relation with concrete realities, I am not wholly inclined to dispute its wisdom. But I should push it farther, & say that it would be well to give up all philosophy, & abandon the student’s life altogether. Ten days of standing for Parliament gave me more relations to concrete realities than a lifetime of thought.”
Nevertheless, beyond these superficial differences, both Kierkegaard and Russell understood, each in his own way, that the easy impulse must be resisted. A passage from Bertrand Russell that I previously quoted in The Overview Effect in Formal Thought makes this point for formal rigor:
“The fact is that symbolism is useful because it makes things difficult. (This is not true of the advanced parts of mathematics, but only of the beginnings.) What we wish to know is, what can be deduced from what. Now, in the beginnings, everything is self-evident; and it is very hard to see whether one self-evident proposition follows from another or not. Obviousness is always the enemy to correctness. Hence we invent some new and difficult symbolism, in which nothing seems obvious. Then we set up certain rules for operating on the symbols, and the whole thing becomes mechanical. In this way we find out what must be taken as premiss and what can be demonstrated or defined.”
Bertrand Russell, Mysticism and Logic, “Mathematics and the Metaphysicians”
“There is a good deal of importance to philosophy in the theory of symbolism, a good deal more than at one time I thought. I think the importance is almost entirely negative, i.e., the importance lies in the fact that unless you are fairly self conscious about symbols, unless you are fairly aware of the relation of the symbol to what it symbolizes, you will find yourself attributing to the thing properties which only belong to the symbol. That, of course, is especially likely in very abstract studies such as philosophical logic, because the subject-matter that you are supposed to be thinking of is so exceedingly difficult and elusive that any person who has ever tried to think about it knows you do not think about it except perhaps once in six months for half a minute. The rest of the time you think about the symbols, because they are tangible, but the thing you are supposed to be thinking about is fearfully difficult and one does not often manage to think about it. The really good philosopher is the one who does once in six months think about it for a minute. Bad philosophers never do.”
Bertrand Russell, Logic and Knowledge: Essays 1901-1950, 1956, “The Philosophy of Logical Atomism,” I. “Facts and Propositions,” p. 185
For Russell, the use of symbols in reasoning constitutes a reformulation of the intuitive in a counter-intuitive form, and this makes it possible for us to struggle toward the truth without being distracted by matters that seem so obvious that our cognitive biases lead us toward deceptive obviousness instead of toward the truth. There is another name for this, defamailiarization (which I previously discussed in Reversing the Process of Defamiliarization). Great art defamiliarizes the familiar in order to present it to us again, anew, in unfamiliar terms. In this way we see the world with new eyes. Just so, the reformulation of intuitive thought in counter-intuitive forms presents the familiar to us in unfamiliar terms and we see our reasoning anew with the mind’s eye.
Intuitions have their place in formal thought. I have in the past written of the tension between intuition and formalization that characterizes formal thought, as well as of the place of intuition in philosophical argument (cf. Doing Justice to Our Intuitions: A 10 Step Method). But if intuitions have their place, they also have their limitations, and the making of easy things difficult is a struggle against the limitations of intuition. What Kierkegaard and Russell have in common in their conception of rigor is that of making something ordinarily easy into something difficult in order to overcome the limitations of the natural and the intuitive. All of this may sound rather arcane and confined to academic squabbles, but it is in fact quite directly related to the world situation today.
I have often written about the anonymity and anomie of life in industrial-technological civilization; this is a familiar theme that has been worked through quite extensively in twentieth century sociology, and one could argue that it is also a prominent element in existentialism. But the human condition in the context of our civilization today is not only marked by anonymity and anomie, but also by high and rising standards of living, which usually translates directly into comfort. While we are perhaps more bereft of meaning than ever, we are also more comfortable than ever before in history. This has also been studied in some detail. Occasionally this combination of a comfortable but listless life is called “affluenza.”
Kierkegaard’s defamiliarization of (institutionalized and inauthentic) Christianity was intended to make Christianity difficult for bourgeois worldlings; the militant Islamists of our time want to make Islam difficult and demanding for those who would count themselves Muslims. It is the same demand for existential rigor in each that is the motivation. If it is difficult to understand why young men at the height of their prowess and physical powers can be seduced into extremist militancy, one need only reflect for a moment on the attraction of difficult things and the earned honors of existential rigor. The west has almost completely forgotten the attraction of difficult things. What remains is perhaps the interest in “extreme” sports, in which individuals test themselves against contrived physical challenges, which provides a kind of existential rigor along with bragging rights.
Extremist ideologies offer precisely the two things for which the individual hungers but cannot find in contemporary industrialized society: meaning, and a challenge to his complacency. An elaborately worked out eschatological conception of history shows the individual his special place within the grand scheme of things (this is the familiar ground of cosmic warfare and the eschatological conception of history), but this eschatological vision is not simply handed for free to the new communicant. He must work for it, strive for it, sacrifice for it. And when he has proved himself equal to the demands placed upon him, then he is rewarded with the profoundly satisfying gift of an earned honor: membership in a community of the elect.
This view is not confined to violent extremists. We meet with this whenever someone makes the commonplace remark that we don’t value that which is given away for free, and Spinoza expressed the thought with more eloquence: “All noble things are as difficult as they are rare.” Anyone who feels this pull of difficult things, who desires a challenge, who wants to be tested in order to prove their worth in the only way that truly counts, is an existentialist in action, if not in thought, because it is the existentialist conception of authenticity that is operative in this conception of existential rigor.
We have tended to think of pre-modern societies, mostly agrarian-ecclesiastical civilization, with their rigid social hierarchies and inherited social positions, as paradigmatic examples of inauthentic societies, but we have managed to create a thoroughly inauthentic society in the midst of our industrial-technological civilization. This civilization and its social order may have its origins in the overturning of the inauthentic social order of earlier ages, but, after an initial period of social experimentation, the present social order ossified and re-created many of the inauthentic and hierarchical forms that characterized the overthrown social order.
Inauthentic societies are awash in unearned unearned advantages. I wrote about this earlier in discussing the urban austerity of Simone Weil, the wilderness austerity of Christopher McCandless (also known as Alexander Supertramp), and comparing the two in Weil and McCandless: Another Parallel:
“…the accomplishments of the elite and the privileged are always tainted by the fact that what they have attained has not been earned. But it is apparent that there are always a few honest individuals among the privileged who are acutely aware that their position has not been earned, that it is tainted, and the only way to prove that one can make it on one’s own is to cut one’s ties to one’s privileged background and strike out on one’s own.”
There is a certain sense in which the available and ample comforts of industrial-technological civilization transformed the greater part of the global population into complacent consumers who accept an inauthentic life. There is another name of this too; Nietzsche called such individuals Last Men.
. . . . .
. . . . .
. . . . .
. . . . .
25 December 2012
Prior to the advent of civilization, the human condition was defined by nature. Evolutionary biologist call this initial human condition the environment of evolutionary adaptedness (or EEA). 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.
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 (though it is called by another name, or no name at all). 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 (which I call agrarian-ecclesiastical 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. Intrinsically arithmetical realities could now be given a rigorous mathematical exposition.
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. Foucault was instrumental in pointing out salient aspects of this, which he called biopower, and which, I suggest, will eventually issues in technopower.
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.
. . . . .
. . . . .
. . . . .
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.
. . . . .
. . . . .
. . . . .