Monday


area_51

There is a mode of fallacious argumentation, related to argumentum ad ignorantiam yet sufficiently distinct from it, that I am going to call the appeal to embargoed evidence (and which could also be called the appeal to sequestered evidence). The appeal to embargoed evidence occurs when someone makes the claim that some open question has been definitely answered, but that the evidence that settles the question is not available to public inspection. The evidence may be missing, or hidden, or suppressed — but whatever its status, it cannot be produced. One is supposed to take the speaker’s assurances on the existence and nature of the evidence.

I have personally experienced the appeal to embargoed evidence many times, as, for example, when a reader responded to my posts Of Distinctions, Weak and Strong and Of Distinctions, Principled and Otherwise with this comment:

May I recommend lunch with a scientist working in nano technology. The ‘mind body problem’ you speak of was “solved” in a lab I worked in years ago. Sadly it’s classified due to ESP Rsrch. Such musings with regard to mind, now seem like Claudius Ptolemy lecturing about his epicycles.

I responded in turn:

If it’s classified, don’t you suppose that I would have a difficult time getting anyone to talk? Also, I would insist on writing about it, which would endanger both myself and my source.

Is anyone persuaded or convinced by claims of evidence that cannot be produced? I can only conclude that the appeal to embargoed evidence must be at least occasionally effective, or I would not run across it as often as I do.

The appeal to embargoed evidence is encountered most frequently today in the form of claims of government suppression of UFOs and alien bodies, or private industry suppression of particular technologies that would adversely affect established business models (e.g., the idea of the 100 MPG car). While the appeal to embargoed evidence is most commonly encountered in discussions of conspiracy theories, it is also to be found in high culture in the form of suppressed books or manuscripts. It is not unusual to hear that a missing or hidden manuscript by some famous author has been glimpsed by an individual, who in virtue of this privileged access to otherwise inaccessible materials has a special insight into the author in question, or maintains that “everything we think we know about x is false” because the speaker is “in the know” about matters denied to the rest of us.

The conspiratorial dimension of the appeal to embargoed evidence appears when it is stated or implied that an omnipotent government entity, or even a non-governmental entity possessed of uncommon power and influence, is able to suppress all, or almost all, evidence relating to certain knowledge kept secret, whether for the good of the public (which is not ready for the knowledge, on which cf. below) or in order to more effectively act upon some comprehensive social engineering project that would presumably be derailed if only the public knew what was really going on.

A moral dimension is added to the appeal to embargoed evidence when it is stated or implied that evidence has been embargoed because you (the individual asking for evidence) are not worthy of seeing it, or, more comprehensively, that the world at large is not ready for some Earth-shattering secret to be revealed, with the implication being that only the elect are allowed to share in the evidence at present, but the world at large will eventually reach a level of maturity when then evidence can be made public without danger.

The appeal to embargoed evidence gives the appearance of respecting scientific canons of knowledge, because it recognizes that evidence is crucial to knowledge, but it represents a fundamental violation of the scientific method because evidence is invoked rather than produced. Scientific knowledge is in principle reproducible by anyone who has the time and cares to make the effort to confirm experimental results for their own satisfaction. Since embargoed evidence cannot be inspected, tested, or made the object of any scientific experimentation, no putative knowledge or proposed theory solely based on embargoed evidence can be considered scientific.

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Fallacies and Cognitive Biases

An unnamed principle and an unnamed fallacy

The Truncation Principle

An Illustration of the Truncation Principle

The Gangster’s Fallacy

The Prescriptive Fallacy

The fallacy of state-like expectations

The Moral Horror Fallacy

The Finality Fallacy

Fallacies: Past, Present, Future

Confirmation Bias and Evolutionary Psychology

Metaphysical Fallacies

Metaphysical Biases

Pernicious Metaphysics

Metaphysical Fallacies Again

An Inquiry into Cognitive Bias by way of Memoir

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Sunday


Science often (though not always or exclusively) involves a quantitative approach to phenomena. As the phenomena of the world are often (though not always or exclusively) continuous, the continuum of phenomena must be broken up into discrete chunks of experience, however imperfect the division. If we are to quantify knowledge, we must have distinctions, and distinctions must be interpolated at some particular point in a continuum.

The truncation principle is the principled justification of this practice, and the truncation fallacy is the claim that distinctions in the name of quantification are illegitimate. The claim of the illegitimacy of a given distinction is usually based on an ideal standard of distinctions having to be based on a sharply-bounded concept that marks an exhaustive division that admits of no exceptions. This is an unreasonable standard for human experience or its systematization in scientific knowledge.

One of my motivations (though not my only motivation) for formulating the truncation principle was the obvious application to historical periodization. Historians have always been forced to confront the truncation fallacy, though I am not aware that there has previously been any name for the conceptual problems involved in historical periodization, though it has been ever-present in the background of historical thought.

Here is an implicit exposition of the problems of the truncation principle by Marc Bloch, one of the most eminent members of the Annales school of historians (which also included Fernand Braudel, of whom I have written on many occasions), and who was killed by the Gestapo while working for the French resistance during the Second World War:

“…it is difficult to imagine that any of the sciences could treat time as a mere abstraction. Yet, for a great number of those who, for their own purposes, chop it up into arbitrary homogenous segments, time is nothing more than a measurement. In contrast, historical time is a concrete and living reality with an irreversible onward rush… this real time is, in essence, a continuum. It is also perpetual change. The great problems of historical inquiry derive from the antithesis of these two attributes. There is one problem especially, which raises the very raison d’être of our studies. Let us assume two consecutive periods taken out of the uninterrupted sequence of the ages. To what extent does the connection which the flow of time sets between them predominate, or fail to predominate, over the differences born out of the same flow?”

Marc Bloch, The Historian’s Craft, translated by Peter Putnam, New York: Vintage, 1953, Chapter I, sec. 3, “Historical Time,” pp. 27-29

Bloch, then, sees times itself, the structure of time, as the source both of historical continuity and historical discontinuity. For Bloch the historian, time is the truncation principle, as for some metaphysicians space (or time, for that matter) simply is the principle of individuation.

The truncation principle and the principle of individuation are closely related. What makes an individual an individual? When it is cut off from the rest of the world and designated as an individual. I haven’t thought about this yet, so I will reserve further remarks until I’ve made an effort to review the history of the principium individuationis.

The “two attributes” of continuity and change are both functions of time; both the connection and the differences between any two “arbitrary homogenous segments” are due to the action of time, according to Bloch.

The truncation principle, however, has a wider application than time. To express the truncation principle in terms of time invites a formulation (or an example) in terms of space, and there is an excellent example ready to hand: that of the color spectrum of visible light. There is a convention of dividing the color spectrum into red, orange, yellow, green, blue, indigo, and violet. But this is not the only convention. Because the word “indigo” is becoming almost archaic, one now sees the color spectrum decomposed into red, orange, yellow, green, blue, and purple.

Both decompositions of the color spectrum, and any others that might be proposed, constitute something like, “arbitrary homogenous segments.” The decomposition of the color spectrum is justified by the truncation principle, but the principle does not privilege any one decomposition over any other. All distinctions are equal, and if any one distinction is taken to be more equal than others, it is only because this distinction has the sanction of tradition.

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Radical Rigor

18 September 2011

Sunday


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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Sunday


Dated futurism is one of my guilty pleasures, and I have written about this previously in A Hundred Years of Futurism. Recently I’ve been reading a number of mid-twentieth century futurist works for some research I am doing. These are not the wide-eyed adolescent takes on the future, but intended to be sober analyses of what one book calls The Most Probable World. This is a project in the spirit of George Friedman’s The Next 100 Years, which I have discussed several times (cf. Ecological Succession in Cultural Geography).

The wide-eyed enthusiasm for possible futures is pure fun, but the serious attempts to try to understand a likely future constitute futurism of another order, and it deserves to be treated separately, if only because of the intentions of the author. While the science fiction scenarios have sometimes come closer to the truth than some overly-serious attempts to futurism (the latter at times approaching self-parody), this kind of nearly-chance correspondence bears some resemblance to the Gettier paradox, which can be intuitively understood as the fact that a non-functioning clock is precisely correct twice a day, but when a stopped clock is correct in indicating the time, it is not correct for the right reason.

Some of these “serious” (for lack of a better term) works of futurism are more sociological than futurist in character, and can only be called futurist in virtue of their discussion of present trends with a strong implication that the trend under discussion will be a central thread in the developments of the immediate future. In this sense, the sort of sober “futurist” works to which I am here referring needn’t even mention the future or prediction. The future is understood to be embodied in the pregnant present, if only we can recognize the inchoate future in embryo.

I would like to suggest that these works of sober futurism are distinct from works of enthusiasm because they are based on a method, however imperfectly put into practice, and this is the method of the historical a priori imagination. In several previous posts I have had occasion to refer to R. G. Collingwood’s conception of the historical a priori imagination. This is given in the Epilogomena to his The Idea of History, as follows:

“I have already remarked that, in addition to selecting from among his authorities’ statements those which he regards as important, the historian must in two ways go beyond what his authorities tell him. One is the critical way, and this is what Bradley has attempted to analyse. The other is the constructive way. Of this he has said nothing, and to this I now propose to return. I described constructive history as interpolating, between the statements borrowed from our authorities, other statements implied by them. Thus our authorities tell us that on one day Caesar was in Rome and on a later day in Gaul ; they tell us nothing about his journey from one place to the other, but we interpolate this with a perfectly good conscience.”

“This act of interpolation has two significant characteristics. First, it is in no way arbitrary or merely fanciful: it is necessary or, in Kantian language, a priori. If we filled up the narrative of Caesar’s doings with fanciful details such as the names of the persons he met on the way, and what he said to them, the construction would be arbitrary: it would be in fact the kind of construction which is done by an historical novelist. But if our construction involves nothing that is not necessitated by the evidence, it is a legitimate historical construction of a kind without which there can be no history at all.”

“Secondly, what is in this way inferred is essentially something imagined. If we look out over the sea and perceive a ship, and five minutes later look again and perceive it in a different place, we find ourselves obliged to imagine it as having occupied intermediate positions when we were not looking. That is already an example of historical thinking ; and it is not otherwise that we find ourselves obliged to imagine Caesar as having travelled from Rome to Gaul when we are told that he was in these different places at these successive times.”

“This activity, with this double character, I shall call a priori imagination; and, though I shall have more to say of it hereafter, for the present I shall be content to remark that, however unconscious we may be of its operation, it is this activity which, bridging the gaps between what our authorities tell us, gives the historical narrative or description its continuity. That the historian must use his imagination is a commonplace; to quote Macaulay’s Essay on History, ‘a perfect historian must possess an imagination sufficiently powerful to make his narrative affecting and picturesque’; but this is to underestimate the part played by the historical imagination, which is properly not ornamental but structural. Without it the historian would have no narrative to adorn. The imagination, that ‘blind but indispensable faculty’ without which, as Kant has shown, we could never perceive the world around us, is indispensable in the same way to history: it is this which, operating not capriciously as fancy but in its a priori form, does the entire work of historical construction.”

The Idea of History, Epilegomena: 2: The Historical Imagination, R. G. Collingwood, Oxford: Oxford University Press (1946)

This is more than I have quoted from Collingwood previously, because I wanted to give a better sense of his exposition. Collingwood calls his method “constructive” (in contradistinction to being “analytic”), but from a formal point of view it is the antithesis of constructive, it is a non-constructive inference of what must be, made on the basis of what is known to be the case.

But I think that Collingwood wanted to call his method “constructive” because he wanted to bring attention to the essentially conservative and traditional aspect of historical thought that he felt himself to be describing. It is one of the remarkable aspects of Collingwood’s conception that it is both metaphysically bold and methodologically conservative. As Collingwood notes, we have no scruples in deducing that when Caesar traveled from Rome to Gaul that he covered the intervening geographical region. This is, in a sense, a necessary truth, and in so far as it is a necessary truth, it is an a priori truth — furnished by imagination.

In works of history, we can make logical deductions as to what must have happened on the basis of connecting two points in history separated by the discrete period of time. In works of futurism, we cannot do this. We have only one point at which the facts are know, and this is the present. And often the present is known far more imperfectly than we would like to admit. As time passes, and we learn more and more about the past, we realize how little we knew of the present when it was in fact present.

Thus futurism labors under a double burden of knowing only half of what is needed to logically extrapolate the historical a priori imaginative narrative, as well as knowing this half highly imperfectly. Despite these substantial handicaps, we can still stand on the firm ground of methodological naturalism in making necessary deductions about the future.

We know that the future must follow from the present as the present has followed from the past. We know furthermore that there will be some future, and that it will be filled with some content, even if we don’t know what that content is. This makes futurism profoundly non-constructive.

Beyond these logical deductions from the very structure of time itself, we know empirically and inductively that things never quite develop as we expect things to develop, meaning that trends that seem to be important in the present often come to nothing, while world-historical events often seem to emerge suddenly if not violently from subtle trends in the present that are often evident only in hindsight.

A better appreciation of non-constructivism as a method of formal reasoning, as well as of subtle trends in the present that are neglected in favor of more obvious trends, would give us a better picture of the content of history that will shape the future. Both of these are highly difficult intellectual undertakings. Despite the fact (which you will know if you are familiar with the literature of formal reasoning) that constructivism is considered a marginal if not ideological mode of thought, I find it remarkable that constructivism has been given several systematic expositions, for example, in the work of Brouwer, Heyting, Dummett, and Beeson, among many others, while non-constructivism, the default form of formal reasoning that makes no special stipulations, has been given no explicit formulation. This is an ellipsis that not only is felt in formal thought, but as we can see here is also felt in historical thought.

As for the empirical and inductive dimension of futurism, a thorough and dispassionate survey of the present, undertaken in a frame of mind informed by parallels with past neglected trends, might reveal a number of threads of historical trends in the present which might hold the key to unexpected developments in the future.

While futurism remains marginal, it is not beyond hope in being given a firmer intellectual basis than it has enjoyed to date. What I have suggested above may be taken as a research program for putting futurism on a more solid footing.

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Tuesday


Aristotle as portrayed by Raphael

Aristotle claimed that mathematics has no ethos (Metaphysics, Book III, Chap. 2, 996a). Aristotle, of course, was more interested in the empirical sciences than his master Plato, whose Academy presumed and demanded familiarity with geometry — and we must understand that for the ancients, long before the emergence of analytical geometry in the work of Descartes (allowing us to formulate geometry algebraically, hence arithmetically), that geometry was always axiomatic thought, rigorously conceived in terms of demonstration. For the Greeks, this was the model and exemplar of all rigorous thought, and for Aristotle this was a mode of thought that lacked an ethos.

Euclid provided the model of formal thought with his axiomatization of geometry. Legend has it that there was a sign over the door of Plato's Academy stating, 'Let no one enter here who has not studied geometry.'

In this, I think, Aristotle was wrong, and I think that Plato would have agree on this point. But the intuition behind Aristotle’s denial of a mathematical ethos is, I think, a common one. And indeed it has even become a rhetorical trope to appeal to rigorous mathematics as an objective standard free from axiological accretions.

In his famous story within a story about the Grand Inquisitor, Dostoyevsky has the Grand Inquisitor explain how “miracles, mystery, and authority” are used to addle the wits of others.

In his famous story within a story about the Grand Inquisitor, Dostoyevsky has the Grand Inquisitor explain how “miracles, mystery, and authority” are used to addle the wits of others.

Our human, all-too-human faculties conspire to confuse us, to addle our wits, when we begin talking about morality, so that the purity and rigor of mathematical and logical thought seem to be called into question if we acknowledge that there is an ethos of formal thought. We easily confuse ourselves with religious, mystical, and ethical ideas, and since the great monument of mathematical thought has been mostly free of this particular species of confusion, to deny an ethos of formal thought can be understood as a strategy to protect and defend of the honor of mathematics and logic by preserving it from the morass that envelops most human attempts to think clearly, however heroically undertaken.

Kant famously said that he had to limit knowledge to make room for faith.

Kant famously stated in the Critique of Pure Reason that, “I have found it necessary to deny knowledge in order to make room for faith.” I should rather limit faith to make room for rigorous reasoning. Indeed, I would squeeze out faith altogether, and find myself among the most rigorous of the intuitionists, one of whom has said: “The aim of this program is to banish faith from the foundations of mathematics, faith being defined as any violation of the law of sufficient reason (for sentences). This law is defined as the identification (by definition) of truth with the result of a (present or feasible) proof…”

Western asceticism can be portrayed as demonic torment or as divine illumination; the same diversity of interpretation can be given to ascetic forms of reason.

Though here again, with intuitionism (and various species of constructivism generally), we have rigor, denial, asceticism — intuitionistic logic is no joyful wisdom. (An ethos of formal thought need not be an inspiring and edifying ethos.) It is logic with a frown, disapproving, censorious — a bitter medicine justified only because it offers hope of curing the disease of contradiction, contracted when mathematics was shown to be reducible to set theory, and the latter shown to be infected with paradox (as if the infinite hubris of set theory were not alone enough for its condemnation). Is the intuitionist’s hope justified? In so far as it is hope — i.e., hope and not proof, the expectation that things will go better for the intuitionistic program than for logicism — it is not justified.

Dummett has said that intuitionistic logic and mathematics are to wear their justification on their face:

“From an intuitionistic standpoint, mathematics, when correctly carried on, would not need any justification from without, a buttress from the side or a foundation from below: it would wear its own justification on its face.”

Dummett, Michael, Elements of Intuitionism, Oxford University Press, 1977, p. 2

The hope that contradiction will not arise from intuitionistic methods clearly is no such evident justification. As a matter of fact, empirically and historically verifiable, we know that intuitionism has resulted in no contradictions, but this could change tomorrow. Intuitionism stands in need of a consistency proof even more than formalism. There is, in its approach, a faith invested in the assumption that infinite totalities caused the paradoxes, and once we have disallowed reference to them all will go well. This is a perfectly reasonable assumption, but one, in so far as it is an article of faith, which is at variance with the aims and methods of intuitionism.

And what is a feasible proof, which our ultra-intuitionist would allow? Have we not with “feasible proof” abandoned proof altogether in favor of probability? Again, we will allow them their inconsistencies and meet them on their own ground. But we shall note that the critics of the logicist paradigm fix their gaze only upon consistency, and in so doing reveal again their stingy, miserly conception of the whole enterprise.

“The Ultra-Intuitionistic Criticism and the Antitraditional program for the foundations of Mathematics” by A. S. Yessenin-Volpin (who was arguing for intellectual freedom in the Soviet Union at the same time that he was arguing for a censorious conception of reason), in Intuitionism and Proof Theory, quoted briefly above, is worth quoting more fully:

The aim of this program is to banish faith from the foundations of mathematics, faith being defined as any violation of the law of sufficient reason (for sentences). This law is defined as the identification (by definition) of truth with the result of a (present or feasible) proof, in spite of the traditional incompleteness theorem, which deals only with a very narrow kinds [sic] of proofs (which I call ‘formal proofs’). I define proof as any fair way of making a sentence incontestable. Of course this explication is related to ethics — the notion fair means ‘free from any coercion or fraud’ — and to the theory of disputes, indicating the cases in which a sentence is to be considered as incontestable. Of course the methods of traditional mathematical logic are not sufficient for this program: and I have to enlarge the domain of means explicitly studied in logic. I shall work in a domain wherein are to be found only special notions of proof satisfying the mentioned explication. In this domain I shall allow as a means of proof only the strict following of definitions and other rules or principles of using signs.

Intuitionism and proof theory: Proceedings of the summer conference at Buffalo, N.Y., 1968, p. 3

What is coercion or fraud in argumentation? We find something of an illustration of this in Gregory Vlastos’ portrait of Socrates: “Plato’s Socrates is not persuasive at all. He wins every argument, but never manages to win over an opponent. He has to fight every inch of the way for any assent he gets, and gets it, so to speak, at the point of a dagger.” (The Philosophy of Socrates, Ed. by Gregory Vlastos, page 2)

According to Gregory Vlastos, Socrates used the kind of 'coercive' argumentation that the intuitionists abhor.

What appeal to logic does not invoke logical compulsion? Is logical compulsion unique to non-constructive mathematical thought? Is there not an element of logical compulsion present also in constructivism? Might it not indeed be the more coercive form of compulsion that is recognized alike by constructivists and non-constructivists?

The breadth of the conception outlined by Yessenin-Volpin is impressive, but the essay goes on to stipulate the harshest measures of finitude and constructivism. One can imagine these Goldwaterite logicians proclaiming: “Extremism in the defense of intuition is no vice, and moderation in the pursuit of constructivist rigor is no virtue.” Brouwer, the spiritual father of intuitionism, even appeals to the Law-and-Order mentality, saying that a criminal who has not been caught is still a criminal. Logic and mathematics, it seems, must be brought into line. They verge on criminality, deviancy, perversion.

Quine was no intuitionist by a long shot, but as a logician he brought a quasi-disciplinary attitude to reason and adopted a tone of disapproval not unlike Brouwer.

The same righteous, narrow, anathamatizing attitude is at work among the defenders of what is sometimes called the “first-order thesis” in logic. Quine sees a similar deviancy in modal logic (which can be shown to be equivalent to intuitionistic logic), which he says was “conceived in sin” — the sin of confusing use and mention. These accusations do little to help us understand logic. We would do well to adopt Foucault’s attitude on these matters: “leave it to our bureaucrats and our police to see that our papers are in order. At least spare us their morality when we write.” (The Archaeology of Knowledge, p. 17)

Foucault had little patience for the kind of philosophical reason that seemed to be asking if our papers are in order, a function he thought best left to the police.

The philosophical legacy of intuitionism has been profound yet mixed; its influence has been deeply ambiguous. (Far from the intuitive certainty, immediacy, clarity, and evident justification that it would like to propagate.) There is in inuitionism much in harmony with contemporary philosophy of mathematics and its emphasis on practices, the demand for finite constructivity, its anti-philosophical tenor, its opposition to platonism. The Father of Intuitionism, Brouwer, was, like many philosophers, anti-philosophical even while propounding a philosophy. No doubt his quasi-Kantianism put his conscience at rest in the Kantian tradition of decrying metaphysics while practicing it, and his mysticism gave a comforting halo (which softens and obscures the hard edges of intuitionist rigor in proof theory) to mathematics which some have found in the excesses of platonism.

L. E. J. Brouwer: philosopher of mathematics, mystic, and pessimistic social theorist

In any case, few followers of Brouwer followed him in his Kantianism and mysticism. The constructivist tradition which grew from intuitionism has proved to be philosophically rich, begetting a variety of constructive techniques and as many justifications for them. Even if few mathematicians actually do intuitionistic mathematics, controversies over the significance of constructivism have a great deal of currency in philosophy. And Dummett is explicit about the place of philosophy in intuitionistic logic and mathematics.

The light of reason serves as an inspiration to us as it shines down from above, and it remains an inspiration even when we are not equal to all that it might ideally demand of us.

Intuitionism and constructivism command our respect in the same way that Euclidean geometry commanded the respect of the ancients: we might not demand that all reasoning conform to this model, but it is valuable to know that rigorous standards can be formulated, as an ideal to which we might aspire if nothing else. And and ideal of reason is itself an ethos of reason, a norm to which formal thought aspires, and which it hopes to approximate even if it cannot always live up the the most exacting standard that it can recognize for itself.

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Studies in Formalism

1. The Ethos of Formal Thought

2. Epistemic Hubris

3. Parsimonious Formulations

4. Foucault’s Formalism

5. Cartesian Formalism

6. Doing Justice to Our Intuitions: A 10 Step Method

7. The Church-Turing Thesis and the Asymmetry of Intuition

8. Unpacking an Einstein Aphorism

9. Methodological and Ontological Parsimony (in preparation)

10. The Spirit of Formalism (in preparation)

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Grand Strategy Annex

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Saturday


The arch-atheist Jean-Paul Sartre

Despite having posted on this twice recently in A Note on Sartre’s Atheism and More on Sartre’s Atheism, I haven’t yet finished with this (as though one could ever be finished with an idea!).

I have, in a couple of posts, quoted a line from Sartre’s “Existentialism is a Humanism” lecture that ends with I must confine myself to what I can see:

I do not know where the Russian revolution will lead. I can admire it and take it as an example in so far as it is evident, today, that the proletariat plays a part in Russia which it has attained in no other nation. But I cannot affirm that this will necessarily lead to the triumph of the proletariat: I must confine myself to what I can see.

For corroboration from a fellow Frenchman and a fellow novelist consider this from Balzac’s Louis Lambert (not his most admired novel, but perhaps his most philosophical novel), delivered by the novel’s protagonist:

“To think is to see,” he said one day, roused by one of our discussions on the principle of human organization. “All science rests on deduction, — a chink of vision by which we descend from cause to effect returning upward from effect to cause; or, in a broader sense, poetry, like every work of art, springs from a swift perception of things.”

Honoré de Balzac, Louis Lambert, translated by Katharine Prescott Wormeley, Boston: Roberts Brothers, 1889, p. 39

Fellow Frenchman and philosopher Descartes offers more than corroboration: he stands at the foundation of the tradition from which both Balzac and Sartre come. In his most systematic work, the Principles of Philosophy (Book I, ix), Descartes presents an all-encompassing conception of thought, as is appropriate for the philosopher who is the locus classicus of the cogito:

By the word thought, I understand all that which so takes place in us that we of ourselves are immediately conscious of it; and, accordingly, not only to understand (INTELLIGERE, ENTENDRE), to will (VELLE), to imagine (IMAGINARI), but even to perceive (SENTIRE, SENTIR), are here the same as to think (COGITARE, PENSER). For if I say, I see, or, I walk, therefore I am; and if I understand by vision or walking the act of my eyes or of my limbs, which is the work of the body, the conclusion is not absolutely certain, because, as is often the case in dreams, I may think that I see or walk, although I do not open my eyes or move from my place, and even, perhaps, although I have no body: but, if I mean the sensation itself, or consciousness of seeing or walking, the knowledge is manifestly certain, because it is then referred to the mind, which alone perceives or is conscious that it sees or walks.

On the one hand, one can view these accounts as tributes to the visible and the tangible, except that Descartes, who stands at the origin of the tradition, can in no way be assimilated to materialism. On the other hand, and more interestingly, all of these accounts can be understood as expressions of various degrees of constructivism — mostly unconsciously formulated constructivism, but nevertheless an awareness that our thought must be disciplined by experience in a rigorous way if it is not to go terribly wrong. This is also a Kantian orientation, as we observed in Temporal Illusions, and Kant is counted as an ancestor of contemporary constructivism.

Skeptics have always demanded that truths be exhibited. We saw this in our previous posts about Sartre’s atheism, taking Doubting Thomas as the paradigm of the skeptic, who must needs touch the wounds of Christ with his own hands before he will believe that it is the same Christ who was crucified and subsequently risen.

It is a feature of constructivist thought, and most especially intuitionism, to reject the law of logic that is called (in Latin) tertium non datur or the Law of the Excluded Middle (LEM, or just EM). This simply states that, of two contradictory propositions, one of them most be true (“P or not-P“). Intuitively, it seems eminently reasonable, except that we all know of instances in ordinary experience that cannot be adequately described in a black-or-white, yes-or-no formulation. Non-constructive reasoning makes unlimited use of the law of the excluded middle, and as a consequence holds that all propositions have definite truth values even if we haven’t yet determined the truth value or even if we can’t determine the truth value. This can lead to strange consequences, like the famous Aristotelian example of the sea fight tomorrow: either there will be a sea battle tomorrow or there will not be a sea battle tomorrow. We don’t know at present which is true, but if we accept the logic of non-constructive reasoning, we will acknowledge that one of these propositions is true while the other is false.

The law of the excluded middle implies the principle of bivalence — the principle that there are two and only two logical values, namely true and false — and bivalence in turn implies realism. Realism as a philosophical doctrine stands in opposition to constructivism. Plato is the most famous realist philosopher, and believed that all kinds of things were real that common sense and ordinary experience don’t think of as being “real,” while at the same time disbelieving in the reality of the material world. Thus Plato is something of an antithesis to the kind insistence upon the tangibility and visibility upon which the skeptic and the materialist rely.

It is interesting, then, in the context of Sartre’s atheism and his insistence upon relying upon the seen, which we have now come to recognize as a kind of constructivism, to contrast the very different viewpoint represented by William James. One of James’ most famous essays is “The Will to Believe” in which he lays down the criteria for legitimate belief even where sufficient evidence is lacking. William James offers, “a defence of our right to adopt a believing attitude in religious matters, in spite of the fact that our merely logical intellect may not have been coerced.” Among the criteria that James invokes is when a choice is forced, which he describes like this:

…if I say to you: “Choose between going out with your umbrella or without it,” I do not offer you a genuine option, for it is not forced. You can easily avoid it by not going out at all. Similarly, if I say, “Either love me or hate me,” “Either call my theory true or call it false,” your option is avoidable. You may remain indifferent to me, neither loving nor hating, and you may decline to offer any judgment as to my theory. But if I say, “Either accept this truth or go without it,” I put on you a forced option, for there is no standing place outside of the alternative. Every dilemma based on a complete logical disjunction, with no possibility of not choosing, is an option of this forced kind.

Logical disjunction is another name used for the law of the excluded middle. Here James reveals himself as a realist, if not a Platonist, in matters of the spirit, just as we saw that Sartre revealed himself as a constructivist, if not an intuitionist, in matters of the spirit. The point I am making here is that this is not merely a difference of belief, but a difference in logic, and a difference in logic and reaches up into the ontology of each and informs an entire view of the world. People tend to think of logic, if they think of logic at all, as something recondite and removed from ordinary human experience, but this is not the case. Logic determines the relationship that we construct with the world, and it organizes how we see the world.

Nietzsche wrote in a famous line (or, perhaps I should say, a line that ought to be more famous than it perhaps is) that the nature and degree of an individual’s sexuality reaches into the highest pinnacles of his spirit. I agree with this, but I would add that the nature and kind of an individual’s logic — be it constructivist or non-constructivist — also reaches into the highest pinnacles of his spirit and indeed informs the world in which his spirit finds a home… or fails to find a home.

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Tuesday


Luigi Nono

For the six months or so that I have been posting to this forum I have been quite preoccupied with intensely practical questions in history, economics, politics, diplomacy and how these spheres of activity are related in a substantive way to human nature.

When Heilbronner wrote his famous book about economists he called them the “the worldly philosophers,” which invites an implicit comparison to unworldly or otherworldly philosophers who do not concern themselves with the ordinary business of life. Be that as it may, it is possible, I think, to be both — thinking both the worldly and the unworldly by turns. Thus is the thinker likely to experience a dialectic not only within the realms of thought, but also as part of the ordinary business of his life.

I find myself today thrown back onto the most abstruse and obscure points of technical philosophy in my attempt to clarify my understanding of very worldly concepts that attempt to elucidate what Marshall called “the ordinary business of life.” I find that I am once again taking down my reference works on ontology, epistemology, and philosophy of logic from my bookshelves, and this, I think, is a good thing. The cross-fertilization of thought, whether inter-disciplinary or intra-disciplinary, is usually a source of fruitful meditation. In particular, I find myself working on the idea of constructivism.

Constructivism means many different things to many different persons. It would almost seem that a sense of “constructivism” has been defined for every conceivable special field of inquiry or endeavor. There is constructivism in the visual arts, and a constructivism in music, and a constructivism in sociology, and, what most concerns me, a constructivism in the philosophy of logic and mathematics.

Dr. David C. F. Wright quoted his friend British composer Reginald Smith Brindle regarding a visit to Luigi Nono:

I went there mostly while he was composing Il Canto Sospeso, a politically orientated work of choral-orchestral character which involved the most abstruse constructivism I have ever come across. Mathematics governed every detail of the composition … the pitch of the notes, their duration, volume and sound character. In his study, there was a wall entirely covered with successions of numbers, notes and performance details and from this he extracted all the details of the composition. It seemed to me that all his intense constructivism was a certain formula for the creation of non-music, yet from recordings of his music, I got the impression of a highly sensitive artistry.

What Brindle describes is more commonly known as integral serialism or total serialism. The relation between constructivism and serialism is an interesting question in itself, but one that I will not address here. And while I don’t have a CD of Il Canto Sospeso, I did have a recording by the Arditti Quartet of Nono’s fragmente – stille, an idiotma, so I put this on as my theme music for constructivism.

I regard the philosophy of mathematics as the ultimate proving ground for all philosophical theories. One finds philosophical theories applied to the philosophy of mathematics in their purest form, and it is in their purest form that theories are seen in their nakedness, revealed to all the world for what they are. This is especially true for constructivism, but while constructivism is best tested by the austere ontology of logic and mathematics, it has universal implications.

Constructivism is a methodological concept, and the distinction between constructive methods and non-constructive methods recapitulates the ancient division between idealism and realism in ontology. One could say that constructivism is idealism put into practice as a method. What, then, is the method of idealism?

At present I am only trying to get clear about the concept of constructivism, its proper scope as a concept. I sent off an e-mail to the phil-logic discussion listserv and got some replies both on-list and off-list that provided some initial stimulation. It is, however, extraordinarily difficult to develop a sympathetic discussion on an e-mail listserv. Even when others are the list are interested in the idea, the tone of discussion can be brutal at times. There is a value in brutal honesty and openness of discussion, but there is also a value in having someone with whole one can share inchoate ideas and help to bring out what is valuable in them without destroying a fragile thought. However, I have no one to act as my intellectual second (i.e., kaishakunin, 介錯人) and thus I pour it out here instead.

It takes a true friend to perform the office of kaishakunin.

It takes a true friend to perform the office of kaishakunin.

I found an interesting discussion of constructivism in Detlefsen’s contribution to the Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, in which Detlefsen does try to formulate the theoretical unity of constructivism, although he never touches on predicativism. Should predicativism be considered something utterly different? There is also a great discussion of constructivism by Michael Hallett in the Handbook of Metaphysics and Ontology edited by Hans Burkhardt and Barry Smith, published by Philosophia Verlag (Hallett’s article is “mathematical objects”). While these two discussions are a great starting point, they don’t get to the essence of the question that is troubling me at the moment.

detlefsen

The many varieties of constructivism are different not only in detail but also importantly different in conceptual scope. Intuitionism, finitism, predicativism, and other conceptions that might generally be called constructivistic in tendency all restrict classical formal reasoning, but there does not seem to be any prima facie unity in virtue of which all deserve to be called constructivist. One of my off-list responses from the phil-logic listserv suggested that there would be “push back” at any attempt to classify intuitionism as a form of constructivism.

handbook of metaphysics and ontology

James Robert Brown’s The Philosophy of Mathematics: An Introduction to the World of Proofs and Pictures has a section on “Constructivist Approaches” that quotes from Errett Bishop’s Foundations of Constructive Analysis. I don’t have a copy of Bishop’s book, so this is helpful. Bishop, at least, explicitly identifies his approach as constructivist, unlike Brouwer or Heyting, Poincare or Weyl, Yesnin-Volpin and Gauthier, Kielkopf and Wittgenstein. This self-ascribed constructivist identity carries more weight than all the other uses of “constructivist” combined.

James Robert Brown

Perhaps constructivism in its pure form should be defined more narrowly, strictly in terms of the avoidance of pure existence proofs, for example. But if we define constructivism more narrowly, then it would seem that there is still a need for a concept under which would fall all those theories of formal reason that restrict what Torkel Franzen called “classical eclectism,” and which would include a narrowly defined constructivism as well as other doctrines previously called constructivist. What concept could we use to cover all instances of principled restrictions upon formal reasoning, and is there any unity of motive in formulating and propounding principled limitations on formal reasoning?

The obvious course of action would be to elucidate the principles embodied in all such doctrines, loosely called “constructivist” up until now, and seek to systematically interrelate them. In every police drama one sees on television, the detectives on a difficult case assemble a large bulletin board upon which they display symbols for clues, and then map the interrelations between clues in an attempt to find a pattern that will solve the case. We need the conceptual equivalent of this in order to understand constructivism.

Two other obvious courses of action present themselves: simultaneously driving down into the foundations of constructivist doctrines while also extrapolating their consequences to the utmost limit. A convergence or divergence of either development would point to fundamental commonality or fundamental incommensurability.

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A Non-Constructive World

19 April 2009


An imaginary illustration of Protagoras teaching.

An imaginary illustration of Protagoras teaching.

Further Ontological Ruminations

In yesterday’s Ontological Ruminations: Six Protagorean Propositions on the Nature of Man and the World I laid down several ontological principles of a Quasi-Protagorean bent. Protagoras (ca. 490– 420 BC; Greek: Πρωταγόρας), you will recall, was one of the greatest of the Presocratics, and was famous for having said, “Man is the measure of all things: of things which are, that they are, and of things which are not, that they are not.” This is one of those rare philosophical quotations that is sufficiently famous to have survived more than two thousand years and is recognized even by those with no interest in philosophy. So when I noted yesterday that the presence of man in the world as an agent constituting the world, it was, in essence, a Protagorean observation.

Beyond the immediate Protagorean interest, my six propositions of yesterday also suggested the non-constructive character of the world, but this requires explanation to make any sense whatsoever. When I speak of constructive and non-constructive in this forum, I mean the terms in their logical, and not their social, signification. What is the logical significance of constructive and non-constructive? That is difficult to easily sum up.

Constructivism in logic, mathematics, and formal thought generally is an embarras de richesse: there are a remarkable number of distinct formulations of constructivism as well as degrees of constructivism. I was just skimming several philosophical reference works for a simple and comprehensive definition of constructivity that would cover its various manifestations, and I couldn’t find anything satisfying.

There are logical approaches to constructivism, some of which involve logic without the Law of the Excluded Middle and others of which forbid the use of “existence” axioms that posit an entity without giving a method for constructing the entity, and there are finitist approaches to constructivism that deny or limit infinitistic propositions or methods, or which confine legitimate thought to finite assertions, and there are again predicative forms of constructivism that proscribe the use of impredicative definitions and methods.

Hopefully from the above (which is admittedly rather compressed and inexact) it should at least emerge that constructivists generally place limits on formal (or ontological) thought that would not otherwise be observed.

Constructive thought is pervasively influential today for a variety of reasons, ranging from essentially constructive nature of computer science, which makes itself felt in our lives in countless ways today because of the role of computers in our lives, to the increasingly constructivistic character of the sciences.

Physics has been turned into a constructivist undertaking without much notice of this profound change in perspective, yet it retains patently idealistic strains within the generally constructive drift—especially the presumption of the rationality of the world, i.e., its amenability to rational explanation, and mathematization of physics and its consequent idealizations and simplifications. Take, for example, the claim of the impossibility of travel at the velocity of light — if a philosopher deduced properties of the world from mathematical equations he would be a laughing stock, but physicists do so with impunity.

Physicists have taken the mantle of speculation from philosophers; science today is much more speculative than philosophy ever was, and the careful pedanticism of contemporary philosophers looks like a parody of scientific method intended to elicit laughter.

Moreover, the world itself seems constructive. Indeed, the constructivity of the world on a quantum scale is dramatically demonstrated by the failure of the law of the excluded middle and bivalence for quantum states: the logic of quantum theory is a logic without tertium non datur.

We see the extent to which the world is constructive when we contemplate the gradual, piecemeal way in which any actuality would need to approach any infinity. Just as we cannot reach aleph null by adding one repeatedly to any arbitrarily large number, so we cannot attain infinite mass by adding increments of mass to any arbitrarily large mass, nor can we shrink any arbitrarily small but finite quantity to nothing by gradually reducing it in size by a finite number of steps.

In mathematics, these limits have not the same function that they have in physics, because we can conduct thought experiments in which time and temporal processes have no place. But all that it subject to the laws of physics is also subject to the laws of time, and time will not allow us more than a constructive infinity of successively adding discrete quantities a finite number of times. This process can only yield an infinite result after the passage of an infinite quantity of time.

For all that, the world is still non-constructive, and even incorrigibly non-constructive.

The particular non-constructive aspect of the world that featured in yesterday’s exposition was the impredicativity of the world. An impredicative definition defines a given entity in terms of a whole of which it is a part. Impredicative reasoning makes use of impredicative definitions, and such are not terribly unusual. Any definition of an individual man that refers to all men is an impredicative definition, since the class of all men includes the individual so defined. And, more to the point in the present context, the world constructed by an agent who is part of that world is a non-constructive conception.

Not only is the world non-constructive and impredicative, but it is also indefinable in the traditional Aristotelian sense. In Aristotle, a term is defined by citing its genus and differentia. But the world has neither genus or differentia, and therefore cannot be defined. The world is a totality the eludes capture in formal thought. Or, as I put it in my Variations on the Theme of Life: “The world” is a metaphor for a concept that cannot be made literal.

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