20 January 2014
Studies in Formalism:
The Synoptic Perspective in Formal Thought
In my previous two posts on the overview effect — The Epistemic Overview Effect and The Overview Effect as Perspective Taking — I discussed how we can take insights gained from the “overview effect” — what astronauts and cosmonauts have experienced as a result of seeing our planet whole — and apply them to over areas of human experience and knowledge. Here I would like to try to apply these insights to formal thought.
The overview effect is, above all, a visceral experience, something that the individual feels as much as they experience, and you may wonder how I could possibly find a connection between a visceral experience and formal thinking. Part of the problem here is simply the impression that formal thought is distant from human concerns, that it is cold, impersonal, unfeeling, and, in a sense, inhuman. Yet for logicians and mathematicians (and now, increasingly, also for computer scientists) formal thought is a passionate, living, and intimate engagement with the world. Truly enough, this is not an engagement with the concrete artifacts of the world, which are all essentially accidents due to historical contingency, but rather an engagement with the principles implicit in all things. Aristotle, ironically, formalized the idea of formal thought being bereft of human feeling when he asserted that mathematics has no ethos. I don’t agree, and I have discussed this Aristotelian perspective in The Ethos of Formal Thought.
And yet. Although Aristotle, as the father of logic, had more to do with the origins of formal thought than any other human being who has ever lived, the Aristotelian denial of an ethos to formal thought does not do justice to our intuitive and even visceral engagement with formal ideas. To get a sense of this visceral and intuitive engagement with the formal, let us consider G. H. Hardy.
Late in his career, the great mathematician G. H. Hardy struggled to characterize what he called mathematically significant ideas, which is to say, what makes an idea significant in formal thought. Hardy insisted that “real” mathematics, which he distinguished from “trivial” mathematics, and which presumably engages with mathematically significant ideas, involves:
“…a very high degree of unexpectedness, combined with inevitability and economy.”
G. H. Hardy, A Mathematician’s Apology, section 15
Hardy’s appeal to parsimony is unsurprising, yet the striking contrast of the unexpected and the inevitable is almost paradoxical. One is not surprised to hear an exposition of mathematics in deterministic terms, which is what inevitability is, but if mathematics is the working out of rigid formal rules of procedure (i.e., a mechanistic procedure), how could any part of it be unexpected? And yet it is. Moreover, as Hardy suggested, “deep” mathematical ideas (which we will explore below) are unexpected even when they appear inevitable and economical.
It would not be going too far to suggest that Hardy was trying his best to characterize mathematical beauty, or elegance, which is something that is uppermost in the mind of the pure mathematician. Well, uppermost at least in the minds of some pure mathematicians; Gödel, who was as pure a formal thinker as ever lived, said that “…after all, what interests the mathematician, in addition to drawing consequences from these assumptions, is what can be carried out” (Collected Works Volume III, Unpublished essays and lectures, Oxford, 1995, p. 377), which is an essentially pragmatic point of view, in which formal elegance would seem to play little part. Mathematical elegance has never been given a satisfactory formulation, and it is an irony of intellectual history that the most formal of disciplines relies crucially on an informal intuition of formal elegance. Beauty, it is often said, in the mind of the beholder. Is this true also for mathematical beauty? Yes and no.
If a mathematically significant idea is inevitable, we should be able to anticipate it; if unexpected, it ought to elude all inevitability, since the inevitable ought to be predictable. One way to try to capture the ineffable sense of mathematical elegance is through paradox — here, the paradox of the inevitable and the unexpected — in way not unlike the attempt to seek enlightenment through the contemplation of Zen koans. But Hardy was no mystic, so he persisted in his attempted explication of mathematically significant ideas in terms of discursive thought:
“There are two things at any rate which seem essential, a certain generality and a certain depth; but neither quality is easy to define at all precisely.
G. H. Hardy, A Mathematician’s Apology, section 15
Although Hardy repeatedly expressed his dissatisfaction with his formulations of generality and depth, he nevertheless persisted in his attempts to clarify them. Of generality Hardy wrote:
“The idea should be one which is a constituent in many mathematical constructs, which is used in the proof of theorems of many different kinds. The theorem should be one which, even if stated originally (like Pythagoras’s theorem) in a quite special form, is capable of considerable extension and is typical of a whole class of theorems of its kind. The relations revealed by the proof should be such as to connect many different mathematical ideas.” (section 15)
And of mathematical depth Hardy hazarded:
“It seems that mathematical ideas are arranged somehow in strata, the ideas in each stratum being linked by a complex of relations both among themselves and with those above and below. The lower the stratum, the deeper (and in general more difficult) the idea.” (section 17)
This would account for the special difficulty of foundational ideas, of which the most renown example would be the idea of sets, though there are other candidates to be found in other foundational efforts, as in category theory or reverse mathematics.
Hardy’s metaphor of mathematical depth suggests foundations, or a foundational approach to mathematical ideas (an approach which reached its zenith in the early twentieth century in the tripartite struggle over the foundations of mathematics, but is a tradition which has since fallen into disfavor). Depth, however, suggests the antithesis of a synoptic overview, although both the foundational perspective and the overview perspective seek overarching unification, one from the bottom up, the other from the top down. These perspectives — bottom up and top down — are significant, as I have used these motifs elsewhere as an intuitive shorthand for constructive and non-constructive perspectives respectively.
Few mathematicians in Hardy’s time had a principled commitment to constructive methods, and most employed non-constructive methods will little hesitation. Intuitionism was only then getting its start, and the full flowering of constructivistic schools of thought would come later. It could be argued that there is a “constructive” sense to Zermelo’s axiomatization of set theory, but this is of the variety that Godel called “strictly nominalistic construtivism.” Here is Godel’s attempt to draw a distinction between nominalistic constructivism and the sense of constructivism that has since overtaken the nominalistic conception:
…the term “constructivistic” in this paper is used for a strictly nominalistic kind of constructivism, such that that embodied in Russell’s “no class theory.” Its meaning, therefore, if very different from that used in current discussions on the foundations of mathematics, i.e., from both “intuitionistically admissible” and “constructive” in the sense of the Hilbert School. Both these schools base their constructions on a mathematical intuition whose avoidance is exactly one of the principle aims of Russell’s constructivism… What, in Russell’s own opinion, can be obtained by his constructivism (which might better be called fictionalism) is the system of finite orders of the ramified hierarchy without the axiom of infinity for individuals…”
Kurt Gödel, Kurt Gödel: Collected Works: Volume II: Publications 1938-1974, Oxford et al.: Oxford University Press, 1990, “Russell’s Mathematical Logic (1944),” footnote, Author’s addition of 1964, expanded in 1972, p. 119
This profound ambiguity in the meaning of “constructivism” is a conceptual opportunity — there is more that lurks in this idea of formal construction than is apparent prima facie. That what Gödel calls a, “strictly nominalistic kind of constructivism” coincides with what we would today call non-constructive thought demonstrates the very different conceptions of what is has meant to mathematicians (and other formal thinkers) to “construct” an object.
Kant, who is often called a proto-constructivist (though I have identified non-constructive elements on Kant’s thought in Kantian Non-Constructivism), does not invoke construction when he discusses formal entities, but instead formulates his thoughts in terms of exhibition. I think that this is an important difference (indeed, I have a long unfinished manuscript devoted to this). What Kant called “exhibition” later philosophers of mathematics came to call “surveyability” (“Übersichtlichkeit“). This latter term is especially due to Wittgenstein; Wittgenstein also uses “perspicuous” (“Übersehbar“). Notice in both of the terms Wittgenstein employs for surveyability — Übersichtlichkeit and Übersehbar — we have “Über,” usually (or often, at least) translated as “over.” Sometimes “Über” is translated as “super” as when Nietzsche’s “Übermensch“ is translated as “superman” (although the term has also been translated as “over-man,” inter alia).
There is a difference between Kantian exhibition and Wittgensteinian surveyability — I don’t mean to conflate the two, or to suggest that Wittgenstein was simply following Kant, which he was not — but for the moment I want to focus on what they have in common, and what they have in common is the attempt to see matters whole, i.e., to take in the object of one’s thought in a single glance. In the actual practice of seeing matters whole it is a bit more complicated, especially since in English we commonly use “see” to mean “understand,” and there are a whole range of visual metaphors for understanding.
The range of possible meanings of “seeing” accounts for a great many of the different formulations of constructivism, which may distinguish between what is actually constructable in fact, that which it is feasible to construct (this use of “feasible” reminds me a bit of “not too large” in set theories based on the “limitation of size” principle, which is a purely conventional limitation), and that which can be constructed in theory, even if not constructable in fact, or if not feasible to construct. What is “surveyable” depends on our conception of what we can see — what might be called the modalities of seeing, or the modalities of surveyability.
There is an interesting paper on surveyability by Edwin Coleman, “The surveyability of long proofs,” (available in Foundations of Science, 14, 1-2, 2009) which I recommend to the reader. I’m not going to discuss the central themes of Coleman’s paper (this would take me too far afield), but I will quote a passage:
“…the problem is with memory: ‘our undertaking’ will only be knowledge if all of it is present before the mind’s eye together, which any reliance on memory prevents. It is certainly true that many long proofs don’t satisfy Descartes-surveyability — nobody can sweep through the calculations in the four color theorem in the requisite way. Nor can anyone do it with either of the proofs of the Enormous Theorem or Fermat’s Last Theorem. In fact most proofs in real mathematics fail this test. If real proofs require this Cartesian gaze, then long proofs are not real proofs.”
For Coleman, the received conception of surveyability is deceptive, but what I wanted to get across by quoting his paper was the connection to the Cartesian tradition, and to the role of memory in seeing matters whole.
The embodied facts of seeing, when seeing is understood as the biophysical process of perception, was a concern to Bertrand Russell in the construction of a mathematical logic adequate to the deduction of mathematics. In the Introduction to Principia Mathematica Russell wrote:
“The terseness of the symbolism enables a whole proposition to be represented to the eyesight as one whole, or at most in two or three parts divided where the natural breaks, represented in the symbolism, occur. This is a humble property, but is in fact very important in connection with the advantages enumerated under the heading.”
Bertrand Russell and Alfred North Whitehead, Principia Mathematica, Volume I, second edition, Cambridge: Cambridge University Press, 1963, p. 2
…and Russell elaborated…
“The adaptation of the rules of the symbolism to the processes of deduction aids the intuition in regions too abstract for the imagination readily to present to the mind the true relation between the ideas employed. For various collocations of symbols become familiar as representing important collocations of ideas; and in turn the possible relations — according to the rules of the symbolism — between these collocations of symbols become familiar, and these further collocations represent still more complicated relations between the abstract ideas. And thus the mind is finally led to construct trains of reasoning in regions of thought in which the imagination would be entirely unable to sustain itself without symbolic help.”
Thinking is difficult, and symbolization allows us to — mechanically — extend thinking into regions where thinking alone, without symbolic aid, would not be capable of penetrating. But that doesn’t mean symbolic thinking is easy. Elsewhere Russell develops another rationalization for symbolization:
“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”
Russell formulated the difficulty of thinking even more strongly in a later passage:
“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
Alfred North Whitehead, coauthor of Principia Mathematica, made a similar point more colorfully than Russell, which I recently in The Algorithmization of the World:
“It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle: they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.”
Alfred North Whitehead, An Introduction to Mathematics, London: WILLIAMS & NORGATE, Chap. V, pp. 45-46
This quote from Whitehead follows a lesser known passage from the same work:
“…by the aid of symbolism, we can make transitions in reasoning almost mechanically by the eye, which otherwise would call into play the higher faculties of the brain.”
Alfred North Whitehead, An Introduction to Mathematics, London: WILLIAMS & NORGATE, Chap. V, pp. 45
In other words, the brain is saved effort by mechanizing as much reason as can be mechanized. Of course, not everyone is capable of these kinds of mechanical deductions made possible by mathematical logic, which is especially difficult.
Recent scholarship has only served to underscore the difficulty of thinking, and the steps we must take to facilitate our thinking. Daniel Kahneman in particular has focused on the physiology effort involved in thinking. In his book Thinking, Fast and Slow, Daniel Kahneman distinguishes between two cognitive systems, which he calls System 1 and System 2, which are, respectively, that faculty of the mind that responds immediately, on an intuitive or instinctual level, and that faculty of the mind that proceeds more methodically, according to rules:
Why call them System 1 and System 2 rather than the more descriptive “automatic system” and “effortful system”? The reason is simple: “Automatic system” takes longer to say than “System 1” and therefore takes more space in your working memory. This matters, because anything that occupies your working memory reduces your ability to think. You should treat “System 1” and “System 2” as nicknames, like Bob and Joe, identifying characters that you will get to know over the course of this book. The fictitious systems make it easier for me to think about judgment and choice, and will make it easier for you to understand what I say.
Daniel Kahneman, Thinking, Fast and Slow, New York: Farrar, Straus, and Giroux, Part I, Chap. 1
While such concerns do not appear to have explicitly concerned Russell, Russell’s concern for economy of thought implicitly embraced this idea. One’s ability to think must be facilitated in any way possible, including the shortening of names — in purely formal thought, symbolization dispenses with names altogether and contents itself with symbols only, usually introduced as letters.
Kahneman’s book, by the way, is a wonderful review of cognitive biases that cites many of the obvious but often unnoticed ways in which thought requires effort. For example, if you are walking along with someone and you ask them in mid-stride to solve a difficult mathematical problem — or, for that matter, any problem that taxes working memory — your companion is likely to come to a stop when focusing mental effort on the work of solving the problem. Probably everyone has had experiences like this, but Kahneman develops the consequences systematically, with very interesting results (creating what is now known as behavioral economics in the process).
Formal thought is among the most difficult forms of cognition ever pursued by human beings. How can we facilitate our ability to think within a framework of thought that taxes us so profoundly? It is the overview provided by the non-constuctive perspective that makes it possible to take a “big picture” view of formal knowledge and formal thought, which is usually understood to be a matter entirely immersed in theoretical details and the minutiae of deduction and derivation. We must take an “Über” perspective in order to see formal thought whole. We have become accustomed to thinking of “surveyability” in constructivist terms, but it is just as valid in non-constructivist terms.
In P or not-P (as well as in subsequent posts concerned with constructivism, What is the relationship between constructive and non-constructive mathematics? Intuitively Clear Slippery Concepts, and Kantian Non-constructivism) I surveyed constructivist and non-constructivist views of tertium non datur — the central logical principle at issue in the conflict between constructivism and non-constructiviem — and suggested that constructivists and non-constructivists need each other, as each represents a distinct point of view on formal thought.
In P or not-P, cited above, I quoted French mathematician Alain Connes:
“Constructivism may be compared to mountain climbers who proudly scale a peak with their bare hands, and formalists to climbers who permit themselves the luxury of hiring a helicopter to fly over the summit …the uncountable axiom of choice gives an aerial view of mathematical reality — inevitably, therefore, a simplified view.”
Conversations on Mind, Matter, and Mathematics, Changeux and Connes, Princeton, 1995, pp. 42-43
In several posts I have taken up this theme of Alain Connes and have spoken of the non-constructive perspective (which Connes calls “formalist”) as being top-down and the constructive perspective as being bottom-up. In particular, in The Epistemic Overview Effect I argued that in additional to the possibility of a spatial overview (the world entire seen from space) and a temporal overview (history seen entire, after the manner of Big History), there is an epistemic overview, that is to say, an overview of knowledge, perhaps even the totality of knowledge.
If we think of those mathematical equations that have become sufficiently famous that they have become known outside mathematics and physics — (as well as some that should be more widely known, but are not, like the generalized continuum hypothesis and the expression of epsilon zero) — they all have not only the succinct property that Russell noted in the quotes above in regard to symbolism, but also many of the qualities that G. H. Hardy ascribed to what he called mathematically significant ideas.
It is primarily non-constructive modes of thought that give us a formal overview and which make it possible for us to engage with mathematically significant ideas, and, more generally, with formally significant ideas.
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Studies in Formalism
9. The Overview Effect in Formal Thought
10. Methodological and Ontological Parsimony (in preparation)
11. Einstein’s Conception of Formalism (in preparation)
12. The Spirit of Formalism (in preparation)
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27 November 2013
Immanuel Kant, in an often-quoted passage, spoke of, “…the starry heavens above me and the moral law within me.” Kant might have with equal justification spoken of the formal law within and the starry heavens above. There is a sense in which the formal laws of thought are the moral laws of the mind — in logic, a good thought is a rigorous thought — so that given sufficient latitude of translation, we can interpret Kant in this way — except that we know (as Nietzsche put it) that Kant was a moral fanatic à la Rousseau.
However we choose to interpret Kant, I would like to quote more fully from the passage in the Critique of Practical Reason where Kant invokes the starry heavens above and the moral law within:
“Two things fill the mind with ever new and increasing admiration and awe, the oftener and the more steadily we reflect on them: the starry heavens above and the moral law within. I have not to search for them and conjecture them as though they were veiled in darkness or were in the transcendent region beyond my horizon; I see them before me and connect them directly with the consciousness of my existence. The former begins from the place I occupy in the external world of sense, and enlarges my connection therein to an unbounded extent with worlds upon worlds and systems of systems, and moreover into limitless times of their periodic motion, its beginning and continuance. The second begins from my invisible self, my personality, and exhibits me in a world which has true infinity, but which is traceable only by the understanding, and with which I discern that I am not in a merely contingent but in a universal and necessary connection, as I am also thereby with all those visible worlds. The former view of a countless multitude of worlds annihilates as it were my importance as an animal creature, which after it has been for a short time provided with vital power, one knows not how, must again give back the matter of which it was formed to the planet it inhabits (a mere speck in the universe). The second, on the contrary, infinitely elevates my worth as an intelligence by my personality, in which the moral law reveals to me a life independent of animality and even of the whole sensible world, at least so far as may be inferred from the destination assigned to my existence by this law, a destination not restricted to conditions and limits of this life, but reaching into the infinite.”
Immanuel Kant, Critique of Practical Reason, 1788, translated by Thomas Kingsmill Abbott, Part 2, Conclusion
This passage is striking for many reasons, not least among them them degree to which Kant has assimilated the Copernican revolution, acknowledging Earth as a mere speck in the universe. Also particularly interesting is Kant’s implicit appeal to objectivity and realism, notwithstanding the fact that Kant himself established the tradition of transcendental idealism. Kant in this passage invokes the starry heavens above and the moral law within because they are independent of the individual …
Moreover, Kant identifies both the starry heavens above and the moral law within not only as objective and independent realities, but also as infinitistic. Just as Kant the idealist looks to the stars and the moral law in a realistic spirit, so Kant the proto-constructivist invokes the “…unbounded extent with worlds upon worlds” of the starry heavens and the moral law as, “…reaching into the infinite.” I have earlier and elsewhere observed how Kant’s proto-constructivism nevertheless involves spectacularly non-constructive arguments. In the passage quoted above both Kant’s proto-constructivism and his non-constructive moments are retained in lines such as, “exhibits me in a world which has true infinity,” which by invoking exhibition in intuition toes the constructivist line, while invoking true infinity allows a legitimate role for the non-constructive.
When it comes to constructivism, we can see that Kant is conflicted. He’s not the only one. One might call Aristotle the first constructivist (or, at least, the first proto-constructivist) as the originator of the idea of the potential infinite, and here (i.e., in the context of the above discussion of Kant’s use of the infinite) Aristotelian permissive finitism is particularly relevant. (Aristotle would likely not have had much sympathy for intuitionistic constructivism, which its rejection of tertium non datur.)
The Greek intellectual attitude to the infinite was complex and conflicted. I have written about this previously in Reason in Moderation and Salto Mortale. The Greek quest for harmony, order, and proportion rejected the infinite as something that transgresses the boundaries of good taste and propriety (dismissing the infinite as apeiron, in contradistinction to peras). Nevertheless, Greek philosophers routinely argued from the infinity and eternity of the world.
Here is a famous passage from Democritus, who was perhaps best known among the Greek philosophers in arguing for the infinity of the world, making the doctrine a virtual tenet among ancient atomists:
“Worlds are unlimited and of different sizes. In some worlds there is no Sun and Moon, in others, they are larger than in our world, and in others more numerous. … Intervals between worlds are unequal. In some parts there are more worlds, in others fewer; some are increasing, some at their height, some decreasing; in some parts they are arising, in others failing… There are some worlds devoid of living creatures or plants or any moisture.”
…and Epicurus on the same theme of the infinity of the world…
“…there is an infinite number of worlds, some like this world, others unlike it. For the atoms being infinite in number, as has just been proved, are borne ever further in their course. For the atoms out of which a world might arise, or by which a world might be formed, have not all been expended on one world or a finite number of worlds, whether like or unlike this one. Hence there will be nothing to hinder an infinity of worlds.”
Epicurus, Letter to Herodotus
There were also poetic invocations of the idea of the infinity of the world, which demonstrates the extent to which the idea had penetrated popular consciousness in classical antiquity:
“When Alexander heard from Anaxarchus of the infinite number of worlds, he wept, and when his friends asked him what was the matter, he replied, ‘Is it not a matter for tears that, when the number of worlds is infinite, I have not conquered one?'”
Plutarch, PLUTARCH’S MORALS, ETHICAL ESSAYS TRANSLATED WITH NOTES AND INDEX BY ARTHUR RICHARD SHILLETO, M.A., Sometime Scholar of Trinity College, Cambridge, Translator of Pausanias, LONDON: GEORGE BELL AND SONS, 1898, “On Contentedness of Mind,” section IV
Like poetry, history had particular prestige in the ancient world, and here the theme of the infinity of the world also occurs:
“…Constantius, elated by this extravagant passion for flattery, and confidently believing that from now on he would be free from every mortal ill, swerved swiftly aside from just conduct so immoderately that sometimes in dictation he signed himself ‘My Eternity,’ and in writing with his own hand called himself lord of the whole world — an expression which, if used by others, ought to have been received with just indignation by one who, as he often asserted, laboured with extreme care to model his life and character in rivalry with those of the constitutional emperors. For even if he ruled the infinity of worlds postulated by Democritus, of which Alexander the Great dreamed under the stimulus of Anaxarchus, yet from reading or hearsay he should have considered that (as the astronomers unanimously teach) the circuit of whole earth, which to us seems endless, compared with the greatness of the universe has the likeness of a mere tiny point.
Ammianus Marcellinus, Roman Antiquities, Book XV, section 1
Like the quote from Kant quoted above, this passage is remarkable for its Copernican outlook, which shows that the ancients were not only capable of thinking in infinitistic terms, but also in more-or-less Copernican terms.
Lucretius was a follower of Epicurus, and gave one of the more detailed arguments for the infinity of the world to be found in ancient philosophy:
It matters nothing where thou post thyself,
In whatsoever regions of the same;
Even any place a man has set him down
Still leaves about him the unbounded all
Outward in all directions; or, supposing
moment the all of space finite to be,
If some one farthest traveller runs forth
Unto the extreme coasts and throws ahead
A flying spear, is’t then thy wish to think
It goes, hurled off amain, to where ’twas sent
And shoots afar, or that some object there
Can thwart and stop it? For the one or other
Thou must admit; and take. Either of which
Shuts off escape for thee, and does compel
That thou concede the all spreads everywhere,
Owning no confines. Since whether there be
Aught that may block and check it so it comes
Not where ’twas sent, nor lodges in its goal,
Or whether borne along, in either view
‘Thas started not from any end. And so
I’ll follow on, and whereso’er thou set
The extreme coasts, I’ll query, “what becomes
Thereafter of thy spear?” ‘Twill come to pass
That nowhere can a world’s-end be, and that
The chance for further flight prolongs forever
The flight itself. Besides, were all the space
Of the totality and sum shut in
With fixed coasts, and bounded everywhere,
Then would the abundance of world’s matter flow
Together by solid weight from everywhere
Still downward to the bottom of the world,
Nor aught could happen under cope of sky,
Nor could there be a sky at all or sun-
Indeed, where matter all one heap would lie,
By having settled during infinite time.
Lucretius, De rerum natura
The above argument is one that is still likely to be heard today, in various forms. If you go to the edge of the universe and throw a spear, either it is stopped by the boundary of the universe, or it continues on, and, as Lucretius says, For the one or other, Thou must admit. If the spear is stopped, what stopped it? And if it continues on, into what does it continue?
The contemporary relativistic cosmology has a novel answer to this ancient idea: the universe is finite and unbounded, so that space is wrapped back around on itself. What this means for the spear-thrower at the edge of the universe is that if he throws the spear with enough force, it may travel around the cosmos and return to pierce him in the back. There is nothing to stop the spear, because the universe is unbounded, but since the universe is also finite the spear will eventually cross its own path if it continues to travel. I do not myself think that the universe is finite and unbounded in precisely the way the many modern cosmologists argue, but I am not going to go into this interesting problem at the present time.
Other than the response to Lucretius in terms of relativistic cosmology, with its curved spacetime — a material response to the Lucretian argument for the infinity of the world — there is another response, that of intuitionistic constructivism, which denies the law of the excluded middle (tertium non datur) — i.e, a formal response to Lucretius. Lucretius asserted that, For the one or other, Thou must admit, and this is exactly what the intuitionist does not admit. As with the relativistic response to Lucretius, I do not myself agree with the intuitionist response to Lucretius. Consequently, I believe that Lucretius argument is still valid in spirit, though it must be reformulated in order to be applicable to the world as revealed to us by contemporary science. Consequently, I take it as demonstrable that the universe is infinite, taking the view of ancient natural philosophers.
Within the overall context of Greek thought, within its contending finitist and infinitistic strains, Greek cosmology was non-constructive, and the Greeks asserted (and argued for) the infinity of the world on the basis of non-constructive argument. Perhaps it would even be fair to say that the Greeks assumed the universe to be infinite in extent, and they at times sought to justify this assumption by philosophical argument, while at other times they confined themselves to the sphere of the peras.
Much of contemporary science is constructivist in spirit, though this constructivism is rarely made explicit, except among logicians and mathematicians. By this I mean that the general drift of science ever since the scientific revolution has been toward bottom-up constructions on the basis of quantifiable evidence and away from top-down argument. I made this point previously in Advanced Thinking and A Non-Constructive World, as well as other posts, though I haven’t yet given a detailed formulation of this idea. Yet the emergence of a “quantum logic” in quantum theory that does away with the principle of the excluded middle is a clear expression of the increasing constructivism of science.
In A Non-Constructive World I also made the point that the world appears to have both constructive and non-constructive features. In several posts about constructivism (e.g., P or not-P) I have argued that constructivism and non-constructivism are complementary perspectives on formal thought, and that each needs the other for an adequate account of the world.
In so far as contemporary science is essentially constructive, it lacks a non-constructive perspective on the phenomena it investigates. This is, I believe, intrinsic to science, and to the kind of civilization that emerges from the application of science to the economy (viz. industrial-technological civilization). By the constructive methods of science we can attain ever larger and ever more comprehensive conceptions of the universe — such as I described in my previous post, The Size of the World — but these constructive methods will never reach the infinite universe contemplated by the ancient Greeks.
How could the logical framework employed by a scientist have any effect over what they see in the heavens? Well, constructive science is logically incapable of formulating the idea of an infinite universe in any sense other than an Aristotelian potential infinite. No one can observe the infinite (in the philosophy of mathematics we say that the infinite is “unsurveyable”). And if you cannot produce observational evidence of the infinite, then you cannot formulate a falsifiable theory of an infinite universe. Thus the infinity of the world is, in effect, ruled out by our methods.
No one should be surprised at the direct impact the ethos of formal thought has a upon the natural sciences; one of the fundamental trends of the scientific revolution has been the mathematization of natural science, and one of the fundamental trends of mathematical rigor since the late nineteenth century has been the arithmetization of analysis, which has been taken as far as the logicization of mathematics. Logic and mathematics have been “finitized” and these finite formal methods have been employed in the rational reconstruction of the sciences.
I look forward to the day when the precision and rigor of formal methods employed in the natural sciences again includes infinitistic methods, and it once again becomes possible to formulate the thesis of the infinity of the world in science — and possible once again to see the world as infinite.
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2 March 2013
Arthur C. Clarke is best remembered for this science fiction stories, but many of his dicta and aphorisms have become common currency and are quoted and repeated to the point that their connection to their source is sometimes lost. (Clarke’s thought ranged widely and, interestingly, Clarke identified himself as a logical positivist.) Recently I quoted one of Clarke’s well-known sayings in Happy Birthday Nicolaus Copernicus!, as follows:
“Two possibilities exist: either we are alone in the Universe or we are not. Both are equally terrifying.”
quoted in Visions: How Science Will Revolutionize the Twenty-First Century (1999) by Michio Kaku, p. 295
In so saying, Clarke asserted a particular case of what is known as the logical law (or principle) of the excluded middle, which is also known as tertium non datur: the idea that, given a proposition and its negation, either one or the other of them must be true. This is also expressed in propositional logic as “P or not-P” (“P v ~P”). The principle of tertium non datur is not to be confused with the principle of non-contradiction, which can be formulated as “~(P & ~P).”
Even stating tertium non datur is controversial, because there are narrowly logical formulations as well as ontological formulations of potentially much greater breadth. This, of course, is what makes the principle fascinating and gives it its philosophical depth. Moreover, the principle of the excluded middle is subtly distinct from the principle of bivalence, though the two usually work in conjunction. Whereas the law of the excluded middle states that of a proposition and its negation, one of the other must be true, the principle of bivalence states that there are only two propositional truth values: true and false.
To get started, here is the principle of the excluded middle as formulated in The Cambridge Dictionary of Philosophy edited by Robert Audi:
principle of excluded middle, the principle that the disjunction of any (significant) statement with its negation is always true; e.g., ‘Either there is a tree over 500 feet tall or it is not the case that there is such a tree’. The principle is often confused with the principle of bivalence.
THE CAMBRIDGE DICTIONARY OF PHILOSOPHY second edition, General Editor Robert Audi, 1999, p. 738
And to continue the Oxbridge axis, here is the formulation from Simon Blackburn’s The Oxford Dictionary of Philosophy:
excluded middle, principle (or law) of The logical law asserting that either p or not-p. It excludes middle cases such as propositions being half correct or more or less right. The principle directly asserting that each proposition is either true or false is properly called the law of bivalence.
The Oxford Dictionary of Philosophy, Simon Blackburn, Oxford University Press, 1996, p. 129
For more partisan formulations, we turn to other sources. Mario Bunge formulated a narrowly syntactical conception of the law of the excluded middle in his Dictionary of Philosophy, which is intended to embody a scientistic approach to philosophy:
EXCLUDED MIDDLE A logical truth or tautology in ordinary (classical) logic: For every proposition p, p v ~p.
Dictionary of Philosophy, Mario Bunge, Prometheus Books, 1999, p. 89
By way of contrast, in D. Q. McInerny’s Being Logical: A Guide to Good Thinking we find a strikingly ontological formulation of the law of the excluded middle:
“Between being and nonbeing there is no middle state. Something either exists or it does not exist; there is no halfway point between the two.”
D. Q. McInerny, Being Logical: A Guide to Good Thinking, Part Two, The Basic Principles of Logic, 1. First Principles, p. 26
What these diverse formulations bring out for us is the difficulty of separating logical laws of how formal systems are to be constructed from ontological laws about how the world is constructed, and in so bringing out this difficulty, they show us the relation between the law of the excluded middle and the principle of bivalence, since the logical intuition that there are only two possible truth values of any one proposition — true or false — is so closely tied to our logical intuition that, of these two values, one or the other (but not both, which qualification is the principle of non-contradiction) must hold for any given (meaningful) proposition.
The powerful thing about Clarke’s observation is that it appeals to this admixture of logical intuitions and empirical intuitions, and in so doing seems to say something very compelling. Indeed, since I am myself a realist, and I think it can be shown that there is a fact of the matter that makes propositions true or false, I think that Clarke not only said something powerful, he also said something true: either there are extraterrestrial intelligences or there are not. It is humbling to contemplate either possibility: ourselves utterly alone in a vast universe with no other intelligent species or civilizations, or some other alien intelligence out there somewhere, unknown to us at present, but waiting to be discovered — or to discover us.
Although these logical intuitions are powerful, and have shaped human thought from its earliest times to the present day, the law of the excluded middle has not gone unquestioned, and indeed Clarke’s formulation gives us a wonderful opportunity to explore the consequences of the difference between constructive and non-constructive reasoning in terms of a concrete example.
To formulate the existence or non-existence of extraterrestrials in the form of a logical law like the law of the excluded middle makes the implicit realism of Clarke’s formulation obvious as soon as we think of it in these terms. In imagining the possibilities of our cosmic isolation or an unknown alien presence our terror rests on our intuitive, visceral feeling of realism, which is as immediate to us as the intuitions rooted in our own experiences as bodies.
The constructivist (at least, most species of constructivist, but not necessarily all) must deny the validity of the teritum non datur formulation of the presence of extraterrestrials, and in so doing the constructivist must pretend that our visceral feelings of realism are misleading or false, or must simply deny that these feelings exist. None of these are encouraging strategies, especially if one is committed to understanding the world by getting to the bottom of things rather than denying that they exist. Not only I am a realist, but I also believe strongly in the attempt to do justice to our intuitions, something that I have described in two related posts, Doing Justice to Our Intuitions and How to Formulate a Philosophical Argument on Gut Instinct.
In P or not-P (as well as in subsequent posts concerned with constructivism, What is the relationship between constructive and non-constructive mathematics? Intuitively Clear Slippery Concepts, and Kantian Non-constructivism) I surveyed constructivist and non-constructivist views of tertium non datur and suggested that constructivists and non-constructivists need each other, as each represents a distinct point of view on formal thought. Formal thought is enriched by these diverse perspectives.
But whereas non-constructive thought, which is largely comprised of classical realism, can accept both the constructivist and non-constructivist point of view, the many varieties of constructivism usually explicitly deny the validity of non-constructive methods and seek to systematically limit themselves to constructive methods and constructive principles. Most famously, L. E. J. Brouwer (whom I have previously discussed in Saying, Showing, Constructing and One Hundred Years of Intuitionism and Formalism) formulated the philosophy of mathematics we now know as intuitionism, which is predicated upon the explicit denial of the law of the excluded middle. Brouwer, and those following him such as Heyting, sought to formulate mathematical and logic reasoning without the use of tertium non datur.
Brouwer and the intuitionists (at least as I interpret them) were primarily concerned to combat the growing influence of Cantor and his set theory in mathematics, which seemed to them to license forms of mathematical reasoning that had gone off the rails. Cantor had gone too far, and the intuitionists wanted to reign him in. They were concerned about making judgments about infinite totalities (in this case, sets with an infinite number of members), which the law of the excluded middle, when applied to the infinite, allows one to do. This seems to give us the power to make deductions about matters we cannot either conceive or even (as it is sometimes said) survey. “Surveyability” became a buzz word in the philosophy of mathematics after Wittgenstein began using it in his posthumously published Remarks on the Foundations of Mathematics. Although Wittgenstein was not himself an intuitionist sensu stricto, his work set the tone for constructivist philosophy of mathematics.
Given the intuitionist rejection of the law of the excluded middle, it is not correct to say that there either is intelligent alien life in the universe or there is not intelligent alien life in the universe; to meaningfully make this statement, one would need to actually observe (inspect, survey) all possible locations where such alien intelligence might reside, and only after seeing it for oneself can one legitimately claim that there is or is not alien intelligence in the universe. For am example closer to home, it has been said that an intuitionist will deny the truth of the statement “either it is raining or it is not raining” without looking out the window to check and see. This can strike one as merely perverse, but we must take the position seriously, as I will try to show with the next example.
Already in classical antiquity, Aristotle brought out a striking feature of the law of the excluded middle, in a puzzle sometimes known as the “sea battle tomorrow.” The idea is simple: either there will be a sea battle tomorrow, or there will not be a sea battle tomorrow. We may not know anything about this battle, and as of today we do not even know if it will take place, but we can nevertheless confidently assert that either it will take place or it will not take place. This is the law of the excluded middle as applied to future contingents.
One way to think of this odd consequence of the law of the excluded middle is that when it is projected beyond the immediate circumstances of our ability to ascertain its truth by observation it becomes problematic. This is why the intuitionists reject it. Aristotle extrapolated the law of the excluded middle to the future, but we could just as well extrapolate it into the past. Historians do this all the time (either Alexander cut the Gordian Knot or Alexander did not cut the Gordian Knot), but because of our strong intuitive sense of historical realism this does not feel as odd as asserting that something that hasn’t happened yet either will happen or will not happen.
In terms of Clarke’s dichotomy, we could reformulate Aristotle’s puzzle about the sea battle tomorrow in terms of the discovery of alien intelligence tomorrow: either we will receive an alien radio broadcast tomorrow, or we will not receive an alien broadcast tomorrow. There is no third possibility. One way or another, the realist says, one of these propositions is true, and one of them is false. We do not know, today, which one of them is true and which one of them is false, but that does not mean that they do no possess definite truth values. The intuitionist says that the assertion today that we will or will not receive an alien radio broadcast is meaningless until tomorrow comes and we turn on our radio receivers to listen.
The intuitionists thus have an answer to this puzzling paradox that remains a problem for the realist. This is definitely a philosophical virtue for intuitionism, but, like all virtues, it comes at a price. It is not a price I am willing to pay. This path can also lead us to determinism — assuming that all future contingents have a definite truth value implies that they are set in stone — but I am also not a determinist (as I discussed in The Denial of Freedom as a Philosophical Problem), and so this intersection of my realism with my libertarian free willist orientation leaves me with a problem that I am not yet prepared to resolve. But that’s what makes life interesting.
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6 August 2012
Brouwer and Wittgenstein were contemporaries, with the whole of Wittgenstein’s years contained within those of Brouwer’s (Wittgenstein lived 1889 to 1951 while Brouwer lived the longer life from 1881 to 1966). It is mildly ironic that even as Brouwer’s followers were playing down his mysticism and trying to extract only the mathematical content from his intuitionist philosophy (even the faithful Heyting distanced himself from Brouwer’s mysticism), Wittgenstein’s writings reached a much larger public which resulted in the mystical content of Wittgenstein’s works being played up and the early Wittgenstein himself, very much the logician following in the tradition of Frege and Russell, presented as a mystic.
Not only were Brouwer and Wittgenstein contemporaries, but we also know that Brouwer played a little-known role in Wittgenstein’s return to philosophy. After having written the Tractatus Logico-Philosophicus and then disappearing into the mountains of Austria to become a village schoolmaster in Trattenbach, some of those philosophers that continued to seek out Wittgenstein in his self-imposed exile convinced him to go to a lecture in Vienna in March 1928. The lecture was delivered by Brouwer (Brouwer gave two lectures; Wittgenstein is said to have attended one of them). Wittgenstein was said to have listened to the lecture with a surprised look on his face (sort of like G. E. Moore saying that Wittgenstein was the only student that looked puzzled at this lectures). So it may be the case that Brouwer played a pivotal role in the transition from the thought of the early Wittgenstein to the thought of the later Wittgenstein. (Matthieu Marion has argued this thesis.)
Wittgenstein’s distinction between saying and showing, a doctrine that dates from the Tractatus (cf. sections 4.113 and following), is often adduced in expositions of his alleged mysticism. According to Wittgenstein’s distinction, some things can be said but cannot be shown, while other things can be shown but cannot be said. While to my knowledge Wittgenstein never used the term “ineffable,” that which can be shown but cannot be said would seem to be a paradigm case of the ineffable. And since Wittgenstein identified a substantial portion of our experience as showable although unsayable, the ineffable seems then to play a central role in his thought. This puts Wittgenstein firmly in the company of figures like, say, St. Symeon the New Theologian (also, like Wittgenstein, an ascetic), which makes the case for his mysticism.
An extract from St. Symeon on the ineffable: “The grace of the all-holy spirit is given as earnest money of the souls pledged in marriage to Christ. Just as a woman without a pledge has no certainty that the union with the groom will occur within a certain length of time, so does the soul have no firm assurance that it will be re-united to its God and Master for all eternity. The soul cannot be certain that it will achieve this mystic, ineffable union nor that it will enjoy its inaccessible beauty if it does not have the pledge of His grace and does not consciously have that grace within.” (Krivocheine, Basil and Gythiel, Anthony P., In the Light of Christ: Saint Symeon, the New Theologian 949–1022, St. Vladimir’s Seminary Press, 1986, p. 367)
Brouwer was a bit more explicit in his doctrine of ineffability than was Wittgenstein, and he repeatedly asserted that the language of mathematics was a necessary evil that approximated but never fully captured the intuitive experience of mathematics, which he understood to be a free creation of the human mind. This comes across both in his early mystical treatise Life, Art, and Mysticism, which is pervaded by a sense of pessimism over the evils of the world (which include the evils of mathematical language), and his more technical papers offering an exposition of intuitionism as a philosophy of mathematics. But, like Wittgenstein, Brouwer does not (to my limited knowledge) actually use the term “ineffable.”
There is another ellipsis common to both Brouwer and Wittgenstein, and that is despite Brouwer’s openly professed intuitionism, which can be considered a species of constructivism (this latter is a point that needs to be separately argued, but I will only pass over it here with a single mention), and despite the strict finitism of the later Wittgenstein, which can also be considered a species of constructivism, neither Brouwer nor Wittgenstein employ Kantian language or Kantian formulations. No doubt there are implicit references to Kant in both, but I am not aware of any systematic references to Kant in the work of either philosopher. This is significant. Both Brouwer and Wittgenstein were philosophers of the European continent, where the influence of Kant remains strong even as his reputation waxes and wanes over the generations.
Kant was an early constructivist, or, rather, a constructivist before constructivism was explicitly formulated, and therefore sometimes called a proto-constructivist — although I have pointed out an obvious non-constructive dimension to Kant’s thought despite his proto-constructivism (which I do not deny, notwithstanding Kant’s non-constructive arguments in the first Critique). Kant’s classic proto-constructivist formulation is that the synthetic a priori truths of mathematics must be constructed, or “exhibited in intuition.” It is this latter idea, of a concept being exhibited in intuition, that has been particularly influential. But what does it mean? Obviously, a formulation like this has invited many interpretations.
The approaches of Brouwer and the later Wittgenstein could be considered different ways of exhibiting a concept in intuition. Brouwer, by casting out the law of the excluded middle from mathematics (at least in infinitistic contexts), assured that double negation was not equivalent to the truth simpliciter, so that even if you know that it is not the case that x is false, you still don’t know that x is true. (On the law of the excluded middle cf. P or not-P.) The later Wittgenstein’s insistence upon working out how a particular term is used and not merely settling for some schematic meaning (think of slogans like “don’t ask for the meaning, ask for the use” and “back to the rough ground”) similarly forces one to consider concrete instances rather than accepting (non-constructive) arguments for the way that things putatively must be, rather than how they are in actual fact. Both Wittgenstein’s finitism and Brouwer’s intuitionism would look with equal distaste upon, for example, proving that every set can be well-ordered without actually showing (i.e., exhibiting) such an order — also, the impossibility of exhaustively showing (i.e., exhibiting in intuition) that every set can be well-ordered if one acknowledges an infinity of sets.
I give this latter example because I think it was largely the perceived excesses of set theory and Cantor’s transfinite number theory that were essentially responsible for the reaction among some mathematicians that led to constructivism. Cantor was a great mathematical innovator, and his radical contributions to mathematics spurred foundationalists like Frege (who objected to Cantor’s methods but not his results) and Russell to attempt to construct philosophico-mathematical justifications, i.e., foundations, that would legitimize that which Cantor had wrought.
The reaction against infinitistic mathematics and foundationalism continues to the present day. Michael Dummett wrote in Elements of Intuitionism, a classic textbook on basic intuitionistic logic and mathematics, that:
“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
In other words, mathematics would show its justification; in contrast, the foundationalist project to assure the legitimacy of the flights of non-constructive mathematics was wrong-headed in its very conception, because nothing that we say is going to change the fact that non-constructive thought that derives its force from proof, i.e., from what is said, does not show its justification on its face. Its justification must be established because it does not show itself. This is what “foundations” are for.
Note: There is also an element of intellectual ascesis in Dummett’s idea of a conservative extension of a theory, and this corresponds to the asceticism of Wittgenstein’s character, and, by extension, to the asceticism of Wittgenstein’s thought — asceticism being one of the clear continuities between the earlier and the later Wittgenstein — like the implicit development of constructivist themes.
But it was not only the later Wittgenstein who reacted with others against Cantor. It seems to me that the saying/showing distinction of the Tractatus is a distinction not only between that which can be said and that which can be shown, but also a distinction between that which is established by argument, possibly non-constructive argument, and that which is exhibited in intuition, i.e., constructed. If this is right, Wittgenstein showed an early sensitivity to the possibility of constructivist thought, and his later development might be understood as a development of the constructivist strand within his thinking, making Wittgenstein’s development more linear than is often recognized (though there are many scholars who argue for the unity of Wittgenstein’s development on different principles). The saying/showing distinction may be the acorn from which the oak tree of the Philosophical Investigations (and the subsequently published posthumous works) grew.
For the early Wittgenstein, the distinction between saying and showing was thoroughly integrated into his idea of logic, and while in the later sections of the Tractatus the mysticism of what which can only be shown but cannot be said becomes more evident, it is impossible to say whether it was the logical impulse that prevailed, and served as the inspiration for the mysticism, or whether it was the mystic impulse that prevailed, and served as the pretext for formulating the logical doctrines. But the logical doctrines are clearly present in the Tractatus, and serve as the exposition of Wittgenstein’s ideas, even up to the famous metaphor when Wittgenstein says that the propositions of the Tractatus are like a ladder than one must cast away after having climbed up and over it.
Just as there is a mathematical content to Brouwer’s mysticism, so too there is a logical content to Wittgenstein’s mysticism. It is, in fact, likely that Wittgenstein’s distinction between saying and showing was suggested to him by what is now called the “picture theory of meaning” given an exposition in the Tractatus. Few philosophers today defend Wittgenstein’s picture theory of meaning, but it is central to the metaphysics of the Tractatus. For Wittgenstein, the logical structure of a proposition can be shown but not said. Since for Wittgenstein in his Tractarian period, “The facts in logical space are the world” (1.13), and “In the proposition the thought is expressed perceptibly through the senses” (3.1) — i.e., the proposition literally exhibits its structure in sensory intuition — thus, “The proposition is a picture of reality.” (4.01) One might even say that a proposition exhibits the world in intuition.
Today these formulations strike us as a bit odd, because we think of anything that can be formulated in logical terms as a paradigm case of something that can be said, and very possibly also something that may not be showable. For us, logic is a language is among languages, and one way among many to express the world; for the early Wittgenstein, on the contrary, logic is the structure of the world. It shows itself because the world shows itself, and after showing itself there is nothing more to be said. The only appropriate response is silence.
As we all know from the final sentence of the Tractatus, whereof one cannot speak, thereof one must remain silent. According to the Wittgenstein of the Tractatus, all scientific questions can be asked and all scientific questions can be answered (shades of Hilbert’s “Wir müssen wissen. Wir werden wissen.” — which Per Martin-Löf has called Hilbert’s solvability axiom, and which is the very antithesis of Brouwer’s rejection of the law of the excluded middle), but even when we have answered all scientific questions, the problems of life remain untouched.
As implied by the early Wittgenstein’s insistence upon the solvability of all scientific questions, the metaphysics of Brouwer and Wittgenstein were very different. Their common constructivism does not prevent their having fundamental, I might even say foundational, differences. Also, while Wittgenstein comes across in a melancholic fashion (a lot like Plotinus, another philosophical mystic), he is not fixated on the evils of the world in the same way that Brouwer was. If both Brouwer and Wittgenstein can be called mystics, they are mystics belonging to different traditions. Brouwer was a choleric mystic while Wittgenstein was melancholic mystic.
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22 July 2012
A couple of days ago in describing my pilgrimage to Kinn I suggested that the phenomenon of pilgrimage is a Wittgensteinian “form of life,” and as a form of life we may understand it better if we confine ourselves to the material infrastructure while setting aside the formal superstructure that surrounds the form of life we call pilgrimage. But in a fine-grained account of pilgrimage we must distinguish between those forms of pilgrimage that, when taking the long view of the big picture, become conflated.
As I attempted to show, in different ways, in Epistemic Orders of Magnitude and P or not-P, both la longue durée and the fine-grained view have their place in our epistemic development — respectively, and roughly, they represent the non-constructive and the constructive perspectives on experience — and we ought to be equally diligent in exploring the consequences of each perspective, since we have something important to learn from each.
I tried to suggest a similarly comprehensive synthesis yesterday in A Meditation upon the Petroglyphs of Ausevik, when remarking that an extrapolation of a personal philosophy of history, when drawn out to a sufficient extent coincides with the history of the world entire. In other words, non-constructivism represents the furthest reach of constructivist thought, which immediately suggests the contrary perspective, i.e., that constructivism represents the furthest reach of non-constructive thought. Constructivism is non-constructivism in extremis; non-construtivism is constructivism in extremis. To translate this once again into historico-personal terms, the history of the world entire coincides with an intimately personal philosophy of history when the former is extrapolated to the greatest extent of its possible scope.
In a fine-grained account of pilgrimage (in contradistinction to pilgrimage understood in outline, in the context of la longue durée), at the level of personal experience that is constructive because every detail is of necessity immediately exhibited in intuition and nothing whatsoever is demonstrated, we can distinguish many forms of pilgrimage. There are religious pilgrimages, such as the Sunnivaleia, there are personal pilgrimages, such as my pilgrimage to Kinn, there are aesthetic pilgrimages, such as when the custom dictated the young gentlemen of good families and fortune would take the “Grand Tour” of Europe, there are political pilgrimages, as when a candidate for office visits a politically significant place — and there are even philosophical pilgrimages. I have previously made some minor philosophical pilgrimages, as when I sought out Kierkegaard’s grave in Copenhagen and similarly visited Schopenhauer’s grave in Frankfurt. Today I made another philosophical pilgrimage, by visiting the small town of Skjolden, where Wittgenstein spent time working on the ideas that would later becomes the Tractatus Logico-Philosophicus.
In the letters that Wittgenstein subsequently exchanged with his acquaintances in Skjolden (which have, of course, been published along with the rest of his correspondence), the people of Skjolden almost always close their letters by observing that Skjolden is as it always was and ever will be, essentially unchanged in the passage of time. I wrote about this previously in The Charms of Small Town Norway. It seems to be true that life changes very slowly, almost imperceptibly, in the fjord country of Norway, as life always changes slowly in isolated, mountainous regions the world over. The peoples who retreat from the onrushing advance of civilization to the margins of the world where they will not be bothered, are not the kind of peoples who wish to indulge in change for the sake of change. It is this latter attitude that typifies industrial-technological civilization, which is still largely confined to the regions of the world fully given over to agricultural civilization. The margins of the world before industrialization largely coincide with the margins of the world after industrialization.
Wittgenstein, I think, left little impact upon Skjolden. He didn’t make waves, as it were, and didn’t want to make waves. Life in Skjolden is probably little changed in essentials from when Wittgenstein isolated himself in a small, bare hut at the end of a fjord in order to think and write about logic. I think that Wittgenstein would have liked this — or, at least, that he would have preferred this near absence of influence. The fjords are unchanged since Wittgenstein lived here, even if life has been modernized, and they still provide a refuge for those who would seek a world largely untouched by what Wittgenstein in his later years would call, “the main current of European and American civilization,” from which he felt profoundly alienated.
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23 January 2012
A short distance north of Nazca, along the Panamericana, and situated between the designs of the “hands” (“manos“) and the “tree” (“arbol“), there is a tower (the “Torre Mirador”) that can be climbed, probably about 40 or 50 feet in height, in order to view some part of the lines of Nazca without flying over them. This close-up view of the lines clearly reveals the construction methods that I quoted yesterday (in Lines in the Desert) from Mason’s The Ancient Civilizations of Peru — stones have been removed from within geometrically defined areas and the removed stones have been piled at the edges of the designs. The piled stones not only represent the space cleared, but the piles themselves serve to make the demarcation between cleared and non-cleared areas all the more obvious, making the distinction more visually striking.
This construction technique was also used at nearby Palpa, and continues to be effective in the present day, as driving along the Panamericana (once outside the archaeologically preserved area) one sees a variety of messages spelled out in the desert, from the initials and names of individuals to fairly elaborate advertisements for small roadside stores.
In my naïveté I though that any intrepid visitor of sufficient curiosity might walk out into the desert and and look at the construction of the lines for themselves, but the desert has been fenced off along the Panamericana to prevent further damage to the lines, and once made aware of the threat it becomes immediately obvious how damaged many of the lines and figures are, which accounts for some of the difficulty in seeing some of the patterns from the air. Some — but not all.
Much is revealed by a close inspection (as one can gain from the tower along the Panamericana) that is lost in a distant view from the air, just as much is revealed in a distant inspection from the air that is close in the close-up view from near the ground. This is a perfect concrete illustration of what I was recently writing about in relation to the distinction between constructive and non-constructive thought (in P or not-P). In this post (on my other blog) I employed an image taken from Alain Connes to illustrate the constructive/non-constructive distinction such that the constructive perspective is like that of a mountain climber while the non-constructive perspective is like that of a visitor who flies over the summit of a mountain laboriously climbed by the other.
Any thorough investigation will want to make use of both perspectives in order to obtain the most comprehensive perspective possible — even though each perspective has its blind spots and its shadows that compromise our perspective on the whole. Indeed, it is precisely because each perspective incorporates deficits specific to the perspective that one will want to supplement any one perspective without another perspective with a different set of specific deficits. Between two or more fundamentally different perspectives on any one state-of-affairs there is the possibility of constructing the comprehensive conception that is excluded by any one perspective in isolation.
The two perspectives offered on the Nazca lines by the tower and an airplane flyover also reminded me of a point that I imperfectly attempted to make in my post on Epistemic Orders of Magnitude, in which I employed aerial photographs of cities in order to demonstrate the similar structures of cities transformed in the imagine of industrial-technological civilization. This similarity in structure may be masked by one’s experience of an urban area from the perspective of passing through the built environment on a human scale — i.e., simply walking through a city, which is how most people experience an urban area.
Now, in light of what I have subsequently written about constructivism, I might say that our experience of a built environment is intrinsically constructive, except for that of the urban planner or urban designer, who must see (or attempt to see) things whole. However, the urban planner must also inform his or her work with the street-level “constructive” perspective or the planning made exclusively from a top-down perspective is likely to be a failure. Almost all of the most spectacular failures in urban design have come about from an attempt to impose, from the top down, a certain vision and a certain order which may be at odds with the organically emergent order that rises from the bottom up.
This reflection gives us yet another perspective on utopianism, which I have many times tried to characterize in my attempts to show the near (not absolute) historical inevitability of utopian schemes transforming themselves upon their attempted implementation into dystopian nightmares — the utopian planner attempts to design from a purely non-constructive perspective without the benefit of a constructive perspective. This dooms the utopian plans to inevitable blindspots, shadows, and deficits. The oversights of a single perspective then, in the fullness of time, create the conditions for cascading catastrophic failure.
Historically speaking, it is not difficult to see how this comes about. After the astonishing planned cities of early antiquity, many from prehistoric societies that have left us little record except for their admirably regular and disciplined town plans, Europeans turned to a piecemeal, organic approach to urbanism. Once this approach was rapidly outgrown when cities began their burgeoning growth with the advent of the Industrial Revolution, it was a natural response on the part of Haussman-esque planners to view organic urbanism as a “failure” that necessitated replacement by another model that envisioned the already-built environment as a tabula rasa to be re-built according to rational standards. Cities henceforth were to be wholly planned to address to inadequacies of the medieval pattern of non-planning, which could not cope with cities with populations that now numbered in the millions.
I have observed elsewhere (in my Political Economy of Globalization) that many ancient prehistoric societies were essentially utopian constructions over which a god-king presided as a living god, present in the flesh among his people, and indeed some of the most striking examples of ancient town planning date from societies that exhibited (or seem to have exhibited) this now-vanished form of order. For only where a god-king is openly acknowledged as such can a social order based upon living and present divinity within the said social order be possible.
Nazca, however, does not seem to have been based on this social plan of a divinely-sanctioned social order which can bring utopian (and therefore likely non-constructive, top-down) planning into actual practice because of the physical presence of the god in the midst of his people. The book that I cited yesterday, The Ancient Civilizations of Peru by J. Alden Mason, has this to say of Nazca society:
“…the general picture seems to be one of a sedentary democratic people without marked class distinctions or authoritarianism, possibly without an established religion. There is less difference in the ‘richness’ or poverty of the graves, and women seem to be on an equality with men in this respect. The apparent absence of great public works, of extensive engineering features, and of temple pyramids implies a lack of authoritarian leadership. Instead, the leisure time of the people seems to have been spent in individual production, especially in the making of quantities of perfect, exquisite textiles and pottery vessels. This seems to indicate a strong cult of ancestor-worship. Cloths on which an incredible amount of labor was spent were made especially for funerary offerings and interred with the dead. The orientation seems to have been towards individualized religion rather than towards community participation, dictation, coercion, and aggression.”
J. Alden Mason, The Ancient Civilizations of Peru, Penguin Books, 1968, p. 85
Such egalitarian societies focused on the satisfaction of consumer demands were rare in the ancient world, but we should not be surprised that it was an egalitarian society, organized constructively from the bottom up, that produced the astonishing lines in the desert of the Nazca. Without an aerial perspective, the making of these lines was a thoroughly constructivistic undertaking, not even counter-balanced by a non-constructive perspective, which has only been obtained long after the Nazca civilization has disappeared, leaving only traces of itself in the dessicated sands of the desert.
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While I am posting this a couple of days after the fact, this entire account was written in longhand on the day here described.
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27 December 2011
Yesterday in The Philosophy of Fear I quoted Descartes from his Discourse on Method, from the section in which he introduces an implicit distinction between the theoretical principles he will use to guide his philosophical activities and the practical moral principles that he will employ in his life while he is going about his theoretical activity. Here is his exposition of his four theoretical principles:
● The first was never to accept anything for true which I did not clearly know to be such; that is to say, carefully to avoid precipitancy and prejudice, and to comprise nothing more in my judgement than what was presented to my mind so clearly and distinctly as to exclude all ground of doubt.
● The second, to divide each of the difficulties under examination into as many parts as possible, and as might be necessary for its adequate solution.
● The third, to conduct my thoughts in such order that, by commencing with objects the simplest and easiest to know, I might ascend by little and little, and, as it were, step by step, to the knowledge of the more complex; assigning in thought a certain order even to those objects which in their own nature do not stand in a relation of antecedence and sequence.
● And the last, in every case to make enumerations so complete, and reviews so general, that I might be assured that nothing was omitted.
Anyone who knows Descartes’ works will recognize that he has here stated, much more simply and compactly, the principles that he was working on in his unfinished manuscript Rules of the Direction of Mind. Here, by way of contrast, is a highly condensed version of Descartes’ practical and provisional moral principles:
● The first was to obey the laws and customs of my country, adhering firmly to the faith in which, by the grace of God, I had been educated from my childhood and regulating my conduct in every other matter according to the most moderate opinions, and the farthest removed from extremes, which should happen to be adopted in practice with general consent of the most judicious of those among whom I might be living.
● My second maxim was to be as firm and resolute in my actions as I was able, and not to adhere less steadfastly to the most doubtful opinions, when once adopted, than if they had been highly certain; imitating in this the example of travelers who, when they have lost their way in a forest, ought not to wander from side to side, far less remain in one place, but proceed constantly towards the same side in as straight a line as possible, without changing their direction for slight reasons, although perhaps it might be chance alone which at first determined the selection; for in this way, if they do not exactly reach the point they desire, they will come at least in the end to some place that will probably be preferable to the middle of a forest.
● My third maxim was to endeavor always to conquer myself rather than fortune, and change my desires rather than the order of the world, and in general, accustom myself to the persuasion that, except our own thoughts, there is nothing absolutely in our power; so that when we have done our best in things external to us, all wherein we fail of success is to be held, as regards us, absolutely impossible: and this single principle seemed to me sufficient to prevent me from desiring for the future anything which I could not obtain, and thus render me contented…
Descartes wrote a lot a extremely long run-on sentences, so that one must cut radically in order to quote him (except for his theoretical principles, above, which I have quoted entire), but I have tried to include enough above to give a genuine flavor of how he expressed himself. Although Descartes did not himself make this distinction between theoretical and practical principles explicit, although the distinction is explicitly embodied in his two sets of explicitly stated principles, he does provide a justification for the distinction:
“…as it is not enough, before commencing to rebuild the house in which we live, that it be pulled down, and materials and builders provided, or that we engage in the work ourselves, according to a plan which we have beforehand carefully drawn out, but as it is likewise necessary that we be furnished with some other house in which we may live commodiously during the operations, so that I might not remain irresolute in my actions, while my reason compelled me to suspend my judgement, and that I might not be prevented from living thenceforward in the greatest possible felicity, I formed a provisory code of morals, composed of three or four maxims, with which I am desirous to make you acquainted.”
After I quoted this in The Philosophy of Fear I realized that it constitutes a perfect antithesis to the conception of the rational reconstruction of knowledge embodied in the image of Neurath’s ship, which I have quoted several times.
Rational reconstruction was an idea that fascinated early twentieth century philosophers, especially the logical positivists, whose philosophical tradition would eventually mature and transform itself into mainstream analytical philosophy. It was logical positivism that gave us an enduring image of rational reconstruction, as related by Otto Neurath:
“There is no way of taking conclusively established pure protocol sentences as the starting point of the sciences. No tabula rasa exists. We are like sailors who must rebuild their ship on the open sea, never able to dismantle it in dry-dock and to reconstruct it there out of the best materials. Only the metaphysical elements can be allowed to vanish without trace.”
Quine then used this image in his Word and Object:
“We are like sailors who on the open sea must reconstruct their ship but are never able to start afresh from the bottom. Where a beam is taken away a new one must at once be put there, and for this the rest of the ship is used as support. In this way, by using the old beams and driftwood the ship can be shaped entirely anew, but only by gradual reconstruction.”
These two epistemic paradigms — what I will call Descartes’ house and Neurath’s ship — represent antithetical conceptions of the epistemological enterprise. Neurath’s ship is usually presented as an anti-foundationalist parable, which would suggest that Descartes’ house is a foundationalist parable. There are certain problems with this initial characterization. The logical positivists who invoked Neurath’s ship with approval were often foundationalists in the philosophy of mathematics while being anti-foundational in other areas.
There is a sense in which it is fair to call Descartes’ house a foundationalist parable: Descartes is suggesting a radical approach to the foundations of knowledge — utterly tearing down our knowledge in order to construct entirely anew on the same ground — and he attempted to put this into practice in his own philosophical work. He doubted everything that he could until he arrived at the fact that he could not doubt his own existence, and then on the basis of the certainty of his own existence he attempted to reconstruct the entire edifice of knowledge. The result was not radical, but actually rather conventional, but the method certainly was radical. It was also total.
Whether or not Neurath’s ship is anti-foundational, it is certainly incrementalist. If we were to attempt to rebuild a ship while at sea, we would need to proceed bit by bit, and very carefully. Nothing radical would be attempted, for to attempt anything radical would be to sink the ship. There is a sense in which we could identify this effort as essentially constructivist in spirit, though not exclusively constructivist: constructivism is certainly not the only motivation for Neurath’s ship, and many who invoked it employed non-constructive modes of reasoning.
Are Descartes’ house and Neurath’s ship mutually exclusive? Not necessarily. We do remodel houses while living in them, although when we do we need to keep some basic functions available during our residency. And we can demolish certain parts of a ship at sea; as long as the hull remains intact, we can engage in a radical reconstruction (as opposed to a rational reconstruction) of the masts and the rigging.
One ought not to push an image too far, for fear of verging on the ludicrous, but it can be observed that, while living in a house, we can tear down half of it to the ground and rebuild that half from scratch while living in the other half, and then repeat this process in the half we have been living in. In fact, I know people who have done this. There will, of course, be certain compromises that will have to be made in wedding the two halves together, so that the seam between the two has the incrementalist character of Neurath’s ship, while each half has the radical and total character of Descartes’ house.
It is difficult to imagine a parallel for the above scenario when it comes to Neurath’s ship. The hull of the ship can only be rebuilt incrementally, although almost everything else can be radically reconstructed. And it may well be that some parts of epistemology must be approached incrementally while other parts of epistemology may be radically reconstructed almost with impunity. This seems like an eminently reasonable conclusion. But it is no conclusion — at least not yet — because there is more to say.
What underlies the image of Descartes’ house and Neurath’s ship is in each case a distinct metaphor, and that metaphor is for Descartes the earth, the solid ground upon which we stand, while for Neurath it is the sea, to which we must go down in ships, and where we cannot stand but must swim or be carried. So, we have two epistemic metaphors — of what are they metaphors? Existence? Being? Human experience? Knowledge? If the house or the ship is knowledge, then the ground or the sea must be that upon which knowledge rests (or floats). This once again suggests a foundationalist approach, but points to very different foundations: a house stands on dirt and stones; a ship floats on water.
Does knowledge ultimately rest upon the things themselves — the world, existence, or being, as you prefer — or upon human experience of the world? Or is not knowledge a consequence of the tension between human experience and the world, so that both the world and human experience are necessary to knowledge?
Intuitively, and without initially putting much thought into this (although I will continue to think about this because it is an interesting idea), I would suggest that the metaphor of the earth implies that knowledge ultimately is founded on the things themselves, while the metaphor of the sea implies that knowledge ultimately is founded on the ever-changing tides of human experience.
Therefore, if knowledge requires both the world and human experience, either the metaphor of Descartes’ house or Neurath’s ship alone, in isolation from the other, is inadequate. We need something more, or something different, to illustrate our relation to knowledge and how it changes.
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14 August 2011
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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28 January 2011
Further to my recent posts on fractals and the Banach-Tarski Paradox (A Question for Philosophically Inclined Mathematicians, Fractals and the Banach-Tarski Paradox, A visceral feeling for epsilon zero, and Adventures in Geometrical Intuition), I realized how the permutations of formal methodology can be schematically delineated in regard to the finitude or infinitude of the number of iterations and the methods of iteration.
The Banach-Tarski Paradox involves a finite number of steps, but for the Banach-Tarski paradox to work the sphere in question must be infinitely divisible, and in fact we must treat the sphere like a set of points with the cardinal of the continuum. Each step in Banach-Tarski extraction is infinitely complex because it must account for an infinite set of points, but the number of steps required to complete the extraction are finite. This is schematically the antithesis of a fractal, which latter involves an infinite number of steps, but each step of the construction of the fractal is finite. Thus we can see for ourselves the first few iterations of a fractal, and we can use computers to run fractals through very large (though still finite) numbers of iterations. A fractal only becomes infinitely complex and infinitely precise when it is infinitely iterated; before it reaches its limit, it is finite in every respect. This is one reason fractals have such a strong hold on mathematical intuition.
A sphere decomposed according to the Banach-Tarski method is assumed to be mathematically decomposable into an infinitude of points, and therefore it is infinitely precise at the beginning of the extraction. The Banach-Tarski Paradox begins with the presumption of classical continuity and infinite mathematical precision, as instantiated in the real number system, since the sphere decomposed and reassembled is essentially equivalent to the real number system. There is a sense, then, in which the Banach-Tarski extraction is platonistic and non-constructive, while fractals are constructivistic. This is interesting, but we will not pursue this any further except to note once again that computing is essentially constructivistic, and no computer can function non-constructively, which implies that fractals are exactly what Benoît Mandelbrot said that they were not: an artifact of computing. However, the mathematical purity of fractals can be restored to its honor by an extrapolation of fractals into non-constructive territory, and this is exactly what an infinite fractal is, i.e., a fractal each step of the iteration of which is infinite.
Once we see the schematic distinction between the finite operation and infinite iteration of fractals in contradistinction to the infinite operation and finite iteration of the Banach-Tarski extraction, two other possibilities defined by the same schematism appear: finite operation with finite iteration, and infinite operation with infinite iteration. The former — finite operation with finite iteration — is all of finite mathematics: finite operations that never proceed beyond finite iterations. All of the mathematics you learned in primary school is like this. Contemporary mathematicians sometimes call this primitive recursive arithmetic (PRA). The latter — infinite operation with infinite iteration — is what I recently suggested in A visceral feeling for epsilon zero: if we extract an infinite number of spheres by the Banach-Tarski method an infinite number of times, we essentially have an infinite fractal in which each step is infinite and the iteration is infinite.
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Fractals and Geometrical Intuition
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