22 February 2013
Much of what I write here, whether commenting on current affairs to delving into the depths of prehistory, could be classed under the general rubric of philosophy of history. One of my early posts to this forum was Of What Use is Philosophy of History in Our Time? (An echo of the title of Hans Meyerhoff’s widely available anthology Philosophy of History in Our Time.) It could be argued that my subsequent posts have been attempts to answer this question (that is to say, to answer the question what is the use of philosophy of history in our time), to demonstrate the usefulness of bringing a philosophical perspective to history, contemporary and otherwise. The reader is left to judge whether this attempt has been a success (partial or otherwise) or a failure (partial or otherwise).
In several recent posts — as, for example in The Science of Time, Addendum on Big History as the Science of Time, and Human Agency and the Exaptation of Selection, inter alia — I have been writing a lot about the philosophy of history from the perspective of big history, which is a contemporary historiographical school that comes to history from the perspective of the big picture and primarily proceeds according to scientific naturalism. This latter condition makes of big history a particular species of naturalism.
In many posts to this forum I have emphasized my own naturalistic perspective both in philosophy generally speaking as well as more specifically in the philosophy of history. For example, in posts such as Natural History and Human History, The Continuity of Civilization and Natural History, and An Existentialist Philosophy of History, I have emphasized the continuity of human history and natural history, especially making the attempt to place civilization in a natural historical context.
This emphasis on big history and naturalism has meant that I have spent very little time writing about alternatives to naturalistic historical thought — with a certain exception, which the reader may well not immediately recognize, so I will point it out explicitly. In several posts — The Ethos of Formal Thought, Foucault’s Formalism, Cartesian Formalism, and Formal Strategy and Philosophical Logic: Work in Progress among them — I have discussed the possibility of formal thought in relation to historical understanding, i.e., topics not usually discussed from a formal perspective (which is usually confined to logic, mathematics, and some branches of science). Formalism represents a certain kind of countervailing intellectual influence to naturalism, and it has probably served roughly that function in my thought.
I have previously mentioned Darren Staloff’s lectures on the philosophy of history, The Search for a Meaningful Past: Philosophies, Theories and Interpretations of Human History. One of the motifs running through Staloff’s lectures is a contrast between what he calls naturalism and idealism. He sums up this motif in the final lecture, in which he adopts the perspectives of naturalism and idealism in turn, trying give the listener a sense of the claims of each tradition. I found Staloff’s exposition of idealism less persuasive that his exposition of naturalism, and so I found the motif of a contrast between naturalism and idealism a bit strained, since it seemed to me that idealism really couldn’t carry its own weight in the way that it might have been able to in the past.
Recently I’ve encountered an approach to the philosophy of history that could be called “idealist” (at least in a certain sense), and this is much more persuasive to me that Staloff’s analytical representatives of the idealist tradition, like R. G. Collingwood. I have found this idealist perspective in the work of Ludwig Landgrebe, who was one of Husserl’s research assistants.
The casual reader of this blog might well have picked up on the amount of contemporary continental philosophy that I have read, but it unlikely to have realized the extent to which Edmund Husserl and phenomenology have been an influence on my thought. Nevertheless, that influence has been profound, to the point that many of Husserl’s expositors and commentators have also influenced my thinking. Recently I have been reading some essays by Ludwig Landgrebe, and this has started to give me another perspective on the philosophy of history.
Landgrebe wrote at least two papers on the philosophy of history, as well as one chapter of his book, Major Problems in Contemporary European Philosophy, from Dilthey to Heidegger. No doubt there is more material, but this is what I have found translated into English. (Landgrebe wrote an entire book on the phenomenological philosophy of history, Phänomenologie und Geschichte, but this has not been translated into English.) The two papers are “Phenomenology as Transcendental Theory of History” (which can be found in the collection of essays Husserl: Expositions and Appraisals, edited by Elliston and McCormick, University of Notre Dame Press, 1977. pp. 101-113) and “A Meditation on Husserl’s Statement: ‘History is the grand fact of absolute Being’” (The Southwestern Journal of Philosophy, Vol. 5, Issue 3, Fall 1974, pp. 111-125).
It is well known that Husserl’s last work, The Crisis of European Sciences and Transcendental Phenomenology: An Introduction to Phenomenological Philosophy, assembled posthumously from his papers, is the work in which Husserl placed phenomenology in historical context (for all practical purposes, for the first time), and considered the emergence of Western scientific thought in historical context. As such, this has been the point of departure of much historically-oriented phenomenological research, and the Crisis (as it has come to be known) and its supplementary texts were clearly influential for Landgrebe.
Landgrebe, however, as Husserl’s research assistant, was more than conversant with Husserl’s logical thought also. Husserl’s Experience and Judgment: Investigations in a Genealogy of Logic was a text assembled by Landgrebe from Husserl’s notes. Landgrebe consulted with Husserl throughout this project, and the original texts are all due to Husserl, but the structure of the book is entirely Landgrebe’s doing. Landgrebe brings the kind of rigor one learns in studying logic to his very compact essays on the philosophy of history. In this way, Landgrebe’s formulations have a formal character that makes them very congenial to me. Landgrebe’s approach is essentially that of a formal phenomenological theory of history, and this perspective allows me to assimilate Landgrebe’s insights both to idealistic historiography as well as my long-standing interest in formal thought.
If I were now to revise my speculative syllabus If I Lectured on the Philosophy of History (lecture 13 of which I had already assigned to phenomenology), I would definitely showcase Landgrebe’s philosophy of history as the most sophisticated phenomenological contribution to the philosophy of history.
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23 November 2012
What is the Church-Turing Thesis? The Church-Turing Thesis is an idea from theoretical computer science that emerged from research in the foundations of logic and mathematics, also called Church’s Thesis, Church’s Conjecture, the Church-Turing Conjecture as well as other names, that ultimately bears upon what can be computed, and thus, by extension, what a computer can do (and what a computer cannot do).
Note: For clarity’s sake, I ought to point out the Church’s Thesis and Church’s Theorem are distinct. Church’s Theorem is an established theorem of mathematical logic, proved by Alonzo Church in 1936, that there is no decision procedure for logic (i.e., there is no method for determining whether an arbitrary formula in first order logic is a theorem). But the two – Church’s theorem and Church’s thesis – are related: both follow from the exploration of the possibilities and limitations of formal systems and the attempt to define these in a rigorous way.
Even to state Church’s Thesis is controversial. There are many formulations, and many of these alternative formulations come straight from Church and Turing themselves, who framed the idea differently in different contexts. Also, when you hear computer science types discuss the Church-Turing thesis you might think that it is something like an engineering problem, but it is essentially a philosophical idea. What the Church-Turing thesis is not is as important as what it is: it is not a theorem of mathematical logic, it is not a law of nature, and it not a limit of engineering. We could say that it is a principle, because the word “principle” is ambiguous and thus covers the various formulations of the thesis.
There is an article on the Church-Turing Thesis at the Stanford Encyclopedia of Philosophy, one at Wikipedia (of course), and even a website dedicated to a critique of the Stanford article, Alan Turing in the Stanford Encyclopedia of Philosophy. All of these are valuable resources on the Church-Turing Thesis, and well worth reading to gain some orientation.
One way to formulate Church’s Thesis is that all effectively computable functions are general recursive. Both “effectively computable functions” and “general recursive” are technical terms, but there is an important different between these technical terms: “effectively computable” is an intuitive conception, whereas “general recursive” is a formal conception. Thus one way to understand Church’s Thesis is that it asserts the identity of a formal idea and an informal idea.
One of the reasons that there are many alternative formulations of the Church-Turing thesis is that there any several formally equivalent formulations of recursiveness: recursive functions, Turing computable functions, Post computable functions, representable functions, lambda-definable functions, and Markov normal algorithms among them. All of these are formal conceptions that can be rigorously defined. For the other term that constitutes the identity that is Church’s thesis, there are also several alternative formulations of effectively computable functions, and these include other intuitive notions like that of an algorithm or a procedure that can be implemented mechanically.
These may seem like recondite matters with little or no relationship to ordinary human experience, but I am surprised how often I find the same theoretical conflict played out in the most ordinary and familiar contexts. The dialectic of the formal and the informal (i.e., the intuitive) is much more central to human experience than is generally recognized. For example, the conflict between intuitively apprehended free will and apparently scientifically unimpeachable determinism is a conflict between an intuitive and a formal conception that both seem to characterize human life. Compatibilist accounts of determinism and free will may be considered the “Church’s thesis” of human action, asserting the identity of the two.
It should be understood here that when I discuss intuition in this context I am talking about the kind of mathematical intuition I discussed in Adventures in Geometrical Intuition, although the idea of mathematical intuition can be understood as perhaps the narrowest formulation of that intuition that is the polar concept standing in opposition to formalism. Kant made a useful distinction between sensory intuition and intellectual intuition that helps to clarify what is intended here, since the very idea of intuition in the Kantian sense has become lost in recent thought. Once we think of intuition as something given to us in the same way that sensory intuition is given to us, only without the mediation of the senses, we come closer to the operative idea of intuition as it is employed in mathematics.
Mathematical thought, and formal accounts of experience generally speaking, of course, seek to capture our intuitions, but this formal capture of the intuitive is itself an intuitive and essentially creative process even when it culminates in the formulation of a formal system that is essentially inaccessible to intuition (at least in parts of that formal system). What this means is that intuition can know itself, and know itself to be an intuitive grasp of some truth, but formality can only know itself as formality and cannot cross over the intuitive-formal divide in order to grasp the intuitive even when it captures intuition in an intuitively satisfying way. We cannot even understand the idea of an intuitively satisfying formalization without an intuitive grasp of all the relevant elements. As Spinoza said that the true is the criterion both of itself and of the false, we can say that the intuitive is the criterion both of itself and the formal. (And given that, today, truth is primarily understood formally, this is a significant claim to make.)
The above observation can be formulated as a general principle such that the intuitive can grasp all of the intuitive and a portion of the formal, whereas the formal can grasp only itself. I will refer to this as the principle of the asymmetry of intuition. We can see this principle operative both in the Church-Turing Thesis and in popular accounts of Gödel’s theorem. We are all familiar with popular and intuitive accounts of Gödel’s theorem (since the formal accounts are so difficult), and it is not usual to make claims for the limitative theorems that go far beyond what they formally demonstrate.
All of this holds also for the attempt to translate traditional philosophical concepts into scientific terms — the most obvious example being free will, supposedly accounted for by physics, biochemistry, and neurobiology. But if one makes the claim that consciousness is nothing but such-and-such physical phenomenon, it is impossible to cash out this claim in any robust way. The science is quantifiable and formalizable, but our concepts of mind, consciousness, and free will remain stubbornly intuitive and have not been satisfyingly captured in any formalism — the determination of any such satisfying formalization could only be determined by intuition and therefore eludes any formal capture. To “prove” determinism, then, would be as incoherent as “proving” Church’s Thesis in any robust sense.
There certainly are interesting philosophical arguments on both sides of Church’s Thesis — that is to say, both its denial and its affirmation — but these are arguments that appeal to our intuitions and, most crucially, our idea of ourselves is intuitive and informal. I should like to go further and to assert that the idea of the self must be intuitive and cannot be otherwise, but I am not fully confident that this is the case. Human nature can change, albeit slowly, along with the human condition, and we could, over time — and especially under the selective pressures of industrial-technological civilization — shape ourselves after the model of a formal conception of the self. (In fact, I think it very likely that this is happening.)
I cannot even say — I would not know where to begin — what would constitute a formal self-understanding of the self, much less any kind of understanding of a formal self. Well, maybe not. I have written elsewhere that the doctrine of the punctiform present (not very popular among philosophers these days, I might add) is a formal doctrine of time, and in so far as we identify internal time consciousness with the punctiform present we have a formal doctrine of the self.
While the above account is one to which I am sympathetic, this kind of formal concept — I mean the punctiform present as a formal conception of time — is very different from the kind of formality we find in physics, biochemistry, and neuroscience. We might assimilate it to some mathematical formalism, but this is an abstraction made concrete in subjective human experience, not in physical science. Perhaps this partly explains the fashionable anti-philosophy that I have written about.
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9 April 2012
Geopolitics and Geostrategy
as a formal sciences
In a couple of posts — Formal Strategy and Philosophical Logic: Work in Progress and Axioms and Postulates of Strategy — I have explicitly discussed the possibility of a formal approach to strategy. This has been a consistent theme of my writing over the past three years, even when it is not made explicit. The posts that I wrote on theoretical geopolitics can also be considered an effort in the direction of formal strategy.
There is a sense in which formal thought is antithetical to the tradition of geopolitics, which latter seeks to immerse itself in the empirical facts of how history gets made, in contradistinction to the formalist’s desire to define, categorize, and clarify the concepts employed in analysis. Yet in so far as geopolitics takes the actual topographical structure of the land as its point of analytical departure, this physical structure becomes the form upon which the geopolitician constructs the logic of his or her analysis. Geopolitical thought is formal in so far as the forms to which it conforms itself are physical, topographical forms.
Most geopoliticians, however, have no inkling of the formal dimension of their analyses, and so this formal dimension remains implicit. I have commented elsewhere that one of the most common fallacies is the conflation of the formal and the informal. In Cartesian Formalism I wrote:
One of the biggest and yet one of the least recognized blunders in philosophy (and certainly not only in philosophy) is to conflate the formal and the informal, whether we are concerned with formal and informal objects, formal and informal methods, or formal and informal ideas, etc. (I recently treated this topic on my other blog in relation to the conflation of formal and informal strategy.)
Geopolitics, geostrategy, and in fact many of the so-called “soft” sciences that do not involve extensive mathematization are among the worst offenders when it comes to the conflation of the formal and the informal, often because the practitioners of the “soft” sciences do not themselves understand the implicit principles of form to which they appeal in their theories. Instead of theoretical formalisms we get informal narratives, many of which are compelling in terms of their human interest, but are lacking when it comes to analytical clarity. These narratives are primarily derived from historical studies within the discipline, so that when this method is followed in geopolitics we get a more-or-less quantified account of topographical forms that shape action and agency, with an overlay of narrative history to string together the meaning of names, dates, and places.
There is a sense in which geography and history cannot be separated, but there is another sense in which the two are separated. Because the ecological temporality of human agency is primarily operational at the levels of micro-temporality and meso-temporality, this agency is often exercised without reference to the historical scales of the exo-temporality of larger social institutions (like societies and civilizations) and the macro-historical scales of geology and geomorphology. That is to say, human beings usually act without reference to plate tectonics, the uplift of mountains, or seafloor spreading, except when these events act over micro- and meso-time scales as in the case of earthquakes and tsunamis generated by geological events that otherwise act so slowly that we never notice them in the course of a lifetime — or even in the course of the life of a civilization.
The greatest temporal disconnect occurs between the smallest scales (micro-temporality) and the largest scales (macro-temporality), while there is less disconnect across immediately adjacent divisions of ecological temporality. I can employ a distinction that I recently made in a discussion of Descartes, that between strong distinctions and weak distinctions (cf. Of Distinctions Weak and Strong). Immediately adjacent divisions of ecological temporality are weakly distinct, while those not immediately adjacent are strongly distinct.
We have traditionally recognized the abstraction of macroscopic history that does not descend into details, but it has not been customary to recognize the abstractness of microscopic history, immersed in details, that does not also place these events in relation to a macroscopic context. In order to attain to a comprehensive perspective that can place these more limited perspectives into a coherent context, it is important to understand the limitations of our conventional conceptions of history (such as the failure to understand the abstract character of micro-history) — and, for that matter, the limitations of our conventional conceptions of geography. One of these limitations is the abstractness of either geography or history taken in isolation.
The degree of abstractness of an inquiry can be quantified by the ecological scope of that inquiry; any one division of ecological temporality (or any one division of metaphysical ecology) taken in isolation from other divisions is abstract. It is only the whole of ecology taken together that a truly concrete theory is possible. To take into account the whole of ecological temporality in a study of history is a highly concrete undertaking which is nevertheless informed by the abstract theories that constitute each individual level of ecological temporality.
Geopolitics, despite its focus on the empirical conditions of history, is a highly abstract inquiry precisely because of its nearly-exclusive focus on one kind of structure as determinative in history. As I have argued elsewhere, and repeatedly, abstract theories are valuable and have their place. Given the complexity of a concrete theory that seeks to comprehend the movements of human history around the globe, an abstract theory is a necessary condition of any understanding. Nevertheless, we need to rest in our efforts with an abstract theory based exclusively in the material conditions of history, which is the perspective of geopolitics (and, incidentally, the perspective of Marxism).
Geopolitics focuses on the seemingly obvious influences on history following from the material conditions of geography, but the “obvious” can be misleading, and it is often just as important to see what is not obvious as to explicitly take into account what is obvious. Bertrand Russell once observed, in a passage both witty and wise, that:
“It is not easy for the lay mind to realise the importance of symbolism in discussing the foundations of mathematics, and the explanation may perhaps seem strangely paradoxical. 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. For instance, the whole of Arithmetic and Algebra has been shown to require three indefinable notions and five indemonstrable propositions. But without a symbolism it would have been very hard to find this out. It is so obvious that two and two are four, that we can hardly make ourselves sufficiently sceptical to doubt whether it can be proved. And the same holds in other cases where self-evident things are to be proved.”
Bertrand Russell, Mysticism and Logic, “Mathematics and the Metaphysicians”
Russell here expresses himself in terms of symbolism, but I think it would better to formulate this in terms of formalism. When Russell writes that, “we invent some new and difficult symbolism, in which nothing seems obvious,” the new and difficult symbolism he mentions is more than mere symbolism, it is a formal theory. Russell’s point, then, is that if we formalize a body of knowledge heretofore consisting of intuitively “obvious” truths, certain relationships between truths become obvious that were not obvious prior to formalization. Another way to formulate this is to say that formalization constitutes a shift in our intuition, so that truths once intuitively obvious become inobvious, while inobvious truths because intuitive. Thus formalization is the making intuitive of previously unintuitive (or even counter-intuitive) truths.
Russell devoted a substantial portion of his career to formalizing heretofore informal bodies of knowledge, and therefore had considerable experience with the process of formalization. Since Russell practiced formalization without often explaining exactly what he was doing (the passage quoted above is a rare exception), we must look to the example of his formal thought as a model, since Russell himself offered no systematic account of the formalization of any given body of knowledge. (Russell and Whitehead’s Principia Mathematica is a tour de force comprising the order of justification of its propositions, while remaining silent about the order of discovery.)
A formal theory of time would have the same advantages for time as the theoretical virtues that Russell identified in the formalization of mathematics. In fact, Russell himself formulated a formal theory of time, in his paper “On Order in Time,” which is, in Russell’s characteristic way, reductionist and over-simplified. Since I aim to formulate a theory of time that is explicitly and consciously non-reductionist, I will make no use of Russell’s formal theory of time, though it is interesting at least to note Russell’s effort. The theory of ecological temporality that I have been formulating here is a fragment of a full formal theory of time, and as such it can offer certain insights into time that are lost in a reductionist account (as in Russell) or hidden in an informal account (as in geography and history).
As noted above, a formalized theory brings about a shift in our intuition, so that the formerly intuitive becomes unintuitive while the formerly unintuitive becomes intuitive. A shift in our intuitions about time (and history) means that a formal theory of time makes intuitive temporal relationships less obvious, while making temporal relationships that are hidden by the “buzzing, blooming world” more obvious, and therefore more amenable to analysis — perhaps for the first time.
Ecological temporality gives us a framework in which we can demonstrate the interconnectedness of strongly distinct temporalities, since the panarchy the holds between levels of an ecological system is the presumption that each level of an ecosystem impacts every other level of an ecosystem. Given the distinction between strong distinctions and weak distinctions, it would seem that adjacent ecological levels are weakly distinct and therefore have a greater impact on each other, while non-adjacent ecological levels are strongly distinct and therefore have less of an impact on each other. In an ecological theory of time, all of these principles hold in parallel, so that, for example, micro-temporality is only weakly distinct from meso-temporality, while being strongly distinct from exo-temporality. As a consequence, a disturbance in micro-temporality has a greater impact upon meso-temporality than upon exo-temporality (and vice versa), but less of an impact does not mean no impact at all.
Another virtue of formal theories, in addition to the shift in intuition that Russell identified, is that it forces us to be explicit about our assumptions and presuppositions. The implicit theory of time held by a geostrategist matters, because that geostrategist will interpret history in terms of the categories of his or her theory of time. But most geostrategists never bother to make their theory of time explicit, so that we do not know what assumptions they are making about the structure of time, hence also the structure of history.
Sometimes, in some cases, these assumptions will become so obvious that they cannot be ignored. This is especially the case with supernaturalistic and soteriological conceptions of metaphysical history that ultimately touch on everything else that an individual believes. This very obviousness makes it possible to easily identify eschatological and theological bias; what is much more insidious is the subtle assumption that is difficult to discern and which only can be elucidated with great effort.
If one comes to one’s analytical work presupposing that every moment of time possesses absolute novelty, one will likely make very different judgments than if one comes to the same work presupposing that there is nothing new under the sun. Temporal novelty means historical novelty: anything can happen; whereas, on the contrary, the essential identity of temporality over historical scales — identity for all practical purposes — means historical repetition: very little can happen.
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Note: Anglo-American political science implicitly takes geopolitics as its point of departure, but, as I have attempted to demonstrate in several posts, this tradition of mainstream geopolitics can be contrasted to a nascent movement of biopolitics. However, biopolitics too could be formulated in the manner of a theoretical biopolitics, and a theoretical biopolitics would be at risk of being as abstract as geopolitics and in need of supplementation by a more comprehensive ecological perspective.
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17 October 2010
Famed French mathematician Benoît Mandelbrot has passed away a few days ago at the age of 85. When someone dies who has lived such a productive life, it would sound a little odd to say that we have “lost” him. We haven’t lost Benoît Mandelbrot. His contribution is permanent. The Mandelbrot set will take its place in history beside Euclidean geometry and Cantorian transfinite numbers. And when I say “history” you might think that I am consigning it to the dead past, but what I mean by “history” is an ongoing tradition of which we are a part, and which spills out into the future, informing lives yet to be lived in unpredictable and unprecedented ways.
Mandelbrot will be best remembered for having invented (discovered? formulated?) fractals and fractal geometry. By an odd coincidence, I have been thinking quite a bit about fractals lately. A few days ago I wrote The Fractal Structure of Exponential Growth, and I had recently obtained from the library the NOVA documentary Fractals: Hunting the Hidden Dimension. As with many NOVA documentaries, I have watched this through repeatedly to try to get all that I can out of it, much as I typically listen through recorded books multiple times.
There are a couple of intuitive definitions of fractals to which I often refer when I think about them. You can say that a fractal is an object that retains its properties under magnification, or you can say that a fractal is an object that possesses self-similarity across orders of magnitude. But while these definitions informally capture some important properties of fractals, the really intuitive aspect of fractals is their astonishing appearance. Mandelbrot was unapologetically interested in the appearance of fractals. In the NOVA documentary he says in an interview, “I don’t play with formulas, I play with pictures. And that is what I’ve been doing all my life.”
It is often said that mathematicians are platonists during the week and formalists on the weekend. In other words, while actively working with mathematics they feel themselves to be engaged with objects as real as themselves, but when engaged in philosophical banter over the weekend, the mathematician defends the idea of mathematics as a purely formal activity. This formulation reduced mathematical formalism to a mere rhetorical device. But if formalism is the mathematician’s rhetoric while platonism is the philosophy that he lives by in the day-to-day practice of his work, the rhetorical flourish of formalism has proved to be decisive in the direction that mathematics has taken. Perhaps we could say that mathematicians are strategic formalists and tactical platonists. In this case, we can see how the mathematician’s grand strategy of formalism has shaped the discipline.
It was in the name of such formalism that “geometrical intuition” began to be seriously questioned at the turn of the nineteenth to the twentieth century, and several generations of mathematicians pursued the rigorization of analysis by way of taking arithmetization as a research program. (Gödel’s limitative theorems were an outgrowth of this arithmetization of analysis; Gödel produced his paradox by an arithmetization of mathematical syntax.) It is to this tacit research program that Mandelbrot implicitly referred when we said, “The eye had been banished out of science. The eye had been excommunicated.” (in an interview in the same documentary mentioned above) What Mandelbrot did was to rehabilitate the eye, to recall the eye from its scientific banishment, and for many this was liberating. To feel free to once again trust one’s geometrical intuition, to “run with it,” as it were, or — to take a platonic figure of thought — to follow the argument where it leads, set a new generation of mathematicians free to explore the visceral feeling of mathematical ideas that had gone underground for a hundred years.
It is the astonishing appearance of fractals that made Mandelbrot famous. Since one could probably count the number of famous mathematicians in any one generation on one hand, this is an accomplishment of the first order. Mandelbrot more-or-less singlehandedly created a new branch of mathematics, working against institutionalized resistance, and this new branch of mathematics has led to a rethinking of physics as well as to concrete technological applications (such as cell phone antennas) in widespread use within a few years of fractals coming to the attention to scientists. Beyond this, fractals became a pop culture phenomenon, to the degree that a representation of the Mandelbrot set even appeared as a crop circle.
While the crop circle representation of the Mandelbrot set would seem to have brought us to the point of mere silliness, so much so that we seem to have passed from the sublime to the ridiculous, it does not take much thought to bring us back around to understanding the intellectual, mathematical, scientific, and technological significance of fractal geometry. This is Mandelbrot’s contribution to history, to civilization, to humankind.
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Fractals and Geometrical Intuition
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3 August 2010
Aristotle claimed that mathematics has no ethos (Metaphysics, Book III, Chap. 2, 996a). Aristotle, of course, was more interested in the empirical sciences than his master Plato, whose Academy presumed and demanded familiarity with geometry — and we must understand that for the ancients, long before the emergence of analytical geometry in the work of Descartes (allowing us to formulate geometry algebraically, hence arithmetically), that geometry was always axiomatic thought, rigorously conceived in terms of demonstration. For the Greeks, this was the model and exemplar of all rigorous thought, and for Aristotle this was a mode of thought that lacked an ethos.
In this, I think, Aristotle was wrong, and I think that Plato would have agree on this point. But the intuition behind Aristotle’s denial of a mathematical ethos is, I think, a common one. And indeed it has even become a rhetorical trope to appeal to rigorous mathematics as an objective standard free from axiological accretions.
Our human, all-too-human faculties conspire to confuse us, to addle our wits, when we begin talking about morality, so that the purity and rigor of mathematical and logical thought seem to be called into question if we acknowledge that there is an ethos of formal thought. We easily confuse ourselves with religious, mystical, and ethical ideas, and since the great monument of mathematical thought has been mostly free of this particular species of confusion, to deny an ethos of formal thought can be understood as a strategy to protect and defend of the honor of mathematics and logic by preserving it from the morass that envelops most human attempts to think clearly, however heroically undertaken.
Kant famously stated in the Critique of Pure Reason that, “I have found it necessary to deny knowledge in order to make room for faith.” I should rather limit faith to make room for rigorous reasoning. Indeed, I would squeeze out faith altogether, and find myself among the most rigorous of the intuitionists, one of whom has said: “The aim of this program is to banish faith from the foundations of mathematics, faith being defined as any violation of the law of sufficient reason (for sentences). This law is defined as the identification (by definition) of truth with the result of a (present or feasible) proof…”
Though here again, with intuitionism (and various species of constructivism generally), we have rigor, denial, asceticism — intuitionistic logic is no joyful wisdom. (An ethos of formal thought need not be an inspiring and edifying ethos.) It is logic with a frown, disapproving, censorious — a bitter medicine justified only because it offers hope of curing the disease of contradiction, contracted when mathematics was shown to be reducible to set theory, and the latter shown to be infected with paradox (as if the infinite hubris of set theory were not alone enough for its condemnation). Is the intuitionist’s hope justified? In so far as it is hope — i.e., hope and not proof, the expectation that things will go better for the intuitionistic program than for logicism — it is not justified.
Dummett has said that intuitionistic logic and mathematics are to wear their justification on their face:
“From an intuitionistic standpoint, mathematics, when correctly carried on, would not need any justification from without, a buttress from the side or a foundation from below: it would wear its own justification on its face.”
Dummett, Michael, Elements of Intuitionism, Oxford University Press, 1977, p. 2
The hope that contradiction will not arise from intuitionistic methods clearly is no such evident justification. As a matter of fact, empirically and historically verifiable, we know that intuitionism has resulted in no contradictions, but this could change tomorrow. Intuitionism stands in need of a consistency proof even more than formalism. There is, in its approach, a faith invested in the assumption that infinite totalities caused the paradoxes, and once we have disallowed reference to them all will go well. This is a perfectly reasonable assumption, but one, in so far as it is an article of faith, which is at variance with the aims and methods of intuitionism.
And what is a feasible proof, which our ultra-intuitionist would allow? Have we not with “feasible proof” abandoned proof altogether in favor of probability? Again, we will allow them their inconsistencies and meet them on their own ground. But we shall note that the critics of the logicist paradigm fix their gaze only upon consistency, and in so doing reveal again their stingy, miserly conception of the whole enterprise.
“The Ultra-Intuitionistic Criticism and the Antitraditional program for the foundations of Mathematics” by A. S. Yessenin-Volpin (who was arguing for intellectual freedom in the Soviet Union at the same time that he was arguing for a censorious conception of reason), in Intuitionism and Proof Theory, quoted briefly above, is worth quoting more fully:
The aim of this program is to banish faith from the foundations of mathematics, faith being defined as any violation of the law of sufficient reason (for sentences). This law is defined as the identification (by definition) of truth with the result of a (present or feasible) proof, in spite of the traditional incompleteness theorem, which deals only with a very narrow kinds [sic] of proofs (which I call ‘formal proofs’). I define proof as any fair way of making a sentence incontestable. Of course this explication is related to ethics — the notion fair means ‘free from any coercion or fraud’ — and to the theory of disputes, indicating the cases in which a sentence is to be considered as incontestable. Of course the methods of traditional mathematical logic are not sufficient for this program: and I have to enlarge the domain of means explicitly studied in logic. I shall work in a domain wherein are to be found only special notions of proof satisfying the mentioned explication. In this domain I shall allow as a means of proof only the strict following of definitions and other rules or principles of using signs.
Intuitionism and proof theory: Proceedings of the summer conference at Buffalo, N.Y., 1968, p. 3
What is coercion or fraud in argumentation? We find something of an illustration of this in Gregory Vlastos’ portrait of Socrates: “Plato’s Socrates is not persuasive at all. He wins every argument, but never manages to win over an opponent. He has to fight every inch of the way for any assent he gets, and gets it, so to speak, at the point of a dagger.” (The Philosophy of Socrates, Ed. by Gregory Vlastos, page 2)
What appeal to logic does not invoke logical compulsion? Is logical compulsion unique to non-constructive mathematical thought? Is there not an element of logical compulsion present also in constructivism? Might it not indeed be the more coercive form of compulsion that is recognized alike by constructivists and non-constructivists?
The breadth of the conception outlined by Yessenin-Volpin is impressive, but the essay goes on to stipulate the harshest measures of finitude and constructivism. One can imagine these Goldwaterite logicians proclaiming: “Extremism in the defense of intuition is no vice, and moderation in the pursuit of constructivist rigor is no virtue.” Brouwer, the spiritual father of intuitionism, even appeals to the Law-and-Order mentality, saying that a criminal who has not been caught is still a criminal. Logic and mathematics, it seems, must be brought into line. They verge on criminality, deviancy, perversion.
The same righteous, narrow, anathamatizing attitude is at work among the defenders of what is sometimes called the “first-order thesis” in logic. Quine sees a similar deviancy in modal logic (which can be shown to be equivalent to intuitionistic logic), which he says was “conceived in sin” — the sin of confusing use and mention. These accusations do little to help us understand logic. We would do well to adopt Foucault’s attitude on these matters: “leave it to our bureaucrats and our police to see that our papers are in order. At least spare us their morality when we write.” (The Archaeology of Knowledge, p. 17)
The philosophical legacy of intuitionism has been profound yet mixed; its influence has been deeply ambiguous. (Far from the intuitive certainty, immediacy, clarity, and evident justification that it would like to propagate.) There is in inuitionism much in harmony with contemporary philosophy of mathematics and its emphasis on practices, the demand for finite constructivity, its anti-philosophical tenor, its opposition to platonism. The Father of Intuitionism, Brouwer, was, like many philosophers, anti-philosophical even while propounding a philosophy. No doubt his quasi-Kantianism put his conscience at rest in the Kantian tradition of decrying metaphysics while practicing it, and his mysticism gave a comforting halo (which softens and obscures the hard edges of intuitionist rigor in proof theory) to mathematics which some have found in the excesses of platonism.
In any case, few followers of Brouwer followed him in his Kantianism and mysticism. The constructivist tradition which grew from intuitionism has proved to be philosophically rich, begetting a variety of constructive techniques and as many justifications for them. Even if few mathematicians actually do intuitionistic mathematics, controversies over the significance of constructivism have a great deal of currency in philosophy. And Dummett is explicit about the place of philosophy in intuitionistic logic and mathematics.
Intuitionism and constructivism command our respect in the same way that Euclidean geometry commanded the respect of the ancients: we might not demand that all reasoning conform to this model, but it is valuable to know that rigorous standards can be formulated, as an ideal to which we might aspire if nothing else. And and ideal of reason is itself an ethos of reason, a norm to which formal thought aspires, and which it hopes to approximate even if it cannot always live up the the most exacting standard that it can recognize for itself.
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Studies in Formalism
9. Methodological and Ontological Parsimony (in preparation)
10. The Spirit of Formalism (in preparation)
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