## The Right of the People to Alter or to Abolish

### 4 July 2014

**Friday **

**O**n this, the 238^{th} anniversary of the signing of the ** Declaration of Independence**, I would like to recall what is perhaps the centerpiece of the document: a ringing affirmation of what would later, during the French Revolution, be called “The Rights of Man,” and how and why a people with “a decent respect to the opinions of mankind” should go about securing these rights:

We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness. — That to secure these rights, Governments are instituted among Men, deriving their just powers from the consent of the governed, — That whenever any Form of Government becomes destructive of these ends, it is the Right of the People to alter or to abolish it, and to institute new Government, laying its foundation on such principles and organizing its powers in such form, as to them shall seem most likely to effect their Safety and Happiness. Prudence, indeed, will dictate that Governments long established should not be changed for light and transient causes; and accordingly all experience hath shewn, that mankind are more disposed to suffer, while evils are sufferable, than to right themselves by abolishing the forms to which they are accustomed.

**T**he famous litany of *life, liberty, and the pursuit of happiness* names certain specific instances, we note, among the unalienable rights of human beings (a partial, and not an exhaustive list of such rights), and in the very same paragraph the founders have mentioned *the Right of the People to alter or to abolish* any form of government that becomes destructive to these ends. This is significant; the right of the people to alter or abolish a government that is destructive of unalienable rights is itself an unalienable right, though qualified by the condition that established governments should not be lightly overthrown. The Founders did not say that long established governments should *never* be overthrown, since they were in the process of overthrowing the government of one of the oldest kingdoms in Europe, but that such an action should not be undertaken *lightly*.

**I**n keeping with with a prioristic language of self-evident truths, the Founders have formulated the right to alter or to abolish in terms of *forms of government*. In other words, the right to alter or abolish is framed not in terms of particular tyrannical or corrupt regimes, but on the *form* of the regime. This is political Platonism, pure and simple. The Founders are here recognizing that there are a few distinct forms of government, just as there are a few distinct unalienable rights. For the political Platonism of the Anglophone Enlightenment, forms of government and unalienable rights are part of the furniture of the universe (a phrase I previously employed in **Defunct Ideas** and some other posts).

**I**t has always been the work of revolutions to alter or to abolish forms of government, and this is still true today, although we are much less likely to think in these platonistic terms about the forms of governments and unalienable rights. To be sure, the idea of rights has become absolutized to a certain extent in the contemporary world, but it is a conflicted absolute idea, because it is an absolute idea stranded in a society that no longer believes in absolute ideas. In just the same way, the governmental tradition of the US is a “stranded asset” of history — an anachronistic relic of the Enlightenment that has survived through several post-Enlightenment periods of history and still survives today. The language of the Enlightenment can still speak to us today — it has a perennial resonance with human nature — but if you can get a typical representative of our age to engage in a detailed conversation about political ideals, you will not find many proponents of Enlightenment ideals, such as the perfectibility of man, throwing off past superstitions, the belief in progress, the dawning of a new world, and a **universalist conception of human nature**. These are, now, by-and-large, **defunct ideas**. But not entirely.

**I**f you *do* find these Enlightenment ideals, you will find them in a very different *form* than the form that they took among the Enlightenment Founders of the American republic — and note here my use of “form” and again the Platonism that implies. Those today who most passionately believe in the Enlightenment ideals of progress, perfectibility, and a new world on the horizon are, by and large, transhumanists and singulatarians. They believe (often enthusiastically) in an optimistic vision of a better future, although the future they envision would be, for some among us, a paradigm of **moral horror** — human beings altered beyond all recognition and leading lives that have little or no relationship to human lives as they have been lived since the beginning of civilization.

**T**ranshumanists and singulatarians also believe in the right of the people to alter or to abolish institutions that have become destructive of life, liberty, and the pursuit of happiness — but the institutions they seek to alter or abolish are none other than the institutions of the human body and the human mind (or, platonistically speaking, the *form* of the human body and the *form* of the human mind), far older than any form of government, and presumably not to be lightly altered or abolished. Looking at the contemporary literature on transhumanism, with some arguing for and some arguing against, it is obvious that one of the great moral conflicts in the coming century (and perhaps for some time after, until some settlement is reached, or until we and our civilization are so transformed that the question loses its meaning) is going to be that over transhumanism, which is, essentially, a platonistic question about what it means to be human (and the attempt to define the distinction between the human and the non-human, which **I recently wrote about**). For some, what it means to be human is already fixed for all time and eternity; for others, what it means to be human is not fixed, but is subject to continual change and revision, taking in the whole of human prehistory and what we were before we were human.

**I**t is likely that the coming moral conflict over transhumanism (both the conflict and transhumanism itself have already started, but they remain at the shallow end of an exponential growth curve) will eventually make itself felt as social and political conflict. The ethico-religious conflict in Europe from the advent of the Reformation to the end of the Thirty Years’ War brought into being the political institution of the nation-state and even created the conditions for the Enlightenment, as a reaction against the religious excesses the Reformation and its consequences. Similarly, the ethico-social conflict that will follow from divisions over transhumanism (and related technological developments that will blur the distinction between the human and the non-human) may in their turn be the occasion of the emergence of revolutionary changes in social and political institutions. Retaining the right of the people to alter or abolish their institutions means remaining open to such revolutionary change.

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*Happy 4 ^{th} of July!*

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## A Note on Fractals and Banach-Tarski Extraction

### 28 January 2011

**Friday **

**F**urther 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.

**T**he 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.

**O**nce 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

2. A Question for Philosophically Inclined Mathematicians

3. Fractals and the Banach-Tarski Paradox

4. A visceral feeling for epsilon zero

5. Adventures in Geometrical Intuition

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## A visceral feeling for epsilon zero

### 30 October 2010

**Saturday **

**I**n many posts to this forum, and most recently in a couple of posts about fractals — **A Question for Philosophically Inclined Mathematicians** and **Fractals and the Banach-Tarski Paradox** — I have discussed the cultivations of novel forms of intellectual intuition that allow us to transcend our native intuitions which make many demonstrable truths counter-intuitive. The cultivation of intuition is a long and arduous process; there is no royal road to it, just as Euclid once informed a king that there was no royal road to geometry.

**T**he good news is that the more people work on difficult ideas, the easier they can make them for others. That is why it is often said that we see farther because we stand on the shoulders of giants. I have pointed out before that the idea of zero was once very advanced mathematics mastered by only a select few; now it is taught in elementary schools. People who are fascinated by ideas are always looking for new and better ways to explain them. This is a social and cultural process that makes difficult and abstract ideas widely accessible. Today, for example, with the emphasis on visual modes of communication, people spend a lot of time trying to come up with striking graphics and diagrams to illustrate an idea, knowing that if they can show what they are saying in an intuitively clear way, that they will make their point all the better.

**W**hat is required for this intuitivization of the counter-intuitive is a conceptual effort to see things in a new way, and moreover a new way that appeals to latent forms of intuition that can then be developed into robust forms of intuition. Every once in a while, someone hits upon a truly inspired intuitivization of that which was once counter-intuitive, and the whole of civilization is advanced by this individual effort of a single mind to understand better, more clearly, more transparently. By the painfully slow methods of autodidacticism I eventually came to an intuitive understanding of ε_{0}, though I’m not sure that my particular way of coming to this understanding will be of any help to others, though it was a real revelation to me. Someplace, buried in my notebooks of a few years ago, I made a note on the day that I had my transfinite epiphany.

**M**y recent discussion of the Banach-Tarski Paradox provides another way to think about ε_{0}. I don’t know the details of the derivation, but if the geometrical case is anything like the arithmetical case, it would be just as easy to extract two spheres from a given sphere as to extract one. I’ve drawn an illustration of this as a branching iteration, where each sphere leads to two others (above but one). Iterated to infinity, we come to an infinite number of mathematical spheres, just as we would with the one-by-one iteration illustrated above. But, if for technical reasons, this doesn’t work, we can always derive one sphere from every previous sphere (I have also attempted to illustrate this (immediately above), which gives us a similar result as the branching iteration.

**N**otice that the Banach-Tarski Paradox is called a *paradox* and not a *contradiction*. It is strange, but it in no way contradicts itself; the paradox is paradoxical but logically unimpeachable. One of the things are drives home how paradoxical it is, is that a mathematical sphere (which must be infinitely divisible for the division to work) can be decomposed into a *finite* number of parts and *finitely* reassembled into two spheres. This makes the paradox feel tantalizingly close to something we might do without own hands, and not only in our minds. Notice also that fractals, while iterated to infinity, involve only a finite process at each step of iteration. That is to say, the creation of a fractal is an infinite iteration of finite operations. This makes it possible to at least *begin* the illustration of fractal, even if we can’t finish it. But we need not stop at this point, mathematically speaking. I have paradoxically attempted to illustrate the unillustratable (above) by showing an iteration of Banach-Tarski sphere extraction that involves extracting an infinite number of spheres at each step.

**A**n illustration can *suggest*, but it cannot *show*, an infinite operation. Instead, we employ the ellipsis — “…” — to illustrate that which has been left out (which is the infinite part that can’t be illustrated). With transfinite arithmetic, it is just as each to extract an infinite number of arithmetical series from a given arithmetical series, as it is to extract one. If the same is true of Banach-Tarski sphere extraction (which I do not know to be the case), then, starting with a single sphere, at the first iteration we extract an infinite number of spheres from the first sphere. At the second iteration, we extract an infinite number of spheres from the previously extracted infinite number of spheres. We continue this process until we have an infinite iteration of infinite extractions. At that point, we will have ε_{0} spheres.

**I**n my illustration I have adopted the convention of using “**ITR**” as an abbreviation of “iteration,” each level of iteration is indicated by a lower-case letter *a*, *b*, *c*, …, *n*, followed by a subscript to indicate the number of spheres extracted at this level of iteration, *1*, *2*, *3*, …, *n*. Thus **ITR***a*_{n}*b*_{n} refers to the n^{th} sphere from iteration *b* which in turn is derived from the n^{th} sphere of iteration *a*. I think this schemata is sufficiently general and sufficiently obvious for infinite iteration, though it would lead to expressions of infinite length.

**I**f you can not only get your mind accustomed to this, but if you can actually *feel* it in your bones, then you will have an intuitive grasp of ε_{0}, a visceral feeling of epsilon zero. As I said above, it took me many years to achieve this. When I did finally “get it” I felt like Odin on the Day of the Discovery of the Runes, except that my mind hung suspended for more then nine days — more like nine years.

**I** will also note that, if you can see the big picture of this geometrical realization of epsilon zero, you will immediately notice that it possesses self-similarly, and therefore constitutes an infinite fractal. We could call it an infinite explosion pattern. All fractals are infinite in so far as they involve infinite iteration, but we can posit another class of fractals beyond that which involve the infinite iteration of infinite operations. We can only generate such fractals in our mind, because no computer could even illustrate the *first* step of an infinite fractal of this kind. This interesting idea also serves as a demonstration that fractals are not merely artifacts of computing machines, but are as platonically ideal as any mathematical object sanctioned by tradition.

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Fractals and Geometrical Intuition

2. A Question for Philosophically Inclined Mathematicians

3. Fractals and the Banach-Tarski Paradox

4. A visceral feeling for epsilon zero

5. Adventures in Geometrical Intuition

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## Benoît Mandelbrot, R.I.P.

### 17 October 2010

**Sunday **

**F**amed 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.

**M**andelbrot 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.

**T**here 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.”

**I**t 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.

**I**t 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.

**I**t 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.

**W**hile 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

2. A Question for Philosophically Inclined Mathematicians

3. Fractals and the Banach-Tarski Paradox

4. A visceral feeling for epsilon zero

5. Adventures in Geometrical Intuition

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## Defunct Ideas

### 11 January 2010

**Monday **

**A** few days ago in **Ideas: Blindness and Illusion** I discussed the adequacy of conceptual schemes and ontological inventories of the furniture of the universe. There I suggested that a rigorous adherence to both the principle of parsimony — avoiding ontological indulgences — and the principle of adequacy — avoiding ontological impoverishment — would yield us the most accurate result in terms of seeing the world for what it is, neither more nor less than what it is.

**T**his, of course, is a great over-simplification and obviously inadequate. For starters, there is no definite number of things in the world, including the fact that there is no definite number of ideas in the world. I can think of at least three reasons for this (there may be others, but this is what occurs to me as I write). A rigorous definition of what constitutes an individual would be necessary to be a rigorous inventory of the number and kind of individuals that exist. It is by no means obvious what is an individual and what is not. This is a function of vagueness. Also connected with vagueness, even given a definition of what constitutes an individual, there will be many cases that, due to vagueness, are ambiguous. Lastly, and certainly not least, the inventory of the world is not static, but dynamic.

**T**he world is not frozen in any one state of affairs. If we consider populations of biological entities, for example, we know that there are always some individuals being born while other individuals are dying. Populations can be stable, but they must be viewed statistically in terms of averages and approximations. There is no Platonic form of the number of people on the planet. You can make that number precise by formulating a number of conventions, but the number would be constantly changing and the conventions adopted would in some cases be arbitrary.

**W**hat holds for human bodies also holds for ideas, with some exceptions (you can count this as an example of **naturalism’s minimalist materialism** that I have written about on several occasions). There is no fixed number of ideas, but ideas grow in number disproportionately to populations of non-abstract objects. Plato was nearly right on this, at least: ideas, once they emerge, are nearly eternal.

**T**here are probably a few cases when ideas have emerged in history, played a role in human societies, and then disappeared, but we cannot prove this. Once an idea enters circulation, even if it is later abandoned, any record of the idea preserves that idea. In the earliest portions of human history, especially in prehistory (when, by definition, there are no written records that might preserve the ideas that enjoyed currency in early societies), an idea may have been conceived and subsequently lost to the vast stretches of time that have since intervened. It is likely that ideas were lost to history during the **Greek dark ages**, when the art of writing disappeared in places and had to be reinvented. Societies at this stage of development (think of the heroic world of Homer) had reached a stage of sophistication and complexity that many ideas would have been in circulation, but these societies hadn’t yet reached the stability or resilience of contemporary civilization, and thus much may have been lost to history.

**S**ince the beginning of the historical period proper, few ideas have been lost to history once they entered general circulation. If an individual hits on a great idea but forgets it, or does not communicate it, or the papers upon which he wrote it were scattered, burned, or lost upon his death, countless ideas of this sort may be and are still being lost to history. But once an idea enters **Popper’s third world** and takes on a life of its own, beyond the mind of a particular individual, our present information technologies will preserve such an idea indefinitely. There is no reason to believe that, if civilization lasts (and maintains its continuity) for another ten thousand or even a hundred thousand years, some future individual would not be able to educate themselves in the ideas in circulation, for example, in Elizabethan England.

**W**hich, at last, brings us to *defunct ideas*. Just because an idea has been preserved, and any interested, sentient party (not even necessarily human) can, through appropriate research, form an adequate conception of the idea, does not mean that that idea is a living option. An idea that has fallen out of use, no longer widely circulates in current human societies, and which is no longer a force in shaping the lives of individuals or populations, I call a *defunct idea*. (We could, alternatively, call them *dead* ideas, by analogy with dead languages, i.e., languages of which there is scholarly knowledge but which are no longer spoken by a population.)

**T**here are many defunct ideas. For example, I would say that the idea of the divine right of kings (and *some* of the ideas clustered with it, like royal absolutism) is a defunct idea. Certainly there are those who still believe in individuals being divinely anointed for some purpose in life, and certainly there are still kings that rule countries (though not many any more), but as a topic that has the power to move men to passionate debate and armed conflict, the divine right of kings is no longer a “mover and shaker” in the world of ideas. All we need do is compare it to an idea that truly has currency — like democracy, communism, or revolution — to understand the difference.

**S**ome of the ideas closely clustered with the idea of the divine right of kings, such as constitutionalism, are not defunct. In fact, they are very much alive because they won out in the historical contest between a dying idea and an emerging idea. In the early modern period, royal absolutism was an old idea, and while still an idea that would later reach its apogee under Louis XIV in France, it was nevertheless already a dying idea. At the same time, constitutionalism, as an alternative to monarchical government, was a new and exciting idea, at times as weak and as defenseless as a new-born babe, but soon enough to grow into its maturity and to replace the dying ideas of the medieval past.

**W**e need to here distinguish further between ideas that are properly defunct and ideas that are points of reference for contemporary societies (ideas that are, in other words, ** embodied ideas**) but are not made explicit. Such implicit ideas would include classic Enlightenment conceptions such as the

*perfectibility of man*. If you asked the typical man-in-the-street today about the perfectibility of man, he probably wouldn’t know what you were talking about. But if you explained the idea, he would almost certainly have an opinion on it one way or the other. In other words, even if it is not given an explicit formulation, the perfectibility of man is a live option in today’s world. People not only think about it, they respond to it and may well have strong feelings about it.

**A**s civilization continues in existence, retaining its continuity of tradition, the list of defunct ideas will and must grow. It is not likely that once an idea becomes defunct that it can reenter circulation, but is possible, over the long term, that societies could change so dramatically that a once-defunct idea could again rise from the dead and take an active role in society. It could be argued that the emergence of religious fundamentalism at the end of the twentieth century represents the recrudescence of a defunct idea. Many scholars of fundamentalism insist upon the modernity of this historical development, and I have some sympathy with this position, but it could be argued that the idea behind fundamentalism is that of fideism, and that fideism is a perennial idea.

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## Extrapolating Plato’s Definition of Being

### 8 August 2009

**Saturday **

**I**n the venerable **Jowett translation of Plato’s Sophist** dialogue, the complete text of which can be found online at

**Project Gutenberg**(and can also be found at Google Books, p. 379), we find the following exchange between Theaetetus and the Eleatic Stranger:

STRANGER: Let us push the question; for if they will admit that any, even the smallest particle of being, is incorporeal, it is enough; they must then say what that nature is which is common to both the corporeal and incorporeal, and which they have in their mind’s eye when they say of both of them that they ‘are.’ Perhaps they may be in a difficulty; and if this is the case, there is a possibility that they may accept a notion of ours respecting the nature of being, having nothing of their own to offer.

THEAETETUS: What is the notion? Tell me, and we shall soon see.

STRANGER: My notion would be, that anything which possesses any sort of power to affect another, or to be affected by another, if only for a single moment, however trifling the cause and however slight the effect, has real existence; and I hold that the definition of being is simply power.

**T**he Greek text of the Eleatic Stranger’s crucial formulation is as follows:

**Ξένος:** λέγω δὴ τὸ καὶ ὁποιανου̂ν [τινα] κεκτημένον δύναμιν [247e] εἴτ’ εἰς τὸ ποιει̂ν ἕτερον ὁτιου̂ν πεφυκὸς εἴτ’ εἰς τὸ παθει̂ν καὶ σμικρότατον ὑπὸ του̂ φαυλοτάτου, κἂν εἰ μόνον εἰς ἅπαξ, πα̂ν του̂το ὄντως εἰ̂ναι: τίθεμαι γὰρ ὅρον [ὁρίζειν] τὰ ὄντα ὡς ἔστιν οὐκ ἄλλο τι πλὴν δύναμις.

**T**his I shall simply call *Plato’s definition of being*.

**T**his memorable definition of being in Plato — the *power* to affect or be affected (Sophist, 247e) — has been construed by Abbagnano in terms of possibility: being is the *possibility* to affect or be affected. The Greek term that Jowett translated as “power” — “δύναμις” — Abbagnano translated as “possibility.” While this isn’t quite as creative as some Heideggerian “translations” of Greek, it is an unusual translation. Despite this, I count the transformation from *power* to *possibility* as justified, since the Platonic account of being does not demand acting or suffering in actuality, but only the *possibility* of acting or suffering. Abbagnano, to my mind, has retained the essence of Plato’s meaning. And the Platonic definition, such as it is, seems reasonable to me, as we can scarcely credit anything with being if it has no relation whatsoever to us (or to the world). Thus the Platonic definition of being so construed also provides us with a definition of non-being.

**T**his Platonic conception of being immediately suggests a tripartite division among beings:

1) those which both act and suffer,

2) those which act only but do not suffer, and

3) those which suffer only but do not act.

**T**here remains, obviously, a fourth possibility, but I noted above that the schematization of the Platonic definition of being introduces a tripartite division *among beings*, while the fourth possibility excludes all beings:

4) those which neither suffer nor act.

**T**his fourth possibility — those which neither act nor suffer — represents, by definition, non-being and may be considered the *null permutation* (analogous to the empty set). There is an implicit paradox here, since we seem to be referring to beings that neither act nor suffer, which of course are impossible. Does non-being consist of impossible beings? And there is, as well, the ontologically interesting question of the possibility of the individuation of non-beings. Nothingness has always been a philosophical puzzle, and the above approach allows us a novel perspective on nothingness. It is difficult if not impossible to imagine the utter oblivion of non-being. Even the sinners trapped in the ice of the frozen Cocytus are able to converse with Dante (or, at least, are merely seen encased in ice), and therefore seem possess being to some degree in virtue of this interaction.

**I**t is difficult to resist observing that the ontological “inertness” which has of late been ascribed to mathematics by Jody Azzouni (and we may generally suppose this to hold for all beings, if such they be, which constitute the formal sciences) would seem to indicate that mathematics is concerned not with beings but with non-being: we do not affect the objects of the formal sciences, and they do not affect us. But this is ultimately much too simplistic: the objects of the formal sciences — numbers, propositions, etc. — affect us in so far as we conceive them, and, depending upon the philosophy of mathematics that one advocates, there remains the possibility that might affect the objects of the formal sciences, for example, by creating and conceiving them (as in a constructivist philosophy of mathematics).

**I**f man is the measure of all things, as according to Protagoras, then mathematics has no measure since mathematics is resolutely anti-anthropocentric. If, as Russell held, mathematics is the study in which we never know what we are talking about nor whether what we are saying is true, this is entirely justified, because we are, as it were, talking about nothing, and what we are saying is neither true nor false. But Russell’s philosophy of mathematics exemplified a classically anti-anthropocentric position, and Russell himself was willing to formulate some of his positions in terms of non-being (as in, for example, his “no classes” theory of classes), so in this particular context Russell’s modern formulation is a re-statement of traditional view, and in his thought no suggestion of a constructivist alternative is ever made. Nevertheless, Azzouni’s account of ontological inertness does closely correspond to the definition of non-being in terms of neither suffering nor acting, and it would be worthwhile to follow up on this correspondence in a systematic way. But another time.

**A**nd there is more. Within each ontological division outlined above we may adopt a hierarchy of being, securely based upon quantification of the number of beings which a given being affects or by which it is affected. The measure of ontological power — ontological potency, as it were — is a being’s scope of acting and suffering. Now, we may assume that those beings that both act and suffer possess a greater scope of being than those one-sided beings which act only or suffer only. Whether either one of these two inferior forms of being — viz. acting only or suffering only — ought to be superior to the other is a greater problem, and we will not address it here.

**W**e can go further than this by defining acting and suffering within regional ontologies (to invoke a formulation of Husserl). Regional ontologies themselves admit a scope of possible acting and suffering. They overlap and intersect (to invoke a formulation of Wittgenstein).

**A** further note: construing the Platonic definition of being in terms of the possibility of affecting or being affected, as Abbagnano does, suggests a distinction between so construing Plato and, in contrast, interpreting the Platonic definition of being in terms of *actually* affecting or being affected. This latter interpretation would require specifying a scope of time during which a putative existent’s affecting or being affected would be relevant, for not everything affects or is affected by everything else at any one moment. But this too suggests a further division (perhaps the narrowest formulation of being) in terms of which only that which affects or is affected by a given being at a given instant contributes to its reality: call it, if you will, *being at an instant*. More of this at another time.

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