Friday


Stephan Banach and Alfred Tarski

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.

Many three dimensional fractals have been investigated, but I don't know of any attempts to show an infinite fractal such that each step of the interation involves an infinite process. One reason for this as no such fractal could be generated by a computer even in its first iteration. Such a fractal can only be seen in the mind's eye. Among the factors that led to the popularity of fractals were the beautifully detailed and colored illustrations generated by computers. Mechanized assistance to intuition has its limits.

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.

Are fractals a mere artifact of computing technology? Certainly we can say that computers have been crucial to the development of fractals, but fractals need not be limited by the finite parameters of computing.

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

1. Benoît Mandelbrot, R.I.P.

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

6. A Note on Fractals and Banach-Tarski Extraction

7. Geometrical Intuition and Epistemic Space

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Grand Strategy Annex

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Saturday


In 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.

The 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.

What 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 ε0, 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.

My recent discussion of the Banach-Tarski Paradox provides another way to think about ε0. 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.

Notice 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.

An 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 ε0 spheres.

In 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 ITRanbn refers to the nth sphere from iteration b which in turn is derived from the nth 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.

If 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 ε0, 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.

Odin was suspended for nine days upon the world tree Yggdrasil in his quest to know the secret of the Runes.

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

1. Benoît Mandelbrot, R.I.P.

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

6. A Note on Fractals and Banach-Tarski Extraction

7. Geometrical Intuition and Epistemic Space

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Grand Strategy Annex

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Friday


In true Cartesian fashion I woke up slowly this morning, and while I tossed and turned in bed I thought more about the Banach-Tarkski paradox, having just written about it last night. In yesterday’s A Question for Philosophically Inclined Mathematicians, I asked, “Can we pursue this extraction of volume in something like a process of transfinite recursion, arriving at some geometrical equivalent of ε0?” The extraction in question is that of taking one mathematical sphere out of another mathematical sphere, and both being equal to the original — the paradox that was proved by Banach and Tarski. I see no reason why this process cannot be iterated, and if it can be iterated it can be iterated to infinity, and if iterated to infinity we should finish with an infinite number of mathematical spheres that would fill an infinite quantity of mathematical space.

All of this is as odd and as counter-intuitive as many of the theorems of set theory when we first learn them, but one gets accustomed to the strangeness after a time, and if one spends enough time engaged with these ideas one probably develops new intuitions, set theoretical intuitions, that stand one in better stead in regard to the strange world of the transfinite than the intuitions that one had to abandon.

In any case, it occurred to be this morning that, since decompositions of a sphere in order to reassemble two spheres from one original does not consist of discrete “parts” as we usually understand them, but of sets of points, and these sets of points would constitute something that did not fully fill the space that they inhabit, and for this reason we could speak of them as possessing fractal dimension. On fractal dimension, the Wikipedia entry says this of the Koch curve:

“…the length of the curve between any two points on the Koch Snowflake is infinite. No small piece of it is line-like, but neither is it like a piece of the plane or any other. It could be said that it is too big to be thought of as a one-dimensional object, but too thin to be a two-dimensional object, leading to the speculation that its dimension might best be described in a sense by a number between one and two. This is just one simple way of imagining the idea of fractal dimension.”

The first space filling-curve discovered by Giuseppe Peano (the same Peano that formulated influential axioms of arithmetic, though the axioms seem to ultimately derive from Dedekind) already demonstrated a way in which a line, ordinarily considered one dimensional, can be two dimensional — or, if you prefer to take the opposite perspective, that a plane, ordinarily considered to be two dimensional, can be decomposed into a one dimensional line. A fractal like the Koch curve fills two dimensional space to a certain extent, but not completely like Peano’s space-filling curve, and its fractal dimension is calculated as 1.26.

Hilbert's version of a space filling curve.

The Koch curve is a line that is more than a line, and it can only be constructed in two dimensions. It is easy to dream up similar fractals based on two dimensional surfaces. For example, we could take a cube and construct a cube on each side, and construct a cube on each side of these cubes, and so on. We could do the same thing with bumps raised on the surface of a sphere. Right now, we are only thinking of in terms of surfaces. The six planes of a cube enclose a volume, so we can think of it either as a two dimensional surface or as a three dimensional body. In so far as we think of the cube only as a surface, it is a two dimensional surface that can only be constructed in three dimensions. (And the cube or sphere constructions can go terribly wrong also, as if we make the iterations too large they will run into each other. Still, the appropriate construction will yield a fractal.)

This process suggests that we might construct a fractal from three dimensional bodies, but to do so we would have to do this in four dimensions. In this case, the fractal dimension of a three dimensional fractal constructed in four dimensional space would be 3.n, depending upon how much four dimensional space was filled by this fractal “body.” (And I hope you will understand why I put “body” in scare quotes.)

I certainly can’t visualize a four dimensional fractal. In fact, “visualize” is probably the wrong term, because our visualization capacity locates objects in three dimensional space. It would be better to say that I cannot conceive of a four dimensional fractal, except that I can entertain the idea, and this is a form of conception. What I mean, of course, is a form of concrete conception not tied to three dimensional visualization. I suspect that those who have spent a lifetime working with such things may approach an adequate conception of four dimensional objects, but this is the rare exception among human minds.

Just as we must overcome the counter-intuitive feeling of the ideas of set theory in order to get to the point where we are conceptually comfortable with it, so too we would need to transcend our geometrical intuitions in order to adequately conceptualize four dimensional objects (which mathematicians call 4-manifolds). I do not say that it is impossible, but it is probably very unusual. This represents an order of thinking against the grain that will stand as a permanent aspiration for those of us who will never fully attain it. Intellectual intuition, like dimensionality, consists of levels, and even if we do not fully attain to a given level of intuition, if we glimpse it after a fashion we might express our grasp as a decimal fraction of the whole.

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A seasonally-appropriate illustration of the Banach-Tarski paradox.

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Fractals and Geometrical Intuition

1. Benoît Mandelbrot, R.I.P.

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

6. A Note on Fractals and Banach-Tarski Extraction

7. Geometrical Intuition and Epistemic Space

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signature

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Grand Strategy Annex

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Friday


Given the astonishing yet demonstrable consequence of the Banach-Tarski paradox, it is the sort of thing that one’s mind returns to on a regular basis in order to savor the intellectual satisfaction of it. The unnamed author of the Layman’s Guide to the Banach-Tarski Paradox explains the paradox thus:

The paradox states that it is possible to take a solid sphere (a “ball”), cut it up into a finite number of pieces, rearrange them using only rotations and translations, and re-assemble them into two identical copies of the original sphere. In other words, you’ve doubled the volume of the original sphere.

The whole of the entry at Wolfram Mathworld runs as follows:

First stated in 1924, the Banach-Tarski paradox states that it is possible to decompose a ball into six pieces which can be reassembled by rigid motions to form two balls of the same size as the original. The number of pieces was subsequently reduced to five by Robinson (1947), although the pieces are extremely complicated. (Five pieces are minimal, although four pieces are sufficient as long as the single point at the center is neglected.) A generalization of this theorem is that any two bodies in R3 that do not extend to infinity and each containing a ball of arbitrary size can be dissected into each other (i.e., they are equidecomposable).

The above-mentioned Layman’s Guide to the Banach-Tarski Paradox attempts to provide an intuitive gloss on this surprising result of set theory (making use of the axiom of choice, or some equivalent assumption), and concludes with this revealing comment:

In fact, if you think about it, this is not any stranger than how we managed to duplicate the set of all integers, by splitting it up into two halves, and renaming the members in each half so they each become identical to the original set again. It is only logical that we can continually extract more volume out of an infinitely dense, mathematical sphere S.

Before I read this today, I’d never come across such a clear and concise exposition of the Banach-Tarski paradox, and in provides food for thought. Can we pursue this extraction of volume in something like a process of transfinite recursion, arriving at some geometrical equivalent of ε0? This is an interesting question, but it isn’t the question that I started out thinking about as suitable for the philosophically inclined mathematician.

When I was thinking about the Banach-Tarski paradox today, I began wondering if a sufficiently generalized formulation of the paradox could be applied to ontology on the whole, so that we might demonstrate (perhaps not with the rigor of mathematics, but as best as anything can be demonstrated in ontology) that the world entire might be decomposed into a finite number of pieces and then reassembled into two or more identical worlds.

With the intuitive gloss quoted above, we can say that this is a possibility in so far as the world is ontologically infinitely dense. What might this mean? What would it be for the world to be ontologically dense in the way that infinite sets are infinitely dense? Well, this kind of question goes far beyond intuition, and therefore lands us in the open-texture of language that can accommodate novel uses but which has no “natural” meaning one way or the other. The open-texture of even our formal languages makes it like a quicksand: if you don’t have some kind of solid connection to solid ground, you are likely to flail away until you go under. It is precisely for this reason that Kant sought a critique of reason, so that reason would not go beyond its proper bounds, which are (as Strawson put it) the bounds of sense.

But as wary as we should be of unprecedented usages, we should also welcome them as opportunities to transcend intuitions ultimately rooted in the very soil from which we sprang. I have on many occasions in this forum argued that our ideas are ultimately derived from the landscape in which we live, by way of the way of life that is imposed upon us by the landscape. But we are not limited to that which our origins bequeathed to us. We have the power to transcend our mundane origins, and if it comes at the cost of occasional confusion and disorientation, so be it.

So I suggest that while there is no “right” answer to whether the world can be considered ontologically infinitely dense, we can give an answer to the question, and we can in fact make a rational and coherent case for our answer if only we will force ourselves to make the effort of thinking unfamiliar thoughts — always a salutary intellectual exercise.

Is the world, then, ontologically infinitely dense? Is the world everywhere continuous, so that it is truly describable by a classical theory like general relativity? Or is the world ultimately grainy, so that it must be described by a non-classical theory like quantum mechanics? At an even more abstract level, can the beings of the world be said to have any density if we do not restrict beings to spatio-temporal beings, so that our ontology is sufficiently general to embrace both the spatio-temporal and the non-spatio-temporal? This is again, as discussed above, a matter of establishing a rationally defensible convention.

I have no answer to this question at present. One ought not to expect ontological mysteries to yield themselves to a few minutes of casual thought. I will return to this, and think about it again. Someday — not likely someday soon, but someday nonetheless — I may hit upon a way of thinking about the problem that does justice to the question of the infinite density of beings in the world.

I do not think that this is quite as outlandish as it sounds. Two of the most common idioms one finds in contemporary analytical philosophy, when such philosophers choose not to speak in a technical idiom, are those of, “the furniture of the universe,” and of, “carving nature at its joints.” These are both wonderfully expressive phrases, and moreover they seem to point to a conception of the world as essentially discrete. In other words, they suggest an ultimate ontological discontinuity. If this could be followed up rigorously, we could answer the above question in the negative, but the very fact that we might possibly answer the question in the negative says two important things:

1) that the question can, at least in some ways, be meaningful, and therefore as being philosophically significant and worthy of our attention, and…

2) if a question can possibly be answered the negative, it is likely that a reasonably coherent case could also be made for answering the question in the affirmative.

The Banach-Tarski paradox is paradoxical at least in part because it does not seem to, “carve nature at the joints.” This violation of our geometrical intuition comes about as a result of the development of other intuitions, and it is ultimately the clash of intuitions that is paradoxical. Kant famously maintained that there can be no conflict among moral duties; parallel to this, it might be taken as a postulate of natural reason that there can be no conflict among intellectual intuitions. While this principle has not be explicitly formulated to my knowledge, it is an assumption pervasively present in our reasoning (that is to say, it is an intuition about our intiutions). Paradoxes as telling as the Banach-Tarski paradox (or, for that matter, most of the results of set theory) remind us of the limitations of our intuitions in addition to reminding us of the limitations of our geometrical intuition.

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Fractals and Geometrical Intuition

1. Benoît Mandelbrot, R.I.P.

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

6. A Note on Fractals and Banach-Tarski Extraction

7. Geometrical Intuition and Epistemic Space

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signature

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Grand Strategy Annex

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