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


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.

. . . . .

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

. . . . .

signature

. . . . .

Grand Strategy Annex

. . . . .

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