Friday, March 25, 2011

The Abyss and the Tao

After Cantor recovered from a stint in the insane asylum for mental breakdowns suffered while trying to prove some (we now know) intractable problems in set theory, he had a stroll with his good friend and colleague Dedekind. Dedekind asked Cantor what he pictured in his mind when he thought of sets. Dedekind remarked that when he thought about a set, he pictured a clear bag containing objects inside which are the set's elements. Cantor responded that when he thought about a set, he pictured an abyss.

In a previous post I remarked that some abstract objects occupy space and have relevant properties of concrete material objects and thus there may not be any philosophical (epistemological or otherwise) problems in dealing with them such as knowing them or having them cause and be caused by physical phenomenon. In mathematics, we have pure sets built from the null set from iterative or otherwise “mental operations.” The old joke that the mathematical universe is an entire infinite universe begotten from nothing (the contents of the null set) is illustrative of the counter-intuitiveness of this idea.

Thus, as the story goes, from the null set, we have everything we need for mathematics. But since these objects of math are all purely abstract without concrete properties (as opposed to abstract objects with concrete elements) which may take part in the causal epistemological chain, how do we know them if e.g., the causal theory of knowledge is correct? If mathematics is built on such foundations, we cannot appeal to impure sets such as sets built on singletons of everyday objects to resolve these epistemological lacunae. Mathematics foundationally built in such a way thus may be metaphysically and epistemologically counter-intuitive to many precisely because of the knowledge problem or some other problems with causally inert properties of pure sets or the begetting of things from nothing. Metaphysically, it is problematic as the old joke suggests, it is a infinitely high house of cards built on top the foundations of an abyss. I want to ask here if there is another alternative system that is built on firmer and more intuitive metaphysical foundations.

Now consider the classical Chinese conception of numbers. Many philosophers of the school of names and the neo-Taoist school inspired by the I-Ching (such as Wang Bi) thought that from some object, we may, by some mental operation, form the class (they did not have a modern notion of set obviously but did have a notion of what we would term 'class') containing that object. Now we have two objects, namely the object and its singleton class. One can go on forming more and more objects this way until one has all the objects needed for one's numbers and thus to furnish one's mathematical universe. This would generate the positive integers instead of the natural numbers including zero as modern mathematics would have it. There is something to be said about the Chinese system; it is iterative like the modern conception of numbers but it starts off from one (the singleton set of some concrete object) instead of zero (the null set). This system is ontologically well-founded on something as opposed to nothing and is concrete "from the start." Specifically, what the Taoist founds his theory of numbers on, the original object on which all his classes are built on, is just the Tao.

More formally, we have according to the Taoist conception: Tao, {Tao}, {Tao, {Tao}}, {Tao, {Tao}, {Tao, {Tao}}}. Now, either the Tao or {Tao} may be arbitrarily designated as the number one.

Let's say that we choose the former. One is unlike the other numbers in that 1. it is not a class (set). 2. for all numbers n other than one contains n elements while one, not being a set contains no elements.

Or one can designate the Tao's singleton {Tao} as the number one. But then all subsequent numbers will contain the Tao which is not a number to generate all successive numbers.

Alternatively, one may have 1=Tao, 2={Tao}, 3={Tao, {Tao}}.

This last interpretation seems to be the one Wang Bi favored.

Significantly, as Wang Bi makes the point in both his Yijing and Laozi commentaries, in this sense “one” is not a number but that which makes possible all numbers and functions. In the latter (commentary to Laozi 39), Wang defines “one” as “the beginning of numbers and the ultimate of things.” In the former (commentary to Appended Remarks, Part I), he writes, “In the amplification of the numbers of heaven and earth [in Yijing divination] … ‘one’ is not used. Because it is not used, use [of the others] is made possible; because it is not a number, numbers are made complete. This indeed is the great ultimate of change.”

However, here, each number will contain n-1 elements instead of the Von Neumann formulation of modern mathematics which has for each number n, n elements.

Thus Chinese mathematics can be impure and would have no problems with the epistemological problem of knowing abstract objects built on the singleton of some concrete object. Everything that can be proven in the modern system of mathematics can be proven in the Chinese system per Lowenheim-Skolem theorem or one of its corollary theorems (since the two number systems are equinumorous and isomorphic). Zero thus is redundant for mathematics. There remains only orthographic problems of how to write numbers and mathematical formula down and the technique the classical Chinese used was to use a non referential “placeholder” in place of '0' or the cipher.

But this system is just as counter-intuitive as the modern for there are problems of its own. If I start off from the singleton set containing my computer and you start off from the singleton set containing your right foot, we would have two different notions of the numbers one and thus all subsequent numbers built on it. But 1=1 is true and it is necessarily true. Modern mathematics does not have this problem because it is easy to prove that there is one and only one null set and thus it and all numbers built on it are identical.

We may be able to obstruct this problem by introducing a single “arbitrary object” which can stand in for any object and starting from the singleton of it and define it as the number one. However, is this arbitrary object concrete or abstract? It might make sense to say of an arbitrary object that it is concrete or abstract with causal properties. I don't know. But if concrete where is this object? It may not have a specific location but since it could “stand in” for any concrete object, it may make some sense to ascribe to it causal properties even if it may be abstract much as the singletons of concrete objects. Kit Fine has defended arbitrary objects has having such common properties of concrete objects.

Alternatively, as pointed out above, Wang Bi and some of the other classical Chinese philosophers thought that the original object which one is is just the Tao, which in turn is itself not strictly definable and may be a ineffable "primitive" (I guess maybe like the term 'nothing' or perhaps 'arbitrary object') or its singleton. The Tao certainly seems to have causal properties and may certainly be said to influence causally the world on their conception whatever it may be.

Still, the Chinese system may not wholly escape some of the counter-intuitiveness associated modern math. It is only slightly less counter-intuitive than the modern system because instead of begetting the whole mathematical universe from nothing, it begets it from one thing. An infinitely high house of cards is built on the flimsiest of foundations instead of on an abyss.