The Ontological Status of the Laws of Mathematics According to the Chazon Ish

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The Ontological Status of the Laws of Mathematics According to the Chazon Ish

Posted on 10/8/2012

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The Ontological Status of the Laws of Mathematics According to the Chazon Ish

I was asked an interesting question, and I am bringing it before you for your enjoyment and amusement.

The Chazon Ish, Emunah u’Bitachon, ch. 1, sec. 9, addresses a proof for the existence of God. There he identifies a flaw in the reasoning of many who challenge the physico-theological proof. This is, in itself, a fascinating philosophical discussion, especially when it appears in a book like this and from a figure like the Chazon Ish. I should preface by saying that this argument and its rejection are not the subject of this thread, and I therefore recommend refraining from opening a discussion about it.

The proof states that every existent has a maker, and therefore there must be something/someone that created the world. To this they ask that if every existent has a maker, then this maker too must have another maker. In other words, the hypothesis of God does not solve the problem.

I have already written here more than once (see, for example, the parallel thread on my remarks on YNET) that this question is based on a mistake, because the assumption does not deal with every existent, but only with an existent that resembles, in some sense, the data of our experience (in his terms: one possessing some sort of finite dimensions). I was pleased to see that the Chazon Ish makes a similar claim (though not an identical one). He argues that this applies only to objects that have extension, measure, or area.

He brings an example for his claim from the laws of mathematics, such as 2+2=4, which exist eternally because they have no body, no measure, and the like. His claim is that “they exist necessarily, and their nonexistence cannot be conceived.”

Several points arise here:

1. This passage does not appear in the standard editions of Emunah u’Bitachon, but only in the version printed in his halakhic work at the end of the Taharot volume. Does anyone know, or does anyone have an idea, why?

2. The implication is that, in the Chazon Ish’s view, the laws of mathematics have existence. These are entities for which it is meaningful to discuss existence and nonexistence. Seemingly, this is mathematical Platonism. Is it really?

3. I was asked whether, and how, this fits with the vacuity of the analytic. Here I will elaborate a bit. A common assumption is that the analytic is vacuous, meaning that a logical argument (a tautology) adds nothing new, since the conclusion is contained in the premises. If it were not contained there, then it could not be proved from them. For example, if all human beings are mortal and Socrates is a human being, then Socrates is mortal. Why does the conclusion of the argument necessarily follow from its premises? Because it is included in them. (If I know that all human beings are mortal, and I also know that Socrates is one of them, then I already know that he is mortal. The statement “All human beings are mortal” is nothing but shorthand for a collection of statements: “Jacob is mortal,” “Socrates is mortal,” and “Ahmed is mortal,” and so on.

And this is what I wrote to him in reply:

You are asking a fascinating question. Seemingly, the Chazon Ish is expressing here a Platonist position that sees ideas as having existence. Moreover, he expresses mathematical Platonism, since he also relates to the laws of mathematics as ideas (=entities). There is not much novelty in that. What you are asking is whether mathematical Platonism contradicts the vacuity of the analytic.

Why do you see a contradiction here? If I understood your words correctly, your point is that if the analytic is vacuous, this means that the laws of logic are nothing but determinations of identity (=or inclusion) between the premises and the conclusion, and therefore they should not be regarded as existing entities.

But if you do in fact accept mathematical Platonism (there is debate about this among philosophers of mathematics), then I do not see why this must follow. The logical or mathematical law is an existing idea, even though it establishes a necessary relation (a tautology).

However, it seems more correct to me that the mathematical law establishes a relation among mathematical entities (planes, lines, or points, or numbers, spaces, and the like). The relations are something like properties of those entities, but they themselves are not entities in the ordinary sense. Just as a person’s kindness does not exist as an entity, even though it describes something in reality (=a property of the person).

It may be that the Chazon Ish sees the existence of God as akin to the existence of a mathematical property, wholly abstract, and therefore not beginning or ending in time. I am not sure that this example really expresses a substantive conception of his regarding the laws of mathematics/logic. He intended to bring an analogical example of something abstract and to explain that you cannot speak of the cessation or beginning of such a thing.

And two further comments:

1. According to Kant, 2+2=4 is synthetic a priori, and not analytic. It is indeed not vacuous. The famous example he gave was 5+7=12.

2. At the abstract level, every application of a mathematical law in the real world is a physical law, not a mathematical one, and therefore it is synthetic and not analytic. I discussed this in my book E-lohim Mesachek BeKubiyot, in the first half of the fourth chapter, and I will explain briefly here. When one examines a situation in which a force of magnitude 10 acts on some body toward the north and another force of magnitude 10 acts toward the east, the resultant force is not 20 (but a little more than 14). Does this contradict the mathematical law 10+10=20? Certainly not. It contradicts the physical law that the combination of forces is described by algebraic addition. That is not correct, and therefore vector arithmetic is required. Theoretically, if I were to conduct an experiment and add oranges to a basket, putting in 2 and then another 2, and in the end count and find 5 oranges in the basket, this would not refute the algebraic law but the physical law that adding oranges to a basket is described by algebraic addition. Likewise, Einstein’s theory does not refute Euclidean geometry, but only shows that our real world is not Euclidean but curved.

Source (forum “Atzor Kan Choshvim”): http://www.bhol.co.il/forums/topic.asp?topic_id=2971653&forum_id=1364