Mathematical and Theological Realities: Responses to Two Comments

I have had two very substantial comments bearing on my January 2, 2008, posting, Religious Experience and Mathematical Experience I, the first from Craig Looney and the second from Michael Mascolo. They both deserve fuller responses than I think the comments section is best suited for. In this posting, I will first and more fully respond to Michael’s comments, because I think I see in them a greater difference from my views than I see in Craig’s.

You begin, Michael, by saying:

In my view, an issue arises when Polkinghorne makes reference to a “nonmaterial reality”. It makes it seem as though the object of a discipline has to be a kind of “thing”. Physics and chemistry and biology deal with tangible physical or organic “things” — look: We can see them; mathematics also deals with things, but non-material things.

My response: I don’t think that a reference to a “nonmaterial reality” “makes it seem as though the object of a discipline has to be a kind of ‘thing’, if a “thing” is understood to be a “tangible physical or organic” “thing.” While I think we have every reason to hold that the object of empirico-mathematical science has to be physical, as of the present writing I know of no reason to hold that the object of any discipline whatsoever has to be a kind of physical thing or reality. Were I to be presented with a demonstration that there is a divine being (or reality; the word “thing” doesn’t fit well here), then I’d be perfectly happy with seeing it as the object of a discipline.

 

You next say:

I don’t believe that it is helpful to say that mathematics deals with things. If we want to see sets as “things” or “objects”, at best they are “objects” of consciousness, in the intentional sense — as objects of an act. Mathematics, I would suggest, is constructed through action and symbolic action; I don’t believe it is helpful to say, for example, that mathematicians “discover” sets and groups and all of that as if these words refer to extant realities or non-material forms that exist in the world (or in heaven).

In response, I will quote again the passage from Alfred North Whitehead’s little classic An Introduction to Mathematics (London, Oxford, and New York: Oxford University Press, 1948) that I quoted in my November 21, 2007, posting, Further Specification of this Blog’s Philosophical Rationalism, and in which he tells us (p. 2):

The first acquaintance which most people have with mathematics is through arithmetic. That two and two make four is usually taken as the type of a simple mathematical proposition which everyone will have heard of. Arithmetic, therefore, will be a good subject to consider in order to discover, if possible, the most obvious characteristic of the science. Now, the first noticeable fact about arithmetic is that it applies to everything, to tastes and to sounds, to apples and angels, to the ideas of the mind and to the bones of the body. The nature of things is perfectly indifferent, of all things it is true that two and two make four. Thus we write down as the leading characteristic of mathematics that it deals with the properties and ideas which are applicable to things just because they are things, and apart from any particular feelings, or emotions, or sensations. This is what is meant by calling mathematics an abstract science.

That is, at least in the case of classical arithmetic and the classical mathematics that is that arithmetic’s elaboration, the objects of the science are real things, i.e., beings or realities. To take up my own stock example: if I have two and only two pennies in my right front pants pocket and two and only two pennies in my left front pants pocket, if no penny in my right front pants pocket is identical with any penny in my left front pants pocket and vice versa, and if  I have no other pennies, then I have four penies. In this sense, I have no problem with recognizing sets of pennies and in general, sets of things.

 Moreover, given what I have said above, I have no reason to hold that the things that constitute a set have to be physical things; sets of immaterial thoughts, if such thoughts do exist, do not trouble me. I don’t, however, think it necessary to hold that there exist sets over and above the things of which there are sets. Given the four pennies in my pockets just mentioned, there is no need to postulate the existence of sets over and above the pennies, whether the sets would be in in my pockets or in some realm of Platonic idealities. The same is true of numbers: while I have a number of pennies in the one pocket and another number of pennies in the other pocket, there is there is no need to postulate the existence of numbers over and above the pennies, whether in my pockets or in some realm of Platonic idealities.

The pennies constitute and exhaustively constitute the sets and the numbers, just as John and Mary Doe-Smith constitute and exhaustively constitute the couple that is the Doe-Smiths.

You go on to say:

A small thought experiment might help me make my point. (This comes from Piaget). A child is counting pebbles. She counts the pebbles and finds that there are 8. Then, starting with a different pebble, she counts again, and comes up with 8. And then again! She performs three different acts of counting. In yet another constructive act, she then abstracts what is common to all of these acts to construct the idea of conservation of number — the number remains the same despite how one counts them.

I agree that, errors or an addition or subtraction of pebbles aside, the three different acts of counting will yield the same answer. And I agree that the acts of counting are constructive acts, as long as we understand that the construction is not of either the objects being counted or their number, i.e., the number that they constitute; rather, it is the conceptual apparatus, the ideas, by means of which we think and count that the acts construct.

 

I don’t, however, think that she goes on to construct “the idea of conservation of number.” The number remains the same because, well, the number of pebbles remains the same and she counted them correctly all three times.

You then go on to ask:

Is the notion of conservation a property of the world? Of a rational mind? Is it an “object” that exists in Platonic Heaven? I would suggest that the notion of conservation of number is a product of action-on-the-world; it is produced through what Piaget called “reflective abstraction”. Note: I’m not abstracting over objects; I’m abstracting over my actions-on-objects — a crucial difference!

The notion of “action-on-the-world” troubles me, if it means anything other than that in counting the pebbles, she is engaged in a mental acts and the object of her action is the set of pebbles in front of her. If it means that the mental action of counting as such effects a real change in the pebbles or their number, I’ll have to say that that does not correspond to the experience that I have had in counting pennies or pebbles.

And finally you conclude:

And so, mathematics is very much grounded, I would say. It is grounded in the regularities that we find in our actions and symbolic actions. (This, actually is also the case for sciences, but that’s a somewhat different story.)

Mathematics is different from theology!

I agree that “mathematics is very much grounded,” but grounded in the real things, beings, or realities populating the universe, and in our actions only insofar as they too are realities populating the universe.

Now on to you, Craig. You state that:

Theology starts with the existence of “God” (or some supernatural concept) and attempts to understand the world (or parts of it) in terms of this “axiom”. Mathematics (often) starts from axioms and works out the implications to see if anything interesting arises, often with little regard to whether they have any bearing on the “real” world. So theology and mathematics are similar in that they use axioms.

Interestingly enough, in their The Mathematical Experience, Davis and Hersh draw attention to Georg Cantor’s arithmetic of infinite numbers and go on to say:

Mathematics, the, asks us to believe in an infinite set. What does it mean that an infinite set exists? Why should one believe it? In its formal presentation this request is institutionalized by axiomatization. Thus, in Introduction to Set Theory, by Hrbacke and Jech, we read on page 54:

     “Axiom of Infinity: An inductive (i.e. infinite) set exists.”

Compare this against the axiom of God as presented by Maimonides (Misneh Torah, Book I, Chapter 1):

     The basic principle of all basic principles and the only pillar of all the sciences is to realize that there is a First Being who brought every existing thing into being.

They go on to say:

Mathematical axioms have the reputation of being self-evident, but it might seem that the axioms [sic] of infinity and that of God have the same character as far as self-evidence is concerned. Which is mathematics and which is theology?

I think it clear that, if axioms are expected to be self-evident, then the two “axioms” at hand are not really axioms. It would be better to call them postulates. Or, equivalently, we can formulate them as the antecedents of conditional statements such as, say:

If an infinite set N exists, then, for any number of objects M, N + M =  N.

Or:

If a First Being exists, then an ontologically simple being exists.

Unless and until we know that the antecedent conditions are true, our theories are but hypothetical. When and only when we have proofs that, in the one case, there is an infinite set, and, in the other, that there is a First Being, can we have categorical theories, theories, as you say, “bearing on the ‘real’ world.”

Tying my responses to your comments together with my responses to Michael’s: the arithmetics of my pennies and that of her pebbles are, in the terminology that I have just used, categorical and not hypothetical.