Two Matters

1. Evolution in the Political Context Again: A Reference

In the January 11 edition of Inside Higher Ed’s “Views,” one Jason R. Wiles has written a nice piece, “The Huckster’s Artful Dodging on Evolution,” reviews Mike Huckabee’s “record on the teaching of evolution in the public schools — an issue that is not specific to higher education, but that ultimately can have a major impact on science education policy and the nature of intellectual debate in the United States.”

Neatly correlative to that summary of the history of Huckabee on evolution is an equally helpful comment by one R.W. Hoyer, “Why Mike Huckabee Should Not Be President,” presenting us with some pause-provoking data on the views of the American public on the topic. To give you a sense of the latter, Royer tells us that “according to a 1987 Newsweek survey just 700 of 480,000 life and earth scientists (0.14%) assign any credence at all to so-called creation-science, yet 87 million adult Americans (47%) support that perspective.”

I have already, in previous postings, devoted more of my very finite intellectual energy to Huckabee’s views than they deserve, at least on their own intrinsic merits. So I’ll content myself here with just giving you the URL that should take you to the article and comment: http://insidehighered.com/views/2008/01/11/wiles.

2. More on Theology and Mathematics: an Epi-Digression

With the matter of the relationship between mathematics and religion or theology on my mind, itself a digression from the matter of the relationship between science and religion or theology that I have been focusing on, I found myself reading a relevant review in the January 13, 2008, The New York Times: Jim Holt’s “Proof,” a review of John Allen Paulos, Irreligion: A Mathematician Explains Why the Arguments for God Just Don’t Add Up (Hill & Wang).

The review’s opening paragraph immediately caught my eye.

A physicist, a biologist and a mathematician walk into a bar. Bartender says, “Any of you believe in God?” Which of the three is most likely to say yes? Answer: the mathematician. Mathematicians believe in God at a rate two and a half times that of biologists, a survey of members of the National Academy of Sciences a decade ago revealed. Admittedly, this rate is not very high in absolute terms. Only 14.6 percent of the mathematicians embraced the God hypothesis (versus 5.5 percent of the biologists).

In my January 2, 2008, posting, “Religious Experience and Mathematical Experience I,” I quoted a statement from the The Mathematical Experience of Philip J. Davis and Reuben Hersh (p. 111-112) that sees the theologian’s belief in a nonmaterial reality as making it easier for a mathematician to believe in “mathematical objects which are simply one particular kind of nonmaterial object.”

Belief in a nonmaterial reality removes the paradox from the problem of mathematical existence, whether in the mind of God or in some more abstract and less personalized mode. If there is a realm of nonmaterial reality, then there is no difficulty in accepting the reality of mathematical objects which are simply one particular kind of nonmaterial object.

The results of the National Academy of Sciences survey suggest that the reverse may alos be the case: the mathematician’s belief in a nonmaterial reality may make it easier for a theologian to believe in a divine reality which is simply one particular kind of nonmaterial object.

I found Holt’s second paragraph equally arresting, for it bears on the Platonism of many mathematicians, including Davis and Hersh, that I noted in the January 6, 2008, posting, “Mathematical and Theological Realities: Responses to Two Comments.”

But here is something you probably didn’t know. Most mathematicians believe in heaven. Not a heaven with angels, but one populated by the abstract objects they devote themselves to studying: perfect spheres, infinite numbers, the square root of minus one and the like. Moreover, they believe they commune with this realm of timeless entities through a sort of extrasensory perception. Mathematicians who buy into this fantasy are called “Platonists,” since their mathematical heaven resembles the realm of the Good and the True described in Plato’s “Republic.” Some years ago, while giving a lecture to an international audience of elite mathematicians in Berkeley, I asked how many of them were Platonists. About three-quarters raised their hands. So you might say that mathematicians are no strangers to belief in the unseen. (Of course, mathematicians don’t drag their beliefs into the public square, let alone fly planes into buildings.)

Noting, parenthetically, that (Paulos, of course, is a mathematician who does drag his beliefs into the public square), let me go on to use Paulos’s own exposition (on pp. 46-47) to illustrate just how Platonist mathematicians can be.

Another example of creating something, the positive whole numbers, literally out of nothing is the mathematician John von Neuman’s recursive definition of them. Two preliminary notions are needed: The first, the union of two sets A and B, is the set of elements in one or the other or both of the two sets. It is symbolized as A È B. The second, the empty set, is the set with no elements. It is sometimes symbolized by a pair of empty braces: {}. The number 0 von Neuman simply defines to be the empty set. Then he takes the number 1 to be the union of 0 and the set containing 0. The number 2 he takes to be the union of 1 and the set containing 1, and the number 3 he takes to be the union of 2 and the set containing 2, and so on. Each number is thus the union of all its predecessors and derives ultimately from the empty set.

And so on infinitely.

Let me close this post by (1) asking why, if a mathematician can thus define numbers into existence, a theologian cannot similarly define God into existence and (2) stating that I can’t believe that the act of definition is up to the task in either case.

Addendum: The Wikipedia article (http://en.wikipedia.org/wiki/John_von_Neumann) on von Neumann begins:

John von Neumann (Hungarian Margittai Neumann János Lajos) (December 28, 1903 – February 8, 1957) was a mathematician who made major contributions to a vast range of fields including set theory, functional analysis, quantum mechanics, ergodic theory, continuous geometry, economics and game theory, computer science, numerical analysis, hydrodynamics (of explosions), and statistics, as well as many other mathematical fields. He is generally regarded as one of the foremost mathematicians of the 20th century.

Also, for what it is worth, the article points out that:

Along with Edward Teller and Stanislaw Ulam, von Neumann worked out key steps in the nuclear physics involved in thermonuclear reactions and the hydrogen bomb.

Postscript: I promise to do my best, in the next posting, to get back to the point I was trying to make with Davis and Hersh and the notion of mathematical experience and then, in the next or nearly next one, to return to my discussion of Polkinghorne’approach to science and theology. But I’ve been having so much fun.

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.

Towards a Definition of Religion

I have to confess to and apologize for what has to be, for a self-professed rationalist and a resolute one at that, a grave lapse; I was tempted to say, “a grave sin.” That is, I have been posting to this blog for a couple of months now and have said a number of things about religion without having offered a definition of religion.

That a definition of religion is important is evident. Take, for example, the first amendment to the U. S. Constitution, which I brought up in my December 8^th (Mitt Romney on Faith in America) discussion of Mitt Romney’s speech, “Faith in America.” The amendment reads:

Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances.

As I noted in the earlier posting, the clauses bearing on religion present us with two distinct theses:

Congress shall make no law respecting an establishment of religion.

Congress shall make no law prohibiting the free exercise of religion.

But one can only begin to determine whether or not a law is one respecting an establishment of religion if one knows what a religion is. Similarly, one can only begin to determine whether or not a law is one prohibiting the free exercise of religion if one knows what a religion is.

What the law has to say about the nature of religion is something that I hope to have, and fear that I will have, occasion to delve into in the coming weeks and months. For now I want to post a first draft of a definition and take note in its light of an obvious recalibration of what a resolute rationalist should be thinking of religion.

As for the first draft: a religion is:

(1) a set of beliefs bearing on that which is the primary and/or ultimate reality or mode of reality (or those which are …);

 

(2) a set of correlative attitudes and valuations; and

 

(3) a set of correlative approaches to practice and action.

As for the recalibration: it seems to me that we all, even the resolute rationalists among us and the physicalists or materialists among the resolute rationalists, have such beliefs: an absolute materialist takes physical reality to be the primary and the ultimate reality, prior to and beyond which there is nothing else, and furthermore has a set of correlative attitudes and valuations and a set correlative approaches to practice and action

 It needs to be understood, however, that this recalibration is not a backing away from the rationalism motivating this blog; it remains that my aim is, as I said in my very first posting, “to persuade those readers who might be needing persuasion to adopt the point of view of philosophical rationalism and so to abandon fideism and indeed all forms of willingness to adhere to beliefs out of proportion to the evidence in their favor.” 

That is, rationalism has no dispute with religion as defined above, that is, with rational religion. Its dispute is with fideistic or non-rational religion, which includes among its beliefs, whether explicitly or implicitly, one that says it is permissible to to adhere to a belief out of proportion to the evidence in its favor; indeed, it is typically not just permissible but virtuous or even obligatory to do so.

Religious Experience and Mathematical Experience I

 In my November 29, 2007, posting, “Theology and Science? What about Philosophy?”, I asked why it was that just the two disciplines of science and theology were being singled out for treatment in John Polkinghorne’s Science and Theology, because I saw two other theories or disciplines that were at least as pertinent to theology as Polkinghorne’s science: philosophy and mathematics. In that posting I argued for the pertinence of philosophy, noting that the title for the work that I would have then written would have been Theology, Philosophy, and Science.  I also said that, in the near future, I would turn to the role of mathematics and noted that the title for the work that would at that time come to my mind would be Theology, Philosophy, Mathematics, and Science. I will at least begin to consider the relationship or relationships that may obtain between mathematics and theology in today’s posting.

As I have noticed at least once or twice in earlier postings, Polkinghorne’s Science and Theology recognizes (p. 18) an important difference between science and theology:

Just as the object of scientific enquiry is the physical world, so the object of theological enquiry is God.

But he also views them as having an equally important similarity (p.20), in that they both “originate in interpreted experience.”

As with science, so even more with theology, the search for verisimilitudinous knowledge is subtle and manifold. Its character is cannot be reduced to a simple, flat description. For both disciplines, critical realism provides a concept that both acknowledges that there is a truth to be found and also recognizes that the finding of that truth is not achievable through the application of some straightforward and specifiable technique. Both disciplines are concerned with the search for motivated belief and their understandings originate in interpreted experience.

I am quite sceptical about the analogy, in particular about the implied thesis, that theologians “encounter” divine reality or somehow experience the divine. For while, as I said in my December 12, 2007, posting, “The Parallelism of Science and Theology,” it is evident in sensory experience that the physical reality of science exists, it is just not evident in sensory experience that a divine reality exists.I do not think, then, that we have a religious or theological experience, from which theology originates, that is analogous to the scientific experience from which science originates. That is the reason why religions like Judaism, Christianity, and Islam adopt versions of fideism, the thesis that we can have, but only can have, a genuine knowledge of the divine through faith, faith dispensing with evidence.

My reflections on Polkinghorne’s understanding that there is such a religious experience brought to mind a book I read a couple of decades ago, Philip J. Davis and Reuben Hersh’s The Mathematical Experience (Boston: Houghton Mifflin Company, 1981). 

The title is significant and I will take up the notion of mathematical experience in a subsequent posting or two, for one might well think that the understanding that Davis and Hersh have of mathematical experience might suggest a way of supporting Polkinghorne’s thesis. In this posting, however, I want to lay some groundwork for that discussion.

Let’s start by taking note that, just as Polkinghorne sees there to be a parallelism between science and theology, so Davis and Hersh (p. 111-112) see there to be one between mathematics and theology or religion, of which theology is the theoretical component. First, they see a similarity between the subject realms of the two disciplines, in that both subject realms are nonmaterial.

Belief in a nonmaterial reality removes the paradox from the problem of mathematical existence, whether in the mind of God or in some more abstract and less personalized mode. If there is a realm of nonmaterial reality, then there is no difficulty in accepting the reality of mathematical objects which are simply one particular kind of nonmaterial object.

For now at least I’ll pass by the “simply” in “simply one particular kind of nonmaterial object” because I wish to enjoy the question that Davis and Hersh go on to ask and the breath-taking answer that they give to it.

So far, we have discussed the intereaction between the discipline of mathematics established religions. We might also ask to what extent does mathematics itself function as a religion. Insofar as the “laws of mathematics” are properties possessed by certain shared concepts, they resemble doctrines of an established church. An intelligent observer, seeing mathematicians at work and listening to them talk, if he himself does not study or learn mathematics, might conclude that they are devotees of exotic sects, pursuers of esoteric keys to the universe.

Then, just as Polkinghorne sees there to be a dissimilarity between science and theology, Davis and Hersh note one between mathematics and theology.

Nonetheless, there is remarkable agreement among mathematicians. While theologians notoriously differ in their assumptions about God, still more in the inferences they draw from these assumptions, mathematics seems to be a totally coherent unity with complete agreement on all important questions; especially with the notion of proof, a procedure by which a proposition about the unseen reality can be established with finality and accepted by all adherents. It can be observed that if a mathematical question has a definite answer, then different mathematicians, using different methods, working in different centuries, will find the same answers.

Seeing mathematics thus enjoying that mode of superiority, Davis and Hersh go on to ask:

Can we conclude that mathematics is a form of religion, and in fact the true religion?

I am not sure how far this was asked with tongue in cheek. But it doesn’t matter, because, whether intentionally or not, they have raised a serious point. That is, if we have made the determination that two disciplines both exist as genuine disciplines, then the question of what relationship or relationships there may be between them arises. And one at least logically possible relationship is that of identity; if the two are not distinct, they are identical. Here, Davis and Hersh have directed our attention to the at least logical possibility that mathematics and the true religion, or true theology, are identical.

If we hold that mathematics is the science of nondivine sets and numbers, while theology is the science of God, we’ll not be inclined to identify the two disciplines. If, however, we hold that sets and numbers are the divine realities, we’ll be inclined to join the ranks of the Pythagoreans and identify mathematics and theology.

To return to science and theology: if we have made the determination that the two disciplines exist both exist as genuine disciplines, then the question of what relationship or relationships there may be between them arises. And one at least logically possible relationship is that of identity; if the two are not distinct, they are identical. Here again, Davis and Hersh have, albeit indirectly, directed our attention to the at least logical possibility that science and theology are identical.

If we hold that science is the theory of nondivine physical reality, while theology is the science of God, we’ll not be inclined to identify the two disciplines. If, however, we hold that physical reality is the divine reality, we’ll be inclined to join the ranks of the disciples of some Spinoza and identify science and theology.

One last thought, one that I enjoy thinking about even though I am not in the least tempted to go in its direction: if mathematics were to be identical to theology and science too were to be identical to theology and, then, of course, all three would be identical.