Below is a fanciful ancient Greek dialogue constructed by the brothers F. LeRon and Benjamin P. Shults. The dialogue is in response to an apocryphal lecture on mathematics by the Apostle Paul given on Mars Hill in Athens in the first century. The interlocutors are Philo, a skeptical philosopher, Priscilla, a Jewish theologian, Damaris, a theoretical mathematician, the addressee of the letter, one Theophilus, and the unnamed author who witnessed this lively discussion on theology and mathematics. Fortunately these interlocutors are privy to two thousand years of debate about the meaning of mathematics, including the likes of Russel, Frege, Cantor, and Goedel. Priscilla, the Jewish theologian, concludes:

"... an abductive inference to the best explanation of the conditions that support the possibility of doing mathematics at all, conditions that appeal to an inaccessible cardinal fulfill. The analogy suggests that as in mathematics, so in theological discourse, the =8Cproof=B9 is not possible withi= n the conditions of the system. For mathematics, the system includes finite and (mathematically not metaphysically) infinite numbers. For theology, th= e system is the whole of creation including finite intelligent beings who search for the infinite Creator."

F. LeRon and Benjamin P. Shults, Ph.D. is Professor of Theology at Bethel Theological Seminary and Benjamin P. Shults, Ph.D. is Assistant Professor o= f Computer Science at Western Carolina University.

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Mathematics on Mars Hill By F. LeRon Shults and Benjamin P. Shults

Most excellent Theophilus:

It has been nearly two millennia since I wrote my two letters about the lif= e and times of Jesus Christ and the history of his early followers, especiall= y the Apostle Paul. I am considering writing a comprehensive third letter about the ups and downs the church has had since then, but I simply had to jot you this brief note to tell you about the most fascinating conversation I heard recently at Mars Hill. The occasion was the banquet at a plenary session of an interdisciplinary conference on Mathematics, Philosophy, and Theology. Because Greece was the host country, and the location was at a conference center on the very hill upon which Paul made his famous original address, the steering committee invited him back for this special Athenian reunion. Here is an abstract of his plenary speech after the banquet: =20 "I perceive that in every way you are very mathematical. For as I passed along, and observed the objects of your study, I found also an axiom with this inscription, "There is an inaccessible cardinal." What therefore you strive for in order to secure the consistency of your theories, this I proclaim to you. The transcendent God, who made the universe and everythin= g in it, does not live in mathematical or scientific theories defined by you, nor is God served by logical proofs, as though God needed anything, since God gives to all people consistency and existence and everything. =20 "God asserted you so that you would try to prove your consistency, in the hope that you might feel after and find God. Yet, at the same time, God is not inaccessible to each one of you, but constitutively pervades the logic of your inquiry by the divine Logos. So God is not an inaccessible cardinal, yet, in God, truly, all of our greatest theories have a model. A= s even some of your own logicians have said, "If it is assumed that there exists an inaccessible cardinal, then the consistency of our simpler theories follows. =20 "Your simpler theories being then consistent, you ought not to think that the Deity is like an inaccessible cardinal, a representation of your axioms and deduction systems. Because of the resurrection of Jesus Christ, we may now say that the one true God is the source of the congruence between human intelligence and the intelligibility of the world, and that his coming will render that judgment obvious to all."

Well, Theophilus, things were going well until Paul mentioned the particularity of the Christian truth claims about the historical Jesus, which led to a most interesting dialogue at my table. I will try my best t= o reconstruct the conversation among my three new colleagues. Philo is a philosopher, a skeptical type who is much enamored by deconstructivist hermeneutics; actually, he reminds me of the sophists. Priscilla is a Jewish theologian, who amazed me with her knowledge of the religious traditions. Damaris is a professional theoretical mathematician, and it wa= s her task to help us understand some of the more technical language in the speech. As you know, Paul has this thing about becoming all things to all people, and in this setting he became a mathematician. This is how things went after the polite applause died down...

What is the babbler talking about, we all heard Philo sigh, not quite under his breath. On the contrary, remarked Damaris, far from babbling he seemed up to date on some of the current philosophical issues surrounding foundations of mathematics and the use of inaccessible cardinals in this debate. I would like to hear more from this man on the subject. Oh, please!, replied Philo, the speech was simply one more attempt to valorize =
a Euro-centric meta-narrative by positing some abstract universal Truth that allegedly explains and so controls human mental activity, simply another instance of white male hegemonic appeal to a putative absolute Logos, in this case buttressed by a patriarchal vision of mathematics.

Interesting criticism, Priscilla noted, then paused. But, she wondered aloud, why is mathematics so powerful an explanatory tool? After all it works not only for males erecting their architectural monuments, but also for females counting their cross-stitch or following a recipe, not only for Anglo-European academics, but also for aboriginal farmers working their crops. As Einstein once said, the most incomprehensible thing about the universe is its comprehensibility. It seems to me that mathematics, like philosophy, inevitably raises the question of the source of intelligibility and of existence, namely, God, a question taken up by theology. For my part, Priscilla continued, I must admit that I have heard some of this teaching about Jesus of Nazareth before, and I find it strangely attractive= . But this idea of an inaccessible cardinal, which seemed central to his argument, is foreign to me. I assume we are not talking about reclusive scarlet birds here. Perhaps you could help me out?

I would be happy to do so, replied Damaris, if you two would return the favor by expounding on your reactions to the possible philosophical implications of our speaker's claims. Her interlocutors agreed, so after asking the wait staff for more coffee (regular) she began. First, she reminded us, a cardinal measures the size or numerality of a set of things. So there are finite cardinals like 1, 2, 3, and so on, and there are infinite cardinals such as (1, (2, (3, and so on.

You seem to be saying that there are different infinite numbers, Philo interjected. Quite so, Damaris continued, although this seems counterintuitive on the surface, reflection on the continuum (the real numbers) forces mathematics to speak of the cardinality of some infinite numbers as larger than other infinite numbers. It seems not only counterintuitive, but simply contradictory, noted Philo. Can you give us a= n example? =20

Surely, offered Damaris. Consider the basic concept of "number." We say that two sets of objects, call them A and B, have the same numerality if it is possible to pair up members of A with members of B in such a way that each member of the set A is associated with exactly one member of the set B and there are no members of B left out. You see, we say that I have the same number of digits on each of my hands because I can pair them up, like so. Here she put her palms together. We say that any set of things has five members if I can pair up the members of the set with the members of th= e set of digits on one of my hands. Now, let us consider infinite sets. Would you guess that the set of positive integers is larger than the set of positive even integers in this sense or that they have the same size?

My first reaction, offered Priscilla, is to think that the set of all positive integers must be larger. On the other hand, suggested Philo, if they are both infinite, then it would appear that they are the same size. They are the same size, Damaris responded, but that does not follow from th= e fact that they are both infinite. The reason they are both the same size i= s that it is possible to pair up the positive integers with the positive even integers as follows: for each member, n, of the set of positive integers, w= e associate the number 2n in the set of even positive integers. Notice that every member of the first set is associated with exactly one member of the second set and there are no members of the second set left out of the pairing. That is why we say that those two sets have the same size or the same cardinality.

Can you give an example, asked Philo, of an infinite set that is not the same size as the set of positive integers? Certainly, replied Damaris. Sh= e went on to argue convincingly that the set of real numbers between zero and one cannot be paired up with the set of positive integers. She concluded that the set of reals between zero and one is larger than the set of positive integers. This proof can be found in various texts and so I omit it from my letter. Consider the set of real number between 0 and 1. This set can be described as the set of all decimal numbers of the form 0.n1, n2= , n3 ..., where each nk is a digit. Now I will demonstrate that there is no way to pair up this set of numbers with the set of positive integers. My argument will proceed by reductio ad absurdum. If I begin by assuming "suc= h a pairing exists" and derive a contradiction, then the assumption is false (i.e., "no such pairing exists"). For of course any assumption that logically implies a contradiction is false. So, starting with the assumption that we have such a pairing, let us construct a table with two columns. In the left-hand column, I will list the positive integers in order, one in each row. On each row of the right-hand column, I will put the real number between 0 and 1 that is associated by our pairing with the integer in the left-hand column of that row. Our current situation, then, is that, under the assumption only that I have such a pairing, I now have a table, albeit infinite, that contains in its right-hand column, each of the real numbers between 0 and 1.

Now, Damaris continued, I will demonstrate the contradiction by producing for you a real number between 0 and 1 that is not on this list. I will cal= l this new number d. The number d will have the form d = 0.d1 d2 d3... Here is how I construct d. To choose the first digit after the decimal point, i.e., d1, I look at the first digit after the decimal point of the number i= n the right-hand column of the first row on our table. I simply chose d1 to be some other digit. So far, we know that my number is not going to be identical to the first number on the list because it disagrees with that number in the first position after the decimal point. Now, how do I determine the next digit, i.e., d2? I look at the second digit in the second number on the table and I choose any other digit to hold the place o= f d2 in the new number I am forming. And so on for the third decimal place, etc. In general, I chose the digit dk by selecting a number that differs from the kth digit of the kth number in our list. If I continue this process, then the number d that I construct will differ from the kth number on the table in the kth digit. Therefore, the number I constructed is not on the table. I have contradicted the fact that we can find a pairing between the positive integers and the real numbers between 0 and 1. You see, I can go through this process for any such alleged pairing of numbers.= ]
She concluded by saying that the set of positive integers is not the same size as the set of real numbers between 0 and 1. Indeed, the latter is the larger of the two.

That does make sense, Philo commented thoughtfully, if you are using the term "infinite" in this mathematical sense. Philosophers are accustomed to discussing "the infinite" differently. What do you mean by "the" infinite, Philo, asked Damaris - why the definite article? The traditional philosophical meaning of the infinite, Philo explained, is related to its conceptual opposition to the finite; it denotes the concept of the unlimite= d and unlimitable. Then we must ask, is it really even a concept? For a concept is inherently limited in so far as it is grasped and differentiated from other concepts. So we have the problem Hegel called the "bad infinite." When we think of the infinite, we are naming something that we take to be distinct from the finite, i.e., not finite. However, we can onl= y name it or experience it as it is mediated through the conditions of finitude. But the problem with stopping there is that this makes the "infinite" a thing, an object, set over against and opposed to the finite. By definition, a finite thing is something that has limits; a thing is finite because it is not something else. The problem is that we easily fal= l into defining the infinite in a similar way. Why is this a problem? If we define the infinite merely in terms of what it is not, then we have just made it finite (de-fined it). We then think of it as limited by its relation to the finite, and so not truly infinite (not unlimited). For the infinite to be `true infinite,' it must be beyond the distinction between finite and infinite; it must not be merely a thing opposed to the finite.

Sounds like this inevitably brings up the idea of divinity, Damaris responded. Indeed, said Philo, for Hegel it did bring up the need for an Absolute Spirit. That is why many philosophers reject it as a task of philosophy. Much more popular these days is simply defining philosophy as the play of hermeneutics. All we have are language games, and the pluralit= y of interpretations must simply be accepted; we must give up the chimerical dream of finding "the truth."

Well, giving up is certainly one option, remarked Priscilla, but surely not the only one, even if it is quite popular. Perhaps the infinite, in the philosophical sense, is not a concept like other concepts, but rather the condition for the forming of all finite concepts. Here, I believe, one inevitably treads the soil of theological reflection. In his third "Meditation" Descartes saw the dependence of the concept of a finite self o= n the concept of the infinite. The former presupposes the latter as a condition for its existence. But perhaps his so-called "ontological" argument has not fared well because he tried to make it do too much, and th= e "infinite" became one existent concept among others. A major stream in the Christian tradition resists this tendency with its "apophatic" or negative approach to language about the infinity of God. The transcendence of God i= s beyond the finite concepts of transcendence that presuppose the polarities of near/far and small/large. Paul mentioned something like that in his speech. In my tradition, the medieval philosopher Maimonides contributed greatly to a similar insistence on the incomprehensibility of the divine essence.=20

So you admit, exclaimed Philo, that you theologians know nothing about God. Then why should you keep talking? And you accused me of giving up! Please note, replied Priscilla, that I did not say that we know nothing, but that we recognize our knowledge of God (as infinite) is unlike our knowledge of finite objects. It is true that we cannot draw a line around (or conceptually grasp) the infinite God. But it may be that we are ourselves grasped and known by this God, and so can "know" as we are known by this God. At this point, Theophilus, I pointed out that Paul had written something similar elsewhere. Priscilla asked me in which journal it had appeared and I promised to explain later.

Then Priscilla continued. Theology explores the ultimate conditions that determine the human search for the True, the Good, and the Beautiful. Although we falter, we are drawn toward these ideals, humbly admitting that they are not within our "grasp." We know that the source of these ideals i= s not susceptible of rational definition in the same way as are particular things that are true, good, and beautiful. But this is not a failure of th= e human mind, simply a recognition that reason has gone as far as it can go. Apophatic theology cannot stand alone as mere negation, but must be complemented by kataphatic theology, a humble speaking of God that recognizes its limits.

Then human language cannot capture God, ventured Damaris. This brings to mind some relatively recent results in algorithmic information theory. It has been proved that, in any interesting theory, there are truths that cannot be deduced from any principles simpler than the statement of the truths themselves. In other words, in any useful and sound linguistic system, there are infinitely many truths that we must accept without being able to break them down to simpler axioms. When we talk about something as complex as the universe, we sometimes have to resign ourselves to saying, "it is what it is," because there is no way to reduce it to simpler axioms.

Can you give some details about this so that we can understand it better, asked Priscilla. Consider the sentence, "This sentence cannot be proved." Is that sentence true? What does it mean for a sentence to be true? interrupted Philo. Well, a sentence is true if the interpretation of the sentence in the universe lines up with reality. So the sentence "This sentence cannot be proved" is true if, in reality, it cannot be proved. So the question about whether that sentence is true comes down to the question of whether that sentence can be proved. Philo asked, How is being provable different from being true? Well, a sentence in some logical system can be proved if we can begin with the assumptions accepted in the field of study and apply the rules of logic to those assumptions and build up to the sentence in question. In a sound logical system, it is impossible to prove a sentence that is not true. However, I will show you in a moment that the sentence in question is true but it cannot be proved.

If we could prove the sentence "This sentence cannot be proved" then that sentence would be false. Thus if we were able to prove it, then it would b= e false and that would imply that the logical system we used to prove it must be unsound. We can conclude that the sentence "This sentence cannot be proved" cannot be proved in a sound logical system in which the meaning of that sentence is understood. Since we know now that the sentence "This sentence cannot be proved" cannot be proved, it follows that the sentence i= s true because, in reality, it cannot be proved and that is what the sentence says. So here we have an example of a sentence that is true but cannot be proved. Thus, in any sound logical system expressive enough to make statements about provability and sentences there are true sentences that cannot be proved. This exemplary sentence is, perhaps, not an interesting sentence in any sense other than its use in this example. However, there are interesting sentences, both simple and complex, with the same property: they are true but not provable.

But the situation is worse for most of these true, unprovable sentences. I= t is worse due to the fact that we cannot be sure that the sentence is true a= s we could with the example I mentioned above. The sentences are interesting and sometimes related to infinite or complex structures. Since they cannot be proved and they are too complex, interesting, and disputable to be clearly true, we are stuck with uncertainty. The generalized continuum hypothesis is an example of such a sentence in the logical system of set theory. The axiom of the inaccessible cardinal is another example. These sentences cannot be proved from the other axioms and they cannot be refuted= . But are they true? We can't be sure. However, since they cannot be refuted, they don't contradict our other axioms and so it is safe to add sentences like this to our set of axioms.

Interesting, mused Priscilla. That reminds me of the name God offered to Moses at the burning bush. But why should this idea surprise us? What is language, after all? It is a human tool, a culturally constructed symbol system that serves to define one finite object over against others. So for the term "coffee" to have meaning, this semantic symbol must function to point out that which we experience as "not tea" and "not sugar," etc. Language is made for the realm of finitude; this is its natural element, it= s homeland as it were. The problem is that when you have such a linguistic hammer, everything (including the infinite) begins to look like a finite nail. =20

It is no surprise, then, concluded Priscilla, that this tool does not help us conceptually grasp infinity. Yet it does help in a way; though it does not get us there, it gropes for the infinite. Perhaps, mused Damaris, that is what Paul meant when he said that we were made to seek after God. Perhaps he was making a theological analogy for the striving for mathematical consistency. Would you mind, asked Priscilla, explaining something about the mathematics behind that?

A theory is "consistent," began Damaris, if there is no sentence in the language that can be both proved and refuted in the theory. Obviously, thi= s is a desirable property. In the late 19th century, Cantor and Frege believed that they had successfully grounded all of mathematics on a very simple logical system that we now call "naive set theory." The universe over which this theory was interpreted was the universe of things and sets of things. Given any predicate, P, one was allowed to consider the set of things that satisfy the predicate P. This was all very good until Russell found a contradiction, that is, an inconsistency in this very simple logica= l system. This means that the universe mentioned earlier, now called "Cantor's Paradise," cannot exist. This inconsistency caused a stir among those logicians concerned with the foundations of mathematics, because they could not believe that the mathematical structures they had built up since Euclid were all to collapse. These logicians immediately went to work on constructing a new logical system, perhaps a bit more complex, that would b= e sufficient for mathematical reasoning and also consistent. That is, a theory that could be interpreted in a universe complex enough to hold the structures that mathematicians had been building up for thousands of years.

The mathematician Hilbert, continued Damaris, believed that the ideal theor= y would have done all this and admitted in itself a proof of its own consistency. So it would be a theory in which one could construct an interpretation under which the theory itself would be true. Various new logical systems and theories were developed, each of which allowed mathematics to be grounded, and no contradictions have been found in these theories to this day. But in the 1930s, Goedel (along with Turing and Church) showed that this ideal of proving the consistency of such an interesting theory within the theory itself was impossible unless the theor= y was inconsistent. But that does not help because one can prove absolutely anything one likes in an inconsistent theory, including "true = false."

Isn't that the "incompleteness theorem," asked Priscilla. Right, answered Damaris. Yes, added Philo, and this is connected to the growing sense of the failure of the Enlightenment with its apotheosis of human reason. If even "hard" mathematics cannot be proved, then we should give up such modernist ideals. I also think that foundationalist modernism has failed, agreed Priscilla, but this does not mean that all ideals are outlawed, or that the search for conditions of intelligibility must stop. It certainly has not stopped in mathematics. Turning to Damaris, she asked whether this is in fact precisely where the idea of the inaccessible cardinal comes in.

Exactly, responded Damaris. Let me briefly explain the idea of an inaccessible cardinal, and get your feedback. The goal is still to render intelligible the consistency of mathematics in general. One of the most popular new theories that developed after Russel found a contradiction in Cantor's paradise is now simply called "set theory." This refers to a set of axioms of first-order logic. (Actually, there are various collections o= f axioms each of which seems suitable for the task, one due to Zermelo and Fraenkel, called ZF, one derived from the work on von Neumann, Goedel, and Bernays, called NGB, and another popular one constructed by Morse and Kelley.) For the sake of discussion, let us say that we settle on a set of seven axioms of set theory. These axioms do everything we can expect. The= y give us a logical foundation for classical mathematics and we have not foun= d any inconsistency in them. Really we don't expect to find any because they have been so reliable for so long and we have a notion of a model for them as well. But the fact remains that we cannot prove the consistency of set theory using only those axioms. However, if we use these axioms with the addition of the axiom of an inaccessible cardinal, then we can prove the consistency of the original seven axioms. This is so because the axiom of the inaccessible cardinal essentially asserts that there is a model for the other seven axioms.

So, interjected Philo suspiciously, we end up being able to prove the consistency of this mathematical system after all? Well, yes and no, smile= d Damaris sheepishly. We cannot prove the consistency of the seven axioms without adding an eighth. But we cannot prove the consistency of the eight= h axiom without adding a ninth, and so on. With this eternal positing of unknowable realities, Philo objected, it sounds like you are siding with th= e Platonists against the formalists in the debate over the foundations of mathematics. Surely logic and mathematics are simply useful games we play, pragmatic inventions for constructing our lives together. Are you appealin= g to some ethereal external reality, some divinely secured foundation for mathematics?

I won't pretend, Damaris responded, to be able to solve this meta-mathematical debate, but actually Goedel's findings were a blow to the formalists. In my opinion, formalism is great pragmatically but cannot solve all of the philosophical issues. I am not satisfied by Platonism her= e either. We need some other philosophical framework to solve these problems= , or at least render the activity plausible.

Perhaps this is where a theological framework must also be introduced, suggested Priscilla. Your description of inaccessible cardinality reminds me of Cantor's idea of the Absolute Infinite. I know that his 19th century mathematical contributions have been superceded, but it seems that he was trying to do something similar, although the theological dimension was more clear in his case. Wasn't his proposal for an Absolute Infinite intended t= o explain the ultimate context for all transfinite numbers and sets? Is that related to this inaccessible cardinal?

They are related, confirmed Damaris. Both are attempts to develop a foundation for mathematics within the field of set theory. While Cantor's description of transfinite numbers (or transfinite cardinals) is still a very important part of mathematics, the notion of the Absolute Infinite quickly fell off of the radar of mainline mathematics. Even in Cantor's mind, I think that notion was more theological than mathematical.

I guess that is my point, said Priscilla. The fact that Cantor was driven by his mathematical reflections into the theological domain reminds me of what Paul said about God giving us a drive that leads us to search for the divine. There does seem to be an analogy, although not a perfect one. In the case of mathematics, the inaccessible cardinal provides the conditions for the consistent operation of particular mathematical models, although it cannot be "proved" within the system. In the case of theology, the claim that we are constituted by the infinite God illuminates the operation of ou= r inherent desire for existential intelligibility.

At this point, Philo became quite agitated and nearly knocked over a waitperson who was finishing the vacuuming around our table. Surely we are past proofs of God's existence, he proclaimed - didn't we learn from the 18th century that this is not possible? Slow down, Priscilla interrupted, =
I was not trying to move from a finite analogy to the infinite as, for example, Hume's character Cleanthes did in his famous Dialogues Concerning Natural Religion. Are you familiar with that text? More than you know, Philo sighed. =20

What I am suggesting, Priscilla continued, is more like an abductive inference to the best explanation of the conditions that support the possibility of doing mathematics at all, conditions that appeal to an inaccessible cardinal fulfill. The analogy suggests that as in mathematics= , so in theological discourse, the "proof" is not possible within the conditions of the system. For mathematics, the system includes finite and (mathematically not metaphysically) infinite numbers. For theology, the system is the whole of creation including finite intelligent beings who search for the infinite Creator. =20 But postmodernity has shown, Philo argued, that we cannot use logic to achieve neutral truth. Truth can be well defined, objected Damaris, in a formal logical system. However, I don't believe this is what you are reall= y objecting to. No, Philo is worried that I am still holding onto Enlightenment notions of truth, Priscilla began again, but I agree with him that human reason cannot grasp "neutral truth," because there is no such thing. But just because truth is not "neutral" does not mean there is no truth. Even the skeptic presupposes some conception of "true" in order to engage in dialogue.

It is important to note, Priscilla continued, that in dealing with postmodernity we have to do with a complex and ambiguous set of phenomena. Most students of culture agree that postmodernism (whatever else it may be) includes a challenge to the Enlightenment modernist ideals of absolute truth, universal reason, autonomous subjectivity, and inevitable progress. Scholars differ widely, however, in the way they respond to this perceived challenge. We can classify these responses broadly into three types. A de-constructive response fully affirms the postmodern challenge and concludes that because there is no neutral knowledge we must be content wit= h a plurality of interpretations. A paleo-constructive response would reject or ignore the challenge of postmodernity and appeal to an earlier pre-moder= n era in which truth and knowledge were allegedly unproblematic. Finally a re-constructive response attempts to distinguish the positive from the negative contributions of postmodernity and aim for a reconfiguration of th= e task of epistemology.

The task then, Priscilla continued, in our present discussion is to determine whether the human drive for understanding, which is exemplified i= n all of our academic disciplines, is best explained by positing that the multiplic

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Published   2003.07.16