[Written March 29, 2001. Presented March 26. Senior Seminar notes. This is not a specific faith/discipline presentation. It is a framework to stimulate your thinking. Besides the references cited at the end of the talk, see my handout on Christianity and Computer Science from my Organization of Programming Languages class last Fall. This is a rough draft only. Please do not distribute until I have had a chance to revise it. I focus on papers that I've written, because one of my goals in writing this paper is to find common threads in my previous faith/discipline work.]
I describe three approaches to Christianity/Computer Science integration. I allude to a fourth synthesizing view, an Anglican view, which I am not prepared to develop for Computer Science, although I have developed it for an easier case study for my President's Scholar talk in 1997 on Christianity and Homosexuality. I set each in the larger context of faith/discipline integration. I call the three approaches the "Incarnational" approach, the "Incompleteness" approach, and the "Imago Dei" approach--although I must admit I picked the third title like a good preacher because I wanted them all to begin with I, after the first two naturally did, not because it fits as tightly as I'd like.
They could also be called the Lutheran, the Calvinistic, and the Anabaptist approaches, for reasons you'll see. In a longer version of this paper, I would like to develop what I regard to be an Anglican approach, and also a Catholic approach. The Anglican view would synthesize the others. The Catholic view--as represented by say Thomas Aquinas--is also more synthetic than the other views.
I. Incarnational integration: Who are we?
In 1983 when I was writing [8], I corresponded with Lynn Arthur Steen of the Lutheran St. Olaf College about integration of Mathematics and Christianity at St. Olaf. Steen replied with a photocopied statement from the St. Olaf faculty about their stand. In summary, it says that they don't think that there is a faith/discipline integration that is uniquely Christian. Later Steen became president of the Mathematical Association of America. In his inaugural address, he defined Mathematics as--to quote the title--"The Science of Patterns" (Science, 240, 29 April 1988, pp. 611-616). Steen integrates his Christian faith with his discipline of Mathematics by noticing the beauty and the order there, and giving credit to God as Creator for that beauty.
Most working scientists don't think much about the foundations of their disciplines. They think instead of how they interact with the subject matter. For example, Dr. Gerald Hess, a Biology professor at Messiah College, made this the focus of his faith/discipline paper. It could be summarized by the sentences, "I am a scientist, and I am a Christian. The integration takes place in me, not in the subject matter." Even Christian mathematicians, who are genuinely concerned about foundational issues, are sometimes called "Sunday Platonists." That is, they would argue because they think it is the right "theology" of Mathematics that mathematical concepts are real in some sort of Platonic way, but it doesn't make any more difference in the way in which the mathematician goes about doing her daily mathematics than church matters to a Sunday-go-to-meeting Christian.
Let me put the most positive spin on this view that I can. When God wanted to get our attention as we were careening away from Him, He didn't send us an argument, He sent his Son. God's method has always been to incarnate truth, not to discuss it. Jesus does not claim to bring the truth, He claimed to be the Truth.
How can I apply this to Computer Science? In the workplace, I will model virtue. In worship I will praise God for beauty, and for coming to us in Jesus.
I called this the Lutheran view not just because St. Olaf College holds the view. Dr. Richard Hughes in a lecture to California Lutheran University said, "The Lutheran vision never seeks to superimpose the kingdom of God onto the world . . . Lutherans seek to bring the world and the kingdom of God into dialogue." At Lutheran Valparaiso University one department chair says with a sneer, "Show us what Lutheran economics is and we'll see if we can find a Lutheran economist who can teach it." As for Luther's own view, it is closer to the Anglican view that I espouse in IV below--as some Lutheran scholars have pointed out--remembering his famous challenge to the Pope, "Unless you convince me by reason and by Scripture, I will not recant." Luther was interested in balancing the subjective of personal experience with the objective of revelation and reason.
The advantage of the incarnational approach is that it addresses the "so what" question. Who cares how faith and my discipline are related? No one, unless they see that it makes a difference in me.
The disadvantage of the incarnational approach is that it doesn't seem to be pursuing integration, rather just letting it happen. As one wag put it, " 'I don't have a philosophy of life,' is a philosophy of life." The approaches that follow are more intentional.
II. Incompleteness: What don't we know?
In the 1930s, mathematician Kurt Gödel proved a remarkable theorem, called the Incompleteness Theorem. It says
For any mathematical system S strong enough to do addition and multiplication of integers, the consistency of that system, Con(S), cannot be proven within the system.
This is a delicate result, since for example Pressburger proved that if you delete the words "and multiplication," the system can be proved consistent. Since we hope that computers will at least be able to do multiplication, Gödel's theorem seems to be a limitation on what computers can do.
It may surprise you that it's even possible to state Con(S) in S. We do that by a trick that Gödel came up with that we now call gödel numbering, honoring him with the lower case g in the phrase. For each symbol, represent it by a prime:
g(x)=2 g(y)=3 g( ( )=5 g( ) )=7 g(+)=11 g(*)=13
g(0)=17 g(1)=19 g(2)=23 ...
Since there are an infinity of primes, we can even number an infinity of variable names. Then represent each statement as a number gotten by raising a product of primes to the powers of the individual terms
g(x*(y+21))=2g(x)3g(*)5g(()7g(y)11g(+)13g(2)17g(1)
And finally represent a proof as sequence of statements, again by raising a product of primes to the gödel numbers of statements. So the single statement Con (S) has some gödel number, and so does any alleged proof of it, a proof being a sequence of statements of a certain kind.
Some philosophers see in Gödel's Incompleteness Theorem a more general argument that "humanity is not the measure of its own meaning." The argument is that if we can't even prove simple arithmetic is logically sound from within a formal mathematical system, how can we ever hope to have certainty about anything based only on the limited human mind, which can certainly do addition and multiplication? The first such argument was by J. R. Lucas in "Minds, Machines and Goedel." (Philosophy 36:112-127. 1961.) and it introduced a firestorm of debate.
By the way, if we know that Con(S) can't be proven in S, then why not simply create a new axiom system consisting of S Con(S)? The answer is that if we do, then Con(S Con(S)) cannot be proven, so we haven't gotten anywhere. The reason that I believe that Lucas is on to something is that Gödel-like incompleteness results depend on finitary reasoning, and I believe that we are capable of reasoning about infinity.
There are other paradoxes in the logical foundations of Mathematics. I applied one called "Skolem's Paradox" in a paper on predestination and free-will for example [9].
Are there incompleteness results in Computer Science (CS)? Yes. I claim that the the semantics of a computer program cannot be represented entirely in a computer program. This is trivial to prove if our semantics involve infinite sets, since computer programs are finite. It is impossible to prove if we leave open the future possibilities of what the semantics of a computer program could be. But let me offer one specific example. We want recursive data structures like linked lists to have a meaning. It is hard to see how there could be a finite model of arbitrary linked lists, the lengths of which are unbound.
One way that I have personally done "incarnational integration" of the kind that I mentioned in Part I above was to footnote how my Christian philosophy of mathematics--what I call a "complementarist" view" [7]--affected a purely secular paper on programming language semantics, "What does a computer program mean?" In that way, when I read the secular paper at Dickinson College in Pennsylvania and at Salisbury State University in Maryland, I could witness to God's character. The current version, [3], is the one that I read at Taylor University.
Are there other incompleteness results that point to what we can't know in Computer Science? Yes, here are two more.
Alan Turing in 1937 proved that the Halting Problem could not be solved. That is to say, he proved the following theorem.
No computer program U can be written which receives as input a program P and its input I, and returns true if P halts on input I, but returns false if P loops infinitely on input I.
If we are trying to decide whether a computer can think, one way to ask the question is to ask whether humans can solve the halting problem for some class of problems for which a computer cannot. This would be a kind of essential creativity. Yes, there are some problems for which a computer cannot solve the halting problem but humans can. Computers can only have access to a finite amount of information at any time; the problems are of the kind that a human can solve by infinitary reasoning.
This seems to settle the question of this section, "What don't we know?" Computers can't reason in all the ways that a human can--at least the ways in which a human with a Ph.D. in mathematical logic can reason! It does leave open the question of whether a computer can reason in all important ways for some narrower purpose, just as mathematicians do a lot of mathematics without reasoning about the foundations of mathematics needed to prove the consistency of the mathematics that they use.
A third problem has not been solved in Computer Science: Is P=NP? A problem of size n is said to be "(deterministic) Polynomial" or P if there is an algorithm to solve the problem that can be run in time O(nk) for the same fixed k for all n. A problem is said to be "Non-deterministic Polynomial" if an answer to the problem can be checked for correctness in a polynomial amount of time. Are all non-deterministic polynomial problems really polynomial? That is, are there any problems for which a solution can be verified in polynomial time, but the solution cannot be discovered in polynomial time? Surely we can test whether an array is sorted in O(n), but we require O(n lg n) time to sort it. O(n lg n) is polynomial, because O(n lg n) < O(n2). [At Messiah College we use O in contexts where it would be more precise to use Θ. I will have to help us to be more careful about this.]
Many problems that are NP have been proven to be equivalent in the sense that if one of them were P, then they would all be. One such problem is the Traveling Salesman Problem (TSP). The TSP is the problem of visiting, say, n towns of the United States with a path that is the shortest possible path. No P algorithm for the TSP is known. In fact, on the face of it one has to check all n! possibilities, and O(n!)=O(2n). But a supposed shortest path can be verified to be or not to be minimal in polynomial time. So it is not known whether P=NP for the TSP. A large class of problems have been proven to be equivalent to the TSP in terms of whether P=NP for those problems. For example, for packing a bin with as many as possible of n given blocks of various sizes and shapes in a way that leaves the least amount of empty space, we don't know whether P=NP. But we do know that if P=NP for the TSP, then P=NP for the bin-packing problem. Problems that are provably equivalent to the TSP in this respect are called "NP complete."
Is P=NP? Optimists say that P=NP is either true or false, and we will find out. Pessimists say that P=NP is undecidable using the rules of mathematical logic that we know about so far.
The "incompleteness" view of integration of Christianity and CS, then, focuses on the opinion that we should not be surprised that there are things beyond our grasp, using the tools that we have, because our tools are so meager!
How can we apply this view to our Christian commitment? We praise God for His greatness. We remember with the second half of Ecclesiastes 3:11 that God "has also set eternity in the hearts of men; yet they cannot fathom what God has done from beginning to end" and so we are actually encouraged by the way in which our deep investigations into science reveal how much like children we are. But what about the first half of that verse? We quote from the last verse of a great hymn,
Infinity - 10000 = Infinity
or to put it in English, "When we've been there 10,000 years, bright shining as the sun,/ We've no less days to sing His praise than when we first begun."
The advantage of this approach is the sense of humility, awe, and mystery that we have before a great God. John Templeton coined the phrase "humility theology" to describe the activities of this kind of faith/discipline integration. The disadvantage of this approach is that it has a "god of the gaps" feel to it. Will this view of faith/discipline integration be of diminishing importance as we learn more and more, as the "gaps" disappear?
If the "incarnational" approach is Lutheran, then the "incompleteness" approach is Calvinistic. The centerpiece of Calvinistic faith/discipline discussions is Genesis 1:28, the "creation mandate" or "cultural mandate" to subdue the earth. It goes far beyond cultivation of the soil. It extends to cultivating our minds. But for the Calvinist, this mandate is handicapped by another principle: Our minds are fallen so that we cannot rightly reason, just as our spirits are dead without Christ, so we cannot rightly obey God. At least, until redeemed by God, we cannot reason rightly.
III. Imago Dei: What do we know?
Jesus spoke in parables, as a way of communicating eternal truths by way of models common to human experience. A parable, a metaphor, or a model is merely a partial representation of reality. In that respect, when we think of models in Computer Science and we think of models in our Christian faith, we again face how little we know. But by thinking in terms of models, we focus instead on what we do know. We build from the ground up, rather than tearing down.
There are many "modeling" approaches that we could take to faith/discipline integration. The late Christian scholar Donald MacKay called his approach "Complementarity." In his IV Press book, The Clockwork Image, MacKay invited us to think about the physics of a sign flashing lights on a marquee, and then to think of the information-processing aspects of the sign. Suppose that the sign said
••••• ••••• •••• ••••
• • • • • •
••••• • • • ••••
• • • • •
••••• • •••• •
MacKay imagines us to suppose that a complete description of that sign could be given in principle from the point of view of Physics, in terms of electrons flowing and hot filaments glowing. Even granting that, he argues, the message of the sign is not at all described, the message to "stop." Likewise in his 1985 talk to the Association of Christians in the Mathematical Sciences (see in the Proceedings for that conference, just before my appreciation for him at [6]), MacKay argues that an artificial life modeling in every way the functions of human thinking in no way threatens issues of faith and responsibility before God, since those can only be described by a different level. The two levels are not contradictory. They complement each other.
Physicists accept the fact that light can act as a wave in some measurements and as a particle in other measurements. They regard the wave description and the particle description as complementary. Just so, a modeling approach to questions of faith and science can live with contradictions in view of the fact that the frames of reference may be different. (See my "Complementarity as a Christian Philosophy of Mathematics" [7] for an extended defense of this point of view.)
Let me get at this by telling a personal story. When I studied Chemistry, I enjoyed it more than Biology because it seemed to be a foundation on which Biology rested. Then I studied Physics, and enjoyed it even yet more, because I found it to be the foundation on which Chemistry rested. But Mathematics was the foundation of Physics, and hence all of science, and so I finally decided that I should profess Mathematics. For the same reason, within Mathematics foundational issues interested me most.
It was only some time later that my reasoning went full circle. I discovered, as I hinted above, that there was no certainty in the foundations of Mathematics. Instead, Philosophy and Psychology loomed as foundations of Mathematics. And within them, the hardest problem, the mid-body problem, brought me full circle to questions of Biology. But is this a vicious circle? I assure you that it is not.
Biology Chemistry
Physics
Philosophy and
Psychology
Mathematics
If one is looking for certainty, it seems as though there is no foundation on which to stand here either. Have I not replaced an unsure foundation with a circular foundation? Yes, if we view the circle as looking for a foundation, but no if we view the circle as the interplay of metaphors, as models each of which captures a part of reality, as stories which are complete at their level, but which miss other levels completely.
Contrast this with a diagram by mathematical physicist Roger Penrose. (As a mathematician, Penrose is known for the invention of Penrose tiles, for example). Penrose draws a similar circle that is a vicious circle. It is a depressingly pessimistic view, in contrast with my optimistic circle.
Mathematics
Physics
Mind
(Presented at the Joint Mathematics Meeting, Washington DC, in January 2000; and very briefly on the videotape by Michael Barnsley, "Is God a number?: Maths that mimic the mind"; or in his book ..., which I have not yet read.).
Penrose's diagram proposes that Mathematics is entirely a product of part of the Mind, but that Mind is entirely a product of part of Physics, which is in turn a product of only part of Mathematics. This is a strong reductionism quite incompatible with MacKay's multiple perspective view. It is engulfed by a triple pessimism. First, Penrose's strong physicalism says Mind is little more than its physics. Second, Penrose's strong Pythagorean idealism says that Physics is little more than the equations which describe it, as though we live in one great big wave function. And thirdly, Penrose has no explanation of how he can be both physicialist and idealist.
Contrast this thrice-pessimistic view with the optimistic view of MacKay, which says that even when we know fully at one level, we only know in part. This is not pessimism, because it holds out the greatest possible hope for a full understanding at any level, given the tools of that level. Instead of focusing on incompleteness, MacKay's complementarist view focuses on models that contribute to our understanding, building from the ground up rather than tearing down.
Let's apply this to Computer Science (CS). I've often said that I could teach CS without computers, because CS is not about computers--about electrons in semiconductors--, but it is a conceptual framework that includes what I have called the "dozen big ideas" of CS: abstraction, hierarchy, time-space tradeoff, parallelization, and others. (Compare <a href= http://www.cs.caltech.edu/~andre/general/computer_science.html>Andre DeHon's list</a>.)
To see how to apply our Christian faith to this view of CS as metaphors, we should ask how Jesus used metaphors--that is, parables--to point us to God. Parables are analogies. Analogical thinking is not deductive (general to specific), nor it is inductive (specific to general). If anything it is transductive in <a href= http://www.google.com/search?q=piaget+induction+deduction+transduction>Piaget's</a> sense, reasoning from specific to specific. Piaget recognizes this as a mode of reasoning that comes earlier in a child's development than either induction or deduction. In Artificial Intelligence, borrowing a term from philosopher Charles Peirce, it is called <a href= http://kwetal.ms.mff.cuni.cz/~gj/abduction.html>abduction</a>.
To pick just one kind of parable, Jesus uses universal social relationships to talk about who God is:
Father / Mother Master / Employer
Judge Friend
Husband Shepherd / Vineyard Owner
We often assume that Jesus' point was to teach us about God by reminding us about familiar relationships. C. S. Lewis points out that the analogy works the other way as well. God is indeed the prototypical Father, after whom all the fathers on the earth are named. It is God's Fatherhood that is the standard; my fathering that pales in comparison with His. Likewise for all of the other relationships. Marriage is an analogy of our primary relationship, our relationship with God. If you are adopted into a human family (as I said in [1]), that is just a small picture of what it means to be adopted by God, which all Christians are.
[[These two paragraphs are slide 22, which is out of order. I should either change the slides or change the order of these paragraphs]]
Next to these models, models in CS seem so sterile. An analogy is often made between software and Thomas Aquinas's view of the soul. Usually the analogy treats Aquinas's "soul" as a given, and looks for ways to view software as related to it, if we are Christians. Or, if we are thoroughly secular, we might use the analogy in the other direction to debunk the Christian view of a soul by saying that it's "nothing but" the software for our brain. MacKay criticizes this kind of reductionism--he calls it "nothing-buttery"--as being neither helpful nor nuanced. Reducing the soul to software is simply dismissive out of hand of five millennia of human history affirming that humans have souls that are more than just algorithms. ('Soul' is 'anima' in Latin, hence De Anima as the title of a book by Aristotle on the soul, and of another book by Aquinas of the same title.) Philosopher John Haldane of St. Andrews University in Scotland answers the title question of his lecture <a href=http://www.abdn.ac.uk/~phl002/as2.htm>"Could the soul be software?"</a> with a resounding "no." Insofar as Aquinas is representative of Catholicism, and that Haldane's argument holds, Catholicism does not see a close connection between Christianity and CS when it comes to software as an analogy.
MacKay would be more gracious. Models need only be partial representations of reality. Don't confuse the map with the territory, he would say. Complementarity can live with contradiction. Contradictions are different from incomprehension. Contradictory models can be like a craftsman picking up first one tool and then another to do a job, neither adequate alone, both necessary.
[[End of out of order slides or paragraphs]]
[[This paragraph did not make the slides]]
How do we come to know anything? The arguments under "Incompleteness" above focus on factual knowledge: "scio," I know (saber, kennen). The argument of this section is that there is another kind of knowing: "cognosco," I know (conocer, wissen) by familiarity with, by association, with wisdom. A modeling view of integrating faith and discipline points us to multiple perspectives of an environment that is not our adversary to subdue, but our home in which to live. We explore the familiar as we would get to know a friend.
[[End of paragraph that did not make the slides]]
Often when we say that we are made "imago Dei"--in the image of God--we think that we should reason, because God is Reasoner, or we should be creative because God is Creator. I submit that we are in the image of God in all of the metaphors about us that Jesus uses: as father or mother, as employer or judge, as vineyard keeper or shepherd. Each additional metaphor does not diminish us further, but enriches our understanding.
This view of faith and learning has an important advantage: It reminds us that we are at home in the universe.
As to the disadvantage, it is easy for us to slip into a relativism that says since there are many tools and many levels of discourse, there are many ways to view reality, and so no view is to be preferred. Why does this not lead to deconstructionism, which is ultimately a prison not unlike solipsism?
To explain that, I borrow from the Anglican view.
This third view of integration as our acting as in God's image--"imago Dei"--is thoroughly Anabaptist. One way to characterize Anabaptism is to say that it privileges fellowship over doctrine. Jesus says (John 13:36), "By this shall all men know that you are my disciples: if you love one another." The NT elsewhere provides doctrinal tests: To "believe in Jesus' name," to be able to affirm that "Jesus came in the flesh" (John 3:18; I John 4:2). But in the "love chapter," I Corinthians 13, we are reminded that we all see through a glass darkly, or as the NIV puts it, we see ourselves--in God's image, for sure--but distorted. At least until we are in heaven, where things might likely clear up.
[[This part that follows is actually better developed on the slides than here.]]
IV. A synthesis. [Pick a better title, maybe one beginning with I and then a colon and then a question!]
The Anglican quadrilateral, sometimes called the Wesleyan quadrilateral, explains how four ways of knowing complement each other, in terms of objective and subjective, and in terms of religious and scientific knowing.
...
Revelation Reason Experience Tradition
Experiment
[1] <a href=http://www.messiah.edu/acdept/depthome/mathsci/courses/sensem/manifesto.htm> Faith-Discipline Musings</a>. Web posting of March 15, 2001 about Intellectual Property.
[2] "Universe resists science." Harrisburg, PA: Patriot News, 9 August 1996 [op ed opinion essay, the title supplied by newspaper]
[3] "What does a computer program mean? An introduction to denotational semantics." A tenth conference on Mathematics from a Christian perspective. Proceedings of the biennial conference of the Association of Christians in the Mathematical Sciences held at Taylor University May 31-June 3, 1995. Robert L. Brabenec, Editor.
[4] "How has Christian theology furthered Mathematics?" An eighth conference on Mathematics from a Christian perspective. Proceedings of the biennial conference of the Association of Christians in the Mathematical Sciences held at Wheaton College May 29-June 1, 1991. Robert L. Brabenec, Editor. Later published as a chapter of Facets of Faith and Science; Volume II The Role of Beliefs in Mathematics and the Natural Sciences; an Augustinian Perspective. Ed. Jitse M. van der Meer. Lanham, MD: University Press of America. 1997.
[5] Seventh Conference on Mathematics from a Christian Perspective. Proceedings of the Biennial Conference of the Association of Christians in the Mathematical Sciences held at Messiah College May 31- June 2, 1989. Wheaton College Press, 1990. Editor.
[6] "Selected bibliography of Donald MacCrimmon MacKay." A Fifth Conference on Mathematics from a Christian Perspective. Proceedings of the Conference Held at the King's College May 29-June 3, 1985: 31-33. Robert L. Brabenec, Editor.
[7] "Complementarity as a Christian philosophy of Mathematics." In Harold Heie and David Wolfe, The Reality of Christian Learning: Strategies for Faith-Discipline Integration. Eerdmans, 1987.
[8] Bibliography of Christianity and Mathematics: 1910-1983. Dordt College Press, 1983. With Calvin Jongsma.
[9] "Skolem's paradox and the predestination/free-will discussion." A Christian Perspective on the Foundations of Mathematics, Proceedings, 28-30 April 1977.