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A fuzzy argument for the existence of God
Fritz Ward, Jr., Ph.D.

By Way of an Introduction.
This piece was written as a letter to a friend who is a devout atheist. I describe him as devout because atheists are some of the most religious people in the world: they have great faith (far more than I do) in their own innate ability to reason, and "science," and the explanations it provides for the world. Many atheists, my friend Tom among them, are also very evangelical, hoping to convert the rest of us to their perspective with rational arguments or, when that fails, by heaping ridicule upon those with religious conviction. Happily, Tom is not of the latter sort. He is very respectful, polite, and considerate of others. This may be due to a Christian upbringing, and it may also be due to his deep concern with questions of ethics, justice, and truth. But he has lost his faith in God, and now believes in science and logic.

I find this very fascinating, because my own life story is very different. I was brought up as an atheist, sent (sentenced?) to a "Unitarian" Church on Sundays, and taught that religion is basically superstition, although figures like Jesus had some very beautiful ideas. As I grew older, I lost my faith in "science" and "rationalism" and eventually found I was willing to believe anything, or at least not reject anything out of hand. Not even Christianity. And that was all the opening God needed in my life. But as I've continued my Christian journey, I have been amazed at how much of what is popularly thought to be "science" by fundamentalists of both the Christian and Atheist variety is, at best, only a crude representation of what science actually does. Indeed, far from being the enemy of faith, science is in fact vindicating many of the philosophical assumptions of religion generally, and Christianity in particular. This is not to say that the Bible is "proven" or even that one can be saved through "science." The latter is more a collection of works in progress in various fields, with fairly limited objectives. Nothing can substitute for a relationship with Christ Jesus, and as a fairly devout high church Episcopalian (read near Catholic here) I feel compelled to add, few things can add more to a relationship to Christ than some devotion to his Blessed Mother, but that is another story.

This particular letter to Tom grows out of a series of debates we have had over whether atheism is a valid perspective. Interestingly enough, the debate prior to this letter found me writing on the scientific evidence pointing to the existence of God, and Tom responding with traditional logic. It is pleasing, in a way, that atheists can no longer rely on science to defend their position, and they are left with putting together trite syllogisms like: If God is all powerful and all knowing and all good, then evil cannot exist. But evil does exist, therefore God cannot. Implicit in this argument is the idea that the attributes some theologians (and many Christians) ascribe to God contradict, and this "proves" God does not exist. And frankly, its a pretty silly argument. One really should first ask what evidence suggests there is a creator before assigning any attributes, contradictory or otherwise, to him.

In my last response to Tom, prior to this letter, I suggested that not only is science no longer a bastion of atheism, but that traditional logic had problems as well. I was already vaguely familiar with fuzzy set mathematics, and I knew it presented a serious challenge to traditional logic. Little did I know just how much. This letter is my first attempt to fully come to grips with what the implications of fuzzy logic are, and why they offer such profound insights to people of faith. They also, incidentally, destroy traditional logic, and with it, the last remaining refuge of atheism.

I hope to post a series of letters to Dr. Smith's website in the future. Most are letters to Tom, my atheist friend, but all are meant for anyone who is sincerely interested in the intellectual underpinnings of Christianity. The old claim that Christians lacked any real basis for their ideas is simply wrong. Advances in math, physics, astronomy and health all provide support for many Christian beliefs. On a suggestion from Dr. Smith, I have titled this essay, "In the Beginning was Math: A Fuzzy Argument for God." I hope you enjoy this essay and those that follow.

Sincerely in Christ, Fritz Ward

Dear Friends:

In my last public exchange with Tom, I noted that science is no longer a refuge for atheists, and logic is not far behind. For the last century or so, beginning with Bertrand Russell, the foundations for traditional logic: in particular the bivalent claim that one cannot have both A and Not A simultaneously has crumbled. It now appears, in fact, that not only can one have both A and Not A, but that the law of the excluded middle doesn't apply anywhere in the real world, and does not, in fact, even apply in mathematics. The simply reality is that our world is fuzzy, that is, almost everything in it has elements of both A and Not A. So, far from the discovery of contradictions precluding the existence of anything, as Tom so desperately hopes, the fact is that a lack of contradictions might be more compelling evidence for the nonexistence of something.

What I intend to do in this piece is give some practical examples of Fuzzy Sets, sets that are both A and Not A, and show how common they are. Indeed, the only place that they don't (apparently) exist is in the realm of abstract mathematics. Next, I will show (following Russell) that even the very foundations of math, set theory, is fuzzy. Finally, I want to review what Fuzzy truth means, how it can be falsified, and present a fuzzy argument for God. I confess right here that most of this material comes from a terrific book, 'Fuzzy Thinking: The New Science of Fuzzy Logic' by Bart Kosko. I like his presentation, which actually does not require a deep math background. My own introduction to fuzzy logic, from Dr. Roy Goetschel, one of the most brilliant mathematicians I have ever met, gave me the basics, but truth be told, I didn't fully understand much of his work, although it also influences my presentation here. I should add that while I like Kosko's book, I can't fully endorse it: in particular, I dislike his conclusions about morality, which he grounds in a very Hobbesian social contract. As something of an expert in that field, I found his treatment here superficial though quite amusing.

What is a fuzzy set? It is a set of all objects that contain characteristics of objects both within and outside the defined boundaries of the set. This will immediately seem like a "contradiction" to Tom, and he is correct. It is. But it is also true that virtually all sets of real (ie material) objects have precisely these sort of characteristics. Consider the set of all tables: an analogy that my professor liked to use. We all have in mind some set of objects that are "tables." But what objects are actually in that set? How about a large flat rock? We don't usually thing of a slab of granite as a table, but it might be. As a long distance hiker, I have used it as one on many occasions. What about a desk. The desk my computer sits on is also used as a table. (Because I'm stupid.) I've used school desks, both student and teacher ones as tables too. Indeed, the set is fuzzy, because it includes things in the set (are what are obviously, to us, in the set) and things that are outside the set: rocks, desks, chairs, etc. So a fuzzy theorist, confronted with the statement, this is a table, does not assign a True or False statement to the claim, but rather assigns a truth value to the claim. The value is somewhere between zero and one. An object very like a table is closer to one: less like a table, closer to zero.

But wait, I can hear the objection already. The problem with the above set is that it is simply not well defined enough. If you define table clearly, then other objects which may have features like a table but are not are excluded. This is the cry of the Aristotelians, who desperately want to avoid contradictions. If only we define a set carefully enough, then there will be no fuzziness to it. But the problem is that you can always find fuzzy sets on the border of any collection of objects, no matter how well defined. The first thinker to point this out was Zeno, but Russell and others have also remarked on the same point. Suppose some imaginary antagonist to Fuzzy sets, say someone with a real stake in Aristotelian logic, let's call him Tom, provided a definition of tables to the fuzzy theorist that clearly excluded, in good Aristotelian fashion, all flat rocks, computer desks etc, one would still have to ask, if "Tom" presented a table, whether is was still a table with one molecule of matter removed. "Tom" might answer yes or no, but in either case, if the experiment is repeated long enough, there would eventually be no table, and somewhere in between the truth value to the statement, "This is a table" would become "fuzzy." And the simple fact of the matter is, this easy experiment could be repeated on any class of real objects. Kosko likes to give an example of a non-fuzzy set by asking the question to audiences, "Are you married?" The answer is either yes or no, and this divides his audience into two clearly defined sets: A and Not A, Married and not Married. Actually, as I will shortly demonstrate, he's wrong even on this claim (or rather, the truth value of this claim is not near as high as he thinks it is). But, he then produces a fuzzy set by asking the same audience: "How many of you are happy with your jobs?" Some are, Some are not, most are a little of both: a fuzzy set. Philosophers like to ignore the second set: it after all fits into their category of "subjective" which is quite different from "objective" sets. Ie, subjective is fuzzy, so traditional logic ignores it by denying that the claim "I am happy with my job" (or partly happy, or somewhat happy) is a statement at all. But what of the marriage claim? Surely it is objective. But I'm not convinced. Ethel and I are married, though we never applied for a state license. We had the sacrament with an orthodox priest. Bill is married too, and lacking my strong libertarian feelings, he did get a license from the state. But his wedding was not sacramental (or at least would not be recognized as such by the traditional church) a matter that bothers him not one whit. Is it still a marriage if one partner cheats? How about if both partners are celibate? Traditional Catholic theology says the latter is not: consumatum ratum defines true marriage, and that's that. Others might not agree. The set of married people is very fuzzy indeed. In fact, the only reason Kosko is able to obtain a non fuzzy result is because, in fact, his question on marriage is just as subjective as his question on job happiness: He is really asking, do you think you are married: some do, some don't, but it's clearly a matter of their personal opinions, and as a result, even supposedly objective statements like "Ellie Mae is married" which traditional logic claims can be definitely determined, are also opinions, and not statements at all.

So the question is, are there any non-fuzzy sets? We've come a long way from Tom's contention that God cannot exist because of contradictory attributes, and my listing several things that do exist with contradictory aspects. We are now asking whether there are any non fuzzy sets: sets that lack contradictory aspects, and the answer is a resounding yes: in limited areas of mathematics there are apparently non fuzzy sets: the set of all even numbers, or the set of all prime numbers, or the set of all equilateral triangles, for example. Interestingly enough, however, these sets don't apparently exist in the real world. I've never seen an equilateral triangle, and neither, I might add, has Tom, or anyone else. Geometry, it has been said, is the application of perfect reasoning to imperfectly drawn objects (A bad paraphrase of George Polya.) More importantly, as Einstein has noted, "So far as the laws mathematics refer to reality, they are not certain [ie, they are fuzzy]. And so far as they are certain, they do not refer to reality. So, one could argue that math deals with non fuzzy sets, but only to the extent that math does not deal with reality.

But actually, even that statement is not true (again, read here "the truth value of this statement is less than 1, ie, the claim that math is not fuzzy is itself a fuzzy claim), because at the very heart of set theory is a contradiction, and on that contradiction, all other math is built. A very frightening thought when you recognize that in traditional logic, contradictions imply everything.* Here is Russell's contradiction in sets, as described by Kosko. (I read Russell on this subject today at Barnes and Noble, before buying a book on Number theory instead, and I still prefer Kosko's version. Basically, Russell is interested in sets of sets. Traditional logic accommodates this very well. The set of apples is part of the set of fruits, as are tomatoes. But what if, says Russell, we consider the set of all sets that are not members of themselves: Apples, Pears, etc, are not members of themselves, while the set of all sets is a member of itself. So we know which groups belong to the "set of all sets that are not members of themselves" or do we? Is the Set of all Sets that are not members of themselves a member of itself? If yes, then no, and if no, then yes. Happily, Russell put this problem into traditional English with his famous barber paradox: the barber shaves all men (and only those men) who do not shave themselves. So who shaves the barber? We tend to smile at paradoxes. They are cute. And we ignore them. But in math, particularly in set theory, a paradox, implying a contradiction, destroys the whole system. And the only way out is to simply get rid of the excluded middle. Ie., to accept a fuzzy element of sets, and ignore the distinction of traditional logic since Aristotle, between A and not A.

Fuzzy sets, as you might well guess, has challenged the foundations of traditional math and logic, and not everyone is happy with it. When Dr. Goetschel was interviewed for a job by my father at Boise State, there were several in the department who disparaged his work because it dealt with controversial material (you might not think math has much in the way of controversial material, but fuzzy sets were for years). They are more generally accepted now, in part because that is the technology behind all sorts of neat electronic devices: washing machines that adjust to their load, self focusing cameras, and even simulations on how to back up a truck. As you know, I've backed up a lot of them, and you may be surprised to learn that no one has ever been able to model a math equation to do it. But using fuzzy computer technology, you can easily simulate this. Fuzzy set and Fuzzy logic are winning out in math and the sciences not because the bivalent crowd are rolling over and playing dead, but because they work, and traditional based math and logic simply don't. Not coincidentally, Kosko is an electrical engineer, a field much more receptive to fuzzy thought than the math of Dr. Goetschel, now retired.

Before going to discuss Kosko's argument for the existence of God, I want to briefly not what fuzzy truth is, and what our atheist friends need to do to provide fuzzy falsification. All claims of fuzzy truth (or falsity, as the case may be) are represented by a number between 0 and 1, in much the same way that probability is represented. Thus, the claim that 2 belongs to the set of even numbers is fuzzy to the 0 degree, and its truth value is 1. However, the claim that the diagram below (inside the square) is a circle is fuzzy. There are no perfect circles, still less this one. A close examination of it (blow it up) will reveal that not every point is equidistant from the center. Nonetheless, it approximates a circle, so its fuzzy value is not 1, but is still above 0.

For now, let say that its fuzzy value is .95. I will show in a minute how to get fairly precise fuzzy values.

Now, suppose we were to assign the letter P to the statement "The above figure is a circle." In traditional logic, to negate the statement, we would say, "The above figure is not a circle." On a truth table, we would represent these claims with a "T" for when the statement is "true" and an "F" for when the statement is false. A good Aristotelian distinction between A and Not A. But in fuzzy logic, if we assign a truth value of say, .95 to the statement, "The above figure is a circle." then the negation of that statement would be to say its truth value is only .05.

With this in mind, I want to follow Kosko's argument closely for why the Universe exists, and then examine his argument for God. On the latter point, I will diverge slightly from his presentation, though I think the ideas I add to it are implicit in his text.

Several modern philosophers, especially positivists, want to argue that the question of why there is something instead of nothing is a meaningless question, and therefore cannot be asked. This conveniently allows them to avoid questions that strongly imply the existence of God, while simultaneously claiming that "Science" has "proven" that God does not exist. They argue it is meaningless because we cannot conceive of what "nothing" would be. But math has for years had a way of conceiving of nothingness, and it is the empty set, often represented as { } or . But what does this mean? Kosko's greatest innovation in this book, and the thing I found so exciting about it, is that he is able to provide a mathematical answer to this question. All one has to do is to recognize that the universe itself is a fuzzy set. And then one simply has to ask the question, "How fuzzy is it?" The answer to that question, and indeed the answer to all questions about the degree to which something is fuzzy is found in the formula


Or in English, the intersection of the sets A and Not A divided by the Union of the Sets A and Not A yields the fuzziness of any given set. And one can in fact get a fairly precise answer to such a fuzzy question. So back to nothingness, the empty set, the lack of the universe that philosophers can't talk about (especially when they are atheists). What can we ask of this? Well, one thing we can ask is how fuzzy the empty set is, and this yields a fascinating result. The fuzziness of nothing is 0/0: a mathematical impossibility! In other words, and this I think is just brilliant, math is impossible in the absence of the universe. Nothingness, the empty set, involves the great singularity of math. Not coincidentally, physicists also refer to the Big Bang as the great singularity. The creation of the Universe and the creation of Math are linked.

But how are they linked? (Here I begin departing a little from Kosko.) Does the creation of the universe involve the creation of math? Or does the creation of math involve the creation of the universe? This question is very important, because it has some strong implications. Even atheists now accept the creation of the universe in the Big Bang, though as I have argued elsewhere, they don't fully comprehend what it means. But what if math is created first, and the universe is a byproduct of math? Math is an entirely deductive exercise. It involves and requires conscious thought to recognize and create it, starting from a very few premises, or axioms. On the other hand, the universe is discovered, experienced and explored inductively. If the universe is the "primary" creation, and math merely a byproduct, then I don't think atheism is threatened. But atheism is threatened a great deal if the universe is a byproduct of math. In this case, there is intelligent design, and intelligent design of a form that precludes the objections raised to simplistic design arguments like those of William Paley. (Note, I accept that both math and the universe came into existence simultaneously, but am asking about which might be the prime "mover" as it were. Of course neither could be, it may be a case of genuine "coincidence" though, for reasons I will explore shortly, I think not.)

So which is it? I think most of us, myself included, think math comes out of the universe, and not the other way around. But Kosko raises the intriguing possibility that it is the other way around, and provides some startling evidence for it. First, consider that because math and science follow different methods, there is no inherent reason for discoveries in math to track science, or vice versa, but in fact, they do. And, here's what is truly interesting: virtually every major scientific discovery, paradigm shift or advance in scientific fields is preceded by math. In other words, the order or "laws" we find in the universe do not come from Kantian blinkers: we don't perceive order and then try to find math to decipher it. Rather, we discover math through entirely deductive processes, and AFTER we do so, the order of the universe becomes apparent. Examples of this could be compiled ad nausea: Einstein develops an equation to suggest that gravity is an illusion, and matter curves space. LATER experiments verify this. Or again, Claude Shannon develops laws of pure information theory and LATER experiments confirm they work on thermodynamics. The universe, in its own fuzzy way, tracks math, which means design. Or, as Kosko puts it in his own inimitable way, "There may be no God but the Mathmaker, and Science is His Prophet." What does this mean? In brief, it means that one way God could create the universe is to simply think math, and the universe comes into being as a byproduct. The implications of this are astounding, because it suggests not only that the universe is created by God, but how God did it as well.

The neat thing about Kosko's claim is that unlike many claims in theology, it can be falsified. Science tracks math, and has for about 4000years, but tomorrow it might not, and if not, then the claim goes out the window. One cannot say the same thing for the simplistic claims of atheists who argue that God's attributes somehow conflict, and therefore "God cannot exist." But it hasn't happened yet. Even areas of math with no obvious ties to the material world (Cantor's infinite sets come to mind here) still provide observations (in this case, how to do proper correspondence) that become the basis of other very useful discoveries, such as fuzzy sets as a good approximation of the physical universe. So, in the final analysis, one must say based on this argument alone, that the truth value of the statement "God exists" is very high indeed.

And this is where the intellectual bankruptcy of atheism is revealed. For to falsify the above statement, one does not prove that God does not exist. Rather, one proves that the true value is low. But even then, one would not have defended atheism. And what a proof it would be. To fly in the face of such a huge chunk of human experience is an extraordinary claim for an atheist to make. And as Carl Sagan has pointed out, extraordinary claims require extraordinary evidence.

Of course, for Christians, the statement "God exists" has a truth value of 1. That is because, in addition to the evidence of religious experience both personal and collective (preserved for us by Holy Scriptures and Tradition) we have relationship with God, through his Son Jesus, with the intercession of other Christians, including for many of us, the Holy mother of God. So, Tom, my challenge to you is to explore some relationship with God. You told me at one point you wished God would provide you with unambiguous evidence He exists. I think the above argument is about as unambiguous as it gets, particularly when combined with the recent advances in physics we have discussed earlier. But a truth value of 1 for the statement "God exists" really does require relationship. And I would urge you to at least be open to accepting the possibility of such a relationship. It doesn't require a lot of you, but I am not a Calvinist. You seems to want God to do it all. I think you have to at the very least be willing to let him. And I will continue to pray for you that you come into a fuller peace with Him.

God Bless you, and take care, Fritz

* In traditional logic, the direct statement, of the form "If....., then......" is false only when the antecedent is true, and the conclusion is false. But if the antecedent is false, then the statement is always true. Thus, traditional logic claims the statement, "The Door is Open and the Door is closed (A and Not A), therefore Tom is a Dalmatian" is true, since the antecedent in traditional logic must be false. Therefore, one can prove anything. This is why traditional math and logic abhors contradictions. But, as Bertrand Russell notes, a contradiction lies at the very heart of set theory, and therefore, everything is permissible. The only escape is to abandon the idea the A and Not A are mutually exclusive, which is precisely what fuzzy sets have done. And thank God, because Tom is not a Dog, but a wonderful friend instead.

Editor's note: Perhaps there is at least oblique scriptural support for Fritz's argument of math preceeding the physical Creation. I have always marveled at the Proverbs account of Wisdom, as a person, playing THE major role in Creation as interesting. Perhaps Fritz has given a rational reason for such depiction. Are not math and widsom one and the same?

In the following text it would indeed seem wisdom...or math...preceeded Creation.

By wisdom the LORD laid the earth's foundations, by understanding he set the heavens in place; by his knowledge the deeps were divided, and the clouds let drop the dew. (Prov 3:19-20, NIV)

This passage seem to again suggestion wisdom, or perhaps math did in fact come before Creation.

"I, wisdom... was appointed from eternity, from the beginning, before the world began. When there were no oceans, I was given birth, when there were no springs abounding with water; before the mountains were settled in place, before the hills, I was given birth, before he made the earth or its fields or any of the dust of the world. I was there when he set the heavens in place, when he marked out the horizon on the face of the deep, when he established the clouds above and fixed securely the fountains of the deep, when he gave the sea its boundary so the waters would not overstep his command, and when he marked out the foundations of the earth. Then I was the craftsman at his side. I was filled with delight day after day, rejoicing always in his presence, rejoicing in his whole world and delighting in mankind. (Prov 8:12a, 23-31, NIV)

Something to ponder, Norbert Smith, Ph.D. Email your comments, Docgater@aol.com