Hello Kyle,
It's a delight to hear from you again. Here are my views on God and Mathematics.
By 'belief' in a proposition I mean a guess, or a hunch, or a suspicion that the probability of the proposition being true is somewhat greater than 50%. Using this definition, I believe that God exists, where God is defined as a transcendent sentience. That is, God is a thinking and feeling being who (the personal pronoun seems appropriate in light of the sentience) exists, or resides, in part or in whole, outside of the universe that is accessible to humans.
I believe this definition is consistent with most commonly held definitions of God. In fact, it includes a host of beings such as angels, devils, saints, and other demigods in which millions if not billions of ordinary people believe today and have believed for millennia.
My definition is also consistent with St. Anselm's definition: that God is "a being, than which nothing greater can be conceived, and it exists both in the understanding and in reality." Anselm's is more restrictive than mine, however, since it limits the number of Gods to one, and relegates all others who qualify for my definition to the status of demigods. I can accept that distinction in order to keep things simple, and to conform to such edicts as, "there is no God but Allah" and so I'll distinguish between 'God' and 'demigods'. I believe in both.
So far, I don't think my beliefs would have gotten me burned at the stake. But my next belief about God definitely would have. I believe that God is not almighty. That is, I believe that God is finite, limited, imperfect, and neither omnipotent, nor omniscient, nor omnipresent. I think the evidence in our world for that belief is overwhelming. Moreover, I think St. Anselm makes a good argument for the finiteness of God in his *Proslogium*.
Now, let me turn to Mathematics. You can think of Mathematics in several ways. I think a comparison with music might be useful to separate out a couple of these ways.
What is music? The dictionary I just grabbed lists five definitions plus another archaic one. Music can mean the art and science of producing emotion-laden sounds. It can mean the sounds themselves. And it can mean the compositions.
Now, we could ask, "Do you believe that [music] exists 'out there' independent of us?" thus paraphrasing your question about Mathematics. We would answer, "Well, yes and no". Certainly there are no CDs or books of sheet music "out there" which were not created by humans. On the other hand, we might say that if not frogs and crickets, then at least birds and whales, were probably making music before humans even arrived on the scene. The existence of the Music of the Spheres may even still be an open question to some extent, so music as the art and science might even have an existence that transcends human involvement.
So, if we consider Mathematics to be the content of the mathematical literature, then it is clear that mathematics is nothing more than "an abstraction of human consciousness". However, if we consider mathematics to be the relationships among the elements of the theorems, then it is clear that many of those relationships have been present among the elements of our universe, again long before humans arrived on the scene. So, in that sense we would conclude that "mathematics exists "out there" independently of us".
I don't think it is worth much time debating this particular question. It is simply a semantic distinction and depends solely on what you mean by 'mathematics'.
Now, to join the two ideas, if we suppose that there is a God, who has the ability to think, it is reasonable to wonder whether or not God "does mathematics". That is, can, or does, or did, God prove some of the same theorems you would find in the human mathematical literature? I think, that given my premise, it is reasonable to conclude that, yes, God does mathematics. After all, if God is "a being, than which nothing greater can be conceived" he/she could easily produce anything humans can produce. My guess is that God worked out some kind of body of "mathematical literature" which probably included at least all of what humans have been able to work out so far, and that this exists "out there" somehow. This, I think, definitely makes me a Platonist.
Now, let me go into some detail on my views of what legitimately constitutes a "mathematical literature", whether you are talking about the human or the Platonic literature. These views are a result of what I was taught in the process of earning B.S. and M.S. degrees in Mathematics, and a considerable amount of reflection and self study on the Foundations of Mathematics since that time. Not that I am an expert, by any means, but I have formed some pretty definite opinions.
The first requirement is consistency. Nothing inconsistent is knowingly allowed to become part of the body of mathematics. So, for the remainder of this essay, let me define 'the body of mathematics' to be a set of consistent theorems, along with the symbology, the primitives, the definitions, and the axioms which provide a consistent context for the theorems.
The current, modern, process for incorporating new theorems into the body of mathematics is somewhat different from the historical process. That difference being that it is acknowledged today that there is no such thing as a self-evident truth. Historically, it was believed that some propositions were true *a priori*. These were called 'axioms' to distinguish them from the postulates which were simply agreed to be accepted as true even if they might not "really" be true. Today we use the term 'axiom' to mean the same thing as 'postulate' did previously. There is no truth express or implied or acknowledged in an axiom.
It is in the process of accepting theorems into the body of mathematics that gets to the heart of your questions, Kyle. This is where I think it gets interesting.
The process involves two sentient beings. (Of course, it is possible for a single sentient being to play both roles in this process.) One of them nominates, or proposes, a statement which is a candidate for inclusion as a theorem. This statement is naturally called a proposition. Along with the proposition which is destined to become a theorem, the first sentient being must accompany the proposition with a series of other propositions which can be seen and understood by the second sentient being, to be consistent with the current body of mathematics. This series of propositions constitutes the "proof" of the theorem.
Among the propositions of the proof there are typically found definitions of new terms. These are nothing more than proposals for an agreement between the two sentient beings that a certain symbolism will henceforth be used to mean, i.e. be considered the same as, some combination of previously developed and accepted terms.
The most shocking and revealing exercise to me in my mathematical education was when the professor spent six weeks on the definition of the term 'number'. This was the watershed that separated out those who could grasp what was going on in the Foundations of Mathematics, and those who could not. Of course failing to grasp this set of concepts did not prevent those others from using very high powered math in physics and engineering applications.
Without going into detail about how 'number' is defined, suffice it to say that numbers are defined in a succession of ever more comprehensive sets, starting with the natural numbers and proceeding through the integers, the rationals, the reals, and the complex numbers. One of the hardest parts of this process is defining the first few natural numbers. That is, defining exactly what is meant by the number one, two and so on. As it turns out, the definition of the number one is the set of all singletons, where a singleton is defined to be a set consisting of a single element. The number two is likewise defined to be the set of all pairs. Three is the set of all triples, and so on.
This might seem weird, but that is the best way to define numbers, and I think Russell proved that it is the only consistent way to do it. (I'm not really sure about that claim, but I have that impression from somewhere.)
Now, if you imagine our two sentient beings in the process of defining the natural numbers early on in the process of developing a body of mathematics, it seems clear to me that the process can never lead to an infinite number of numbers. The propositions defining each distinct number must be set out in succession and at each and every point of this development, only a finite number of numbers will have been defined.
Modern mathematicians understood this problem and solved it (or in my view skipped over it) by including the Axiom of Choice The way I understood it when the professor explained it to us, was that the axiom of choice allowed you to assume that some kind of automatic process would go on independent of sentient involvement and without actually documenting the resulting definitions, but which would generate an infinite number of numbers.
There was a great debate about a hundred years ago on whether this should be allowed into the body of mathematics. Cantor showed how it could be done in a meaningful way, while Kronecker and Brouwer among others opposed the inclusion. Cantor won the debate and I still maintain that Kronecker and Brouwer should have.
Cantor's development of the nature of infinite sets immediately demonstrated antinomies, which in my opinion, should have immediately given the victory to Kronecker. But instead, mathematicians somehow accepted the nonsense.
Some twenty or thirty years later, Goedel proved that any mathematical system which contains Arithmetic is either inconsistent or incomplete and you can't prove which. In my opinion, this proof should have been the final nail driven into the coffin of the acceptance of the axiom of choice. Goedel's Theorem in essence says that if you allow infinity into your system in the hopes that it will make the system complete, first of all you can't know if you succeeded, but secondly, you can be sure that if you did, then you have introduced inconsistencies into your system.
The condition, that the system must contain Arithmetic, is, in my opinion, all too frequently overlooked. People seem to quote Goedel's Theorem as if it applies to all mathematics, which it does. But for finite systems, there is no question of whether the system is incomplete or inconsistent. All finite systems are by their nature incomplete. Knowing that, it is possible to know that some finite systems are consistent. It is only the systems that admit a concept of infinity in which we can't know whether the system is incomplete or inconsistent.
So it is obvious to me that in order to completely rule out the possibility of inconsistency, the system of mathematics must remain finite. Thus, I believe that if the world, i.e. reality, is consistent, it can be described or modeled using a finite system of mathematics.
Furthermore, to the extent that reality is consistent, it can be described in language, and it is the business of science to come up with such language descriptions. And, finally, it is my belief that Dick Stafford has proved that if reality is consistent, i.e. describable in language, it must necessarily conform to a set of constraints which are tantamount to the laws of physics. Thus, among the limitations of God, is the limitation that if he/she wants to create a consistent universe, then it must operate very much like the one we find ourselves in.
I have said nothing here that I haven't already said in this forum many times before, and I know that people think I am wrong, or crazy, or from another planet because of these views. But to them, I would say that, while I appreciate their opinions, what I am really interested in hearing are specific arguments that point out errors in the reasoning that has led me to these conclusions.
Thanks for asking about my views, Kyle, and thanks for reading this.
Warm regards,
Paul |