Thursday, February 21, 2013

Down the rabbit hole--infinity, paradoxes, and weirdness, Part 5

Now it is time to reveal the humanness of mathematics and bring ourselves to the gate of yet another deep and unsettled debate in the philosophy of mathematics, a debate at least two and perhaps three millennia old. Here, I want to talk about axioms--which given recent scientific findings about the Higgs Boson (don't call it the "God particle") might really be the only way we can even have a concept of infinity to talk about.

Although I've gone on about them without really clearing up what they are, axioms are statements that we are unable to prove and yet that we assume the validity of in order to get started with the logical process, be it deductive or inductive. This is why I say claiming that God exists is an axiom, a thought reinforced by claims of apologists like William Lane Craig that "in order to get to God, we have to start with God." We can cry circular reasoning all we want (legitimately--one does not seek to prove axioms, only to see if they can be proven from a different set of axioms), but if they want to define God into abstract "existence" that way, I don't really care. Axioms undergird the entire process of doing logical reasoning: a set of axioms together with a particular logical framework creating an axiomatic system within which we are able to determine the "truth" of various statements and propositions. "Truth" is in quotes because we are actually only able to determine the truth values of certain statements with respect to the axiomatic system, where the logical framework itself contains, in part, the number and kinds of different truth values as well as how to determine which is being exhibited.

If this sounds like a rather manufactured endeavor, it's because in some respects, it is. Indeed, my position is that it's entirely manufactured--that reality has determined our methods of doing logic and not the other way around. Then, I'm not a Platonist. Platonists think otherwise, and this debate gets pretty ugly. As a relevant aside, it is worth noting that this Platonism is related at its fundamental level to the Platonism that underlies the philosophy from which Christian theology is built.

It's easy, if you're not a Platonist, to reject the Platonic position as being rather silly, but it's not so easy to do it. Indeed, we don't choose our axiomatic systems willy-nilly (unless we're theologians...), but rather we attempt to choose both the logical framework and the axioms that we will use within it by observing reality and trying to make maximally basic "self-apparent" statements about it. For the Platonist, this leaves open the argument that it is a human failure to understand the proper logic and axioms, instead using crude approximations and erroneous guesses, that causes the problems (usually paradoxes), with the further assumption that with better axioms and the proper logical framework, these problems will not be so glaring--and questions we cannot answer now will have answers in the superior framework.

It is actually more complex than this already complex mess, though. In the first half of the twentieth century, a logician named Kurt Gödel proved his famous "incompleteness theorems," which assert that every (nontrivial) axiomatic system has the limitation of being either incomplete or inconsistent. Incompleteness means that there are statements that can be made from within the axiomatic system but whose truth values cannot be determined from within the system. Inconsistency implies that there are paradoxes afoot. The ultimate theme of this post is to illustrate that Gödel's most famous theorems force us (again, humanness playing a clear role) to make choices about what the fundamental logico-axiomatic systems are that we want to use for various purposes.

Before getting to that, it is very tempting to say "hey, wait a minute... doesn't Gödel's work kind of cast some doubt on the Platonist position?" Well, it can be read  that way, but to really complicate this messy affair, Gödel himself was a Platonist.

What does this have to do with infinity?

Infinity is supposed to be the central theme of this series of essays, and so I need to point out that infinity isn't exactly just an example of this problem at play--it is the example. Dating into antiquity, debates about the infinite (and their inverses, infinitesimals) have been a major issue in the philosophy of mathematics. Archimedes rejected infinity outright, and so we call the rejection of the infinite (itself an axiom) the Archimedean principle. Others developed complex and intractable notions about it, given the mathematical machinery of the day. It took another mathematical camp, formalism, to come along to start to "shed light" on this problem, but that's rather the exact opposite of what happened.

Number theory has been a very active field of mathematical research and development since before the Renaissance. For a long while, a set of axioms that were formalized in the 19th century by Giuseppe Peano, and thus bear his name, underwrote the field of number theory. These axioms, in and of themselves, are not terribly controversial and appeal to reality very nicely: zero is a number, equals means what equals means, etc., and that the successor (add one) of any number is also a number.

Okay, so if we have that adding one to any number provides another, bigger number, then we predict that the total number of numbers is not finite (I've purposefully not said "infinite" here). Thus, we could argue that the Peano axioms seem to suggest that "infinity" is a meaningful concept--that which is characteristic in size of the entire collection of natural numbers, for example. The issue is that when we go to formulate set theory using the Peano axioms, we end up with the Archimedean principle, i.e. the rejection of infinite sets. The Peano axioms suggest that "infinity" means something but disallow infinite sets. This was the topic of the first part of this series (Link).

This became majorly debate-worthy because it happened before Gödel did the work that explained what was going on--indeed, this debate is why Gödel was working on this problem in the first place, as it was part of more than one of Hilbert's famous list of essential problems for mathematics to solve in the twentieth century, written and delivered in 1900. The problem is that the Peano axioms form a system that is either incomplete or inconsistent, and so we are left with some choices. Critically, the Peano axioms themselves cannot tell us which choice is correct.

To complicate matters further, still ahead of Gödel, Cantor was able to do history-changing work a few decades after Peano that really opened up the thinking about infinity and forced a new axiomatic construction to reformulate set theory entirely. Infinity got very contentious and was rocking the entire foundation of the philosophy of mathematics as the nineteenth century evolved into the twentieth--and again it's because looking closely at the questions raised by the ancient concept of the infinite was making a mess of things.

The choices seem to complicate things

There are at least four choices that we can choose in the situation presented by the question raised about infinity by following the logic of the Peano axioms.
  1. We can accept the axiom of infinity, and with some work, number theory can be rewritten in terms of the set theory that results (called Zermelo-Fraenkel set theory, which is the foundation for the most commonly used axiomatic conception of mathematics today).
  2. We can accept the negation of the axiom of infinity, which leaves us with unanswerable questions about infinity.
  3. We can chalk up the question of the axiom of infinity to "incompleteness," leaving it unanswered and unanswerable, i.e. accept the Peano axioms as they are without adding more axioms to it, at the cost of being unable to answer very intuitive and natural questions.
  4. We can use it to decide that the Peano axiomatic system is inherently flawed and start over.
All four of these endeavors have been followed, and different mathematics erupt from each choice. These different systems occasionally give contradictory answers to the same question, making statements true in one system and false in another.

I would suggest that formalists would say that it doesn't particularly matter which path we choose, or that we can follow all of them for what they're worth, but Platonists cannot accept this. Platonists essentially believe that there is a correct logic, Logos, and our endeavors in these veins are seeking to discover what it is. For the Platonist, one or more of the above choices, since they have some overlap, is the right choice--probably choice four, though most proceed with accepting the axiom of infinity in the resulting axiomatic framework (shared with choice one).

Let us now suppose that we do as mathematicians did following Cantor's work and refomulate the axioms that underlie mainstream mathematics according to choice one--which really, in practice, was choice four since it sought to build number theory from set theory and not the other way around. The Zermelo-Fraenkel axioms (ZF) resulted, and as Gödel would later be able to explain, the same problem arose on another unanswerable question, the question about the "continuum hypothesis."

Since it relates to infinity, a short aside about the continuum hypothesis: The acceptance of ZF set theory, as Cantor showed, is that there isn't just one size of infinite (requiring a more sophisticated definition of "infinite" than "not finite"). Indeed, there are infinitely many sizes of infinity under ZF! The natural question, then, is "which infinity tells us how many infinities there are?" The continuum hypothesis essentially states that the first infinity, the number of natural numbers, is the number of infinities. No one can prove that, though, from within ZF, although a large number of statements that are equivalent to it exist. This led to the question about another axiom: the axiom of choice, which states loosely that it is possible to select one object from each of infinitely many non-empty bins.

At this point, we get to make a choice with at least four possibilities:
  1. Accept the axiom of choice into ZF, which gives us the system ZFC, the most commonly used axiomatic system in modern set-theoretic mathematics.
  2. Reject the axiom of choice, maintaining ZF, which gives us the system ZF~C.
  3. Leave it alone, carrying on with ZF and leaving the question about C undecided (which actually speaks to how I was actually taught mathematics at the post-graduate level, with the proviso that most mathematicians accept C and note explicitly that they are doing so).
  4. Decide that ZF is inherently broken and reformulate from the ground up.
Secretly, there's a fifth one here, a variant on choice four, that pitches first-order logic in order to reformulate ZFC in second-order logic, known as the von Neumann-Bernays-Gödel system (NBG), which starts with "everything true in ZFC is true in NBG").

Which one is right? As Gödel tells us, nothing in ZF can tell us. Most mathematicians accept ZFC, but it carries with it a couple of fairly non-intuitive (and perhaps undesirable) facts if we do so:
  1. If we accept the axiom of choice, then it is possible to construct non-measurable sets, e.g. the Vitali set, courtesy of another eponymous Giuseppe. Without getting into the gobbledegook of it, measurability is the abstract extension of the concept of being able to determine the length (of intervals) and has been the key foundation of modern analysis, which is essentially calculus grown up.
  2. Accepting the existence of non-measurable sets enables the famous Banach-Tarski paradox, which says that we can take a three dimensional object like a sphere, decompose it into a finite number of non-overlapping pieces, and put those pieces back together without changing their "shapes" to yield two identical copies of the original sphere. In the countingest sense, then, the Banach-Tarski paradox can be seen, in a way, as suggesting that 2=1, and that's not good.
How are we to tell which one of these things is the right way to go? You tell me. I'm mostly a formalist, I think, so I essentially take the position that all of this is abstraction that attempts to give us the ability to understand nature by another abstraction that maps our calculations and models onto what we see. A Platonist, however, has to deal with it differently. There are at least five more key positions in the debate about which foundational position of mathematics is correct.

To make it more fun, Gödel tells us that whatever we decide to do to fix the problem, we still end up with a system that is either inconsistent or incomplete. At some point, we have to embrace that realistic limitation on our systems, but absolutely nothing provided by Gödel's theorem suggests where that line is. Some people draw it at the axiom of infinity. Other people do not.

What the hell, then?

Yeah, exactly. For me, all of this mish-mash just suggests that whatever shape reality actually has, our logico-axiomatic systems wrap themselves around it in our ongoing attempt to find the best approximation of understanding of it that we can have. If insights are available under one system and a contradictory system (e.g. ZFC and ZF~C), I'm not particularly fussed about it because I don't actually think ZFC or ZF~C or any other logical framework actually matches reality.

I see this as having a pretty big upshot in the debate about religion, especially about Christianity. Christianity, thanks to the Gospel of John and the leanings of many of the Church Fathers (including whoever wrote the Gospel of John, apparently) were Platonists or Neoplatonists. Particularly, they believe that a significant aspect of their three-in-one God is the Logos: The first chapter of the Gospel of John is an homage to how the Logos predated all and became flesh in Jesus. The upshot we get from this whole discussion, then, is that while the Bible claims this is how it was, we can see that there are distinct interpretations of the matter that have very solid meaning without having to accept the Platonist view. In my experience, in fact, although I cannot say this with certainty, very few mathematicians today are Platonists, knowing what they know about these things.

Hence, when Christians want to argue with me that logic itself proves that God exists as the giver of logic, not only am I glad to pin them to that definition of God that they've given (and cannot effectively connect to others as they need to), I'm comfortable in saying that that statement itself is on shaky philosophical ground and utterly without empirical weight. In other words, I have better reasons not to have to accept it than simply that it's a bald assertion--it's a bald assertion against other more plausible possibilities.

Another small upshot is that, as I mentioned, it allows us to see their God, which has no empirical evidence for it and which always seems to be defined as an (abstract) idea, as an axiom within an axiomatic system. That axiomatic system, then, is necessarily either trivial, incomplete, or inconsistent. My guess is that they side with inconsistency over triviality or completeness, but then it raises the question of why that choice, instead of the others, is being made. All of this makes it very difficult to see God as a being that does anything.

What do you think?


For the other parts of this series, follow these links: Part 1, Part 2, Part 3, Part 4 and Related: On Reality and Logic.



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