Mathematics and the Search for Perfection: the Inner Meaning of Tron

The problem with philosophy in mathematics is that it is no longer part of the high school curriculum (at least not here in Ontario). It used to be, but in the last reorganization of the curriculum, it was sadly removed.

Most of us don't get to study this in university either. Most science majors are directed into some form of applied math (frequently calculus, the Bane of Geologists*; otherwise statistics). Consequently, most don't think about the differences between scientific reasoning and that of the mathematical kind. I know I never used to.

Scientific reasoning is inductive; meaning that whatever truths there are in the Universe have to be inferred from observations. You never know if your perception of the truth is accurate, however, all you can say about it is that it appears to be consistent with observations.

Mathematical reasoning is different because mathematics is a human construct. A system of mathematics is composed of a few true statements and a set of rules (rules of inference) for transforming those few true statements (axioms) into other true statements (theorems). The rules are chosen carefully so that the statements which sprout from them are true, provided they were seeded with truths.

But we get to decide what rules to apply. We don't always have to agree; but like playing a game, if we change the rules, we change the game. That doesn't mean that other games are less enjoyable than the one we normally play.

For Euclidean geometry, one of the axioms states that given a line and a point not on the line, there is only one line parallel to the first which passes through the point. However, if you want to play a game where there are no parallel lines, then you are pursuing elliptic geometry. Or if you prefer to have more than one parallel line, in which case you are pursuing hyperbolic geometry. Arguing about which one is true is meaningless.

What defines a great mathematical system? Well, first of all, it should be consistent--meaning that as long as you apply the rules of inference to a true statement, only true statements result. In other words, all provable statements are true.

Secondly, we would like the system to be complete, meaning that it is possible to prove all true statements. In other words, all true statements are provable.

Both of these criteria have different meanings, despite their similarity. A major goal for mathematicians has been to develop a system which is both consistent and complete, and in their Principia Mathematica, Whitehead and Russell believed they had accomplished this.

Consistency is easy--the key is that the rules of inference tend to be simple (e.g., if a = b, then a + c = b + c). But completeness is really difficult. Can we know every true statement? Can we know they are all true?

Consider Fermat's last theorem. First proposed (as far as we know) in 1637, it was not successfully proven until 1994. So for over 350 years, the truth of this theorem was unknown (it was then called Fermat's Conjecture). Once it was proven, we knew it was true. Prior to its proof, it was also a true statement, but we didn't know it was true. There may be other true statements out there that remain unproven. There are probably other conjectures which may be false, but our inability to prove them is not enough, by itself, to know that they are false.

So completeness is a very difficult hurdle to cross.

Spoiler alert--this discussion was rendered moot by Godel, who in 1931 showed that all non-trivial mathematical systems are either incomplete or inconsistent. We normally insist on consistency--the price is completeness. There are always unprovable truths.

Which brings us to Tron: Legacy (2010). There may be some spoilers ahead.

In this film, we have the character Flynn, who is the Creator of a virtual universe, his creation (Clu), his son (Sam), and his apprentice (Quorra). It's an old story--the one about the creation turning on his creator. Flynn has been seeking to create a 'perfect universe'; unfortunately, what is meant by perfect is not adequately described. Flynn does reference freedom and openness a lot.

The essential conflict in the story is between Flynn and his creation. Through flashbacks, we learn that as Flynn is building the virtual world, something unexpected happens. A new group of programs, isomorphic algorithms ('Isos') appears. When asked how they appeared, Flynn says, "They manifested . . . the conditions were right and they just appeared."

We might say that the Isos were emergent properties of the system. But here Flynn and Clu disagree about the meaning of their appearance. Both regard these as unexpected phenomena in their 'perfect' system. But Flynn looked at them in a positive manner, unexpectedly emerging from a system he imagined he'd controlled. From the lyrical way he describes them, we can see that he believed they represent a key to life and creation--manifestations of a blinding truth, which was inexplicable within the rules of the system. The beauty of this unexpected creation brings Flynn to the realization that there are states better than perfection.

Clu, by contrast, only sees the Isos as inconsistencies. Unfortunately, there is no positive spin you can put on an inconsistency in a system. It means a complete tear-down is necessary. And of course, the inconsistencies have to go.

Even worse, Clu sees that his Creator had become corrupted by these inconsistencies, having declared his intentions to protect them. So not only must the Isos be destroyed, but Flynn has to go too.

Flynn and his buddy Tron, moments before Clu stages his coup

Clu: "Do you still want me to create the perfect system?"

Flynn: "Yeah . . ."

And the coup begins. Flynn escapes into exile within the digital world, while Clu destroys the Isos and remakes the world into his image of perfection.

Flynn struggles to regain control of the world, but it turns out the more he fights, the stronger Clu becomes. He realizes his only hope is to do nothing. Clu is thus free to remake the world, but is unable to find Flynn, who therefore remains a potential threat, somewhere at the edge of the world. A kind of stalemate ensues, which Clu tries to break by introducing another piece to the board--luring Sam into the virtual world.

This is where the movie starts--we see Sam's attempts to understand the new world in which he has unexpectedly arrived. Facing death by gladiatorial combat, Sam is rescued by Quorra, and reunited with his father.

The remainder of the film is a fairly linear storyline that eventually brings Flynn and Clu back together again, and for a moment it looks as if some sort of reconciliation may be possible. But Flynn, as a human, has been able to add to his understanding, which is why he can so easily abandon his quest for perfection. Clu, the construct, is frozen at the level of understanding that Flynn had when he created him--and he simply cannot abandon his pursuit of perfection. The final resolution is somewhat devastating, although Sam and Quorra are transmitted to the real world.

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* While in grad school I used to occasionally ask acquaintances why they studied geology as opposed to, say, engineering or chemistry, and a very common answer was, "Because I couldn't pass calculus". I loved the stuff, but then I was actually a geophysicist.

The meaning of Schools in the Sciences--and its Implications for the Dismal One

Schools and paradigms in science

Nowadays, most people think of scientists as being relatively uniform in thought and deed. They may read of a "scientific consensus" about issues like global warming, acid rain, and so forth. Any scientist who does not agree with the scientific consensus is presumably one of those mad scientists out to destroy the world or remove fluoride from the drinking water. (that is intended to be ironic).


But science wasn't always this way. Eighteenth century geologists were divided into schools, each of which had their own unique way of looking at rocks and interpreting the history of the world. The Neptunists, led principally by Abraham Werner, believed that all rocks were formed in the sea, either by deposition of sediments, or by precipitation from solution (both well-known processes at the time). By contrast, the Plutonists believed that rocks were formed from magma.

Most students of geology learn about these two schools (as well as two other competing schools--the catastrophists and uniformitarians) in introductory geology courses, but what is never discussed is the deeper philosophical significance of having these competing schools in science.

After all, we don't have competing schools in geology now. That is not to say that everyone agrees about all topics all the time--nothing could be farther from the truth (trust me on this--I go to conferences all the time). But there are no disagreements on the scale of requiring an entire school of geology which interprets all observations in one way as opposed to another school which will interpret the same observations differently. It seems impossible to have such structures in a science that is based on attempting to learn about the Earth through observation. What we have now . . . well, I'll get to that later.

The fundamental difference between geology now and geology in the 18th century is that the 18th century schools of geology were really based on formal systems. They were based on a logical system consisting of a set of irreducible statements, or axioms, which represented fundamental truths; and a series of rules of inference, which allowed new theorems to be proposed. Observations were used occasionally to justify one of these proven theorems. Anyone who accepted the axioms and rules of inference would logically move from one conclusion to another, and as the axioms were true statements, and rules of inference were designed to generate true statements from other true statements, any theorem generated by the formal system had to be true.

The difficulty in these systems is usually in the truth of the axioms. Rules of inference are almost always logical and work as desired (but see this story by Lewis Carroll for an argument against even the simplest rule of inference). As the statements were irreducible, they were unprovable. They had to be accepted on faith. Frequently, observations which disproved some theory would simply be explained away.

Why was science practiced this way? The scientific method was known at the time, and bodies such as the Royal Society strongly favoured this approach to science. My thinking is that the interpretation of geological observations was not sophisticated at the time, making the ideal approach to science impossible.


The practice of geology has changed fundamentally since then. And yet . . . you can still say there are schools in science. The only difference is that virtually everyone practices within the same school.

We call such schools paradigms. They differ somewhat from the older schools in that they are not based on formal systems--there are no axioms in which its practitioners keep faith. But a paradigm still places limits on scientific endeavour--it limits the types of questions asked; the way in which they are asked; and the means by which they are answered. In principle, any paradigm can be overturned by a single observation; but in reality overturning a paradigm is something that can take dozens of scientists decades (and oftimes professionally painful decades) to accomplish.

In geology, the plate tectonic paradigm is the framework within which field observations are fitted. Frequently there are some observations that don't agree with the paradigm as it is currently understood, but the paradigm concept is flexible enough to allow for a little bit of tinkering. The trouble may start when the inconsistencies start to do violence to the paradigm (see The Scientific Method and the Human Condition). Defenders of the paradigm will point out that there are literally thousands of peer-reviewed papers that have been published demonstrating that some local field observation can be explained by plate tectonic theory--strong support indeed. However, since the papers were written to fit the paradigm, then logically they should not be considered to be honest support for the paradigm.

The biggest paradigm of all in science is the Newtonian paradigm--the mechanistic view of the world. It is so large and pervasive that probably most scientists don't really think about it. Yet it just may be that this paradigm is beginning to shift.

The Dismal Science

Economics as a science still seems to be divided into numerous schools, each of which has varying successes in predicting economic behaviour. There are actually more schools than I had believed possible considering that all there is to economics is trying to figure out how humans are going to choose between alternatives in a resource-limited world.

The principal differences between the schools are in their axioms--in particular, each school has axioms which state which criteria are important when a group of humans has to choose between alternative courses of action. Different schools posit that humans base their choices on different criteria, or possibly weigh them differently.

For instance, some schools of economics consider only the opportunity and input costs for a project against the benefits. Other schools would add the costs to the environment of the project, and may well arrive at a different conclusion for the viability of the same project.

To argue about which of these schools is right is impossible, especially for an outsider--all you can do is choose the one which is closest to your own values.

To a geologist (me), this freedom to choose your reality makes it difficult for me to consider economics to be a science. At the same time, I see it as a logical way to make an economic decision. If, for instance, I decide to open a mine in Ghana, I have to decide which economic method to apply when I compare the benefits to the costs. I may have decided to include the environmental cost in my calculations of mine viability, and there are consequences to that decision--it limits the location of the mine and the style and scale of operations. Possibly another geologist would not consider the environmental costs, and would arrive at a different model for the mine. Yet both of these are viable alternatives within these different economic views.

Just don't expect me to consider economics to be a science.

Volcker-Bernanke puzzle no puzzle

In a recent article in the Asian Times, Hossein Askari and Noureddine Krichene, who I can only assume pass for economists, talk about the effects of interest rates on unemployment. They propose the existence of the "Volcker-Bernanke puzzle", which I will leave to them to describe. . .


"Assuming Fed chairman Ben Bernanke succeeds in reverting the US economy to full employment and rapid growth, then economic historians will be facing a difficult puzzle that could be coined the Volcker-Bernanke puzzle. Paul Volcker, Fed chairman from August 1979 to August 1987, got the US economy out of 11-12% unemployment by pushing money market rates to 19%. Bernanke pushed unemployment from 4% to 10% through aggressive monetary policy with near-zero interest rates, massive monetary injection, and buying all toxic bank loans; however, Bernanke, if he does succeed by his indicated path, will have pulled the US out of 10% unemployment by even more monetary stimulation."

Well, that certainly is a conundrum! But wait a minute. . . isn't that first assumption a little presumptuous?


"Somehow, either extreme, very tight or very loose monetary, could be followed by policymakers to solve the unemployment problem and propel the economy back to prosperity. It makes no difference which extreme is adopted!"


Amazing! No matter what we do, the economy will recover! Of course that doesn't explain how we got into trouble in the first place.


Maybe our problem was that after the fall of Volcker and before the rise of Bernanke, our interest rates were too moderate. They should have been either much higher or much lower! Now who was the guy in charge of setting interest rates back then?

I'm still not getting the puzzle though. On one hand, raising interest rates ultimately resulted in lower unemployment. On the other, lowering interest rates "pushed unemployment from 4% to 10%".

I still don't see a problem. One method produces lower unemployment, and the other produces high unemployment. Logically, you simply choose the method which produces the desired outcome. Given all the Ph.D. economists working on this, we can only assume that that indeed is what is happening.

So what is the puzzle? Our favourite economists again . . .

"Assuming Fed chairman Ben Bernanke succeeds in reverting the US economy to full employment and rapid growth, then economic historians will be facing a difficult puzzle that could be coined the Volcker-Bernanke puzzle."

Well always assuming that of course. We might also add  . . .

Assuming leprechauns exist, Obama's new economic plan wherein Federal agents are to be tasked with chasing rainbows in order to seize pots of gold at their ends will balance the Budget by 2012 at the latest.

It's hard to know on what basis this assumption is being made (the Askari-Krichene assumption, that is*). I hereby formally name this assumption after them with the sincere hope that it leads to their lasting fame.

Formally the Askari-Krichene assumption goes as follows:

"Assuming Fed chairman Ben Bernanke succeeds in reverting the US economy to full employment and rapid growth (which is precisely the opposite of what we are empirically observing), then economic historians will be facing a difficult puzzle that could be coined the Volcker-Bernanke puzzle."

We could add to their fame by creating an entire genre of logical statements that could be referred to as Askari-Krichene logical positions as follows . . .


If A is observed, then assume not A.

Which can be reduced to the Askari-Krichne Rule of Inference for Keynesian Economics:

If A then not A.



*The assumption about leprechauns follows from careful empirical observations.