Thursday, January 6, 2011

A Note on Mathematics and the Importance of Gödel’s Theorem

There is a fundamental difference between mathematics and all the other branches of science. Besides the fact that it seems that all the other (or at least most of the other) sciences make use of the “Queen” of science, there is another big difference.

Theories in Physics

In other branches, such as physics, discoveries and theories are often expressed as formulas. When these new discoveries pass certain scientific tests, such as explaining current phenomenon and measurements, and typically predicting a new result that is then verified, the scientific theory is accepted. A good example is the “law” of gravity and the formulas developed by Sir Isaac Newton (1643-1727). Using Newton’s equations the movement of the planets and moons were all analyzed and verified.

In fact, the formulas, when used to examine the orbit of Uranus, predicted the existence of another planet even further from the Sun. This planet was later discovered by telescope and name Neptune. Still the orbits didn’t exactly match Newton’s formulas, and Pluto was later found to better match the math. (Pluto is really a Mickey Mouse planet, and was lately demoted to an asteroid or break-away moon.)

Newton’s formulas also didn’t match Mercury's orbit, the inner most planet in the Solar System. Due to the success of the discovery of the new outer planets, some astronomers supposed a planet closer to the Sun than Mercury and even named it “Vulcan.” But it was never found. So, a little more than 200 years after Newton’s brilliant work, Einstein provided an adjustment to the formulas required in the case of massive gravity or high velocity. (As the closest planet, Mercury moves the fastest in its orbit and “relativistic corrections” have to be applied.)

So a scientific theory is just that, just a theory. Even the theories that are so well accepted they are called “laws.” They are subject to correction by future discoveries. Mathematical theories (or better termed theorems) are the result of “proofs” and barring an error in the proof, a later discovery will not change a current proof and they can not later be deemed inaccurate or wrong. That is a unique fact of mathematics based on the axiomatic nature of a mathematical proof.

Theorems in Mathematics

For example, an ancient math problem was to “square the circle.” That is, to create a square using only a straight edge and compass (the tools of plane geometry) that has the same area as a given circle. After centuries of attempts, mathematicians realized that pi (a mathematical constant used to calculate area of a circle) is irrational. That is, it can’t be expressed as a fraction of integers. Once that was know, it was clear that there is no way to “square the circle” using the standard geometric tools.

To this day, math journals receive proofs on squaring the circle. The editor throws them right in the trash. There is no way it can be done and that is PROVEN!! Little else in our lives can be so certain; no other area of science has that perfection and finality. So does that mean that mathematics has no limits?

Limits in Science

Realize that other sciences have limits. In physics there is the Heisenberg Uncertainty Principle which limits the ability to measure atomic particles. It is not a weakness of our instruments or techniques, but a very real limit to what can be measured no matter how advanced science may become (assuming that Werner Heisenberg's (1901-1976) theory is correct). I always found that interesting, that there are known limits to atomic measurements, no matter how advanced the instruments. Even more interesting, it turns out that mathematics has a similar theoretical limit.

But how can that be? After all we create math out of pure thought stuff, and certainly there is no limit if we could just think hard enough. Yes, it is true that once a mathematical theorem is proven, that pretty much settles the issue in perpetuity, but it was recently discovered that math has its limits too. (Well, 1931, pretty recent compared to Euclid.)

Axiomatic Systems

Let's start at the begining. The early Greek philosophers began with basic axioms or postulates. That is what is behind Euclid’s “Elements” (300 B.C.). The ancients thought these axioms such as “all right angles are equal to each other” and “the whole is greater than the part” were intuitively obvious and true by their very nature. They then began constructing mathematics by creating proofs that built from these basic axioms and previously proven facts into the wonders of geometry and other maths.

Later a more nuanced interpretation developed that fundamental axioms and postulates are just statements and starting points. In fact, they are un-provable statements; they are not necessarily true. What mathematics entails are consistent and non contradictory structures. Whether they are true or actually represent the universe is not the point. The goal of mathematics has been to build logical and consistent structures on the foundation of a minimum of basic assumptions. These are called "axiomatic systems," and that is the basis of all “proofs” in mathematics.

Whether the basic assumptions are minimum is one of the questions mathematicians will ask. For example, let’s take Euclid’s Parallel Postulate: “That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.” Basically it defines parallel lines and says they never meet or cross. (Read it again!)

Euclid and others were always concerned that this postulate could be derived from the more basic statements, and many mathematicians tried to show that. Eventually Bernhard Riemann (1826-1866) and others attempted to derive the parallel postulate by assuming the parallel lines do cross. If that assumption turned out to be logically flawed, then that would be a step to deriving the basic assumption. Low and behold, this did not lead to an inconsistency, but a perfectly consistent mathematical structure different from Euclid's that we now call non-Euclidian geometry. (In point of fact it led to several non-Euclidean geometries.)

Non-Euclidian Geometry

If you assume the earth is a flat plane (it is actually the surface of a sphere) you will see the longitude lines all cross the equator at right angles which would make them parallel. Yet they meet at the poles. Therefore, treating the earth’s surface as a flat “map” is actually non-Euclidean geometry. We assumed that Euclid's system describes nature, and non-Euclidian were just mathematical odditities.

Later Einstein and his theory of curved space show that non-Euclidean geometry actually matches the natural universe better than the Greek’s classic work. Our three dimensional space is no more “flat” than the surface of the earth is flat. Of course, just as the surface of a sphere is three dimensions and the flat map is only two dimensions, so Einstein’s universe has four dimensions. It is a little more complicated than that, as the fourth dimension is time, but the analogy basically holds. Our three dimensional world has warps in the fourth dimension just as our two dimensional flat earth has warps in the third dimension.

In fact, a lot of modern physics including string theory holds the universe actually has 26 dimensions. Now that is really NON-Euclidean!!

So, math is just the development of a logically consistent system based on some basic ideas called axioms or postulates and the whole structure builds up from there. (Please note that the goal of mathematics is not necessarily to agree with nature and physics, but that math is a powerful tool for understanding nature and physics.)

Mathematical Paradox

Now what? Well, certain philosophers such as Bertrand Russell (1872 – 1970), found problems with that. These were the so called paradoxes. “Who shaves the barber in the town where the barber shaves everyone who does not shave themselves?” If the barber does it, then he shaves someone who does shave himself. If someone else does it, then the barber does not. It seems like there is no logical end to this conundrum.

I remember a Star Trek episode where they shut down the computer mind of these androids with such a paradox. (It is called the "lier's paradox.") One character said that Dr. McCoy always lied. McCoy then said he never tells the truth. So if he always lies then it is a lie that he always lies so he must tell the truth, but then … See the “strange loop” you get into following that paradox. Wait, strange loops … chaos theory … no, we don’t have time!

Incompleteness Theorem

Then along came Kurt Gödel, 1906 – 1970, and he proves that all logically consistent systems (including mathematical systems) contain at least one proposition that can’t be determined within the system. This is commonly called Gödel’s Incompleteness Theorem. That means that all sets of rules, no matter how perfect and self-consistent, must contain concepts that can’t be identified as true or false within the system, and you have to move outside this system to a “meta-system” to evaluate. Of course, the meta-system will require a meta-meta- system and so on to infinity. For those curious, I suggest a little visit to Wikipedia and search on Gödel. It is really fun, and you don’t need a Master’s in Math, although it does help!

This has great impact in computer science on issues like computability and the Halting Theorem. (Yet another visit to Wikipedia.) I just take it as God having a little fun with us. We’re pretty smart, but not as smart as God. There are limits to everything we do, even mathematics.

Back in the seventies I read (and reread often) a great book by the Computer Scientist Douglas Hofstadter called “Gödel, Escher, and Bach – The Eternal Golden Braid.” No coincidence that I’m such a fan of Bach’s music (what mathematician isn’t) and Escher’s thought provoking drawings. Hofstadter even uses Greek classic dialog between Achilles and the Tortoise (from the Tortoise and the Hare) to explain the idea … but, again, I digress. (Hofstadter later took over the great “Scientific American” column, “Mathematical Games” from Martin Gardner. I own several books by both authors and they are all very thought provoking and recommended to the mathematician and non-mathematician alike.)

Math is Hard

I think you all know that algebra, in a sense, is just generalized arithmetic. That is, we take the operations from arithmetic, such as adding and multiplying, and their inverses, subtraction and division, and apply them to variables which basically represent “all numbers.” Well, what do you suppose generalized algebra would be? Now we take the operations of add and multiply and generalize them to any binary operations.

For example, there is clock math. Think about hours on a clock. They are never more than twelve. If you add four hours to ten o’clock, you don’t get fourteen o’clock, you get two o’clock. That is a consistent mathematical system.

Or take matrices and matrix math. (A matrix is a lot like a tic-tac-toe or crossword structure with rows and columns of numbers that are operated on as a whole, very important in physics.) One interesting aspect of Matrix math is that multiplication is not commutative. That is, A * B is not equal to B * A. So not all the rules from standard algebra apply.

I studied “abstract algebra” in graduate school and it was the first time that math made my head ache. I had to study very hard to understand how to extend basic algebra into the abstract, just as some have to work hard to understand regular algebra. That was a very meaningful point in my life that I tried hard to understand what seemed at the time to be the non-understandable. It really humbled me that the great “A” student would have a problem with anything in school.

The Library

That lead me to the local library seeking peace and quiet to concentrate (at the time Mike and Mark were young, and noisy boys), which led to my tenure on the local library board and ultimate election as Library Board President. In that position I was influential getting a new Longmont Library built, but that is another story for another time.

Since then I’ve been humbled many times, but still I’m arrogant. Obviously, regardless of my GPA, I’m a slow learner. Whenever I get too big for my britches and start telling people what I think as if that is the ultimate answer, I think of Gödel.

We could all use a little humility, especially those who are so quick to explain just how “you” are wrong and “they” are right. Know anyone like that?


I end with the mathematical conclusion that love = . Proof left to the reader.