Elements of Mathematical Thinking

Mathematical thinking is based on several key elements:

1. Learn to ask good questions

Great discoverers are also great questioners. Their original insights resulted from a habit of asking the questions others failed to consider. Good questioning skills awakens an awareness of the patterns upon which possible answers can be based.

2. Develop pattern awareness

The discovery of universal patterns is the fundamental insight that makes mathematics reasonable and possible. If there were no predictable patterns, there would no need for mathematics, which is is a sense the language of patterns.

3. Use algorithms

Once a significant pattern for solving a problem is discovered, short cuts (or algorithms) can be used each time the same pattern is encountered. Such repeatable methods become generally accepted techniques to be used again, and can be easily passed on to future generations (who may or may not use the hard-earned knowledge wisely). The memorization of such techniques makes up the bulk of a typical, standardized, fact-oriented math program. It is here that far too many were taught to believe the study of mathematics ends. Actually, the memorization of patterns discovered by previous civilizations is only the beginning--or better stated, only the past...

One famous example of an ancient and still very valuable discovery is the Pythagorean theorem. (It was not actually discovered by Pythagoras, but that's another story.) The Pythagorean Theorem can be described with words, but is far more elegantly demonstrated using Algebra and Geometry.

"In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle)."

4. Thou shalt trust in predictability

Unfortunately, the patterns discovered by open and honest exploration do not always conform to our most cherished beliefs. When a theory will not conform to our beliefs, we are most likely to abandon the theory rather than our beliefs. Many wars have been fought and civilizations destroyed in the vain attempt to defend some hopeless idea in the face of predictability. Honest questioners must accept the heart-breaking experience of seeing their most cherished theories and beliefs crushed upon the hard rocks of reality. It is the rare questioner who is willing to accept the unforgiving evidence of reality.

Due mainly to humanity's habit of clinging desperately to cherished beliefs, the discovery of universal patterns has been a long and difficult project, often coming into direct conflict with powerful institutions whose authority may rest upon the very beliefs that new ideas are proving false.

Those who trust the evidence of predictability have an easier time letting go of obsolete ideas and embarrassing the new. Such rare individuals are often recognized (sometimes long after their murder by the mob) for their great contributions to humanity, but...

An oft ignored fact is that modern scientific is based on a single, unprovable, and absolutely massive act of faith. That is the faith that the patterns we see around us are in fact meaningful and real, and can therefore be relied upon as the basis of empirical knowledge.

The reason we can't prove the "truth" of predictability is that we use predictability itself is to establish our proofs. It's not only scientists who rely on a faith in predictability. We all rely on evidence from predictable patterns to support our common--and not so common--beliefs. To do otherwise often seems insane. However, if faith in predictability is mistaken, then the our faith in the truth of all empirical knowledge is an illusion (and you are probably not reading this). Many today believe in the power of predictable evidence, but those who believe in magic, luck, miracles and divine intervention have a very different view of reality, and there is no scientific proof that they are wrong.

All this talk of predictability may seem obvious and self-evident. Clearly it's easy to predict that opening a door before walking through the door frame is easier than knocking one's head against a closed door. Unfortunately, the plot gets thicker. Modern science is pushing at the edge of it's own faith in predictability.

While writing Principia Mathematica, Bertrand Russell discovered a paradox (previously discovered by Ernst Zermelo) that shook the foundations of mathematical certainty.

Kurt Gödel follow with two incompleteness theorems. The more famous incompleteness theorem states that, "for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (Peano arithmetic), there are true propositions about the naturals that cannot be proved from the axioms." To prove this theorem, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers.

The discoveries of quantum physics further challenge the basis of all scientific understanding. See for example, The Heisenberg Uncertainty Principle. Perhaps not surprisingly, even this challenge to all empirical knowledge can be stated mathematically:

5. Face the unknown

When the patterns we seek do not appear, we may call this Chaos, Noise, Randomness, The Void, The Devil, or The Unknown. We seem to have an inborn dislike for a lack of patterns. For example, music--one of our finest arts--is essentially an attempt to convert random noise into pleasingly predictable sequences of tones and rhythms. Few can explain why we like music; we just know that we do. Humans seem to seek harmony and beauty in all it's forms. It's hard to describe harmony, but we know it when we see, feel, or hear it, and more often than not the harmonies we seek have a mathematical basis. See for example The Golden Ratio.

6. Seek ultimate truth

Scientific "knowledge" is always open to debate, testing and correction. Most scientists and mathematicians are well aware that scientists actually know very little. Science's greatest achievement has been in highlighting just how much we don't know.

When an important scientific theory seems to always conform to predictable patterns, it may be elevated to the status of a Law of Nature. Scientists work exceedingly hard to reduce such laws to a single, clear mathematical equation. Surprisingly, we have discovered very few such laws. For example we have the Laws of Thermodynamics, but only a Theory of Relativity.

One example of cutting-edge inquiry is String Theory. I won't attempt to clearly define it. That task is left to future generations. But a rough description of the current state of string theory will illustrate just how far real mathematics is from the boring factoids taught in many classrooms:

"Since the string theory is widely believed to be a consistent theory of quantum gravity, many hope that it correctly describes our universe, making it a theory of everything. There are known configurations which describe all the observed fundamental forces and matter but with a zero cosmological constant and some new fields. There are other configurations with different values of the cosmological constant, which are metastable but long-lived. This leads many to believe that there is at least one metastable solution which is quantitatively identical with the standard model, with a small cosmological constant, which contains dark matter and a plausible mechanism for inflation. It is not yet known whether string theory has such a solution, nor how much freedom the theory allows to choose the details."

Source: http://en.wikipedia.org/wiki/String_theory

If you found that interesting, you may be interested in other unsolved problems. There's plenty to do! More career-oriented folk may want to seriously consider the Millennium Prize Problems. Of the original ten problems, six still remain unsolved. Each solution is worth $1,000,000 USD.

7. Predict the unknown

Another stage of mathematical discovery is the insight that methods developed long ago to solve earlier problems can help solve completely new and unforeseen problems. Thus, besides enabling us to model the known world, mathematics is amazingly successful at modeling the infinitely larger world of the not-yet-even-imagined. As far as mathematicians are aware, there is no obvious reason why this is should be so. This almost miraculous seeming situation has lead many a philosopher to seriously ponder the unreasonable effectiveness of  mathematics.

8. Connect it all

As humanity learns to ask ever deeper questions about our world, mathematics proves to be the preeminent language for accurately modeling our evolving understanding. This leads to a more sophisticated appreciation for the apparent mathematical basis of our shared reality. The consequences of these discoveries can be shattering, liberating, terrifying, and awe-inspiring. Powerful, long established worldviews have fallen or been transformed by new mathematical insights. Through it all mathematics--our universal language--has survived and grown.

Mathematical reasoning begins with--and is inspired by--such insights. When they are carefully considered by students, mathematics can become elegant, mysterious, and beautiful, much like this awesome and mysterious world it seems to model so well.