Discovery

  This page is part of Math on the XO project.

Conventional education is almost always focused on getting the approved "right" answer. But in the real world, there is almost never a single right answer. Multiple answers may be appropriate to the same question, and sometimes the standard answer turns out to be wrong.

It is commonly supposed that a theorem in mathematics, once proven, stays proven and remains a theorem forever, apart from occasional mistakes. This turns out not always to be the case, although exceptions are fairly rare. The best-known examples are supposed proofs that Euclidean geometry is the only one possible, that there can be no consistent arithmetic containing infinitesimals, and that there is only one structure satisfying the Peano axioms for the natural numbers. There have been extended periods of confusion in mathematics: plane geometry from Euclid to Gauss, Riemann, Bolyai, and Lobachevsky; calculus, from the time of Newton to the rigorous constructions of Dedekind, Weierstrass, and Cauchy; mathematical logic and set theory, from the time of the discovery of the great paradoxes to the creation of non-standard mathematics and model theory.

Another kind of problem has been much more common. The mathematical profession has often objected to the introduction of new notations (Arabic numerals, for example), and new objects (zero, negative numbers, complex numbers, infinite sets, infinitesimals) into mathematics, or to particular methods of proof (non-constructive proofs, use of the Axiom of Choice and its equivalents).

Thomas Kuhn proposed a theory of scientific discovery in which a single contrary fact meant doom for a theory. This turns out not to be true for science in general, and even for mathematics. This is fortunate for us, considering how many paradoxes of logic and set theory there were early in the 20th century, and how many physical theories failed to match up with experiment. A single verified, thoroughly tested fact may require the creation of a more general theory or a new branch of a science, but the old theory is usually kept on in the domain where it is adequate. Examples include celestial navigation (as though the Earth stands still), and the use of Newtonian mechanics and Clerk-Maxwell electromagnetism where none of Special Relativity, General Relativity, and Quantum Mechanics is required. Mendelian genetics is still taught as the introductory version, although our understanding has gone much further. Valence is still taught in introductory chemistry, even though chemical bonds are due to the structure of quantum mechanical electron orbitals.

In general, science advances fastest when practitioners get the wrong answers according to the prevailing theory, and math advances fastest when current models fail to support the demands of new constructions.

In the schools, discovery has long been suppressed, both as content and as activity. Children are capable of discovering laws of nature and of mathematics far beyond the level generally expected, and of learning how to validate them properly, whether through mathematical proof or through theoretical modeling and experiment in science. See, for example, the work of Caleb Gattegno, Jerome Bruner, Seymour Papert, and Alan Kay.

Thinking Out of the Box

The original box puzzle that led to this common phrase is this: Take nine dots arranged in a square at equal distances. Thus they form three rows and three columns of dots.

. . .
. . .
. . .

Now connect these dots with a continuous polygonal line of the fewest possible segments. You may think of this as drawing a path through the dots without lifting your pencil from the paper, and only moving the pencil in connected straight line segments. Five segments is trivial, in either a squared-off 5 or 2 shape, or a squared-off spiral or G shape, somewhat like the diagrams below.

 __    __
 __|  | _
|__   |__|

Four segments is the canonical solution. The segments in this solution extend outside the square of the dots; hence, thinking outside the box. There: I have told you the secret, but you still have to understand how to apply it in order to get the solution.

But there are also solutions in three segments and in one segment, that children have discovered. Recognizing these possibilities requires two more leaps of understanding. I will give you clues: For the solution in three lines, there is one word in the definition of the problem that almost everybody makes an incorrect assumption about. For the solution in one line, there is another word that you may need to reinterpret.

It is much more satisfying to discover all three solutions yourself, even if it takes days or weeks, rather than to be told the answers.

Kinds of Discovery

On first looking into Chapman's Homer

Much have I travel'd in the Realms of Gold,
And many goodly fiefs and kingdoms seen.
'Round many Western islands have I been
Which bards in fealty to Apollo hold.

Oft of one wide expanse had I been told,
Which deep-brow'd Homer ruled as his demesne,
But never had I breath'd its pure serene
'Til I heard Chapman speak out loud and bold.

Then felt I like some watcher of the skies
When a new planet swims into his ken,
Or like stout Cortes, when with eagle eyes
He stared at the Pacific; and all his men
Look'd at each other with a wild surmise,
Silent, upon a peak in Darien.

John Keats