Impossible constructions (John D. Barrow, Impossibility: The Limits of Science and the Science of Limits, 1998)

Most architects think by the inch, talk by the yard, and should be kicked by the foot. PRINCE CHARLES

There are many areas of human activity where we set ourselves exercises which must be achieved in the face of some constraint. Blindfold chess, race-walking, and handicapped horse racing are all examples of activities where some deliberate constraint is applied in order to make a goal more challenging to achieve. Sports are not alone in liking this formula for making life more interesting, Mathematicians have long had an interest in finding out whether it is possible to do things using only particular tools. The Greeks' love of practical geometry led them to investigate what things were possible by the use of a straight edge {sometimes referred to as a 'ruler', but no use can be made of the markings to measure lengths) and compasses alone. A ruler enables you to draw a straight line between two points; a pair of compasses enables you to draw an arc or a circle and to mark out equal distances. These were the basic tools of architects at the time, and this whole problem clearly had the serious practical purpose of discovering the procedures they should follow to carry certain routine constructions when drawing up their building plans. Indeed, one can find essentially the same problems posed and solved {in the same way) in other advanced ancient cultures, like that of early India, where these constructions were required for the construction of altars and for religious ceremonies!

Consider the simplest problem of this sort: how do you bisect a line? The midpoint of any line can be found using the compasses by drawing two arcs, one centred on each end of the line { of radius greater than half the length of the line, so that they intersect). Now draw a straight line between the two points where the two arcs have intersected each other. It passes through the midpoint of the line (Fig. 7.4).

The next step after the bisection of a line is to ask if it is possible to bisect an angle with these tools (Fig. 7.5). Draw any angle, then put the compass point at the corner of the angle, A; next, draw any arc that cuts the two lines. Now we just need to find the midpoint of the arc joining these two intersections. Draw two arcs, centred at B and C; then draw a line with the rule from the point where these arcs intersect to the corner A. This line bisects the angle.

Fig. 7.4 Dividing a line in half by ruler and compass' constructions. Set the pair of compasses to an arc of radius greater than half the length of the horizontal line. Drawa circle of this size centred on each of the end points of the line. They will intersect at two points, above and below the line. The vertical straight line joining these two points divides the horizontal line in half.
           

Fig. 7.5 Dividing an angle in half. Place the compass point at the corner of the angle, A. Draw any arc that intersects both lines forming the angle at B and C. Now draw two arcs of equal radius from each of the two points of intersection. The straight line from the point where these two arcs intersect to the corner of the angle divides the angle in half.


All this was child's play to the early Greek geometers. They kept exploring the scope of their method of' ruler-and-compass construction, firm in the belief that anything could be achieved if one was ingenious enough. Their interest crystallized around one problem that they could not crack: how to trisect an angle. This problem remained unsolved until 1837, when Pierre Wantzel proved that such a ruler-and-compass construction is impossible. Curiously, Wantzel remains virtually unknown as a mathematician, and received surprisingly little acclaim for solving a two-thousand-year-old problem even in his own day. Wantzel achieved his proof by changing the problem into one of algebra. Important developments had been made in this subject by Ruffini and Abel, which Wantzel used to establish the impossibility of trisection. Mathematicians viewed the algebraic developments as being deeper and more fundamental than their application to the trisection problem and, as a result, Abel became far more renowned for his role in establishing this field. Ruffini and Abel showed that no algebraic equation of degree greater than four allows us to find its solutions by a formula. Again, this problem had some history. The solution for equations of degree one was trivial; degree two had been known for thousands of years; degrees three and four had been solved by the Renaissance mathematicians Scipio del Ferro, in 1515, and Ferrari, in 1545. Competition had been intense amongst mathematicians ever since to solve the next case for perhaps some clever rule might solve them all at one fell swoop.

Abel, aided by the work of Galois, finally established an impossibility theorem. Later, he discussed the general question of solubility in mathematics and, sounding a little like Hilbert many years later, he realized that any attempt at complete understanding of mathematical problem must have two means of attack; one to find explicit solutions; the other, to discover whether solution is possible or not. Only in this way could a problem be closed, because

To arrive infallibly at something in this matter, we must therefore follow another road. We can give the problem such a form that it shall always be possible to solve it, as we can always do with any problem. Instead of asking for a relation of which it is not known whether it exists or not, we must ask whether such a relation is indeed possible. ..When a problem is posed in this way, the very statement contains the germ of the solution and indicates what road must be taken; and I believe there will be few instances where we shall fail to arrive at propositions of more or less importance, even when the complication of the calculations precludes a complete answer to the problem.

Interestingly, Abel at first thought he had found the solution for the degree five problem. But before his paper could be published he found a mistake and, as a result, started to see the problem in a completely different light. This vital change of perspective led ultimately to his proof of the impossibility of the very result that he thought he had once established.

It appears that Abet's work did not give rise to deep philosophical and theological speculations about why it was that solubility stopped at degree 4. Clearly, it could have done. Equations of higher degree certainly have solutions. We can solve some of them by inspired guesswork, approximations, and so forth (as could mathematicians in Abel's day) but Abel's proof seemed to open up a gap between what human reasoning could achieve and what was true in the transcendental world of mathematical truths, or in the mind of God.

Many of the philosophical issues raised by Godel's theorem could have been stimulated by these discoveries that there are limits to our ability to solve algebraic equations and to the scope of ruler-and-compass construction, but they were not. There are many analogies between the two lines of inquiry. Both Abel and Godel attacked problems that everyone expected could be solved. Both displayed remarkable flexibility of mind in establishing an impossibility theorem: Abel did a last-minute about-turn after thinking he had got a 'possibility' theorem, and Godel had actually been proving the completeness of smaller logical systems than arithmetic (this was his doctoral thesis work) just months before announcing his impossibility theorem for arithmetic.

Godel established a correspondence between statements of mathematics and statements about mathematics (metamathematics). He did this by using prime numbers to encode each ingredient of a logical or mathematical statement. The product obtained by multiplying the prime numbers together then defines the whole statement. This number is now called its Godel number. Moreover, since any number can be expressed as a product of prime numbers in one and only one way (for example, 51 = 3 X 17,  54 = 2 X 3³,  9000 = 2³ X 3² X 5³) the correspondence is unique: to each Godel number there corresponds a logical statement. In this way every Godel number corresponds to some logical statement about numbers (not necessarily a very interesting one) and each statement about numbers corresponds to some Godel number. For example, the Godel number 243,000,000 = 2⁶ X 3⁵ X 5⁶. The logical sentence is defined by the powers of the prime numbers taken in order, that is 656. The symbol 6 corresponds to the arithmetic object zero, 0, while 6 corresponds to = , and so this Godel number represents the rather uninteresting arithmetical formula 0 = 0.

Godel decisive step is to consider the statement

        The theorem possessing Godel number X is undecidable.

He calculates its Godel number and substitutes that value for X in the statement. The result is a theorem that establishes its own unprovability.

The essential feature that make the incompleteness argument work is the possibility of self-reference: the correspondence between arithmetic and statements about arithmetic. This is possible only in logical systems which are complicated enough to allow statements about them to be coded uniquely and completely within the systems themselves, so that if each possible ingredient of a logical statement is ascribed to a different prime number, then any complete statement can be represented by a Godel number which can be factorized uniquely to give the statement about arithmetic to which it corresponds. Some logical theories, like geometries, do not contain enough machinery to allow statements about themselves to be encoded within them in this way. These theories cannot display incompleteness.