Once Before Time - Martin Bojowald [149]
The procedure in such a situation is, however, different for physicists than it was for moral theorists. While many theoretical physicists still keep up their hopes that further developments of the theory will once result in additional constraints on the solutions, strongly reducing the size of the solution space, others turn necessity into a virtue by simply declaring the investigation of this enormous set to be a new discipline. Physical predictions can no longer be obtained from a concrete solution? Let us then appeal to probability arguments. If we can only find sufficiently many solutions with a certain property, let us suggest this property also for our universe! Implicitly, we make an additional assumption: that our universe is typical among all those mathematically possible. This is again a physically untestable proposition, for we have but one universe, and what should we compare its type with? Unfortunately, many of the probability arguments, with their giant total number of solutions, come dangerously close to an act of Zenoic desperation: Instead of dealing, as usual, with the concrete instances of the universe given and accessible to us, a great number of possible worlds is introduced. Problems of our world, solely in its theoretical description to be sure, disappear in a shoreless sea of utopias, never to be solved.
11. THE LIMITS OF SCIENCE AND THE NOBILITY OF NATURE
This noble metaphysical delusion is added to science as its instinct and time and again leads it to its limits, where it must turn into art: the actual aim of this mechanism.
—FRIEDRICH NIETZSCHE, The Birth of Tragedy
The idea (or fantasy?) of a theory of everything is ancient and powerful, and so it is not surprising that it often formed the background of secret societies such as the Pythagoreans or the Rosicrucians. It has entered science only recently, even as it may again take the appearance of a secret society due to the impenetrability of the subject. Sometimes, central statements of theories take the form of a dense web of interlocking (but unproven) conjectures, difficult to see through even for insiders. Especially in such cases, there is a considerable danger that the whole edifice will collapse like a house of cards, should someone only make a serious attempt to check it.
According to Pythagorean thought, unbroken integer numbers were deemed to take the role of elementary quantities. This surely is no theory of everything, but a guiding principle dominating further considerations. Should this hypothesis be correct, everything in the world must be expressible by integer numbers or their ratios (fractions). Building on this, the Pythagoreans developed impressive mathematical results, even though the precise attribution of individual bits (such as the “Pythagorean theorem”) is historically uncertain. In spite of everything, the theory of the Pythagoreans had a fundamental flaw: Not everything can be expressed as a ratio of integers. For instance, the diagonal of a square with a side length of one meter has a length in meters represented by the square root of two: not a ratio of integers but an irrational number. Even in Pythagorean times this flaw was recognized—a shock from which that school could not recover despite apparent attempts to cover up the unfavorable result.
Hypotheses and unproven conjectures play a large role in science because they can, if sufficiently solid, stimulate further investigations. Even if they eventually turn out to be false, they crucially contribute to the progress of knowledge. It is, for instance, a truly masterly deed, one not to be underestimated, to prove the irrationality of the square root of two—or even just to question its representability as a fraction. The very emphasis placed on fixed principles often spurs critical scientists to disprove