Once Before Time - Martin Bojowald [36]
Most important, the value of the Planck length, which is very small but, after all, nonzero, shows that some cosmological situations such as the big bang can only be understood with a quantum theory of gravity. If the whole extension of the universe is supposed to vanish at that time, even the Planck length would be gigantic by comparison. With such a theory one could analyze the singularity problem and the possible existence of stabilizing repulsive forces. While no completely formulated theory yet exists, there are several concrete expectations and many encouraging signs.
3. AN INTERLUDE ON THE ROLE OF MATHEMATICS
THE WORLD IN AN EQUATION?
I had to learn thus with a frown:
No thing can be where words break down.
—STEFAN GEORGE, The Word
General relativity and quantum mechanics, taken separately, result in numerous confusing phenomena or even apparent paradoxa, not just for inexperienced observers but also for longtime researchers. A clear view is possible only thanks to their mathematical formulation, without which evaluating and understanding these theories would be impossible. It takes significant effort to learn and develop these theories—in particular when one aims at quantum gravity, in which the underlying mathematical concepts of general relativity and quantum mechanics are to be combined. Once learned, however, mathematics is incorruptible and lays out strict recipes to answer questions on which Intuition would refuse to take a stand or would even entangle herself in paradoxa.
Ever since Galileo Galilei, the mathematification of physics has been responsible for its unprecedented success. It leads us to entirely new worlds—allegorical worlds of new phenomena, literal new worlds in cosmology, or “brave new worlds” with a critical or prescriptive touch. According to Galileo, mathematics is the language of nature, indispensable for its understanding. Mathematical formulas are nature’s sentences, and its words are mathematical variables and operations. “No thing can be where words break down”: Without mathematics, scientific knowledge of nature is confined to objects directly accessible by the senses. At the singularities of space-time as they occur in general relativity, things, even entire worlds, are literally lost. The breakdown of the most important sentence of the theory, Einstein’s equation, prevents any access to the world before the big bang or deep into the heart of black holes. New mathematical vocabulary is still to be coined.
Mathematically predicted phenomena bring a new quality into physics. On the one hand, physical theories can explain sensory perceptions, such as the aurora borealis or the blue of the sky. But theories also give rise to new impressions experienced only abstractly, thus extending the realm of our senses. Examples can be found especially in cosmology, in the phenomena of the big bang or black holes. Such impressions exist only in the mathematics underlying the theories, but they must be tested by observations in the real universe even if those tests may be indirect rather than based on direct sensory imprints. It is this combination of strict mathematics and undeluded confirmation in nature that elevates physics above mere speculation or myths. Only this makes it trustworthy.
MATHEMATICS FOR ITS OWN SAKE?
MATHEMAGIC
Pfuehl was one of those theoreticians who so love their theory that they lose sight of the theory’s object—its practical application. His love of theory made him hate everything practical, and he would not listen to it. He was even pleased by failure, for failures resulting from deviations in practice from the theory only proved to him the accuracy of the theory.
—LEO TOLSTOY, War and Peace
In recent times, mathematics has found a life of its own in physics—especially in quantum gravity research, which is not yet subject to policing observations and in which mathematical consistency serves as the only selection criterion by which to choose theories. This is indeed a strong condition in quantum gravity, whose