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who, as a student of Parmenides of Elea, tried to support his teacher’s ontological theory. (In addition to this theory, Parmenides had provided explanations for the phases of the moon as well as of other astronomical phenomena, which makes him one of the first important cosmologists.) Zeno’s problem was the boldness of Parmenides’ theory, which claimed that no motion exists and that any perception of motion is illusion. In the defense of this theory, Zeno employed several arguments, in their modern formulation all based on the concept of infinity. He constructed, for instance, a footrace between Achilles and a tortoise. As a fair sportsman, Achilles grants his weaker competitor a head start. Once he begins to run, Achilles quickly reaches the tortoise’s starting point, which in the meantime has inched forward a bit. Achilles needs some time to get to this place, but by then the tortoise has again crawled ahead. Repeating this process over and over again, Achilles never catches up with the tortoise. If Achilles cannot overtake the slow-moving tortoise, in contrast to our expectation for motion, motion itself can only be illusion.

Zeno tries to ensnare us. He certainly knows that Achilles will catch the tortoise after a certain amount of time, for he surely had watched runners compete, or even competed himself. (He was, perhaps, being petty enough to attempt a mental revenge for a defeat.) Now he subdivides the time between start and catch-up into infinitely many smaller intervals, changing single-handedly, but only mentally, the flow of time. Instead of allowing time to run as usual, he jumps from one interval to the next. With intervals becoming shorter and shorter, time in his thoughts passes differently; it is slowed down more and more compared to the conventional flow. He thus transforms a finite range—the time required for Achilles to catch up with the tortoise—into an infinite one. Mathematically speaking, he performs a coordinate transformation, mapping the finite time of the catch-up to an infinite value. His argument takes place in the new time, in which the infinite value is indeed never reached. But he overlooks, or tries to seduce us into doing so, the fact that in our actual perception of the competition only the original, conventional time is significant, which requires just a finite period for the tortoise to be overtaken.

The concept of infinity is dangerous, but often too enticing to resist. Especially in moments of Zenoic desperation—situations in which one despondently tries to construct a proof in the face of better knowledge—infinity is time and again abused in physics. Comparable thoughts can, for instance, be found in the context of our main problem, that of singularities in general relativity. Many of the proposed arguments can be reduced to a transformation of time, mapping the finite duration between the big bang singularity and the present to an infinite one. Viewed in this time, the big bang happened an infinite amount of time ago, thus at no finite point in time, or never. What is overlooked here is, of course, that the newly introduced concept of time, which plays only a mathematical role, is irrelevant, unlike physical (also called proper) time as perceived by us. An astronaut falling into a black hole would hardly be consoled by the fact that his small, finite remaining lifetime can mathematically be mapped to an infinite range.

ON HYPOTHESES AND THEORIES:

CODE OF CONDUCT FOR A THEORIST

I have not been able to discover the cause of those properties of gravity from phenomena, and I frame no hypothesis.… It is enough that gravity does really exist, and acts according to the laws which we have explained.

—ISAAC NEWTON

Despite all seductions, mathematics cannot be banned from physics. Depending on the degree of mathematical formulations, one distinguishes between hypotheses and theories. Hypotheses are of speculative character and usually mark the beginning of investigations in a new range of problems and the buildup of ideas. Theories are, in contrast to the vernacular use of