Last modified: 7:58 AM Saturday, 14 January 2017

‘In a cat’s eye!’

The limits of math and why time won’t end

Most mathematicians, physicists and cosmologists suffer from an anomalous mental condition: infiniphobia (so much so, in one case that I’ve seen, as to induce a physics forum moderator to end a discussion invoking the question in an inferable panic lest non-“mainstream” ideas be examined): an irrational aversion to infinities. I suspect that this is in part because our mathematical system doesn’t yet contain appropriate tools for handling what I have termed “limited infinities.”

Cat's Eye Nebula

The Cat’s Eye Nebula, as photographed by the Hubble Space Telescope, resembles what our sun may look like five
billion years hence, some astrophysicists say.
[ Image Source ]

Without such tools, infinities existing within other infinities yield paradoxes rather than useful values in many calculations. For example, according to current probability theory, given an infinite multiverse, anything that can happen will happen infinite times. This sounds like Murphy’s Law Squared, and that is apropos, for it means that such probabilities cannot be determined or even approximated — they are all, by definition, infinite — and for a mathematician this is the worst of all possible outcomes. This fact combines with a human tendency to avoid the intellectual vertigo induced by looking too deep into the abyss of nested infinities, or merely to evade examining what our mental vision cannot illumine and encompass, to scatter conventional mathematicians and scientists in panic flight.

For so long as orthodox, mainstream professionals continue to flee the questions posed by limited infinities, certain vital calculations about our universe and the infinite, all-inclusive “multiverse” of which it is a part must remain forever beyond our reach. But if, in the spirit of the sciences, we aspire to a deeper and more accurate understanding of the true nature of our environment in its most inclusive sense, then it is precisely into those vertiginous depths that we must now plunge until we have plumbed them so far as to yield ways to represent and operate upon specific values for each of the limited infinities that we will meet. And this will call for what orthodox scientists are trained to distrust, and what an unorthodox scientist named Albert Einstein called more important than knowledge: imagination.

By the imagination and intuition of genius, science has not plodded sedately from discovery to discovery: It has overleapt the stars to examine the outermost boundaries of our universe and peered within the atom to discern in its parts the architecture that shapes and the laws that govern the whole. It is this talent, this capacity to fuse the findings of disparate disciplines, rather than the specialist’s tunnel-vision and the millipede march of the stolid incrementalist, that science most needs. For only imagination can stare into — and from — the eyes of the ineffable.

How many dimensions there are is a question disputed: Answers range from the merest physical three that we can perceive and measure — or four, if time be considered a dimension — to 10 or 26 under different flavors of “string theory”; others have begun to speculate that the number of such dimensions may itself turn out to be infinite. The larger numbers are called for, say some astrophysicists, to account for such phenomena as “dark matter” and “dark energy” and the apparent constant appearance and disappearance — seemingly from and to nothing — of “virtual particles”; in string theory and its relatives, all of these anomalies are accounted for by the interaction of our dimension with others that we can’t directly observe, perhaps because they are so tightly involute, so nearly infinitely curved in upon themselves, as to occupy no measurable space. (If so, then we may reasonably suspect that to any conscious entity native to such dimensions, our own set of them would likely also appear infinitely involute and therefore proof against direct observation.)

But every theory agrees in regarding each dimension as infinite within itself: Thus, one dimension would be a line of zero height or breadth, but of infinite length. This, of course, is a prime example of a limited infinity. It is also a paradox, for the volume of a single dimension is zero times infinity; and this yields exactly the kind of paradox that brings froth to the lips of mathematicians. Nonetheless, as in the classic demonstration offered by The Time Traveller in HG Wells’ famed 19th-century work of science-fiction, The Time Machine, absent duration, any object, irrespective of volume, must have only a virtual existence; therefore, as current physical theory now holds, time may be regarded as a dimension as essential as the visible three.

Thus, in a multiverse that must, by definition, be infinite, time can be nothing less than eternal. For it, too, candesces with the vital spark of limited infinity that appears to suffuse the cosmos.

Originally published as a review of a National Geographic article on the end of time.

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