Bridges to Infinity — Michael Guillen on the Boundlessness of Life and Discovery

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Bridges to Infinity — Michael Guillen on the Boundlessness of Life and Discovery

Definite yet unending

To challenge our worldview, we are sometimes encouraged to push our reasons and presumptions to their logical extremes. The hope is that we will uncover something new amidst all the bending and stretching and stressing we do. But perhaps nothing is more extreme or ultimate, both in notion and reality, than the concept of infinity. After all, “infinity is the name [we give] to something that is bigger than our minds can imagine”, writes the physicist Michael Guillen. It is “whole and definite”, yet “unending”. Today, mathematicians talk not only about the infinite, but about the transfinite and absolute infinite—a vast gallery of truly enormous, unimaginable quantities.

Living with infinity

Yet what is just as bewildering about infinity is its omnipresence in the physical universe. For one, the electric force between two electrically charged bodies, Guillen notes, follows Charles-Augustin de Coulomb’s inverse square law—an empirical law of physics that extends over an infinite range, and tells us that as the distance between the two bodies reduces to zero, the force of attraction or repulsion between them, depending on their charge signs, approaches infinity. This relation is no trivial thing. The electric force is more powerful than gravity, and involved in just about everything, including the interactions between atomic nuclei.

For infinities at a cosmic scale, we need only look to dying stars, says Guillen. Indeed, general relativity tells us that when the innards of certain gas giants are no longer able to withstand the onslaught of gravity, they may collapse into an infinitely small but infinitely dense point. What results is a “one-way exit”, a black hole—a swallower of light and breaker of time. Thousands of them litter our galaxy right now. And at the center of the Milky Way is Sagittarius A, a supermassive black hole whose mass is so large that it dwarfs our humble Sun some four million times to one. The strange and the infinite are deeply intertwined. 

Convergence and constants

Still, infinities present themselves in endlessly diverse ways. In arithmetic, Guillen notes, for example, that the sum of the infinite series, 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 and so on, is exactly equal to 1. On first encounter, the result is perplexing as we attempt to compute an unending stream of shrinking fractions in our heads. Yet the result is nonetheless true. The proof is an elegant reminder that dealing with the infinite is not always beyond the reach of the mathematician. 

Elsewhere, in geometry, Archimedes’ constant or pi (π), the ratio that relates a circle’s circumference to its diameter, turns out to be an irrational number. Like other irrationals, we cannot express pi as a ratio of two integers. Its decimals run into perpetuity with no obvious pattern in sight. In fact, many interesting mathematical constants are just as irrational. They include Euler’s number (e) and the Golden Ratio (φ).

To me, it is peculiar that the fundamental relations of geometry and nature ought to be this way. But I am not alone in this feeling. Long ago, Pythagoras of Samos and his acolytes wanted to believe that the Universe was orderly and numerically rational—that everything could be expressed by the ratio of whole integers. To their horror, they discovered that even the seemingly innocuous square root of two—now dubbed Pythagoras’ constant—is actually an irrational number. The result was so unnerving to the Pythagoreans that they conspired to keep their discovery a secret.

The youth of discovery

We should remember, however, that our understanding of math and science is a newborn by historical standards. While the Jain mathematicians of early India (400 to 200 BCE) were aware of innumerable and infinite numbers, we owe much of our conception of infinity to Georg Cantor’s nineteenth century work on set theory—the mathematical study of collections of objects. Cantor demonstrated formally that infinity is not just a single sized thing. He proved instead that there is “a hierarchy of infinities that [continues] infinitely.” They come in different types and sizes—an “infinity of infinities”, if you can imagine. (In fact, Cantor’s revelations are so otherworldly that even his peers rebuked it for some time—rejections that led to Cantor’s nervous breakdowns.)

But infinities are not the only infants in the room. Take negative numbers, for instance. While there were early representations of debits in the first millennium, negative numbers did not acquire its modern interpretation until John Wallis invented the number line in the seventeenth century in his Treatise of Algebra. Similarly, while the Sumerians could represent the “absence of a number” (like the ‘0’ in ‘101’ or ‘110’) as early as five thousand years ago, the “notion of nothingness” in arithmetic was not formally conceived until the seventh century by Hindu mathematicians. Their name for zero took the Sanskrit word śūnya, which comes from the Hinduism concept of śūnyatā—the state of emptiness or voidness.

Organon and the Elements

And only most recently was our conviction in mathematical certainty upended. Before the twentieth century, mathematicians of a bygone era believed in the coherence of their work—that somewhere in the winding roads of logic must exist a proof of truth or falsehood to every mathematical conjecture we imagine. The job of the mathematician then was to find the right roads and turns to get us there.

The roots of mathematical certainty date back more than two thousand years ago to Aristotle’s Organon and Euclid’s Elements. As Guillen explains, “Aristotle [had] reduced the ill-defined process of deductive reasoning to fourteen rules and a few canons by which conclusions could properly be derived from assumptions.” Some decades later, “Euclid followed the principles of deductive reasoning in deriving hundreds of theorems in geometry from only ten assumptions.”

Russell’s paradox

But something was amiss. In 1901, when Gottlob Frege was completing his second volume on the Fundamental Laws of Arithmetic, Bertrand Russell wrote to him of a logical paradox he encountered. In his thinking about the lists and classes of things, Russell wondered, as Guillen retells, about an “unimaginably huge class that contains all the classes [of things] that are not members of themselves.” Such a monster class would include, for instance, a list of teaspoons, a list of birds, and so on. But Russell wondered if such a class-of-classes would include itself. 

Russell realized, however, that answering this question in a logical way was paradoxical. Indeed, how can a class of classes that does not include itself include itself, or not include itself? Frege was equally troubled by Russell’s observation, for he could not resolve it. At the time, other logicians appealed to a “vicious circle principle”. The principle states, as Guillen explains, that “whatever involves all of a collection must not be one of the collection.” This allowed mathematicians to sidestep Russell’s paradox altogether.  

Logic incriminates logic

Twentieth century mathematicians soon learned that the bandaid was not enough. The edifice for mathematical certainty fell apart in 1931 when Kurt Godel proved, as Natalie Wolchover explains, “that any set of axioms you could posit as a possible foundation for math will inevitably be incomplete; there will always be true facts about numbers that cannot be proved by those axioms.” Godel’s feat was to devise a numbering and mapping procedure that gave rise to contradictory metamathematical expressions along the lines of ‘this statement is not provable’. 

Godel’s discovery is no trifle curiosity. It tells us that there are mathematical statements we cannot prove true or false, and not for a lack of trying. “What made Godel’s achievement even more noteworthy”, Guillen writes, “is that he had used logic to incriminate logic.” Guillen himself shares Morris Kline’s analogy in The Loss of Certainty—that mathematicians are like gardeners in an overgrown forest. They free their patch of woes only to find even more “wild beasts lurking” in between. Unprovable truths may exist, forever beyond the reaches of humanity. 

Unreasonable effectiveness

What then can we make of our worldview given the peculiarities, infinities and uncertainties that seem to persist in the physical and conceptual? It’s hard to say. In philosophy, the Platonists argue, for instance, that mathematics, while abstract, exists independently of human imagination and language. To them, “mathematical truths are discovered, not invented.” The Formalists, by contrast, suggest that mathematics is not an abstraction of reality, but “more akin to a game” of Chess or Go. The mathematical results we derive are simply a consequence of the rules that we imagine for ourselves. 

In the end, “the only certainty in this world”, Guillen reminds, “is change.” While Homo sapiens have walked this Earth for around three hundred thousand years, our conceptions of just about everything have undergone radical transformations in just the last few millennia alone. Our mathematical and scientific understanding may or may not continue along similar transitions in the centuries to come. Who’s to say?

As Guillen writes:

“Ever since Cantor’s reverie, part of us has been liberated from even the far limits set by Archimedes’ myriads, and now we roam freely beyond the ordinary infinity of the ponderable universe.”

Michael Guillen. (1988). Bridges to Infinity.

Sources and further reading

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