The awards and rewards of grasping infinity
Shift in thought
Two mathematicians who made a breakthrough in understanding infinity were recently given a medal. Their work itself reflects an unbounded progress in explanations of reality.
—With so much news being about the scarcity of things, it may be easy to overlook news about infinity, or rather our understanding of it. In July, two scholars were awarded one of the highest honors in math for solving a problem that has stumped mathematicians for seven decades: whether two variations of infinity expressed in sets of numbers are the same. It turns out they are. Not only was the proof a surprise and an elegant one, it may bring practical applications.
The award, called the Hausdorff medal, was given to Maryanthe Malliaris of the University of Chicago and Saharon Shelah of the Hebrew University of Jerusalem and Rutgers University for a 2016 paper in the Journal of the American Mathematical Society. Their breakthrough, proved over 60 pages of complex calculations, was in applying one field known as model theory to another field called set theory. This allowed them to overturn conventional understanding about the sizes of infinite sets.
While the discovery was in theoretical math, it illustrates the steady recognition among scholars and other thinkers that infinity in all its aspects may be knowable in thought despite the limitations of the physical senses. By its very nature, infinity is inexhaustible and has been a source of wonder since ancient times. The desire to grasp infinity has contributed to progress in many fields, from science to religion. In fact, the ability to come up with new understandings about reality may itself be infinite.
That was a key point in a 2011 book titled “The Beginning of Infinity: Explanations That Transform the World,” by British physicist David Deutsch. He argues that progress will not end but is unbounded. Every fundamental field of knowledge is on a journey of discovery, or “the quest for good explanations” that are universal.
He writes: “From each such field we learn that, although progress has no necessary end, it does have a necessary beginning: a cause, or an event with which it starts, or a necessary condition for it to take off and to thrive. Each of these beginnings is ‘the beginning of infinity’ as viewed from the perspective of that field.”
As mathematicians and other scholars try to understand infinity and other aspects of reality, he says, they do so with “the infinite reach” of new explanations. “If unlimited progress really is going to happen, not only are we now at almost the very beginning of it, we always shall be,” Mr. Deutsch writes.
Humanity’s struggle to explain infinity, in other words, deserves far more attention – perhaps more so than its struggles with limitations. And that is why it should be bigger news when two scholars win an award for a new discovery about infinity.