Today's lesson focuses on a small word wrapped in a big package...Infinity.
Indeed, the term infinity has many distinct interpretations depending on the subject matter, and often, popular usages run counter to its more scientific meanings.
On the lighter side, for example, we have the battle cry of Disney's animated space action figure Buzz Lightyear: "To Infinity...and beyond!"
And, of course, there is the tongue-in-cheek observation of English dramatist Tom Stoppard: Infinity is "...a terrible thought...where's it all going to end?"
(Stoppard, Tom, Rosencrantz and Guildenstern Are Dead, Act 2, Faber, London, 1967)
Generally speaking, we define infinity as a state of endlessness, with no limits of space, time, or any other measurable quantity.
The concept has been hounding scholars for years, as scientists and mathematicians vigorously debate whether to outlaw the idea or to embrace it.
"Is (infinity) part of the problem or part of the solution?" questions author John Barrow in his book, The Infinite Book: A Short Guide to the Boundless, Timeless and Endless. Are infinities just a signal that we have not found enough pieces of the puzzle, or are they a vital part of (solving) ultimate problems like the beginning and end of the Universe...?"
(Barrow, John D., The Infinite Book - A Short Guide to the Boundless, Timeless and Endless, 2005, p. xiv)
A recent Wall Street Journal column by Sharon Begley gives us an idea of infinity's complexity. In her article, "Definition of Infinity Expands for Scientists and Mathematicians, "she begins with a thought experiment called "Hotel Infinity," which was devised many years ago by German Mathematician David Hilbert.
At this hotel, it is impossible to overbook and customers are never refused a room. Here's how it works:"If you arrive with a reservation and find that the hotel's infinite number of rooms are all occupied, the manager simply moves the guest in Room 1 to Room 2, the guest in Room 2 to Room 3, and on and on until every guest has a room and you get Room 1. In an ‘infinite set,' such as the rooms at the Hotel, whatever you thought was the highest-numbered member of that set, isn't."
"If thinking of infinities makes your head spin, you're in good company," Begley adds..."Georg Cantor, the early-20th-century mathematician, who did more than anyone (at that time) to explore infinities, suffered a nervous breakdown..."
Indeed, there was a time when mathematics was simpler. Long before humans could read and write, their only thought was in keeping track of property and following the passage of time. Light meant day and dark meant night, and primitive languages had no words for high numbers. Likely, the only way to express quantities was "one" and "many."
Just as confusing and possibly more illogical is infinity's cousin, zero. According to author Charles Seife, "Within Zero, there is the power to shatter the framework of logic." Seife wrote the book, Zero: The Biography of a Dangerous Idea.
(Seife, Charles, Zero: The Biography of a Dangerous Idea, Chapter Zero, p. 5)
"The biggest questions in science...are about nothingness and eternity..." he states. Zero and infinity "...are equal and opposite, yin and yang. They are equally paradoxical and troubling."
"...through all its history, despite the rejection and the exile, zero has always defeated those who opposed it. Humanity could never force zero to fit its philosophies. Instead, zero shaped humanity's view of the universe..."
Some mathematicians also point out that zero not only represents nothing, it can represent the sum of everything if an infinite quantity of positive and negative numbers are added together.
"Today, however, infinities aren't just a central part of mathematics. More surprising, says cosmologist John Barrow, (a research professor at the University of Cambridge)... scientists who study the real world are having to take infinities seriously, too."
"Not long ago, if the solution to an equation included an infinity, alarms went off. In particle physics, for instance, ‘the appearance of an infinite answer was always taken as a warning that you had made a wrong turn.'"
"Suspecting just that, some scientists now see infinities ‘as an essential part of the physical description of the universe.'"
"In addition to coming around to the view that infinities might be real...some cosmologists suspect that infinities at the beginning and end of time ‘have quite different structures.'"
In other words, not all infinities are created equal!
Yet the concept of infinity remains intriguing. "It lies at the heart of all sorts of fundamental human questions," writes Barrow. "Can you live forever? Will the Universe have an end? Did it have a beginning? Does the Universe have an ‘edge' or is it simply unbounded in size?"
"Although it is easy to think about lists of numbers or sequences of clock ‘ticks' that go on forever, there are other sorts of infinity that seem to be more challenging. What about an infinite temperature or an infinite brightness - can such physical things actually be infinite? Or is infinity just a shorthand for ‘finite but awfully big'?"
According to Barrow, cosmologists must contemplate "...the possibility of an infinite future. Does the Universe seem to be on course to last forever? What does ‘forever' mean? Can life in any form continue forever? And, at the more human level, what would it mean - socially, personally, mentally, legally, materially and psychologically...?"
Indeed, these questions should keep scientists and mathematicians busy for centuries, but it is these unknowns and others that continue to drive our thirst for understanding. And therein lies the power of science and the magic of universities.
It lies in the relentless pursuit of truth, in the progressive discovery of knowledge, in the connectedness of life, and in the sense that we can, and we must, advance our common future.
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