The Infinity Illusion New Scientist 17.08.2013.

Over 40 years ago I had a long discussion with a maths teacher about infinity. It culminated in his conceding that the series of whole numbers, although apparently unbounded in both directions, was not an example of true infinity. Now, in 2013, more than 40 years on, could the scientific/mathematical community be moving in that direction? Imagine my elation when I saw, in the New Scientist an article suggesting that some physicists were trying to remove infinity from scientific – and even mathematical – calculations.

Max Tegmark, it seems, regards infinity as “…the ultimate untested assumption.” He says: “All of our problems with inflation and the measure problem come immediately from our assumption of the infinite.”

This must raise the question: How can we test for infinity? Surely such a test would require an infinite amount of information.

Later in the article we are told that “The largest number of informational bits a universe of our size can hold is 10^122.”

Nobel laureate, David Wineland, says: “Certainly we need nothing like that number of bits to record the outcome of experiments.” Is he saying that experimental science has no need for infinity? The article points out that “…even the best device will not measure with infinite accuracy….”.

Raphael Bousso says: “I don’t think anyone likes infinity. It’s not the outcome of any experiment.”

Mathematician Norman Wildberger says of “potential” infinity: “This type of infinity allows us to add 1 to any number without fear of hitting the end of the number line, but is never actually reached itself. That is a long way from accepting “actual” infinity – one that has already been reached and conveniently packaged as a mathematical entity we can manipulate in equations.”

I doubt that Wildberger is, here, criticising Cantor who appears to have achieved precisely that “packaging” in set theory. However, the fact remains that Cantor’s infinities are “mathematical” infinities, and run into problems with “absolute” infinity”.

“For the past decade he [Wildberger] has been working on a new, infinity-free of trigonometry and Euclidian geometry.” He is working to counter the fact that angles are related, via circles, to pi, with its endless digits following the decimal point. I lack the maths to evaluate Wildberger’s work, but Doron Zeilberger say of it: “Everything is made completely rational. It’s a beautiful approach.”

If Wildberger’s approach is revolutionary, surely, Zeilberger’s must be calculated to set the cat among the pigeons.

Zeilberger, it seems, wants, not only to get rid of infinity; he “wants to dispose of potential infinity as well.” Zeilberger believes there is a largest number. “Start at 1 and just keep on counting and eventually you will hit a number you cannot exceed – a kind of speed of light for mathematics.” Zeilberger’s answer to the question as to what happens if you add 1 to this number is that it acts like a computer which has a maximum number it can process. If you add 1 to it, it either gives you an “error” message, or resets the number to zero. Apparently, Zeilberger favours the latter option.

My own feeling about this is that it draws too heavily on the analogy between a computer and the Universe. Who, or what, would generate the “error” message, or reset the number? Zeilberger’s assertion that this largest number is “…so big you could never reach it” seems a bit of a cop out. Consider what he has actually said: “…eventually you will hit a number you cannot exceed” and “…you could never reach it”.

Tegmark points out that “the calculations and simulations that physicists use to check a theory against the hard facts of the world can all be done on a finite computer. “That” he says “already shows that we don’t need the infinite for anything we’re doing.” He continues: “There’s absolutely no evidence whatsoever that nature is doing it any differently, that nature needs to process an infinite amount of information.”

Physicist Seth Lloyd points out that “We have no evidence that the universe behaves as though it were a classical computer, and plenty of evidence that it behaves like a quantum computer.” This reintroduces infinity, because: “If you really wanted to specify the full state of one qubit, it would require an infinite amount of information.”

Obviously, we do not have an infinite amount of anything at our disposal, so the problem seems academic.

Set theorist, Hugh Woodin suggests separating the two issues of physical and mathematical infinities. He says: “It may well be that physics is completely finite, but in that case, our conception of set theory represents the discovery of a truth that is somehow far beyond the physical universe.”

Obviously there is more in this article than I have mentioned here, but hopefully others will read it and raise different points. I have added a few of my own thoughts above, but I would be delighted to have some other people’s input before risking swamping the subject with my own crackpottery.