Science a GoGo's Home Page Forums General Discussion Physics Forum Retiring Infinity from science

Max Tegmark (Physicist, MIT; Researcher, Precision Cosmology; Scientific Director, Foundational Questions Institute)

Infinity

I was seduced by infinity at an early age. Cantor's diagonality proof that some infinities are bigger than others mesmerized me, and his infinite hierarchy of infinities blew my mind. The assumption that something truly infinite exists in nature underlies every physics course I've ever taught at MIT, and indeed all of modern physics. But it's an untested assumption, which begs the question: is it actually true?

There are in fact two separate assumptions: "infinitely big" and "infinitely small". By infinitely big, I mean the idea that space can have infinite volume, that time can continue forever, and that there can be infinitely many physical objects. By infinitely small, I mean the continuum: the idea that even a liter of space contains an infinite number of points, that space can be stretched out indefinitely without anything bad happening, and that there are quantities in nature that can vary continuously. The two are closely related because inflation, the most popular explanation of our Big Bang, can create an infinite volume by stretching continuous space indefinitely.

The theory of inflation has been spectacularly successful, and is a leading contender for a Nobel Prize. It explained how a subatomic speck of matter transformed into a massive Big Bang, creating a huge, flat and uniform universe with tiny density fluctuations that eventually grew into today's galaxies and cosmic large scale structure, all in beautiful agreement with precision measurements from experiments such as the Planck satellite. But by generically predicting that space isn't just big, but truly infinite, inflation has also brought about the so-called measure problem, which I view as the greatest crisis facing modern physics. Physics is all about predicting the future from the past, but inflation seems to sabotage this: when we try to predict the probability that something particular will happen, inflation always gives the same useless answer: infinity divided by infinity. The problem is that whatever experiment you make, inflation predicts that there will be infinitely many copies of you far away in our infinite space, obtaining each physically possible outcome, and despite years of tooth-grinding in the cosmology community, no consensus has emerged on how to extract sensible answers from these infinities. So strictly speaking, we physicists are no longer able to predict anything at all!

This means that today's best theories similarly need a major shakeup, by retiring an incorrect assumption. Which one? Here's my prime suspect: ∞.

A rubber band can't be stretched indefinitely, because although it seems smooth and continuous, that's merely a convenient approximation: it's really made of atoms, and if you stretch it too much, it snaps. If we similarly retire the idea that space itself is an infinitely stretchy continuum, then a big snap of sorts stops inflation from producing an infinitely big space, and the measure problem goes away. Without the infinitely small, inflation can't make the infinitely big, so you get rid of both infinities in one fell swoop—together with many other problems plaguing modern physics, such as infinitely dense black hole singularities and infinities popping up when we try to quantize gravity.

In the past, many venerable mathematicians expressed skepticism towards infinity and the continuum. The legendary Carl Friedrich Gauss denied that anything infinite really existed, saying "Infinity is merely a way of speaking" and "I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics." In the past century, however, infinity has become mathematically mainstream, and most physicists and mathematicians have become so enamored with infinity that they rarely question it. Why? Basically, because infinity is an extremely convenient approximation, for which we haven't discovered convenient alternatives. Consider, for example, the air in front of you. Keeping track of the positions and speeds of octillions of atoms would be hopelessly complicated. But if you ignore the fact that air is made of atoms and instead approximate it as a continuum, a smooth substance that has a density, pressure and velocity at each point, you find that this idealized air obeys a beautifully simple equation that explains almost everything we care about: how to build airplanes, how we hear them with sound waves, how to make weather forecasts, etc. Yet despite all that convenience, air of course isn't truly continuous. I think it's the same way for space, time and all the other building blocks of our physical word.

Let's face it: despite their seductive allure, we have no direct observational evidence for either the infinitely big or the infinitely small. We speak of infinite volumes with infinitely many planets, but our observable universe contains only about 1089 objects (mostly photons). If space is a true continuum, then to describe even something as simple as the distance between two points requires an infinite amount of information, specified by a number with infinitely many decimal places. In practice, we physicists have never managed to measure anything to more than about 17 decimal places. Yet real numbers with their infinitely many decimals have infested almost every nook and cranny of physics, from the strengths of electromagnetic fields to the wave functions of quantum mechanics: we describe even a single bit of quantum information (qubit) using two real numbers involving infinitely many decimals.

Not only do we lack evidence for the infinite, but we don't actually need the infinite to do physics: our best computer simulations, accurately describing everything from the formation of galaxies to to tomorrow's weather to the masses of elementary particles, use only finite computer resources by treating everything as finite. So if we can do without infinity to figure out what happens next, surely nature can too—in a way that's more deep and elegant than the hacks we use for our computer simulations. Our challenge as physicists is to discover this elegant way and the infinity-free equations describing it—the true laws of physics. To start this search in earnest, we need to question infinity. I'm betting that we also need to let go of it.

Bill S will no doubt have a view

Unlike the linear Schrödinger equation, the NLSE never describes the time evolution of a quantum state (except hypothetically, as in some early attempts, in the 1970s, to explain the quantum measurement process).

if there is a real infinity nothing measurable/observable or real exists there.

Essentially this would be an infinity of nothingness. Is that even a scientific concept?

Presumably it could not exist in the Universe. and as it would be nothing, would it be meaningful to talk about "where" it might be?

Exactly correct so you invoke occam's razor and the universe is finite.

There seems to be divided opinion among scientists as to whether or not one should try to apply Occam's razor to anything outside the Universe. Personally, I think it is justifiable, precisely because it is just a guideline. It points in a rational direction, but makes no dogmatic statement.

BTW; the questions remain unanswered.

"Can there ever have been nothing?"

"If ever there was nothing, could there be something now?"

Now here is another slippery slope can I define nothing because if nothing is a hard fixed the answer is no and obviously so.

However lets take money scenario you can have nothing but you can borrow some create more pay back the original and then have money.

The real problem you are chasing I think Bill S is not nothing but can one borrow in physics and does borrowing imply that there is not nothing behind it

If there is nothing, there is nothing, there is nothing to borrow and no one to do the borrowing or lending

Now can I reverse the problem why do you feel the need to even consider an absolute nothing? If you really feel absolute nothing as you have defined is essential then you need to take up a religion there isn't another choice and I am serious.

That is called "Absolute nothing" and it's not a science term

Absolute nothing as per your definition can't exist in science it's not possible because there is an instant falsification because you exist.

Now can I reverse the problem why do you feel the need to even consider an absolute nothing? If you really feel absolute nothing as you have defined is essential then you need to take up a religion there isn't another choice and I am serious.

So make a choice Bill S .... A or B .... no-one is ever going to give you any others and can't because of what absolute nothing implies.

Orac, I don't really have a big problem with infinity and nothing the way Bill S. does

That then raises the question of what is outside the universe. That would presumably be absolutely nothing. So there is a bit of a conflict there.