Posted on 02/20/2015 6:01:20 PM PST by LibWhacker
Infinity Is a Beautiful Concept – And It’s Ruining Physics
I was seduced by infinity at an early age. Georg 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 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 assumptions 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 explains how a subatomic speck of matter transformed into a massive Big Bang, creating a huge, flat, 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 and the BICEP2 experiments. But by 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 there will be infinitely many copies of you, far away in our infinite space, obtaining each physically possible outcome; and despite years of teeth-grinding in the cosmology community, no consensus has emerged on how to extract sensible answers from these infinities. So, strictly speaking, we physicists can no longer predict anything at all!
This means that today’s best theories 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 far, 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 were skeptical of infinity and the continuum. The legendary Carl Friedrich Gauss denied that anything infinite really exists, 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’ll find that this idealized air obeys a beautifully simple equation explaining almost everything we care about: how to build airplanes, how we hear them with sound waves, how to make weather forecasts, and so forth. 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 world.
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 seventeen 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 need the infinite to do physics. Our best computer simulations, accurately describing everything from the formation of galaxies 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.
I’m almost 62 now and I have yet to have an answer to my question. It is something that Mankind will never really know, unless the next Star Trek Movie explains it of course.
How anyone can look up at the Night Sky and think all of this is just some sort of coincidence baffles me to this day.
The odds are beyond staggering, but it doesn’t matter to some people. They think the Science is settled, LOL.
The question is a contradiction in terms. "Where" assumes the existence of space. That's what she ought to have said while you were sitting on the front stoop.
Cordially,
However, the set of all real numbers is uncountably infinite.
His whole point is that he expects us to discover there is a finitely smallest unit of space. In finding that we will resolve a number of the conundrums that result from quantum mechanics, and especially trying to combine quantum mechanics with relativity.
I just finished reading David Foster Wallace's book "Everything and More". When dealing with infinite quantities mathematicians have to be very careful how they state things. Having an infinity and an equal sign in the same equation is almost always a bad idea.
It's the expression "ten times smaller" that makes me bang my head against the desk. Are they being deliberately ambiguous or really that ignorant?
Good science is good. But ya have great points. Coincidence, Creator or whatever, man want’s to know. It’s the nature of the beast. Answering the infinite question.
Very clever.
“My own suspicion is that the Universe is not only queerer than we suppose, but queerer than we can suppose.”
J. B. S. Haldane (5 November 1892 – 1 December 1964) was a British geneticist and evolutionary biologist.
Similar remarks that seem derived from this have in recent years been attributed to Arthur Stanley Eddington, as well as to Haldane, but without citations of an original source.
I like the queer version.
Yeah, I caught that after my post. See #34.
“Inside the museums, infinity goes up on trial”
B.Dylan
If you know L'Hospital's Rule, you can easily construct limits which superficially appear to be indeterminate forms like ±∞/±∞, 0/0, 1∞, ∞0 which can be constructed from actual limits which are +∞, -∞, 0, 1, or any real number.
For example:
limx→∞ (1+x)α/x. Superficially, this looks like 1∞. But, for every N, no matter how large it may be 1N = 1. This is pretty much a textbook definition that the idea of 1∞ = 1. However [you can verify this very quickly with a calculator or by plugging numbers into the google web page search] You can make the above limit any positive real number by choosing α correctly. For example, if α = 1, then limx→∞ (1+x)1/x = e. If α = 0, that limit = 1, If α = -1, it's 1/e. If α = ln(π) limx→∞ (1+x)α/x = π.
You can do the same thing with any indeterminate form that looks like ∞/∞. Just tune the limit properly and it can be 0, ∞, or any other number you like, including the OP's "1." That's why indeterminate forms are symbols, which have no meaning.
The number of reals is infinite. The number of integers is infinite. The statement [number of reals]/[number of integers] has no meaning. The cardinality of the reals is greater than that of the integers. There is no bijective function [1-1 and onto] that maps the integers to the reals. There is a "sense" in which there are more reals than integers, but that "sense" is not translatable in terms of "the number of objects in the sets," because both are infinite.
Infinities are to mathematicians as high current live wires are to electricians: There are tools for handling them, and they won't harm you as long as you're careful. Rigor is the key.
This physicist may want to get rid of the concept of infinity in physics, but he can't. The reason he can't is that he cannot get away from the concept of zero. And as long as you are talking about zero, infinity is always on the other side of that coin.
Because, to be "infinitely small" means EXACTLY this: no matter how small it is, there is something smaller. We have a name for that: it is zero. A thing which is smaller than all other things has no existence.
It's been half a lifetime since Calculus; could you remind me what is L'Hopital's Rule?
If you try to find the limit of liberals’ stupidity, you’ll see that infinity does indeed exist.
Hasn’t quantum mechanics abolished the concept of infinitely small already? Isn’t the planck length the point at which the universe “pixilates” — that is, at which no smaller thing could exist? Or does the theory of inflation suggest that the size of the planck length itself has grown?
Also: How weird would it be if there isn’t enough matter in the universe to cause a Big Crunch, but there is something else altogether which would limit the universe’s size before Heat Death? Some force or property of space which limits expansion, the way you can only stretch a waistband so far before it resists further stretching?
Hi-di-ho ho-di-hi
modern kids:
https://www.youtube.com/watch?v=gcq4Tq68KJk
old school:
https://www.youtube.com/watch?v=z7at9X_ympQ
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