What is a singularity?

Not Von Neumann's urinal, but a
model of an essential singularity.
[Source: Wikipedia Commons.]

I recently read a bit around about the technological singularity, but it’s hard. It’s hard because I have to endure sentences like this:

“Singularity is a term derived from physics, where it means the point at the unknowable centre of a black hole where the laws of physics break down.”

Ouch. Or this:

“[W]e cannot see beyond the [technological] singularity, just as we cannot see beyond a black hole's event horizon.”

Aargh. Then I thought certainly they must have looked up the word in a dictionary, how difficult can it be? In the dictionary, I found this:

“sin-gu-lar-i-ty
noun, plural sin-gu-lar-i-ties for 2–4.

1. the state, fact, or quality of being singular.
2. a singular, unusual, or unique quality; peculiarity.
3. Mathematics, singular point.
4. Astronomy (in general relativity) the mathematical representation of a black hole.”

I don’t even know where to start complaining. Yes, I did realize that black holes and event horizons made it into pop culture, but little did I realize that something as seemingly simple as the word “singularity” is surrounded by such misunderstanding.

Von Neumann.

Let me start with some history. Contrary to what you read in many places, it was not Vernor Vinge who first used the word “singularity” to describe a possible breakdown of predictability in technological development, it was von Neumann.

Von Neumann may be known to you as the man behind the Von Neumann entropy. He was a multiple talented genius, one of a now almost extinct breed, who contributed to many disciplines in math and physics, and what are now interdisciplinary fields like game theory or quantum information.

In Chapter 16 (p 157) of Stanislav Ulam’s biography of Von Neumann, published in 1958, one reads:

“One conversation centered on the ever accelerating progress of technology and changes in the mode of human life, which gives the appearance of approaching some essential singularity in the history of the race beyond which human affairs, as we know them, could not continue.”

The term “singularity” was then picked up in 1993 by Vinge who coined the expression “technological singularity”. But let us dwell for a moment on the above Von Neumann quote. Ulam speaks of an “essential singularity”. You may be forgiven mistaking the adjective “essential” as a filler, but “essential singularity” is a technical expression, typically found in the field of complex analysis.

A singularity in mathematics is basically a point in which a function is undefined. Now it might be undefined just because you didn’t define it, but it is possible to continue the function through that point. In this case the singularity is said to be removable and, in some sense, just isn’t an interesting singularity, so let us leave this aside.

What one typically means with a singularity is a point where a function behaves badly, so that one or several of its derivatives diverge, that is they go to infinity. The ubiquitous example in school math is the poles of inverse powers of x, which diverge with x to zero.

However, such poles are not malign, you can remove them easily enough by multiplying the function with the respective positive power. Of course this gives you a different function, but this function still carries much of the information of the original function, notably all the coefficients in a series expansion. This procedure of removing poles (or creating poles) is very important in complex analysis where it is necessary to obtain the “residuals” of a function.

Some singularities however cannot be removed by multiplication with any positive power. These are those cases in which the function contains an infinite number of negative powers, the most commonly used example is exp(-1/x) at x=0. Such a singularity is said to be “essential”. Please appreciate the remarkable fact that the function itself does not diverge for x to zero, but neatly goes to zero! So do all its derivatives!!

So what did von Neumann mean with referring to an essential singularity?

From the context it seems he referred to the breakdown of predictability at this point. If all derivatives of a function are zero, you cannot make a series expansion (neither Taylor nor Laurent) around that point. If you hit that point, you don’t know what happens next, basically. This is a characteristic feature of essential singularities. (The radius of convergence cannot be pushed through the singular point.)

However, predictability of the laws of nature that we have (so far) never breaks down in this very sense. It breaks down because the measurement in quantum theory is non-deterministic, but that has for all we know nothing to do with essential singularites. (Yes, I’ve tried to make this connection. I’ve always been fond of essential singularities. Alas, not even the Templeton Foundation wanted anything to do with my great idea. So much about the reality of research.)

Geodesic incompleteness.
Artist's impression.

The other breakdown of predictability that we know of are singularities in general relativity. These are not technically essential singularities if you ask for the behavior of certain observables – they are typically poles or conical singularities. But they bear a resemblance to essential singularities by a concept known as “geodesic incompleteness”. It basically means that there are curves in space-time which end at finite proper time and cannot be continued. It’s like hitting the wall at 32km.

The reason for the continuation being impossible is that a singularity is a singularity is a singularity, no matter how you got there. You lose all information about your past when you hit it. (This is why, incidentally, the Maldacena-Horowitz proposal to resolve the black hole information loss by putting initial conditions on the singularity makes a lot of sense to me. Imho a totally under-appreciated idea.)

A common confusion about black holes concerns the nature of the event horizon. You can construct certain quantities of the black hole spacetime that diverge at the event horizon. In the mathematical sense they are singular, and that did confuse many people after the black hole space-time was first derived, in the middle of the last century. But it was quickly understood that these quantities do not correspond to physical observables. The physically relevant singularity is where geodesics end, at the center of the black hole. It corresponds to an infinitely large curvature. (This is an observer independent statement.) Nothing special happens upon horizon crossing, except that one can never get out again.

The singularity inside black holes is widely believed not to exist though, exactly because it implies a breakdown of predictability and causes the so paradoxical loss of information. The singularity is expected to be removed by quantum gravitational effects. The defining property of the black hole is the horizon, not the singularity. A black hole with the singularity removed is still a black hole. A singularity with the horizon removed is a naked singularity, no longer a black hole.

What has all of this to do with the technological singularity?

Nothing, really.

To begin with, there are like 17 different definitions for the technological singularity (no kidding). None of them has anything to do with an actual singularity, neither in the mathematical nor in the physical sense, and we have absolutely no reason to believe that the laws of physics or predictability in general breaks down within the next decades or so. In principle.

In practice, on some emergent level of an effective theory, I can see predictability becoming impossible. How do you want to predict what an artificial intelligence will do without having something more powerful than that artificial intelligence already? Not that anybody has been able to predict what averagely intelligent humans will do. Indeed one could say that predictability becomes more difficult with absence of intelligence, not the other way round, but I digress.

Having said all that, let us go back to these scary quotes from the beginning:

“Singularity is a term derived from physics, where it means the point at the unknowable centre of a black hole where the laws of physics break down.”

The term singularity comes from mathematics. It does not mean “at the center of the black hole”, but it can be “like the center of a black hole”. Provided you are talking about the classical black hole solution, which is however believed to not be realized in nature.

“[W]e cannot see beyond the [technological] singularity, just as we cannot see beyond a black hole's event horizon.”

There is no singularity at the black hole horizon, and predictability does not break down at the black hole horizon. You cannot see beyond a black hole horizon as long as you stay outside the black hole. If you jump in, you will see - and then die. But I don’t know what this has to do with technological development, or maybe I just didn’t read the facebook fineprint closely enough.

And finally there’s this amazing piece of nonsense:

“Singularity: Astronomy. (in general relativity) the mathematical representation of a black hole.”

To begin with General Relativity is not a field of astronomy. But worse, the “mathematical representation of a black hole” is certainly not a singularity. The mathematical representation of a (classical) black hole is the black hole spacetime and it contains a singularity.

And just in case you wondered, singularities have absolutely nothing to do with singing, except that you find both on my blog.