Singularities

Hal 9000, from the film
2001, a space odyssey

A singularity is a mathematical concept applied to a function of a variable that reaches an infinite value for one or several finite values of its independent variable.

For example, the function y=1/x presents a singularity for x=0, since it is often said that 1 divided by zero is equal to infinity.

Función y=1/x

In fact, saying that a function f(x) reaches infinite value for x=a is a way of saying that for values of x increasingly closer to “a”, the absolute value of f(x) grows without limit.

There are functions with more than one singularity. For example, the function y=tangent(x) presents infinite singularities for x equal to any odd multiple of π/2.

Función y=tangente(x)
en las proximidades de π/2

The exponential function y=e^x does not present any singularity, because y tends to infinity only when x tends to infinity.

In physics we also talk about singularities. Physicists often take a dim view of an infinite value, which at first glance does not seem should be applicable to any variable in nature. However, the following possible infinities have not been excluded:

• Infinity of space or time in our universe. The infinity of time from the past is usually excluded, because taking into account the second principle of thermodynamics, if our universe had had an infinite duration, it would already be in the state of heat death. But there remains the infinity of time towards the future, and the infinity of space. I talked about this in another post.
• Infinite density at the center of a black hole. Since all the matter would be contained in a volume equal to zero, its density would be infinite. In another post I commented on one of the proposals made to eliminate this infinity.
• Infinite density and temperature at the time of the Big Bang. In 1970, Roger Penrose and Stephen Hawking proved that Einstein’s equations of General Relativity require that there must have been a singular point in the universe, at which density and temperature became infinite.

In the world of technology, there is also talk about singularity. The idea is based on an extrapolation of Moore’s law, which in its original form said that the number of transistors on each chip doubles approximately every two years. But this corresponds to exponential growth, which never reaches singularity. And to top it off, Moore himself, in a lecture given at the IBM Academy in 2003, said this:

Exponential growth does not exist

By which he meant that all natural, physical and biological processes, which at first seem to grow exponentially, actually follow a logistic curve, and end up approaching a horizontal asymptote, never a vertical one, not even an exponential.

Curva logística

For a technological singularity to be possible, it would be necessary for technology to grow at a hyper-exponential rate (greater than the exponential function), such as the tangent function mentioned above, something that has never occurred, even approximately, in the natural world.

But supporters of technological singularity usually define this concept differently:

The technological singularity will take place when artificial intelligence surpasses human intelligence: when we manage to build machines more intelligent than us (superintelligence).

In my opinion, this is an abuse of language. The word singularity does not make sense in this context, since it does not refer to an infinite value. But it is introduced because at the same time it is stated that technological advances are going to be so enormous during this century that they will tend to infinity and will make it possible to achieve anything that is possible to be achieved.

The hypothetical overcoming of human intelligence by artificial intelligence, if it were possible (which is not evident), does not require an infinite value. It does not represent a vertical asymptote, but rather the point of intersection of two curves, and it would be more logical to call it a critical point, rather than a singularity. But it looks like this second name has more appeal, and that is why it is used instead of the first, even though incorrect.

Besides which, those who believe in the hyper-exponential growth of technology forget that the resources necessary to achieve it (energy, manpower, economic resources) should also grow at the same rate. I find it unbelievable that, just when we have realized that our planet’s resources are not inexhaustible, we are predicting infinite technological development. This shows that human folly knows no limits. Perhaps this variable (human folly) is actually growing at a hyper-exponential rate.

The same post in Spanish

Thematic Thread about Natural and Artificial Intelligence: Previous Next

Manuel Alfonseca