The concept of singularity is interdisciplinary. After all, it is present in both physics and mathematics. This is not so surprising when we consider the very close relationship developed between these two areas of knowledge. Galileo Galilei, the great Italian scientist, is said to have said that "mathematics is the alphabet with which God wrote the universe. And indeed, when weremember the sciences, physics, chemistry, and even biology seen in school, mathematics can be understood as a constant presence when human beings try to describe nature.

The truth is that there is more than one philosophical perspective as to the reasons for the relationship between modern science and mathematics. But in fact, mathematics is a fundamental tool for making, for example, theoretical predictions in physics. Or for developing models that describe reality. And on these, we can test them from experimental data. Relativity, for example, to bea recognized theory as it is today, had to be proven by observations of the universe.

The notion of singularity, however, although present in theories built from mathematics, is more complicated. This is because it does not represent, in fact, something physical, but a point in space where physics as we know it stops working.

The mathematics of black holes

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Any first-year student in a mathematics course, even if they don't know it yet, often finds singularities. In calculus courses, it is common, for example, to find expressions like y = 1/x. And, as x approaches zero, y increases, tending to infinity. The problem is that infinity cannot, in fact, be taken as a number. But rather, something more general, like amathematical concept. After all, simplifying the matter, you can't count to infinity. So even if this equation in question describes something physical, the point at which x = 0 is impossible in nature. That is, a singularity.

In physics, singularities appear, for example, in general relativity. And Karl Schwarzschild, a famous astrophysicist, applied Einstein's ideas to a system of spherical mass, such as a star. The solution to the math, then, contained these points at the center of the body and at a certain distance from it (the so-called Schwarzschild radius). Thus, if we compress the mass of a body inside aSchwarzschild radius, what occurs is a process of collapse due to gravity. The object, therefore, ends up being reduced to an infinitesimally small point. That is, the point of the central singularity, which remains there even if we change the reference coordinates for the calculations.

That's what happens with the famous black holes. That is, they are bodies, in this case collapsed stars, whose density at the center is infinite. That is, a singularity. And it is this infinitesimally small region of space that we cannot describe well with our physics. For that, it is likely that we would need a new model for the attraction of gravity. To be more specific, one that understandsvery intense forces on tiny length scales. So a quantum theory of gravity.

The singularity of the Big Bang

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The Big Bang theory caused quite a controversy in its early moments, both for its philosophical implications, and even in the scientific community. A curious fact, in fact, is that the creator of the term "Big Bang" proposed, in fact, an alternative model. Fred Hoyle, one of the great responsible for our understanding about the production of chemical elements and energy in stars,Over time, however, evidence began to appear about the expansion of the universe. And so the Big Bang, through its own observation of the universe, was becoming the most consolidated theory about the origin of time and space.

The problem is that the point where the Big Bang occurred (i.e., the "big bang") is, in itself, a singularity. Thus, even though it is a very successful theory, the theory is not complete in understanding the universe over time. And a theoretical possibility to solve the issue could be in the causal set theory. In it, space-time, already united by relativitygeneral, it would not be a continuum. Rather, it would break up into pieces. Like, for example, "space-time atoms". And nothing in the universe would be smaller than them, which would make the existence of singularities impossible.

The challenge is to try to describe the initial moments of the Big Bang with these theoretical tools. After that, the universe would become giant and "well-behaved." And then the continuum would become a good approximation. In any case, there is no universal solution to fill this gap in our understanding of things. That is, there is a lot of work ahead. We can be sure, however, that allthose efforts will be fascinating.