Aren't singularities a math theory problem, not a science problem?
No, singularities are
both a math theory problem
and a science problem. Mathematics expands & strengthens science by providing a logical & explanatory foundation that connects seemingly unconnected observations. Science expands & strengthens mathematics by providing a physical or empirical interpretation of mathematical equations. In this context, nothing meaningful is either purely mathematics or purely science, but rather everything is to some extent both mathematical & scientific.
The singularity is a math theory problem because the equations are literally undefined at the singularities. But they are also a science problem because there is nothing for science to interpret. So, either you need to find new equations that have no singularities and can thus be interpreted by science, or you need a scientific observation that proves the mathematical equations are wrong. In the absence of either of these, then you simply don't know what's happening, either mathematically or scientifically.
With that in mind, consider ...
Nature doesn't have a problem with existing. Math has a problem describing the universe.
As a general rule, the 2nd sentence is absurd. Math has no problem describing the universe. In fact, quite the opposite is true. Mathematics is
too successful at describing the universe for the point to be taken lightly; see, for instance, Max Tegmark's
The Mathematical Universe & citations thereto, or Eugene Wigner's 1960 essay "
The Unreasonable Effectiveness of Mathematics in the Natural Sciences" (
Communications in Pure and Applied Mathematics 13(1): 1-14 (1960).
The observations of radiation from a highly luminous object are "science" except you cant do an experiment to confirm your hypothesis. There arent too many astronomers that go into the lab to perform experiments.
This is quite misleading, and for the most part wrong. What is an "astronomer"? if by that you mean somebody who only looks through telescopes & etc., you are right. They don't personally do experiments in a laboratory because that's not the manner of science they do. But so what? Maybe you think that laboratory scientists and astronomers never talk to each other, never interact at any level? Astronomers use the Zeeman effect to derive magnetic fields from spectral line measurements. But the Zeeman effect is calibrated for them, in real laboratories by real physicists. Who cares that the astronomers themselves did not actually do the experiments? Astronomers look at highly luminous things like really bright stars and use those observations to derive physical properties (temperature, composition, mass & etc.). But they make use of extensive laboratory studies of nuclear astrophysics. So when astronomers tell you that stars are powered by nuclear reactions rather than electric currents, they have an
extensive laboratory experience to support them. Does it really matter that nuclear physicists did the experiments, rather than the astronomers themselves? And so it goes down the line ... plasma physics, thermodynamics, radiative transfer physics, all subject to
extensive laboratory experience, and all that experience is exported by the laboratory physicists and imported by the astronomers. Your suggestion that astronomy lacks laboratory support & confirmation is absurd.
The laws of physics don't break down at a singularity, the mathematical description breaks down, assuming it really is a place where there is "alot of gravity" made by ultra dense matter.
This much is correct. We assume that the laws of physics exist independent of us, and therefore do not break down anywhere. But our understanding of the laws of physics most certainly does break down completely at the singularity. That's because, without being able to observe inside the event horizon of a black hole, we rely on the mathematics to provide us with something to physically interpret. With neither mathematics nor observation to interpret, we are left with nothing. The physical interpretation of the mathematical singularity as literally a point of infinite mass density is unreasonable, and exists only in popular level discussions to avoid complications. But if you look at the real science, nobody says that. Rather, everybody says the obvious, that we simply have no idea what physically happens at the singularity because it is a mathematical entity, not a physical entity. A quantum physical theory of gravity (string theory, loop quantum gravity & etc.) will allegedly provide equations which do not have singularities where the general relativistic equations do. That will in turn mean that we will have something mathematical for which we can then provide a physical interpretation, eliminating the physical (i.e., scientific) problem of the singularity.
Now with all that in mind, ...
You're free to think the black hole has infinite density at the singularity, but the theory of relativity doesn't actually say that. ... Some people, including Hannes Alfvén and Michael Mozina, have attacked general relativity by complaining about the non-sensicality of infinite density at the singularity. That's a bit of projection on their part, abetted by the popular accounts from which they learned about the singularities. They may think the density is infinite at the singularity, but that conclusion is theirs, not general relativity's.
W.D.Clinger is right on the money. Let me repeat myself ... "
The physical interpretation of the mathematical singularity as literally a point of infinite mass density is unreasonable, and exists only in popular level discussions to avoid complications. But if you look at the real science, nobody says that. Rather, everybody says the obvious, that we simply have no idea what physically happens at the singularity because it is a mathematical entity, not a physical entity."