There's little doubt that mathematics rules the reality roost when it comes to the laws, principles and relationships within the sciences in general and the physical sciences in particular. Further, mathematics plays a dominant role when it comes to the purely economic aspects of our lives and where would sports be without statistics? However, when it comes down to brass tacks, how much of really real reality is actually reflected in our mathematics?
The Reality of Mathematics.
Mathematics is just a shorthand mental concept that simulates reality, or approximates reality or a possible reality or even an imaginary / impossible 'reality'. Mathematics is NOT reality itself. You can mathematicatically manipulate the challenged extra dimensions in String Theory but that does not mean that necessity that these extra dimensions actually exist.
Mathematics is a tool that at first approximation attempts to reflect upon the nature of really real reality. Mathematics is not reality itself. Further, our mathematics are structured to reflect our version of reality based on our observations not of necessity what really happens. The perfect example is Quantum Mechanics. For example, we may not know, even can not know even in principle, exactly where a particle is as well as at the same time where it is going with 100% precision. So we invent a form of probability mathematics like the Schrodinger Equation or the equation that governs the Heisenberg Uncertainty Principle. Those equations are for our edification but they do not alter the really real reality fact that the particle has actual coordinates and is going from A to B. Probability in Quantum Mechanics, and the mathematical equations associated with it, are just reflections on the limits of the human observer and human instrumentation, not a reflection on Mother Nature's really real reality. Our Quantum Mechanical equations are imposed approximations to really real reality much like Newton's equation for gravitational attraction was really only in hindsight an approximation.
There can be multiple models of reality, each based on mathematics, but they can not all be right. Cosmology is a case in point.
The phrase "but the mathematics works" means absolutely nothing. Just because mathematics predicts the possibility of some kind of structure and substance, or some law, relationship or principle that the Cosmos might have, does not of necessity make it so. A prime example where the mathematics worked but the Cosmos did not go along for the ride was the ad-hoc piling on those epicycles upon epicycles in order to explain the motion of the planets. It finally got so unwieldy that the baby was thrown out with the bathwater and a new baby conceived, that being that the Earth was just another planet and not at the center of life, the Universe and everything. Once it was postulated that the Earth went around the Sun, planetary motion fell into place – mathematically into place as well.
Take a more modern example. The mathematics works in String Theory, but to date String Theory remains a theorists' theoretical dream (accent or emphasis on the word "dream").
Probability theory is that branch of mathematics that interposes itself between the macro human and human comprehension and abilities and the micro world of quantum mechanics. That has way more to do with the macro than with the micro since the absolutes of the micro are not visible in the realm of the macro; they are beyond the realm of the macro to resolve through no fault by the way of human comprehension or abilities.
A prime example is that there is no probability in quantum mechanics, only probability introduced by the limitations of the conscious mind to get down and dirty to the level of detail required to eliminate the concept of probability from quantum mechanics.
Mathematics serves no purpose, useful or otherwise, outside of the context of the human mind (specifically) or outside of the intellectual conscious minds of other sentient species (in general), thus making allowances for ET and possibly the terrestrial great apes; whales and dolphins; and perhaps other advanced minds – sometimes elephants as well as some birds.
In the absence of any conscious minds, what use has the Universe for arithmetic, geometry, trigonometry, calculus, topology, statistics and the multi other branches of mathematics? Now 1 + 1 = 2 may be universally the case and logically true even in the absence of any conscious mind, or before any life form ever came to pass, but so what? That cuts no mustard with the Universe! There was nobody around to conceive of that or to make use of that or to equate the manipulation of numbers as a reflection of universal reality (or even even non-reality *). There was no conscious or intellectual mind around to appreciate any mathematical utility or usefulness or beauty or elegance.
Mathematics in fact is not a reflection on or of reality, only that reality as observed or defined once having been filtered through sensory apparatus that pondered over by the conscious mind. Reality as perceived in the mind is several transitional layers of processing removed from whatever pure external reality there happens to be. There's even an additional layer if instrumentation is a middleman. So the conscious mind is thus limited in terms of its ability to come to terms with the full scope of really real reality.
Mathematics is the interface between humans and human comprehension, understanding, etc. of the Cosmos at large. Mathematics can tell you in actuality or theoretically the 'what' but never the 'how' or the 'why'. For example, there's Newton's Law of Gravity, but even he realized that that equation just told you 'what', not 'how' or 'why'.
The Non-Reality of Mathematics.
The following examples are some of what I term the non-realities of mathematics.
* Hyper-cubes are a nice abstract concept that mathematics / geometry can incorporate. However, while you might be able to play with real cubes, like dice, hyper-cubes will be forever beyond you.
* Stephen Hawking's concept of negative time. Since IMHO time is just change and change is just motion, then negative time would have to be negative change and negative motion. That does not make any sense at all. So while Hawking's negative time may be useful in a mathematical sense, it has no bearing on our reality and can safely be ignored.
* Lots of quantum mechanical equations yielded up infinities so a sleight-of-hand concept called re-normalization was invented to deal with those cases involving infinities. That strikes me as dealing cards from under the table or otherwise known as a insertion a "fudge factor". Does re-normalization represent really real reality?
* The mathematics of singularities inherent at the moment of the Big Bang or in Black Holes goes down the rabbit hole in that the laws, principles and relationships inherent in the physical sciences that are so otherwise adequately described mathematically now break down when trying to describe singularities and so so does the accompanying mathematics that are involved as well. So what actually is the really real reality behind singularities?
* Mathematics are perfectly capable of dealing with alleged extra dimensions inherent in String Theory. However, that does not make String Theory a reality, does it make a half-dozen extra and hidden dimensions a reality.
* Mathematics is perfectly capable of dealing with an inverse cube law that has no correspondence with our physics. Just because a mathematical equation works does not mean that there is a one-on-one correspondence to the real physical world.
* Mathematics are perfectly capable of dealing with zero, one and two dimensions yet these are just mental concepts that can not actually be constructed and then have no real real reality.
* Space-Time: Since space is just an immaterial mental concept (that imaginary container that actual physical stuff has to tear in) and since time is also just an immaterial mental concept (our way of coming to terms with change which is just motion – which is also an immaterial mental concept since motion itself is not composed of anything physical), then space-time has to be an immaterial mental concept. Either space nor time nor space-time is actually composed of any material substance and the trilogy has no material 3-D structure. However, the mathematics incorporating the concept of space-time are a useful tool in describing reality, but not actually really real reality itself.