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 mathematically manipulate the alleged extra dimensions in String Theory but that doesn’t mean of necessity that these extra dimensions actually exist.

Mathematics is a tool that at first approximation tries 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 cannot 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 don’t 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't 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 didn't 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 aren't 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 E.T. and maybe the terrestrial great apes; whales and dolphins; and perhaps other advanced minds - perhaps 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 might 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 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 thus 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’.