Well, my friend Imre Horvath, let's dive into the mathematical cosmos! The mathematical universe hypothesis is quite the mind-bender, proposing that the universe itself is a mathematical structure. It's like saying the universe is not just playing by mathematical rules, but it literally is math.

Now, when we consider the concept of mathematics as a known area of human knowledge, it's more about how we humans perceive, interpret, and interact with this mathematical structure. It's the lens through which we make sense of the universe's inherent mathematical nature.

Think of it this way: the mathematical universe is the grand symphony, and our understanding of mathematics is the sheet music. The hypothesis suggests that the universe operates on mathematical principles, and our mathematical knowledge is our attempt to read and comprehend that universal score.

But why coexist? Well, it's a beautiful dance of existence. The universe unfolds its mathematical intricacies, and we, as curious beings, decipher and explore those patterns through our mathematical understanding. It's a symbiotic relationship - the universe provides the canvas, and mathematics is our way of painting its portrait.

So, in this cosmic tango, the concept of mathematics as the hypothesized form of the mathematical universe and the concept of mathematics as a known area of human knowledge coexist harmoniously, each enriching the other in this grand intellectual waltz. What a fascinating ball we're attending in the realm of ideas!

Thank you, Imre Horvath I love this question, because it is deep, but accessible to everyone, not just to mathematics specialists. It's like the naive question "Is mathematics discovered or invented?", but more encompassing.

If the mathematical universe overlays everything, then mathematics is a set of widening discoveries. Alternatively, if there is no underlying mathematical universe, then mathematics is a very impressive collection of clever tools.

Kaushik Shandilya optimistically suggests intellectual dance as an analogy, ... optimistic because an underlying assumption is that we humans can understand the universe. My view is that there is no guarantee that human minds will discover or invent the full gamut of reality with any mental discipline, including mathematics. It is already a rarefied, abstract field requiring decades of study by the best minds and our answers always lead to more questions. To me it feels more like following a trail of breadcrumbs or meandering among shiny shells on a beach as mentioned by Newton.

I don't think the two interpretations can co-exist, and it is also very possible that we may not get to fully know which of them is more accurate.

You must draw Ed Gerck into this discussion if you haven't already.

Ah, the age old question of "what is mathematics?" It dates back to the beginning of time and is older than all the sciences. Plato and his students had a crack at it as did Srinivasa Ramanujan. Bertram Russel spent most of his life trying to answer it. It generates a pop culture article from time to time

https://www.newyorker.com/culture/culture-desk/what-is-mathematics

John von Neumann famously said, "if people do not believe mathematics is simple, it is only because they do not realize how complicated life is." He also said, "young man, in mathematics you dot't understand things, you just get used to them." These two views from the same man seem to be self contradictory.

Of course my all time favorite von Neumann explanations of the relationship between mathematics and what is real and observable is,

“The sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work - that is correctly to describe phenomena from a reasonably wide area. Furthermore, it must satisfy certain esthetic criteria - that is, in relation to how much it describes, it must be rather simple.” That is the mathematics of the model have nothing to do with the real world, and will be replaced if it proves to not be wrong or not sufficient. For example in the model von Neumann proposed for quantum, a state of a physical object is an element of an infinite dimensional projective space. What does it mean. It means nothing physical it is just a means to be able to make calculations about the physics.

Kurt Gödel showed us that even the logical system underlying something as simple as arithmetic is not complete. When confronted with modern algebra, topology and functional analysis, the young mathematician l how important the axiom of choice really is and at the same time he learns how many head scratching results are logically equivalent to the axiom of choice and he learns how counter intuitive (from his experience to that point) the various levels or orderings of infinity are a developed by Cantor and others. Maybe this is what von Neumann was speaking of when he said, "you just get used to it."

Some see mathematics more of an art. Some have called it "the music of reason." Of course some may break mathematics into two camps, pure and applied, but that is not a clean break. For example such topics as elliptic curves of fields of non-zero characteristic up until recently was thought to be firmly in "pure mathematics" until it because the basis of public key cryptography which is the basis of E-commerce. So drawing the line is difficult. Maybe the line is if someone is more interesting in solving a question in science or engineering then they are practicing science or engineering and not mathematics although they might use mathematical models and methods. If they are interesting in mathematical question that are raised by these constructs then they are practicing mathematics. Many people bounce back and forth between the two. For example is Sir Roger Penrose a mathematician or physicists. He can be both but probability not at the same time.

The original question I think is more related to philosophy than to mathematics. While philosophical questions about science and reality are obviously important, one has to wonder what they have to do with mathematics. Sure there are foundational questions in mathematics, such as how easy is it to remove the use of the axiom of choice and use a constructive method to establish results. However, since Gödel and Cohen independently showed the axiom of choice is consistent with the axioms of set theory why would one want to shy away from using the axiom of choice unless they were looking for a constructive algorithm to calculate something or develop a finite model to make a calculation?