Math is discovered and not invented

Pro's logic is fatally flawed.Their argument may be syllogized as follows:Premise 1: Axioms are discovered. [Their argument 1.]Premise: Math is axiomatic. [Their argument 2.]Conlcusion: Math is discovered.Both of Pro's premises are flawed -- and moreover, poorly supported in the first place.PREMISE 1: AXIOMS ARE DISCOVEREDPro's argument for this premise is very, very weak. Pro merely asserts that:No matter where you are, an object is the same as that object. A human does not need to state that an apple is an apple. The concept is there independent of the mind.Syllogizing this statement shows its absurdity:P1: Humans know what apples [and axioms in general] are.P2: If humans know what something is, then the idea of that something is independent of the mind.C1: The idea of apples [and axioms in general] is independent of the mind.It should be obvious that P2 is weak. Just because most humans innately understand what an apple is, does not mean that the concept of an apple is itself "innate". Let me show you two disproofs.[1] First: Machine learning is incredibly good at recognizing objects like cars, mammals, buildings, and so on. (Search "Google Deep Dream" if you don't believe me.) Yet machine learning is not conscious, but is merely recognizing patterns of information and fitting mathematical functions to those patterns. If nonconscious algorithms can recognize things like apples, this suggests that it's possible to recognize things like apples without having an "idea" of an apple in the first place. And since human brains are known to function heuristically -- ie, somewhat machine-like -- it's reasonable to think that human brains have similarly, through the course of evolution, come to be able to recognize patterns like apples and faces and cars (and identify new patterns as well). And via Occam's Razor, we should presume the simpler idea -- humans can learn -- over the more complex idea -- ideas are ethereal and beyond physical. TLDR: If deterministic electrons can do it, you don't need a magical "idea"-land.P1: If entities that have no ideas can identify apples, then the existence of a metaphysical "idea of an apple" is unnecessary.P2: (Occam's Razor) If the existence of a metaphysical "idea of an apple" is unnecessary, then we should presume that it is false unless evidence is presented for it.P3: No evidence is presented for the existence of a metaphysical "idea of an apple".C1: The existence of a metaphysical "idea of an apple" should be presumed false.[2] Second: Proof via counterexample. It is known that people's perceptions of colors are different. What you describe as "purple" might be "reddish blue" to me. Similarly, people's conceptions of "an apple" will be different. One person will think apples are red; another will think apples are green. One will think apples are big; another will think apples are small. This suggests that people don't all obtain their definition of appleness from a singular unified idea of appleness.P1: If people's don't derive their idea of appleness from a singular "idea of an apple", then the existence of a metaphysical "idea of an apple" is unnecessary.P2: (Occam's Razor) If the existence of a metaphysical "idea of an apple" is unnecessary, then we should presume that it is false unless evidence is presented for it.P3: No evidence is presented for the existence of a metaphysical "idea of an apple".C1: The existence of a metaphysical "idea of an apple" should be presumed false.Moreover, Pro did nothing to support P2 in the previous round. As such, Pro provides no reason to believe Pro that axioms are independent of the mind. The null hypothesis would be that axioms are merely human ideas -- since "people have ideas" is vastly simpler than "people have ideas AND ideas exist outside of people". As such, Pro has failed their burden of proof (BOP) to support the premise that axioms are independent of the mind.And if axioms aren't independent of the mind, then necessarily they cannot be "discovered" -- they are just human thoughts that humans happen to think are useful. PREMISE 2: MATH IS AXIOMATICPro's argument is that math is purely axiomatic. However, Pro fails to realize two fatal things:FIRST: Math as a discipline is not produced in an axiomatic vacuum. It is a science, and people get things wrong. Euler's original theorems were based upon too many assumptins to be rigorous. Famed mathematician William Shanks famously calculated π wrong [1]. Many times, math that was widely accepted was overturned. Kempe's proof of the four color theorem not proven wrong for a decade. Lagrange's work was almost entirely based upon the idea that all real continuous functions on real intervals can be expanded as a power series except at a finite number of isolated points -- but Dirichlet's proper definition of "function" killed it. Similarly, Ampere "proved" any continuous function is differentiable except as finitely many points. Weirestrass found a counter example with a continuous but nowhere differentiable function.My personal favorite: Daniel Biss, mathematics superstar, published a paper that was fatally flawed. After he published a retraction, he went from math to politics within the year (2011) [2].If all the gibberish seems incomprehensible, the takeaway is this: math, like science, takes steps forward -- and sometimes steps back. Both are human endeveaurs, with human successes and failures. To treat mathematics as some sort of hyperlogical axiom-game is straight up wrong.SECOND: Pro illogically conflates "not changing" with "being axiomatic". Even if the value of π is universal -- which Pro has not proven -- this does not prove that it exists outside of the minds of smart mathematicians. If we had used the wrong axioms to define π, then that value of π would not be constant. Consider: If we removed the axiom for division/multiplication -- if it was not possible to divide/multiply in our mathematical system -- then the value of π could not exist at all. Instead, properly defining π requires a system that can adequately describe geometric shapes and do prescribed funcitons with them.Imagine that aliens have discovered a third basic function beyond add/subtract and multiply/divide. They might have whole concepts that never would have occurred to us -- because we had never adopted that mathematical function as an axiom.In other words: math is the successful selection of axioms, not the axioms themselves. When math chooses wrong, it gets things wrong. When math chooses right, it gets things right.The axioms themselves are just ideas that our minds mush together to produce logical inferrences -- no more, no less.SUMMARYThere's no reason to think that metaphysical ideas exist in the first place. Pro hasn't presented evidence for it, and it's very reasonable to presume it's untrue.And math itself is demonstrably a human endeavor about the choosing of axioms, not about "discovering" new axioms.Finally: the burden of proof is on Pro, and they have not fulfilled it.REFERENCES[1] http://www.americanscientist.org...[2] https://en.wikipedia.org...