Mathematics: In Nature or in the Human mind?

This question has been discussed and debated by mathematicians as well as by lay-thinkers for a long time. My own view is that in this context it is useful to distinguish between mathematics as a system, and particular instances of mathematics.

As a system mathematics is a framework which includes the following elements:

( Pure logical deduction. Pure logical deduction corresponds to strict causality (If this, then that) in an abstract realm. This abstract realm has its mapping in the human brain (this is a mystery). Thus, all normally functioning human brains are conditioned to the basic rules of logic.

( Integers. Any world of multiplicity incorporates integers which are subject to certain rules of combination, like three objects with four objects making seven objects. The integers and rules of combination are given names by the human mind.

(c Symmetries. There are any countless symmetries in the physical world: brightness and darkness, up and down, an object and its reflection in a mirror, are examples. These symmetries are recognized and explored by the human mind.

(d Patterns. There are many patterns in natural phenomena. For example, orbits of planets, and the behavior of aspects of physical world which we call laws of nature.

(e Relationships. Natural phenomena are interconnected in complex ways, often in precise and tractable modes. These interconnections exist as recognizable mathematical (functional) connections.

Thus, (assuming the existence of an objective reality which the human mind grasps and interprets) the basic ingredients of mathematical thought permeate that world of reality. These are reflected in the human mind which casts them in its linguistic modes.

In this sense, the mathematics we use is not an invention but a discovery that is formulated in a human-cerebral language. Whether there is a human mind or not, planets and electrons will move in elliptical orbits, the inverse-square law will govern gravitation and electromagnetic, and the symmetries of group theory will be implicit in elementary particle physics, etc.

One may compare the situaion to colors in the universe. etc. Does the human brain invent or discover colors? What the brain does is to interpret electromagnetic waves wihin a certain range of frequencies as colors. It discovered colors in this sense.

Having said this, it should be added that there are branches of mathematics (such as matrices, differential geometry, number theory, complex analysis, and quaternion) which are largely human inventions. On the other hand most mathematicians would agree that the Mandelbrot Set (whose boundary is a fractal) was a discovery.