@Harp I’m not so sure that the innate pattern-grasping ability of our minds is entirely “phenomenological” or all in our heads. It may well be that the distinction between the “noumenal” (the world independent of our senses) and the “phenomenological” (the world as we perceive it) breaks down at the point of consciousness.

Consciousness, what we take to be our individual consciousness, may actually be a physical process, like photosynthesis (which scientists now believe is a quantum-level phenomenon). It may well be that consciousness creates a kind of mathematical space (a subset of Hilbert space), somewhat the same way that a calculator creates a computational space for all the possible arithmetic operations allowable in the space of it’s display window. (And what we see from that window is more or less constrained by our position in what biologists call “fitness space,” determined by the evolutionary trajectory of our physical and cultural evolution.)

Everywhere we look in the universe, we see fractal geometry . In fact, that may be the underlying mathematics of the cosmos, complete in it’s entirety. If so, holographic perception and memory may be coded and decoded in our experience by a kind of “fractal compression” algorithm of the sort described by the mathematician Michael Barnsley. Alternatively, since fractals are self-replicating, and have essentially the same pattern whether they are very, very, very small or the unimaginably huge, we may all literally carry a complete microcosm of the fractal universe in our heads.

This would apply not only to us, but the same mathematics of consciousness would apply to all of life. Or, to put it another way, all of life and consciousness may be deeply “mathematical.” We should be able to use such a mathematics to construct a 3D version of television, or a Star Trek holodeck someday. It might even become a medium of communication with distant alien species.

Now to the question at hand: I think that maths are *discovered* in the sense that when we explore mathematical space, we stumble onto relatively permanent features, like pi, phi, zero, infinity, numbers, fractals, arithmetic, ratios, etc., which exist as patterns not only form the deep structures of consciousness but “noumenal” reality as well—in everything from the distribution of the orbital rings of planets, to the distribution of capillaries and synapses. But, I think mathematics are *invented* to the extent that we develop a notation and a language for talking about these patterns, a set of axioms which allow for mathematically acceptable proofs, and all the other social and cultural aspects of mathematics.