The fact that we are intelligent points to a few perplexing tautologies. We arose as part of the natural world, so life leads to intelligence. You could say that intelligence is inherent in living systems, that intelligence naturally blooms from within life. This tautology becomes the anthropic principle at the cosmological level: our universe is the livable one, the one which naturally bubbles forth intelligence. So, physics leads to intelligence!
There is a deeper form of this tautology, applied to mathematics: the nature of mathematics allows and creates intelligence. Suppose we had a different set of axioms, the rules at the foundation of mathematics — would we see the same ‘Edge of Chaos’ interactions which seem vital to intelligence? While some combinations of axioms might produce complex dynamics like our brains (or tarnished silver), many arrangements of axioms produce system dynamics which are self-contradictory or structurally uninteresting. We actually have a few closely related axiomatic foundations, yet they share the essential features that make emergent complex behavior possible. Our mathematics is the one which creates complex systems, the one birthing intelligence into both biological and silicon shells.
We already possess multiple axiomatic systems which overlap well enough that they all produce the general ‘mathematical behaviors’ making intelligence possible. Perhaps there are other self-consistent sets of axioms, which still manage to generate systems at an ‘Edge of Chaos’, systems with intelligence.
We could call the list of every ‘intelligence-generating’ axiomatic set The Intelligent Maths, while axiomatic sets which are inconsistent are The Unintelligible Maths, and sets that cannot create chaotic systems with self-organizing complexity are The Boring Maths.
I would guess that most conglomerations of axioms are Unintelligible — their axioms stumble over each other, at some point. Far along the web of theorems derived from these axioms, we would uncover contradiction and crumble into uncertainty. Meanwhile, most of the remaining axiomatic sets are Boring Maths — the theorems which can be proven under these axioms are linear elaborations of their predecessors, adding only to the number of parentheses without increasing complexity. And, I suspect that only a minute proportion of axioms can combine to make something that functions like our brains. Intelligent Maths are rare. I wouldn’t be surprised if our standard axiomatic sets are in fact the smallest maths which can produce intelligent behavior.
Which brings me to the perplexity of this tautology. There is a perennial puzzle, growing only more puzzling as science advances: why is it that so much of the universe is so well described by mathematics? There is no reason for the universe to follow maths, they are entirely different things. Mathematics do not require a universe; mathematics are true even if there was no universe! So, why would the universe seem to need mathematics?
Is it possible that all arrangements of axioms ‘exist in superposition’? Every combination of starting rules lends its weight (none, if those rules led to a contradiction…) so that the universe is composed entirely of the overlap of Intelligent and Boring Maths. In such a universe, each Boring Math would make arguments which are completely repetitive within itself, yet each Boring Math would repeat something entirely different from the other Boring Maths. As a result, their summed weights have no overlap, and become insubstantial due to this dilution. Meanwhile, the Intelligent Maths would overlap in significant regions. Aspects of behavior observed with one axiomatic system would arise in many other axiomatic systems; their accumulated weight is non-zero for a few particular behaviors.
From this perspective, the universe follows mathematics because it is made of all mathematics. The universe follows Intelligent Maths because the Unintelligible ones weighed nothing and the Boring ones averaged out. Only Intelligent Maths displayed overlapping behavior, so only they are observed.
If this woolly hypothesis could ever be proven, it would grant us the deepest tautology: In the nature of all mathematics and the universe, there is a potential and a pressure for intelligence. This would be true not only in this universe, as compared to imagined other universes, but in all possible universes, together. All maths, together! Intelligence would be innate, fated, no matter how we began. (I think, therefore the universe must?)
Yet, at the same time: meh… whatever.