At any rate, this video might serve as a quick introduction to Penrose's three world idea for those interested.
Personally, I do think that the immaterial world of ideas must be primary—at least certain aspects of mathematics seem so necessary that they’d be discovered by intelligent life, no matter the galaxy… or simulation…
For some speculative philosophical fiction that explores related ideas I highly recommend Neal Stephenson's Anathem.
https://www.greekbible.com/philippians/3/8
Edit: This also seems like a decent opportunity to bring up the scatological Luther.
https://www.wilsonquarterly.com/quarterly/summer-2012-americ...
In alchemy and western esoterica, excrement is associated with the tenth sephirah, the 10s of the Tarot minor arcana, and symbolizes the end result of a process and any remaining waste byproducts, for obvious reasons. In The Holy Mountain's (1973) depiction of the alchemical magnum opus, The Fool's excrement is transmuted into gold, symbolizing the awakening of unconscious, reactive matter into fully enlightened and integrated, free willed, egoic man.
I don't know why but your comment made me remember a novel[1] I read thirty-some years ago about a temple found deep in the sand of the Sahara desert. Sometime later, an archeologist gave himself permission to defecate in a corner of the temple, only for his wastes to be absorbed by the temple in a few hours, which told him the temple was actually a living biological structure.
1: https://www.daliaf.com/oeuvres/etrange-monument-du-desert-ly...
As one who has had occasion to tidy up after wasps who were little accustomed, though palpably interested, quite so closely to share human habitation, you make me wish I read French. Do you happen by chance to know if the work has had a worthy English translation?
In fact, by pretty much any measure, most life does not defecate (because they have no digestive tract).
My take away was that he sees a mystery in the connections between these things (physical world, consciousness, ideas) that hints at some missing ideas in our conceptions of these things. But he clearly wants to avoid that mystery allowing what he calls out as "vague" answers to the question (mostly religious dogmatic certainties).
Unfortunately it needs a definition of "idea" which isn't recursive, so...
As for math - it's a conceit to believe that the mechanisms we call math aren't just a patchwork of metaphors that build up from experience.
There's some self-insight in the sense that after a while you start making meta metaphors like category theory.
But it's a very bold claim to suggest that any of this has to be universal, especially when the structures math uses can't be proved from the ground up.
Or that completely different classes of metaphors we can't imagine - because we evolved in a certain way with certain limitations - might not play an equivalent role.
Does the universe know what pi is? Or an integer? Or a manifold?
Does it need to?
> it's a conceit to believe that the mechanisms we call math aren't just a patchwork of metaphors that build up from experience.
I'm not sure it is a conceit as much as a commitment to a metaphysic. If one believes that experience is a definite relationship with an external reality (a phenomenological view) then the fact that experience is structured is suggestive that external reality is structured. If one believes that experience is primarily interior then one could assume that the internal mechanism of cognition is structured and external reality is something entirely different.
However, I'm not sure how anyone could hold the latter view without a deep solipsism. One would presumably have to account for the perception of billions/trillions of other living creatures behaving as if the external world was structured. I mean, we seemingly all did evolve from the same single cell structure, so it is possible this perceptual quirk is based on some shared ancestry, so I suppose that is another possible view than solipsism.
What I mean to say is, I can imagine my perception of a fundamentally unstructured reality is a perception that falsely presents itself as structured to my own experience as a result of my limitations. However, I would have to extend that exact same flawed perception to all other life forms that seem to act the same as I do. So either every single living creature has the exact same flawed perception or the structure is inherent in the external world.
> Does the universe know what pi is?
No one is suggesting an epistemological view, the question is ontological. As Penrose mentions in the video, the set of possible mathematical structures is vastly larger than the actual structures we see in the universe. So even if one has a purely idealist view, one has to account for why our perception only experiences a nearly infinitesimally small fraction of that set of possibilities.
Of course, a weak anthropic principle is one answer. One could posit that all possibilities are manifest in a vast multiverse and this little corner of that multiverse just happens to be finely tuned enough to allow for limited creatures like ourselves to perceive anything at all. But that just shifts the question to the limitations necessary for perception/experience/consciousness, which is a valid enough topic to address on its own. The questions then becomes "why do these particular structures result in conscious experience", which is exactly the kind of question that a guy like Penrose is ultimately searching for (as he heavily implies in the linked video).