Can space be both Euclidean and non-Euclidean, "at the same time"?

Question

It is often said (or implied) that Kant "dropped the ball" when he said that our knowledge of physical space as Euclidean is given to us a priori; but others come to his defense and say that at least by attributing a sort of abstract contingency to major principles of Euclidean geometry, Kant was theoretically open to the logical possibility of non-Euclidean systems (and at any rate, he had occasion to obscurely consider other "open-minded" options, e.g. multidimensional time).

Still, is it impossible for Kant to be substantially right, but also for modern physics to be substantially right, together? Meaning: is it possible for space/time to sustain multiple geometries coevally?

Intentionality and the imagination. Say there was a quantum field (QF) whose particles did not interact via gravitons (~G). So whether gravity is itself a function of causal set theory causing itself to become causal-dynamical triangulation going on to evolve into a set of worldcrystals, or is strictly continuous, it makes no difference to the special QF as such. So the ~GQF particles would not, I think, have any mass, and would use no energy during their interactions (or the exchanges would be uniformly virtual, or mediated only by raw entanglement; but on that last note, continue reading). Or so they wouldn't move at all, not even with gravitonic spacetime curvature. However, we suppose that when some other particle types collide with them, they do passively couple to the particles crashing into them, and become entangled with them to some degree.

Now anyway, so the ~GQF particles seem rather "epiphenomenal," so we will take to calling them epiparticles (not a term I coined, though I didn't see that it was used exactly here as it was elsewhere).

So consider that it seems possible, in intentional imagination/the imagining of intentions, to have perfectly straight lines. I.e., merely by internally "stipulating" that one is visualizing such a line, one does so visualize. Yet if intentions are physicalistic in nature, such as to occupy regions of spacetime, shouldn't their lines of presence and action be curved?

This is not even a hypothesis, but suppose that epiparticles uniquely couple to leptons, as leptons pass through/over them, and so the epiparticle field is a consciousness field. But rather than consciousness having a strong role in physical causation as such, it seems as if it would be quite epiphenomenal, just as per the name of its particles and the actual origins of the word used for that name. Consciousness would mostly just passively experience its inputs; perhaps quantum flux and the theoretical aftermath of the Dirac sea would yield a space for microscopic apriority, but we will not dwell on what we as yet do not "know." The point is just that epiparticles are imaginary "possible examples" of a form of QF-theoretic matter that can form exactly straight lines in spacetime, by virtue of its field not coupling to the gravity field. And this is highly imaginary, as yet.

But still, then, can there be forms of matter, or of spacetime proper, or whatever, that either oscillate between Euclidean and non-Euclidean geometries (either some specific example(s) of those, or all of them, no less), or even perpetually occupy both domains? If I had studied physics instead of set theory, maybe I would know whether this would imply that the problem of reconciling QFT and general relativity will involve a mathematization in which background geometries are themselves mediated by the operations in play. I say this in the sense that one gets the impression, a little bit, sometimes, maybe, or at least I get the impression, that if photons, weak-force carriers, and gluons were "left to their own devices," they would proceed according to exactly straight lines (e.g., gluons would forge non-curved triangles from quarks). So instead, we might have to show, eventually, how non-Euclidean gravitational geometry can be integrated with Euclidean QFT possibilities.

Now the only information I've found so far by Googling for it, that seems possibly relevant to the question, is a write-up talking about embedding non-Euclidean geometries into Euclidean ones, but this seems to make it out that the Euclidean structures subsume the other ones as the "process" unfolds, so if I were to imagine the oscillation picture mentioned above, I might have to imagine that whenever spacetime becomes Euclidean, it eventually becomes non-Euclidean again, and back and forth, except the write-up at issue looks to be saying that the "process" is a matter of higher and higher dimensions, so we'd end up with a picture of spacetime's dimensionality increasing upward, which might not be the kind of conclusion I ought to try to come to based on the available evidence more broadly.

EDIT: To try to avoid a lack of clarity in the above, I want to re-emphasize the initial subquestion about a QF that doesn't couple to/via the gravity field. If gravity causes spatiotemporal/dynamical curvature, then wouldn't "gravitational dark matter" be able to avoid being curved? And wouldn't a QF with this nature not have to avoid being Euclidean? Or rather, if QFT is the best ambient model of "the world" as such, then does anything in QFT logically privilege the notion that all quantum fields must couple to/via the gravity field? Or then to assume that the universe only obeys One True Geometry, no less.

Answer 1

As a great statistician once said: “All models are wrong, but some are useful.”

Euclidean Geometry is a way of approximating reality. It is useful under some circumstances and unhelpful in others. Any model we construct for reality is an approximation and will have limits to its applicability.

So depending on what you want to know and how precisely you want to know it, yes, you could probably reach useful approximations using both Euclidean and non-Euclidean methods for the same problem.

Answer 2

The universe is this way, already. Far away from any gravitational influences, space is locally flat. Close to a strong gravity field, it is significantly curved.

Note also that a space which is strictly euclidean would have no gravity, since that arises from spacetime curvature.

Answer 3

Photons do always travel in straight lines - but across curved space.

Closed curvature spaces can be embedded in Euclidean spaces. Open curvature spaces, 'saddle shaped' ones, cannot.

Remember as well this is not only about space, but time is get warped by strong gravitational fields too.

Asking about oscillation between is like asking, 'In a world of hills and valleys can there be flat places?' Like, of course there can. In regions of space with minimal gravity fields, & low relative accelerations, it's approximately Euclidean - Special Relativity is based on this.

The nature of gravity at small scales, is much less well understood than people realise, because it is so weak it's very hard to measure. At particle scales and inside atoms, little is known.

In summary, no, time and space are relative and emergent. Absolute Euclidean space even as a convenient fiction, is dead. In fact leading theorists say spacetime is doomed.

Answer 4

I’d turn to Sellars. I think upon learning about Gallilean Relativity my manifest image changed. (Before, my thought experiments violated physics!) What Conifold said about local geometry only requiring Euclidean processing might be contrasted to our language processing centers which seem less restricted and more adaptable. Instead, our manifest image including perceptual space might actually be adaptable if we train ourselves on those parts of the universe (larger scales, contemplating the twin experiment, contemplating hyperbolic or spherical global geometries).

I know the common refrain we can’t actually see in 5 dimension, but maybe that is simply because those parts of the universe, if they exist, are hard to make present by being remote and exotic. And excursions into them might be adaptive.

The brain might be capable of “evolution” and it is the environment of universe that adapts it. We know the brain physically “brings in” our outside world as neural connections.

Mathematically these embedding would be quite easy so we can say the are mathematically possible. Physically we have already have flat (your room) embedded in spherical (Earth) embedded in flat again (geometry of space), a in loose sense I’m abusing the terms slightly.

So from this vantage there are no epiparticles. Those can be eliminated. But I don’t think Kant can be right either then.