The question in another thread about whether or not mathematics is a science was an interesting one to contemplate. Beyond the basic answer I gave in that earlier thread, thinking about the question got me to thinking about geometry, but I thought sharing those thoughts in the other thread would be too much off topic, so I decided it might be better to start another thread rather than have even the small risk of spinning too much of the discussion off away from the primary theme of that thread.

There are many kinds of geometries. Each kind has its own unique set of axioms (though there may be a large number of subsets of axioms that are the same or very similar amongst various geometries). Many, if not all, geometries are logically inconsistent with each other because they have some different fundamental axioms. Mathematicians have no problem with the inconsistencies amongst geometries because they see mathematics as a field of understanding that is about the logical application of axioms related to quantities and is, in its most dogmatic form, "agnostic" about any metaphysical truth or falsity being affixed to one set of axioms just because it is logically inconsistent with another set of axioms. Pure mathematics treats all systems of axioms as equal in the application of basic rules of logic; e.g., the rules of any axiomatic system must be consistent with the law of non-contradiction or it will be deemed mathematically illogical.

Significantly for purposes of this discussion, mathematics has no necessary empirical factor against which to test a system of axioms, because pure mathematics is "agnostic" with respect to any relationship between mathematical axioms and physical reality (events).

Science, on the other hand, is anything but "agnostic" about its application of logic to the hunt for patterns in observations of physical events. Science is empirical. To make a scientific hypothesis is to make a claim about the presence of a pattern in physical events. The hypothesis must not only be logically valid (which is the minimal requirement for a mathematical system), it must also be physically tested to see if it properly predicts relevant physical events.

So...

there are many geometries, but if Einstein is right, there is only one geometry for space-time. Mathematics and science are fairly distinct spheres of thought, but there is a deep conceptual connection between the two due to the natures of mathematics being about quantities and science being about measuring quantities when observing to try to verify a hypothesized pattern in physical events. Certain axiomatic systems about quantities in mathematics work to predict patterns in physical events (and other mathematical axiomatic systems do not).Oftentimes, the hunt for patterns in physical events inherently involves a hunt for mathematical equations derived from axiomatic systems that "match" observed patterns.

I think it's interesting to note, though, that in mathematics you can have an "ironclad" proof, but not in science. In science, any conclusion about what is most likely to be the correct identification of a pattern in physical events is a conclusion that is tentative. Nothing in science is proven in the "ironclad" sense. I think that is a reflection of the fundamentally different natures of mathematics and science. Mathematical axiomatic systems are closed loops of logic - though the loopz can expand and contort in the most intriguing ways. When something is mathematically proven, a logical loop is closed. Until there stops being the experience of new events, though, the logic of science can never be closed - it must always be open to the possibility of more accurately finding patterns in physical events. Science can't prove anything (in the "ironclad" sense); the loop is always open.