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Comment by fooker | Hacker News Reader

Now draw the same triangle on a curved surface. No matter how the idea of area comes from logic, the fundamental assumption for the commonly known formula to calculate area is that we are doing euclidean geometry.

We do not yet know if there is some similar assumption hiding behind our ignorance of physics.

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Euclidian geometry would be the same in any universe though, as it does not depend on objective physical reality.

It could be argued that the Euclidean plane is its own universe.

The Euclidean plane is a conceptual model also not based on physics.

It is based on observing (and needing to measure) plots of land.

The abstraction came later.

Didn't Hume already prove we cannot invent anything not related to our prior experience (which ultimate goes to the universe's natural laws we experience).

> Didn't Hume already prove we cannot invent anything not related to our prior experience (which ultimate goes to the universe's natural laws we experience).

Do you mind elaborating on this, or pointing to some resources where I can read more?

The map is not the territory. It was always an abstraction.

how is general relativity related to any prior experience of the 1920s?

The maths that underpin GR were developed before Einstein. Without the maths you have no ontology, no way to evaluate the truth of a scientific claim.

https://en.wikipedia.org/wiki/Tullio_Levi-Civita

the math doesn't prove anything. we didn't get experimental results until decades later

Trains and clocks, mostly!

that is special relatively , not general

Yes yes. But you can ask that question, and we can answer it, regardless of whether our space is curved or flat. PI is the same in a flat space in any universe where you ask the question, because we're doing a mental exercise, not measuring it locally.

> PI is the same in a flat space in any universe

pi is the same in _any_ space, flat or not.

Indeed, the space we are in is probably not flat, but pi is unchanged.

True but Pi (mathematical constant Pi) isn't necessarily the ratio between the circumference and the diameter in those spaces.

Even in a non-Euclidean geometry, wouldn't π still emerge (or be approached in the limit) in "flat" areas?

With regular spherical and hyperbolic geometry, yes, it is flat-ish if you look at a small enough area. On the other hand, if you have a different distance metric, the ratio of a circle's diameter to circumference can be a different constant. For example, in taxicab geometry, it is 4

In Uber Geometry, it can vary from 5 to 14 during surge pricing.

In London geometry, the distance depends where you are in the one-way system.

I know this is sort of a low quality joke post, but I don’t think I’d ever thought about a directed taxi-cab distance before. I’m not yet convinced it’s a useful model outside GIS, but I may spend considerable time today thinking about it.

Thank you.

Happy to be of service, in filth it will be found!

TIL that Taxicab circles are squares with sides oriented at a 45° angle to the coordinate axes.

It depends if you define a circle as "constant distance from centre point" or "minimum (circumference/area)".

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