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Most people, if asked whether they know any geometry, will answer no; but most, if not all, can recognise a straight line, a right angle, or a circle; of course they will not be able to define them as a mathematician does: a straight line is the shortest curve between two points etc. So, it appears that their answer reflects their understanding of geometry as it stands in immediate relationship to themselves, rather than an understanding of pure mathematics.

Now: Does this mean that humans have an innate sense of geometry, or is this acquired knowledge?

Does Kant suggest this? Is geometric knowledge a priori?

We know now that these concepts are contingent. That is, there are geometries that are non-euclidean. Of course locally, i.e. in our immediate environment, they are euclidean. In fact these geometries are called manifolds in mathematics, and it is the property of local euclideaness that defines them.

This means that although there are such geometries, because as human beings we have only our immediate environment to purvey, that is, our spatial knowledge is local, what is a straight line or circle in the standard sense remains effective. It does not have to be acquired, but can be innate.

But when Kant suggests we know geometry a priori does he mean this in a deeper sense - i.e. we are spatially aware? That we have an intuition of what space is, which stands between our immediate sensory input and our conscious knowledge of space?

According to both Kant and (later) Frege geometry is an example of the

synthetic a priori. – Dennis – 2013-05-27T01:07:14.850@Dennis: Geometry is term that is so imbued with mathematical connotations that its difficult to get away from it. It surely cannot mean for example that people know pythagorases theorem. The question I'm asking is what do they mean by geometry here. The synthetic a priori angle is interesting too, not least because of how is that at all possible. I didn't know Frege said that - was he essentially agreeing with Kant, or was it an independent discovery? – Mozibur Ullah – 2013-05-27T01:43:55.947

He was largely agreeing with Kant. Towards the end of his career he returned to his Kantian roots. The piece to read here is "Numbers and Arithmetic". He turned away from his earlier logicist proposal in the foundations of arithmetic and argued that arithmetic had geometrical foundations. You can read about the development of his views on this matter here.

– Dennis – 2013-05-27T05:36:23.6372

It is not clear to me that our geometric intuitions are necessarily Euclidean in nature, although our intuitions do seem to have a strong inclination in that direction somehow. On the other hand, there's a pretty rich Philosophy of Mind literature that investigates the possibility that even our visual perception is not Euclidean. This cue is taken from some optical illusions that seem impossible of we visually intuit a Euclidean space. See Suppes for one example ( http://goo.gl/CzBOl )

– Addem – 2013-05-27T05:47:01.230@Addem: I'm talking locally to us. For example we easily notice when two lines are parallel to us. But of course if they are extended then they (appear) to converge. This doesn't happen in Euclidean geometry. Optical illusions are interesting, but I'm not sure here they're appropriate as they

gameour perceptual system. We don't see them in nature. From your article Berkeleys ideas about vision are interesting, and I didn't know that Euclid wrote on optics. – Mozibur Ullah – 2013-05-27T13:23:43.540Even in Euclidean geometry, parallel lines may appear to converge depending on how you model an Euclidean observer--that's not exactly the way in which optical illusions are exploited in those arguments. But as for their appropriateness, the very fact that the senses

canbe gamed (in the particular ways indicated in those arguments) seems to at least suggest that our perceptions are not Euclidean. – Addem – 2013-05-28T04:01:19.973@MoziburUllah Somewhat related (particularly to some of the comments) is the

– None – 2013-08-06T13:27:35.540firstversiononlyof this answer of mine on Physics SE. (You can see the first version by clicking on the "1".)I wonder if technologically primitive cultures, lacking much in the way of straight lines and planes in their environment, would have as much "innate" understanding of these concepts that technologically immersed cultures do. Seems easy enough to determine through field research. – obelia – 2013-08-15T03:21:06.523