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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 May 27 '13 at 1:07
  • @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 May 27 '13 at 1:43
  • 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 May 27 '13 at 5:36
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    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 ( goo.gl/CzBOl ) – Addem May 27 '13 at 5:47
  • @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 game our 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 May 27 '13 at 13:23
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There is always some special arrangement of the neurons that makes the difference between a brain and a mass of neurons. We can even say we associate words to shapes in an innate (culturally independent) way, as the Bouba/kiki effect demonstrates. But from that substrate there are many things that we learn.

To measure how much we learn we can take a look at children, as in the answer from cartomancer, or we can consider people that have different culture, capabilities, etc.

For instance we can consider blind people. Blind people are in general are more used to a three-dimensional world where two dimensional objects do not make so much sense. [They are still normal for them (more than 4D objects) as a wire can take any of those shapes, for instance.] The point is that our experience and perception shapes our understanding and comparing with haptic perception helps to understand where do our prototypes and concepts for shapes come from.

There are many notable blind mathematicians specialized in geometry allegedly due to this difference in perception of the world, shapes and geometry, that at the same time influences a difference in the understanding of geometry. I would personally say that a posteriori influence is what makes a difference in the a priori substrate.

To finish up I'd like to call the attention over a specific paragraph in the original question:

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.

The point about innate as being more effective does not make much sense. The same can be said about colors, for instance, however colors mean nothing to blind people and are perceived in different ways depending on color-blindness. That point suggests an intelligent design that creates people in ways that are efficient, however that is not the way evolution works and to the best of our knowledge that [efficiency] is not a good reason to think humans are in one way or another. We should be very careful about assumptions that are introduced inadvertently in such ways.

PD: Actually evolution pushes human beings to have the least amount of innate knowledge, the brain is not mature at the moment of birth due to limitations to enable birth in a bipedal species like humans.

Also, it may be interesting to consider spiders, and probably other animals. Spiders can make webs that can be perceived as complex to the human eye. We can debate about whether that is innate knowledge or knowledge at all. Spiders sure don't have an explicit knowledge that they can transfer to any other being, or reason about. Probably it is just a feeling, about what feels right at a given time making a web, and that feeling is altered by drugs. In this case I'd say there is a substrate and an emerging pattern through some kind of "spider feelings", but the spider never knew anything in an explicit way.

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Humans appear to have some innate sense of geometry, visible at as early as 5.5 months.

CONCLUSIONS

Although previous research has shown that infants are sensitive to geometric cues, this sensitivity is often demonstrated in highly simple contexts. The findings from this study show that by around 5.5 months of age, humans can use the relevant geometric cues from an enclosed layout under variable viewing conditions to distinguish among the corners of the layout. It makes a great deal of adaptive sense that evolution would select for sensitivity to geometry. Geometric information about shape is arguably one of nature’s most enduring properties (Gallistel, 1990), and the sensitivity to this information might serve as the foundation for abilities that require the use of geometry, such as identifying objects and determining location (Dehaene, Izard, Pica, & Spelke, 2006).

Children also seem to grasp simple mathematical concepts at about 5 months.

Even in the cradle, babies as young as 5 months have a rudimentary ability to add and subtract, according to a study being published today.

The study seems to show that infants know when simple calculations like one plus one or two minus one are done correctly or incorrectly. The infants in the study indicated awareness that a wrong answer was given by staring longer at the unexpected results.

This does not rule out the possibility of geometric knowledge being a priori, but it provides reasons to think that at the most basic level, geometric knowledge is a posteriori.

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Kant's synthetic a priori (in the case of geometry made out of our geometric intuitions, let's call it M) means that we could, given enough time, deduce everything that can be deduced from M (including things like c^2 = a^2 + b^2)

Whether (and how much) M corresponds to reality is a different issue. Kant doesn't claim that "if a geometric principle P holds in nature, it must hold in M".

If M doesn't hold in nature (which is probably the case), it still holds in, for example, computer graphics, and there is no inherent reason for preffering theories that can be used in CG versus those that can be used in general physics.

It's synthetic because axioms' truth doesn't follow trivially by elementary logic, or from the way axioms are structured.

It's a priori because you don't need experience to understand why e.g. c^2 = a^2 + b^2.

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What does it mean for anything to be innate?

I think the best way to sum it up is to say it is the opposite of learned.

What is geometry?

According to wikipedia Geometry is described as being "...concerned with questions of shape, size, relative position of figures, and the properties of space".

Is geometry innate?

Well geometry is a branch of mathematics so in the shortest answer I think, it's no, geometry as a mathematical discipline is learned... unless you're Euclid.

The longer answer I think would be: If the underlying aspects of geometry were not innate, would "Geometry" even exits?

In this way I think it is innate, even for a dog, albeit not as refined as to make it mathematical in the case of a pet.

How can we define a right angle? The angle between two lines which are perpendicular.

Even without so much as the language you're reading now to express this, a dog knows the difference between moving forward and moving "that way" to the left/right.

A circle other than being a geometric figure with one side where all points are an equal distance from it's centers, is simply round. We may even go as far as to say, perfectly so.

If the ability to discern shape, size, relative position and space is not innate, then I don't know what is.

when Kant suggests we know geometry a priori does he mean this in a deeper sense - i.e. we are spatially aware?

I think you got it.

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Absolutely one hundred percent yes. We all have innate geometry (a priori, from start).

In fact, question goes even deeper... there are no other known geometries - since all of them infinitesimally similar to ours (Euclidean). Other geometries is a myth, simply a mathematical construction. They exist only inside Euclidean and thus are not NEW geometries.

For example - any construction which leads to description of Non-Euclidean geometry is intrinsically euclidean and even done in Euclidean space.

You are dissecting too much. Everything what we see is innate to us. There is nothing outside (at least until we are ready to face outside). There was philosopher who developed this topic well -- Jung or Hume, don't remember.

If we can have reaction on something (what people call experience) it means that inside of us there is already a "detector" which knows that such a process is possible. This basically means that everything is a priori.

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