going to do some legitimate thinking aloud here. Im not really sure if this is going to come towards a coherent point, or if its just going to be some babbling.

I have a Youtube channel where I talk about philosophy and I end up in discussions with a variety of people about philosophical debates and issues I discuss on the video. Last fall I put up a video about Immanuel Kant’s Transcendental Aesthetic in his Critique of Pure Reason which talks about how arithmetic is the science of time and how geometry is the science of space as well as the facts that intuitions of time and space are a priori and synthetic.

A user who has since deleted his account Omnicron777 asked something that I find interesting. Kant states that intuitions of space are synthetic a priori. More importantly and simply for the current purposes, intuitions of space are a priori. Furthermore, Kant states that the pure intuition of space is a prerequisite for a posteriori experience. The pure intuition of space allows one even before experience to understand that this is close to that or that Im here and thats there.

Euclid had 5 postulates of geometry. Four of those postulates are what are understood as analytic or what Russell would call logically and psychologically primitive, i.e. hard data. The fifth one is something epistemologists and geometers have picked on for a long time and this postulate has led to different thought in analytic philosophy as well as new streams in geometry.

Riemann and Lobachevsky are geometers who gave birth to non-Euclidean geometry. Given the idea of a geometry different from Euclid’s, what we know and understand about space might be different.

What Omnicron777 suggested was that the fact that space is an a priori intuition might not be true given non-Euclidean geometry. I can show how this might be so…

The Fifth Postulate or the Parallel Postulate is illustrated like this:

The two lines that go from being solid into dashes are important. The sold/dashed line that makes the α angle we will call line A, and the solid/dashed line that makes the β angle we will call line B. The other line which is laid across A and B we can call C. Lines A and B come together as dashed lines at a point. Line C is viewed with respect to this proposed point made by A and B. The fifth postulate states that there will be only one line that goes through the point made by A and B which will be parallel to line C.

The other postulates of Euclid are thought of as easily understood without psychological or logical processes but this postulate is not so. One cannot understand it without logical and psychological processes.

Riemannian and Lobachevskian geometries are non-Euclidean geometries which use the notion in philosophy of language of implicit definitions to make the conclusion that there are either no parallel lines going through that point to C or infinite parallel lines going through the point to line C.

Riemannian geometry is used given a sphere:

Riemannian geometry uses implicit definitions. What is meant by implicit definitions is a predicate being defined based upon other predicates and other parts of a sentence. For example, implicitly defining a point would be “that sort of thing that lines go through.” Defining a line: “that sort of thing that is made up by points”. Whats parallel: “That sort of thing that can happen when lines don’t create points with each other.,… and the chain goes on. In this kind of language no concrete definitions are given allowing the fifth postulate to be twisted and manipulated to either give a no parallel or infinite parallel conclusion.

Riemannian geometry uses a sphere where the same kind of geometrical situation is done but, the point made by A and B is made at one of the poles of the sphere. The idea of a line in this case is a great circle or the line that is the largest possible line that goes all the way around the sphere and covers the most distance. On a sphere there are only two. Line C here is a great circle and A and B are line segments that end when they come to points with C. Given the sphere no line can be created through the point at the pole that would be parallel with C. Riemannian geometry changes the fifth Euclidean postulate to being the No Parallel Postulate.

Lobachevskain geometry is done with a 2d disk:

Above is not a disk but Lobachevskian geometry concludes with the Many Parallel or Infinite Parallel Postulate. Shown in the figure there can be made an infinite amount of lines that go through point C that given the implicit definitions of points, lines etc… are parallel to line AB. All that Lobachevskian geometry needs to give a parallel line through C to AB is that there is a segmented part of the line that is parallel and the whole line need not be given the manipulative implicit definitions.

Given these two non-Euclidean geometries it has been shown that the fifth postulate can be used to give conclusions other than the one parallel postulate.

The reason these geometries would possibly contradict Kant’s statement that space is a pure a priori intuition is that the fifth postulate being shown not necessarily true at all times given non-Euclidean geometries shows that space may not be much different from facts of a posteriori experience.

Using terms of Russell, if space is a pure a priori intuition, it is hard data or logically primitive and psychologically primitive things. It cannot upon the “solvent influence of critical reflection” (Hard and Soft Data by Bertrand Russell) be doubted. Space and the nature of it cannot become doubtful upon reflection and contemplation. Also, you do not need to experience things to understand the nature of space.

These relatively recent non-Euclidean geometry might lead to the suggestion that intuitions of space are a posteriori and not a priori.

Thinking of an a priori intuition is difficult to conceive of anyway. I guess the best way to think about such a thing is to think about whether or not you would have to had some prior experience to understand and comprehend the perception. Time seems to me to be, according to arithmetic, an a priori intuition. I don’t think you need to have any prior experience to understand that a couple seconds of time passed by even if you do not know of the predicate “second.” And I don’t think you need to have prior experience to understand that if I have one shoe and another shoe, I have two shoes even if you do not know of the term “2”.

Space on the other hand seems to me to be a bit more complicated and that you just might have to have experience before you really really understand why “Im here and that over there is over there and not over here.”

Language is an integral part of any discipline whether it be science, mathematics, epistemology or whatever. The only reason non-Euclidean geometries were able to be made is due to changes in semantics and definition. Thus, language pointing out that something might be more complicated than previously thought would give probable cause to investigate that further and not write the complex thing off as a priori intuition.

In summary, I don’t think space is an a priori intuition. Applied geometry shows that as well as just thinking about basic understandings of space and what it might take to have such understandings.

…yes this was a bit of a lot of incoherent thought….