I’d say in mathematics there are fundamentally three or four fundamental forms or structures.
[- We could say three. - The fourth would questionably be relevant for it.]
- Or perhaps I should say in geometry. - But it hardly seems to matter. - To me.
The first is the [single] point. - That would correspond to materialism.
- I am not looking for a definition, [for a single point] and I doubt whether one could be given or is necessary.
The second is the [- infinite, - i.e.] straight line.
Seeing it as reaching the infinite and piercing through it, - (- which is the way it is usually viewed in modern mathematics, - fwiw) would mean it is actually the same as a circle of an infinite radius. - For it is a closed line the convexity of never changes. [- That is to say - if you see it as able to connect to itself through the infinite. - The view is not as if there is a different infinite at each direction you might trod in, - but as if whichever direction you move in there is the same single one [point at the] infinite you encounter, - and cross - following this view - returning from the opposite side. [- i.e. - in the same direction]]
- Though again, - according to the common view - it does have any point you could call its center.[- * -]
That would correspond to idealism.
- That is to say the point would be a geometrical symbol of materialism, - expressing its portion of the spirit in that way; - and the straight line would - in the same way - be an expression of idealism.
The third one would be the graphical description of the function ex - in the mathematical system or structure known as the real quaternions.
- The real quaternions [- often just referred to as “the quaternions”, - as far as I can remember] form a four dimensional space.
Therefore the form regarded would be an eight dimensional form.
- Therefore, - of course - it can not be imagined. - Not in any way that I could manage to imagine, - at least. - And quite certainly not by very very nearly the whole of humanity.
- What I am about to say - as you might easily predict or expect - is that it corresponds to realism.
Still it might be somewhat meaningless if you haven’t got the faintest idea of what it looks like.
- Still - I will not try to describe.
Its form in the [single dimensional] real field may be quite well known. - However, - on the other dimensions of the quaternion space it might be quite different.
- One who could contribute a link on the comments section where partial views of it could be visualized it might be welcome. - I suppose a software by which 3, - or even 4, - dimensional cuts of it could be displayed could be easily fashioned - but I haven’t come across any such thing that seemed worthy of posting in my view.
- Anyway - I do say this form or structure is the third fundamental form - unique among all others no less than the two others here earlier mentioned - I believe, - and with equivalent importance - both for mathematics itself and as a symbol of philosophical ideas and systems, - and that it would correspond to realism and express it as the two other preliminary ones would express the other two matching views.
I do not know what its value for mathematics would or might be, - and I don’t wish to guess because if I’m wrong it might lead some to assume I was wrong altogether.
The fourth would not be very significant for mathematics, - it is beyond the field of mathematics, - it seems: - It would be just the whole universe as it is. - Simple and dry. - Beyond the sphere of all ideas and the embodiment and inherence of them all. - Not conforming to expectations, but in a way in full accord with them all.
So far as for that. I suspect that different parts of the exponential function (ex - the third form I mentioned, - the eight dimensional one) might correspond to different parts of realistic philosophy, - but I could say nothing about that.
I can not rationally justify my idea. - Not in its entirety at least. Not for now anyway.
I rely mainly on an inner conviction. - But I believe some of the readers can - and will - grasp my notion and share my confidence.
- Even at the absence of an explicit illustration.
- Still - one other thing: -
- Christianity has its cross. - This is known to all. - Islam has its crescent moon. - Judaism - the Star of David. - Hindu might have its Aum symbol.
- But as far as I recall - from-I-can’t-remember-how-long-ago, - Buddhism did not have a sign or a symbol. - It seems the wheel used today to indicate Buddhism is rather a recent idea, - probably chosen artificially as to liken it to other religions; - perhaps seeing it as necessary - either in a trivial or in a somewhat essential way.
- However - what I want to point here - is - that the third form or shape I related to earlier - is the true [- i.e.: - natural -] sign of Buddhism.
Of course it cannot be drawn on a two dimensional paper. Or presented on a computer screen.
- I also suspect the eight dimensions of this form could be related to the eight lanes of the eight fold path.
- Again - I don’t know how. - Though it seems the four dimensions of the quaternions could easily fit “the four world views”, - or the four dimensions of the world we live in.
[- * -] - If you observe the way a circle is usually (chosen to be) “defined”, - as the set of points at an equal distance from a certain one single point, - and if you then choose to examine what would a circle of an infinite radius than be - then for each-point-which-does-not-lie-at-the-infinite (as its center) it would give the point at the infinite exclusively itself alone, - which might still make some sense; - (- though than you would get a circle every-point-in-the-interior-of- which is a center of - a circle with an infinite number of centers - each qualifying for your chosen “definition”) but if we are still to ask what would a circle of an infinite radius be if its center is at the infinite, - than it would follow that such a circle would constitute of all other points in the plain: - the whole of the plain minus the point at the infinite which is said to be its center.
- Given that this is hardly what we would expect or want a circle to be, - it may certainly be that the common definition is faulted and does not reflect the correct and natural view.