This is a note where I begin to develop my own geometry based on the example of Euclid, but with an eye towards addressing its shortcomings - what hubris?! But seriously, I’m pretty sure my geometry will pale in comparison. The purpose of the exploration is not to show off, but to learn more about the world and in this regard, will be interesting and educational.

Gotta start somewhere!

Start from first principles - the evidents.

Shapes exist and are perceived. That is, we see and describe the world around us in abstract terms and shapes are abstract. As we abstract away details and try to describe things that exist, we highlight some things / aspects of things, and disregard others. We strive to arrive at useful ways of referring to groups of / sets of objects. Geometry, which started as a way of measuring land, has evolved into an ever-abstracting study of shape. What is shape? Wow, so hard to establish the bare necessities. Maybe a shape is a collection of points? What is a collection and what is a point?

Let’s just say (posit) that a collection is just what it sounds like, one or more than one of something, or nothing at all (for convenience, later). In this case, a collection of points. A point being the idea of the smallest thing (or not thing), imaginable. We represent this idea generally with a dot drawn on a page. Keep in mind that a point is an idea and doesn’t have width, height, or any physical measure, whereas the dot does.

Maybe think of the dot as a graph (physical representative) of the point and imagine zooming in on the dot, imagine it never changing in appearance, whereas the area around it expands, endlessly.

We have our first postulates:

  • a point is an idea that can be represented by a dot. It has no measure.
  • a collection (set) is a collection of zero or more points.

What can we do with our postulates? Not much, at this “point” :). We can put points into a collection and take them out again. We can imagine that our collection, thus created, refers to different points. But, if a point has no measure, or other attribute, how can two be different?

Now we can posit something new about points, relationship. Points can be different. If they are different, they cannot be the same point :). This is an intuition related to the nature of information - any difference that makes a difference to a knowing subject. But again, how are they different? We are in desperate need of a property or aspect of points with which to differentiate them. Naming them provides one basis - we can call our points by name. Let’s say A is one of our imaginary points and B is another - difference! But, that’s of limited utility in the context of shapes, so we look for another. How about location. Wow! That “maps” to our intuition pretty well, points are located. What this means will be expanded on, but for now, just know that location simply means that point A is not in the same place as B, there is difference. In space, this would mean that points A and B do not occupy the same space, and that’s reasonable and seems consistent with reality, but in this case we are only suggesting that A and B are different and that difference is location.

This brings us to our next postulate:

  • location - objects (in our world, there are collections and points only) have difference.

Location is tricky to describe, but I mean that points are located (have difference) somewhere, even if its only in our mental world. We will represent our idea by referring to the idea of physical space, in the real world, that is the 3 dimensions we normally consider space.

How are points related locationally (is that even a word)? Suppose that points can be next to each other. Indeed, if two points exist (even in the mind), in order for them to be different, by definition (see above), they have different locations and one is next to the other (if is isn’t clear, then here they are so defined, as being next to each other).

The hard part comes when we add a third point. Where are the points in relation to each other. If points have no measure, then why can’t they all be next to each other? If they are all next to each other, then they can’t not be next to each other, right? But, we know this isn’t how the real world works, things can be between each other… ick, so hard to explain betweeness. Why? Well, if a point has no measure, then it can’t be put between two points. Unless, perhaps, we establish the idea that there is order on the universe. Let’s do that. The universe is ordered, so mote it be. If that’s the case, then we have another postulate:

  • order - the universe has order

That is, one thing can be said to be ordered (precede, or follow) with respect to another. Order is imposed, so, without knowing the established order, one can only detect it and not know what the origin or orientation of that order is in its entirety.

Now, betweeness becomes evident as the established order of three objects (points). We say, determine, decree, demand, etc. that a point is between two others. This may sound silly and needlessly complex, but it is what it is and betweeness is our next postulate:

  • betweeness - the property of a point that indicates it precedes a particular point and follows another particular point

Now we can speak about dimensionality. We can define dimensions thusly, a single dimension exists when a point is brought into existence. Another, the second dimension, when three points are realized. Why three and not two, well, I’m not sure, with two points we can speak about the relationship between the points, A is not B, B is not A, A exists, B exists, A and B exist, but that’s pretty much it. In order to go further, we really need at least 3 points (this is an exploration of the mind, not definitive). With three points, we can say much, much more. We can say is everything we can say about two, plus some more.

Let’s say that A and B and C are points in our universe. A is next to B and B is next to C, but C is not next to A. Then, B is between A and C.

Here’s a thought exercise to be integrated into the discussion as it is determined where it should go…

Think of a light filled universe.

Bring a point, A, into existence. If the point had measure, and the light had a source, this would change the universe in a measurable way, as it stands this is not the case. It just is.

Bring another point B, into existence. No change... or is there? There is difference, but not measurable.

Bring a third point C, into existence, such that we say B is between A and C. Again, difference, but not measurable. Why not? Because points have no measure!

But when we bring our points into the real world, and of necessity given them measure, voila! There's going to be measurable differences. But, very local. Perspective is hinted at here - we need it in order to appreciate our new universe. If we were to place our consciousness at a point and look, as it were, around, what would we "see". If we were at A, we would see B, if we were at C, we could also see B, but from A, we could not see C and from C we could not see A. Wow!

But, for now, we’re sticking with betweeness being nothing more than established by fiat and being definitional in nature. B is between A and C because I say it is and beyond establishing order between A, B, C, has no material effect on anything.

Ah, but let’s give another name to that order of points and introduce another postulate:

  • line - a line is an ordered set of points

But, it’s tricky again, the line may be a set of points, but surely, we can’t refer to a line by it’s points, that’s nuts. So, we’ll say that a line can be named by any two points, belonging to the set, where the set established the order. Line AB therefore is part of ABC. As is AC and BC.

Let’s say that the order we write the points matches their ordering, this means AB is not the same ordering as BA, and ABC means that B is between A and C, with A preceding and C following.

To recap what we have so far:

  • point
  • collection
  • location (difference)
  • order
  • betweeness
  • line

The point is fairly standard to think about in this way, collection or set is what we would expect, location is a bit different :),, order is meta, betweeness is necessary, but I’m no geometry expert, so I could be naive on this. Line is way different from what it is usually described as, but this is because I’m working through it from the ground up.

… enough for today. Too many gaps, too many doubts, need to consider for a bit :).

– will

post edited 2023-12-19 13:58:00 -0600