[I apologize. I am publishing my rough draft. I edited it and cleaned it up, but when I tried to save it everything got lost. I cannot be bothered to repeat those efforts.]
The stage is set. It is the 4th century BCE. It is ancient Greece. People are thinking about ideas. Possible explanations for how the world works, how nature operates, are being entertained. A Greek mathematician, Euclid, decides to essentially compile all known geometric and mathematical knowledge known in the world at that point. His work, the elements, is the foundation for much of the mathematics and geometry to move throughout history in human thought.
Now imagine. Place yourself in this time. This is the beginning of creating a system of math and geometry, of putting it to language in written form and to have the ability to send out to the world for people to consume and acquire. It is a hypothetical truth statement, it is not a book of false statements. They assert their own validity and truth value. Nothing of this sort had existed up until then (at least that we have empirical evidence of, and of such vastness). Now, what reason might a person who picks up this book ever decide to accept it as truth at all? Imagine, the very first argument for mathematics or geometry. The first time someone ever said “the sum of all angles within a triangle is 180 degrees” or that “the area of a triangle = 1/2 the length of the base multiplied by the length of the height of the triangle. What criteria would someone who has never heard that before have to accept it as truth?
This question can be answered for both how any child consumes and acquires mathematics, and for how the people of the day, grown men and women, who had lived their entire lives without knowledge of geometric and mathematical proofs and methods, are to accept mathematics…
The child does so simply by accepting it as true by convention. They are told this is the way things are, this is the truth, you memorize it, and you will be tested on it later to make sure that you still remember, you have acquired these truth statements. Perhaps this is why many kids do not do well with mathematics. They are not taught the first principals of which they are based off of. Instead, they are asked simply to believe, on faith, via convention, via authority, that what is being told of them is true. Others acquire it easier because they find it aesthetically pleasing. It makes sense to them. They see how it fits together. It is an aesthetic event and phenomena. They see something and conceive of something that someone else cannot. They see connections and relations where the other person cannot. Perhaps this is why memorization is not optimal.
But imagine a grown man, in the 4th or 3rd century BCE. Or whatever century copies of this book, The Elements, finds its way to you, to your country, and translated in your language (which requires someone who knows both languages, and is in contact with a scribe of sorts, and the means to produce the book). There was no internet, no data transfer, no email, no planes. Animals, carts, people, and testimony (gossip/news).Now, you receive this copy of the book, and you are now told all these things. What is your criteria for accepting it? Do you simply accept all foreign and different truth statements.
Geometry is not quite the same as farming or making cloths. Though farming could very well be an aesthetic and phenomenal model of geometry. Geometry could be intuited from farming, once one saw land, related sizes of one plot with the size of the plants growing, inferred that double that size would double the size of crop production. Found some system of denoting the size/shape/geometry/area of the crop (number of steps, lengths of cow, whatever), then made the inference that if that same area is repeated a similar crop would be produced. There is a correlation made. A connection between two things that would not have normally been there before. A new concept. And that concept is that of identity or relation. If this = that, and you make another this, then you should get another that.
It is not necessarily true, but it is an inference, a hypothesis, an assumption, a guess, an axiom.
And then we do that. Then once successful, or a relation is made, then desire can come into play. If I know I can manipulate nature, I can make an action in the world and a causal effect would come about that I wanted to, then now the only question I should ask is, what is it that I want?
And so a food quota could be produced. If I reason that we need a certain amount of food, and there are this many of us, we will need to produce this much. Now that I know how much I want, I can solve for x, x being the size of the plot of land necessary to produce the amount of food that I desire. I need this amount of land.
I am on a tangent.
So, what reason would that person have to accept it as true? So mathematics has to give you a reason to accept it. Geometry has to provide a reason to accept it. Mathematics/geometry come up to you and say “I am the truth”, you would want to ask “why?” or “how can I know that you are the truth?”. Mathematics would say “Here are my assumptions, the starting point from which I necessarily arrive at these truths. I call these assumptions axioms. I assume them, I take them to be the case because they are self-evident and do not require any proof of their own. In fact, they cannot be proven. That is just the way it is.
So, let’s make believe. Let’s pretend you are that person from over 2300 years ago. You open the book,first you see a list of definitions.
- A point is that which has no part
- A line is a breadthless length
- A straight line is a line which lies evenly with the points on itself
- A surface is that which has length and breadth only
- A plane surface is a surface which lies evenly with the straight lines on itself
- A figure is that which is contained by any boundary or boundaries
- A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another
And more. Now, it is you, the reader, to decide whether or not you choose to accept this as true. What does it mean for a point to have no part? Can something with no parts exist? A straight line is made up of points. And so on. All resting on a point is that which has no part. A point (something with existence) is that which has no part (is not made up of anything).
Now to the the assumptions, the axioms you see are:
- Things which are equal to the same thing are also equal to one another.
- If equals be added to equals, the wholes are equal.
- If equals be subtracted from equals, the remainders are equal.
- Things which coincide with one another are equal to one another.
- the whole is greater than the part.
So, we have if A=B, and B=C, then A=C, x + y = x + y, x – y = x – y, if A=B then B=A, and if A and B are both things (not zero, not non-existing), then A + B > (is greater than) A, and A + B > B.
Those are up to you to decide if you accept them as true or not. This is a book you just got and have nobody in the world to talk to about if you accept or not. You have never read this book yet, though perhaps have heard the ideas (or not).
From these axioms Euclid moves to his 5 postulates. Let it be postulated that:
- A line can be drawn from two points.
- Any line can be extended along that straight line indefinitely.
- For any straight line, a circle can be drawn having the line as the radius and one end point the center of the circle.
- All right angles are congruent.
- If two lines that intersect a third line such that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.
The first seems fairly straight forward. Pick any two points and a straight line can be drawn. We have the definition of what points and a straight line are above. This seems fine. The second also seems fine. You have a line, you can just keep making it longer, and it will still be that same straight line. You can make a circle by taking a line, keeping one end constant, and rotating it around in any direction along the other end of that line. All lines that form at 90 degrees/perpendicular are the same. But the last one… what does that look like?
There are two lines (one ones going more horizontally) and they intersect a third, any third line, in any orientation. If the two internal angles (alpha and beta) are less than two right angles, then if you extend those lines, eventually they will meet on that same side. we can use the opposite side, (the left side of the picture where the lines are moving away) and choose to see if those are smaller than two 90 degrees either, it doesn’t matter. But when we DO find these angles to be less than two 90 degrees, the lines will necessarily touch if those lines are extended, eventually.
And so, from just this hundreds of pages of proofs for propositions were deduced, using 100% only what was stated above, and no other influence or knowledge. Just those definitions, those axioms and those propositions (and any deductions that came along the way).
From that a form of geometry, a single form, came into existence. This form of geometry stayed in existence as the only form for roughly 2000 years. There was only Euclidean geometry. From acquiring this geometry we could now see things using these concepts, see relations using these concepts. We used geometry to model our reality, our technologies, our understanding of the world. We used these geometries to describe forces, to describe physics, the laws that physics and the sciences are based from.
Historically, the 5th postulate of Euclid has always been the one that stands out. It simply has. It is a longer postulate, yes. But there was something about it that caused it to stick out from since day one. In the middle ages Islamic philosophers attempted to disprove the 5th postulate.
Then in the 19th century non-Euclidean geometries came into acceptance as valid concepts.
For any given system, as long as the same starting definitions and assumptions are held to be true, no matter what logical deduction and interpretation that can be arrived at is valid. So, if it is possible to start from the same basic statements (definitions and axioms) and only necessarily true (deductive) statements are made from that logical system with the same set of rules, then whatever is deduced, however it is interpreted, will be true. There can be no means to say one interpretation is more true or valid than the other.
Euclidean geometry is true when the lines drawn are on a flat planar surface, but surfaces are not defined such that they have to be flat. They are just two dimensional figures. In fact, there is no reason why we should even infer that a plane SHOULD be flat. But that was just how Euclid saw things, and that is just how everyone else saw things. And that was what was accepted, by convention, as geometry.
But almost no surface is flat. Surfaces are curved. When you zoom in enough, there actually is no such thing as a flat surface. Flat surfaces can only exist in a universe where lines are made of points and points are made of nothing (no parts), which is to say, no dimensions. But in a physical universe of stuff, of matter, everything has a part, everything takes up space, has volume, has three dimensions. So a mathematically, geometrically flat surface is not even possible.
We see more often in the world curved surfaces. Such as the entire surface of your body, the surface of a tree, of a leaf, of the earth.
And so if we look at the geometry, as defined by Euclid, on the surface of a sphere (such as the Earth), we get straight lines forming a grid such as our longitude, and our latitude. The unique thing is that any given straight line on the surface makes a complete circle around it along the longest possible route (the circumference). Each line of longitude pass through two points, the poles. Now, if we take a look at the 5th postulate using this spherical surface:
We will see the example again of the 5th postulate. Take any two lines and a third that intersects them both. Like these 3 lines:
You will see that any two straight lines that can be drawn parallel from the equator of a sphere (or any longitudinal line), will eventually meet at a single point (the pole of the sphere). Euclids 5th postulate states that that occurs when the sum of the two interior angles are LESS THAN the sum of two right angles. Yet, in the spherical geometry we see that the sum of the two internal angles are equal to two right angles, because they both are right angles, and all right angles are the same (congruent).
So the 5th postulate is not true under this spherical geometry (non-flat planar surface).
and interesting thing to see is that under this spherical geometry the sum of the angles of a triangle necessarily must be GREATER than 180 degrees. This is very different than any geometry we normally believe to be true. And yet, empirically it is true for our planet.
If you were to go buy a piece of land and the gps coordinates of that land were such that along a given line of latitude (ie. the equator), and you were to draw two straight lines heading to a single point (meaning the shape of the land was a triangle), you could literally get out your ruler and protractor and measure it out, and the sum total of the angles on the inside of that triangle would add up to more than 180 degrees. It is the geometry of our planet. It is empirically observable. It is true.
Yet we don’t see geometry that way. We see it as flat and in a plane.
We could do the same thing for other surfaces. For hyperbolic geometries, geometries of any shape.
So what? What does this matter to me? How does geometry matter?
Geometry matters because it is directly related to how we perceive in the universe, how we perform science, how science and math is used to explain the universe, and our entire body of scientific “knowledge”.
Here is why:
Because both sets of geometry are both 100% true, both 100% equally possible, either can be adopted. Now, science operates under the belief that the truth of the universe, the true way the universe is and can be explained and modeled, is objective. Meaning there is one single truth to be discovered and elucidated.
That means that there can only be one geometric shape of the universe. Is the universe a sphere? Is it planar? Is it hyperbolic? Is it _____ (insert any possible geometric surface curvature here)?
So science and scientific knowledge depends on there being only one single true geometric shape of the universe.
Now, science is a study of empirical data. To determine something scientifically it must be testable. You must be able to refute the claim made by science, the theory.
An experiment could be made. You could take the surface of whatever it is you want test to determine its curvature. And you can draw along that surface grids that pertain to all the different geometries. Euclidean or non-Euclidean (spherical, hyperbolic, etc) grids on the same one single surface (ie. the floor of the room you are in now). And the experiment could be done that you will roll a ball on that surface along a straight line, and you could then watch the path of that ball and see which grid it traverses. Then you would know the answer to the surface curvature (the geometry of it).
You can do this experiment because it rests on the belief/assumption/axiom/Newtonian “law” that all of physics is based off, of inertia. That something when put in motion will stay in motion along a straight line until some external force prevents this. This is your basic law of inertia. Self evident. It needs no proof. It is an axiom of science.
So, keeping this in mind you do the experiment. The ball traverses the floor and you can see and plot out its path empirically. Now, it will follow a straight line in SOME grid system used (and each grid system used represents a distinct possible geometry that follows Euclids rules of geometry). Now, here is the thing. From the results of this experiment it is impossible to judge which geometry the surface actually is (objectively). You cannot determine an “absolute” geometry.
The reason for this is because you can interpret the results in any way, and none are necessarily true. If the path of the ball followed along the grid of straight lines that represented a flat surface geometry, and not of a curved surface, you could say one of two things, all based on the law inertia:
- The surface is flat, and no forces were acting on the ball.
- The surface is curved, and a force was acting on it such that it moved off the straight line of the curved surface and so it appeared to move on a flat surface.
Likewise, if the ball followed along the grad of lines that represented a curved (ie. spherical or hyperbolic geometry) surface, you could say one of the two things, all based on the law of inertia:
- The surface is curved, and no forces were acting on the ball.
- The surface is flat, and a force was acting on the ball such that it moved off the straight line and so it appeared to move on a curve.
No matter what the outcome, a necessarily true interpretation cannot be found. All interpretations are valid.
The inference of forces stems from objects moving out of an inertial frame (a straight line given that accepted geometry). So the two are co-dependent. With that said, depending on which geometry you decide to accept, this will change the existence of forces. Some disappear and new ones appear under different geometries of space, of the universe.
So when we hold a “scientific fact” to be the absolute and objective truth of the way things are, we should keep things like this in mind. That we cannot truly know for certain, as an objective truth. Even our very fundamental concept of the geometry of things comes into question. An alien from another part of the universe that evolved such that their ancient mathematicians and philosophers first devised a mathematics and geometry which was conceptually spherical will have deduced different laws to explain the same universe. They could create the same technologies and everything. The only difference is the model, the aesthetic of the universe, the allegory for truth changes. What is being represented by the model will still be accurate in the way an analogy is accurate.
And so meeting that alien we might have different laws of physics, different equations and conceptual schemas of what exists, what is in the universe, what laws govern it, etc. But both will be capable of the exact same things.