the foundations of maths were established by the Babylonians.
All mathematicians lie
a bit of advertising to hook you in. this piece mostly not about mathematicians, but about mathematics. And in particular only those bits of mathematics. that is claimed to be proven true by geometry.
I have been told that I need to start a piece of writing such as this with a short summary of the points to come. And why you should be interested. So the next paragraph outlines some examples of where I think mathematics does not quite work. Then I will look at the underlining fault that I think undermines much of mathematics and at how it arose.
After that, I try to identify possible solutions. Then a bit of a moan about the type of constants of proportionality that are only there to make the sums give the right answer. As to why you should be interested? If you don't use higher mathematics for a living. The only reasons I can think why you should read on are; curiosity to see if I can make a good case, or because you have already decided I am wrong and want to pick up the sticks to beat me with.
There are some simple sums that do not give a satisfactory answer. If you ask a mathematician to tell you numerically and to the last decimal place, what is the square root of two? The answer you will probably get is that the square root of two is an irrational number. It sounds reasonable. But it is not a number. It might be more honest to say the square root of two is a sun that does not have a precise numerical answer. A sum without an answer!
Then there are imaginary numbers, I am not sure that they really exist? But if they don't relate to real objects, then counting non-existent objects to make your sums add up. It makes me think there's something rotten in the state of mathematics.
Now a real-world example that hints at what the underlining problem is. Two airplanes take off from London, one goes southeast to Paris, The other sets off northwest to New York. At about the latitude of the Scottish border, it is flying east to west, and it arrives at New York from the northeast. To see this as a picture follow this link. https://www.greatcirclemap.com/?routes=LHR-JFK%2C%20LHR-CDG%2C%20CDG-JFK
On planet Earth, the shortest distance you can travel between London and New York is a curve, not a straight line.
Even if the above has not convinced you that there is something suspect at the core of mathematics, perhaps the following argument might.
Any proof in mathematics can only be as sound as the earlier proofs that it is based on. This goes all the way back to the proof of the simplest geometric definitions. "the shortest distance between any two points is a straight line." In flat two-dimensional Euclidean geometry it is. In the real world, it is not!
The assumption that the two-dimensional Euclidean is the correct one to use, as no proof! That out of all of the possible geometries that could be used to prove the basic definitions of mathematics, there is no obvious evidence to show anyone is better than any other. If solid spherical geometry had been used, then the shortest distance between any two points would be an ark of a grate circle.
To repeat. It is my opinion that this assumption that underpins most mathematics is unproven, unprovable, and probably wrong. If you know of a proof. Please publish it so it can be peer-reviewed. If you are reading this on WDC and you think you know of proof, I would be grateful if you would let me know.
Of all of the possible geometries, flat two-dimensional geometry has the distinctive property that it does not exist, anywhere on a large scale, in what we are told is a curved universe. In case someone thinks that the Milky way is flat, I give you the following quote "it bulges in the middle 16 000 lightyears thick, but out by us it's just 3000 lightyears wide." It would not matter if it is only 3 feet wide, it's still three-dimensional.
The reason why this problem arose is that the earliest foundations of mathematics were established by the early Babylonians, who quite reasonably, believed that the world was flat. By the Greek area, it was known that the world was round. But mathematics based on the mistaken belief that it was flat, was allowed to become the basis of modern maths.
The solution would build new mathematics on the correct geometry. But what is the correct geometry?
The next best solution is simple in principle. But will be very difficult to do in practice. Starting with first principals to build complete new mathematics. And to do this for as many possible geometries as are likely to generate useful mathematics. In the hope of finding one, that is the right one.
As an aside, while mathematics is putting its self in order. Physics can take the opportunity to choose new values for units. To replace the arbitrary set of units that exist for historical reasons, with a rational system of units. That is to change E=MC2 which says energy is proportionate to mass, to E=M. This will leave only three types of constants in physics. Values of forces that are fixed. Situations where there are a large number of factors that are necessary for a complete understanding. But only a few are needed for pragmatic use, the rest might be bundled up as a constant if in practice they sum up to a regular value. Both of those are sound science. The type of constant that I want to complain about, and would like to see hounded out of physics completely are constants of proportionality. Those numbers that physicists add to their sums to make give the right answer. In case you do not believe that it is an acceptable practice for scientists to fiddle their figures, I give you, Einstein's cosmological constant. Originally invented by Albert Einstein to make his sums add up to a static universe. When it was proven that the universe is not static, the cosmological constant was doped. Now it has been brought back to make the sums add up to a universe whose expansion is accelerating. As an aside to this aside.
I personally think that the reason matter feels the effect of gravity, a tiny bit less each day. Is because there is less mass each day, and therefore less gravity!
To conclude. Mathematics is a house built on quicksand! And all mathematicians lie: whether it is peacefully in their bed. Fitfully in their sleep. In a cold grave. Or on a taxidermist slab. Sooner or later all mathematicians lie down.