Ok work with me here. How do we achieve the assumption that numbers are a finite field?

Feel free to refer me to beginner level reading to save you typing it all out.

As for a "beginner level" introduction, I'm not really aware of any that isn't gritty rigorous Math. And in that case any would be as good as any other I suppose, since the definitions are always the same (except for perhaps the symbols used). If you feel like digging deeper I'm sure you'll find any number of resources on Google. All my notes and books for these topics are in non-English though, so I can't recommend you any reference here. But alas.

Assumptions are created out of more or less thin air. Generally based on past experiences (I'm not sure if we can arrive at any assumption without making references to previous ones, which is a problem in and of itself). It's also not exactly that numbers "are" a finite field, they can be depending on how many you have to play with. The numbers we usually use are not a finite field, you always have a unique "+1". However.

In the case of finite fields, if you want to go from the "human experience" side you could think of it like this.

If you think that infinity exists, then there's not much reason to argue here since you can just keep adding numbers and always get a unique new one, so 1+1 will always be 2 in that world.

If you however believe that infinity can't exist in reality, then the only conclusion is that any field of numbers must be finite, because you'll eventually run out. And in that world you eventually come full circle, otherwise you can't have a number field that functions in the way we understand numbers. It is the only way to formalize our natural understanding of numbers (that I am currently aware of).

As for what fields are, here's a very brief overview that leaves out a fair deal of the gritty parts that are necessary to formalize this. But this should hopefully give somewhat of a more formal intuition that you could compare against our natural intuition.

A field can be any set of numbers that satisfies a few properties. The most important ones for the topic we're having are the existence of a set of elements (or numbers), the existence of two operations (e.g. addition and multiplication), the existence of a neutral element for each operation, and the existence of an inverse element for each operation.

Let's take {0,1} as the set of numbers. And addition and multiplication as its two operations. The operation "o" (+ or *) now has to ensure that each element has a unique inverse element. So if you take any element X , there must be another element Y such that X o Y spits out the neutral element regarding "o". For addition this is 0 (you add 0, shit happens), for multiplication it's 1 (you multiply by 1, nothing happens). If you think about it, whenever you invert a number you get its neutral, 3 * 1/3 = 1, 3 + (-3) = 0.

If these properties are violated you can somehow show that the whole natural intuition of numbers just breaks down, but I can't think of a good example on the spot. Been too long since I've done anything in this area.

In the usual fields the inverse element would just be -X for addition and 1/x for multiplication. In this finite field you can't do this, because "1/1" clearly does not exist as neither does an element called "-1".

However, with the circular arrangement you can quickly see that:

0+0 = 0

1+1 = 0

0*1 = 0

1*0 = 0

1*1 = 1

Hence, each element has an inverse regarding multiplication and addition, and our intuition still works. This way to look at it satisfies the requirements of a field. It

*just so happens to be* finite. This curiously doesn't work for any set of numbers either, the number of elements has to be infinite, a prime, or a prime power.

If you want to get technical, then "2" in the way we normally understand it won't give you "2+2 = 0", but there is an abstract field that extends {0,1} in a way such that the elements that you could "call '2'" would satisfy 2+2 = 0. You can easily get 2+3=0 for {0,1,2,3,4} with the usual addition though (check this yourself if you'd like as an exercise).

As for 'why' this works, I doubt anybody knows. Emergent properties?

Essentially, fields are merely a formalization of the way in which we naturally came to use numbers based on our experience of reality.

**And the formalization ***naturally* gives rise to both *finite and infinite* fields. There are also weird fields that have polynomials as their "numbers" and where "1" is suddenly a polynomial (the constant polynomial 1). But you would never expect either of these by not carefully thinking about the fundamentals, what you already know, the implications of either, or what other ways you could view what you already know in.

These weird mysteries are why I'm against quickly rushing to conclusions on any subject and prefer looking for as many explanations and vantage points as I am currently capable of. The universe has a way to always screw us when we think we "know", and to reward us with new exciting experiences if we remain open.