Have any attempts been made to define numbers through "up-to-down-method"?

[Norman]

We can say what natural numbers are an then, with them, define an ordered set of <a, b>, where a is "+" and b is any natural number, which becomes an integer-like-definition (+a, -a). Then, we can go on defining other classes on top of them. Imagine, you have the integers and define rational numbers like <a, b> with a and b being integers such that a/b. You can go on with this structure for quite some time, I believe, until you hit the reals. There, suddenly a new strategy to create numbers is needed, but that's also done already. We can go on: complex numbers are <a, b> with a and b in R and so on and so forth. This may go on forever, defining new numbers with tuples and so on.

That's quite an interesting approach, I believe. We can say that we can do this for ever, defining new sets of numbers from previous ones. Let's say we call those numbers "dimensions" (for a lack of a better word in my mind).

But was there ever any attempt to do it the other way 'round? Let's suppose we have an ∞-dimensional number (the kind of dimension would be kind of interesting, but let's just assume a naïve interpretation of infinity), and now we may be able to, from this type of number with, I assume* detectable properties, we can define things about all numbers and may get a deeper insight in the nature of numbers.

Has this ever been tried before?

* I'm not a mathematician, sadly, just some hobbyist, and my knowledge about what exactly or even how those proofs may look like is very small. I'm just suggesting this here as a more-or-less-thought-experiment.

 

[HallsofIvy]

We can say what natural numbers are an then, with them, define an ordered set of <a, b>, where a is "+" and b is any natural number, which becomes an integer-like-definition (+a, -a). Then, we can go on defining other classes on top of them. Imagine, you have the integers and define rational numbers like <a, b> with a and b being integers such that a/b. You can go on with this structure for quite some time, I believe, until you hit the reals. There, suddenly a new strategy to create numbers is needed, but that's also done already. We can go on: complex numbers are <a, b> with a and b in R and so on and so forth. This may go on forever, defining new numbers with tuples and so on.

That's quite an interesting approach, I believe. We can say that we can do this for ever, defining new sets of numbers from previous ones. Let's say we call those numbers "dimensions" (for a lack of a better word in my mind).

Actually what you say here, if I understand correctly, is what is normally done! Given the natural numbers, we define the "integers" as the equivalence classes of pairs of natural numbers, {(a, b)}, with equivalence relation (a, b) is equivalent to (c, d) if and only if a+ d= b+ c. For example, the pair (3, 2) is equivalent to (4, 3) because 3+ 3= 4+ 2. Similarly, the pair (2, 6) is equivalent to (3, 7) because 2+ 7= 6+ 3. It is easy to see that if (a, b) is equivalent to (c, d), so that a+ d= b+ c, with a> b then c> d and a- b= c- d. (I chose a> b in order to stay within the natural numbers.) In that case, we can associate the positive integer "n" with the class of pairs (a, b) such that a- b= n. If b> a and (c, d) is equivalent the c> d and b- a= d- c. In that case we associate the negative integer "n" with -(a- b) (again staying in the natural numbers.

Similarly, we can define the set of all rational numbers as the equivalence class of pairs of integers, (a, b) (with b not equal to 0), with equivalence relation defined by "(a, b) is equivalent to (c, d) if and only if ad= bc".

The real numbers can't be defined "algebraically" but require a "limit" procedure. One method is to use "Dedekind Cuts", https://en.wikipedia.org/wiki/Dedekind_cut. Another, simpler in concept, is to define the real numbers as equivalence classes of sequences of rational numbers with the equivalence relation defined by "\(\displaystyle a_n\) is equivalent to \(\displaystyle b_n\) if and only if the sequence \(\displaystyle (a_n- b_n)\) converges to 0. For example, the real number, \(\displaystyle \pi\) is the equivalence class the contains the sequence "3, 3.1, 3.14, 3.141. 3.1415, etc."

The complex numbers can be defined as pairs of real numbers, (a, b) (NOT an equivalence class of pairs), with addition defined by (a, b)+ (c, d)= (a+ c, b+ d) and multiplication by (a, b)*(c, d)= (ac- bd, bc+ ad). We write complex number (a, b) in the simpler notation a+ bi where i is the pair (0, 1).

But was there ever any attempt to do it the other way 'round? Let's suppose we have an ∞-dimensional number (the kind of dimension would be kind of interesting, but let's just assume a naïve interpretation of infinity), and now we may be able to, from this type of number with, I assume* detectable properties, we can define things about all numbers and may get a deeper insight in the nature of numbers.

Has this ever been tried before?

You would first have to say what you mean by "∞-dimensional number".

* I'm not a mathematician, sadly, just some hobbyist, and my knowledge about what exactly or even how those proofs may look like is very small. I'm just suggesting this here as a more-or-less-thought-experiment.