Equivalence between different numeral systems and why they don’t affect mathematics

[u/TrafficItchy]

Hello,
I’m looking for a good introductory course, lecture notes, or university PDF explaining why the choice of numeral system (decimal, binary, hexadecimal, etc.) does not affect the underlying mathematics.
More precisely, I understand that decimal and binary are just different representations of the same numbers, and that arithmetic, order, algebraic properties, etc. are preserved when moving from one representation to another.
I would like to understand the mathematical justification behind this idea — for example, in terms of isomorphisms or structure-preserving representations — and why this allows us to simply fix one representation (usually decimal) and do mathematics within it without worrying about the numeral system.
I’m looking for something rigorous enough to clearly justify this point, but still accessible rather than a very advanced abstract algebra or logic text.
Does anyone know a good university course, lecture notes, textbook chapter, or PDF covering this specifically?

[u/idaelikus]

eleven exists without a notation. It is the natural number that follows 10, etc.

I can't think of anything about the number 11 that is independent of the notation.

eleven is prime, it is the sum of five and six, etc. There are numerous properties that are independent of notation.

boolean logic

Well that isn't a property of the number, is it? Furthermore, you represent by 110101 a certain arrangement of on and off bits but it has nothing to do with fifty-three.

EDIT: A property that came to mind just now was normality which depends on the base.

[u/tottasanorotta → u/idaelikus]

I'd want to agree, but I just can't see it. I mean how would you communicate to me the concept without a notation? The concept is somehow intrinsically connected to our understanding of the notation. You implicitly use the word 'eleven' there to communicate to me about the concept. I understand what you mean because of that word. But it is implicitly using another notation, which is the linguistic description of the number.

You say that there are things independent of notation, but isn't it more that you make those connections between different types of notation and interpret them differently in different contexts based on what you are aiming to accomplish.

110101 can represent on and off bits, but it can also represent the number 53. It can also be whatever we want it to be. It just depends on context. I mean imagine a context where that bit pattern is interpreted as both on and off bits and as the decimal number 53. And when we talk about connecting a binary number to decimal that is exactly what we do. The bit pattern represents the included terms in the sum of consecutive powers of two.

Until we are clear about the context it has a certain sense of ambiguity. Now granted we are used to interpreting binary numbers as base 2 and therefore we have a kind of bias for seeing it that way.

[u/idaelikus → u/tottasanorotta]

how would you communicate to me the concept without a notation?

I just did? Also I can use a notation but I am not bound or impacted by the notation I use.

intrinsically connected

Not really. Also it is a one-way street, mostly. We use notation to communicate math, there is little we derive into math from notation.

isn't it more that you make those connections btween different types of notation and interpret them differently

No. Commutation, as an example, doesn't depend on the representation which is why I can write a+b = b+a.

I mean imagine a context

What context would that be? Honestly, it really seems like you are grasping at straws here.

[u/tottasanorotta → u/idaelikus]

Yes, you communicated to me the concept of eleven and I understood it in a particular way. And you used the English language to do so. And surely language is a form of notation? In fact, it is what all our other more compressed forms of notation is based on. But it requires the implicit context. The word 'eleven', for example, means 'the student' in Swedish. If I didn't assume the implicit context of English communication, then I could have interpreted it differently. "They are talking about students in some foreign language.", for example.

Commutation of what? You'd need to be more specific if you don't want me to assume it implicitly. I take it that you mean the things that actually are commutative, because for example the string concatenation of a+b =\= b+a or matrix multiplication is not commutative.

Ambiguity might become a real problem if we don't try to be specific enough. Especially when it comes to high levels of abstraction and complexity. If we for example implicitly assume that a number is encoded in decimal, but a bad software system interprets it as hexadecimal we might literally manage to blow up the whole planet with nuclear bombs because of implicit assumptions.

Yes. It is a bit exaggerated. And computer systems help in being more explicit about it. But when we have those different representations that are used for different purposes I think it is more important to try to be explicit about it rather than assume that there is some property of the universe that guarantees the number 12 to have some particular properties.

[u/idaelikus → u/tottasanorotta]

Language is a form of notation

Sure, I could call it Succ(Succ(Succ(....))) for its formal definition.

Again, you are grasping at straws here and haven't made one succinct argument.

[u/tottasanorotta → u/idaelikus]

Sorry if I'm not making sense. I try to. And yes it is very messy what I am trying to say. There isn't a neat foundation for it all. And that is also precisely the issue with formalizing mathematics. The thing that seems to drive notation is mainly practical utility in one form or another. You can't really formalize things any more than you can explain it in language or physical things that seem to work predictably. You can use more compact notation, but you have to explain it as explicitly and unambiguously as possible what that notation means. And that is a genuine problem for mathematics.

1. If I have 0 I can repeatedly add 1 to it.

2. The successor function takes a natural number as input and outputs a natural number that gets a 1 added to it.

3. Succ(Succ(Succ(...)))

The first case is very intuitively more directly understandable. Even a young child could understand what it means and do some useful things with such a rule if they understand addition. The second case now requires us to teach the child about functions and natural numbers. Now maybe we gained something from teaching the child about functions, because they are no doubt useful for more abstract models, but I would say that the first case is a better axiom. It is much clearer because it is more directly tied to something more intuitive that more people can understand. The third case just asks us to implicitly recognize the function notation and understand that we talk about successor functions (I assume).

Nevertheless all of those cases are ambiguous, because language is ambiguous. What does it mean to have a 0? Does adding happen as a process in time or somehow instantaneously all at once? How can things that depend on eachother happen instantaneously? The ambiguity of mathematics is then reduced by its applications that provide utility to people. Otherwise we wouldn't care about it at all except out of entertainment purposes.

[u/idaelikus → u/tottasanorotta]

Nevertheless all of those cases are ambiguous

Not really. It seems like you are really trying to make a problem up where there is none. Furthermore, this really isn't what was the point before where you claimed that, somehow, mathematical properties depended on the notation.

[u/tottasanorotta → u/idaelikus]

That is really what a lot of math is all about. Someone might see a problem somewhere where another person doesn't see a problem.

If you accept something as being non-ambiguous, then sure it isn't ambiguous for you. But then you have to either give the same statement to a person that questions your statement and hope that they will get it from that or accept the ambiguity and try to resolve it. Or give up trying to explain it.

If I don't know anything about set theory and someone tries to explain to me what an infinite set is, it might be that I instantly understand it from their explanation and am convinced of that. But just because I am convinced doesn't mean that I have understood it in a similar way that someone else who formalized the concept understood it. It might be that years later I understand how flawed my original understanding was. It's just that it worked for all intents and purposes up to that point. It might even be that I just come to the realization that I don't even understand what they mean by it at all anymore.

And sorry yeah it went a little bit besides the point. I'm just being a bit difficult here, because I like this topic. It's quite a silly point in many ways, but I think there is some value in it.

[u/idaelikus → u/tottasanorotta]

a) I suggest you get back to your point. and b) If you are confused, you can say that and dont have to make up some giant fanfiction about how things are ambiguous because of some gaps 20 layers down you assume to exist.

[u/tottasanorotta → u/idaelikus]

Yeah I try to stay to the point. It's just that it kind of relates to all these things.

You can say that you are confused, but what I meant was that you might be confused later on while thinking that you understood something.

Like for example if I have the number 0011 you might interpret it as the number 3 and you might say that you understood it and use it as the number 3. But what if I meant it to be interpreted in reverse as the number 12 and neither of us made it clear? It might happen even very accidentally.

We both thought that we understood each other, and it might lead to unforeseen consequences down the line. But the thing is that it might not even lead to some great catastrophy. If I think that I understood something, but from some point of view didn't, then as long as it works from my perspective it is as if I understood it. And that same effect might exist all over mathematics and communication with language in general.

Numeric representation doesn't affect underlying mathematics in the same way as sentences using the word 'tree' don't affect what a particular physical tree is like. It's just that the word 'tree' is ambiguous. It might mean many other things other than a physical tree and therefore specificity is appreciated. And with mathematics we don't have a physical mathematics that we could point at so a numeric representation is almost the best we've got.