Interpretation and symbolic notations of operations in mathematics

Question

As we know a mathematical operation is a function. And a function is a special type of relation. Ultimately a relation is a set of ordered pairs. For example, what is addition of natural numbers? At the most abstract level, addition is a functional relation=correspondence between an ordered set (a,b) of two natural numbers and another number c. So in a rigorous way addition is just a set containing elements of the form ((a,b),c) for example: +={....; ((1,2),3); ((2,3),5); ((7,0),7),...} Here we can say that c is the sum of a and b in this order (the order should not matter because of commutativity )

So why it is addition viewed by almost everyone as a "process" with a "result"? What in the world is a "process" in mathematics? Can be the notion of "mathematical process" be rigorously defined? I don;t see how... And if we want, we could even critique the writing "1+2=3"?? We know very well what "3" is, but what is "1+2"?? What type of mathematical object is that? How can a mathematical object be equal to a combination" of two other objects? Oh, we say 1+2 is just a "symbolic expression" or just another "name" for 3... i find this a bit unsatisfying from a highly rigorous point of view. Even some mathematicians when seeing "1+2=3" think like this: well we have a process of adding 1 and 2 and we get three... i don;t like this interpretation. At all! Others even say "that is the performing of a computation"...What the heck? So on the left side we have an "unperformed computation" and on the right side a result?? What's a performed and unperformed computation?? How can be something unperformed equal to other something?? How can a single object be equal to a combination of other two objects?? As in 1+2=3 And then, shouldn't 1+2=3 be different than "1+2=3+0"?? In the latter we have an equality of two "combinations", in the first we have an equality between a single combination and its result... from a highly rigorous point of view the tqo equal signs "Act" a bit different.

So if we want complete rigor in math, statements as "1+2=3" lack it a bit... because "1+2" and "3" should be of different types, so how can they be equal?

So my questions are: 1) Speaking from a purely SYNTACTICAL point of view, how should we rigorously define what a "combination" of two mathematical objects is, for example two numbers? If 1,2,3,... are a "type" of math objects, shouldn't "1+2" be of another type??

2) How can two things be equal if they are of different types? For example in 1+2=3, LHS is a "combination" of two numbers and RHS is a single number... how can they be treated as equal, having different types?

Answer 1

See First-order arithmetic for the syntactical specifications of the formal language dealing with natural numbers.

The basic arithmetical symbols are : 0 (an individual constant denoting the number zero), the unary successor function s(x), and two binary functions : + and x for the numerical operations of sum and product.

The axioms are the first-order version of Peano axioms.

The usual numerical symbols (the numerals) : 1, 2, 3, ... are introduced as abbreviations :

1=s(0), i.e. "1 is the name for the (unique) successor of the number zero";

2=s(1)=s(s(0)), and so on.

Using the axioms, we prove that :

1+2=3 [which abbreviates : s(0)+s(s(0))=s(s(s(0)))].

Answer 2

I've found two interesting papers about this issue: 1) Gottlob Frege's "On sense and denotation" https://faculty.washington.edu/smcohen/453/FregeDisplay.pdf

2)Alonzo Church Introduction to mathematical logic (in the introductory chapter about "Names") http://www.hist-analytic.com/ChurchIntroductionIntroduction.pdf

They treat exactly what i asked in a very nice way.

Thanks to @John Forkosh who suggested above the "sense-denotation " duality. It's been a long time since i've reflected about this philosophical issue, and it seemed untractable. But know i've cleared it.

Answer 3

Heres my attempt at a simpler answer, without the need for math:

Your problem arises from your interpretation of equality. You are interpreting = to mean "The thing on the left of the sign is equal in all qualities to the thing on the right, while in math a = sign only signifies that the things on both sides of the sign represent the same quantity.

Likewise, in other fields of math, the equal-sign only compares specific qualities of the things representated by the notation (e.g. that two sets contain the same elements).