origin of syntax for mathematical equations

Question

Bear with me, I don't have any formal training in mathematics. I wonder if there is something that accounts for the syntax of mathematical equations, some deeper logic or reasons why I know that "1+1=2" is a valid expression (regardless of its truth value) and the expression "+1+" is nonsensical, or has invalid syntax.

I learned a while ago there are ways to prove that 1+1=2, which I would otherwise have suspected would have been a fundamental basis, or starting point for mathematics. If you had asked me if there were a way to prove it, I would have said no, it's natural and obvious, and it's the starting point from which all else follows. However mathematicians have delved into it and devised ways of proving it from yet more fundamental logic.

Likewise it seems natural and obvious that "1+1=2" is sensical while certain other expressions are not, such as "=1+" or "++2" -- i.e., that there is a syntax to mathematics. If someone were to ask me where this syntax arises from, I would say that it is fundamental and the natural starting point, like I would have about supposed proof that 1+1=2, before I heard of Peano's axioms.

Is there something, perhaps in logic, that accounts for the syntax of mathematics? I've looked at ring theory, and what little I understand doesn't seem to account for the order of symbols in mathematical expressions.

Answer 1

The basics of what you're asking deal with operations. Addition is a binary operation (uses two terms), and is typically indicated with the symbol $+$ between the two terms.

The same symbol can also be a unary operation. For example, $+1$ is not addition per se, even though it uses the same symbol as for addition. In this case, it's making explicit the fact that we're talking about positive one.

The equals symbol $=$ designates an assertion that the expression on one side of it represents the same value as the expression on the other side of it. Other assertions you may have seen are congruency ($\cong$) and equivalence ($\iff$).

In any case, the syntax and notation are meant to state specific, precise mathematical ideas in a consistent shorthand that should be able to be read in whatever verbal language you want. So,

$$1+1=2$$

is read as "One plus one equals two," just as

$$\int_0^1 x dx = \frac{1}{2}$$

is read as "The definite integral of $x$, with respect to $x$, evaluated from $x=0$ to $x=1$, equals one-half."

Answer 2

The short answer is that the symbols and syntax work the way they do because that's how we've defined them.

This may not seem satisfying, but it makes a certain amount of sense, considering that mathematics is based on axioms and definitions. As for why they are defined that way in particular, there are a number of different reasons, depending on which aspect of your question we look at.

Expressions such as "+1+" don't make sense because addition is a binary operator. You need two things to add together for addition to make sense. Note that in informal contexts, these can be implied, such as when "+1" in a comment means that one vote has been added to an answer's total. Some operators have multiple meanings, depending on context, like "$-$" which can mean subtraction (a binary operator) or negation (unary).

As to why we put the symbols in the order that we do (by which I mean infix notation, putting operators between their operands), I suspect that has more to do with language and how we think than with logic. There are notational systems that order things differently, like Polish and Reverse Polish (prefix and postfix) notations. Postfix notation puts operators after the operands, so "$1+2$" would be written "$1\ 2 +$," and "$3\times (5-1)$" would be "$5\ 1-\ 3\times$". This form makes sense logically and is easy to process with a computer, but it is harder to understand intuitively than "normal" infix notation.

All in all, you should remember that for most of history, mathematics was not developed particularly formally by modern standards. People were doing arithmetic for millenia before we even had a rigorous definition of the natural numbers. People thought, as you did and many others still do, that "$1+1=2$" is fundamental and obvious, and built on things like that. It wasn't until relatively recently that people went back and formally proved them. So, historically, the reason that we write "$1+1=2$" (or "one and one is two") is that, as you said, it seems natural and obvious.