Math philosophy:about arithmetic operations and equality

Question

This is some philosophic stuff about arithmetical operations that bothers me in some sense. During first grades, pupils are taught to perceive the arithmetical operations as "a process with a result". For , let's say, a $8-9-10$ year old, $3+4=7$ means exactly "when we perform the operation of addition between numbers $3$ and $4$ , we get the result 7". And "=" means "that is the result of the operation". So this way of thinking about "3+4=7" is very natural, intuitive and accesible to lower ages. Ultimately, this is what we do when we use addition in practice, for example when we compute how much we have to pay at the store: we make a process of adition of the prices of the objects bought and get a result, and we pay that result. So, everytime a child sees $"4+5= "$ he means "compute, find the result" of the additon. Every operation has a result. So this is the way we see operations and equality in primary grades. In fact, at this primary grades age, a child may be surprised if he sees something like $"7=3+4"$, because this cannot be interpreted in the "operation-process-result" as "7 evaluates to 3+4" .

So this is the most simple "interpretation" of "3+4=7", in fact it is the first interpretation we work with in school, and is very very tied to practice. But, for formal mathematics, this interpretation is not that satisfactory, for a number of reasons: 1) First of all, how do we define in rigorous/axiomatic/formal math the term of "process" and the term of "result"? It is not that simple i think, if possible. For example, the addition 3+4 has a result of 7. 2) Equality in formal math cannot have the meaning "evaluates to/find the result". In formal math, 3+4=7 must mean one and only thing :that, "3+4" and "7" are the same mathematical concept. So, "3+4" and "7" are two "names" for the same mathematical abstraction. But then we can ask ourselves another question "what's a mathematical name"? Can we rigorously formalize what a "mathematical name" is? So we have two kinds of stuff in mathematics: -mathematical abstract concepts/objects, such as numbers, sets, functions, etc (yes, ultimately they are all just sets...) -mathematical "names" for those abstract concepts. So i'm interested about these "mathematical names". In fact, a very big part of mathematics focuses just on this: giving names to abstract concepts and then finding if these names refer to the same abstract object or not. Solving equations means exactly this: finding for which values of the unknown, the LHS and RHS of the equations are just two different names for the same number.

3) I think that, for the primary grades kid, the construction "3+4" does not even represent/denote a number! At all! It denotes only a process, and only the result, in our case, "7" is a number. "3+4" is just the description of the process. So, 3,4,7 are numbers, but "3+4" is not a number , it is the description of a process. But,of course this is unacceptable for formal math! In formal math "3+4" is , in fact, a number, even though we don't "compute the result". The second we have written/mentioned it, it is a well-respected number, and we can work with it without the need to "compute the result". Or don't want to. Or simply cannot compute the result, as in the case of variables. "a+b" is of course a real number, whatever a and b are, even though we cannot compute "the result" as long as we don't know the particular values of a and b. But even if we know them, it can be hard to compute the result, as in the case of irrational numbers. In formal/higher math we work with sums without bothering what the "result" is.

So what do you think about this "conflict" between intuitive, practical view of an operation and the formal interpretation?

Answer 1

Comment

You are right: in First-order theory of arithmetic (see : Peano axioms) a term can be :

a variable, a constant ($0$) or a "complex" term built up with functional symbols : $S, +, \times$.

Some examples of arithmetical terms :

$0, S(0), S(x), S(0)+0, \ldots$

A term acts as a "name" for a number. We define :

$1=S(0), 2=S(1)=S(S(0)), \ldots$

With these definitions and the axioms for sum :

for all $n, \ n+0=n$

for all $n$ and $m, \ n+S(m)= S(n+m)$

we can prove arithmetical theorems :

$1+1=S(0)+S(0)=S(S(0)+0)=S(S(0))=2$.

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With the above example we can try to summarize some answers :

In formal math, $1+1=2$ must mean one and only thing : that, "$1+1$" and "$2$" are two names for the same mathematical (abstract) object (and we can rigorously formalize what a "mathematical name" is).

At the same time, the formalized language of arithmetic can formalize also the "process" of evaluating an equality (an expression with arithmetical terms) in terms of "find the result", i.e. performing an arithmetical computation.

Answer 2

You have solved a mystery for me. I teach mathematics in a university, to bright, motivated students. They almost all have great difficulty in understanding the concept of equality, and in substituting expressions.

For instance, if you write $f(z) = \frac{e^{iz} - e^{-iz}}{2i}$ and then ask them what is $f(3+4i)$, they often don't understand that you mean to ask the value of $\frac{e^{i(3+4i)} - e^{-i(3+4i)}}{2i}$.

They also use the equals sign incorrectly as punctuation, to mean "therefore I will compute..." For example, they might write $f(z) = z^2 + 2z = f'(z) = 2z + 2$.

I find I have to train them to use the equals sign whenever two quantities have the same value, and not in any other situation.

The students are very intelligent, and capable of understanding in some depth the most advanced mathematics that we teach them, yet they cannot manipulate mathematical expressions in the way that all mathematicians beyond high school level need to.

Thanks to your post, I now know that it is the fault of Math educators at lower levels, who have misunderstood mathematical notation, and codified their misunderstanding as part of a system which they are teaching to students. It seems that in order to preserve their empire and to create jobs for themselves, they have invented a parallel formalisation of mathematics which exists only in high school, and is of no use in preparing students for further study.

Am I being too judgemental here ?

Answer 3

There's also a concept of 'simplicity' involved here, I think. From the purely formal point of view, it suffices to know (or be able to prove) that, for example, the operation '$+$' sends two natural numbers to a third natural number, as is the case with $3$ and $4$, which are 'sent' by the '$+$' operation to $7$. By using some properties of such an operation, you can find out more about the way such an operation acts on the natural numbers -- for example, you can show that if you perform $k + m$ and $l + m$, and you know that $k > l$, then it follows that the result of '$k+m$' (whatever it may be) is larger than the result of '$l+m$'. This in turn can be used to find out more about the properties of the set of objects it acts on, in this case the natural numbers: think about how the study of the 'multiplication' operation quickly leads to the classification of prime numbers.

My point is that the actual 'answer' is not really central in this approach, only that it exists and has certain properties. Now, if you write '$3+4=7$', this really gives an answer to the question 'what is the result of addition applied to the concrete numbers $3$ and $4$?'. This might seem somewhat besides the point, but consider the famous Euler's identity \begin{equation} e^{i\,\pi} = -1. \end{equation} Here, we have the operations 'multiplication' and 'taking the power of', applied to a combination of two transcendental numbers and the imaginary unit. From formal theory on these operations on the complex numbers, you know that all this is allowed and produces another complex number. What's really surprising, is that this number is a) real and even b) an integer!

To my mind, the insight gained by studying the result of these operations performed in this particular way on the complex numbers $i$, $\pi$ and $e$, is the fact that the result is particularly 'simple', in terms of complexity of the underlying (algebraic) field. Of course, this again provides insight in the structure of $\mathbb{C}$ and the relations between $i$, $e$ and $\pi$, but the beauty of Euler's identity is its simplicity.