It is often said that in mathematical equations, such as ‘2+2=4’, the symbol ‘=’ represents the fact that on both sides of the equation there is the same thing. That’s wrong: on the one side there are three symbols, while on the other there is just one.

Even in the equation ‘1=1’ the same-thing-on-both-sides-reading is wrong, because there there are two separate things (the ‘1’ on the left and the ‘1’ on the right).

So how should we understand the ‘=’-symbol? It might seem reasonable to say that in the equation ‘2+2=4’ we should read ‘=’ as meaning that the things that the numbers refer to are the same (in number). If the numbers in the equation refer to apples, for example, then the equation merely states that if you add two apples to two apples you end up with four apples. The ‘=’-symbol tells you that two apples added to two apples is the same as four apples.

But this view is problematic. Mathematics is usually understood as a language or formalism that is independent of reality. Mathematics is, to put it disrespectfully, merely a bunch of tautologies, whose truth depends on the definitions that we ourselves choose. We can of course apply mathematical language/formalism to things we see, but then we are no longer doing mathematics – we’ve turned into physicists.

We can’t just say that the number of apples referred to by the symbols are the same, because mathematical terms have no meaning outside of mathematics.

Philosopher of physics at Amsterdam University College and Utrecht University, managing editor for Foundations of Physics and international paraclimbing athlete

Apropos your interesting apples, it seems to me that your ‘=’ is a relation between apples that have been so abstracted that each is reduced to the number 1. And certainly, my mother would not buy such a thing. Rather, knowing that no two physical objects are the same, her focus would be on such properties as mass, color, odor, freshness, firmness, etc — before considering the cost — with this free advice to all: it is essential to distinguish carefully between concrete and abstract objects.

She (like me) would also compliment you on your question! But she would also point out that the mathematical terms in her bank account DO have a meaning outside of mathematics.

I don’t really understand the problem here. We assume in set theory that we have a notion of equality: We can determine when two elements of a given set are equal. I’m not a set theorist, but I think we have a basic rules for this (It is an equivalence relation for example: a=b and b=c implies that a=c etc.). We have a function (addition) +:N\times N\rightarrow N, that takes as input two natural numbers and outputs their sum. We write a+b for +(a,b). Then the question if a+b=c is the equality of the elements of the set N. This has nothing to do with the expression “a+b=c”.

Dear Fedde,

Apropos your interesting apples, it seems to me that your ‘=’ is a relation between apples that have been so abstracted that each is reduced to the number 1. And certainly, my mother would not buy such a thing. Rather, knowing that no two physical objects are the same, her focus would be on such properties as mass, color, odor, freshness, firmness, etc — before considering the cost — with this free advice to all: it is essential to distinguish carefully between concrete and abstract objects.

She (like me) would also compliment you on your question! But she would also point out that the mathematical terms in her bank account DO have a meaning outside of mathematics.

With best regards; G

I don’t really understand the problem here. We assume in set theory that we have a notion of equality: We can determine when two elements of a given set are equal. I’m not a set theorist, but I think we have a basic rules for this (It is an equivalence relation for example: a=b and b=c implies that a=c etc.). We have a function (addition) +:N\times N\rightarrow N, that takes as input two natural numbers and outputs their sum. We write a+b for +(a,b). Then the question if a+b=c is the equality of the elements of the set N. This has nothing to do with the expression “a+b=c”.