When people say “2+2=4”, what do they mean? Well, “=” is a standard symbol in logic (IIRC it can be derived from purely syntactical rules). But 2 and 4 and + aren’t standard; they are defined as part of the model you’re working with. For instance, if your model is boolean algebra, there are no 2 or 4, there are only ‘true’ and ‘false’, and “2+2=4” isn’t valid or invalid, it’s syntactically meaningless.
Depending on the model you choose, the sentence “2+2=4” may be true (as for the Peano integers), or undefined (as for boolean algebra), or even false (exchange the usual meanings of the strings “2“ and “4”). The latter case would be human-perverse but mathematically-sound. But once you choose a a model, every sentence—including “2+2=4”—is either a logical truth (is valid) or it is not.
Now, there are some standard (in the sense of widely used) models where 2 and 4 and + are used as symbols. These include the integers, the real numbers, etc. Usually when people say “2+2=4”, they are thinking of one of these. And in these standard models, “2+2=4″ is indeed a logical truth.
So given a model, we are not assuming 2+2=4. Our choice of a standard model dictates that “2+2=4” is true, without extra assumptions. A different model might dictate that “2+2=4″ is false, or undefined.
But we are relying on our shared assumption about what model we’re talking about—which is a different kind of assumption; it’s not about whether “2+2=4” is true (valid), but about what the string means—how to read it. It’s similar to the assumption that we’re speaking English, rather than a different language in which all the words just happen to mean something else.
When people say “2+2=4”, what do they mean? Well, “=” is a standard symbol in logic (IIRC it can be derived from purely syntactical rules). But 2 and 4 and + aren’t standard; they are defined as part of the model you’re working with. For instance, if your model is boolean algebra, there are no 2 or 4, there are only ‘true’ and ‘false’, and “2+2=4” isn’t valid or invalid, it’s syntactically meaningless.
Depending on the model you choose, the sentence “2+2=4” may be true (as for the Peano integers), or undefined (as for boolean algebra), or even false (exchange the usual meanings of the strings “2“ and “4”). The latter case would be human-perverse but mathematically-sound. But once you choose a a model, every sentence—including “2+2=4”—is either a logical truth (is valid) or it is not.
Now, there are some standard (in the sense of widely used) models where 2 and 4 and + are used as symbols. These include the integers, the real numbers, etc. Usually when people say “2+2=4”, they are thinking of one of these. And in these standard models, “2+2=4″ is indeed a logical truth.
So given a model, we are not assuming 2+2=4. Our choice of a standard model dictates that “2+2=4” is true, without extra assumptions. A different model might dictate that “2+2=4″ is false, or undefined.
But we are relying on our shared assumption about what model we’re talking about—which is a different kind of assumption; it’s not about whether “2+2=4” is true (valid), but about what the string means—how to read it. It’s similar to the assumption that we’re speaking English, rather than a different language in which all the words just happen to mean something else.