Mathematics

Our professor today told us that the expression "7 + 5" is a single entity and a number, just like 12, and not an operation or otherwise importantly different from 12. The context was an attempt to understand Plato's aviary analogy in Theaetetus, where our professor tried to have us imagine one bird being the "7 + 5" bird and two others being the "11" and "12" birds. This seems bizarre; while 12 is obviously the result of 7 + 5, it seems that saying they are the same is like saying a cake is the same thing as its recipe. So which is it? Is a simple mathematical equation like 7 + 5 identical to its result, or is it a different kind of thing where the similarity lies only in the numeric value the two have?

Comments

Perhaps it will help to distinguish between what "7+5" refers to and how it does the referring. The expressions "7+5," "8+4,", "2x6," "36/3" and countless others all refer to the number 12. (Though not everyone agrees that there really are numbers, we'll set that issue aside here.) But they do it in different ways. Compare:

"The 44th President of the United States is Barack Obama"

This is true, and it's true because "The 42nd President of the United States" refers to the same person as "Barack Obama." Barack Obama is the same person as the 42nd President of the United States, just as the number 12 is the same number as 7+5. (Of course, the process of adding two numbers is not a number, but "7+5 = 12" doesn't say it is.)

The sense of confusion here comes from the fact that there can be more to the meaning of a referring expression than just what it refers to. The description "The 42nd President of the United States" refers to Barack Obama, as does the description "The first African American President of the United States." But the two descriptions don't have the same meaning; meaning isn't exhausted by reference.

That said, the arithmetic case raises some more complicated issues. The meaning of "The first African American President of the United States" doesn't guarantee that it refers to the same person as "The 44th President of the United States." But the meaning of "7+5" does guarantee that it refers to the same thing as "12." Just how to account for this while taking account of the fact that there is an apparent difference in meaning between the two expressions is something I'll leave to those who work on such issues. But whatever the best answer, what "7+5+ refers to is the same number that "12" refers to.

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