So, I'm writing this paper for my philosophy class. It's about the distinction between analytic and synthetic statements. Specifically, I argue that all mathematical truths are analytic statements. Kant gives the following definitions. An analytic statement is a statement in which the predicate adds no new content to the concept of the subject. A synthetic statement is a statement in which the predicate adds new content to the concept of the subject. He also claims that the statement "7+5=12" is synthetic. That is, he claims that the concept of "12" is not entailed in the concept of "7+5". I would disagree for a few reasons (I know, bold, right? Disagreeing with Kant... sheesh), but there's a point I'd like to brainstorm a bit, instead. What does Kant mean by "concept of the subject"? Why didn't he just say "adds new content to the subject"? What is the difference between "the subject" and "the concept of the subject"?
After a lot of thought, I think that what Kant is saying is that "7+5" contains the concepts "7", "5", and "+", whereas "12" contains only the concept "12". Specifically, the idea of addition is not contained within "12", whereas it is in "7+5". I would disagree, and here's my argument: Conceptualize the integer n. Now, conceptualize the integer immediately following n. Of course, you probably conceptualized n+1. In doing so, you conceptualized addition. I would claim, then, that all numbers require the conceptualization of addition, multiplication, or exponentiation. For example, take the number 352. We use the decimal system, so we have ideas of "1", "10" and "100". ("Ah, a regress! How do we have those concepts?" you say, to which I politely reply, "See my inductive argument above for the concept of the number 10, and so on.") So, when we conceptualize "352", we really conceptualize "3*10^2+5*10^1+2*10^0". We have no concept of what "352" is without the concepts of addition, multiplication, and exponentiation. Or, at least, that's how it seems to me. So, in short, if Kant's reasoning for the synthetic status of "7+5=12" is that "7+5" contains addition and "12" does not, I would heartily disagree. Again, that's how it seems to me, that's how I look at numbers. I wouldn't know what 352 is without basic operations. For that matter, I wouldn't know what 7 is without the ideas of "1" and "addition". Maybe I'm crazy...
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I don't disagree with your point, but I think the "meaning" of 352 is entirely subjective. For example, if you had to endure 352 days without human interaction, that would probably seem like a very long time; if you were told you only had 352 days left to live, it would probably seem like a very short time. Similarly, if you found $352 on the ground it would seem like a lot of money; if you only had $352 to live on for the year it would seem like very little money! So is the concept of 352 (or any number for that matter) completely relative? Probably way off topic from your paper, but a fun thought nonetheless :)
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