Pondering a Proof

[QED Stepfiles]

Originally Posted by devonin

The problem is that if you're only concerned with the actual statement 1*1=1, the simple expression of that as "one group of objects wherein each group contains one object is a set that contains one total object across all groups" is obvious not because of the nature of integers, not because of the way set theory works, but because of the lingusitic definitions of the words "one" and "group"

The only way this is complicated is if you somehow want to question the linguistic definitions involved.

This isn't really a concern of linguistics. We are not talking about "one" and "group" in a physical sense - indeed, the generalization of mathematics to something more tangible is a bit irrelevant here. Instead, we are treating "one" and "group" as mathematical concepts, and as purely ideal mathematical objects. You're making another assumption in that 1*1=1 is a simple expression that relates to some physical set of objects. There is still a major concern of whether or not such a leap is even allowed (and we have no reason to believe that it is not, but it still is a leap that should not just be treated so callously as to say it's "obvious").

And MrRubix, you can legitimately say that 1+1=1 if you define what you mean by "1." If, for example, I use 1 to refer to an equivalence class of integers modulo 1, then it is correct to say that 0=1=2=3.... etc. Of course, if we are purely working in the system of integers, then such a statement is impossible, but then again we have to axiomatically define the integers. And it certainly is not rigorously obvious to jump from the mathematical concept of "1" to having "one apple."

Once again, it's important to treat "1" as a purely mathematical object, or else this thread really doesn't have a point. We're not allowed to assume that "1" has any physical meaning attached to it unless we actually show that such a physical meaning is justified. As of yet, nobody has really posted anything without making a boatload of assumptions...

This really isn't an issue we can resolve within the space of one forum thread (indeed, it's been a hot topic in mathematics for much, much longer than this thread has been running). Still, I feel it's important to see what the issue is here. It's not a matter of linguistics, or physical intuition, but rather a matter of mathematics as a pure system, and whether or not that system can be defeated at ANY angle.

PS: Linguistics really isn't a great way to study mathematics. In fact, the English language has so many ambiguities in it that simply using the English language to define mathematical concepts is completely unrigorous. Take the following example, for instance (which I have taken from Munkres, Toplogy, 2nd Ed):

Compare the following two statements:

(1) "Miss Smith, if any student registered for this course has not taken a course in linear algebra, then he has taken a course in analysis."

(2) "Mr. Jones, if you get a grade below 70 on the final, you are going to flunk this course."

In statement (1), the logical flow is that if student A has not taken linear algebra, then student A has taken analysis. However, if student A has taken linear algebra, then he may or may not have taken a course in analysis.

In statement (2), the logical flow is that if Jones receives a grade less than 70, he will flunk the course. However, it is understood from context that if he does not get a grade below 70 on the final, he will not flunk the course. This is, in fact, the converse of the statement.

In other words, statement (1) reads "if P, then Q", whereas statement (2), albeit it also reads "if P, then Q," is understood to mean "if and only if P, then Q." Mathematics does not allow this. If a statement reads "If P, then Q," then the converse (if Q, then P) of the statement definitely does not have to hold true.

The bottom line is that we should really not try to analyze mathematics in the context of linguistics. There are a lot of logical problems about the English language, and many of them need to be fixed before we can even try to make sense of math.