PHL and Group Theory Midterm Re-Hash

I was reunited with two midterms today, and what glorious reunions they were! The first was PHL, Methods and Concepts. Here are some of the things that I missed, and some things that the prof liked in particular:

1. A regress is a fallacy in which the logic of the argument requires the existence of a prior, or meta, object, which, in turn, requires the further existence of a prior or meta object, ad infinitum. Then, there is no stable, logical truth for the argument to be based on, so it cannot prove anything. Apparently, this definition, which I put forward in my exam, was not satisfactory. I believe that this is because of the ambiguity of the phrase "stable, logical truth for the argument to stand on," which employs the metaphors of stability and standing. However, the definition itself is correct, just slightly unclear.
   
2. The prof apparently enjoyed my comments about the masked man fallacy: "Are knowledge, beliefs, and perceptions of an object essential qualities of an object?" This is the key of the masked man fallacy; in fact, knowledge, beliefs, and perceptions are not essential qualities. As I wrote in my exam, "an object is identical to another object regardless if I know it is. An object's identity or essence is not related to how that object is perceived." I think I rocked the masked man fallacy.
   
3. Category mistakes were the worst of it. A category mistake is a fallacy in which an object is taken to mean something other than what it actually means; more formally, it is a fallacy in which an object is attributed qualities that cannot be applied to it. The category mistake in the statement, "The average employee of Golman Sachs made $563447 last year", then, is this: the average employee is not an actual employee, there to greet you at the door, nor is it some ghostly spirit-employee. It is a mathematical construction based on statistics. Simple as that.
   

Very pleased with how that turned out. I'm going to post some ruminations about my upcoming paper here in a bit: Mathematical Truths as A Priori Analytic Truths.

The second exam I got back today was Group Theory. I anticipated a low-mid A... and I was not disappointed in my assessment of my work. Again, here are the problems in which I lost points:

1. I lost one point because I just didn't read the problem fully and forgot to determine whether the permutation A was even or odd. A=(29)(24)(13)(17)(15). Because there are an odd number of 2-cycles, A is an odd permutation. Pretty easy, just forgot it. I think he was pretty lenient in only taking off one point, for which I am thankful.
   
2. I made a leap-o'-logic in the one-to-one part of the following proof. Let G be a finite Abelian group of order 12. Claim: the mapping P: G->G, defined by P(x)=x^5, is an automorphism. Pf: P is an automorphism iff P is one-to-one, onto, and operation preserving. One-to-one: Let a,b be in G and P(a)=P(b). Then, a^5=b^5. This implies that (a^5)(b^-5)=e=(ab^-1)^5. Then, the order of (ab^-1) must be 5; however, by Lagrange's Theorem, the order of the group (12, in this case) is divisible by the order of the element. Because 5 does not divide 12, the order of (ab^-1) cannot be 5. The only other option, then, is that (ab^-1)=e, implying that a=b. So P is one-to-one. Onto: Since G is finite and one-to-one, then it is onto as well. [Note: I did a little proof of onto-ness, but I really didn't need to; I'll keep this little fact in mind for next time.] Operation Preserving: P(ab)=(ab)^5=(a^5)(b^5)=P(a)P(b). So P is operation preserving. So, P is an isomorphism.
   
3. Question: Are the groups Z(2)#Z(5) and Z(10) isomorphic? If so, provide an isomorphism, If not, explain why. Solution: Yes, Z(2)#Z(5) and Z(10) are isomorphic. To see this, note that they have the same numbers of each element. [Here's where I messed up; I miscalculated the order of the element (0,3) in Z(2)#Z(5) as 10 instead of 5.] Because they are isomorphic, a generator maps to a generator. So, P((1,1))=1, since (1,1) and 1 are generators of Z(2)#Z(5) and Z(10), respectively. Then, a possible isomorphism is P(n(1,1))=n.
   

Overall, I'm very pleased with how I did on this exam. I shouldn't have miscalculated the order of (0,3), but, even if I had decided that they were isomorphic, I'm not sure I could have figured out that isomorphism in the allotted time. Maybe I'm not giving myself enough credit. As for the other missed points... Well, the first was a stupid mistake and the second was a problem in my understanding of one-to-one proofs, and it has been rectified.
Ok, time to send a few emails, then I'll be back to write about my PHL paper.