Good Morning, Proof!

Here is the third proof (or, rather, set of proofs) I'll be expected to know.
Properties of Elements and Subgroups Under Homomorphisms
Let G and G' be groups, where P maps G into G', and H is a subgroup of G.

1. Claim: P maps the identity of G to the identity of G'.
   Pf: This proof is the same as the proof for property 1 of the previous set of proofs. But I'll do it again for practice. P(e)=P(ee)=P(e)P(e), and e'P(e)=P(e). So, e'P(e)=P(e)P(e), implying that e'=P(e). qed
2. Claim: P(g^n)=[P(g)]^n.
   Pf: Again this is the same as property 2 of the previous set of properties. If n is positive, then the claim follows by mathematical induction and by the definition of homomorphisms. If n=0, then the claim follows by the previous property. If n is negative, then -n is positive. Then, P(e)=P((g^n)(g^-n))=P(g^n)P(g^-n)=P(g^n)[P(g)]^-n. Then, P(g^n)=[P(g)]^n. qed
3. Claim: If |g| is finite, then |P(g)| divides |g|.
   Pf: Let |g|=n. Then, g^n=e. Then, e=P(e)=P(g^n)=[P(g)]^n. Then, by Thm 4.2, |P(g)| divides n=|g|. qed
4. Claim: KerP is a subgroup of G.
   Pf: By property 1, KerP is nonempty. Let a,b be in KerP. Then, P(a)=P(b)=e. Then, P(ab^-1)=P(a)P(b^-1)=P(a)[P(b)]^-1=ee^-1=e. Then, ab^-1 is in KerP, so KerP is a subgroup of G. qed
5. Claim: P(a)=P(b) iff aKerP=bKerP.
   Pf: I will prove the statement "If P(a)=P(b), then aKerP=bKerP" first. Assume that P(a)=P(b). Then, e=P(a)[P(b)]^-1=P(a)P(b^-1)=P(ab^-1), which is in KerP. By property 5 of the Lemma of Cosets, iff ab^-1 is an element of a subgroup H, then aH=bH. Then, the claim follows. Now, conversely, assume that aKerP=bKerP. Then, by the same property of the Lemma of Cosets, P(ab^-1) is in KerP, and so on backwards to P(a)=P(b). qed

There is a bunch more, but I think I get the general idea. Besides, I'll be asked to prove one of several, and I doubt I'd choose this one. I'll probably choose the set of properties in the last posting. Ok, back to PHL.

Another Group Proof

Need a breather from PHL, so here's another group theory proof.
Properties of Isomorphisms Acting on Elements and Subgroups
Let G and G' be groups, and let P map G to G'. Let H be a subgroup G.

1. Claim: P maps the identity of G to the identity of G'
   Pf: Because e=ee, P(e)=P(ee)=P(e)P(e). Also, because P(e) is in G', P(e)=e'P(e). Then, e'P(e)=P(e)P(e) --> e'=P(e). qed
2. Claim: For every integer n and for every group element a in G, P(a^n)=[P(a)]^n.
   Pf: By definition of isomorphisms and by mathematical induction, if n is positive, then the claim is true. If n=0, P(a^n)=P(a^0)=P(e) and [P(a)]^n=[P(a)]^0=e. If n is negative, then -n is positive, and we have from the first claim and mathematical induction that e=P(e)=P([g^n][g^-n])=P(g^n)P(g^-n)=P(g^n)[P(g)]^-n =e. Then, P(g^n)=[P(g)]^-(-n)=P(g)^n. qed
3. Claim: For any elements a and b in G, a and b commute iff P(a) and P(b) commute.
   Pf: I will prove the claim "P(a) and P(b) commute if a and b commute" first. Assume that a and b commute. Then, ab=ba. P(ab)=P(a)P(b) and P(ab)=P(ba)=P(b)P(a), so P(a)P(b)=P(b)P(a). Now I will prove the converse. Assume that P(a) and P(b) commute. Then, P(a)P(b)=P(b)P(a). Then, P(a)P(b)=P(ab)=P(b)P(a)=P(ba), so P(ab)=P(ba). Because P is one-to-one, ab=ba. qed
4. Claim: G='a' iff G'='P(a)'.
   Pf: I will prove the claim "G'='P(a)'" first. Let G='a'; then, by closure, 'P(a)' is a subset of G'. Because P is onto, for any element b in G', there is an element a^k in G such that P(a^k)=b. Thus, b=[P(a)]^k, so b is in 'P(a)'. Then, G'='P(a)'. Now I will prove the converse. Suppose that G'='P(a)'. 'a' is a subset of G (by closure, etc.). For any b in G, we have P(b) is an element of 'P(a)'. Then, for some integer k, we have P(b)=[P(a)]^k=P(a^k). Because P is one-to-one, b=a^k. Then, G='a'. qed
5. Claim: Isomorphisms preserve orders.
   Pf: Let g be in G, where |g| is n. Then, P(g^n)=P(e)=e and P(g^n)=[P(g)]^n. So, [P(g)]^n=e. So, |P(g)|=|g|. qed
6. Claim: If G is finite, then G and G' have exactly the same number of elements of every order.
   Pf: By property 5, elements of order n map to elements of order n. Then, G and G' must have the same number of elements of each order. qed
7. Claim: G is Abelian iff G' is Abelian.
   Pf: By property 3, this property follows.
   
8. Claim: G is cyclic iff G' is cyclic.
   Pf: By property 4, this property follows.
   
9. If K is a subgroup of G, then P(K) is a subgroup of G'.
   Pf: Let a,b be in K, and P(a)=c and P(b)=d. I need to show that cd^-1 is in P(K). By definition, c and d are in P(K). P(ab^-1)=P(a)P(b^-1)=P(a)[P(b)]^-1=cd^-1, which is in P(K) by closure.
   

That was fun. Now, back to PHL.

Here's a proof to kick off my monster study sess

So I'm supposed to be studying crazy hard for this PHL final, but I'm not really feelin' it. So I'm going to spend a little time doing a proof or two for Group first, to get me motivated. Here goes:
The Fundamental Theorem of Cyclic Subgroups
Claim: Every subgruop of a cyclic group is cyclic. Moreover, if |'a'|=n, then the order of any subgroup of 'a' is a divisor of n; and, for each positive divisor k of n, the group 'a' has exactly one subgroup of order k -- namely, 'a^(n/k)'.
Pf: Let G='a' and suppose that H a subgroup of G.
To prove the first claim (that every subgroup of a cyclic group is itself cyclic), we need to show that H is cyclic. Assume that H does not equal just the identity e. I claim that H contains an element of the form a^t, where t is positive. Since G='a', every element of H has the form a^t; and, when a^t belongs to H with t<0, h="'a^m'." g="'a'," b="a^k" k="mq+r," k="a^(mq+r)="(a^mq)(a^r)-">a^r=(a^-mq)(a^k); since a^k and a^-mq are both in H, a^r is in H. However, m is the least positive integer such that a^m is in H, but r is less than m and greater than or equal to 0. So, r=0, implying that b=a^k=a^-mq=(a^m)^q which is in 'a^m'. Then, the arbitrary element b is in the cyclic subgroup of H 'a^m', so H='a^m', so H is cyclic. This proves the first claim.
To prove the second claim (if |'a'|=n, then the order of any of 'a' will be a divisor of n), suppose that |'a'|=n and H is a subgroup of 'a'. H is cyclic, so let H='a^m', where m is the least positive integer such that a^m is in H. Let b=a^mq again, such that a^mq=e. Then, e=a^mq=b=a^n, implying that mq=n. So, the order of the subgroups of a cyclic group divide the order of the group.
To prove the third and final claim (for each positive divisor k of n, the group 'a' has exactly one subgroup of order k, 'a^(n/k)'), let k be any positive divisor of n. I need to show that 'a^(n/k)' is the one and only subgroup of 'a' of order k. |'a^(n/k)'|=n/gcd(n,n/k)=k. Now, let H be any subgroup of 'a' of order k. I know that H='a^m', where m divides n. Then, m=gcd(n, m) and k=|a^m|=|a^gcd(n,m)|=n/gcd(n,m)=n/m. Thus, m=n/k, and H='a^(n/k)'.
qed

That was pretty confusing. I need to work on that one.
Ok, PHL time. I'll publish another one in a bit.

Three Problems from Discrete

Here are three problems from the review sheet for Discrete Math. They are all counting problems. I'll post other types, too, but these ones are the trickiest, personally.

1. Question: In how many ways can 12 (identical) apples be distributed among 5 (distinct) children so that no child gets more than 7 apples?
   Solution: This is a "bins" problem. Let's look at the total number of ways of distributing the apples, without the qualifier. There are 12 apples and 5 children; thinking of the children as apple-receptacles (bins), there are 4 dividers between the 5 bins. So, there are 12+4=16 total objects -- 12 items and 4 dividers -- so I need to decide where to put 12 apples OR 4 dividers in a total of 16 objects. Then, the total number of ways to distribute the apples is 16C12=16C4. However, this includes the possibility that one kid will have at least 7 apples. Let's now assume that one kid does indeed at least 7 apples; giving him or her 7 apples off the bat reduces our number of objects from 16 to 9. So, there are 9C5=9C4 ways of distributing the remaining 5 apples. However, there are 5 different kids, so each of these combinations can occur 5 different times. So, there are really 5(9C5) ways of distributing the remaining 5 apples. Of course, we want the situation in which this DOESN'T happen, so the real solution is the total distributions minus the unwanted distributions, or 16C4-d(9C4).
   
2. Question: How many arrangements of all of the letters of JUPITER are there with the vowels occurring in alphabetical order (but not necessarily consecutively)?
   Solution: Let's focus just on the vowels first. There are three vowels, and they need to be in the order EIU. There are seven spaces total, so there are 7C3 ways to distribute the vowels; of course, if there were no alphabetical qualifier, then there would be 3(7C3) ways to distribute the vowels, but, since they MUST go in a certain order, there is really 1(7C3) ways. Now, we have 4 remaining letters, each of which must be used once. So, there are 4! ways to distribute them. Then, there are 4!(7C3) possible arrangements.
   
3. Question: At a birthday party, 36 identical balloons are distributed to 8 distinct children. If the balloons are distributed randomly, what is the probability that one child will get 20 or more balloons?
   Solution: This is a probability question, so I'm going to do it in 2 (or 3) steps. First, I'm going to figure out what the denominator will be. In this case, it will be the total ways of distributing the 36 balloons. Again, this is a bins problem, and there are 36 items and 8 bins (with 7 dividers), so there are 43 objects. Then, the total number of distributions is 43C7. This is the denominator. However, we want the ways in which the one kid gets 20 or more balloons. This will be the numerator. Let's give one kid 20 balloons, just give 'em away. Then, there are 23 objects left (36-20+8-1=23). So, there are 23C7 distributions; again, however, there are 8 kids, so the numerator is 8(23C7). Then, the probability is 8(23C7)/(43C7).
   

These are just my answers. I'll edit in corrections when the answers are posted. Tomorrow (or maybe later today), expect some problems concerning recurrence relations, functions, and equivalence relations. For now, I'm off to study PHL.

Group Theory Final Study Guide 1

So, I'll be posting study guides for Group Theory and PHL in chunks over the next few days. I'll also be posting some problems from the Discrete Math review sheet. What can I say? It's crunch time, folks...
I'll be expected to know the definitions of the following terms for my Group Theory final:

1. Group
2. Order of an element
3. Factor group
4. Group homomorphism
5. Kernel of a homomorphism

Ok, here goes. Keep in mind, this is from memory as much as possible.

1. Group: A group G is a set with a binary operation that has the following characteristics: 1) It is transitive; that is, for all a,b,c in G, (ab)c=a(bc). 2) There is an identity element e, such that, for all a in G, ae=ea=a. 3) There are inverses; for all all a in G, there is an element b in G such that ab=ba=e.
   
2. Order of an element: In a group G with element g, the order of g is the smallest positive integer n such that g^n=e.
   
3. Factor group: Define a group G with a normal subgroup H. The factor group G/H is defined as G/H={aH| a in G}, under the operation (aH)(bH)=abH. What does this mean? It means that G/H is the set of all (distinct) cosets of H in G, under the specified operation. It should be noted that the orders |aH| and |a| may not be equal. In fact, even though H contains e, the identity of G/H is NOT e, but, rather H (specifically, it is eH). Further, note that (aH)^2=(a^2)H, NOT (a^2)(H^2). Just remember that G/H is a set of (distinct!) cosets of H in G under the operation aH*bH=abH. And breath deep.
   
4. Homomorphism: A homomorphism P on a group G is a mapping from G into G' that preserves operation; that is, for all a,b in G, P(ab)=P(a)P(b). A few things to note: P maps INTO not ONTO; rather, not necessarily onto.
   
5. Kernel of a homomorphism: Define a group G and a homomorphism P that maps G to G' with the identity element e. Then, the kernel of P is KerP={x in G| P(x)=e}; that is, the kernel is the set of all elements in G that are mapped to the identity of G' by P.
   

Here are some things to remember about homomorphisms:

1. P carries the identity of G to G'.
   
2. P(g^n)=(P(g))^n for all n in Z. This means that...
   
3. If |g| is finite, then |P(g)| divides |g|.
   
4. KerP is a subgroup of G.
   
5. P(a)=P(b) iff aKerP=bKerP. I'll come back to this one, I think...
   
6. If P(g)=g', then Pinv(g')={x in G| P(x)=g'}=gKerP. This means that Pinv(g') is a SET, not necessarily just one element. Also, note that P(KerP)=e and P(gKerP)=g'. Interesting...
   
7. If H is a subgroup of G, then P(H) is a subgroup of G'.
   
8. If H is cylic, then P(H) is cylic.
   
9. If H is Abelian, then P(H) is Abelian.
   
10. If H is normal, then P(H) is normal.
    
11. The order of P(H) divides the order of H.
    
12. KerP is always normal to G.
    
13. If P is onto and KerP={e}, then P is an isomorphism from G onto G'.
    

Here are some things to remember about factor groups:

1. If G/Z(G) is cyclic, then G is Abelian.
   
2. G/Z(G) is isomorphic to Inn(G).
   
3. Cauchy's Theorem for Abelian Groups: Let G be a finite Abelian group and let p be a prime that divides the order of G. Then G has an element of order p.
   

That's all for the moment. The super important things to recall will be the properties of homomorphisms and Cauchy's Theorem. That's a lot of properties to recall about homomorphisms.
I'll post some proofs on Sunday, and some properties and examples and such on Monday or Tuesday. Also, expect some problems from Discrete later tonight.
Ciao.

Here's the final draft of my paper on mathematical statements as analytic statements. I'm very pleased with how it turned out. I didn't spend as much time on how identity statements are analytic as I would have liked, but I think that I did focus on things that were within the scope of a 4-ish page paper. Let me know what you think.

Mathematical Statements as Analytic

I claim that statements of mathematics are analytic. This challenges Kant’s view of mathematical statements. First, I will try to show why Kant’s reasoning may be contradictory. Then, taking each of his criteria independently, I will examine them in relation to mathematical statements. Finally, I will identify and address likely flaws in my arguments.

Allow me to point out a potential contradiction in Kant’s argument. Kant defines analytic and synthetic statements as follows: 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 does add new content to the concept of the subject (Ayer, 77). In the example “7+5=12”, he reasons that, because the reader can conceptualize “7+5” without conceptualizing “12” and vice versa, it must be synthetic. Finally, in the example, “all bodies take up space”, he argues that the truth of “all bodies take up space” is subject to the Law of Non-Contradiction (Ayer, 77). It would seem that, according to Kant, all three of these criteria are equivalent. Here, then, is a contradiction. Assume that “7+5=12” is synthetic by Kant’s reasoning. However, it cannot be true that “7+5=12" and 7+5=(not)12.” So, the truth of the statement is subject to the Law of Non-Contradiction. Then, it must be analytic. However, the statement cannot be both analytic and synthetic. So, if Kant is asserting that his three criteria are equivalent, then he appears to contradict himself (Ayer, 78). For the sake of this paper, I will consider all three of his criteria, in the hopes of showing that, regardless of which is chief among them (which is outside the scope of this paper), mathematical statements will be held analytic.

Let us first examine the psychological criterion: the subject and the predicate can be conceptualized independently in a synthetic statement and not independently in an analytic statement (B/F, 140). What does it mean to conceptualize objects such as “7+5” and “12”? I maintain that the concept of the complex object “7+5” is the union of the concepts of the objects “7”, “+”, and “5”. What, then, is the concept of the object “7”? I claim that we have no way of conceptualizing “7” without conceptualizing fundamental numbers and operations. Conceptualize an integer “n”. Now, conceptualize the integer immediately following “n”. Of course, you probably thought of “n+1”; then, in order to conceptualize any number, we must rely on at least addition and “1”. What about, the reader may protest, very large numbers, say 792? We have no way to conceptualize the object “792” as a value independent of its component values; when we think of “792”, we really think of “”7(10^2)+9(10^1)+2(10^0)”. That is to say, we cannot conceptualize any number without at the same time conceptualizing addition, multiplication, exponentiation, and the numbers “0”, “1”, and “10” (and division, for rational numbers, etc.). Then, by this argument, when we examine the statement “7+5=12”, the ideas of “7”, “+”, and “5” are all entailed in “12”, because we must conceptualize repeated addition of “1” for “7”, “5”, and “12”. It would appear, then, that the statement is analytic.

Next, let us examine the logical criterion: the relation between the subject and predicate is subject to the Law of Non-Contradiction in an analytic statement and not in a synthetic statement (Ayer, 77). Kant gives the statement “all bodies are heavy” as an example of a synthetic statement that fails this criterion, and “all bodies are extended” as an example of an analytic statement that fulfills this criterion. Certainly, it is possible for a body to be heavy and not heavy, whereas it is impossible for a body to take up space and not take up space. However, as I have pointed out, it is impossible by the definitions of the constituent symbols that “7+5=12” and “7+5=(not)12” are both true statements. Then, the statement would again seem to be analytic.

Finally, we come to Kant’s original 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 does add new content to the concept of the subject. This is a criterion of neither psychology nor logic, but, rather, of semantics (B/F, 140). That is to say, what the reader knows is not relevant to this distinction; what are relevant are the definitions attributed to the subject and predicate. Is there content in the predicate that is not present in the subject? What exactly does Kant mean by “content”? I hold that “content” refers to the set of properties attributed to an object. Then, all “bodies” have the property “being extended” but not “being heavy.” How does this help us make sense of the contents of “7+5” and “12”? What sort of property could be attributed to “12” that is not attributed to “7+5” or vice versa? The canny reader might say that “7+5” has the property of being two numbers added together, whereas “12” is only one number. However, since “12=7+5”, “12” has the property of being two numbers added together, as well; similarly, “7+5” has the property of having one value, namely, “12”. I claim that, because “7+5” and “12” are related through an identity, no new content can possibly be added to the subject by the predicate. The set of properties of “7+5” is the same as the set of properties of “12”, so “12” can attribute to “7+5” no new properties. Observe that this is without regard to what the reader comprehends; the reader might be under the impression that “7+5=0”, but, still, by the definitions of the constituent symbols, no new content can possibly be added to “7+5” by “12”. And, again, the statement would appear to be analytic.

My reasoning is not without flaws. In the case of the psychological criterion, one might note that irrational numbers cannot possibly be conceptualized through fundamental numbers and operations, because they cannot be expressed as one integer over another non-zero integer. However, they can be expressed as an infinite sum of rational numbers, and, since (it seems to me) we cannot conceptualize “infinity,” we cannot conceptualize a number that is defined as an infinite sum of numbers. Then, we can only conceptualize approximations of irrational numbers. However, if this is true, is it also true that “pi=3.1415” since our concept of pi is 3.1415 (or more digits, depending on one’s accuracy)? Certainly not, regardless of to how many decimal places you extend the predicate. So, it seems that we have some concept of pi that exceeds the aid of other fundamental numbers and operations, puncturing a hole in my argument that all numbers must be conceptualized as such. In the case of the semantic criterion, it could be noted that, however much I want to eschew the reader, there must be one. A statement cannot be viewed purely in terms of its semantics; the reader always brings conceptualizations to it. We cannot escape psychology. I have little response to this attack. Let us assume that the statement “7+5=12” is presented to a person who does not understand the constituent symbols. By the argument for the semantic criterion, they need not be able to for the statement to be analytic, since the statement is true by definition of those symbols. However, this regresses: there must be someone who knows the definition of the symbols, someone who can conceptualize them, or else the statement would be without meaning. I admit that this is a dire flaw in my argument for the semantic criterion, and leads down epistemic paths beyond the scope of this paper. Finally, my claim stated, “Statements of mathematics are analytic,” yet I have only taken the example “7+5=12.” The mathematically-minded reader should observe that, in order to fully demonstrate the veracity of my claim, I would need to also examine statements of inequality and functions (especially functions that are not one-to-one or onto). However, this is again beyond the scope of this paper.

I believe that I have adequately showed how Kant’s reasoning leads to a contradiction if one assumes that his semantic, logical, and psychological criteria are equivalent. Then, taking them to be separate, I hope to have given the reader cause to believe that mathematical statements are analytic by each criterion. It is outside the range of this paper, however, to explore whether Kant is truly assuming the equivalence of his criteria, and, if he is not, to choose which shall hold priority in making the distinction between analytic and synthetic statements.

Works Cited

Ayer, Alfred Jules. Language, Truth, and Logic. New York: Dover Publications, Inc, 1946.

Baggini, Julian, and Peter S. Fosl. The Philosopher’s Toolkit. Malden: Blackwell Publishing, 2003.

Kant and Stuff

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...

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.