Talk:Majoring in Mathematics
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This is an extremely rough outline for this new article. I will continue to expand it over the next few weeks based on feedback about the appropriateness of the problems and other suggestions for improvements. Please let me know what you think! --MarkGall 23:12, 4 October 2009 (EDT)
- well I was a math major in college many years ago (before Rubic's cube!) Suggest adding statistics.
- Ah, good idea. At my undergrad school mathematics and statistics were separate programs, but I guess this isn't typical. I'll throw in a new section. I'm not much for statistics -- can anyone suggest some good problems? --MarkGall 23:20, 4 October 2009 (EDT)
- well I was a math major in college many years ago (before Rubic's cube!) Suggest adding statistics.
Tremendous work, Mark. I'll try to add something for probability or statistics, which I studied.--Andy Schlafly 23:22, 4 October 2009 (EDT)
- Great, thanks. These are subjects in which I'm inexpert. Now I'm inclined to move Probability/Statistics to a "field" along with algebra, analysis, etc. Please add a nice problem or theorem if you have any in mind! --MarkGall 23:24, 4 October 2009 (EDT)
Terrific analysis on what college mathematics is not. That should be required reading for many seeking to major in math!--Andy Schlafly 22:30, 15 October 2009 (EDT)
Why is there no mention of the bible or theology in this article? It should be mentioned, as it is in Axiom_of_Infinity, that the notion of an infinite set is somewhat blasphemous. Tomkup32 10:58, 9 December 2009 (EST)
For Bezout's theorem, you have some inaccurate claims: the line x+y=0 and x+y=1 never interset. We need to move to projective space for the degree of the intersection to be exactly mn.
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Liberal bias
How can there be liberal bias in math? Does saying 1+1 = 2 have a motive?SusanP 23:29, 9 February 2012 (EST)
- College math is not completely immune to liberal influences that have destroyed other subjects like physics.--Andy Schlafly 23:37, 9 February 2012 (EST)
- Can you give an example of liberal bias in math? --Bogart12
- Possible liberal bias--the study of so-called "homomorphisms" in modern liberal algebra could be part of the gay agenda?--Bogart12
- No, sorry, that's absurd and I think you know it. "Homo" simply means same and is used in a wide variety of contexts that have nothing to do with liberal bias. Is homogenized milk liberal bias? Or homonyms? Homophones? I've already reverted your addition to the article once, so I won't do it again just yet, but please consider removing or modifying it yourself.--JustinD 19:32, 24 February 2012 (EST)
- There is a crucial difference between your examples and my claim. A homophone describes two words sounding the same but being different. Bat is not a homophone with bat, that doesn't make sense. Saying wind and wind are homophones acknowledges that they sound the same but are different words. A homomorphism allows groups that are different, like Z_2xZ_2 and the Klein four group, to be treated as equal. Additionally the global warming obsession is prolific. I didn't mean my writing to sound flippant, and perhaps its tone can be improved, but it was certainly not meant as a joke.
- I'm not sure I see the "crucial difference" you're seeing. I don't know what you mean by saying that homomorphisms "allow[] groups that are different . . . to be treated as equal." What do you mean by "treated as equal"? Obviously, if a homomorphism exists between two groups it shows that there are certain structural similarities between the two groups, but this hardly means that they are equal. The fact that 124 and 490 are both even shows that there are certain similarities between them (namely that each includes 2 in its prime factorization), but that doesn't mean we treat them as equal and it certainly isn't evidence that evenness is the result of some sort of liberal bias. Also, I think you have your examples backwards. Bat (animal) is a homophone with bat (sports) because both are pronounced the same but have different meanings. Wind (air flow) is not a homophone with wind (watches) because the pronunciation differs. They are, however, homographs. --JustinD 01:23, 26 February 2012 (EST)
- Haha you are absolutely right, my apologies. I thought I had picked a word specifically with no homophones. Allow me to elaborate: saying that 124 and 490 are both even emphasizes similarities which is fine. But consider a statement like "there is only one group of order 3." There are clearly many, because you can define one on any set with 3 elements. Nevertheless, you will never hear discussion of the many different groups of order 3, they are all lumped together because of homomorphisms. Compare this with the professor value of refusing to distinguish between men and women. To say men and women are both children of God is to discuss similarities while acknowledging differences, as in saying 124 and 490 are both even. To say men and women are basically the same, as universities today do, is consistent with saying that groups are essentially the same. I hope that clarifies what I was struggling to say before. --Bogart12
- They key point you are missing, however, is that in many, many subjects, such as in mathematical aptitude, there is no functional difference between male and female. So yes, in many situations, women and men are essentially the same.KenShomer 15:15, 26 February 2012 (EST)
- Haha you are absolutely right, my apologies. I thought I had picked a word specifically with no homophones. Allow me to elaborate: saying that 124 and 490 are both even emphasizes similarities which is fine. But consider a statement like "there is only one group of order 3." There are clearly many, because you can define one on any set with 3 elements. Nevertheless, you will never hear discussion of the many different groups of order 3, they are all lumped together because of homomorphisms. Compare this with the professor value of refusing to distinguish between men and women. To say men and women are both children of God is to discuss similarities while acknowledging differences, as in saying 124 and 490 are both even. To say men and women are basically the same, as universities today do, is consistent with saying that groups are essentially the same. I hope that clarifies what I was struggling to say before. --Bogart12
- I'm not sure I see the "crucial difference" you're seeing. I don't know what you mean by saying that homomorphisms "allow[] groups that are different . . . to be treated as equal." What do you mean by "treated as equal"? Obviously, if a homomorphism exists between two groups it shows that there are certain structural similarities between the two groups, but this hardly means that they are equal. The fact that 124 and 490 are both even shows that there are certain similarities between them (namely that each includes 2 in its prime factorization), but that doesn't mean we treat them as equal and it certainly isn't evidence that evenness is the result of some sort of liberal bias. Also, I think you have your examples backwards. Bat (animal) is a homophone with bat (sports) because both are pronounced the same but have different meanings. Wind (air flow) is not a homophone with wind (watches) because the pronunciation differs. They are, however, homographs. --JustinD 01:23, 26 February 2012 (EST)
- If that statement were true, then why are the top achievers on difficult math contests (such as the Putnam exam) more than 90% male, and less than 10% female?--Andy Schlafly 16:32, 26 February 2012 (EST)\
- I can't vouch for this on a national level, but when I took the Putnam at UConn last year, about 90% of the participants were male. If that's true at a national level (and I don't know if it is) then it stands to reason. Gregkochuconn 21:28, 26 February 2012 (EST)
- It's gotten so bad that now they make women put red stickers on the exams so they can graded differently! Women often have separate prizes too.--Bogart12
- Yeah, I know that. But from what I understand, that's more an incentive to get women to participate than it is an effort to give them a handicap, so to speak. Don't get me wrong, I find it ridiculous. But I think you are misunderstanding the intentions of the Putnam committee. Gregkochuconn 11:54, 4 March 2012 (EST)
- If that statement were true, then why are the top achievers on difficult math contests (such as the Putnam exam) more than 90% male, and less than 10% female?--Andy Schlafly 16:32, 26 February 2012 (EST)\
- I don't know why anyone (i.e., liberals) would expect or pretend that God created men and women to have absolutely identical aptitudes in everything. Men and women are physically very different.--Andy Schlafly 01:06, 27 February 2012 (EST)
- Mr. Schlafly, lets assume you represent the average conservative male. We know that conservatives are smarter than liberals. However, you assume that women are somehow poorer at math than a male of comparable upbringing, and conservative women usually listen to their fathers and other good, conservative influences. This subtle (or overt) bias leads to scores of smart, conservative women turning away from math in favor of biology and social science. This leads to the only potential female math majors being liberal, who then reinforce this stereotype. Otherwise, if women were biologically incapable of doing math as well as men can, the percentage of female mathematics PHD holders would remain at a constant 0%, instead of gradually increasing as the years pass on.KenShomer 17:35, 27 February 2012 (EST)
- I don't know why anyone (i.e., liberals) would expect or pretend that God created men and women to have absolutely identical aptitudes in everything. Men and women are physically very different.--Andy Schlafly 01:06, 27 February 2012 (EST)
- I think the problem is a lack of imprecision when using words like "same" or "equal". In mathematics such terms have very precisely defined meanings and so to say that all groups of order three are the same or are equal (or, equivalently, to say that "there is only one group of order 3") is not to say that all such groups are the same in all respects (obviously, this isn't true or they would all have the same name/description and we wouldn't even doubt there were more than one such group), but instead only says that all such groups behave similarly in all the ways that are (currently) important to mathematicians. When dealing with real world situations like mathematical aptitude and gender, it's much easier to speak imprecisely (either mistakenly or maliciously) and thus much more plausible that some suggested equivalences are the result of liberal bias and not some underlying similarities. I just don't see how that is possible, however, in the realm of mathematics, at least with respect to homomorphisms.
- This is getting a bit off topic, but if Andy and you are right and (liberal?) universities improperly equate the aptitude of men and women (by analogy, posit a homomorphism between men and women), we shouldn't conclude that the positing of all equivalences is liberal bias (by analogy, that homomorphisms are the result of liberal bias), but instead should conclude that universities are mistaken and that no such equivalence exists in this case (by analogy, there is no homomorphism between a group of order 3 and a group of order 4, say).--JustinD 17:01, 26 February 2012 (EST)
- Again though, the university claims more than "these sets have similar group structures on them," and goes so far as to insist that there is no reason to distinguish between them. Bogart12
- Okay, so then universities are wrong to equate men and women in this way. What does that have to do with homomorphisms? Are we still disagreeing whether they serve a legitimate mathematical purpose or are instead the product of improper liberal bias?--JustinD 21:54, 26 February 2012 (EST)
- It's central to liberal ideology to view these blasphemous notions as "useful." Next, you'll be defending relativity as useful, and that is a slippery slope my friend. We draw the line here at Conservapedia where ideas that are wrong are rejected, regardless of how useful we have been told they are.Bogart12
- Maybe you weren't joking before, but this is clearly just a goof, right? You don't really intend that as an argument do you?--JustinD 01:25, 27 February 2012 (EST)
- It's central to liberal ideology to view these blasphemous notions as "useful." Next, you'll be defending relativity as useful, and that is a slippery slope my friend. We draw the line here at Conservapedia where ideas that are wrong are rejected, regardless of how useful we have been told they are.Bogart12
- Okay, so then universities are wrong to equate men and women in this way. What does that have to do with homomorphisms? Are we still disagreeing whether they serve a legitimate mathematical purpose or are instead the product of improper liberal bias?--JustinD 21:54, 26 February 2012 (EST)
Examples of Liberal Math
The purported proof of Fermat's Last Theorem would be an example of liberal math.--Andy Schlafly 16:48, 26 February 2012 (EST)
- This may quickly move outside my area of understanding, but could you clarify this at all? Is the concern the use of the axiom of choice and/or non-ZF axioms? I've read our article and vaguely remember a book I read in high school, but other than the use of some non-standard techniques, I wasn't aware there was any serious doubt as to the validity of the proof. --JustinD 17:10, 26 February 2012 (EST)
- What would be an example of Conservative Math? --BradleyS 17:13, 26 February 2012 (EST)
- Godel's Incompleteness Theorems, and liberals ostracized Kurt Godel for it.--Andy Schlafly 17:41, 26 February 2012 (EST)
- What is there that evidence that liberals ostracized GĂ¶del? And what makes the Incompleteness Theorem conservative? --BradleyS 19:12, 26 February 2012 (EST)
- Godel's Incompleteness Theorems, and liberals ostracized Kurt Godel for it.--Andy Schlafly 17:41, 26 February 2012 (EST)
- Liberals kept Godel from obtaining tenure at most university math departments, and ostracized him in other ways. Bertrand Russell, a leading leftist of that time, was humiliated by Godel's insight, which was conservative in how it debunked intellectual arrogance.--Andy Schlafly 19:33, 26 February 2012 (EST)
- Exactly Andy, Whitehead and Russell were both in the middle of trying to "prove" everything when Godel administered a much-needed dose of humility. Bogart12
- Right. A liberal claims to know almost everything. A conservative recognizes that the liberal actually knows almost nothing of value.--Andy Schlafly 22:39, 26 February 2012 (EST)
- When did you become a liberal? DaveF 09:16, 27 February 2012 (EST)
Carbon Dating
Using the most advanced scientific & mathematical calculations, the dating process is still prone to error. The Shroud of Turin is claimed as fabrication by the scientific community but the liberals fail to grasp their errors in calculating exposure to contamination, such as soot. --Jpatt 19:37, 26 February 2012 (EST)
- That's not necessarily mathematics, but, archaeology + applied mathematics.JonM 19:50, 26 February 2012 (EST)
- Good point, but applied math is still math, and the example illustrates liberal use of math to convey an unjustified claim of "proof".--Andy Schlafly 20:10, 26 February 2012 (EST)
- Hammers have been used throughout history for the construction of buildings intended to promote both liberal and conservative ideals. The hammer has no inherent political philosophy. The hammer is a tool, and though it can be used for liberal or conservative purposes, it is inherently a neutral party. Math is the same way. While math can be used to promote liberal ideals, and is often studied by liberals, it is not liberal or conservative. Math is a tool.KenShomer 22:11, 26 February 2012 (EST)
- Good point, but applied math is still math, and the example illustrates liberal use of math to convey an unjustified claim of "proof".--Andy Schlafly 20:10, 26 February 2012 (EST)
- But hammers are not courses of study or an ideology. Math is. And all courses of study and ideologies are susceptible to liberal bias. Engineering, with its grounding in what works, is probably the most immune to liberal bias. Women's studies and English literature are probably the most susceptible to liberal bias.--Andy Schlafly 22:37, 26 February 2012 (EST)
- However, I think the point Ken is trying to make is that engineering can be used by both conservatives and liberals to accomplish their goals. (I.e. Engineering is equally useful whether you are building a church or a gay strip club). Similarly, you can use mathematical concepts for either liberal or conservative purposes, depending on what you want to do. Perhaps it's more directly obvious with mathematics, but I don't think it's nearly as bad as women's studies or English lit. Gregkochuconn 12:00, 4 March 2012 (EST)
- But hammers are not courses of study or an ideology. Math is. And all courses of study and ideologies are susceptible to liberal bias. Engineering, with its grounding in what works, is probably the most immune to liberal bias. Women's studies and English literature are probably the most susceptible to liberal bias.--Andy Schlafly 22:37, 26 February 2012 (EST)
atheist and liberal idiocy extends to math: 2+2=5? (Lawrence Krauss vs William Lane Craig)
2+2=5? (Lawrence Krauss vs William Lane Craig) Conservative 20:33, 26 February 2012 (EST)
Homomorphism
A homomorphism is a function/relation that maps one algebraic structure to another and preserves its structure. It does not equate them, so I removed it because its definition was erroneous. A comparison illustrates why the original definition is incorrect. If f(x) = x + 2, then f(3) = 5. This is a relation that maps to . It does not, however, equate them. Even though f(x) maps 3 to 5, it is not stating that 3 = 5. This is a simple example, but hopefully it illustrates the proper definition. Thank you! Kevin Davis ^{Talk} 09:09, 27 February 2012 (EST)
- This is incorrect, the map you've described is not a homomorphism. A homomorphism satisfies, for all a,b in A, φ(ab) = φ(a)φ(b). I am not talking about claiming that 3=5, I am talking about claiming that the Klein 4-group is the same as Z_2xZ_2.--User:Bogart12
- What precisely do you mean when you say that the "Klein 4-group is the same as Z_2xZ_2." As I've mentioned before, you need to be careful when you use such words as "same" or "equal." If all you mean is there exists a homomorphism between the two groups, well then that's tautological and completely uninteresting. If you mean something stronger, well what then be more specific so we can determine if such a claim is true or rather the result of liberal bias. --JustinD 22:22, 27 February 2012 (EST)
- As another example, we can define four groups as follows
- The group , where S = {a,b,c,d} and is defined according to the table
a | b | c | d | |
---|---|---|---|---|
a | a | b | c | d |
b | b | c | d | a |
c | c | d | a | b |
d | d | a | b | c |
- The group , where consists of the equivalence classes of integers modulo 4, and + is equivalence class addition.
- The group , where T = {1,i, − 1, − i}, the four complex fourth roots of 1, and is standard multiplication of complex numbers.
- By definition, none of these groups are equal, since groups are technically ordered pairs and none of the underlying sets are equal. However, each pair of these groups is isomorphic, which means that there exists a bijective homomorphism. Isomorphic groups have the same group structure. For example, each of these groups is abelian, which means that the group operation is commutative. Also, each group has four elements. Further, each group is cyclic, meaning that it is generated by a single element; the Klein four-group is not cyclic. Therefore, when we say that there are only two groups of order 4 up to isomorphism, we use isomorphisms and homomorphisms because we really don't care about groups that have the same structure but just have the elements relabeled.
- Homomorphisms exist between any pair of groups: the most obvious example is the homomorphism mapping every element of the domain group to the identity element of the target group. Homomorphisms are very interesting for many reasons; the kernel of a homomorphism (the preimage of the identity element) is always a normal subgroup, and free groups F(X) have the property that any map from the set of generators X to any group G can be extended uniquely to a homomorphism from F(X) to G. GregG 21:16, 28 February 2012 (EST)