Talk:Majoring in Mathematics - Revision history

Gregkochuconn: /* Carbon Dating */

2012-03-04T17:00:55Z

‎Carbon Dating

Gregkochuconn: /* Liberal bias */

2012-03-04T16:54:26Z

‎Liberal bias

GregG: /* Homomorphism */ fix again

2012-02-29T02:20:24Z

‎Homomorphism: fix again

GregG: /* Homomorphism */ edit

2012-02-29T02:18:58Z

‎Homomorphism: edit

GregG: /* Homomorphism */ fix table again, unnumber

2012-02-29T02:17:35Z

‎Homomorphism: fix table again, unnumber

GregG: /* Homomorphism */ edit

2012-02-29T02:17:04Z

‎Homomorphism: edit

GregG: /* Homomorphism */ re

2012-02-29T02:16:25Z

‎Homomorphism: re

JustinD: /* Homomorphism */

2012-02-28T03:22:06Z

‎Homomorphism

Bogart12: /* Homomorphism */

2012-02-28T01:46:22Z

‎Homomorphism

KenShomer at 22:42, February 27, 2012

2012-02-27T22:42:13Z

← Older revision		Revision as of 17:00, March 4, 2012
Line 65:		Line 65:

	::::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.--[[User:Aschlafly\|Andy Schlafly]] 22:37, 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.--[[User:Aschlafly\|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. [[User:Gregkochuconn\|Gregkochuconn]] 12:00, 4 March 2012 (EST)

	== atheist and liberal idiocy extends to math: 2+2=5? (Lawrence Krauss vs William Lane Craig) ==		== atheist and liberal idiocy extends to math: 2+2=5? (Lawrence Krauss vs William Lane Craig) ==

@@ Line 29: / Line 29: @@
 ::::::: 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. [[User:Gregkochuconn|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
 ::::::::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.--[[User:Aschlafly|Andy Schlafly]] 01:06, 27 February 2012 (EST)

@@ Line 78: / Line 78: @@
 {| class="wikitable"
 |-
-! \cdot
+! <math>\cdot</math>
 ! <math>a</math>
 ! <math>b</math>

@@ Line 108: / Line 108: @@
 | <math>c</math>
 |}
-:*The group <math>(\mathbb{Z}_4, +)</math>, where <math>\mathbb{Z}_4</math> consists of the equivalence classes of integers modulo 4, and <math>+</math> is equivalence class addition
+:*The group <math>(\mathbb{Z}_4, +)</math>, where <math>\mathbb{Z}_4</math> consists of the equivalence classes of integers modulo 4, and <math>+</math> is equivalence class addition.
-:*The group <math>(T, \cdot)</math>, where <math>T = \{1, i, -1, -i\}</math>, the four complex fourth roots of 1.
+:*The group <math>(T, \cdot)</math>, where <math>T = \{1, i, -1, -i\}</math>, the four complex fourth roots of 1, and <math>\cdot</math> 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 <math>F(X)</math> have the property that any map from the set of generators <math>X</math> to any group <math>G</math> can be extended uniquely to a homomorphism from <math>F(X)</math> to <math>G</math>.  [[User:GregG|GregG]] 21:16, 28 February 2012 (EST)

@@ Line 75: / Line 75: @@
 ::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. --[[User:JustinD|JustinD]] 22:22, 27 February 2012 (EST)
 :As another example, we can define four groups as follows
-:#The group <math>(S, \cdot)</math>, where <math>S = \{a, b, c, d\}</math> and <math>\cdot</math> is defined according to the table
+:*The group <math>(S, \cdot)</math>, where <math>S = \{a, b, c, d\}</math> and <math>\cdot</math> is defined according to the table
-:#:{| class="wikitable"
+{| class="wikitable"
 |-
 ! \cdot
@@ Line 108: / Line 108: @@
 | <math>c</math>
 |}
-:#The group <math>(\mathbb{Z}_4, +)</math>, where <math>\mathbb{Z}_4</math> consists of the equivalence classes of integers modulo 4, and <math>+</math> is equivalence class addition
+:*The group <math>(\mathbb{Z}_4, +)</math>, where <math>\mathbb{Z}_4</math> consists of the equivalence classes of integers modulo 4, and <math>+</math> is equivalence class addition
-:#The group <math>(T, \cdot)</math>, where <math>T = \{1, i, -1, -i\}</math>, the four complex fourth roots of 1.
+:*The group <math>(T, \cdot)</math>, where <math>T = \{1, i, -1, -i\}</math>, the four complex fourth roots of 1.
 :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 <math>F(X)</math> have the property that any map from the set of generators <math>X</math> to any group <math>G</math> can be extended uniquely to a homomorphism from <math>F(X)</math> to <math>G</math>.  [[User:GregG|GregG]] 21:16, 28 February 2012 (EST)

← Older revision		Revision as of 02:16, February 29, 2012
Line 74:		Line 74:
	:This is incorrect, the map you've described is not a homomorphism. A homomorphism <math>\phi:A\rightarrow B</math> satisfies, for all a,b in A, <math>\phi(ab)=\phi(a)\phi(b).</math> 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]]		:This is incorrect, the map you've described is not a homomorphism. A homomorphism <math>\phi:A\rightarrow B</math> satisfies, for all a,b in A, <math>\phi(ab)=\phi(a)\phi(b).</math> 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. --[[User:JustinD\|JustinD]] 22:22, 27 February 2012 (EST)		::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. --[[User:JustinD\|JustinD]] 22:22, 27 February 2012 (EST)
		+	:As another example, we can define four groups as follows
		+	:#The group <math>(S, \cdot)</math>, where <math>S = \{a, b, c, d\}</math> and <math>\cdot</math> is defined according to the table
		+	{\| class="wikitable"
		+	\|-
		+	! \cdot
		+	! <math>a</math>
		+	! <math>b</math>
		+	! <math>c</math>
		+	! <math>d</math>
		+	\|-
		+	\| '''<math>a</math>'''
		+	\| <math>a</math>
		+	\| <math>b</math>
		+	\| <math>c</math>
		+	\| <math>d</math>
		+	\|-
		+	\| '''<math>b</math>'''
		+	\| <math>b</math>
		+	\| <math>c</math>
		+	\| <math>d</math>
		+	\| <math>a</math>
		+	\|-
		+	\| '''<math>c</math>'''
		+	\| <math>c</math>
		+	\| <math>d</math>
		+	\| <math>a</math>
		+	\| <math>b</math>
		+	\|-
		+	\| '''<math>d</math>'''
		+	\| <math>d</math>
		+	\| <math>a</math>
		+	\| <math>b</math>
		+	\| <math>c</math>
		+	\|}
		+	:#The group <math>(\mathbb{Z}_4, +)</math>, where <math>mathbb{Z}_4</math> consists of the equivalence classes of integers modulo 4, and <math>+</math> is equivalence class addition
		+	:#The group <math>(T, \cdot)</math>, where <math>T = \{1, i, -1, -i\}</math>, the four complex fourth roots of 1.
		+	: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 <math>F(X)</math> have the property that any map from the set of generators <math>X</math> to any group <math>G</math> can be extended uniquely to a homomorphism from <math>F(X)</math> to <math>G</math>. [[User:GregG\|GregG]] 21:16, 28 February 2012 (EST)

← Older revision		Revision as of 03:22, February 28, 2012
Line 73:		Line 73:
	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 <math>f(x) = x+2</math>, then <math>f(3) = 5</math>. This is a relation that maps <math>\mathbb{R}</math> to <math>\mathbb{R} + 2</math>. It does not, however, equate them. Even though <math>f(x)</math> maps 3 to 5, it is not stating that <math> 3 = 5</math>. This is a simple example, but hopefully it illustrates the proper definition. Thank you! [[User:KevinDavis\|Kevin Davis]] <sup>[[User talk:KevinDavis\|Talk]]</sup> 09:09, 27 February 2012 (EST)		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 <math>f(x) = x+2</math>, then <math>f(3) = 5</math>. This is a relation that maps <math>\mathbb{R}</math> to <math>\mathbb{R} + 2</math>. It does not, however, equate them. Even though <math>f(x)</math> maps 3 to 5, it is not stating that <math> 3 = 5</math>. This is a simple example, but hopefully it illustrates the proper definition. Thank you! [[User:KevinDavis\|Kevin Davis]] <sup>[[User talk:KevinDavis\|Talk]]</sup> 09:09, 27 February 2012 (EST)
	:This is incorrect, the map you've described is not a homomorphism. A homomorphism <math>\phi:A\rightarrow B</math> satisfies, for all a,b in A, <math>\phi(ab)=\phi(a)\phi(b).</math> 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]]		:This is incorrect, the map you've described is not a homomorphism. A homomorphism <math>\phi:A\rightarrow B</math> satisfies, for all a,b in A, <math>\phi(ab)=\phi(a)\phi(b).</math> 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. --[[User:JustinD\|JustinD]] 22:22, 27 February 2012 (EST)

← Older revision		Revision as of 01:46, February 28, 2012
Line 72:		Line 72:

	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 <math>f(x) = x+2</math>, then <math>f(3) = 5</math>. This is a relation that maps <math>\mathbb{R}</math> to <math>\mathbb{R} + 2</math>. It does not, however, equate them. Even though <math>f(x)</math> maps 3 to 5, it is not stating that <math> 3 = 5</math>. This is a simple example, but hopefully it illustrates the proper definition. Thank you! [[User:KevinDavis\|Kevin Davis]] <sup>[[User talk:KevinDavis\|Talk]]</sup> 09:09, 27 February 2012 (EST)		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 <math>f(x) = x+2</math>, then <math>f(3) = 5</math>. This is a relation that maps <math>\mathbb{R}</math> to <math>\mathbb{R} + 2</math>. It does not, however, equate them. Even though <math>f(x)</math> maps 3 to 5, it is not stating that <math> 3 = 5</math>. This is a simple example, but hopefully it illustrates the proper definition. Thank you! [[User:KevinDavis\|Kevin Davis]] <sup>[[User talk:KevinDavis\|Talk]]</sup> 09:09, 27 February 2012 (EST)
		+	:This is incorrect, the map you've described is not a homomorphism. A homomorphism <math>\phi:A\rightarrow B</math> satisfies, for all a,b in A, <math>\phi(ab)=\phi(a)\phi(b).</math> 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]]

@@ Line 31: / Line 31: @@
 ::::::::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.--[[User:Aschlafly|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 major 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.[[User:KenShomer|KenShomer]] 17:35, 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.[[User:KenShomer|KenShomer]] 17:35, 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.