How should Hermann Weyl's use of terminology in the first pages of “The Classical Groups” be understood?...
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Edit to add: I could now propose a logical (not opinion-based, IMHO) answer to my question, but for the hold placed on it.
I'm trying to read Hermann Weyl's The Classical Groups, Their Invariants and Representations. The opening discussion is confusing to me. Am I correct in my perception that his use of terminology is a bit wanting in some places? For example, on page 2 he states:
In a field of characteristic zero we can form the aliquot part $beta=alpha/n$ of $alpha$ with any integer $n$, i.e. a number $beta$ satisfying the equation $nbeta=alpha$.
I'm not sure if Weyl intends only positive integers in this context. Apparently $nne{0}$ is necessary, so the statement is incorrect. I assume that I am to apply the unstated exclusion of dividing by the integer zero. At this point in the discussion, if $0$ is used a an integer multiplier, then this $0$ is not the same as $0in{k}$ where $k$ is the field of elements called numbers. So we have no stated rule forbidding division of $alphain{k}$ by $0in{mathbb{Z}}.$ But division of $alphain{k}$ by $0in{k}$ has been expressly forbidden.
Am I reading too much into this? Do other's find Weyl's application of terminology in these paragraphs reckless?
I have immense respect for Weyl, and expect this to be one of his best works. But this seems like a shaky start.
abstract-algebra group-theory terminology
asked Dec 5 '18 at 19:16
Steven HattonSteven Hatton
883318
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closed as primarily opinion-based by rschwieb, Lord_Farin, Lord Shark the Unknown, Chinnapparaj R, KReiser Dec 6 '18 at 3:01
Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.
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show 4 more comments
$begingroup$
Edit to add: I could now propose a logical (not opinion-based, IMHO) answer to my question, but for the hold placed on it.
I'm trying to read Hermann Weyl's The Classical Groups, Their Invariants and Representations. The opening discussion is confusing to me. Am I correct in my perception that his use of terminology is a bit wanting in some places? For example, on page 2 he states:
In a field of characteristic zero we can form the aliquot part $beta=alpha/n$ of $alpha$ with any integer $n$, i.e. a number $beta$ satisfying the equation $nbeta=alpha$.
I'm not sure if Weyl intends only positive integers in this context. Apparently $nne{0}$ is necessary, so the statement is incorrect. I assume that I am to apply the unstated exclusion of dividing by the integer zero. At this point in the discussion, if $0$ is used a an integer multiplier, then this $0$ is not the same as $0in{k}$ where $k$ is the field of elements called numbers. So we have no stated rule forbidding division of $alphain{k}$ by $0in{mathbb{Z}}.$ But division of $alphain{k}$ by $0in{k}$ has been expressly forbidden.
Am I reading too much into this? Do other's find Weyl's application of terminology in these paragraphs reckless?
I have immense respect for Weyl, and expect this to be one of his best works. But this seems like a shaky start.
abstract-algebra group-theory terminology
asked Dec 5 '18 at 19:16
Steven HattonSteven Hatton
883318
$endgroup$
closed as primarily opinion-based by rschwieb, Lord_Farin, Lord Shark the Unknown, Chinnapparaj R, KReiser Dec 6 '18 at 3:01
Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.
$begingroup$
See the edited version of the post.
$endgroup$
– Steven Hatton
Dec 5 '18 at 19:24
6
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Yes, this is an error and it should require $n$ to be nonzero. One small oversight like this hardly strikes me as "reckless" though.
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– Eric Wofsey
Dec 5 '18 at 19:25
1
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The style in which the first few pages are written clearly indicates that Weyl expects his reader to already be familiar with everything he is saying. He is just making a quick review for the sake of completeness and to fix some definitions and notations and make sure everyone is on the same page.
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– Eric Wofsey
Dec 5 '18 at 19:26
1
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I don't think you are "reading too much into this." I think you are "making a mountain out of a molehill."
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– rschwieb
Dec 5 '18 at 19:28
1
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As I said, these passages are written for an audience who already knows all of what Weyl is saying. If you are not familiar with these ideas already and able to parse what Weyl means, then you probably do not have the background to understand most of the book.
$endgroup$
– Eric Wofsey
Dec 5 '18 at 20:02
|
show 4 more comments
$begingroup$
Edit to add: I could now propose a logical (not opinion-based, IMHO) answer to my question, but for the hold placed on it.
I'm trying to read Hermann Weyl's The Classical Groups, Their Invariants and Representations. The opening discussion is confusing to me. Am I correct in my perception that his use of terminology is a bit wanting in some places? For example, on page 2 he states:
In a field of characteristic zero we can form the aliquot part $beta=alpha/n$ of $alpha$ with any integer $n$, i.e. a number $beta$ satisfying the equation $nbeta=alpha$.
I'm not sure if Weyl intends only positive integers in this context. Apparently $nne{0}$ is necessary, so the statement is incorrect. I assume that I am to apply the unstated exclusion of dividing by the integer zero. At this point in the discussion, if $0$ is used a an integer multiplier, then this $0$ is not the same as $0in{k}$ where $k$ is the field of elements called numbers. So we have no stated rule forbidding division of $alphain{k}$ by $0in{mathbb{Z}}.$ But division of $alphain{k}$ by $0in{k}$ has been expressly forbidden.
Am I reading too much into this? Do other's find Weyl's application of terminology in these paragraphs reckless?
I have immense respect for Weyl, and expect this to be one of his best works. But this seems like a shaky start.
abstract-algebra group-theory terminology
asked Dec 5 '18 at 19:16
Steven HattonSteven Hatton
883318
$endgroup$
Edit to add: I could now propose a logical (not opinion-based, IMHO) answer to my question, but for the hold placed on it.
I'm trying to read Hermann Weyl's The Classical Groups, Their Invariants and Representations. The opening discussion is confusing to me. Am I correct in my perception that his use of terminology is a bit wanting in some places? For example, on page 2 he states:
In a field of characteristic zero we can form the aliquot part $beta=alpha/n$ of $alpha$ with any integer $n$, i.e. a number $beta$ satisfying the equation $nbeta=alpha$.
I'm not sure if Weyl intends only positive integers in this context. Apparently $nne{0}$ is necessary, so the statement is incorrect. I assume that I am to apply the unstated exclusion of dividing by the integer zero. At this point in the discussion, if $0$ is used a an integer multiplier, then this $0$ is not the same as $0in{k}$ where $k$ is the field of elements called numbers. So we have no stated rule forbidding division of $alphain{k}$ by $0in{mathbb{Z}}.$ But division of $alphain{k}$ by $0in{k}$ has been expressly forbidden.
Am I reading too much into this? Do other's find Weyl's application of terminology in these paragraphs reckless?
I have immense respect for Weyl, and expect this to be one of his best works. But this seems like a shaky start.
abstract-algebra group-theory terminology
abstract-algebra group-theory terminology
asked Dec 5 '18 at 19:16
Steven HattonSteven Hatton
883318
asked Dec 5 '18 at 19:16
Steven HattonSteven Hatton
883318
edited Dec 7 '18 at 1:02
Steven Hatton
asked Dec 5 '18 at 19:16
Steven HattonSteven Hatton
883318
asked Dec 5 '18 at 19:16
Steven HattonSteven Hatton
883318
asked Dec 5 '18 at 19:16
Steven HattonSteven Hatton
883318
883318
closed as primarily opinion-based by rschwieb, Lord_Farin, Lord Shark the Unknown, Chinnapparaj R, KReiser Dec 6 '18 at 3:01
Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.
closed as primarily opinion-based by rschwieb, Lord_Farin, Lord Shark the Unknown, Chinnapparaj R, KReiser Dec 6 '18 at 3:01
Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.
$begingroup$
See the edited version of the post.
$endgroup$
– Steven Hatton
Dec 5 '18 at 19:24
6
$begingroup$
Yes, this is an error and it should require $n$ to be nonzero. One small oversight like this hardly strikes me as "reckless" though.
$endgroup$
– Eric Wofsey
Dec 5 '18 at 19:25
1
$begingroup$
The style in which the first few pages are written clearly indicates that Weyl expects his reader to already be familiar with everything he is saying. He is just making a quick review for the sake of completeness and to fix some definitions and notations and make sure everyone is on the same page.
$endgroup$
– Eric Wofsey
Dec 5 '18 at 19:26
1
$begingroup$
I don't think you are "reading too much into this." I think you are "making a mountain out of a molehill."
$endgroup$
– rschwieb
Dec 5 '18 at 19:28
1
$begingroup$
As I said, these passages are written for an audience who already knows all of what Weyl is saying. If you are not familiar with these ideas already and able to parse what Weyl means, then you probably do not have the background to understand most of the book.
$endgroup$
– Eric Wofsey
Dec 5 '18 at 20:02
|
show 4 more comments
$begingroup$
See the edited version of the post.
$endgroup$
– Steven Hatton
Dec 5 '18 at 19:24
6
$begingroup$
Yes, this is an error and it should require $n$ to be nonzero. One small oversight like this hardly strikes me as "reckless" though.
$endgroup$
– Eric Wofsey
Dec 5 '18 at 19:25
1
$begingroup$
The style in which the first few pages are written clearly indicates that Weyl expects his reader to already be familiar with everything he is saying. He is just making a quick review for the sake of completeness and to fix some definitions and notations and make sure everyone is on the same page.
$endgroup$
– Eric Wofsey
Dec 5 '18 at 19:26
1
$begingroup$
I don't think you are "reading too much into this." I think you are "making a mountain out of a molehill."
$endgroup$
– rschwieb
Dec 5 '18 at 19:28
1
$begingroup$
As I said, these passages are written for an audience who already knows all of what Weyl is saying. If you are not familiar with these ideas already and able to parse what Weyl means, then you probably do not have the background to understand most of the book.
$endgroup$
– Eric Wofsey
Dec 5 '18 at 20:02
$begingroup$
See the edited version of the post.
$endgroup$
– Steven Hatton
Dec 5 '18 at 19:24
$begingroup$
See the edited version of the post.
$endgroup$
– Steven Hatton
Dec 5 '18 at 19:24
6
6
$begingroup$
Yes, this is an error and it should require $n$ to be nonzero. One small oversight like this hardly strikes me as "reckless" though.
$endgroup$
– Eric Wofsey
Dec 5 '18 at 19:25
$begingroup$
Yes, this is an error and it should require $n$ to be nonzero. One small oversight like this hardly strikes me as "reckless" though.
$endgroup$
– Eric Wofsey
Dec 5 '18 at 19:25
1
1
$begingroup$
The style in which the first few pages are written clearly indicates that Weyl expects his reader to already be familiar with everything he is saying. He is just making a quick review for the sake of completeness and to fix some definitions and notations and make sure everyone is on the same page.
$endgroup$
– Eric Wofsey
Dec 5 '18 at 19:26
$begingroup$
The style in which the first few pages are written clearly indicates that Weyl expects his reader to already be familiar with everything he is saying. He is just making a quick review for the sake of completeness and to fix some definitions and notations and make sure everyone is on the same page.
$endgroup$
– Eric Wofsey
Dec 5 '18 at 19:26
1
1
$begingroup$
I don't think you are "reading too much into this." I think you are "making a mountain out of a molehill."
$endgroup$
– rschwieb
Dec 5 '18 at 19:28
$begingroup$
I don't think you are "reading too much into this." I think you are "making a mountain out of a molehill."
$endgroup$
– rschwieb
Dec 5 '18 at 19:28
1
1
$begingroup$
As I said, these passages are written for an audience who already knows all of what Weyl is saying. If you are not familiar with these ideas already and able to parse what Weyl means, then you probably do not have the background to understand most of the book.
$endgroup$
– Eric Wofsey
Dec 5 '18 at 20:02
$begingroup$
As I said, these passages are written for an audience who already knows all of what Weyl is saying. If you are not familiar with these ideas already and able to parse what Weyl means, then you probably do not have the background to understand most of the book.
$endgroup$
– Eric Wofsey
Dec 5 '18 at 20:02
|
show 4 more comments
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$begingroup$
See the edited version of the post.
$endgroup$
– Steven Hatton
Dec 5 '18 at 19:24
6
$begingroup$
Yes, this is an error and it should require $n$ to be nonzero. One small oversight like this hardly strikes me as "reckless" though.
$endgroup$
– Eric Wofsey
Dec 5 '18 at 19:25
1
$begingroup$
The style in which the first few pages are written clearly indicates that Weyl expects his reader to already be familiar with everything he is saying. He is just making a quick review for the sake of completeness and to fix some definitions and notations and make sure everyone is on the same page.
$endgroup$
– Eric Wofsey
Dec 5 '18 at 19:26
1
$begingroup$
I don't think you are "reading too much into this." I think you are "making a mountain out of a molehill."
$endgroup$
– rschwieb
Dec 5 '18 at 19:28
1
$begingroup$
As I said, these passages are written for an audience who already knows all of what Weyl is saying. If you are not familiar with these ideas already and able to parse what Weyl means, then you probably do not have the background to understand most of the book.
$endgroup$
– Eric Wofsey
Dec 5 '18 at 20:02