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edited Dec 4 2009 at 21:25 |
What Are The Most Misleading Alternate Definitions In Taught Mathematicsare the most misleading alternate definitions in taught mathematics?
I suppose this question can be interpreted in two ways. It is often the case that two or more equivalent (but not necessarily semantically equivalent) definitions of the same idea/object are used in practice. Are there examples of equivalent definitions where one is more natural or intuitive? (I'm meaning so greatly more intuitive so as to not be subjective.)
Alternatively, what common examples are there in standard lecture courses where a particular symbolic definition obscures the concept being conveyed.
Community wiki
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edited Dec 4 2009 at 14:42 |
What Are The Most Misleading/Obscuring Misleading Alternate Definitions In Taught Mathematics?
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edited Dec 4 2009 at 8:20 |
(This question is inspired by someone recently asking me to explain the product topology, and after looking at the definition being used:
$T_{X \times Y} = \{ W \subset X \times Y : \forall (x,y) \in W, \exists U \in \tau, V \in \sigma, s.t.\ x \in U, y \in V, U \times V \subset W \}$
I decided that, whilst obviously perfectly sound, is just not as illuminating as to say 'The coarsest topology for which the projection functions are continuous', which they immediately understood.)I suppose this question can be interpreted in two ways. It is often the case that two or more equivalent (but not necessarily semantically equivalent) definitions of the same idea/object are used in practice. Are there examples of equivalent definitions where one is more natural or intuitive? (I'm meaning so greatly more intuitive so as to not be subjective.)
Alternatively, what common examples are there in standard lecture courses where a particular symbolic definition obscures the concept being conveyed.
Community wiki
(Note: I'm not the greatest writer so I thought I should include my example in the question to illustrate the question, but I'll remove it and submit it as an answer if that is more appropriate, and I'll do this after a few submitted answers anyway. Please rephrase my question as you see fit.)
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edited Dec 4 2009 at 8:19 |
(This question is inspired by someone recently asking me to explain the product topology, and after looking at the definition being used:
$T_{X \times Y} = \{ W \subset X \times Y : \forall (x,y) \in W, \exists U \in \tau, V \in \sigma, s.t.\ x \in U, y \in V, U \times V \subset W \}$
I decided that, whilst obviously perfectly sound, is just not as illuminating as to say 'The coarsest topology for which the projection functions are continuous', which they immediately understood.)
I suppose this question can be interpreted in two ways. It is often the case that two or more equivalent (but not necessarily semantically equivalent) definitions of the same idea/object are used in practice. Are there examples of equivalent definitions where one is more natural or intuitive? (I'm meaning so greatly more intuitive so as to not be subjective.)
Alternatively, what common examples are there in standard lecture courses where a particular symbolic definition obscures the concept being conveyed.
Community wiki
(Note: I'm not the greatest writer so I thought I should include my example in the question to illustrate the question, but I'll remove it and submit it as an answer if that is more appropriate, and I'll do this after a few submitted answers anyway. Please rephrase my question as you see fit.)
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edited Dec 3 2009 at 0:50 |
(This question is inspired by someone recently asking me to explain the product topology, and after looking at the definition being used:
$T_{X \times Y} = \{ W \subset X \times Y : \forall (x,y) \in W, \exists U \in \tau, V \in \sigma, s.t.\ x \in U, y \in V, U \times V \subset W \}$
I decided that, whilst obviously perfectly sound, is just not as illuminating as to say 'The coarsest topology for which the projection functions are continuous', which they immediately understood.)
I suppose this question can be interpreted in two ways. It is often the case that two or more equivalent (but not necessarily semantically equivalent) definitions of the same idea/object are used in practice. Are there examples of equivalent definitions where one is more natural or intuitive? (I'm meaning so greatly more intuitive so as to not be subjective.)
Alternatively, what common examples are there in standard lecture courses where a particular symbolic definition obscures the concept being conveyed.
Community wiki
(Note: I'm not the greatest writer so I thought I should include my example in the question to illustrate the question, but I'll remove it and submit it as an answer if that is more appropriate, and I'll do this after a few submitted answers anyway. Please rephrase my question as you see fit.)
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edited Dec 2 2009 at 22:11 |
(This question is inspired by someone recently asking me to explain the product topology, and after looking at the definition being used:
$X T_{X \times Y} = \{ W \subset X \times Y : \forall (x,y) \in W, \exists U \in \tau, V \in \sigma, s.t.\ x \in U, y \in V, U \times V \subset W \}$
I decided that, whilst obviously perfectly sound, is just not as illuminating as to say 'The coarsest topology for which the projection functions are continuous', which they immediately understood.)
I suppose this question can be interpreted in two ways. It is often the case that two or more equivalent (but not necessarily semantically equivalent) definitions of the same idea/object are used in practice. Are there examples of equivalent definitions where one is more natural or intuitive? (I'm meaning so greatly more intuitive so as to not be subjective.)
Alternatively, what common examples are there in standard lecture courses where a particular symbolic definition obscures the concept being conveyed.
Community wiki
(Note: I'm not the greatest writer so I thought I should include my example in the question to illustrate the question, but I'll remove it and submit it as an answer if that is more appropriate, and I'll do this after a few submitted answers anyway. Please rephrase my question as you see fit.)
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edited Dec 2 2009 at 17:32 |
(This question is inspired by someone recently asking me to explain the product topology, and after looking at the definition being used:
$X \times Y = \{ W \subset X \times Y : \forall (x,y) \in UW, \exists U \in \tau, V \in \sigma, s.t.\ x \in U, y \in V, U \times V \subset W \}$
I decided that, whilst obviously perfectly sound, is just not as illuminating as to say 'The coarsest topology for which the projection functions are continuous', which they immediately understood.)
I suppose this question can be interpreted in two ways. It is often the case that two or more equivalent (but not necessarily semantically equivalent) definitions of the same idea/object are used in practice. Are there examples of equivalent definitions where one is more natural or intuitive? (I'm meaning so greatly more intuitive so as to not be subjective.)
Alternatively, what common examples are there in standard lecture courses where a particular symbolic definition obscures the concept being conveyed.
Community wiki
(Note: I'm not the greatest writer so I thought I should include my example in the question to illustrate the question, but I'll remove it and submit it as an answer if that is more appropriate, and I'll do this after a few submitted answers anyway. Please rephrase my question as you see fit.)
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edited Dec 2 2009 at 17:14 |
What Are The Most Misleading/Obscurifying Misleading/Obscuring Alternate Definitions In Taught Mathematics?
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asked Dec 2 2009 at 15:59 |
What Are The Most Misleading/Obscurifying Alternate Definitions In Taught Mathematics?
(This question is inspired by someone recently asking me to explain the product topology, and after looking at the definition being used:
$X \times Y = \{ W \subset X \times Y : \forall (x,y) \in U, \exists U \in \tau, V \in \sigma, s.t.\ x \in U, y \in V, U \times V \subset W \}$
I decided that, whilst obviously perfectly sound, is just not as illuminating as to say 'The coarsest topology for which the projection functions are continuous', which they immediately understood.)
I suppose this question can be interpreted in two ways. It is often the case that two or more equivalent (but not necessarily semantically equivalent) definitions of the same idea/object are used in practice. Are there examples of equivalent definitions where one is more natural or intuitive? (I'm meaning so greatly more intuitive so as to not be subjective.)
Alternatively, what common examples are there in standard lecture courses where a particular symbolic definition obscures the concept being conveyed.
Community wiki
(Note: I'm not the greatest writer so I thought I should include my example in the question to illustrate the question, but I'll remove it and submit it as an answer if that is more appropriate, and I'll do this after a few submitted answers anyway. Please rephrase my question as you see fit.)
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