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Some of the examples given in the question are a careless misuse of the word. Who writes "canonical basis" for $K^n$ when they mean "standard basis", and who writes of a "canonical presentation of a fraction" when they mean a "fraction in its lowest terms", which isn't even canonical unless everything is positive.

In my field, arithmetic geometry, "canonical" has a well-understood meaning even if it is difficult to write down a precise definition. In his 1980 book, Milne was comfortable assuming that his readers would know what it meant (in his later writings, he has switched to using $\simeq$ for "canonically isomorphic"). Roughly, it means that the object can be constructed without making any arbitrary choices. There is a huge difference between saying two objects are isomorphic and saying they are they are canonically isomorphic. Barr botched this in his translation of Grothendieck's Tohoku paper by replacing "=" (meaning canonically isomorphic) with isomorphic.

I agree that the use of "canonical" is problematic in the Langlands program. There are major conjectures saying that some set (of representations) is bijective to some other set. After Serre pointed out that this only means that the two sets have the same cardinality, the word "canonical" was added. It is part of the problem to figure out what that means.

Some of the examples given in the question are a careless misuse of the word. Who writes "canonical basis" for $K^n$ when they mean "standard basis", and who writes of a "canonical presentation of a fraction" when they mean a "fraction in its lowest terms", which isn't even canonical unless everything is positive.

In my field, arithmetic geometry, "canonical" has a well-understood meaning even if it is difficult to write down a precise definition. In his 1980 book, Milne was comfortable assuming that his readers would know what it meant. Roughly, it means the object can be constructed without making any arbitrary choices. There is a huge difference between saying two objects are isomorphic and saying they are they are canonically isomorphic. Barr botched this in his translation of Grothendieck's Tohoku paper by replacing "=" (meaning canonically isomorphic) with isomorphic.

I agree that the use of "canonical" is problematic in the Langlands program. There are major conjectures saying that some set (of representations) is bijective to some other set. After Serre pointed out that this only means that the two sets have the same cardinality, the word "canonical" was added. It is part of the problem to figure out what that means.

Some of the examples given in the question are a careless misuse of the word. Who writes "canonical basis" for $K^n$ when they mean "standard basis", and who writes of a "canonical presentation of a fraction" when they mean a "fraction in its lowest terms", which isn't even canonical unless everything is positive.

In my field, arithmetic geometry, "canonical" has a well-understood meaning even if it is difficult to write down a precise definition. In his 1980 book, Milne was comfortable assuming that his readers would know what it meant (in his later writings, he has switched to using $\simeq$ for "canonically isomorphic"). Roughly, it means that the object can be constructed without making any arbitrary choices. There is a huge difference between saying two objects are isomorphic and saying they are canonically isomorphic. Barr botched this in his translation of Grothendieck's Tohoku paper by replacing "=" (meaning canonically isomorphic) with isomorphic.

I agree that the use of "canonical" is problematic in the Langlands program. There are major conjectures saying that some set (of representations) is bijective to some other set. After Serre pointed out that this only means that the two sets have the same cardinality, the word "canonical" was added. It is part of the problem to figure out what that means.

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anon
anon
  • 184
  • 4

Some of the examples given in the question are a careless misuse of the word. Who writes "canonical basis" for $K^n$ when they mean "standard basis", and who writes of a "canonical presentation of a fraction" when they mean a "fraction in its lowest terms", which isn't even canonical unless everything is positive.

In my field, arithmetic geometry, "canonical" has a well-understood meaning even if it is difficult to write down a precise definition. In his 1980 book, Milne was comfortable assuming that his readers would know what it meant. Roughly, it means the object can be constructed without making any arbitrary choices. There is a huge difference between saying two objects are isomorphic and saying they are they are canonically isomorphic. Barr botched this in his translation of Grothendieck's Tohoku paper by replacing "=" (meaning canonically isomorphic) with isomorphic.

I agree that the use of "canonical" is problematic in the Langlands program. There are major conjectures saying that some set (of representations) is bijective to some other set. After Serre pointed out that this only means that the two sets have the same cardinality, the word "canonical" was added. It is part of the problem to figure out what that means.