A great common divisor

tyuiop said:
Yes, you could say of somebody that he is one of the two oldest siblings in his family, but do you find it appropriate to say that somebody is "an oldest sibling"? I think that you do not. So why do you defend the analogous expression "a greatest common divisor"?
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Yes, I would call myself an oldest sibling. That's implied by calling myself one of two "oldest siblings". In fact, in discussions of birth order, I've called myself a firstborn.

There are many terms in math that slightly deviate from ordinary uses. We use "if" in a slightly twisted way, saying that "if pigs fly, then the moon is made of green cheese" is true. We use "or" in a slightly twisted way, making it always inclusive rather than expressing alternatives. We use "rectangle" in a slightly twisted way, so that a square is a rectangle. We use "multiple" in a slightly twisted way, calling a number a multiple of itself, and we can even "multiply" a number and make it smaller. We extend definitions, making "fraction" no longer mean only a part of something (improper fractions). I'm probably missing the best examples, but you should get the idea. Math is in the habit of stretching the language.

But if you can't accept this as a slight stretch, just call it an idiom. English has enough of those that we should all be used to apparent illogic.

In any case, your "great common denominator" is completely unacceptable.
 
JeffM said:
Because mathematics may create ideas that natural languages never experience outside of mathematics, there is always difficulty in translating such mathematics into natural language. Natural languages can "stretch" to incorporate new ideas, but such stretching needs to occur along the line of least resistance. Implying that two or more distinct things can be described as greater than all others but are not greater than each other while remaining distinct is just not describable in normal English. So the question becomes how to stretch.

Dr. Peterson suggests letting go of the notion that the superlative is necessarily unique. You suggest letting go of the entire concept of degree. I say your idea represents a far greater stretch of normal English usage than does Dr. P's.

Halls and pka point out that the whole problem arises only in the context of an order relationship. If such an order relationship cannot be defined at all or if it cannot be defined uniquely, then we are outside the frame of reference of standard English grammar. PKA suggests a mathematical definition of greatest common divisor that does not necessarily imply an order relationship but that conforms to normal English usage when a unique order relationship does exist. It seems to me that any further argument is superflous. We can recognize the greater imaginative power of mathematics with the least violence to the social norms of the English language.
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I just explained what kind of normal English expression you can use if you want to say that two distinct things are greater than all others but not greater than each other, namely this one: "He is one of the two oldest siblings". We say such things all the time without having to stretch the language. If I think that Mozart and Beethoven are equally great but greater than every other composer, I could say that Beethoven is one if the two greatest composers. I do not have to stretch the language by saying that Beethoven is "a greatest composer".
 
Dr.Peterson said:
Yes, I would call myself an oldest sibling. That's implied by calling myself one of two "oldest siblings". In fact, in discussions of birth order, I've called myself a firstborn.

There are many terms in math that slightly deviate from ordinary uses. We use "if" in a slightly twisted way, saying that "if pigs fly, then the moon is made of green cheese" is true. We use "or" in a slightly twisted way, making it always inclusive rather than expressing alternatives. We use "rectangle" in a slightly twisted way, so that a square is a rectangle. We use "multiple" in a slightly twisted way, calling a number a multiple of itself, and we can even "multiply" a number and make it smaller. We extend definitions, making "fraction" no longer mean only a part of something (improper fractions). I'm probably missing the best examples, but you should get the idea. Math is in the habit of stretching the language.

But if you can't accept this as a slight stretch, just call it an idiom. English has enough of those that we should all be used to apparent illogic.

In any case, your "great common denominator" is completely unacceptable.
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Why would you say such a strange thing as "I am an oldest sibling" when you have the option to use normal, intelligible, well sounding expressions like "I am the oldest sibling" or " I am one of the two oldest siblings"?
 
Then you can say that x is one of the [unknown number of] greatest common divisors! That's exactly what the standard term, "a greatest common divisor", means.

Why are you so insistent on this? Why are you unable to accept it as an idiom (that is, something that is commonly said, though it doesn't make literal sense)? If you speak English (or any other language) you do that all the time. I don't think you have to call this an idiom, but you can.
 
@tyuiop

As you say, it is perfectly idiomatic to say "one of the greatest" about a member of a class the members of which are all equally great. In fact, the language police will not issue you a speaking ticket if the class contains members that are only approximately equal. So there is no need to do any violence to the norms of English by equating "great" with "one of the greatest." You can be mathematically exact and idiomatic at the same time by using "one of the greatest common divisors" in any case where there is a possibility that the number of greatest comon divisors exceeds 1.

As Dr. Peterson says with examples, mathematicians alter natural language all the time. I have no problem with "a greatest common divisor" meaning meaning any member of the class of greatest common divisors. That is how the indefinite article works in English. Now in the majority of cases where the concept of a greatest common divisor is being applied, the class has exactly one member, which makes the definite article appropriate. That is the only reason that "a greatest common divisor" sounds odd.

I am tired of this topic so I shall let you have the last word.
 
JeffM said:
@tyuiop

As you say, it is perfectly idiomatic to say "one of the greatest" about a member of a class the members of which are all equally great. In fact, the language police will not issue you a speaking ticket if the class contains members that are only approximately equal. So there is no need to do any violence to the norms of English by equating "great" with "one of the greatest." You can be mathematically exact and idiomatic at the same time by using "one of the greatest common divisors" in any case where there is a possibility that the number of greatest comon divisors exceeds 1.

As Dr. Peterson says with examples, mathematicians alter natural language all the time. I have no problem with "a greatest common divisor" meaning meaning any member of the class of greatest common divisors. That is how the indefinite article works in English. Now in the majority of cases where the concept of a greatest common divisor is being applied, the class has exactly one member, which makes the definite article appropriate. That is the only reason that "a greatest common divisor" sounds odd.

I am tired of this topic so I shall let you have the last word.
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I hope that you are not tired of the topic because you find me rude or anything. It is stimulating to discuss with you and with Dr. Peterson, who seems to have a Swedish name like me, by the way. Are you sure that the indefinite article works that way in English? Should you not say either "a great" or "the greatest"? And why insist on using the strange expression "a greatest common divisor" when there is no need for it? It would, I think, be natural to call a common divisor of a and b which satisfies the additional criterion that we mentioned "a great common divisor". "Great" would mean in this context that it is a multiple of every common divisor of the two elements.
 
I don't think you are a native speaker of English. (And I didn't know that tyuiop was a Swedish name ;).) If there are multiple "greatests" (and there can be: who would you say were the greatest mathematicians of all time?), then "a" is fully appropriate. Yes, we would more often tend to say "one of the greatest", but that doesn't make "a" wrong. And your issue was not initially about "a", which you didn't even use; it was about "greatest". You don't have to say "a gcd" if you don't want to; just say "one of the gcd's", and everyone should be happy.

But I, too, am tired of the conversation, because you are not listening. I have no more to say.
 
Dr.Peterson said:
I don't think you are a native speaker of English. (And I didn't know that tyuiop was a Swedish name ;).) If there are multiple "greatests" (and there can be: who would you say were the greatest mathematicians of all time?), then "a" is fully appropriate. Yes, we would more often tend to say "one of the greatest", but that doesn't make "a" wrong. And your issue was not initially about "a", which you didn't even use; it was about "greatest". You don't have to say "a gcd" if you don't want to; just say "one of the gcd's", and everyone should be happy.

But I, too, am tired of the conversation, because you are not listening. I have no more to say.
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My point has always been that it is grammatically wrong to use the indefinite article together with an adjective in the superlative form. That is wrong in Swedish and as far as I know also in English. So our discussion is not about English grammar but about grammar in general. Of Gauss you could say that he is "the greatest mathematician of all time" or "one of the greatest mathematicians of all time", but not "a greatest mathematician of all time". And if you think that Archimedes is in fact the greatest mathematician of all time, you could say that Gauss was just "a great mathematician".
 
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I lied. I am back.

No, you were not rude. I tutor at English Learners Stack Exchange and am currently rated above 99.5%. So arguing about English grammar with someone who is not a native speaker is frustrating.

The expectation in English grammar is that a superlative refers to singular item. As such, it takes the definite article.

An indefinite article indicates a single unspecified member of a class, which may or may not have multiple members.

You are positing a case where there are multiple items that can be considered greatest, and you want to talk about one of them without identifying which one. One rule calls for the definite article, and another rule calls for an indefinite article. So, as you yourself said before, there is an idiomatic and exact but slightly verbose way to resolve the issue: "one of the greatest."

If you want to avoid verbosity, the least attractive option is to abandon degree altogether because you thereby lose any sense of what is important about the class: the class is not merely "great," an amorphous term that may mean almost anything other than least; each member of the class is superlative.

To avoid verbosity, you can use ellipsis: "a greatest" as an ellipsis for "a [member of the class of] greatest." (I find that non-native speakers are often a bit perplexed by ellipsis: native speakers fill in the missing words without conscious thought. I suspect the same thing happens in Swedish.) Or you can use the definite article and ignore the fact that there are multiple instances. Or, if there is a community that frequently speaks of such multiplied superlatives, it can agree on a word or phrase that captures the exact meaning. In this specific case, I believe pka has already supplied a technical definition that does that.
 
JeffM said:
I lied. I am back.

No, you were not rude. I tutor at English Learners Stack Exchange and am currently rated above 99.5%. So arguing about English grammar with someone who is not a native speaker is frustrating.

The expectation in English grammar is that a superlative refers to singular item. As such, it takes the definite article.

An indefinite article indicates a single unspecified member of a class, which may or may not have multiple members.

You are positing a case where there are multiple items that can be considered greatest, and you want to talk about one of them without identifying which one. One rule calls for the definite article, and another rule calls for an indefinite article. So, as you yourself said before, there is an idiomatic and exact but slightly verbose way to resolve the issue: "one of the greatest."

If you want to avoid verbosity, the least attractive option is to abandon degree altogether because you thereby lose any sense of what is important about the class: the class is not merely "great," an amorphous term that may mean almost anything other than least; each member of the class is superlative.

To avoid verbosity, you can use ellipsis: "a greatest" as an ellipsis for "a [member of the class of] greatest." (I find that non-native speakers are often a bit perplexed by ellipsis: native speakers fill in the missing words without conscious thought. I suspect the same thing happens in Swedish.) Or you can use the definite article and ignore the fact that there are multiple instances. Or, if there is a community that frequently speaks of such multiplied superlatives, it can agree on a word or phrase that captures the exact meaning. In this specific case, I believe pka has already supplied a technical definition that does that.
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If you want to avoid verbosity, you can just say "a great common divisor", "a maximal common divisor" or something along that line. There is no need to violate grammar in order to avoid verbosity. Since you are a native English speaker, you do not really have any excuse for using grammatically incorrect expressions.
 
tyuiop said:
If you want to avoid verbosity, you can just say "a great common divisor", "a maximal common divisor" or something along that line. There is no need to violate grammar in order to avoid verbosity. Since you are a native English speaker, you do not really have any excuse for using grammatically incorrect expressions.
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No you cannot. Now you have become rude. I am putting you on ignore.
 
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