A great common divisor
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Dr.Peterson
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What definition were you given for "greatest common divisor" in this context?
And the third (or ninth) graders who understand what a "ring" is in mathematical jargon, form what percentage of the set of of third (or ninth) graders? Please provide empirical evidence that the percentages are materially different from zero.
Otis
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Hi tyuiop. It usually means they're discussing something different. Did you post because you need help understanding the definitions?
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I take it that a gcd of a and b is by definition a common divisor of a and b and a multiple of every common divisor of a and b. And in a gcd-domain there are probably always more than one element that is a gcd to any two elements a and b. Therefore it should be called "great common divisor" and not "greatest common divisor". Do you agree?
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pka
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This from from the well used text Topics in Algebra by Herstein: The positive integer \(\bf e\) is said to be the greatest common of \(\bf a~\&~b\) if:
(1) \(\bf e\) is a divisor of \(\bf a\) and \(\bf b\);
(2) any divisor \(\bf a\) and \(\bf b\) is a divisor of \(\bf e\).
(1) \(\bf e\) is a divisor of \(\bf a\) and \(\bf b\);
(2) any divisor \(\bf a\) and \(\bf b\) is a divisor of \(\bf e\).
This is neither good English nor useful math.
If two numbers have a single common divisor, strict English grammar says that there is nothing to compare it with. Saying it is the "great" common divisor is meaningless: great compared to what?
If two numbers have a pair of common divisors, strict English grammar says that one of the pair is greater than the other rather than "greatest than the other." This is one of the few remnants in English of the dual number found for example in Greek.
Only if two numbers have more than two common divisors does strict English grammar recognize the appropriateness of the superlative.
When we think about common divisors in math, we do not know in general whether there are one, two, or more than two. It is an excusable technical violation of strict English grammar to say
If the numerator and denominator have a common divisor, reduce the fraction to lowest terms by dividing both numerator and denominator by the greatest common divisor
rather than
If the numerator and denominator have a single common divisor, reduce the fraction to lowest terms by dividing both numerator and denominator by the common divisor, but if the numerator and denominator have exactly two common divisors, reduce the fraction to lowest terms by dividing both numerator and denominator by the greater common divisor whereas if the numerator and denominator have more than two common divisors, reduce the fraction to lowest terms by dividing both the numerator and denominator by the greatest common divisor.
Although the second version accords with every scruple of English grammar, it never uses the word "great." It also is pedantic nonsense.
Dr.Peterson
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I don't think your issue is "greater" vs. "greatest" (as JeffM is assuming), or even, as you are actually saying, "great" vs. "greatest"; rather, it's that if there can be more than one, you can't properly say the greatest common divisor. I would say "a greatest common divisor", recognizing the possibility of more than one (in your particular context). I see nothing wrong with talking about more than one "greatest", but perhaps that is because mathematicians stretch many terms in the same way, so I am used to it.
"Greatest" just means that there is nothing bigger, rather than that everything else is smaller (in whatever sense is appropriate, in this case divisibility). Using pka's statement of the definition, we can compare [MATH]e | a[/MATH] to [MATH]e \le a[/MATH], which fits my idea. On the other hand, "great" does not communicate the meaning at all! How great does it have to be to be called great? The answer is, "one of the (equally) greatest"!
Of course, when restricting our attention (as at an elementary level) to positive integers, there is only one gcd, and this issue doesn't come up.
But I asked about the definition, expecting that you would have an actual source to quote, because that is essential! And a good author will explain how we can talk about "greatest" in this case. I had to look up "factorial ring" to get your context, as I was unfamiliar with the term, but found that is is the same as "unique factorization domain". And statements of the definition of gcd that I found in that context typically comment on the question of uniqueness; for example, the first Google hit (https://encyclopediaofmath.org/wiki/Greatest_common_divisor) says this:
A greatest common divisor of elements of an integral domain is defined as a common divisor of these elements that is divisible by any other common divisor. In general, a greatest common divisor of two elements of an integral domain need not exist (cf Divisibility in rings), but if one exists, it is unique up to multiplication by an invertible element.
Note the "a" and the comment on uniqueness.
Yes, but the "up to" part of the statement says that such an element does not need to be unique at all, and therefore I think it would be better to call it "a great common divisor" than "a greatest common divisor". The latter expression sounds silly to my ears.
I see the logic, but natural languages are not strictly logical. They have their own rules, and one of the most entrenched is that specifying the degree of adjectives.
The way you want to use "great" may sound appropriate to you in a specific mathematical context, but it does not make sense even in other mathematical contexts, where "greatest" absolutely does make sense. Language is a social construct, not a matter for individual dictation, and it does not get reversed by one very rare case. It might make more sense to come up with a technical noun for this type of object (like super-divisor), Then you can talk about a super-divisor without trying to overturn the whole system of indicating degree in adjectives in English (I have studied four other languages besides my native English, and all have similar structures for indicating degree. This system is very deeply rooted in western Indo-European languages.)
HallsofIvy
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A ring is defined in terms of its operations. There is not necessarily any order defined. One can define "divisors" and "common divisors" without any order. To have a "greatest common divisor" there must be an order defined The question then is whether whatever order is defined on your ring is a linear order or not.
Dr.Peterson
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Frankly, "great" sounds far sillier to me. And I presume I'm at least as much a native speaker of both English and math as you are. I guess we just have to arm wrestle over this. Oh, right -- that's not how language issues are decided!
As to the use of the superlative, consider that I am an identical twin, and as such am one of the two "oldest" siblings in my family. (Yes, technically I am the middle child, but not in practical terms.) Language allows for stretches of this sort, as I said before. (And we wouldn't correct my statement by saying I am "an old sibling"! That just doesn't mean anything like the same thing.)
In any case, we are talking about a definition, and since mathematicians agree on this term, it means what it means.
It's a good point that even "great" is not taken literally here; we are ordering by divisibility, not size, so the word "great" is an analogy.
Yes. I want to clarify that I have not been disagreeing with you. Language does indeed stretch, but there are limits. I have not thought that your suggestion went past those limits.
Dr.Peterson
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No, I've been taking you as an ally, just marching down a parallel road.
pka
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That id exactingly why I like Herstein's definition. Although is does use the word greatest it in way depends upon order. It only depends on his definition of divisor.
pka
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Yes Jeff thanks for the correction.
Try m
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"?
b
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"?
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.