Mathmetician's use of the word "generalize."?

Question

Ever notice that when mathmeticians generalize something they mean "with no exceptions" - e.g. you may prove something for one example, then generalize it to all cases.

But in common speech, when we say "in general" or that we are "generalizing" we mean that we are saying it is for all cases, but we acknowledge that there are a few exceptions.

I think it explains why sometimes students have problems learning proofs. that is, when we take common speech words and modify their meaning a bit.

Thanks for answers so far -
yes, the question is in regard to why do we in math sometimes adopt common words and use some of the lesser definitions.

Keep in mind most people don't consult a dictionary every time they use a word. So whatever it "should" be is a matter of opinion.

GOod point that "to generalize" in math may mean to make "more general than it is now," but I've heard many PhD's in math mean it differently - so it is not consistent.

Answer 1

Sure. Your observation seems correct to me--it took me a while to understand the difference from the colloquial usage. There is another, allied usage: to generalize a problem is to reduce some constraints and so broaden the application. For example, one can generalize a planar geometry problem to three or more dimensions.

Answer 2

If there is a question above I think it must be something like "why do mathematicians use the word generalize in an inconsistent way to which it is used in daily speech". Well my answer to that is, no this is not how it is. The verb "to generalize", in mathematical speech, either in proofs or elsewhere, does not mean you take a statement and you show that, that statement is true for every possible case. Indeed it means you take a theorem, a definition or any similar mathematical expression and you make it work for a case which is more general then the case in its original statement. Not necessarily every possible case, as no theorem is true in every possible case. This of course is what "to generalize" should mean in daily speech and it has of course nothing to do with the use of the word "general" in the sentence "In general college students are as lacking in mathematics as in English".

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Note to Ahab's note: I guess then I should add that many Math PhD's are as lacking in English as college students. Also, of course there is the habit of colloquialism, in which people slightly deviate from the proper grammatical use of a verb to shorten the sentences, in the expectation that the readers (colleagues) will understand the correct thing. However, in mathematical texts written for "general public" or college students, one rarely finds such usages. After all textbook publishing is a big big industry and any college level book publisher would love to claim to publish the most understandable text book.

Answer 3

The problem is you van not be consistent with everyday language .
Because
1. Everyday language itself not consistent.
2. In most of the case there is no exact word for what mathmetician wats to say, you haven't suggested another word for generalize.

So mathmetician have to give some meaning to a word.
In mathmetical proof, what the words mean should be made clear before for rigourousness.

Answer 4

This is a good observation about the common usage of the term.

On the other hand, in mathematics the only exceptions are stated clearly in the statement of the theorem / proposition / lemma. That's the point of the rigor.

Answer 5

Another Word For Generalize

Answer 6

It simply demonstrates an incomplete knowledge of the definition of "generalize", in particular, the transitive form. See

http://dictionary.reference.com/browse/generalize

3a - To make generally or universally applicable.

Answer 7

This is because mathmatecians can't function in reality, math is their crutch.

You should be like me. I don't believe in science, I think it's all huiy.