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I was wondering if there is a good way to "define" what definition means exactly in mathematics. Since the answers may be subjective or philosophical, I want to ask only for references on this topic. So I am looking for references which answer the question

"What does definition mean in mathematics" in a concise and commonly accepted way.

and also for references which discuss philosophical problems connected with this question (if there are any).

I think this question should be community wiki.

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I've made the question community wiki. –  Zev Chonoles Jul 10 '11 at 9:30
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You mean you don't like this: A line is breadthless length. ... or ... A straight line is a line which lies evenly with the points on itself –  GEdgar Jul 10 '11 at 16:53

5 Answers 5

I would summarize my personal views about what "definition" means in mathematics as follows:

"[M]eaning is use — words are not defined by reference to the objects they designate, nor by the mental representations one might associate with them, but by how they are used. (source)

and

"Mathematicians do not study objects, but relations between objects. Thus, they are free to replace some objects by others so long as the relations remain unchanged. Content to them is irrelevant: they are interested in form only." - Henri Poincaré

For example consider the symbol "$\mathbb{Z}$". If I wanted to tell someone what I meant by it, I might write $$\mathbb{Z}=\{\ldots,-2,-1,0,1,2,\ldots\}$$ or if were being highbrow, perhaps $$\mathbb{Z}\text{ is the infinite cyclic group.}$$ or $$\mathbb{Z}\text{ is the initial object in }\mathsf{Ring}.$$ If I spoke another language, or used a different method of writing integers, or used a different notation for sets, these would appear quite different. But such differences are irrelevant; it doesn't even matter if someone else's mental conception of the integers is radically different from my own. What matters is that our usages agree - if it is the case that, any time I write a statement about $\mathbb{Z}$ that I consider true, anyone else agrees that (modulo differences of language / notation) that is a true statement about whatever it is they think of when they see "$\mathbb{Z}$", then functionally, our "definitions" agree. So, I don't think of "definition" as a formal concept in math (I know almost nothing about logic / set theory / metamathematics - I am just expressing my opinions). Even in formal logic, how can we hope to define parentheses? Or "$\in$"? We just start using them, and if people agree what we've written makes sense to them, that's the best we can hope for - we can try to use natural language to convey our mental conceptions to other people, but we can't dive into their heads and check that their mental conception is actually the same. (Obviously, intuitions / mental conceptions are of the utmost importance in doing mathematics - we won't get anywhere with blind manipulation of symbols. I'm just saying that all we can check our agreement on are external expressions such as equations or sentences.)

Finally, I'd just like to add this comic from SMBC: enter image description here

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Minor note: of course in your example, the three meanings are different: the first is $\mathbb{Z}$ as a set, the second as a group, the third as a ring. –  wildildildlife Jul 10 '11 at 10:52
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That's a good point, I was glossing over that detail. To be more precise, if I wrote down the relevant definitions in group theory, ring theory, and category theory necessary to parse those statements, then if someone's definition of $\mathbb{Z}$-as-a-ring (and all the previous definitions) agreed with mine, they would also agree that (whatever they thought of when they saw the phrase) "the additive group of $\mathbb{Z}$" was (whatever they thought of when they saw the phrase) "the infinite cyclic group". –  Zev Chonoles Jul 10 '11 at 11:13
    
Wrt the three definitions of $\mathbb Z$, the first is referred to as Extensional and the last two are called Intensional. In philosophy, this distinction cuts through to the differences between Realism and Nominalism. –  Dactyl Jul 10 '11 at 12:11

To define a word, even the word "define", you need a language with which to define it. Trying to do so in English is difficult, because English is not what we call a formal language. A formal language is a list of symbols and an acceptable grammar for these symbols to follow.

In mathematics, we generally use the formal language of Zermelo-Frankel set theory (or ZFC) to talk to each other (although many alternative ways have been studied). In this language, I would define a definition to be a finitely generated formula (would you accept an infinitely long definition of something?) of set theory that is legitimate according to the grammar whose quantifiers range over previously known results.

For example, in ZFC the definition "A number is an even number if it is a multiple of 2," can be written as "If x is a natural number and there exists another natural number y so that x=2y, then we define x as an even number," in ZFC which, in the scope of set theory, is a legitimate sentence whose quantifier ("all") ranges over the set of natural numbers.

A sentence that isn't definable would be something like "Call a set U universal if it contains all possible sets," because to define it in ZFC, you would need a formula "If U is a set so that for any set X, X is in U, then we call U universal," this formula quantifies over the set of all sets, which is not a set by Russell's Paradox, so this is not a legitimate definition.

Kurt Gödel studied "definable" structures in set theory and came up with the constructible universe, called L, which is a very useful concept in studying models of set theory. L is basically the "set of things definable by a formula of ZFC". Notably, under the assumption that all of the universe of set theory is actually equal to L, one can prove the generalized form of continuum hypothesis, one of the biggest problems in set theory during the 20th century.

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Your remarks about $L$ at the end are not quite accurate, since you are conflating the constructible sets with the definable sets. The constructible sets are those that appear in the constructible hierarchy, which is defined by a very restricted concept of definability. But it is fully consistent with ZFC that there are definable objects not in $L$. Indeed, the set $\mathbb{R}$ is definable, but it is consistent that $\mathbb{R}$ is not in $L$, and it cannot be in $L$ in any model in which CH fails. –  JDH Jul 10 '11 at 17:14

References (of a philosophy-of-mathematics nature):

(Note that Wittgenstein studied under Russell's direction at Cambridge, at Frege's suggestion).

Lastly, for context (and a deeper understanding of the strengths/weaknesses of formal languages), you'll probably want to study a formal language or two, and perhaps additionally study Godel's Incompleteness Theorems.

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Saying that Wittgenstein was "influenced" by Russell (btw, two 'l's) is an understatement, considering that Wittgenstein travelled to Cambridge expressly to study with Russell (by suggestion of Frege). –  Willie Wong Jul 10 '11 at 11:56
    
@Willie Wong: Thanks—suggestions have been incorporated. –  Raeez Jul 10 '11 at 16:25

No references from me, sorry, but allow me a quick and handwaving answer.

The way I understand it, a definition of some object within a given theory is a meaningful shortcut.

"Shortcut", because it provides a name for a bunch of certain properties that an object in the theory may or may not have; then every time the properties have to be refered to, the name is used. And "meaningful", in the sense that the properties being grouped under the name, either already have an important counterpart in the intended model of the theory, or else, they prove to be mathematically (read "technically") useful in the elaboration of the theory.

Of course, this is still a narrow understanding, even mathematically (as opposed to "philosophically" I suppose), at least in that definitions are set forth in multiple levels of everyday mathematical practice--not just within a given theory, but on various meta-levels as well.

In fact, as an aside, I believe this might be a key question to ask before trying out questions of "invention vs discovery" (see Is there any difference between a math invention and a math discovery? for example): do mathematicians choose their definitions, or are these forced upon them? Is there a uniform answer to be applied to all definitions (within a given theory)? Et cetera.

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I was wondering if there is a good way to "define" what definition means exactly in mathematics.

To make the definition of definition exact IMHO you need to make everything exact. This is what tomcuchta is talking about. Use some computer language to do mathematics, look for proof checkers and proof assistants. (If a language is computerized, this guarantees that the language is completely formal.) Then definition is a syntactic construction which binds a name.

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