True in dictionaries but not in mathematics. Definition means making

meaning definite, whether in reference to words or things

Yeah, use-mention confusion is a big problem in math and related areas. But if you look at the formal system, you will find that what a definition amounts to a permissible replacement of expressions (and in programming as shortcut for calling up a subprogram).

<Functions can be defined. Functions are not words. Functions can be

assigned names. Their names may be words or phrases.>

All true, with the possible exception of the first sentence, where it depends on what you mean: mathematicians do talk this way, but their practice belies their words generally.

<This is not the usage of mathematicians.

Examples:

Definition: A point is that which has no part.

Definition: A function is a set of ordered pairs such that no two

pairs have the same first element. >

The first is hopeless as a definition; it is sometiimes used in a very loose way as an axiom (where, except in very specialized contexts, it is still hopeless). The second is a crude way of saying "f is a function" can be replaced by "f is a relation such that if <xy> and <xz> are in f, y=z" or whatever the formalism requires. that is, this is exactly the usage of mathematicians, though they do their best to hide it.

<These definitions define the things referred to by the words, not the

words themselves, which should be understood simply as bound

variables or pronouns. This has nothing to do with dictionary

definition, which explains what words mean in common usage.>

No, these are not dictionary defintions, since they have nothing to do with common usage; these are just abbreviations in a formal system (actual or intended to come some day). I'm not sure what the the "bound variables or pronouns" means here -- some kind of Ramsey sentences? Yes, we could take the whole system as under a bunch of quantifiers and just use variables where we now use words -- or treat the words as such variables -- but the replacements would still go through as before, except now we have one variable replaced by a complex term involving other variables.

<For example, we can select a point in three-dimensional Euclidean

space, and regard all of the lines through that point as points in a

projective geometry, with the lines of the projective geometry being

planes through the designated point in the Euclidean space. This

works because two planes through a point have a line in common, and

two lines through a point have a plane in common. Alternatively, we

can take the planes as the points and the lines as the lines.>

Relevance escapes me; what has this to do with definitions? There are lots of analogies in geometry, especially if you get away from pure Euclidean planes, but an analogy is not a definition. There are God knows how many ways to represent the natural numbers, but that doesn't mean that they are all (or any of them) the natural numbers.

<A function can be defined without naming it by listing pairs, or by

defining a domain, a range, and a rule, or by other means.>

These are specifications, although at least the first alternative (and in many cases the second) will usually turn out to be a replacement rule.

<Exactly right. You specified a domain, a range, and a rule, thereby

defining the function completely. There are a number of areas in

mathematics, including combinatory logic and Lambda calculus, where

nameless functions are extremely important. Also in some programming

languages, including LISP and Functional Programming languages,

functions can be defined and used without being named.>

I suspect some hank-panky on "named" here. Can you really do anything with a function that does not have some rule for computing it? Yes, you can specify it very generally and prove that such a function exists and that it has further properties and the like. You can even just do the computations over and over without every shortcutting by a defined replacement. In lambda and combinatory logic, just about everything is a function name, which happens also to be a computation rule -- that is why it is so handy (well, it often isn't until you introduce shorthands for the horrors).

<> You've described a function (often called "successor') but you

> haven't defined it (it is not a word -- unless, of course, you want

> to get into just what "n+1" is).

No, this is the definition.>

Well, yes it is, but only because of the systemic use-mention confusion of mathematical terminology (what the parenthesis was about).

<

> If you want to use it often, you

> probably won't want to use that description every time, so you will

> name it eventually. Why not at the beginning?

We could, but we don't have to, and sometimes there are reasons not

to.>

Quite true, but practicalities usually require the shortcuts and almost only really strange theoretical purposes are served by doing everything in primitives (you can't build an efficient turing machine with full number notation -- the state count gets too high and subroutines to numerous).

<Confusion of use vs. mention. In a lojban sentence, the x1 sumti is a

piece of text referring to something in the universe of discourse. In

the case of "fancu" the x1 place can be filled with a name, but this

is not required. It can be a description of a function, such as "the

inverse of the natural logarithm". This describes the exponential

function but does not name it. >

I thought you were using "name" very strictly; I was using it perhaps too loosely: I would count a description as a name in the appropriate sense, though this might well not be an abbreviation.

<The inverse of the logarithm function ku cu fancu domain-the-reals

range-the-positive-reals ma'o e^x

(where x is a bound variable representing the argument).>

Presumably {ce'u}.

I think our disagreement is largely terminological, except that you sem to think that functions the things are somehow actually (as opposed to nominally because of the confusion) involved in mathematical definitions. The statements which mathematicians call "definitions" are often at best truths about whatever they are talking about which are true by virtue of an almost identical definition, but that does not make them definitions, but theorems.

{1} =df {0'}

0' = 0' (by reflexivity of identity)

1 = 0' (by definitional replacement).

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