Re: [lojban] Re: [jboske] RE: Anything but tautologies

[Edward Cherlin <edward@webforhumans.com>]

On Friday 15 February 2002 17:46, pycyn@aol.com wrote:
> In a message dated 2/15/2002 5:32:59 PM Central Standard Time,
>
> jjllambias@hotmail.com writes:
> > >Definitions are, by definition, of words.

True in dictionaries but not in mathematics. Definition means making 
meaning definite, whether in reference to words or things.

> > Are functions words? Or are functions not really defined,
> > only their names?

Functions can be defined. Functions are not words. Functions can be 
assigned names. Their names may be words or phrases.

> No, functions are things (relations with cooccurence restrictions,
> say).

This is one of the usages found in mathematics. There are others.

> <Can we define a function?>
>
> No. We can call attention to a function that already exists, by
> naming it or otherwise pointing it out. If we name it, we can
> define what the name (refers to) by describing the function
> involved. This is also how we call attention it, usually.

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. 

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.

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.

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.

> <What we normally mean by defining a function does not necessarily
> involve giving it a name. For example, we can define a function by
> saying that it maps each natural number n to the natural number
> n+1. There, it's defined. Now we can give it a name if we want to,
> but there is no need to. It seems odd that in Lojban we can only
> talk of functions with names, never of nameless functions! Well, we
> can do {zi'o fancu} I suppose>

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.

> 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.

> 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.

> We can, of course
> describe nameless functions, using {zi'o} if necessary -- or just
> leaving them with nebulous names. And then we can talk about the
> functions describe (but in the process the description is liable to
> turn into a name "the so and so" {le du'u makau broda ce'u}).

da poi le fancu x2 x3 x4 cu is-nameless

> My copy of the gismu list does not actually insist that the fancu1
> be a name or a bit of text, but it is probably out of date. I
> agree that that requirement is a mistake, for all sorts of reasons.

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. 

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).

I leave the rest of the translation into Lojban as an exercise for 
the reader.
-- 
Edward Cherlin
edward@webforhumans.com
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