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

From: [email-masked]
Date: Wed, 27 Feb 2002 12:08:04 EST

In a message dated 2/27/2002 4:52:03 AM Central Standard Time, 
[email-masked] writes:

> 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).