Re: Review of Mueckenheims book.

On Mar 9, 9:31 am, David Marcus <DavidMar...@xxxxxxxxxxxxxx> wrote:
MoeBlee wrote:
On Mar 8, 6:55 pm, David Marcus <DavidMar...@xxxxxxxxxxxxxx> wrote:
MoeBlee wrote:
(3) Let a function be a triple <D C f>.

(3) it is.

You just keep asserting. I don't claim that you can't find such a
definition in some advanced areas of mathematics (I always said there
is all kind of terminology out ther). But please show me basic
textbooks in abstract algebra, analysis or topology that claim a
function is a triple.

All the ones that I listed in my previous post do exactly that, if you
know how to read a normal non-formal math book.

First, I ask you to read my post before responding to it line by line.
For example, I ask you not type your response to the first of the
following paragraphs before you're read the second too. (You might do
that anyway; I'm not suggesting that you don't.)

Not one of those examples you mentioned says that a function actually
is a triple. You may choose to read INTO the prose mixed with symbols
to formalize that way, but the prose mixed with symbols does not
actually say that a function is a triple. And I may elect for a triple
also but not take the function itself to be the triple.

It's fine to read above and beyond the text to put it into sharper
formalization. But it is just imperious of you to claim that your
branch in going above and beyond the actual text is standard as
opposed to not taking the function itself to be the triple, even
though yours is explicitly inconsistent with such analysis books as
Browder and Apostol, just for example (while also explicity endorsed
by books such as Hocking & Young).

And, by the way, you just said "all" of the books you had mentioned.
No, actually six of the seven non mathematical logic books (or was it
five of the six?).

Your INTREPRETATION is not standard in general mathematics. Notice
that I never claimed that my interpretation is standard in general
mathematics. I only said that it is found in many books even aside
from set theory, and is actually entailed (as I proved, formula for
formula) by many of the common definitions.

You are being dogmatic by insisting that your INTERPOLATION must be
standard and that mine must result from a lack of "knowing how to read
a math book".

Neither of our interpolations are standard, but I never claimed that
my INTERPOLATION is standard. Your snide "know how to read a math
book" is dismissed.

Beccause of cross posting and my replies getting out of correct
sequence (due to problems with my posting interface), I'm going to put
here some passages I wrote to Virgil:

I recognize that the 'function is a triple' approach (as opposed to
there is a triple of which the function is a coordinate approach) may
be better suited for many mathematical purposes. That is not at issue.
What is at issue is, with you, (1) the standard set theoretical
definition of 'is a function', and, with David, (2) whether the
'function is a triple' approach is standard in general mathematics. On
(1), the standard set theoretic definition is as I mentioned it. On
(2), while the 'function is a triple' approach can be found explicity
in some books (I found it in Hocking & Young last night), it is not
standard in general mathematics, as it is explicitly contradicted in
such analysis texts as Browder and Apostol, and algebra texts such as
Warner, and is inconsistent otherwise in many texts, and even
inconsistent with the way functions are handled in some topology
texts.

Between two triples proposals, David's clashes with set theory and
mine may clash with certain uses where indeed a mathematician wants to
take a function to BE a bundle of a certain relation, domain and
codomain, or other variations.

But I'm not proposing a definition that such mathematicians who want
the "bundle" would have to use. I'm proposing a formalization so that
we don't have to throw out the set theory definition and such that the
formalization is for anyone who is just interested in seeing a
formalization carried out, not necessarily for the everyday purposes
of mathematicians who want to use the "bundle" approach.

Neither route, David's or mine, can satisfy both ends - set theory at
one end and other kinds of ways of speaking as are found in what Ralf
Bader mentioned or in category theory, at least in the way you
described it. A formalization enforces a uniformity of definitions
while in fact the whole of mathematics does not have such a
uniformity. Such is life...

And, no, saying "f is a function from D to C" or
"f is a subset of DxC such that...", etc. is NOT claiming that f is a
triple.

Wrong.

I am amazed that you would say this. Math books are not written
in the formal language of set theory.

You don't need to tell ME that.

This doesn't mean that an
experienced mathematician doesn't know how to translate the usual prose
into such a language, if they feel the need.

What is at issue is that you arrogate that YOUR interpolation must be
standard, even though your INTERPOLATION is explicitly inconsistent
with a good many texts and inconsistent with how those texts handle
the matter in later parts of the book (e.g., see McCarty on topology
in which he speaks of the function f:x->y then goes right ahead to
speak of f itself as the function, not the function being <f x y> or
whatever).

As you would perhaps agree, a math text might use all kinds of
informal ways of speaking that if taken literally would be
inconsistent. And, yes, we as readers reconcile that, either
intuitively or formally. So when a book is going in both directions at
once - sometimes speaking of the function f:x->y and sometimes
speaking of the function f - it is a matter of our choice as to which
branch we want to take for our own mental (or even scribed)
formalization. Your choosing one branch is fine, but that does not
make it standard, especially since your branch is explicity
inconsistent with a fair number of textbooks.

What is nuts about this is that we could resolve this simply:

I recognize that my 'function' is to be spelled 'munction' and you
keep 'function' for certain kinds of triples. Or you recognize that
your 'function' is to be spelled 'dunction' and I keep 'function' for
certain kinds of sets of ordered pairs.

Meanwhile, we both understand the textbooks we read.

And in that light, you egregiously overstep by suggesting that I don't
know how to read a math book.

The fact that some logic/set theory books find it handy to use a
different definition of "function" for technical reasons does not change
the fact that the standard definition in virtually all of mathematics is
a triple (when formalized in set theory).

No, you're just arrogating that YOUR formalization is standard, even
though it is explicity inconistent with a good number of general math
books.

Your proposal just rides roughshod over the simplicity of doing things
like proving that f is a function in the manner Halmos did and as is
quite in keeping with a lot of common mathematical practice.

So, you still insist that your reading of Halmos is correct even after I
posted what Halmos actually wrote and pointed out that you are wrong?

No, I replied and showed that you are incorrect. Halmos EXPLICITY
gives the TEST for determing whether u is a function: That test is
whether u is a relation that is many-one, as is standard in set
theory. Look at it. Right there on page 48.

MoeBlee

.

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