Re: Calculus XOR Probability

imaginatorium@xxxxxxxxxxxxx said:
Tony Orlow wrote:
imaginatorium@xxxxxxxxxxxxx said:

Tony Orlow wrote:
Matt Gutting said:
Tony Orlow wrote:
Matt Gutting said:
Tony Orlow wrote:
Matt Gutting said:
Tony Orlow wrote:
<snip>

Basically, all I'm saying boils down to inductive proof of equality holding for
infinite n. If some relationship between measures of a set holds for all finite
cases greater than some n, then it can be considered to hold for infinite n,
(Matt)
How do you know that there are any infinite n in the first place?
(Tony again)
Because there are sets with infinite numbers of elements, such as any set of
all reals in a finite interval. You cannot have half a real number in your set,
so this infinite number is integral, and therefore part of what I consider the
integers, or hyperintegers. Otherwise, infinite sets cannot have a size, which
makes the "infinite" part kind of meaningless.
But how do you know it's an integer in the first place? In other words, what
makes you so sure that there is an integer describing the size of this set?
Must sizes always be describable by a number? If so, why?

Matt

Because the size of the set is the count of the elements included in it, as far
as I'm concerned. That's why I don't accept a system where you add an infinite
number of elements and the "size" doesn't change. You don't normally have
fractional elements in a set, so this "count" has got to be integral, whether
it's finite or infinite. Of course, if you are using something like fuzzy set
theory, you may very well have set sizes which are not integral, but I don't
think that's what we're discussing, is it?

I'm not talking about fractional set sizes. I'm asking how you know that the
descriptor which describes the size of this set is a number.

Matt


Because that's what a number IS. You have a set of objects, and you ask what
the size is. How do you measure this? For finite sets, you COUNT the objects,
and the answer is a NUMBER.

Right. Which 'NUMBER' in particular? I suggest the one at which the
count stops (because it has reached the end of the finite set). In the
familiar method of counting by reciting a ditty, this answer is thus
the last number shouted out.

Right, so generally if there is no well defined end, there is no well defined
size.

Fine. I think that's an excellent start. I'd prefer to say that there
is no end at all, though a fortiori there is no well-defined end. I'm
not too happy with the idea that it can ever be productive to discuss a
somehow ill-defined something that obviously does not in fact exist.

So let's agree to eschew assigning a "size" to any set for which any
counting is unending. I'm sure you're familiar with the general notion
in software that if you have a property - say a database field called
"size" - that is not always available, it helps to have two sorts of
values in that field: one is just a number, that is the size; the other
is a flag (string, atom, whatever, in the language you're using), such
as "Nosize", meaning that no size is assigned. If you're familiar with
javascript, variables can have numeric values, but they can also have
the value 'NaN', for 'not a number', which means that they are _not_ a
number. However, the set of values that a javascript variable may have
is a very well-defined (finite!) one, and it is perfectly possible to
discuss the mathematical structure of the values including NaN under
arithmetical operators. It is also convenient to invent a name for the
union of the set of numeric values (numbers) with the not-number values
NaN and also the javascript specific value called "Infinity" (which
need not be related to the general i-word); we might call them
numeroids, remembering that at least some are definitely not numbers.

Do you follow this? If so can you not see that several of your
following remarks are, to put it politely, missing the point?
Mathematics is not about "numbers", it is about abstract structures far
more general than "numbers".


Doesn't it occur to you that, no matter what kind of field you're using, it all
boils down to 0's and 1's in the end, and that NaN is simply a numeric value
reserved for such purposes? Do the abstract structures of which you speak go
beyond numbers, or are they encoded as numbers?


So, for infinite sets, you want to claim that size
is something OTHER than a number???

Well, the "size" of an unending sequence can't really be the last
number you shout (oh, or was it 'sing') from the ditty, can it, since
there isn't a last number.

Is that true of all infinite sets? Isn't "1" the last number chanted in the
ditty of reals in [0,1]?

Do remind me how this ditty starts? I mean, I assume "0, ..." but what
follows 0?

(sigh) 0.000...000, 0.000...001, 0.000...010, ... , 0.111...110, 0.111...111,
1.000...000. That's the series (x=0->Big'un: Lil'un*x).


Of course that's not the size, but if you count by
Lil'uns, the last Lil'un is the Big'unth one, and Big'un's the last index
dittied.

And can you define this ditty in the way I suggested a while back?

A while back? You mean right above?


[Pause while you gibber for a bit]
[Pause with you ditty and doodle your way into the haze...]
[Well, I assumed you'd have to chant that mantra bit]

I didn't see a "largest finite" argument. I don't just go around saying "Huyah
huyah" for my health you know (although it helps ;).



Pray tell, what kind of a thing IS the size
of an infinite set, if not some kind of infinite number? If it's not a number,
what is it doing in mathematics? This just seems like a silly question.

See above.

So, it's just a placeholder for where you might have a number, but you don't
have a number, so it's NaN. Real great. What kind of math can you do on a Java
NaN?


Yes, it probably does to you, but then you have not the tiniest clue
what mathematics is nor what it is about. Do you think the elements of
the Klein 4-group are "numbers"?

I am not familiar with the Klein 4-group, but if truth itself is decomposable
into numbers, then what isn't? Of course, something like the size of a set is
generally considered to mean the number of elements in the set, so Klein or
not, set sizes are numbers. I'm not even interested in debating this. It's a
matter of mathematical fundamentalism.

The fact you don't know (even vaguely) what the Klein 4-group is is
diagnostic, I think. You have plainly never read any general
introductory texts to mathematics. (Goodness knows what "truth being
decomposable into numbers" means - sounds like something you caught off
Lester.)

No, it's something I tried to explain to Lester, but of course it met
resistance. He seems to want to explore the nature of logical truth, but not in
any logical way, as far as I can tell. I explained it in another post last
week.

And I wonder what you mean by "mathematical fundamentalism"? I
don't think you have any idea how flexible mathematicians can be (have
to be) - the only real requirement is that something makes sense (hint:
your stuff doesn't), and within that limit, almost anything can be
defined however you like. So you have been told many times already, but
the basic problem is that you simply have no idea how much you are
missing in terms of a basic grasp of this subject you believe you are
about to revolutionise.

Well, that was a really great explanation of a Klein 4-group. Thanks for your
assistance. I can see that you have only the furtherment of knowledge as a life
goal. Keep up the good work.


Brian Chandler
http://imaginatorium.org



--
Smiles,

Tony
.