Re: Calculus XOR Probability

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

Tony Orlow wrote:
<snippolino>

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.

Oh dear. First mistake - assuming you understood something about
software.

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?

No, it doesn't. ("Not-a-number" is not not a number, then?)

Do the abstract structures of which you speak go
beyond numbers, or are they encoded as numbers?

You have no concept of an abstract model for a programming language? Is
that right?
If so, we'll give up on this, I think.

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).

Yes, perhaps it is, but it isn't the reals (not the real reals,
anyway), since there can't be a "second real" after 0, because if there
were, call it p, then p/2 is also a real, contradiction, but I recall
you can't understand reductio ad absurdam. (Incidentally, how do you
pronounce 0.000...000 while you're singing the ditty?)

<snip>
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.

Try http://en.wikipedia.org/wiki/Klein_four-group

The elements of the Klein 4-group are not numbers in any sense that
would be recognisable by a mathematician, but don't let that stop
you...

Brian Chandler
http://imaginatorium.org

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