Re: Calculus XOR Probability

imaginatorium@xxxxxxxxxxxxx said:
Tony Orlow wrote:
imaginatorium@xxxxxxxxxxxxx said:
Tony Orlow wrote:
David R Tribble said:
Tony Orlow wrote: [on NaN]
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?

Pretty much the same arithmetic operations you can do on Math.INFINITY
in Java. An arithmetic operation involving a NaN results in a NaN, and
similarly any operation involving an infinity operand results in either
an infinity or a NaN.

But you're not using Java floating-point arithmetic as a basis to
explain abstract mathematics, are you?

Why don't you ask Brian why he compared infinite set sizes to NaNs in Java?

Sorry, I probably just introduced extra confusion - not something
you're exactly short on, Tony.

You described counting the size of a finite set. OK. No problem.
You suggested that *in the sense of counting a finite set*, an infinite
set does not have such a "size". Also OK - no problem.

In handling numerical calculations in Javascript and other such
languages, it is possible to give a variable any of a (very) large
number of numerical values, and also the non-numerical value
represented by the atomic symbol "NaN". In just the same way one could
identify the size of any finite set as the numerical value obtained
from a counting process, and for any set that is not a finite set, use
a "placeholder" (if you like) value which is not a size (number), but
is an atomic symbol (e.g.) 'NaS' for not-a-size.

I wondered if this might help, but it doesn't look like it.

Well, that sounds almost like what the standard theory does, doesn't it? And,
what I've been advocating is a more numeric approach to infinity. I see
infinity as a quantitative concept, and seek to treat it more consistently with
the rest of math, and not as some kind of magical exception to every rule. So,
thanks for the suggestion, but it's not very satisfying, because NaS seems like
Not an Answer, and answers are there to be found.

OK, can you then explain your comment in the post just up this thread
(here's a googlink):
http://groups.google.com/group/sci.math/browse_frm/thread/c5e8522696fb2b97?scoring=d&;

No, I really can't. I see two posts of mine on that first page, and have no
idea which part of which you want me to comment on. Any clues?


Tony Orlow wrote:
imaginator...@xxxxxxxxxxxxx said:
Tony Orlow wrote:
<snip>
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.

Sometimes you seem to agree that (for example) the sequence of pofnats
(0, 1, 2, ...) has no end, and almost always you agree it has no
well-defined end. You say this means it "has no well defined size", yet
you (now) say you advocate "a more numeric approach to infinity", which
appears to mean you insist it must have a "size". It doesn't bother you
that these two claims appear to contradict each other?

It doesn't bother me that they appear to YOU to contradict each other. I have
repeatedly said that the unboundedness of the finites poses problems for
measuring the set of finite naturals, and that no real size can be attributed
to this set. However, if we say that there is a specific infinite number of
reals in each unit interval, and that there is an equally infinite number of
unit intervals on the real line, then we have a cohesive system consistent with
rules governing finite sets of such values, namely, the Inverse Function Rule.

Just a note to Virgil: You complained that my IFR will only work with
monotonically increasing functions. That covers most quantitative bijections,
unless they use trigonometric functions or some other way to rearrange the
target set out of quantitative order. More work needs to be done to directly
address such situations, but every bijection is an invertible function, and so
this is not something entirely new, but an extension to the notion of a
bijection as a tool for measuring infinite sets.



Brian Chandler
http://imaginatorium.org



--
Smiles,

Tony
.

Relevant Pages

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