Re: infinity
- From: Tony Orlow <aeo6@xxxxxxxxxxx>
- Date: Mon, 17 Oct 2005 16:15:49 -0400
Jonathan Hoyle said:
> >> I suppose I can see that, in theory, but it seems to me that in many
> >> ways the infinite reals and naturals may be treated the same as the
> >> finites, and that these systems shange treatment arbitrarily in ways
> >> that lead to nnsensical conclusions. Wouldn't it be better if we had
> >> one consistent way of dealing with infinite quantities? I think so.
>
> Perhaps, but how is one way better than another. Non-Standard Analysis
> is perhaps the most "natural" in an arithmetic sense, in that it
> perserves most of the known laws of arithmetic.
Yes, I have to dig into Robinson. Almost got the furnace installed....
> On the other hand, the
> most natural from a "size" perspective is cardinality and are (to me)
> the most "real" of the infinite systems, as it is the one that is most
> widely used across different areas of mathematics. The ordinals are an
> interesting phenomenon, but to me that are useful only in as far as a
> bridge to get to cardinal numbers.
Cardinality seems to be the standard and most widely accepted view of infinity,
but it also seems to have a perennial crops of detractors, I think for good
reason. It rubs many the wrong way, intuitively, and that's not acceptable to
everyone.
> The Extended Real line adds the
> ideal point at infinity to give the real line topological equivalence
> to the circle (likewise the Extended Complex plane becomes
> topologically equivalent to the sphere), and has almost no noticeable
> impact upon arithmetic.
It has some interesting implications, even if mostly of a philosophical nature.
It does exactly mirror the 2's complement system, in the limit as the number of
bits goes to oo. Now, doesn't the addition of the imaginary dimension, also
circular, create a toroidal topology, rather than spherical? I had always
thought speheres and hyperspehere made sense as the dimensional extension of
circles, having heard of the universe as the surface of a hypersphere, and yet
it occurred to me a few years ago that they would not preserve the orthogonal
nature of the curved dimensions, and that toroidal topologies would be the
natural result of multiple curved dimensions. This is why I have come to view
the universe as the 3D surface of a 4D toroid, since of course the nature of
all reality is derived from eternal mathematical truth. :D
> I don't think there is a need to choose.
> After all, since there are many finite numbers, why should there be
> only one infinite number?
Huh? I am advocating a plethora, a spectrum of infinities, both below N and
between N and R (aleph_0 and c). There should definitely not be just one
infinite number. Did I say that?
>
> >> What part of my reasoning is inconsistent, specifically, in your
> >> opinion?
>
> First let me point out that I think you can consistently create a
> system using some of your ideas, but most of the inconsistencies come
> (I think) due your adherence to past statements which cannot withstand
> critical examination. For example, you state that the the integers are
> "finite but unbounded", which is clearly a contradiction in terms. Now
> you may wish to ue your own definition of finite, say TO-finite, which
> means something other than the standard definition (Dedekind did the
> same kind of thing). But there is already a definition of "finite" and
> "infinite", and they are not yours. Pulling stunts like that will
> immediately place you into the halls of crankdom.
Perhaps, but it seems to me that the standard definitions are not consistent
with other mathematical notions of finite and infinite. For instance, I think
we can all agree that, to have an infinite number of strings constructed froma
set of symbols, or alphabet, if we have a finite alphabet, and limit our
strings to any finite length, we can only have a finite number of distinct
strings. In order to have an infinite number of strings, we need to have either
an infinite alphabet, or infintely long strings. If we are using a finite
digital system, then our alphabet is equal to the number base, a finite number,
and so we would need infinitely long strings. If we are using these strings to
represent the naturals, and they include infinitely long strings (which are not
all zeroes at infinite digit-places, since that would not change the value an
denote a unique natural number), because of the nature of digital
representation, any nonzero digit infinitely far to the left of the digital
point represents an infinite value. This seems like a perfectly valid argument,
given common notions of how arithmetic operations can produce infinite values.
>
> Secondly, instead of following mathematical procedure (start with
> knowns and prove conclusions based on these), you are playing politics
> and religion (start with your pre-determined conclusions, and then look
> for any arguments which support them, consistent or not). This is not
> how mathematicians operate, nor will any take you seriously when you do
> this.
Actually, I have put forth several starting assumptions, such as the N=S^L
above, upon which I think we can all agree, and derived conclusions from those,
since they seem to provide the application of the peoperties of the elements
which is key to making sense of the set. Of course I am looking for a certain
type of result, and want to create a system which describes exactly how to
achieve that seemingly correct result. If the system is consistent, then it
works. I believe my system is consistent, even if it's incosistent with
standard theory. I am not sure you can say that mathematicians don't go about
trying to prove certain conclusions sometimes anyway. This whole area seems to
have been the source of quite a bit of political, and probably religious,
haggling once upon a time.
> You postulate a system where there is a largest integer and no
> smallest infinity (the opposite of what we have in ZFC), and the
> natural question is: how did you arrive at this?
I don't postulate the existence of a largest finite overall. I advocate the use
of value range to determine relative sizes of infinite sets and absolute sizes
of subsets thereof. There is no largest finite, nor smallest infinity, by the
same logic. For any finite there is a larger one, and for any infinite a
smaller one, because addition or subtraction of a finite will never bridge the
gap. I suggested that you make your system more balanced by adding alpha as the
largest finite, but no one seemed to like that idea. :D
> If the only answer
> is: "I don't like the conclusions of the other system", you are
> engaging in religion, not mathematics. Frankly, there is nothing wrong
> with postulating a system where you end up with a largest integer and
> no smallest infinity, and it certainly will have different properties
> than the current system. But unless you are rigorously following the
> rule of logic to get there, all you have is essentially the
> mathematical equivalent to a fiction novel. Honestly, what reason do
> you think a mathematician would have to consider what you're saying, if
> you are playing so fast and loose with the truth?
Ah, but I am not. It may sound like that, as I am going at things from the
opposite of the normal direction, and working more visually than axiomatically
with the subject at hand, but the generalizations I am working on actually work
in a much more satisfying manner than cardinality/ordinality, and don't have
internal contradictions as far as I can see.
>
> I do believe you could architect a consistent system, using different
> axioms and have surprising conclusions, but what you have so far is (I
> am sorry to say) nothing more than a footnote in newsgroup crankdom. I
> believe you have the intellectual capabilities to break of of this, but
> I am also aware of the psychology behind so desperately clinging to
> this world view, that it may be beyond your emotional capability to
> give up. I don't know, that is a decision for you to decide.
What world view do you think I am clinging to? One where visualizations and
intuition count, and can be used as a starting point for a formal system? I
rather like that one.
>
> People familiar with sci.math know of many cranks here: James Harris,
> Archimedes Plutonium, Nathan the Great, Ross Finlayson and others.
> They will take up a lot of space on this newsgroup, but they will never
> be taken seriously as mathematicians. They won't, simply because they
> are unwilling (or unable) follow mathematical and logical procedure.
Are you sure that there is only one type of procedure that is helpful for the
study of quantity? It seems to me that a topic as complex as infinity should be
viewed from as many perspectives as possible. Perhaps some of them sound
cranky. Perhaps some of the cranks have valid points.
>
> Hope that helps (yeah, I know, I should find a new tag line),
>
> Jonathan Hoyle
>
>
--
Smiles (yeah me too, maybe),
Tony
.
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