Re: Where's respect? was Re: Corrective interpretation of real numbers

From: Eckard Blumschein (blumschein_at_et.uni-magdeburg.de)
Date: 02/11/05 

Date: Fri, 11 Feb 2005 13:20:21 +0100

On 2/10/2005 2:39 PM, Bernd Funke wrote:

>> Of course, I just indicated my intention to stress the reciprocity of
>> large and small.
>
> Which is irrelevant for the \epsilon-\delta-concept.

I did not deal much with the matter. So it is just my guess that the
"silent variable" delta already anticipated Cantor's actual infinity to
some extent. If this is correct, then denial of the actual infinity
corresponds to rejection of any consequence attributed to delta.

>
>
>>>> respectively but it is nonetheless different from genuine zero or
>>>> potential infinity,
>>>
>>> "potential infinity" is not a real number anyway.
>>
>> Factually seen, it plays about the same role as does zero.
>
> The word "factually" looses its meaning when used in conjunction with your
> beloved mollifiers such as "playing about the same role". You state a vague
> opinion, but not a *fact*.

No. I just separate the infinitely small from the understandable biasing
role of zero as a neutral element.

> A fact is e.g.: Addition is associative also
> when 0 is involved as an operand, but it's not with oo.

You got it.

>> I know, the axioms demand the existence of zero but not of infinity.
>
> Which axioms?

I just recall: Inf plagiarises Archimedes axiom and adds the postulated
existence of zero; Fund; perhaps more.

>> Since a "silent variable" delta is arbitrary,
>
> It's *not* arbitrary, it depends on the \epsilon.

The pair epsilon and delta must be considered together.

> But the \delta does *not* do your undefined "border crossing".

That's it. As long as we operate with epsilon delta, we are in the world
of finite numbers. I do not expect a possibility to "define" border
crossing as long as one denies that there is a border.

>
>
>>>> My original point was that IR+ fits to the ln(x) function with the
>>>> neutral element 1, while IR fits to the exp(x) function with the
>>>> corresponding neutral element 0, and they can be replaced by each other
>>>> without any loss.
>>>
>>> Sure, that exp(y) provides a bijection from IR to IR+ is unquestionable.
>>> But what's the point? In an application, like describing the position of
>>> a particle with constant velocity v, i.e. x=v*t, you would rewrite this
>>> as x'=exp(x)=exp(v*t)=exp(v*ln(t'))=t'^v just to have x' and t' in IR+?
>>> Where is the advantage?
>>
>> I didn't refer to simple application.
>
> But to what else? Who should when restrict himself to IR+?

Maybe we can resume these rather involved issues later.

>> I see this not a mathematical question but the user's business. He does
>> not need unnecessary definitions.
>
> Bending subjects again? Your claim was that "mathematics would be better
> off"! How a user implements or "approximates" infinity in a certain
> application, that's another business. IEEE found one obviously widely
> satisfying way.

Those wo are using mathematics in physics often complain about allegedly
poor models. Meanwhile, I presume serious mathematical inconsistency due
to blind rigorosity, instead.

>> I perfer verbal descriptions,
>
> I know, because then you can be as vague as you like.

Language is more subtle.

> He could have been ahead of his time, divining the Blumscheinian advances in
> math.

Please do not send contradicting signals. Either we respect each other
or we cannot talk seriously.

>
>
>>> Okay, at least I understand that his
>>> "genuine continuum" is something different than what is usually called
>>> continuum in math.
>>
>> Didn't you understand this before? I consider this difference important.
>
> If you consider it important, then *please* take care to clearly specify at
> each instance to *which* continuum you are referring. Either standard
> continuum or P-continuum (i.e. your "genuine continuum").

OK. I prefer "genuine" because P-continuum is not ubiquitously
understandable.

>
>> The question is
>> indeed how to judge so called actual alias "standard" infinity and so
>> called "standard" continuum.
>
> To judge the standard continuum

and actual infinity

with respect to what?

First of all a sound theoretical basis where obvious religious roots are
to be excluded from, at least for my opinion.

Please check:
>
> [ ] consistency

with logical reasoning: So far a lot of antinomies and of failed proofs.

> [ ] usability

Let's focus on fertility in mathematical practice as compared to the
expectations or promises by Hilbert and Fraenkel:
"Scientific revolution, at least as important as Kopernik, Einstein, and
Planck." Really?

> [ ] self-evidence

How to confirm that? Set-theory (Mengenlehre = ML) is almost as
self-evidend as was Marxism-Leninism.

> [ ] aesthetic

My gut feeling tells me that ML is far from beauty of mathematics for
several reasons.

> [ ] other: __________

I am here because I was unhappy with laws of standard mathematics
contradicting my reasoning. Admittedly, set theory is not to blame for
all of them but it made very old dilemmas more obvious.
>
>
>> Meanwhile I tend to guess Mueckenheim is correct.
>
> Math is not about guessing but about proving,

Well. Cantor already announced a proof for CH but failed to provide it.
Hilbert in 1925 offered a proof that was not tennable. Goedl and Cohen
were also unable to finally shed light into the CH-question.
Mueckenheim

>> You introduced the term silent variable as to describe delta.
>
> And you, being a free man, changed it into "silent limit" and immediately
> dropped its meaning.

Deliberately. I argue: A limit towards infinity makes sense while the
same delta in the meaning of a variable makes the elusive promise to
possibly reach infinity.

>> I got that
>> this is a veiled description of the fact that numbers cannot be extended
>> endlessly in practice of mathematics, neither to the small nor to the
>> large.
>
> As I said: You got it *wrong*, it's *not*, as I already pointed out, some
> sort of precision of the real numbers.

Mathematics assumes numbers to be exact, in other words, to have
infinite precision. This is only then reasonable if on restricts
resolution of representations of these numbers.

>>>> He spoke of "mere potentialities".
>>>
>>> Yes, but what did he tell about the *real numbers*?
>>
>> They do not have a directly approachable place in the sense of a
>> Dedekind cut.
>
> *No*, that's not what he said. AFAIU, he didn't object the
> rationals+irrationals, but he considered the Euclidian line (as *his*
> representative of the "genuine" continuum) to contain even *more* than the
> rationals+irrationals.

I will once again look into the sources. Let me tell my own reasoning.
Goedel questioned CH. He imagined cardinality of standard continuum much
larger. Maybe you, maybe Peirce also felt a desire to exclude any hole
on the line. I confess having adopted the idea that a really infinite
number of points actually constitutes the genuine continuum. For my
understanding, the standard notions of dense and completeness are
misleading. The question whether there are more irrational numbers than
rational ones is accordingly pointless. For delta and epsilon equal
zero, one cannot distinguish between rationals and irrationals. If there
is no number of rationals and no number of irrationals, then the sum of
both has the same quality: it is just not finite.

>> For instance, pi is an exact number but cannot be exactly
>> attributed to a corresponding value within IQ.
>
> Of course \pi is not in IQ (though tough guys as you and Mückenheim can
> place it there by writing it as "314159.../100000...", can't you?)

No. The meaning of the three points is not clear without explanation.
The number pi does not belong to IQ because th sequence 314... never
ends and has no finite period of repetition.

, that
> was the very reason to introduce the irrationals consistently.

Cantor definitely underestimated IQ.
A problem in physics is continuous shift of the scale. It does not work
with the standard continuum.

>
>
>>>>>> However, set theory requires resolved elements.
>>>>>> This dilemma is my central point.
>>>>>
>>>>> "Resolved" is no mathematical term from set theory, you just made it up
>>>>> without being able to give it a precise meaning.
>>>>
>>>> Resolution is clearly understandable to everybody.
>>>
>>> It is an intuitive concept from real life (and can be stated more
>>> precisely in physics) but it is *not defined* in the context of the
>>> continuum formed by reals numbers,
>>
>> Yes, I quote Mainzer in Ebbinghaus: "Und so schreibt Dedekind an
>> Lipschitz besondes mit Blick auf den Vollständigkeitsbegriff: ... Aber
>> Euklid schweigt vollständig über diesen für die Arithmetik wichtigen
>> Punkt"
>
> What's your point? Where does the word "resolution" (="Auflösung") occur in
> this quote?

Vollständigkeit = completeness losses its meaning in case of an infinite
amount because this amount cannot be specified in terms of numbers.
Before checking completeness, one has to resolve the points.

>
>
>>> What do you mean by an *element* being discrete anyway? I know only of
>>> discrete sets.
>>
>> Elements are entities.
>
> What kind of answer is this? I asked you what a *discrete* element should
> be.

I reiterate: The elements of sets are discrete. The number one denotes a
concrete entity. Negative whole numbers including zero and complex whole
numbers can only denote abstract entities. Mathematics creates new
entities by combining elementary ones. In order to leave the realm of
discrete mathematics, one has to perform border crossing by combining an
unlimited (not finite) amount of entities.

>
>
>>> You're wrong about "Kanticity" and infimum.
>>
>> Formally yes, factually not.
>
> Very well, Eckard, using puns to evade reason: What *is* the fact in your
> "factually"? Show to us where Peirce involved the infimum in his
> definition(s) of "Kanticity", please.

My fact is not Peirce. The fact is that I see a logical contradiction if
one demands delta>0 but assumes for IR+ an infimum at 0.

>> I called exactly this source not representative.
>
> Sure, you are the censor.

I gave less biased and more detailed references:
http://www.angelfire.com/super/magicrobin/peirce.htm
http://agora.phi.gvsu.edu/kap/Neoplatonism/

> To me it seems that Peirce turned strongly
> from math to philosophy in his later years.

Mathematics needs philosophy only if it is in a deadlock.

>
>
>>>> I quote Peirce: ?A continuum is precisely that every part of which has
>>>> parts?
>>>
>>> So, is the number (say) 1 a part of the continuum? If yes, what are its
>>> "parts"?
>>
>> You got the point. Numbers are not parts.
>
> So, the number 1 does *not* belong to the P-continuum? Furthermore, *no*
> numbers belong to the P-continuum? Is the P-continuum a set at all?
> Apparently not.

You got it.
>
> So what's your point now? That the P-continuum cannot be described by set
> theory? That's no surprise if it's not a set at all.

You got it. Peirce wrote of mere potentialities.

> Or is your point that the P-continuum is "better" than the standard one?

Yes in the sense that it is a more natural theoretical concept.
Yes in the sense that it should be the basis for corrective
interpretations of real numbers.

> Then, in turn, you should explain how to operate on this non-set.

Here I will try an answer to Herman Jurjus:
Well, the problems with set theory are not new. The only strong and
ultimative argument of proponents is the proclaimed lack of an
alternative. I am a layman. Nonetheless I guess, Brouwer, Heyting,
Robinsohn, and many others did not yet find a much better alternative.
The best alternative will be a critical evaluation of standard theory.
Would omission of alephs etc. really damage mathematics? Given there are
consequences, would they really be unacceptable or possibly even
advantageous to the user of mathematics?
As long as the notion 'number' and in particular 'zero' is not yet
subtle enough, botchering with the axioms might worsen the crisis.

> Any you
> should also point out the shortcomings of the standard continuum, besides
> being less intuitive (to you) than the P-continuum.

Here you got me wrong. I confirm: We definitely need something like the
standard continuum. Does this mean that we need alephs too if they will
proove nonsense?

>
> [Numbers]
>> They merely may be used in order to count parts, e.g. properties.
>
> That's *your* narrowminded interpretation of numbers, not Peirce's.

Peirce wrote of 'mere potentialities'. I am ready to follow the idea
that one can attribute infinitesimal short pieces dx to the line x with
the caveat that merely shrinking dx does not change the scale-marks'
assumed symmetrical position with respect to zero.
Logically, a line can consist of a finite number of small liniear pieces
while only am infinite amount of points constitutes a line. So the
concept of pieces is superior to Dedekind's cuts. A piece corresponds to
a pair of points.

>
>>>> and ?the points on a line not yet actually determined are mere
>>>> potentialities?
>>>
>>> How to "actually determine" the points?
>>
>> Thank you for this forward pass: By means of a "Grenzuebergang" (border
>> crossing) because continuum and numbers are essentially different from
>> each other but mutually complement each other. That's my message.
>
> Numbers are already necessarily different from the *standard* continuum,
> because the latter is a set and the numbers are its elements.

You did not get the fundamental difference between discrete and really
continuous.

> And if the P-continuum does not even *contain* numbers, then it's no
> surprise that it is "essentially different" from numbers.

I appreciate your almost correct understanding. The genuine continuum
might be imagined to contain all numbers in a merged state exhibiting
strange peculiatiies: You can remove or add as many points/numbers as
you like. This will not make any difference, provided that the amount of
 removed/added points/numbers is finite.

>
> Moreover: Your magical "border crossing" when transforming an "element" of
> the P-continuum, i.e. a "mere potentiality" into an actual number, has
> little to do with Weierstrass' concept of limits.

Yes, Weierstrass avoids border crossing. Nonetheless, his atomistic
continuum can approach the genuine one as close as you like.

>> Exact description of the genuine continuum by means of numbers requires
>> border crossing beyond the very small as well as beyond the very lage.
>> Because these border crossings are only possible in fiction, the ideal
>> description is not available. There are just approximations.
>
> Maybe I begin to grasp you dissatisfaction: In the (to you more intuitive)
> P-continuum there is "more" than just real numbers,

No. There is not even more than rational numbers.

> but, being an ultrafinitist,

Why not a heretic? A finitist is just someone who does not share
Cantor's belief.

> you cannot appreciate this "more" since you can only operate
> with numbers on a "digital display".

I return this infamous accusation by reminding of Cantor who also only
could operate with rational numbers even when he intended to demonstrate
that real numbers are uncountable.

> On the other hand, the "users" (i.e.
> mathematicians) of the standard continuum (which "only" contains the
> reals), being non-finitists, don't need to revert to approximations.

No, they are just not aware that they are kept in Cantor's paradise.
A closer look reveals that the standard continuum just embeds the reals.
One cannot exactly approach pi from this side.

> Weierstrass showed how to prove that 0.[9]=1 *without* involving an infinite
> number of operations.

A am absolutely sure that any proof of 0.[9]=1 is bogus unless the
series of nines is not subject to any limitation. Please point me to the
original paper.

> But anyway, you're trying to point out an inconsistency in 0.[9]=1
> by means of an attack from "outside"

Of course.

Eckard, that's *not* an inconsistency.

Is mathematics a matter of belief?

>
>> I confess sharing the finitist
>> position for a compelling reason:
>> You might either cross the border
>> between numbers and continuum or not. In that case, tertium non datur.
>
> Already since you cannot describe how that "borders crossing" works (because
> it's beyond your imagination),

According to Hilbert, infinity is beyond the imagination of anybody.
He referred to infinite sets.
I rather appreciate Aristotele, Newton, and Gauss who clearly
understood: Infinitum actu non datur.
There are limit values of sums or products of endless but convergent series.
Infinity is just a "Facon de parler".

> it's bogus to call comparing it to your
> above allegories "reasoning".

My reasoning itself was not allegorical but compelling and by no means
bogus:
You might either cross the border between numbers and genuine continuum
or not.
Let me clarify that the standard continuum borders the genuine one.
While validity of 0.[9]=1 with a non-ending number of nines requires the
continuum to be genuinely continuous, set theory does not work there.

I see it bogus if one concludes from similarity of the symbol oo with
"other" numbers that oo can be used in calculations like a number.
At least Weierstrass und Poission actually spoke of infinite numbers!
What nonsense.

>
>>>> Focus on the last (least) decimal. Let it
>>>> count step by step. After 8 follows 9, after 9 follows 0 no matter how
>>>> large the supremum is.
>>>> You could argue the number of decimals is infinitely large.
>>>
>>> That's the very meaning of 0.[9]. You can take on the ultrafinitistic
>>> (sp?) point of view by denying the mere existence of 0.[9]. Fine, but
>>> then you cannot produce statements of it being un-/equal to something.
>>
>> I see it the other way round.
>
> That is?

0.[9] existst (merged with 1) within the genuine continuum.
However, it goes beyond the limit set by delta>0. So the standard
continuum just contains approximative values with as many nines as you
like except for the impossible i.e. a non-ending series of decimals.

"The ultrafinitistic viewpoint is correct

If so, why do you not share it?

>
>>>> This would actually imply 0.99...=1.
>>>> However, this would change the discrete
>>>> process of counting into a continuous estimation.
>>>
>>> "change the discrete process of counting into a continuous estimation" is
>>> the usual Blumscheinian utterly vague non-math.
>>
>> Provided you are willing to understand me. Do you have a better
>> description?
>
> No, because it's *your* diffuse idea, and you are refusing to describe it in
> a precise way (hiding behind your ideological pretexts).

Misusing my name is no sign of sublimity. You will perhaps accept that
Stifel and Weyl were more ingenious mathematicians than you. Standing
beyond any suspicion of ideological bias, they used the mathematically
absolutely correct descriptions fog and sauce. Your word 'diffuse' came
close, too. Can you grasp that it may be more precise than any formula?

>
>
>>>> Cantor's paradise is
>>>> doomed to restrict itself to a finite amount of numbers.
>>>
>>> Why should this be true?
>>
>> Because numbers denote repetitious operations.
>
> Your ultrafinitist viewpoint of what numbers "are", does *not* apply "inside
> Cantor's paradis".

I agree that my notion of numbers is less vague.

> You are merely pointing out dissaccords between the ultrafinitist's
> viewpoint and Platonism.

How that? I share the Platonist point of view that mathematics is to be
found rather than botchy "created" as demonstrated by Cantor.

>
>>>> An infinite amount of numbers is a illusion
>>>
>>> Why? (Beyond the fact that *all* mathematical objects are in a way
>>> "illusions", us being unable to touch them.)
>>
>> Mathematical objects should be abstractions rather than illusions.
>
> Look who's talking!

?

>> Let's reject mere illusions like the actual infinity,
>
> What's illusory about it? That it does not fit on a "digital display"? Do
> you call that "abstraction"?

I see it already an illusion that something can be finite and infinite
at a time. Let's ask for what reason one should distinguish between the
unambiguous notion of infinity as something potential and the mere
allowance to approach it. Cantor was correct when he discovered that IQ
behaves like IN in that, it is open-ended. However he failed to provide
any tennable evidence for his Infinitum creatum sive Transfinitum.
Obviously, his continuum hypothesis related to intentions for bridging
the unbridgeable gap between numbers and the genuine continuum. While it
was alost a matter of course to exclude infinity from the reals. Nobody
was willing to abandon its reciprocal zero.

>
>> endlessly leading to indications of
>> inconsistency.
>
> You still were not able to show a single *inconsistency*.
>
I explained a lot of my objections again and again. Perhaps with my
original posting too.
As a layman, I recently even found out that Cantor's proof for reals
actually was performed with just rationals.

> You really are an imperialist.

??

Eckard

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