Mathematics in its relation with the divine has played a special role in the course of history. [...] Mathematics is abstract and it often seems absolute, universal, eternal and pure. More than other kinds of knowledge it possesses characteristics that we associate with the divine. [T. Koetsier, L. Bergmans: "Mathematics and the divine: A historical study", Elsevier (2005) p. 4]

At the moment you say that infinite sets are of the same size when there exists a bijection, then you already introduce some of the choices Cantor made. But those are choices, not proofs. [...] there are at least two choices you can make: a) You may refer to  even when there is no logical system that can list all its elements (this is Cantor's choice). b) You must be aware that  is never complete in your logical system (alternative to Cantor). I don't think choice b is very attractive, but to my opinion it is a way you can try to go. However, I always questioned if saying that || < || has any more meaning than saying that irrational numbers are green. [...] I think the people that don't like Cantor have the greatest problem with the fact that whatever system you use,  contains numbers which can not be expressed. At least I can say about it, it is a fair concern. [L.B. Kruijswijk in "Shannon defeats Cantor = single infinity type", sci.math (11 & 12 Dec 2003)]

The concept of "infinity" is anti-science, and the use of infinity as a constant by mathematicians and scientists, whether they know it or not, plays into the hands of advocates of religions and other forms of the supernatural, who also describe the objects of their beliefs as infinite. [...] Unfortunately, the recognition of infinity in science is also an excellent justification for zealots to convince others that, since science accepts infinity as a reality, deities that are just as infinite and unknowable are also reality. [H. Lubin in "To settle infinity dispute, a new law of logic", Quanta Magazine (20 Jul 2014)]

The line all points of which have a finite distance is itself finite. [T. Pesch: "Institutiones philosophiae naturalis secundum principia S. Thomae Aquinatis", Freiburg, Herder (1880) § 425]

Ha Ha, conceptualism, how outdated.

WM schrieb:
> Mathematics in its relation with the divine has played a special role in the course of history. [...] Mathematics is abstract and it often seems absolute, universal, eternal and pure. More than other kinds of knowledge it possesses characteristics that we associate with the divine. [T. Koetsier, L. Bergmans: "Mathematics and the divine: A historical study", Elsevier (2005) p. 4]
>
> At the moment you say that infinite sets are of the same size when there exists a bijection, then you already introduce some of the choices Cantor made. But those are choices, not proofs. [...] there are at least two choices you can make: a) You may refer to  even when there is no logical system that can list all its elements (this is Cantor's choice). b) You must be aware that  is never complete in your logical system (alternative to Cantor). I don't think choice b is very attractive, but to my opinion it is a way you can try to go. However, I always questioned if saying that || < || has any more meaning than saying that irrational numbers are green. [...] I think the people that don't like Cantor have the greatest problem with the fact that whatever system you use,  contains numbers which can not be expressed. At least I can say about it, it is a fair concern. [L.B. Kruijswijk in "Shannon defeats Cantor = single infinity type", sci.math (11 & 12 Dec 2003)]
>
> The concept of "infinity" is anti-science, and the use of infinity as a constant by mathematicians and scientists, whether they know it or not, plays into the hands of advocates of religions and other forms of the supernatural, who also describe the objects of their beliefs as infinite. [...] Unfortunately, the recognition of infinity in science is also an excellent justification for zealots to convince others that, since science accepts infinity as a reality, deities that are just as infinite and unknowable are also reality. [H. Lubin in "To settle infinity dispute, a new law of logic", Quanta Magazine (20 Jul 2014)]
>
> The line all points of which have a finite distance is itself finite. [T. Pesch: "Institutiones philosophiae naturalis secundum principia S. Thomae Aquinatis", Freiburg, Herder (1880) § 425]
>

On Tuesday, December 28, 2021 at 9:51:07 AM UTC, WM wrote:
> Mathematics in its relation with the divine has played a special role in the course of history. [...] Mathematics is abstract and it often seems absolute, universal, eternal and pure. More than other kinds of knowledge it possesses characteristics that we associate with the divine. [T. Koetsier, L.. Bergmans: "Mathematics and the divine: A historical study", Elsevier (2005) p. 4]
>
> At the moment you say that infinite sets are of the same size when there exists a bijection, then you already introduce some of the choices Cantor made. But those are choices, not proofs. [...] there are at least two choices you can make: a) You may refer to  even when there is no logical system that can list all its elements (this is Cantor's choice). b) You must be aware that  is never complete in your logical system (alternative to Cantor). I don't think choice b is very attractive, but to my opinion it is a way you can try to go. However, I always questioned if saying that || < || has any more meaning than saying that irrational numbers are green. [...] I think the people that don't like Cantor have the greatest problem with the fact that whatever system you use,  contains numbers which can not be expressed. At least I can say about it, it is a fair concern. [L.B. Kruijswijk in "Shannon defeats Cantor = single infinity type", sci.math (11 & 12 Dec 2003)]
>
> The concept of "infinity" is anti-science, and the use of infinity as a constant by mathematicians and scientists, whether they know it or not, plays into the hands of advocates of religions and other forms of the supernatural, who also describe the objects of their beliefs as infinite. [...] Unfortunately, the recognition of infinity in science is also an excellent justification for zealots to convince others that, since science accepts infinity as a reality, deities that are just as infinite and unknowable are also reality. [H. Lubin in "To settle infinity dispute, a new law of logic", Quanta Magazine (20 Jul 2014)]
>
> The line all points of which have a finite distance is itself finite. [T. Pesch: "Institutiones philosophiae naturalis secundum principia S. Thomae Aquinatis", Freiburg, Herder (1880) § 425]

And if it turns out the universe (speaking more broadly than the merely observable universe) turns out to be actually infinite? Or if the many-worlds interpretation of quantum mechanics is correct?

On Tuesday, December 28, 2021 at 4:51:07 AM UTC-5, WM wrote:
> Mathematics in its relation with the divine has played a special role in the course of history. [...] Mathematics is abstract and it often seems absolute, universal, eternal and pure. More than other kinds of knowledge it possesses characteristics that we associate with the divine. [T. Koetsier, L.. Bergmans: "Mathematics and the divine: A historical study", Elsevier (2005) p. 4]
>
> At the moment you say that infinite sets are of the same size when there exists a bijection, then you already introduce some of the choices Cantor made. But those are choices, not proofs.

If only you understood what a finite set was, you would not make such absurd claims. BTW we are STILL waiting for your definition of a finite set, Mucke. What seems to be the problem?

Dan

> And if it turns out the universe (speaking more broadly than the merely observable universe) turns out to be actually infinite? Or if the many-worlds interpretation of quantum mechanics is correct?

Many worlds and uncountable sets. Kindred notions.

Regards, WM

Dan Christensen schrieb am Dienstag, 28. Dezember 2021 um 14:12:48 UTC+1:
> On Tuesday, December 28, 2021 at 4:51:07 AM UTC-5, WM wrote:
> > Mathematics in its relation with the divine has played a special role in the course of history. [...] Mathematics is abstract and it often seems absolute, universal, eternal and pure. More than other kinds of knowledge it possesses characteristics that we associate with the divine. [T. Koetsier, L. Bergmans: "Mathematics and the divine: A historical study", Elsevier (2005) p. 4]
> >
> > At the moment you say that infinite sets are of the same size when there exists a bijection, then you already introduce some of the choices Cantor made. But those are choices, not proofs.
> If only you understood what a finite set was, you would not make such absurd claims.

Classical logic was abstracted from the mathematics of finite sets and their subsets. (The word finite is here to be taken in the precise sense that the members of such set are explicitly exhibited one by one.) [H. Weyl]

Regards, WM

more bogus bad math quotes from WM. Foul.

On 12/28/2021 3:51 AM, WM wrote:
> Mathematics in its relation with the divine has played a special role in the course of history. [...] Mathematics is abstract and it often seems absolute, universal, eternal and pure. More than other kinds of knowledge it possesses characteristics that we associate with the divine. [T. Koetsier, L. Bergmans: "Mathematics and the divine: A historical study", Elsevier (2005) p. 4]

Mathematics and the divine. Conjecture.

>
> At the moment you say that infinite sets are of the same size when there exists a bijection, then you already introduce some of the choices Cantor made. But those are choices, not proofs. [...] there are at least two choices you can make: a) You may refer to  even when there is no logical system that can list all its elements (this is Cantor's choice). b) You must be aware that  is never complete in your logical system (alternative to Cantor). I don't think choice b is very attractive, but to my opinion it is a way you can try to go. However, I always questioned if saying that || < || has any more meaning than saying that irrational numbers are green. [...] I think the people that don't like Cantor have the greatest problem with the fact that whatever system you use,  contains numbers which can not be expressed. At least I can say about it, it is a fair concern. [L.B. Kruijswijk in "Shannon defeats Cantor = single infinity type", sci.math (11 & 12 Dec 2003)]

a Quote from sci.math! Bogus.

>
> The concept of "infinity" is anti-science, and the use of infinity as a constant by mathematicians and scientists, whether they know it or not, plays into the hands of advocates of religions and other forms of the supernatural, who also describe the objects of their beliefs as infinite. [...] Unfortunately, the recognition of infinity in science is also an excellent justification for zealots to convince others that, since science accepts infinity as a reality, deities that are just as infinite and unknowable are also reality. [H. Lubin in "To settle infinity dispute, a new law of logic", Quanta Magazine (20 Jul 2014)]

"infinity ... plays into the hands ... of religions and other forms of the supernatural." Bogus.

>
> The line all points of which have a finite distance is itself finite. [T. Pesch: "Institutiones philosophiae naturalis secundum principia S. Thomae Aquinatis", Freiburg, Herder (1880) § 425]

al finite lines are finite. TRUE !!

WM wrote on 12/28/2021 :
> Dan Christensen schrieb am Dienstag, 28. Dezember 2021 um 14:12:48 UTC+1:
>> On Tuesday, December 28, 2021 at 4:51:07 AM UTC-5, WM wrote:
>>> Mathematics in its relation with the divine has played a special role in
>>> the course of history. [...] Mathematics is abstract and it often seems
>>> absolute, universal, eternal and pure. More than other kinds of knowledge
>>> it possesses characteristics that we associate with the divine. [T.
>>> Koetsier, L. Bergmans: "Mathematics and the divine: A historical study",
>>> Elsevier (2005) p. 4]
>>>
>>> At the moment you say that infinite sets are of the same size when there
>>> exists a bijection, then you already introduce some of the choices Cantor
>>> made. But those are choices, not proofs.
>> If only you understood what a finite set was, you would not make such absurd
>> claims.
>
> Classical logic was abstracted from the mathematics of finite sets and their
> subsets. (The word finite is here to be taken in the precise sense that the
> members of such set are explicitly exhibited one by one.) [H. Weyl]

Philosophy needs to be scraped off the shoes of mathematics.

On Tuesday, December 28, 2021 at 9:03:20 AM UTC-5, WM wrote:
> Dan Christensen schrieb am Dienstag, 28. Dezember 2021 um 14:12:48 UTC+1:
> > On Tuesday, December 28, 2021 at 4:51:07 AM UTC-5, WM wrote:
> > > Mathematics in its relation with the divine has played a special role in the course of history. [...] Mathematics is abstract and it often seems absolute, universal, eternal and pure. More than other kinds of knowledge it possesses characteristics that we associate with the divine. [T. Koetsier, L. Bergmans: "Mathematics and the divine: A historical study", Elsevier (2005) p. 4]
> > >
> > > At the moment you say that infinite sets are of the same size when there exists a bijection, then you already introduce some of the choices Cantor made. But those are choices, not proofs.

> > If only you understood what a finite set was, you would not make such absurd claims.

> Classical logic was abstracted from the mathematics of finite sets and their subsets. (The word finite is here to be taken in the precise sense that the members of such set are explicitly exhibited one by one.

Makes no sense. Sorry, this is just not good enough, Mucke. You need a formal definition stated on the language of set theory. A set is said to be finite iff ____________________________________. (Fill in the blank.)

Dan

Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

On Tuesday, December 28, 2021 at 4:11:53 AM UTC-8, ross.pro...@gmx.com wrote:
> On Tuesday, December 28, 2021 at 9:51:07 AM UTC, WM wrote:
> > Mathematics in its relation with the divine has played a special role in the course of history. [...] Mathematics is abstract and it often seems absolute, universal, eternal and pure. More than other kinds of knowledge it possesses characteristics that we associate with the divine. [T. Koetsier, L. Bergmans: "Mathematics and the divine: A historical study", Elsevier (2005) p. 4]
> >
> > At the moment you say that infinite sets are of the same size when there exists a bijection, then you already introduce some of the choices Cantor made. But those are choices, not proofs. [...] there are at least two choices you can make: a) You may refer to  even when there is no logical system that can list all its elements (this is Cantor's choice). b) You must be aware that  is never complete in your logical system (alternative to Cantor). I don't think choice b is very attractive, but to my opinion it is a way you can try to go. However, I always questioned if saying that || < || has any more meaning than saying that irrational numbers are green. [...] I think the people that don't like Cantor have the greatest problem with the fact that whatever system you use,  contains numbers which can not be expressed. At least I can say about it, it is a fair concern. [L.B. Kruijswijk in "Shannon defeats Cantor = single infinity type", sci.math (11 & 12 Dec 2003)]
> >
> > The concept of "infinity" is anti-science, and the use of infinity as a constant by mathematicians and scientists, whether they know it or not, plays into the hands of advocates of religions and other forms of the supernatural, who also describe the objects of their beliefs as infinite. [...] Unfortunately, the recognition of infinity in science is also an excellent justification for zealots to convince others that, since science accepts infinity as a reality, deities that are just as infinite and unknowable are also reality. [H. Lubin in "To settle infinity dispute, a new law of logic", Quanta Magazine (20 Jul 2014)]
> >
> > The line all points of which have a finite distance is itself finite. [T. Pesch: "Institutiones philosophiae naturalis secundum principia S. Thomae Aquinatis", Freiburg, Herder (1880) § 425]
> And if it turns out the universe (speaking more broadly than the merely observable universe) turns out to be actually infinite? Or if the many-worlds interpretation of quantum mechanics is correct?

It seems one should keep in mind that quantum mechanics' MWI is
an interpretation in a theory, for what after causality, illustrates
the results according to chance. But, there's nothing what must not
eventually result as "real" except that after causality: MWI isn't scientific
in the sense it's not falsifiable, so it's fair to say in conversation as it's
let out as an "interpretation", for what it models in the theory, what
the theory models in its physical interpretation.

As I've often: "mathematics _owes_ physics more and better mathematics
of infinity", is that, for solving some super-classical problem in the classical
in the free field, for what is like a singularity theory (for what are singularities
everywhere or original analysis), there is solving for unipotential as all what
results that the linear to non-linear (disorganized) to highly-non-linear (reorganized),
has that usual least but most expressive system have a label for a constant
for infinity collected near the top of declarations.

I.e., besides that of course symbolically for algebra that "writing a label for
infinity is a concise notation for systems of rates in unboundedness", the
"standard" analysis, and real analysis is always part of foundations, then
also, high-order mathematics of infinity, results from the interplay of the
three definitions of continuity: line, field, and signal.

I.e., what mathematics owes physics for mathematics of infinity is more
for the quantum mechanics themselves, than unfalsifiable interpretations
what follow causality and chance.

Anyways mathematics needs (or, has) now at least three definitions of
continuity: line, field, and signal, most programs of which are aware of
at least one, and often for example only one.

.... That foundations includes from the most direct central notions of
mathematical logic.

On 12/28/2021 8:03 AM, WM wrote:
> Dan Christensen schrieb am Dienstag, 28. Dezember 2021 um 14:12:48 UTC+1:
>> On Tuesday, December 28, 2021 at 4:51:07 AM UTC-5, WM wrote:
>>> Mathematics in its relation with the divine has played a special role in the course of history. [...] Mathematics is abstract and it often seems absolute, universal, eternal and pure. More than other kinds of knowledge it possesses characteristics that we associate with the divine. [T. Koetsier, L. Bergmans: "Mathematics and the divine: A historical study", Elsevier (2005) p. 4]
>>>
>>> At the moment you say that infinite sets are of the same size when there exists a bijection, then you already introduce some of the choices Cantor made. But those are choices, not proofs.
>> If only you understood what a finite set was, you would not make such absurd claims.
>
> Classical logic was abstracted from the mathematics of finite sets and their subsets. (The word finite is here to be taken in the precise sense that the members of such set are explicitly exhibited one by one.) [H. Weyl]
>
> Regards, WM
>
>

more diversion

On Tuesday, December 28, 2021 at 6:06:25 AM UTC-8, Serg io wrote:
> more bogus bad math quotes from WM. Foul.
> On 12/28/2021 3:51 AM, WM wrote:
> > Mathematics in its relation with the divine has played a special role in the course of history. [...] Mathematics is abstract and it often seems absolute, universal, eternal and pure. More than other kinds of knowledge it possesses characteristics that we associate with the divine. [T. Koetsier, L. Bergmans: "Mathematics and the divine: A historical study", Elsevier (2005) p. 4]
> Mathematics and the divine. Conjecture.
> >
> > At the moment you say that infinite sets are of the same size when there exists a bijection, then you already introduce some of the choices Cantor made. But those are choices, not proofs. [...] there are at least two choices you can make: a) You may refer to  even when there is no logical system that can list all its elements (this is Cantor's choice). b) You must be aware that  is never complete in your logical system (alternative to Cantor). I don't think choice b is very attractive, but to my opinion it is a way you can try to go. However, I always questioned if saying that || < || has any more meaning than saying that irrational numbers are green. [...] I think the people that don't like Cantor have the greatest problem with the fact that whatever system you use,  contains numbers which can not be expressed. At least I can say about it, it is a fair concern. [L.B. Kruijswijk in "Shannon defeats Cantor = single infinity type", sci.math (11 & 12 Dec 2003)]
> a Quote from sci.math! Bogus.
> >
> > The concept of "infinity" is anti-science, and the use of infinity as a constant by mathematicians and scientists, whether they know it or not, plays into the hands of advocates of religions and other forms of the supernatural, who also describe the objects of their beliefs as infinite. [...] Unfortunately, the recognition of infinity in science is also an excellent justification for zealots to convince others that, since science accepts infinity as a reality, deities that are just as infinite and unknowable are also reality. [H. Lubin in "To settle infinity dispute, a new law of logic", Quanta Magazine (20 Jul 2014)]
> "infinity ... plays into the hands ... of religions and other forms of the supernatural." Bogus.
> >
> > The line all points of which have a finite distance is itself finite. [T. Pesch: "Institutiones philosophiae naturalis secundum principia S. Thomae Aquinatis", Freiburg, Herder (1880) § 425]
> al finite lines are finite. TRUE !!

Now it would be better I'd ever heard of Kruijswijk, but the point of Shannon or "signal continuity"
is that definitions under continuity result from the rationals establishing the condition of being
indistinguishable from continuous, which is all sorts of the usual results of linear autonomous
systems that recognize them.

I.e., if 2003's "Shannon, Cantor, Nyquist, ..." is agreeable I'd imagine it's in couching
conditions of continuity, into usual results all what follow. (... Here after the countability
of the rationals to the character of a continuous domain.)

On 12/28/2021 10:30 AM, Ross A. Finlayson wrote:
> On Tuesday, December 28, 2021 at 6:06:25 AM UTC-8, Serg io wrote:
>> more bogus bad math quotes from WM. Foul.
>> On 12/28/2021 3:51 AM, WM wrote:
>>> Mathematics in its relation with the divine has played a special role in the course of history. [...] Mathematics is abstract and it often seems absolute, universal, eternal and pure. More than other kinds of knowledge it possesses characteristics that we associate with the divine. [T. Koetsier, L. Bergmans: "Mathematics and the divine: A historical study", Elsevier (2005) p. 4]
>> Mathematics and the divine. Conjecture.
>>>
>>> At the moment you say that infinite sets are of the same size when there exists a bijection, then you already introduce some of the choices Cantor made. But those are choices, not proofs. [...] there are at least two choices you can make: a) You may refer to  even when there is no logical system that can list all its elements (this is Cantor's choice). b) You must be aware that  is never complete in your logical system (alternative to Cantor). I don't think choice b is very attractive, but to my opinion it is a way you can try to go. However, I always questioned if saying that || < || has any more meaning than saying that irrational numbers are green. [...] I think the people that don't like Cantor have the greatest problem with the fact that whatever system you use,  contains numbers which can not be expressed. At least I can say about it, it is a fair concern. [L.B. Kruijswijk in "Shannon defeats Cantor = single infinity type", sci.math (11 & 12 Dec 2003)]
>> a Quote from sci.math! Bogus.
>>>
>>> The concept of "infinity" is anti-science, and the use of infinity as a constant by mathematicians and scientists, whether they know it or not, plays into the hands of advocates of religions and other forms of the supernatural, who also describe the objects of their beliefs as infinite. [...] Unfortunately, the recognition of infinity in science is also an excellent justification for zealots to convince others that, since science accepts infinity as a reality, deities that are just as infinite and unknowable are also reality. [H. Lubin in "To settle infinity dispute, a new law of logic", Quanta Magazine (20 Jul 2014)]
>> "infinity ... plays into the hands ... of religions and other forms of the supernatural." Bogus.
>>>
>>> The line all points of which have a finite distance is itself finite. [T. Pesch: "Institutiones philosophiae naturalis secundum principia S. Thomae Aquinatis", Freiburg, Herder (1880) § 425]
>> al finite lines are finite. TRUE !!
>
> Now it would be better I'd ever heard of Kruijswijk, but the point of Shannon or "signal continuity"
> is that definitions under continuity result from the rationals
no. Shannon never said continuity is result of rationals.

You seem to also believe in uncountable sets, when your intention is:

WM schrieb am Dienstag, 28. Dezember 2021 um 11:50:12 UTC+1:
> The power set axiom requires x⊆ ℕ <==> x ∈ P(ℕ). In words:
either a model of ZF contains all subsets of ℕ or it is not a model of ZF.
https://groups.google.com/g/sci.logic/c/H7KQao__WGo/m/W5d5-1qqAQAJ

Basically you want P(ℕ) not only internally uncountable, but
also externally uncountable? Or you dont know what you are

taling about?

WM schrieb am Dienstag, 28. Dezember 2021 um 15:03:01 UTC+1:
> > And if it turns out the universe (speaking more broadly than the merely observable universe) turns out to be actually infinite? Or if the many-worlds interpretation of quantum mechanics is correct?
> Many worlds and uncountable sets. Kindred notions.
>
> Regards, WM

Oh somebody wrote about conceptualism:

MATHEMATICAL CONCEPTUALISM
NIK WEAVER
https://arxiv.org/pdf/math/0509246.pdf

But its rather a desease than a cure in my opinion.
Example par execellence, WM wrote:
> Why should they, if not forced by the power set axiom?

Thats appeal to ones own ignorance? Why do you think
ink was spilled by many to explain the weakness of the
power set axiom? By simply ignoring these mathematical

facts does not give more strength to the power set axiom.

LMAO!

Mostowski Collapse schrieb am Dienstag, 28. Dezember 2021 um 12:44:45 UTC+1:
> Ha Ha, conceptualism, how outdated.
>
> WM schrieb:
> > Mathematics in its relation with the divine has played a special role in the course of history. [...] Mathematics is abstract and it often seems absolute, universal, eternal and pure. More than other kinds of knowledge it possesses characteristics that we associate with the divine. [T. Koetsier, L. Bergmans: "Mathematics and the divine: A historical study", Elsevier (2005) p. 4]
> >
> > At the moment you say that infinite sets are of the same size when there exists a bijection, then you already introduce some of the choices Cantor made. But those are choices, not proofs. [...] there are at least two choices you can make: a) You may refer to  even when there is no logical system that can list all its elements (this is Cantor's choice). b) You must be aware that  is never complete in your logical system (alternative to Cantor). I don't think choice b is very attractive, but to my opinion it is a way you can try to go. However, I always questioned if saying that || < || has any more meaning than saying that irrational numbers are green. [...] I think the people that don't like Cantor have the greatest problem with the fact that whatever system you use,  contains numbers which can not be expressed. At least I can say about it, it is a fair concern. [L.B. Kruijswijk in "Shannon defeats Cantor = single infinity type", sci.math (11 & 12 Dec 2003)]
> >
> > The concept of "infinity" is anti-science, and the use of infinity as a constant by mathematicians and scientists, whether they know it or not, plays into the hands of advocates of religions and other forms of the supernatural, who also describe the objects of their beliefs as infinite. [...] Unfortunately, the recognition of infinity in science is also an excellent justification for zealots to convince others that, since science accepts infinity as a reality, deities that are just as infinite and unknowable are also reality. [H. Lubin in "To settle infinity dispute, a new law of logic", Quanta Magazine (20 Jul 2014)]
> >
> > The line all points of which have a finite distance is itself finite. [T. Pesch: "Institutiones philosophiae naturalis secundum principia S. Thomae Aquinatis", Freiburg, Herder (1880) § 425]
> >

So what is the desease of conceptualism?
Well here it is expressed:

"The basic premise of mathematical conceptualism is that in
order for a construction to be considered valid it need not be
physically realizable, but it must be conceptually definite,
meaning that we must be able to form a completely clear
**mental picture** of how the construction would proceed."
> MATHEMATICAL CONCEPTUALISM
> NIK WEAVER

Ha Ha, good luck with that, WM has the **mental picture**
that the axiom of power set must deliver full power sets, but
it doesn't. A very interesting form of quak autism indeed.

Mostowski Collapse schrieb am Dienstag, 28. Dezember 2021 um 18:49:24 UTC+1:
> Oh somebody wrote about conceptualism:
>
> MATHEMATICAL CONCEPTUALISM
> NIK WEAVER
> https://arxiv.org/pdf/math/0509246.pdf
>
> But its rather a desease than a cure in my opinion.
> Example par execellence, WM wrote:
> > Why should they, if not forced by the power set axiom?
>
> Thats appeal to ones own ignorance? Why do you think
> ink was spilled by many to explain the weakness of the
> power set axiom? By simply ignoring these mathematical
>
> facts does not give more strength to the power set axiom.
>
> LMAO!
> Mostowski Collapse schrieb am Dienstag, 28. Dezember 2021 um 12:44:45 UTC+1:
> > Ha Ha, conceptualism, how outdated.
> >
> > WM schrieb:
> > > Mathematics in its relation with the divine has played a special role in the course of history. [...] Mathematics is abstract and it often seems absolute, universal, eternal and pure. More than other kinds of knowledge it possesses characteristics that we associate with the divine. [T. Koetsier, L. Bergmans: "Mathematics and the divine: A historical study", Elsevier (2005) p. 4]
> > >
> > > At the moment you say that infinite sets are of the same size when there exists a bijection, then you already introduce some of the choices Cantor made. But those are choices, not proofs. [...] there are at least two choices you can make: a) You may refer to  even when there is no logical system that can list all its elements (this is Cantor's choice). b) You must be aware that  is never complete in your logical system (alternative to Cantor). I don't think choice b is very attractive, but to my opinion it is a way you can try to go. However, I always questioned if saying that || < || has any more meaning than saying that irrational numbers are green. [...] I think the people that don't like Cantor have the greatest problem with the fact that whatever system you use,  contains numbers which can not be expressed. At least I can say about it, it is a fair concern. [L.B. Kruijswijk in "Shannon defeats Cantor = single infinity type", sci.math (11 & 12 Dec 2003)]
> > >
> > > The concept of "infinity" is anti-science, and the use of infinity as a constant by mathematicians and scientists, whether they know it or not, plays into the hands of advocates of religions and other forms of the supernatural, who also describe the objects of their beliefs as infinite. [...] Unfortunately, the recognition of infinity in science is also an excellent justification for zealots to convince others that, since science accepts infinity as a reality, deities that are just as infinite and unknowable are also reality. [H. Lubin in "To settle infinity dispute, a new law of logic", Quanta Magazine (20 Jul 2014)]
> > >
> > > The line all points of which have a finite distance is itself finite. [T. Pesch: "Institutiones philosophiae naturalis secundum principia S. Thomae Aquinatis", Freiburg, Herder (1880) § 425]
> > >

If you put all finites together you get an fundamental infinity.
Unlimited smallness exists as well.

Mitchell Raemsch

Dan Christensen schrieb am Dienstag, 28. Dezember 2021 um 17:09:39 UTC+1:
> On Tuesday, December 28, 2021 at 9:03:20 AM UTC-5, WM wrote:

> > > If only you understood what a finite set was, you would not make such absurd claims.
>
> > Classical logic was abstracted from the mathematics of finite sets and their subsets. (The word finite is here to be taken in the precise sense that the members of such set are explicitly exhibited one by one.) [H. Weyl]
> Makes no sense. Sorry, this is just not good enough

If I have the choice between Weyl and you, I will choose Weyl. If I have the chioce between you and nothing, I will choose nothing.

Regards, WM

On Tuesday, December 28, 2021 at 8:53:28 AM UTC-8, Serg io wrote:
> On 12/28/2021 10:30 AM, Ross A. Finlayson wrote:
> > On Tuesday, December 28, 2021 at 6:06:25 AM UTC-8, Serg io wrote:
> >> more bogus bad math quotes from WM. Foul.
> >> On 12/28/2021 3:51 AM, WM wrote:
> >>> Mathematics in its relation with the divine has played a special role in the course of history. [...] Mathematics is abstract and it often seems absolute, universal, eternal and pure. More than other kinds of knowledge it possesses characteristics that we associate with the divine. [T. Koetsier, L. Bergmans: "Mathematics and the divine: A historical study", Elsevier (2005) p. 4]
> >> Mathematics and the divine. Conjecture.
> >>>
> >>> At the moment you say that infinite sets are of the same size when there exists a bijection, then you already introduce some of the choices Cantor made. But those are choices, not proofs. [...] there are at least two choices you can make: a) You may refer to  even when there is no logical system that can list all its elements (this is Cantor's choice). b) You must be aware that  is never complete in your logical system (alternative to Cantor). I don't think choice b is very attractive, but to my opinion it is a way you can try to go. However, I always questioned if saying that || < || has any more meaning than saying that irrational numbers are green. [...] I think the people that don't like Cantor have the greatest problem with the fact that whatever system you use,  contains numbers which can not be expressed. At least I can say about it, it is a fair concern. [L.B. Kruijswijk in "Shannon defeats Cantor = single infinity type", sci.math (11 & 12 Dec 2003)]
> >> a Quote from sci.math! Bogus.
> >>>
> >>> The concept of "infinity" is anti-science, and the use of infinity as a constant by mathematicians and scientists, whether they know it or not, plays into the hands of advocates of religions and other forms of the supernatural, who also describe the objects of their beliefs as infinite. [...] Unfortunately, the recognition of infinity in science is also an excellent justification for zealots to convince others that, since science accepts infinity as a reality, deities that are just as infinite and unknowable are also reality. [H. Lubin in "To settle infinity dispute, a new law of logic", Quanta Magazine (20 Jul 2014)]
> >> "infinity ... plays into the hands ... of religions and other forms of the supernatural." Bogus.
> >>>
> >>> The line all points of which have a finite distance is itself finite. [T. Pesch: "Institutiones philosophiae naturalis secundum principia S. Thomae Aquinatis", Freiburg, Herder (1880) § 425]
> >> al finite lines are finite. TRUE !!
> >
> > Now it would be better I'd ever heard of Kruijswijk, but the point of Shannon or "signal continuity"
> > is that definitions under continuity result from the rationals
> no. Shannon never said continuity is result of rationals.

Shannon is sometimes given credit for Nyquist theorem,
where it may be so that Shannon and "continuity is result
of rationals" have that Shannon's results in signal theory
and for information theory, for information theory, and,
signal theory, with respect to formalism and here with
formalism in foundations, have that "no, it is not so that
'continuity is result of rationals', it's not to be implied
that it's a point of Shannon's when connecting signal theory
and properties of continuous sampling theorems, except
to illustrate the entire connection here in the abstract."

Then, that "I said so", is basically that mostly I have
studied the formalism of the line continuity and the
field continuity, with respect to the signal continuity,
and continuous functions which is the usual formalism.

On Tuesday, December 28, 2021 at 10:12:41 AM UTC-8, Mostowski Collapse wrote:
> So what is the desease of conceptualism?
> Well here it is expressed:
>
> "The basic premise of mathematical conceptualism is that in
> order for a construction to be considered valid it need not be
> physically realizable, but it must be conceptually definite,
> meaning that we must be able to form a completely clear
> **mental picture** of how the construction would proceed."
> > MATHEMATICAL CONCEPTUALISM
> > NIK WEAVER
>
> Ha Ha, good luck with that, WM has the **mental picture**
> that the axiom of power set must deliver full power sets, but
> it doesn't. A very interesting form of quak autism indeed.
> Mostowski Collapse schrieb am Dienstag, 28. Dezember 2021 um 18:49:24 UTC+1:
> > Oh somebody wrote about conceptualism:
> >
> > MATHEMATICAL CONCEPTUALISM
> > NIK WEAVER
> > https://arxiv.org/pdf/math/0509246.pdf
> >
> > But its rather a desease than a cure in my opinion.
> > Example par execellence, WM wrote:
> > > Why should they, if not forced by the power set axiom?
> >
> > Thats appeal to ones own ignorance? Why do you think
> > ink was spilled by many to explain the weakness of the
> > power set axiom? By simply ignoring these mathematical
> >
> > facts does not give more strength to the power set axiom.
> >
> > LMAO!
> > Mostowski Collapse schrieb am Dienstag, 28. Dezember 2021 um 12:44:45 UTC+1:
> > > Ha Ha, conceptualism, how outdated.
> > >
> > > WM schrieb:
> > > > Mathematics in its relation with the divine has played a special role in the course of history. [...] Mathematics is abstract and it often seems absolute, universal, eternal and pure. More than other kinds of knowledge it possesses characteristics that we associate with the divine. [T. Koetsier, L. Bergmans: "Mathematics and the divine: A historical study", Elsevier (2005) p. 4]
> > > >
> > > > At the moment you say that infinite sets are of the same size when there exists a bijection, then you already introduce some of the choices Cantor made. But those are choices, not proofs. [...] there are at least two choices you can make: a) You may refer to  even when there is no logical system that can list all its elements (this is Cantor's choice). b) You must be aware that  is never complete in your logical system (alternative to Cantor). I don't think choice b is very attractive, but to my opinion it is a way you can try to go. However, I always questioned if saying that || < || has any more meaning than saying that irrational numbers are green. [...] I think the people that don't like Cantor have the greatest problem with the fact that whatever system you use,  contains numbers which can not be expressed. At least I can say about it, it is a fair concern. [L.B. Kruijswijk in "Shannon defeats Cantor = single infinity type", sci.math (11 & 12 Dec 2003)]
> > > >
> > > > The concept of "infinity" is anti-science, and the use of infinity as a constant by mathematicians and scientists, whether they know it or not, plays into the hands of advocates of religions and other forms of the supernatural, who also describe the objects of their beliefs as infinite. [...] Unfortunately, the recognition of infinity in science is also an excellent justification for zealots to convince others that, since science accepts infinity as a reality, deities that are just as infinite and unknowable are also reality. [H. Lubin in "To settle infinity dispute, a new law of logic", Quanta Magazine (20 Jul 2014)]
> > > >
> > > > The line all points of which have a finite distance is itself finite. [T. Pesch: "Institutiones philosophiae naturalis secundum principia S. Thomae Aquinatis", Freiburg, Herder (1880) § 425]
> > > >

"(Woodin’s
actual views on the nature of mathematical truth are somewhat unusual; he has
elsewhere suggested, or seemed to suggest, that elementary number theory has no
canonical model and that if the twin primes conjecture is undecidable in ZFC then
it may have no definite truth value ([21], p. 34). This apparently indicates a dis-
belief in the natural numbers coupled with a belief in the axioms of set theory, an
odd combination.)"

There is an interesting twist. The Nyquist theorem
is related to the twin primes conjecture. If p1 and p2

are the last twin primes then the sampling distance
for all mathematical functions is 1/p1-1/p2.

Thats the Nyquist-Bolabur conjecture.

Ross A. Finlayson schrieb am Dienstag, 28. Dezember 2021 um 21:05:54 UTC+1:
> On Tuesday, December 28, 2021 at 10:12:41 AM UTC-8, Mostowski Collapse wrote:
> > So what is the desease of conceptualism?
> > Well here it is expressed:
> >
> > "The basic premise of mathematical conceptualism is that in
> > order for a construction to be considered valid it need not be
> > physically realizable, but it must be conceptually definite,
> > meaning that we must be able to form a completely clear
> > **mental picture** of how the construction would proceed."
> > > MATHEMATICAL CONCEPTUALISM
> > > NIK WEAVER
> >
> > Ha Ha, good luck with that, WM has the **mental picture**
> > that the axiom of power set must deliver full power sets, but
> > it doesn't. A very interesting form of quak autism indeed.
> > Mostowski Collapse schrieb am Dienstag, 28. Dezember 2021 um 18:49:24 UTC+1:
> > > Oh somebody wrote about conceptualism:
> > >
> > > MATHEMATICAL CONCEPTUALISM
> > > NIK WEAVER
> > > https://arxiv.org/pdf/math/0509246.pdf
> > >
> > > But its rather a desease than a cure in my opinion.
> > > Example par execellence, WM wrote:
> > > > Why should they, if not forced by the power set axiom?
> > >
> > > Thats appeal to ones own ignorance? Why do you think
> > > ink was spilled by many to explain the weakness of the
> > > power set axiom? By simply ignoring these mathematical
> > >
> > > facts does not give more strength to the power set axiom.
> > >
> > > LMAO!
> > > Mostowski Collapse schrieb am Dienstag, 28. Dezember 2021 um 12:44:45 UTC+1:
> > > > Ha Ha, conceptualism, how outdated.
> > > >
> > > > WM schrieb:
> > > > > Mathematics in its relation with the divine has played a special role in the course of history. [...] Mathematics is abstract and it often seems absolute, universal, eternal and pure. More than other kinds of knowledge it possesses characteristics that we associate with the divine. [T. Koetsier, L. Bergmans: "Mathematics and the divine: A historical study", Elsevier (2005) p. 4]
> > > > >
> > > > > At the moment you say that infinite sets are of the same size when there exists a bijection, then you already introduce some of the choices Cantor made. But those are choices, not proofs. [...] there are at least two choices you can make: a) You may refer to  even when there is no logical system that can list all its elements (this is Cantor's choice). b) You must be aware that  is never complete in your logical system (alternative to Cantor). I don't think choice b is very attractive, but to my opinion it is a way you can try to go. However, I always questioned if saying that || < || has any more meaning than saying that irrational numbers are green. [...] I think the people that don't like Cantor have the greatest problem with the fact that whatever system you use,  contains numbers which can not be expressed. At least I can say about it, it is a fair concern. [L.B. Kruijswijk in "Shannon defeats Cantor = single infinity type", sci.math (11 & 12 Dec 2003)]
> > > > >
> > > > > The concept of "infinity" is anti-science, and the use of infinity as a constant by mathematicians and scientists, whether they know it or not, plays into the hands of advocates of religions and other forms of the supernatural, who also describe the objects of their beliefs as infinite. [....] Unfortunately, the recognition of infinity in science is also an excellent justification for zealots to convince others that, since science accepts infinity as a reality, deities that are just as infinite and unknowable are also reality. [H. Lubin in "To settle infinity dispute, a new law of logic", Quanta Magazine (20 Jul 2014)]
> > > > >
> > > > > The line all points of which have a finite distance is itself finite. [T. Pesch: "Institutiones philosophiae naturalis secundum principia S. Thomae Aquinatis", Freiburg, Herder (1880) § 425]
> > > > >
> "(Woodin’s
> actual views on the nature of mathematical truth are somewhat unusual; he has
> elsewhere suggested, or seemed to suggest, that elementary number theory has no
> canonical model and that if the twin primes conjecture is undecidable in ZFC then
> it may have no definite truth value ([21], p. 34). This apparently indicates a dis-
> belief in the natural numbers coupled with a belief in the axioms of set theory, an
> odd combination.)"

On 12/28/2021 1:55 PM, Ross A. Finlayson wrote:
> On Tuesday, December 28, 2021 at 8:53:28 AM UTC-8, Serg io wrote:
>> On 12/28/2021 10:30 AM, Ross A. Finlayson wrote:
>>> On Tuesday, December 28, 2021 at 6:06:25 AM UTC-8, Serg io wrote:
>>>> more bogus bad math quotes from WM. Foul.
>>>> On 12/28/2021 3:51 AM, WM wrote:
>>>>> Mathematics in its relation with the divine has played a special role in the course of history. [...] Mathematics is abstract and it often seems absolute, universal, eternal and pure. More than other kinds of knowledge it possesses characteristics that we associate with the divine. [T. Koetsier, L. Bergmans: "Mathematics and the divine: A historical study", Elsevier (2005) p. 4]
>>>> Mathematics and the divine. Conjecture.
>>>>>
>>>>> At the moment you say that infinite sets are of the same size when there exists a bijection, then you already introduce some of the choices Cantor made. But those are choices, not proofs. [...] there are at least two choices you can make: a) You may refer to  even when there is no logical system that can list all its elements (this is Cantor's choice). b) You must be aware that  is never complete in your logical system (alternative to Cantor). I don't think choice b is very attractive, but to my opinion it is a way you can try to go. However, I always questioned if saying that || < || has any more meaning than saying that irrational numbers are green. [...] I think the people that don't like Cantor have the greatest problem with the fact that whatever system you use,  contains numbers which can not be expressed. At least I can say about it, it is a fair concern. [L.B. Kruijswijk in "Shannon defeats Cantor = single infinity type", sci.math (11 & 12 Dec 2003)]
>>>> a Quote from sci.math! Bogus.
>>>>>
>>>>> The concept of "infinity" is anti-science, and the use of infinity as a constant by mathematicians and scientists, whether they know it or not, plays into the hands of advocates of religions and other forms of the supernatural, who also describe the objects of their beliefs as infinite. [...] Unfortunately, the recognition of infinity in science is also an excellent justification for zealots to convince others that, since science accepts infinity as a reality, deities that are just as infinite and unknowable are also reality. [H. Lubin in "To settle infinity dispute, a new law of logic", Quanta Magazine (20 Jul 2014)]
>>>> "infinity ... plays into the hands ... of religions and other forms of the supernatural." Bogus.
>>>>>
>>>>> The line all points of which have a finite distance is itself finite. [T. Pesch: "Institutiones philosophiae naturalis secundum principia S. Thomae Aquinatis", Freiburg, Herder (1880) § 425]
>>>> al finite lines are finite. TRUE !!
>>>
>>> Now it would be better I'd ever heard of Kruijswijk, but the point of Shannon or "signal continuity"
>>> is that definitions under continuity result from the rationals
>> no. Shannon never said continuity is result of rationals.
>
> Shannon is sometimes given credit for Nyquist theorem,
> where it may be so that Shannon and "continuity is result
> of rationals" have that Shannon's results in signal theory
> and for information theory, for information theory, and,
> signal theory, with respect to formalism and here with
> formalism in foundations, have that "no, it is not so that
> 'continuity is result of rationals', it's not to be implied
> that it's a point of Shannon's when connecting signal theory
> and properties of continuous sampling theorems, except
> to illustrate the entire connection here in the abstract."
So you agree, Shannon never said "continuity is result of rationals".
>
> Then, that "I said so", is basically that mostly I have
> studied the formalism of the line continuity and the
> field continuity, with respect to the signal continuity,
> and continuous functions which is the usual formalism.
>
yes, formal formalism is formation of formalist thought.

Finites are like the universe they go on forever.
But the mathematician needs to start at a size of infinity
that is completed by being built of its fundamental
or calculus unlimited small. This is calculus continuum
hypothesis... what sizes of infinity are made of...

Mitchell Raemsch

On Tuesday, December 28, 2021 at 2:03:01 PM UTC, WM wrote:
> > And if it turns out the universe (speaking more broadly than the merely observable universe) turns out to be actually infinite? Or if the many-worlds interpretation of quantum mechanics is correct?
> Many worlds and uncountable sets. Kindred notions.
>
> Regards, WM

And yet the universe might be actually infinite. If you have a proof that it cannot be then you submit it to a respected peer reviewed journal on cosmology.

Regards

QB

On Tuesday, December 28, 2021 at 4:26:35 PM UTC, Ross A. Finlayson wrote:
> On Tuesday, December 28, 2021 at 4:11:53 AM UTC-8, ross.pro...@gmx.com wrote:
> > On Tuesday, December 28, 2021 at 9:51:07 AM UTC, WM wrote:
> > > Mathematics in its relation with the divine has played a special role in the course of history. [...] Mathematics is abstract and it often seems absolute, universal, eternal and pure. More than other kinds of knowledge it possesses characteristics that we associate with the divine. [T. Koetsier, L. Bergmans: "Mathematics and the divine: A historical study", Elsevier (2005) p. 4]
> > >
> > > At the moment you say that infinite sets are of the same size when there exists a bijection, then you already introduce some of the choices Cantor made. But those are choices, not proofs. [...] there are at least two choices you can make: a) You may refer to  even when there is no logical system that can list all its elements (this is Cantor's choice). b) You must be aware that  is never complete in your logical system (alternative to Cantor). I don't think choice b is very attractive, but to my opinion it is a way you can try to go. However, I always questioned if saying that || < || has any more meaning than saying that irrational numbers are green. [...] I think the people that don't like Cantor have the greatest problem with the fact that whatever system you use,  contains numbers which can not be expressed. At least I can say about it, it is a fair concern. [L.B. Kruijswijk in "Shannon defeats Cantor = single infinity type", sci.math (11 & 12 Dec 2003)]
> > >
> > > The concept of "infinity" is anti-science, and the use of infinity as a constant by mathematicians and scientists, whether they know it or not, plays into the hands of advocates of religions and other forms of the supernatural, who also describe the objects of their beliefs as infinite. [...] Unfortunately, the recognition of infinity in science is also an excellent justification for zealots to convince others that, since science accepts infinity as a reality, deities that are just as infinite and unknowable are also reality. [H. Lubin in "To settle infinity dispute, a new law of logic", Quanta Magazine (20 Jul 2014)]
> > >
> > > The line all points of which have a finite distance is itself finite. [T. Pesch: "Institutiones philosophiae naturalis secundum principia S. Thomae Aquinatis", Freiburg, Herder (1880) § 425]
> > And if it turns out the universe (speaking more broadly than the merely observable universe) turns out to be actually infinite? Or if the many-worlds interpretation of quantum mechanics is correct?
> It seems one should keep in mind that quantum mechanics' MWI is
> an interpretation in a theory, for what after causality, illustrates
> the results according to chance. But, there's nothing what must not
> eventually result as "real" except that after causality: MWI isn't scientific
> in the sense it's not falsifiable, so it's fair to say in conversation as it's
> let out as an "interpretation", for what it models in the theory, what
> the theory models in its physical interpretation.
>
> As I've often: "mathematics _owes_ physics more and better mathematics
> of infinity", is that, for solving some super-classical problem in the classical
> in the free field, for what is like a singularity theory (for what are singularities
> everywhere or original analysis), there is solving for unipotential as all what
> results that the linear to non-linear (disorganized) to highly-non-linear (reorganized),
> has that usual least but most expressive system have a label for a constant
> for infinity collected near the top of declarations.
>
> I.e., besides that of course symbolically for algebra that "writing a label for
> infinity is a concise notation for systems of rates in unboundedness", the
> "standard" analysis, and real analysis is always part of foundations, then
> also, high-order mathematics of infinity, results from the interplay of the
> three definitions of continuity: line, field, and signal.
>
> I.e., what mathematics owes physics for mathematics of infinity is more
> for the quantum mechanics themselves, than unfalsifiable interpretations
> what follow causality and chance.
>
> Anyways mathematics needs (or, has) now at least three definitions of
> continuity: line, field, and signal, most programs of which are aware of
> at least one, and often for example only one.
>
> ... That foundations includes from the most direct central notions of
> mathematical logic.

The MWI is falsifiable (and is hardly unmotivated); to give one illustration: there are interpretations such as Roger Penrose's which make different predictions. Empirical corroboration of these predictions of a Penrose style interpretation would therefore falsify Many Worlds.

Regards

QB