During the renaissance, in particular with Bruno, the actual infinite is carried over from God to the world. The finite world models of present science show clearly, how the superiority of the idea of an actual infinite has come to an end with classical (modern) physics. In this light the inclusion of the actual infinite into mathematics, which explicitly began during the end of the last century with G. Cantor, appears disconcerting. In the intellectual framework of our century – in particular when considering existential philosophy – the actual infinite appears really as an anachronism. [...] In arithmetic [...] there does not exist a motive to introduce the actual infinite. The surprising appearance of an actual infinite in modern mathematics therefore can only be understood by taking geometry into consideration. [...] A finite decimal fraction can be written out, an infinite one can never be written. To talk about a sequence of infinitely many digits is therefore – if it is not nonsense at all – at least a hazardous business. [...] Similar to the elimination of the infinitely small at that time {{in the 19th century when the naive notion of continuity was replaced by the --definition}} now the infinitely great (more precisely the actual infinite) has to be shown dispensable. The motivating force of this reform is not based on a haughty purism but on the wish to restore the absolute safety and reliability of mathematics that presently is in danger to be lost in the set theoretic contradictions or their circumvention by rather arbitrary set theoretic formalisms. [P. Lorenzen: "Das Aktual-Unendliche in der Mathematik", Philosophia naturalis 4 (1957) pp. 3-11]

Much more difficult, it seems to me, will it be to meet the objections raised on the moral level by the formalists, namely that Constructivists should not waste their time, especially that they should not try to persuade other people to waste their time too, with cumbersome and perhaps unusual attempts to reconstruct the achievements of the traditionally given higher parts of mathematics. Instead, they should join the big game of axiomatic set theory: "You will become famous if you please famous people – and all famous mathematicians like axiomatic set theory". [...] Of course, then we would have no text-books with theorems about the hierarchy of transfinite cardinals. But we have also no text-books any more about the hierarchy of angels. No one seriously regrets this – though of course scholastic theology could be formalized. [P. Lorenzen: "Constructive mathematics as a philosophical problem", Compositio Mathematica 20 (1968) p. 137f]

Regards, WM

onsdag 24 november 2021 kl. 10:23:16 UTC+1 skrev WM:
> During the renaissance, in particular with Bruno, the actual infinite is carried over from God to the world. The finite world models of present science show clearly, how the superiority of the idea of an actual infinite has come to an end with classical (modern) physics. In this light the inclusion of the actual infinite into mathematics, which explicitly began during the end of the last century with G. Cantor, appears disconcerting. In the intellectual framework of our century – in particular when considering existential philosophy – the actual infinite appears really as an anachronism. [...] In arithmetic [...] there does not exist a motive to introduce the actual infinite. The surprising appearance of an actual infinite in modern mathematics therefore can only be understood by taking geometry into consideration. [...] A finite decimal fraction can be written out, an infinite one can never be written. To talk about a sequence of infinitely many digits is therefore – if it is not nonsense at all – at least a hazardous business. [...] Similar to the elimination of the infinitely small at that time {{in the 19th century when the naive notion of continuity was replaced by the --definition}} now the infinitely great (more precisely the actual infinite) has to be shown dispensable. The motivating force of this reform is not based on a haughty purism but on the wish to restore the absolute safety and reliability of mathematics that presently is in danger to be lost in the set theoretic contradictions or their circumvention by rather arbitrary set theoretic formalisms. [P. Lorenzen: "Das Aktual-Unendliche in der Mathematik", Philosophia naturalis 4 (1957) pp. 3-11]
>
> Much more difficult, it seems to me, will it be to meet the objections raised on the moral level by the formalists, namely that Constructivists should not waste their time, especially that they should not try to persuade other people to waste their time too, with cumbersome and perhaps unusual attempts to reconstruct the achievements of the traditionally given higher parts of mathematics. Instead, they should join the big game of axiomatic set theory: "You will become famous if you please famous people – and all famous mathematicians like axiomatic set theory". [...] Of course, then we would have no text-books with theorems about the hierarchy of transfinite cardinals. But we have also no text-books any more about the hierarchy of angels. No one seriously regrets this – though of course scholastic theology could be formalized. [P. Lorenzen: "Constructive mathematics as a philosophical problem", Compositio Mathematica 20 (1968) p. 137f]
>
> Regards, WM

can you come with something on your own instead of citing quotes from dead people?

On 11/24/2021 3:23 AM, WM wrote:
> During the renaissance, in particular with Bruno, the actual infinite is carried over from God to the world.

so, who carried over actual infinity from God, and who did they give it too ?

The finite world models of present science show clearly, how the superiority of the idea of an actual infinite has come to an end with classical
(modern) physics. In this light the inclusion of the actual infinite into mathematics, which explicitly began during the end of the last century with G.
Cantor, appears disconcerting. In the intellectual framework of our century – in particular when considering existential philosophy – the actual
infinite appears really as an anachronism. [...] In arithmetic [...] there does not exist a motive to introduce the actual infinite. The surprising
appearance of an actual infinite in modern mathematics therefore can only be understood by taking geometry into consideration. [...] A finite decimal
fraction can be written out, an infinite one can never be written.

wrong. 1/3 in base(1/3) = 1

To talk about a sequence of infinitely many digits is therefore – if it is not nonsense at all – at least a hazardous business. [...] Similar to the
elimination of the infinitely small at that time {{in the 19th century when the naive notion of continuity was replaced by the --definition}} now the
infinitely great (more precisely the actual infinite) has to be shown dispensable.

disagree.

The motivating force of this reform is not based on a haughty purism but on the wish to restore the absolute safety and reliability of mathematics that
presently is in danger to be lost

disagree.

in the set theoretic contradictions or their circumvention by rather arbitrary set theoretic formalisms. [P. Lorenzen: "Das Aktual-Unendliche in der
Mathematik", Philosophia naturalis 4 (1957) pp. 3-11]
>
> Much more difficult, it seems to me, will it be to meet the objections raised on the moral level by the formalists,

disagree, 'moral level' ???

? namely that Constructivists should not waste their time, especially that they should not try to persuade other people to waste their time too, with
cumbersome and perhaps unusual attempts to reconstruct the achievements of the traditionally given higher parts of mathematics.

this is not math, but politics.

Instead, they should join the big game of axiomatic set theory: "You will become famous if you please famous people – and all famous mathematicians like
axiomatic set theory". [...] Of course, then we would have no text-books with theorems about the hierarchy of transfinite cardinals. But we have also no
text-books any more about the hierarchy of angels.

red herring.

No one seriously regrets this – though of course scholastic theology could be formalized. [P. Lorenzen: "Constructive mathematics as a philosophical
problem", Compositio Mathematica 20 (1968) p. 137f]
>
> Regards, WM
>

your book references are bad.

Lorenzen wasn't invited to the Scottish Café, this made him an angry man.
https://en.wikipedia.org/wiki/Scottish_Caf%C3%A9

> The motivating force of this reform is not based on a haughty purism but on the wish to restore the absolute safety and reliability of mathematics that presently is in danger to be lost in the set theoretic contradictions or their circumvention by rather arbitrary set theoretic formalisms. [P. Lorenzen: "Das Aktual-Unendliche in der Mathematik", Philosophia naturalis 4 (1957) pp. 3-11]

I think Hydra games cannot be proved to terminate
in Peano, but in ZFC Hydra games have termination proofs.

Kill the Mathematical Hydra | Infinite Series
https://www.youtube.com/watch?v=uWwUpEY4c8o

Mostowski Collapse schrieb am Mittwoch, 24. November 2021 um 16:21:07 UTC+1:
> Lorenzen wasn't invited to the Scottish Café, this made him an angry man.
> https://en.wikipedia.org/wiki/Scottish_Caf%C3%A9
> > The motivating force of this reform is not based on a haughty purism but on the wish to restore the absolute safety and reliability of mathematics that presently is in danger to be lost in the set theoretic contradictions or their circumvention by rather arbitrary set theoretic formalisms. [P. Lorenzen: "Das Aktual-Unendliche in der Mathematik", Philosophia naturalis 4 (1957) pp. 3-11]

On 11/24/2021 9:28 AM, Serg io wrote:
>
> wrong.  1/3 in base(1/3) = 1
>

Are non-integer bases a valid concept?

On Wednesday, November 24, 2021 at 8:57:05 AM UTC-8, Michael Moroney wrote:
> On 11/24/2021 9:28 AM, Serg io wrote:
> >
> > wrong. 1/3 in base(1/3) = 1
> >
> Are non-integer bases a valid concept?

Not only valid, but useful

https://en.wikipedia.org/wiki/Non-integer_base_of_numeration

On Wednesday, November 24, 2021 at 1:23:16 AM UTC-8, WM interpolated:
> in the 19th century when the naive notion of continuity was replaced by the --definition

Alas, the '--definition [sic]' fails to capture the concept aimed at by that (ridiculed) 'naive notion of continuity', vis. being able to draw it without lifting your pencil from the paper.

The epsilon-delta definition, rather captures the notion of preservation-of-closeness -- captures is rather successfully judging by it's ubiquitousness in the past two centuries' mathematics.

The 'naive notion' is perhaps better modeled by the mathematical concept of connectedness -- but, alas, that leads us into the accursed field of topology and that devil's tool: Dedekind cuts

On Wed Nov 24 11:56:52 2021 Michael Moroney wrote:
> On 11/24/2021 9:28 AM, Serg io wrote:
> >
> > wrong. 1/3 in base(1/3) = 1
> >
>
> Are non-integer bases a valid concept?

*
More generally:

X in base X = 1

for any X.

earle
*

On 11/24/2021 12:22 PM, FredJeffries wrote:
> On Wednesday, November 24, 2021 at 8:57:05 AM UTC-8, Michael Moroney wrote:
>> On 11/24/2021 9:28 AM, Serg io wrote:
>>>
>>> wrong. 1/3 in base(1/3) = 1
>>>
>> Are non-integer bases a valid concept?
>
> Not only valid, but useful
>
> https://en.wikipedia.org/wiki/Non-integer_base_of_numeration
>

good find!

On Wednesday, 24 November 2021 at 10:23:16 UTC+1, WM wrote:
> During the renaissance, in particular with Bruno, the actual infinite is carried over from God to the world. The finite world models of present science show clearly, how the superiority of the idea of an actual infinite has come to an end with classical (modern) physics. In this light the inclusion of the actual infinite into mathematics, which explicitly began during the end of the last century with G. Cantor, appears disconcerting. In the intellectual framework of our century – in particular when considering existential philosophy – the actual infinite appears really as an anachronism. [...] In arithmetic [...] there does not exist a motive to introduce the actual infinite. The surprising appearance of an actual infinite in modern mathematics therefore can only be understood by taking geometry into consideration. [...] A finite decimal fraction can be written out, an infinite one can never be written. To talk about a sequence of infinitely many digits is therefore – if it is not nonsense at all – at least a hazardous business. [...] Similar to the elimination of the infinitely small at that time {{in the 19th century when the naive notion of continuity was replaced by the --definition}} now the infinitely great (more precisely the actual infinite) has to be shown dispensable. The motivating force of this reform is not based on a haughty purism but on the wish to restore the absolute safety and reliability of mathematics that presently is in danger to be lost in the set theoretic contradictions or their circumvention by rather arbitrary set theoretic formalisms. [P. Lorenzen: "Das Aktual-Unendliche in der Mathematik", Philosophia naturalis 4 (1957) pp. 3-11]
>
> Much more difficult, it seems to me, will it be to meet the objections raised on the moral level by the formalists, namely that Constructivists should not waste their time, especially that they should not try to persuade other people to waste their time too, with cumbersome and perhaps unusual attempts to reconstruct the achievements of the traditionally given higher parts of mathematics. Instead, they should join the big game of axiomatic set theory: "You will become famous if you please famous people – and all famous mathematicians like axiomatic set theory". [...] Of course, then we would have no text-books with theorems about the hierarchy of transfinite cardinals. But we have also no text-books any more about the hierarchy of angels. No one seriously regrets this – though of course scholastic theology could be formalized. [P. Lorenzen: "Constructive mathematics as a philosophical problem", Compositio Mathematica 20 (1968) p. 137f]

What a pile of nonsense: actually, infinity is just all over the place, in science and not only...

Thanks for posting, and what a disappointment this Lorenzen: another uber-finitist for the other side of the very same broken coin.

Oh, well...

Julio

On 11/24/2021 2:16 PM, Earle Jones wrote:
> On Wed Nov 24 11:56:52 2021 Michael Moroney wrote:
>> On 11/24/2021 9:28 AM, Serg io wrote:
>>>
>>> wrong. 1/3 in base(1/3) = 1
>>>
>>
>> Are non-integer bases a valid concept?
>
> *
> More generally:
>
> X in base X = 1
>
> for any X.

You meant 10 I assume.

That much I figured, but how could you have a non-integer number of
digits to represent a number? Like what digits could there be to
represent a number in base pi? Base 3 would have 3 unique digits to
represent a number, base 4 has 4 digits, but base pi? How to represent
100 decimal in base pi?

I'll go read the wiki page Fred J mentioned, perhaps already answered.

On Wed Nov 24 19:16:49 2021 Earle Jones wrote:
> On Wed Nov 24 11:56:52 2021 Michael Moroney wrote:
> > On 11/24/2021 9:28 AM, Serg io wrote:
> > >
> > > wrong. 1/3 in base(1/3) = 1
> > >
> >
> > Are non-integer bases a valid concept?
>
> *
> More generally:
>
> X in base X = 1
>
> for any X.
>
> earle
> *

*
Actually 10 = X in base X

Thanks,

earle
*