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Hiilbert and Cantor's paradise

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  Hiilbert and Cantor's paradise David Petry <davidlpetry@gmail.com> - 2018-05-12 13:26 -0700
    Re: Hiilbert and Cantor's paradise "Shobe, Martin" <martin.shobe@yahoo.com> - 2018-05-12 20:24 -0500
      Re: Hiilbert and Cantor's paradise David Petry <davidlpetry@gmail.com> - 2018-05-13 14:59 -0700
        Re: Hiilbert and Cantor's paradise "Shobe, Martin" <martin.shobe@yahoo.com> - 2018-05-13 20:22 -0500
          Re: Hiilbert and Cantor's paradise WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-14 00:54 -0700
            Re: Hiilbert and Cantor's paradise "Shobe, Martin" <martin.shobe@yahoo.com> - 2018-05-14 19:17 -0500
            Re: Hiilbert and Cantor's paradise David Petry <davidlpetry@gmail.com> - 2018-05-14 21:00 -0700
              Re: Hiilbert and Cantor's paradise Alan Mackenzie <acm@muc.de> - 2018-05-15 06:57 +0000
                Re: Hiilbert and Cantor's paradise WM <wolfgang.mueckenheim@hs-augsburg.de> - 2018-05-15 06:04 -0700
                  Re: Hiilbert and Cantor's paradise Alan Mackenzie <acm@muc.de> - 2018-05-15 19:41 +0000
                    Re: Hiilbert and Cantor's paradise "Shobe, Martin" <martin.shobe@yahoo.com> - 2018-05-15 19:54 -0500
                      Re: Hiilbert and Cantor's paradise Python <python@python.invalid> - 2018-05-16 03:04 +0200
                    Re: Hiilbert and Cantor's paradise David Petry <davidlpetry@gmail.com> - 2018-05-16 17:53 -0700
                      Re: Hiilbert and Cantor's paradise "Ross A. Finlayson" <ross.finlayson@gmail.com> - 2018-05-16 20:44 -0700
                      Re: Hiilbert and Cantor's paradise Alan Mackenzie <acm@muc.de> - 2018-05-17 18:32 +0000
                        Re: Hiilbert and Cantor's paradise David Petry <davidlpetry@gmail.com> - 2018-05-18 21:56 -0700
                          Re: Hiilbert and Cantor's paradise Alan Mackenzie <acm@muc.de> - 2018-05-20 18:54 +0000
                            Re: Hiilbert and Cantor's paradise David Petry <davidlpetry@gmail.com> - 2018-05-20 12:13 -0700
                              Re: Hiilbert and Cantor's paradise bursejan@gmail.com - 2018-05-20 14:15 -0700
                                Re: Hiilbert and Cantor's paradise bursejan@gmail.com - 2018-05-20 14:21 -0700
                                Re: Hiilbert and Cantor's paradise bursejan@gmail.com - 2018-05-20 14:26 -0700
                Re: Hiilbert and Cantor's paradise FredJeffries <fredjeffries@gmail.com> - 2018-05-15 08:47 -0700
              Re: Hiilbert and Cantor's paradise "Shobe, Martin" <martin.shobe@yahoo.com> - 2018-05-15 20:09 -0500
                Re: Hiilbert and Cantor's paradise David Petry <davidlpetry@gmail.com> - 2018-05-16 17:57 -0700
                  Re: Hiilbert and Cantor's paradise "Shobe, Martin" <martin.shobe@yahoo.com> - 2018-05-16 20:39 -0500
                    Re: Hiilbert and Cantor's paradise David Petry <davidlpetry@gmail.com> - 2018-05-16 20:02 -0700
                      Re: Hiilbert and Cantor's paradise "Shobe, Martin" <martin.shobe@yahoo.com> - 2018-05-17 06:56 -0500
      Re: Hiilbert and Cantor's paradise "Ross A. Finlayson" <ross.finlayson@gmail.com> - 2018-05-15 13:55 -0700
    Re: Hiilbert and Cantor's paradise bursejan@gmail.com - 2018-05-12 18:50 -0700

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#412998 — Hiilbert and Cantor's paradise

Hilbert is famous for the quote, "mathematicians will never be evicted from the paradise Cantor created for us".  However, ten years later he wrote:

"The infinite is nowhere realized; it is neither present in nature nor admissible as the foundation of our rational thinking" [D. Hilbert: "Über das Unendliche", Math. Annalen 95 (1925) p. 190] 


I think that's interesting because so often people claim that the first quote is that last word in the debate over whether Cantor's ideas about infinity should be accepted as part of mathematics.  But the second quote shows clearly that Hilbert had second thoughts about the issue.

I'd like to thank Mueckenheim for his work in finding such quotes and bringing them to our attention.

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#413012

On 5/12/2018 3:26 PM, David Petry wrote:
> 
> Hilbert is famous for the quote, "mathematicians will never be evicted from the paradise Cantor created for us".  However, ten years later he wrote:
> 
> "The infinite is nowhere realized; it is neither present in nature nor admissible as the foundation of our rational thinking" [D. Hilbert: "Über das Unendliche", Math. Annalen 95 (1925) p. 190]
> 
> 
> I think that's interesting because so often people claim that the first quote is that last word in the debate over whether Cantor's ideas about infinity should be accepted as part of mathematics.  But the second quote shows clearly that Hilbert had second thoughts about the issue.
> 
> I'd like to thank Mueckenheim for his work in finding such quotes and bringing them to our attention.
> 
Quote mining is can be fun. In the same paper.

But analysis alone does not provide us with the deepest insight into the 
nature of the infinite. This insight is procured for us by a discipline 
which comes closer to a general philosophical way of thinking and which 
was designed to cast new light on the whole complex of questions about 
the infinite. This discipline, created by George Cantor, is set theory. 
In this paper, we are interested only in that unique and original part 
of set theory which forms the central core of Cantor's doctrine, viz., 
the theory of transfinite numbers. This theory is, I think, the finest 
product of mathematical genius and one of the supreme achievements of 
purely intellectual human activity. What, then, is this theory?

Martin Shobe

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#413040

Martin Shobe wrote:

> David Petry wrote:

>> Hilbert is famous for the quote, "mathematicians will never be evicted from the paradise Cantor created for us".  However, ten years later he wrote: 
>> 
>> "The infinite is nowhere realized; it is neither present in nature nor admissible as the foundation of our rational thinking" [D. Hilbert: "Über das Unendliche", Math. Annalen 95 (1925) p. 190] 
>> 
>> 
>> I think that's interesting because so often people claim that the first quote is that last word in the debate over whether Cantor's ideas about infinity should be accepted as part of mathematics.  But the second quote shows clearly that Hilbert had second thoughts about the issue. 

> 
> Quote mining is can be fun. In the same paper. 

> But analysis alone does not provide us with the deepest insight into the 
> nature of the infinite. This insight is procured for us by a discipline 
> which comes closer to a general philosophical way of thinking and which 
> was designed to cast new light on the whole complex of questions about 
> the infinite. This discipline, created by George Cantor, is set theory. 
> In this paper, we are interested only in that unique and original part 
> of set theory which forms the central core of Cantor's doctrine, viz., 
> the theory of transfinite numbers. This theory is, I think, the finest 
> product of mathematical genius and one of the supreme achievements of 
> purely intellectual human activity. What, then, is this theory? 
 

Well that's very interesting.

If I'm reading him correctly, he's acknowledging that mathematics has a major role to play in understanding nature, and it has a major role to play in understanding our own rational thinking, but infinity has no role to play in those things.

Furthermore, he asserts that our understanding of infinity does not come from our understanding of mathematics, but rather from "a general philosophical way of thinking".

But then he says that Cantor's theory of infinity is "the finest product of mathematical genius".

If he had said that the theory of infinity is the finest achievement in philosophy and/or art, then I would say that he is being consistent, even though I disagree with him.  But what he wrote suggests that he is actually quite confused.

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#413042

On 5/13/2018 4:59 PM, David Petry wrote:
> Martin Shobe wrote:
> 
>> David Petry wrote:
> 
>>> Hilbert is famous for the quote, "mathematicians will never be evicted from the paradise Cantor created for us".  However, ten years later he wrote:
>>>
>>> "The infinite is nowhere realized; it is neither present in nature nor admissible as the foundation of our rational thinking" [D. Hilbert: "Über das Unendliche", Math. Annalen 95 (1925) p. 190]
>>>
>>>
>>> I think that's interesting because so often people claim that the first quote is that last word in the debate over whether Cantor's ideas about infinity should be accepted as part of mathematics.  But the second quote shows clearly that Hilbert had second thoughts about the issue.
> 
>>
>> Quote mining is can be fun. In the same paper.
> 
>> But analysis alone does not provide us with the deepest insight into the
>> nature of the infinite. This insight is procured for us by a discipline
>> which comes closer to a general philosophical way of thinking and which
>> was designed to cast new light on the whole complex of questions about
>> the infinite. This discipline, created by George Cantor, is set theory.
>> In this paper, we are interested only in that unique and original part
>> of set theory which forms the central core of Cantor's doctrine, viz.,
>> the theory of transfinite numbers. This theory is, I think, the finest
>> product of mathematical genius and one of the supreme achievements of
>> purely intellectual human activity. What, then, is this theory?
>   
> 
> Well that's very interesting.
> 
> If I'm reading him correctly, he's acknowledging that mathematics has a major role to play in understanding nature, and it has a major role to play in understanding our own rational thinking, but infinity has no role to play in those things.

You're not understanding him. He has a role for infinity in 
understanding nature but that doesn't make infinity part of nature. 
Similarly infinity is not part of our notion of proof or rational thought.

> Furthermore, he asserts that our understanding of infinity does not come from our understanding of mathematics, but rather from "a general philosophical way of thinking".
> 
> But then he says that Cantor's theory of infinity is "the finest product of mathematical genius".
> 
> If he had said that the theory of infinity is the finest achievement in philosophy and/or art, then I would say that he is being consistent, even though I disagree with him.  But what he wrote suggests that he is actually quite confused.

The confusion is not his.

Martin Shobe

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#413062

Am Montag, 14. Mai 2018 03:22:45 UTC+2 schrieb Shobe, Martin:
> On 5/13/2018 4:59 PM, David Petry wrote:
> > Martin Shobe wrote:
> > 
> >> David Petry wrote:
> > 
> >>> Hilbert is famous for the quote, "mathematicians will never be evicted from the paradise Cantor created for us".  However, ten years later he wrote:
> >>>
> >>> "The infinite is nowhere realized; it is neither present in nature nor admissible as the foundation of our rational thinking" [D. Hilbert: "Über das Unendliche", Math. Annalen 95 (1925) p. 190]
> >>>
> >>>
> >>> I think that's interesting because so often people claim that the first quote is that last word in the debate over whether Cantor's ideas about infinity should be accepted as part of mathematics.  But the second quote shows clearly that Hilbert had second thoughts about the issue.
> > 
> >>
> >> Quote mining is can be fun. In the same paper.
> > 
> >> But analysis alone does not provide us with the deepest insight into the
> >> nature of the infinite. This insight is procured for us by a discipline
> >> which comes closer to a general philosophical way of thinking and which
> >> was designed to cast new light on the whole complex of questions about
> >> the infinite. This discipline, created by George Cantor, is set theory.
> >> In this paper, we are interested only in that unique and original part
> >> of set theory which forms the central core of Cantor's doctrine, viz.,
> >> the theory of transfinite numbers. This theory is, I think, the finest
> >> product of mathematical genius and one of the supreme achievements of
> >> purely intellectual human activity. What, then, is this theory?
> >   
> > 
> > Well that's very interesting.
> > 
> > If I'm reading him correctly, he's acknowledging that mathematics has a major role to play in understanding nature, and it has a major role to play in understanding our own rational thinking, but infinity has no role to play in those things.
> 
> You're not understanding him. He has a role for infinity in 
> understanding nature but that doesn't make infinity part of nature. 
> Similarly infinity is not part of our notion of proof or rational thought.
> 
> > Furthermore, he asserts that our understanding of infinity does not come from our understanding of mathematics, but rather from "a general philosophical way of thinking".
> > 
> > But then he says that Cantor's theory of infinity is "the finest product of mathematical genius".
> > 
> > If he had said that the theory of infinity is the finest achievement in philosophy and/or art, then I would say that he is being consistent, even though I disagree with him.  But what he wrote suggests that he is actually quite confused.
> 
> The confusion is not his.

In one and the same paper he praises Cantor's "theory of transfinite numbers; this appears to me as the most admirable blossom of mathematical spirit and really one of the supreme achievements of purely intellectual human activity." (p. 167) 

"The infinite is nowhere realized; it is neither present in nature nor admissible as the foundation of our rational thinking" (p. 190)

This excludes the infinite even in axiomatic form. In mathematics axioms are the foundation of our rational thinking.

Therefore the confusion is solely Hilbert's - except in matheologial circles where never a contradiction can be acknowledged since the basic theory is known as theory of Zero Findable Contradictions.

Regards, WM

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#413099

On 5/14/2018 2:54 AM, WM wrote:
> Am Montag, 14. Mai 2018 03:22:45 UTC+2 schrieb Shobe, Martin:
>> On 5/13/2018 4:59 PM, David Petry wrote:
>>> Martin Shobe wrote:
>>>
>>>> David Petry wrote:
>>>
>>>>> Hilbert is famous for the quote, "mathematicians will never be evicted from the paradise Cantor created for us".  However, ten years later he wrote:
>>>>>
>>>>> "The infinite is nowhere realized; it is neither present in nature nor admissible as the foundation of our rational thinking" [D. Hilbert: "Über das Unendliche", Math. Annalen 95 (1925) p. 190]
>>>>>
>>>>>
>>>>> I think that's interesting because so often people claim that the first quote is that last word in the debate over whether Cantor's ideas about infinity should be accepted as part of mathematics.  But the second quote shows clearly that Hilbert had second thoughts about the issue.
>>>
>>>>
>>>> Quote mining is can be fun. In the same paper.
>>>
>>>> But analysis alone does not provide us with the deepest insight into the
>>>> nature of the infinite. This insight is procured for us by a discipline
>>>> which comes closer to a general philosophical way of thinking and which
>>>> was designed to cast new light on the whole complex of questions about
>>>> the infinite. This discipline, created by George Cantor, is set theory..
>>>> In this paper, we are interested only in that unique and original part
>>>> of set theory which forms the central core of Cantor's doctrine, viz.,
>>>> the theory of transfinite numbers. This theory is, I think, the finest
>>>> product of mathematical genius and one of the supreme achievements of
>>>> purely intellectual human activity. What, then, is this theory?
>>>    
>>>
>>> Well that's very interesting.
>>>
>>> If I'm reading him correctly, he's acknowledging that mathematics has a major role to play in understanding nature, and it has a major role to play in understanding our own rational thinking, but infinity has no role to play in those things.
>>
>> You're not understanding him. He has a role for infinity in
>> understanding nature but that doesn't make infinity part of nature.
>> Similarly infinity is not part of our notion of proof or rational thought..
>>
>>> Furthermore, he asserts that our understanding of infinity does not come from our understanding of mathematics, but rather from "a general philosophical way of thinking".
>>>
>>> But then he says that Cantor's theory of infinity is "the finest product of mathematical genius".
>>>
>>> If he had said that the theory of infinity is the finest achievement in philosophy and/or art, then I would say that he is being consistent, even though I disagree with him.  But what he wrote suggests that he is actually quite confused.
>>
>> The confusion is not his.
> 
> In one and the same paper he praises Cantor's "theory of transfinite numbers; this appears to me as the most admirable blossom of mathematical spirit and really one of the supreme achievements of purely intellectual human activity." (p. 167)
> 
> "The infinite is nowhere realized; it is neither present in nature nor admissible as the foundation of our rational thinking" (p. 190)
> 
> This excludes the infinite even in axiomatic form. In mathematics axioms are the foundation of our rational thinking.
> 
> Therefore the confusion is solely Hilbert's - except in matheologial circles where never a contradiction can be acknowledged since the basic theory is known as theory of Zero Findable Contradictions.

No, here the confusion (or more accurately, the dishonesty) is yours.

Martin Shobe

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#413100

Martin Shobe and Wolfgang Mueckenheim each gave slightly different translations of a quote from Hilbert:

Shobe:
 This theory is, I think, the finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity.

Mueckenheim:
 this appears to me as the most admirable blossom of mathematical spirit and really one of the supreme achievements of purely intellectual human activity.


There's actually a very big difference between the two.  In the one given by Mueckenheim, Hilbert is saying that Cantor's theory of infinity is in the spirit of mathematics, and doesn't address the question of whether the theory should be accepted as actually part of mathematics.

The translation Shobe gives us leaves the impression that Hilbert thinks the theory is some kind of major advance in our understanding of mathematics itself, which is bizarre.

Yes, Cantor's theory of infinity is "in the spirit of mathematics"--it consists of axioms, definitions, theorems and proofs, just like mathematics.


But Hilbert also wrote:

"The infinite is nowhere realized; it is neither present in nature nor admissible as the foundation of our rational thinking" 

which suggests he at least understood the reason many mathematicians were refusing to accept Cantor's theory of infinity as part of mathematics;  if we agree that the purpose of mathematics is to help us understand nature and our rational thinking, then infinity doesn't belong in mathematics.


It's interesting how seemingly small differences in two translations can make such a big difference in meaning.

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#413105

David Petry <davidlpetry@gmail.com> wrote:

> Martin Shobe and Wolfgang Mueckenheim each gave slightly different
> translations of a quote from Hilbert:

> Shobe:
>  This theory is, I think, the finest product of mathematical genius and
>  one of the supreme achievements of purely intellectual human activity.

> Mueckenheim:
>  this appears to me as the most admirable blossom of mathematical
>  spirit and really one of the supreme achievements of purely
>  intellectual human activity.

How about quoting the original, so that we can make up our own minds?
:-)

> There's actually a very big difference between the two.  In the one
> given by Mueckenheim, Hilbert is saying that Cantor's theory of
> infinity is in the spirit of mathematics, and doesn't address the
> question of whether the theory should be accepted as actually part of
> mathematics.

> The translation Shobe gives us leaves the impression that Hilbert
> thinks the theory is some kind of major advance in our understanding of
> mathematics itself, which is bizarre.

???  Why "bizarre"?

> Yes, Cantor's theory of infinity is "in the spirit of mathematics"--it
> consists of axioms, definitions, theorems and proofs, just like
> mathematics.


> But Hilbert also wrote:

> "The infinite is nowhere realized; it is neither present in nature nor
> admissible as the foundation of our rational thinking" 

I'm not aware of any mathematical axiom based on the notion of infinity.
That is likely what Hilbert meant.  The infinite arises as the
consequences of "finite" axioms.

> which suggests he at least understood the reason many mathematicians
> were refusing to accept Cantor's theory of infinity as part of
> mathematics;  if we agree that the purpose of mathematics is to help us
> understand nature and our rational thinking, then infinity doesn't
> belong in mathematics.

That's a very big "if".  I wouldn't agree with it, at least not totally.

> It's interesting how seemingly small differences in two translations
> can make such a big difference in meaning.

-- 
Alan Mackenzie (Nuremberg, Germany).

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#413113

Am Dienstag, 15. Mai 2018 08:57:42 UTC+2 schrieb Alan Mackenzie:
> David Petry <davidlpetry@gmail.com> wrote:
> 
> > Martin Shobe and Wolfgang Mueckenheim each gave slightly different
> > translations of a quote from Hilbert:
> 
> > Shobe:
> >  This theory is, I think, the finest product of mathematical genius and
> >  one of the supreme achievements of purely intellectual human activity.
> 
> > Mueckenheim:
> >  this appears to me as the most admirable blossom of mathematical
> >  spirit and really one of the supreme achievements of purely
> >  intellectual human activity.
> 
> How about quoting the original, so that we can make up our own minds?
> :-)

diese erscheint mir aIs die bewundernswerteste Blüte mathematischen Geistes und überhaupt eine der höchsten Leistungen rein verstandesmäßiger menschlicher Tatigkeit.

> > But Hilbert also wrote:
> 
> > "The infinite is nowhere realized; it is neither present in nature nor
> > admissible as the foundation of our rational thinking" 

Das Gesamtergebnis ist dann: das Unendliche findet sich nirgends realisiert; es ist weder in der Natur vorhanden, noch als Grundlage in unserem verstandesmäBigen Denken zulässig - eine bemerkenswerte Harmonie zwischen Sein und Denken.
> 
> I'm not aware of any mathematical axiom based on the notion of infinity.

The axiom of infinity is the basis of ZF. ZF is based upon the infinite.

> That is likely what Hilbert meant.  The infinite arises as the
> consequences of "finite" axioms.

noch als Grundlage in unserem verstandesmäBigen Denken zulässig
> 
> > It's interesting how seemingly small differences in two translations
> > can make such a big difference in meaning.

I would prefer my translation.

Regards, WM

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#413135

WM <wolfgang.mueckenheim@hs-augsburg.de> wrote:
> Am Dienstag, 15. Mai 2018 08:57:42 UTC+2 schrieb Alan Mackenzie:
>> David Petry <davidlpetry@gmail.com> wrote:

>> > Martin Shobe and Wolfgang Mueckenheim each gave slightly different
>> > translations of a quote from Hilbert:

>> > Shobe:
>> >  This theory is, I think, the finest product of mathematical genius and
>> >  one of the supreme achievements of purely intellectual human activity.

>> > Mueckenheim:
>> >  this appears to me as the most admirable blossom of mathematical
>> >  spirit and really one of the supreme achievements of purely
>> >  intellectual human activity.

>> How about quoting the original, so that we can make up our own minds?
>> :-)

> diese erscheint mir aIs die bewundernswerteste Blüte mathematischen
> Geistes und überhaupt eine der höchsten Leistungen rein
> verstandesmäßiger menschlicher Tatigkeit.

I would go with WM's translation as the more accurate; it's practically a
word for word translation.

>> > It's interesting how seemingly small differences in two translations
>> > can make such a big difference in meaning.

But I honestly can't get excited at the difference David sees between
these two translations.  "the finest product of mathematical genius" and
"the most admirable blossom of mathematical spirit" aren't different
things in practice, or in the real world.  What is produced by
"mathematical spirit" is mathematics just as what is produced by amber
spirit is drunkenness.

> I would prefer my translation.

> Regards, WM

-- 
Alan Mackenzie (Nuremberg, Germany).

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#413148

On 5/15/2018 2:41 PM, Alan Mackenzie wrote:
> WM <wolfgang.mueckenheim@hs-augsburg.de> wrote:
>> Am Dienstag, 15. Mai 2018 08:57:42 UTC+2 schrieb Alan Mackenzie:
>>> David Petry <davidlpetry@gmail.com> wrote:
> 
>>>> Martin Shobe and Wolfgang Mueckenheim each gave slightly different
>>>> translations of a quote from Hilbert:
> 
>>>> Shobe:
>>>>   This theory is, I think, the finest product of mathematical genius and
>>>>   one of the supreme achievements of purely intellectual human activity.
> 
>>>> Mueckenheim:
>>>>   this appears to me as the most admirable blossom of mathematical
>>>>   spirit and really one of the supreme achievements of purely
>>>>   intellectual human activity.
> 
>>> How about quoting the original, so that we can make up our own minds?
>>> :-)
> 
>> diese erscheint mir aIs die bewundernswerteste Blüte mathematischen
>> Geistes und überhaupt eine der höchsten Leistungen rein
>> verstandesmäßiger menschlicher Tatigkeit.
> 
> I would go with WM's translation as the more accurate; it's practically a
> word for word translation.

That doesn't necessarily make it more accurate.

Martin Shobe

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#413149

Shobe, Martin wrote:
> On 5/15/2018 2:41 PM, Alan Mackenzie wrote:
>> WM <wolfgang.mueckenheim@hs-augsburg.de> wrote:
>>> Am Dienstag, 15. Mai 2018 08:57:42 UTC+2 schrieb Alan Mackenzie:
>>>> David Petry <davidlpetry@gmail.com> wrote:
>>
>>>>> Martin Shobe and Wolfgang Mueckenheim each gave slightly different
>>>>> translations of a quote from Hilbert:
>>
>>>>> Shobe:
>>>>>   This theory is, I think, the finest product of mathematical 
>>>>> genius and
>>>>>   one of the supreme achievements of purely intellectual human 
>>>>> activity.
>>
>>>>> Mueckenheim:
>>>>>   this appears to me as the most admirable blossom of mathematical
>>>>>   spirit and really one of the supreme achievements of purely
>>>>>   intellectual human activity.
>>
>>>> How about quoting the original, so that we can make up our own minds?
>>>> :-)
>>
>>> diese erscheint mir aIs die bewundernswerteste Blüte mathematischen
>>> Geistes und überhaupt eine der höchsten Leistungen rein
>>> verstandesmäßiger menschlicher Tatigkeit.
>>
>> I would go with WM's translation as the more accurate; it's practically a
>> word for word translation.
> 
> That doesn't necessarily make it more accurate.

A word for word translation coming from a filthy person is
very likely to be a complete lie. "Herr Mueckenheim is a venom".

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#413208

Alan Mackenzie wrote:

> the finest product of mathematical genius" and 
> "the most admirable blossom of mathematical spirit" aren't different 
> things in practice, or in the real world.

This might make things clearer.

It is well know that Kurt Godel "proved" the existence of God.  The "proof" had the axiom-definition-theorem-proof format.  So it was "in the spirit of" mathematics.  But most mathematicians wouldn't accept the proof as a mathematical proof, so we couldn't say it was a product of "mathematical genius".  Although we could say that it was a non-mathematical product of *a* mathematical genius.

Even native speakers of English might not agree with me.

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#413215

"Collected."

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#413269

David Petry <davidlpetry@gmail.com> wrote:

> Alan Mackenzie wrote:

>> the finest product of mathematical genius" and "the most admirable
>> blossom of mathematical spirit" aren't different things in practice,
>> or in the real world.

> This might make things clearer.

No, your attempted derailling of the discussion is mere sophistry.

> It is well know that Kurt Godel "proved" the existence of God.  The
> "proof" had the axiom-definition-theorem-proof format.  So it was "in
> the spirit of" mathematics.

Rubbish.  It might have had the formalism of maths but certainly not its
spirit.

> But most mathematicians wouldn't accept the proof as a mathematical
> proof, so we couldn't say it was a product of "mathematical genius".
> Although we could say that it was a non-mathematical product of *a*
> mathematical genius.

> Even native speakers of English might not agree with me.

It's very hard to agree with such an obscure, irrelevant diversion.

By the way, do you understand proof by contradiction?  You never
clarified that point in the other discussion.

-- 
Alan Mackenzie (Nuremberg, Germany).

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#413371

Alan Mackenzie wrote:

>> It is well know that Kurt Godel "proved" the existence of God.  The 
>> "proof" had the axiom-definition-theorem-proof format.  So it was "in 
>> the spirit of" mathematics. 

> Rubbish.  It might have had the formalism of maths but certainly not its 
> spirit. 


Good point. Really! I've argued the same point myself.  That is, so I argue, mathematics is defined by its purpose, which is to provide a conceptual framework for reasoning about phenomena in the physical world.  And then the "spirit" of mathematics would be equated with its purpose.

But the consensus view of the mathematics community is that mathematics is defined by its form, not its purpose.  They have to take that view in order to justify accepting Cantorian set theory as part of mathematics.  So the consensus view is that the spirit of mathematics is its form.

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#413440

David Petry <davidlpetry@gmail.com> wrote:

> Alan Mackenzie wrote:

>>> It is well know that Kurt Godel "proved" the existence of God.  The
>>> "proof" had the axiom-definition-theorem-proof format.  So it was "in
>>> the spirit of" mathematics. 

>> Rubbish.  It might have had the formalism of maths but certainly not
>> its spirit. 


> Good point. Really! I've argued the same point myself.  That is, so I
> argue, mathematics is defined by its purpose, which is to provide a
> conceptual framework for reasoning about phenomena in the physical
> world.  And then the "spirit" of mathematics would be equated with its
> purpose.

I think you will find yourself quite isolated if that's what you hold the
purpose of mathematics to be.  Most of maths has been developed without
any such purpose, though an appreciable amount of it has later found
service in describing the "physical world".

Maths is its own spirit, its own purpose, just as a piece of great music
or a work of great art is.

> But the consensus view of the mathematics community is that mathematics
> is defined by its form, not its purpose.

I doubt that very much.  Maths will be defined by its substance and its
spirit.  One does not get reams and reams of nonsense in the form of
axioms, reasoning, and so on published by the mathematical journals which
is devoid of substance.

> They have to take that view in order to justify accepting Cantorian set
> theory as part of mathematics.

Oh, here we go again.  You've set up a straw man and now proceed to knock
it down.  Cantor's set theory has passed through the filters of time, of
hostility from people who think as you do, and has prevailed over this
extensive critical analysis.  It is now a part of standard maths.

> So the consensus view is that the spirit of mathematics is its form.

No.

What's the problem with your relationship with proof by contradiction,
that you can't even talk about it?  I've twice asked you a simple factual
question, do you understand proof by contradiction, and both times you've
snipped the question without answering it.  Why?

-- 
Alan Mackenzie (Nuremberg, Germany).

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#413443

Alan Mackenzie wrote:

> do you understand proof by contradiction,

You can't believe impossible things, unless you practice for half an hour every day.

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#413455

I don't think numerical analysis implies there is
no real analysis. Or as Paul Taylor writes:

"Whilst it is widely appreciated that constructivism 
emphasises similar issues to those that arise in 
computational practice [Dav05], classical analysts 
sometimes feel that their constructive colleagues 
want to rob them of their theorems, without replacing 
them with algorithms that are any better than those 
that numerical analysts already know.

When we look more closely into these complaints, we 
find that the two sides are talking at cross-purposes
and even the traditionalists are conflating two different 
theorems of their own. Some mathematicians consider 
the “generality” of classical results to be more 
important than their applications. 

On the other hand, anyone who is genuinely interested 
in solving an equation, i.e. in finding a number, 
will probably already have some algorithm (such as 
Newton’s) in mind, and will be willing to accept the 
pre-conditions that this imposes. We find, on examination, 
that these imply the extra property 

that constructivists require, which is in fact very 
mild: it is satisfied by any example in which you 
might reasonably expect to be able to compute a 
zero. Topologically, these conditions are weaker 
forms of openness, whilst the “general” theorem 
is about continuous functions.

Disclaimer: Did not yet spend a lot of time on this 
quote, but it somehow confirms and shows my gut feeling 
about all the nonsense here on sci.math. Well the
continous functions are less many than all functions from
R to R, but they are uncountable many.

dass die Mächtigkeit von C(X) die Mächtigkeit des 
Kontinuums ist (falls X nicht leer ist). 
https://de.wikipedia.org/wiki/Stetigkeit#Menge_der_stetigen_Funktionen

So to deny the existence of continuum, would mean
also to deny a C(X), the function space of all 
continous functions. This might hold for the computer
since you can anyway only have countable many functions
on the computer. But what about out there?

Am Sonntag, 20. Mai 2018 21:13:37 UTC+2 schrieb David Petry:
> Alan Mackenzie wrote:
> 
> > do you understand proof by contradiction,
> 
> You can't believe impossible things, unless you practice for half an hour every day.

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#413457

Quote is from here:
http://paultaylor.eu/~pt/ASD/lamcra/introivt

Am Sonntag, 20. Mai 2018 23:15:17 UTC+2 schrieb burs...@gmail.com:
> I don't think numerical analysis implies there is
> no real analysis. Or as Paul Taylor writes:
> 
> "Whilst it is widely appreciated that constructivism 
> emphasises similar issues to those that arise in 
> computational practice [Dav05], classical analysts 
> sometimes feel that their constructive colleagues 
> want to rob them of their theorems, without replacing 
> them with algorithms that are any better than those 
> that numerical analysts already know.
> 
> When we look more closely into these complaints, we 
> find that the two sides are talking at cross-purposes
> and even the traditionalists are conflating two different 
> theorems of their own. Some mathematicians consider 
> the “generality” of classical results to be more 
> important than their applications. 
> 
> On the other hand, anyone who is genuinely interested 
> in solving an equation, i.e. in finding a number, 
> will probably already have some algorithm (such as 
> Newton’s) in mind, and will be willing to accept the 
> pre-conditions that this imposes. We find, on examination, 
> that these imply the extra property 
> 
> that constructivists require, which is in fact very 
> mild: it is satisfied by any example in which you 
> might reasonably expect to be able to compute a 
> zero. Topologically, these conditions are weaker 
> forms of openness, whilst the “general” theorem 
> is about continuous functions.
> 
> Disclaimer: Did not yet spend a lot of time on this 
> quote, but it somehow confirms and shows my gut feeling 
> about all the nonsense here on sci.math. Well the
> continous functions are less many than all functions from
> R to R, but they are uncountable many.
> 
> dass die Mächtigkeit von C(X) die Mächtigkeit des 
> Kontinuums ist (falls X nicht leer ist). 
> https://de.wikipedia.org/wiki/Stetigkeit#Menge_der_stetigen_Funktionen
> 
> So to deny the existence of continuum, would mean
> also to deny a C(X), the function space of all 
> continous functions. This might hold for the computer
> since you can anyway only have countable many functions
> on the computer. But what about out there?
> 
> Am Sonntag, 20. Mai 2018 21:13:37 UTC+2 schrieb David Petry:
> > Alan Mackenzie wrote:
> > 
> > > do you understand proof by contradiction,
> > 
> > You can't believe impossible things, unless you practice for half an hour every day.

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