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tech / sci.math / Re: Mathematics, science and Abraham Robinson

SubjectAuthor
* Mathematics, science and Abraham RobinsonDavid Petry
+- Re: Mathematics, science and Abraham Robinsonzelos...@gmail.com
+- Re: Mathematics, science and Abraham RobinsonFromTheRafters
+* Re: Mathematics, science and Abraham Robinsonzelos...@gmail.com
|`* Re: Mathematics, science and Abraham RobinsonDavid Petry
| +- Re: Mathematics, science and Abraham Robinsonsergio
| `- Re: Mathematics, science and Abraham Robinsonzelos...@gmail.com
`* Re: Mathematics, science and Abraham Robinsonsobriquet
 `* Re: Mathematics, science and Abraham RobinsonFromTheRafters
  `* Re: Mathematics, science and Abraham RobinsonFredJeffries
   +* Re: Mathematics, science and Abraham Robinsonsergio
   |`* Re: Mathematics, science and Abraham RobinsonFromTheRafters
   | `* Re: Mathematics, science and Abraham Robinsonsergi o
   |  `* Re: Mathematics, science and Abraham RobinsonFredJeffries
   |   +* Re: Mathematics, science and Abraham RobinsonWM
   |   |+* Re: Mathematics, science and Abraham RobinsonFritz Feldhase
   |   ||`* Re: Mathematics, science and Abraham RobinsonWM
   |   || +- Re: Mathematics, science and Abraham RobinsonFromTheRafters
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   |   |`* Re: Mathematics, science and Abraham RobinsonTimothy Golden
   |   | `* Re: Mathematics, science and Abraham RobinsonRoss A. Finlayson
   |   |  `- Re: Mathematics, science and Abraham RobinsonTimothy Golden
   |   `- Re: Mathematics, science and Abraham RobinsonDavid Petry
   `* Re: Mathematics, science and Abraham Robinsonsobriquet
    `* Re: Mathematics, science and Abraham RobinsonWM
     +* Re: Mathematics, science and Abraham Robinsonsobriquet
     |`* Re: Mathematics, science and Abraham RobinsonWM
     | +- Re: Mathematics, science and Abraham RobinsonFromTheRafters
     | +* Re: Mathematics, science and Abraham RobinsonFritz Feldhase
     | |`* Re: Mathematics, science and Abraham RobinsonWM
     | | +- Re: Mathematics, science and Abraham Robinsonsergi o
     | | +* Re: Mathematics, science and Abraham RobinsonJim Burns
     | | |+- Re: Mathematics, science and Abraham RobinsonFritz Feldhase
     | | |`* Re: Mathematics, science and Abraham RobinsonWM
     | | | +- Re: Mathematics, science and Abraham Robinsonsergi o
     | | | +* Re: Mathematics, science and Abraham RobinsonJim Burns
     | | | |`* Re: Mathematics, science and Abraham RobinsonWM
     | | | | +- Re: Mathematics, science and Abraham Robinsonsergi o
     | | | | +* Re: Mathematics, science and Abraham RobinsonJim Burns
     | | | | |`* Re: Mathematics, science and Abraham RobinsonWM
     | | | | | +- Re: Mathematics, science and Abraham Robinsonsergi o
     | | | | | `* Re: Mathematics, science and Abraham RobinsonJim Burns
     | | | | |  +* Re: Mathematics, science and Abraham RobinsonWM
     | | | | |  |+* Re: Mathematics, science and Abraham RobinsonFritz Feldhase
     | | | | |  ||+* Re: Mathematics, science and Abraham RobinsonWM
     | | | | |  |||+- Re: Mathematics, science and Abraham RobinsonJim Burns
     | | | | |  |||`* Re: Mathematics, science and Abraham RobinsonWM
     | | | | |  ||| +* Re: Mathematics, science and Abraham RobinsonFromTheRafters
     | | | | |  ||| |+- Re: Mathematics, science and Abraham RobinsonTimothy Golden
     | | | | |  ||| |+- Re: Mathematics, science and Abraham Robinsonsergi o
     | | | | |  ||| |`* Re: Mathematics, science and Abraham RobinsonWM
     | | | | |  ||| | `- Re: Mathematics, science and Abraham Robinsonsergi o
     | | | | |  ||| `- Re: Mathematics, science and Abraham RobinsonJim Burns
     | | | | |  ||+* Re: Mathematics, science and Abraham RobinsonWM
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     | | | | |  |||`- Re: Mathematics, science and Abraham Robinsonsergi o
     | | | | |  ||`* Re: Mathematics, science and Abraham RobinsonWM
     | | | | |  || `- Re: Mathematics, science and Abraham Robinsonsergi o
     | | | | |  |+* Re: Mathematics, science and Abraham RobinsonJim Burns
     | | | | |  ||+* Re: Mathematics, science and Abraham RobinsonWM
     | | | | |  |||`* Re: Mathematics, science and Abraham RobinsonJim Burns
     | | | | |  ||| `* Re: Mathematics, science and Abraham RobinsonWM
     | | | | |  |||  +- Re: Mathematics, science and Abraham Robinsonsergi o
     | | | | |  |||  `* Re: Mathematics, science and Abraham RobinsonJim Burns
     | | | | |  |||   `* Re: Mathematics, science and Abraham RobinsonJim Burns
     | | | | |  |||    `- Re: Mathematics, science and Abraham RobinsonRoss A. Finlayson
     | | | | |  ||`* Re: Mathematics, science and Abraham RobinsonWM
     | | | | |  || `- Re: Mathematics, science and Abraham Robinsonsergi o
     | | | | |  |+- Re: Mathematics, science and Abraham Robinsonsergi o
     | | | | |  |+- Re: Mathematics, science and Abraham RobinsonFritz Feldhase
     | | | | |  |+- Re: Mathematics, science and Abraham RobinsonFritz Feldhase
     | | | | |  |+* Re: Mathematics, science and Abraham RobinsonRoss A. Finlayson
     | | | | |  ||`- Re: Mathematics, science and Abraham Robinsonsergi o
     | | | | |  |+- Re: Mathematics, science and Abraham Robinsonzelos...@gmail.com
     | | | | |  |+- Re: Mathematics, science and Abraham Robinsonzelos...@gmail.com
     | | | | |  |+- Re: Mathematics, science and Abraham RobinsonFritz Feldhase
     | | | | |  |+- Re: Mathematics, science and Abraham RobinsonFritz Feldhase
     | | | | |  |+* Re: Mathematics, science and Abraham RobinsonFritz Feldhase
     | | | | |  ||`- Re: Mathematics, science and Abraham Robinsonsergi o
     | | | | |  |+- Re: Mathematics, science and Abraham RobinsonFritz Feldhase
     | | | | |  |+- Re: Mathematics, science and Abraham RobinsonFritz Feldhase
     | | | | |  |`- Re: Mathematics, science and Abraham Robinsonzelos...@gmail.com
     | | | | |  `- Re: Mathematics, science and Abraham RobinsonTimothy Golden
     | | | | `- Re: Mathematics, science and Abraham Robinsonsergi o
     | | | `* Re: Mathematics, science and Abraham Robinsonzelos...@gmail.com
     | | |  `* Re: Mathematics, science and Abraham RobinsonWM
     | | |   +- Re: Mathematics, science and Abraham Robinsonsergi o
     | | |   +* Re: Mathematics, science and Abraham RobinsonFritz Feldhase
     | | |   |`* Re: Mathematics, science and Abraham RobinsonWM
     | | |   | +* Re: Mathematics, science and Abraham Robinsonzelos...@gmail.com
     | | |   | |`* Re: Mathematics, science and Abraham RobinsonWM
     | | |   | | +- Re: Mathematics, science and Abraham Robinsonsergi o
     | | |   | | `- Re: Mathematics, science and Abraham Robinsonzelos...@gmail.com
     | | |   | `- Re: Mathematics, science and Abraham Robinsonsergi o
     | | |   `- Re: Mathematics, science and Abraham Robinsonzelos...@gmail.com
     | | `- Re: Mathematics, science and Abraham RobinsonFritz Feldhase
     | +* Re: Mathematics, science and Abraham Robinsonsergi o
     | `- Re: Mathematics, science and Abraham Robinsonzelos...@gmail.com
     `- Re: Mathematics, science and Abraham Robinsonzelos...@gmail.com

Pages:123456
Mathematics, science and Abraham Robinson

<91ac989e-170a-4b18-a470-300d3bf384bfn@googlegroups.com>

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Subject: Mathematics, science and Abraham Robinson
From: davidlpe...@gmail.com (David Petry)
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 by: David Petry - Thu, 12 May 2022 02:37 UTC

Here's a quote from Abraham Robinson, who is undeniably a legit mathematician (use Wikipedia if you need to know more).

"I think that there is a real need, in formalism and elsewhere, to link our understanding of mathematics with our understanding of the physical world." (A. Robinson)

That's exactly what I have been saying for years, and the response I get from guys like Malum, Messager, and Burns is that the idea is stupid, crackpot, and motivated by evil intentions. There are nothing but crackpots in this newsgroup.

Even serious mathematicians often tell me that they have no interest in that idea, and since mathematics is by definition what mathematicians are interested in, the idea has no role to play in mathematics.

Here's another quote from Mr. Robinson.

"Infinite totalities do not exist in any sense of the word (i.e., either really or ideally). More precisely, any mention, or purported mention, of infinite totalities is, literally, meaningless." (A. Robinson)

Exactly right. Someday, mathematicians will wake up.

Here's a quote from Yuri Manin, another outstanding mathematician.

"Georg Cantor's grand meta-narrative, Set Theory, created by him almost singlehandedly in the span of about fifteen years, resembles a piece of high art more than a scientific theory." (Y. Manin)

I mention that because I've been asked why I usually mention Cantor by name when I say that set theory doesn't really belong in mathematics. I think that he (Cantor) is ultimately responsible.

The mathematics that is undeniably of value to the world is the mathematics that can be understood as a science.

Re: Mathematics, science and Abraham Robinson

<53245815-a055-4519-9c95-2a79210fed0en@googlegroups.com>

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Subject: Re: Mathematics, science and Abraham Robinson
From: zelos.ma...@gmail.com (zelos...@gmail.com)
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 by: zelos...@gmail.com - Thu, 12 May 2022 04:50 UTC

torsdag 12 maj 2022 kl. 04:37:18 UTC+2 skrev david...@gmail.com:
> Here's a quote from Abraham Robinson, who is undeniably a legit mathematician (use Wikipedia if you need to know more).
>
> "I think that there is a real need, in formalism and elsewhere, to link our understanding of mathematics with our understanding of the physical world." (A. Robinson)
>
> That's exactly what I have been saying for years, and the response I get from guys like Malum, Messager, and Burns is that the idea is stupid, crackpot, and motivated by evil intentions. There are nothing but crackpots in this newsgroup.
>
> Even serious mathematicians often tell me that they have no interest in that idea, and since mathematics is by definition what mathematicians are interested in, the idea has no role to play in mathematics.
>
> Here's another quote from Mr. Robinson.
>
> "Infinite totalities do not exist in any sense of the word (i.e., either really or ideally). More precisely, any mention, or purported mention, of infinite totalities is, literally, meaningless." (A. Robinson)
>
> Exactly right. Someday, mathematicians will wake up.
>
> Here's a quote from Yuri Manin, another outstanding mathematician.
>
> "Georg Cantor's grand meta-narrative, Set Theory, created by him almost singlehandedly in the span of about fifteen years, resembles a piece of high art more than a scientific theory." (Y. Manin)
>
> I mention that because I've been asked why I usually mention Cantor by name when I say that set theory doesn't really belong in mathematics. I think that he (Cantor) is ultimately responsible.
>
> The mathematics that is undeniably of value to the world is the mathematics that can be understood as a science.
set theory is part of matematics and very important in it

Re: Mathematics, science and Abraham Robinson

<t5ii51$nf6$1@dont-email.me>

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https://www.novabbs.com/tech/article-flat.php?id=99795&group=sci.math#99795

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From: nom...@afraid.org (FromTheRafters)
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Subject: Re: Mathematics, science and Abraham Robinson
Date: Thu, 12 May 2022 01:56:59 -0700
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 by: FromTheRafters - Thu, 12 May 2022 08:56 UTC

David Petry has brought this to us :
> Here's a quote from Abraham Robinson, who is undeniably a legit mathematician
> (use Wikipedia if you need to know more).
>
> "I think that there is a real need, in formalism and elsewhere, to link our
> understanding of mathematics with our understanding of the physical world."
> (A. Robinson)
>
> That's exactly what I have been saying for years, and the response I get from
> guys like Malum, Messager, and Burns is that the idea is stupid, crackpot,
> and motivated by evil intentions. There are nothing but crackpots in this
> newsgroup.
>
> Even serious mathematicians often tell me that they have no interest in that
> idea, and since mathematics is by definition what mathematicians are
> interested in, the idea has no role to play in mathematics.

Many mathematicians aren't interested in philosophy.

Re: Mathematics, science and Abraham Robinson

<1e87dd23-a8c1-42cc-b887-6acc1dcb912fn@googlegroups.com>

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https://www.novabbs.com/tech/article-flat.php?id=99804&group=sci.math#99804

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Subject: Re: Mathematics, science and Abraham Robinson
From: zelos.ma...@gmail.com (zelos...@gmail.com)
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 by: zelos...@gmail.com - Thu, 12 May 2022 11:43 UTC

torsdag 12 maj 2022 kl. 04:37:18 UTC+2 skrev david...@gmail.com:
> Here's a quote from Abraham Robinson, who is undeniably a legit mathematician (use Wikipedia if you need to know more).
>
> "I think that there is a real need, in formalism and elsewhere, to link our understanding of mathematics with our understanding of the physical world." (A. Robinson)
>
> That's exactly what I have been saying for years, and the response I get from guys like Malum, Messager, and Burns is that the idea is stupid, crackpot, and motivated by evil intentions. There are nothing but crackpots in this newsgroup.

There are plenty of us sane ones, I am and a few others. You, Gabriel, etc are the cranks and crackpots here.

>
> Even serious mathematicians often tell me that they have no interest in that idea, and since mathematics is by definition what mathematicians are interested in, the idea has no role to play in mathematics.

It doesn't because mathematicians do not care about the real world and physical stuff.

>
> Here's another quote from Mr. Robinson.
>
> "Infinite totalities do not exist in any sense of the word (i.e., either really or ideally). More precisely, any mention, or purported mention, of infinite totalities is, literally, meaningless." (A. Robinson)
>
> Exactly right. Someday, mathematicians will wake up.

There is nothing to wake up from. Infinities work in mathematics and mathematicians know it is not a physical thing and guess what? We do not care.

>
> Here's a quote from Yuri Manin, another outstanding mathematician.
>
> "Georg Cantor's grand meta-narrative, Set Theory, created by him almost singlehandedly in the span of about fifteen years, resembles a piece of high art more than a scientific theory." (Y. Manin)
>
> I mention that because I've been asked why I usually mention Cantor by name when I say that set theory doesn't really belong in mathematics. I think that he (Cantor) is ultimately responsible.
>
> The mathematics that is undeniably of value to the world is the mathematics that can be understood as a science.

It has value because of what it is, it lose value if it goes down the path you are proposing.

Re: Mathematics, science and Abraham Robinson

<570b03f7-fdd2-4b70-8e14-84c7a8dbe029n@googlegroups.com>

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Subject: Re: Mathematics, science and Abraham Robinson
From: dohduh...@yahoo.com (sobriquet)
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 by: sobriquet - Thu, 12 May 2022 12:33 UTC

On Thursday, May 12, 2022 at 4:37:18 AM UTC+2, david...@gmail.com wrote:
> Here's a quote from Abraham Robinson, who is undeniably a legit mathematician (use Wikipedia if you need to know more).
>
> "I think that there is a real need, in formalism and elsewhere, to link our understanding of mathematics with our understanding of the physical world." (A. Robinson)
>
> That's exactly what I have been saying for years, and the response I get from guys like Malum, Messager, and Burns is that the idea is stupid, crackpot, and motivated by evil intentions. There are nothing but crackpots in this newsgroup.
>
> Even serious mathematicians often tell me that they have no interest in that idea, and since mathematics is by definition what mathematicians are interested in, the idea has no role to play in mathematics.
>
> Here's another quote from Mr. Robinson.
>
> "Infinite totalities do not exist in any sense of the word (i.e., either really or ideally). More precisely, any mention, or purported mention, of infinite totalities is, literally, meaningless." (A. Robinson)

In a way they do exist. Like if you add up an infinite number of numbers, the result might be
a number (like the sum of numbers obtained from 1/(2^n) for n=0 to n=infinity, which
adds up to 2).
Even though you can't actually obtain the result by summing up an infinite number of
numbers, there are alternative ways to obtain the result that you would get if you could
add up an infinite number of numbers.

Though perhaps this can be refuted with an argument from physics where
we might be able to obtain evidence that any region of space or spacetime can't
be infinitely subdivided (analogous to the way you can't keep subdividing
a quantity of gold or other forms of energy or matter into infinitesimally small
quantities).

>
> Exactly right. Someday, mathematicians will wake up.
>
> Here's a quote from Yuri Manin, another outstanding mathematician.
>
> "Georg Cantor's grand meta-narrative, Set Theory, created by him almost singlehandedly in the span of about fifteen years, resembles a piece of high art more than a scientific theory." (Y. Manin)
>
> I mention that because I've been asked why I usually mention Cantor by name when I say that set theory doesn't really belong in mathematics. I think that he (Cantor) is ultimately responsible.
>
> The mathematics that is undeniably of value to the world is the mathematics that can be understood as a science.

Re: Mathematics, science and Abraham Robinson

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From: nom...@afraid.org (FromTheRafters)
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Subject: Re: Mathematics, science and Abraham Robinson
Date: Thu, 12 May 2022 11:22:49 -0700
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 by: FromTheRafters - Thu, 12 May 2022 18:22 UTC

sobriquet laid this down on his screen :
> On Thursday, May 12, 2022 at 4:37:18 AM UTC+2, david...@gmail.com wrote:
>> Here's a quote from Abraham Robinson, who is undeniably a legit
>> mathematician (use Wikipedia if you need to know more).
>>
>> "I think that there is a real need, in formalism and elsewhere, to link our
>> understanding of mathematics with our understanding of the physical world."
>> (A. Robinson)
>>
>> That's exactly what I have been saying for years, and the response I get
>> from guys like Malum, Messager, and Burns is that the idea is stupid,
>> crackpot, and motivated by evil intentions. There are nothing but crackpots
>> in this newsgroup.
>>
>> Even serious mathematicians often tell me that they have no interest in that
>> idea, and since mathematics is by definition what mathematicians are
>> interested in, the idea has no role to play in mathematics.
>>
>> Here's another quote from Mr. Robinson.
>>
>> "Infinite totalities do not exist in any sense of the word (i.e., either
>> really or ideally). More precisely, any mention, or purported mention, of
>> infinite totalities is, literally, meaningless." (A. Robinson)
>
> In a way they do exist. Like if you add up an infinite number of numbers, the
> result might be a number (like the sum of numbers obtained from 1/(2^n) for
> n=0 to n=infinity, which adds up to 2).
> Even though you can't actually obtain the result by summing up an infinite
> number of numbers, there are alternative ways to obtain the result that you
> would get if you could add up an infinite number of numbers.

That's what many people overlook. Also, I'm hard-pressed to find a real
number which doesn't have at least one convergent series
representation.

Re: Mathematics, science and Abraham Robinson

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Subject: Re: Mathematics, science and Abraham Robinson
From: fredjeff...@gmail.com (FredJeffries)
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 by: FredJeffries - Mon, 16 May 2022 17:39 UTC

On Thursday, May 12, 2022 at 11:23:06 AM UTC-7, FromTheRafters wrote:
> sobriquet laid this down on his screen :
> > On Thursday, May 12, 2022 at 4:37:18 AM UTC+2, david...@gmail.com wrote:

> >> Here's another quote from Mr. Robinson.
> >>
> >> "Infinite totalities do not exist in any sense of the word (i.e., either
> >> really or ideally). More precisely, any mention, or purported mention, of
> >> infinite totalities is, literally, meaningless." (A. Robinson)
> >
> > In a way they do exist. Like if you add up an infinite number of numbers, the
> > result might be a number (like the sum of numbers obtained from 1/(2^n) for
> > n=0 to n=infinity, which adds up to 2).
> > Even though you can't actually obtain the result by summing up an infinite
> > number of numbers, there are alternative ways to obtain the result that you
> > would get if you could add up an infinite number of numbers.
>
> That's what many people overlook. Also, I'm hard-pressed to find a real
> number which doesn't have at least one convergent series
> representation.

Gentlemen, gentlemen. PLEASE.

Even though you can't do something we can still obtain the result that we would get if we could do it?!

What kind of silliness is that? How do we KNOW what we would get if we can't do it? Not only can't do it, but have no idea what it would mean to do it?

We cannot 'add up an infinite number of numbers'. It's not (merely) that there are too many of them. It's that there is no last term in the series. The process of repeated binary addition never stops. Talk about an END of the process or what happens when the process FINISHES is just nonsense.

Finding the limit of the sequence of partial sums of an infinite series is NOT an 'alternative ways to obtain the result that you would get if you could add up an infinite number of numbers'.

It is an entirely DIFFERENT species of task/process altogether.

The 'sum of an infinite series', i.e. the limit of the sequence of partial sums, is the (one, unique) number which best characterizes that series as a whole. It is a 'universal' number for that series. It is the number which best SUMmarizes the series.

Finding the limit of the sequence of partial sums is what we do INSTEAD of adding up all of the terms because the notion of 'adding up all of the terms' is at best currently undefined and at worst complete nonsense. And all talk of 'adding up all of the terms' only gives the cranks and trolls more grist for their mills.

https://en.wikipedia.org/wiki/Universal_property

Re: Mathematics, science and Abraham Robinson

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Subject: Re: Mathematics, science and Abraham Robinson
From: davidlpe...@gmail.com (David Petry)
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 by: David Petry - Tue, 17 May 2022 02:50 UTC

On Thursday, May 12, 2022 at 4:43:47 AM UTC-7, zelos...@gmail.com wrote:
> torsdag 12 maj 2022 kl. 04:37:18 UTC+2 skrev david...@gmail.com:
> > Here's a quote from Abraham Robinson, who is undeniably a legit mathematician (use Wikipedia if you need to know more).

> > "I think that there is a real need, in formalism and elsewhere, to link our understanding of mathematics with our understanding of the physical world." (A. Robinson)


> > That's exactly what I have been saying for years, and the response I get from guys like Malum, Messager, and Burns is that the idea is stupid, crackpot, and motivated by evil intentions. There are nothing but crackpots in this newsgroup.

> There are plenty of us sane ones, I am and a few others. You, Gabriel, etc are the cranks and crackpots here.

Recall what I've said many times: the serious purpose of mathematics is to provide a conceptual framework that helps us reason about the real world.

So, (question for Zelos), how is that different from what Robinson said? And if you admit that there is no difference, then do you think that Robinson was a crank and a crackpot?

Re: Mathematics, science and Abraham Robinson

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Subject: Re: Mathematics, science and Abraham Robinson
Date: Mon, 16 May 2022 22:21:33 -0500
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 by: sergio - Tue, 17 May 2022 03:21 UTC

On 5/16/2022 12:39 PM, FredJeffries wrote:
> On Thursday, May 12, 2022 at 11:23:06 AM UTC-7, FromTheRafters wrote:
>> sobriquet laid this down on his screen :
>>> On Thursday, May 12, 2022 at 4:37:18 AM UTC+2, david...@gmail.com wrote:
>
>>>> Here's another quote from Mr. Robinson.
>>>>
>>>> "Infinite totalities do not exist in any sense of the word (i.e., either
>>>> really or ideally). More precisely, any mention, or purported mention, of
>>>> infinite totalities is, literally, meaningless." (A. Robinson)
>>>
>>> In a way they do exist. Like if you add up an infinite number of numbers, the
>>> result might be a number (like the sum of numbers obtained from 1/(2^n) for
>>> n=0 to n=infinity, which adds up to 2).
>>> Even though you can't actually obtain the result by summing up an infinite
>>> number of numbers, there are alternative ways to obtain the result that you
>>> would get if you could add up an infinite number of numbers.
>>
>> That's what many people overlook. Also, I'm hard-pressed to find a real
>> number which doesn't have at least one convergent series
>> representation.
>
> Gentlemen, gentlemen. PLEASE.
>
> Even though you can't do something we can still obtain the result that we would get if we could do it?!
>
> What kind of silliness is that? How do we KNOW what we would get if we can't do it? Not only can't do it, but have no idea what it would mean to do it?
>
> We cannot 'add up an infinite number of numbers'. It's not (merely) that there are too many of them. It's that there is no last term in the series. The process of repeated binary addition never stops. Talk about an END of the process or what happens when the process FINISHES is just nonsense.
>
> Finding the limit of the sequence of partial sums of an infinite series is NOT an 'alternative ways to obtain the result that you would get if you could add up an infinite number of numbers'.
>
> It is an entirely DIFFERENT species of task/process altogether.
>
> The 'sum of an infinite series', i.e. the limit of the sequence of partial sums, is the (one, unique) number which best characterizes that series as a whole. It is a 'universal' number for that series. It is the number which best SUMmarizes the series.
>
> Finding the limit of the sequence of partial sums is what we do INSTEAD of adding up all of the terms because the notion of 'adding up all of the terms' is at best currently undefined and at worst complete nonsense. And all talk of 'adding up all of the terms' only gives the cranks and trolls more grist for their mills.
>
>

and....

Re: Mathematics, science and Abraham Robinson

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Subject: Re: Mathematics, science and Abraham Robinson
Date: Mon, 16 May 2022 22:25:52 -0500
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 by: sergio - Tue, 17 May 2022 03:25 UTC

On 5/16/2022 9:50 PM, David Petry wrote:
> On Thursday, May 12, 2022 at 4:43:47 AM UTC-7, zelos...@gmail.com wrote:
>> torsdag 12 maj 2022 kl. 04:37:18 UTC+2 skrev david...@gmail.com:
>>> Here's a quote from Abraham Robinson, who is undeniably a legit mathematician (use Wikipedia if you need to know more).
>
>
>>> "I think that there is a real need, in formalism and elsewhere, to link our understanding of mathematics with our understanding of the physical world." (A. Robinson)
>
>
>>> That's exactly what I have been saying for years, and the response I get from guys like Malum, Messager, and Burns is that the idea is stupid, crackpot, and motivated by evil intentions. There are nothing but crackpots in this newsgroup.
>
>> There are plenty of us sane ones, I am and a few others. You, Gabriel, etc are the cranks and crackpots here.
>
>
> Recall what I've said many times: the serious purpose of mathematics is to provide a conceptual framework that helps us reason about the real world.

corrected;

"A serious purpose of mathematics is to provide a conceptual framework that helps us model the real world".

>
> So, (question for Zelos), how is that different from what Robinson said? And if you admit that there is no difference, then do you think that Robinson was a crank and a crackpot?

this one is BS;

"I think that there is a real need, in formalism and elsewhere, to link our understanding of mathematics with our understanding of the physical world."
(A. Robinson)

Re: Mathematics, science and Abraham Robinson

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Subject: Re: Mathematics, science and Abraham Robinson
From: zelos.ma...@gmail.com (zelos...@gmail.com)
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 by: zelos...@gmail.com - Tue, 17 May 2022 05:05 UTC

tisdag 17 maj 2022 kl. 04:50:54 UTC+2 skrev david...@gmail.com:
> On Thursday, May 12, 2022 at 4:43:47 AM UTC-7, zelos...@gmail.com wrote:
> > torsdag 12 maj 2022 kl. 04:37:18 UTC+2 skrev david...@gmail.com:
> > > Here's a quote from Abraham Robinson, who is undeniably a legit mathematician (use Wikipedia if you need to know more).
>
>
> > > "I think that there is a real need, in formalism and elsewhere, to link our understanding of mathematics with our understanding of the physical world." (A. Robinson)
>
>
> > > That's exactly what I have been saying for years, and the response I get from guys like Malum, Messager, and Burns is that the idea is stupid, crackpot, and motivated by evil intentions. There are nothing but crackpots in this newsgroup.
>
> > There are plenty of us sane ones, I am and a few others. You, Gabriel, etc are the cranks and crackpots here.
> Recall what I've said many times: the serious purpose of mathematics is to provide a conceptual framework that helps us reason about the real world.

No one cares what you, a crank, says.

>
> So, (question for Zelos), how is that different from what Robinson said? And if you admit that there is no difference, then do you think that Robinson was a crank and a crackpot?

He might have been a crank. I know for certain that you are. But both of you are wrong here.

Re: Mathematics, science and Abraham Robinson

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 by: FromTheRafters - Tue, 17 May 2022 07:24 UTC

sergio has brought this to us :
> On 5/16/2022 12:39 PM, FredJeffries wrote:
>> On Thursday, May 12, 2022 at 11:23:06 AM UTC-7, FromTheRafters wrote:
>>> sobriquet laid this down on his screen :
>>>> On Thursday, May 12, 2022 at 4:37:18 AM UTC+2, david...@gmail.com wrote:
>>
>>>>> Here's another quote from Mr. Robinson.
>>>>>
>>>>> "Infinite totalities do not exist in any sense of the word (i.e., either
>>>>> really or ideally). More precisely, any mention, or purported mention,
>>>>> of
>>>>> infinite totalities is, literally, meaningless." (A. Robinson)
>>>>
>>>> In a way they do exist. Like if you add up an infinite number of numbers,
>>>> the
>>>> result might be a number (like the sum of numbers obtained from 1/(2^n)
>>>> for
>>>> n=0 to n=infinity, which adds up to 2).
>>>> Even though you can't actually obtain the result by summing up an
>>>> infinite
>>>> number of numbers, there are alternative ways to obtain the result that
>>>> you
>>>> would get if you could add up an infinite number of numbers.
>>>
>>> That's what many people overlook. Also, I'm hard-pressed to find a real
>>> number which doesn't have at least one convergent series
>>> representation.
>>
>> Gentlemen, gentlemen. PLEASE.
>>
>> Even though you can't do something we can still obtain the result that we
>> would get if we could do it?!
>>
>> What kind of silliness is that? How do we KNOW what we would get if we
>> can't do it? Not only can't do it, but have no idea what it would mean to
>> do it?
>>
>> We cannot 'add up an infinite number of numbers'. It's not (merely) that
>> there are too many of them. It's that there is no last term in the series.
>> The process of repeated binary addition never stops. Talk about an END of
>> the process or what happens when the process FINISHES is just nonsense.
>>
>> Finding the limit of the sequence of partial sums of an infinite series is
>> NOT an 'alternative ways to obtain the result that you would get if you
>> could add up an infinite number of numbers'.
>>
>> It is an entirely DIFFERENT species of task/process altogether.
>>
>> The 'sum of an infinite series', i.e. the limit of the sequence of partial
>> sums, is the (one, unique) number which best characterizes that series as a
>> whole. It is a 'universal' number for that series. It is the number which
>> best SUMmarizes the series.
>>
>> Finding the limit of the sequence of partial sums is what we do INSTEAD of
>> adding up all of the terms because the notion of 'adding up all of the
>> terms' is at best currently undefined and at worst complete nonsense. And
>> all talk of 'adding up all of the terms' only gives the cranks and trolls
>> more grist for their mills.
>>
>>
>
>
> and....

I fail to see how this refutes what I said. We have the real numbers
because we want to use the ideas of 'convergence' and 'limit' to define
our numbers. Summation is usually done one way for finite summations
and another way for infinite summations but both are summations.

Re: Mathematics, science and Abraham Robinson

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 by: sergi o - Tue, 17 May 2022 14:29 UTC

On 5/17/2022 2:24 AM, FromTheRafters wrote:
> sergio has brought this to us :
>> On 5/16/2022 12:39 PM, FredJeffries wrote:
>>> On Thursday, May 12, 2022 at 11:23:06 AM UTC-7, FromTheRafters wrote:
>>>> sobriquet laid this down on his screen :
>>>>> On Thursday, May 12, 2022 at 4:37:18 AM UTC+2, david...@gmail.com wrote:
>>>
>>>>>> Here's another quote from Mr. Robinson.
>>>>>>
>>>>>> "Infinite totalities do not exist in any sense of the word (i.e., either
>>>>>> really or ideally). More precisely, any mention, or purported mention, of
>>>>>> infinite totalities is, literally, meaningless." (A. Robinson)
>>>>>
>>>>> In a way they do exist. Like if you add up an infinite number of numbers, the
>>>>> result might be a number (like the sum of numbers obtained from 1/(2^n) for
>>>>> n=0 to n=infinity, which adds up to 2).
>>>>> Even though you can't actually obtain the result by summing up an infinite
>>>>> number of numbers, there are alternative ways to obtain the result that you
>>>>> would get if you could add up an infinite number of numbers.
>>>>
>>>> That's what many people overlook. Also, I'm hard-pressed to find a real
>>>> number which doesn't have at least one convergent series
>>>> representation.
>>>
>>> Gentlemen, gentlemen. PLEASE.
>>>
>>> Even though you can't do something we can still obtain the result that we would get if we could do it?!
>>>
>>> What kind of silliness is that? How do we KNOW what we would get if we can't do it? Not only can't do it, but have no idea what it would mean to do it?
>>>
>>> We cannot 'add up an infinite number of numbers'. It's not (merely) that there are too many of them. It's that there is no last term in the series.
>>> The process of repeated binary addition never stops. Talk about an END of the process or what happens when the process FINISHES is just nonsense.
>>>
>>> Finding the limit of the sequence of partial sums of an infinite series is NOT an 'alternative ways to obtain the result that you would get if you
>>> could add up an infinite number of numbers'.
>>>
>>> It is an entirely DIFFERENT species of task/process altogether.
>>>
>>> The 'sum of an infinite series', i.e. the limit of the sequence of partial sums, is the (one, unique) number which best characterizes that series as
>>> a whole. It is a 'universal' number for that series. It is the number which best SUMmarizes the series.
>>>
>>> Finding the limit of the sequence of partial sums is what we do INSTEAD of adding up all of the terms because the notion of 'adding up all of the
>>> terms' is at best currently undefined and at worst complete nonsense. And all talk of 'adding up all of the terms' only gives the cranks and trolls
>>> more grist for their mills.
>>>
>>>
>>
>>
>> and....
>
> I fail to see how this refutes what I said. We have the real numbers because we want to use the ideas of 'convergence' and 'limit' to define our
> numbers. Summation is usually done one way for finite summations and another way for infinite summations but both are summations.

seems like FredJeffries has never read Newton, nor had a class on convergence, or he forgot it all.
I think he misquoted Robison or did not understand it..

Re: Mathematics, science and Abraham Robinson

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Subject: Re: Mathematics, science and Abraham Robinson
From: dohduh...@yahoo.com (sobriquet)
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 by: sobriquet - Tue, 17 May 2022 17:52 UTC

On Monday, May 16, 2022 at 7:39:56 PM UTC+2, FredJeffries wrote:
> On Thursday, May 12, 2022 at 11:23:06 AM UTC-7, FromTheRafters wrote:
> > sobriquet laid this down on his screen :
> > > On Thursday, May 12, 2022 at 4:37:18 AM UTC+2, david...@gmail.com wrote:
>
> > >> Here's another quote from Mr. Robinson.
> > >>
> > >> "Infinite totalities do not exist in any sense of the word (i.e., either
> > >> really or ideally). More precisely, any mention, or purported mention, of
> > >> infinite totalities is, literally, meaningless." (A. Robinson)
> > >
> > > In a way they do exist. Like if you add up an infinite number of numbers, the
> > > result might be a number (like the sum of numbers obtained from 1/(2^n) for
> > > n=0 to n=infinity, which adds up to 2).
> > > Even though you can't actually obtain the result by summing up an infinite
> > > number of numbers, there are alternative ways to obtain the result that you
> > > would get if you could add up an infinite number of numbers.
> >
> > That's what many people overlook. Also, I'm hard-pressed to find a real
> > number which doesn't have at least one convergent series
> > representation.
> Gentlemen, gentlemen. PLEASE.
>
> Even though you can't do something we can still obtain the result that we would get if we could do it?!
>
> What kind of silliness is that? How do we KNOW what we would get if we can't do it? Not only can't do it, but have no idea what it would mean to do it?
>
> We cannot 'add up an infinite number of numbers'. It's not (merely) that there are too many of them. It's that there is no last term in the series. The process of repeated binary addition never stops. Talk about an END of the process or what happens when the process FINISHES is just nonsense.
>
> Finding the limit of the sequence of partial sums of an infinite series is NOT an 'alternative ways to obtain the result that you would get if you could add up an infinite number of numbers'.
>
> It is an entirely DIFFERENT species of task/process altogether.
>
> The 'sum of an infinite series', i.e. the limit of the sequence of partial sums, is the (one, unique) number which best characterizes that series as a whole. It is a 'universal' number for that series. It is the number which best SUMmarizes the series.
>
> Finding the limit of the sequence of partial sums is what we do INSTEAD of adding up all of the terms because the notion of 'adding up all of the terms' is at best currently undefined and at worst complete nonsense. And all talk of 'adding up all of the terms' only gives the cranks and trolls more grist for their mills.
>
>
> https://en.wikipedia.org/wiki/Universal_property

We know it by mathematical induction . So given an area of, say 2, we know we can subdivide one of its constituent areas into two equal areas of 1 (the base case). Then we apply the assumption that we can further subdivide one of its two smallest constituent areas into two smaller equal areas 1/2 and 1/2 and we continue this step recursively indefinitely.
Either we end up with a number that can't be further subdivided into smaller numbers and we found the smallest possible number, or there is nothing which prevents us from further subdividing one of the two smallest numbers into two equal smaller numbers.

https://en.wikipedia.org/wiki/Mathematical_induction

I see no conceptual difference between the claim that there is no biggest natural number (or biggest prime number) and the claim that there is no smallest number, since we can always add 1 to a number or divide a number by 2.

Infinitely subdividing an area of two is equivalent to the claim that there is no biggest finite number of subdivisions beyond which the area we're subdividing would suddenly no longer sum up to two.

Re: Mathematics, science and Abraham Robinson

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Subject: Re: Mathematics, science and Abraham Robinson
From: askaske...@gmail.com (WM)
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 by: WM - Wed, 18 May 2022 12:42 UTC

sobriquet schrieb am Dienstag, 17. Mai 2022 um 19:52:54 UTC+2:

> I see no conceptual difference between the claim that there is no biggest natural number (or biggest prime number) and the claim that there is no smallest number, since we can always add 1 to a number or divide a number by 2.
>
> Infinitely subdividing an area of two is equivalent to the claim that there is no biggest finite number of subdivisions beyond which the area we're subdividing would suddenly no longer sum up to two.

Both supply potential infinity, i.e., numbers growing larger than any given number and (positive) numbers shrinking smaller than any given positive number. But that is not the infinite of set theory, namely actual infinity.

All numbers you get by induction have ℵo successors before ω:
∀n ∈ ℕ_ind: |ℕ \ {1, 2, 3, ..., n}| = ℵo .
They cannot be exhausted, because they remain always there.

But the set of all natural numbers exhausts also these successors:
|ℕ \ {1, 2, 3, ...}| = 0
or
{0, 1, 2, 3, ..., ω} \ ℕ = {0, ω}.

Regards, WM

Re: Mathematics, science and Abraham Robinson

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Subject: Re: Mathematics, science and Abraham Robinson
From: fredjeff...@gmail.com (FredJeffries)
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 by: FredJeffries - Wed, 18 May 2022 16:14 UTC

On Tuesday, May 17, 2022 at 7:29:54 AM UTC-7, sergi o wrote:
> On 5/17/2022 2:24 AM, FromTheRafters wrote:
> > sergio has brought this to us :
> >> On 5/16/2022 12:39 PM, FredJeffries wrote:
> >>> On Thursday, May 12, 2022 at 11:23:06 AM UTC-7, FromTheRafters wrote:
> >>>> sobriquet laid this down on his screen :
> >>>>> On Thursday, May 12, 2022 at 4:37:18 AM UTC+2, david...@gmail.com wrote:
> >>>
> >>>>>> Here's another quote from Mr. Robinson.
> >>>>>>
> >>>>>> "Infinite totalities do not exist in any sense of the word (i.e., either
> >>>>>> really or ideally). More precisely, any mention, or purported mention, of
> >>>>>> infinite totalities is, literally, meaningless." (A. Robinson)
> >>>>>
> >>>>> In a way they do exist. Like if you add up an infinite number of numbers, the
> >>>>> result might be a number (like the sum of numbers obtained from 1/(2^n) for
> >>>>> n=0 to n=infinity, which adds up to 2).
> >>>>> Even though you can't actually obtain the result by summing up an infinite
> >>>>> number of numbers, there are alternative ways to obtain the result that you
> >>>>> would get if you could add up an infinite number of numbers.
> >>>>
> >>>> That's what many people overlook. Also, I'm hard-pressed to find a real
> >>>> number which doesn't have at least one convergent series
> >>>> representation.
> >>>
> >>> Gentlemen, gentlemen. PLEASE.
> >>>
> >>> Even though you can't do something we can still obtain the result that we would get if we could do it?!
> >>>
> >>> What kind of silliness is that? How do we KNOW what we would get if we can't do it? Not only can't do it, but have no idea what it would mean to do it?
> >>>
> >>> We cannot 'add up an infinite number of numbers'. It's not (merely) that there are too many of them. It's that there is no last term in the series.
> >>> The process of repeated binary addition never stops. Talk about an END of the process or what happens when the process FINISHES is just nonsense..
> >>>
> >>> Finding the limit of the sequence of partial sums of an infinite series is NOT an 'alternative ways to obtain the result that you would get if you
> >>> could add up an infinite number of numbers'.
> >>>
> >>> It is an entirely DIFFERENT species of task/process altogether.
> >>>
> >>> The 'sum of an infinite series', i.e. the limit of the sequence of partial sums, is the (one, unique) number which best characterizes that series as
> >>> a whole. It is a 'universal' number for that series. It is the number which best SUMmarizes the series.
> >>>
> >>> Finding the limit of the sequence of partial sums is what we do INSTEAD of adding up all of the terms because the notion of 'adding up all of the
> >>> terms' is at best currently undefined and at worst complete nonsense. And all talk of 'adding up all of the terms' only gives the cranks and trolls
> >>> more grist for their mills.
> >>>
> >>>
> >>
> >>
> >> and....
> >
> > I fail to see how this refutes what I said. We have the real numbers because we want to use the ideas of 'convergence' and 'limit' to define our
> > numbers. Summation is usually done one way for finite summations and another way for infinite summations but both are summations.
> seems like FredJeffries has never read Newton, nor had a class on convergence, or he forgot it all.
> I think he misquoted Robison or did not understand it..

Inasmuch as no one understood the point I was trying to make, I will only apologize for being such a bad expositor and wasting your time with my semi-rant. I will not pursue the matter further.

On to bookkeeping points: I am not responsible for the Robinson (mis)quote in this thread. That should be obvious not only from the indentations and headings, but because I would have included the source and context of the statement. Most importantly, I would not have omitted the remainder of Robinson's statement (see below).

The statement is from Abraham Robinson's (one of the most important mathematicians of the 20th century, by the way) talk at the 1964 International Congress for Logic, Methodology, and Philosophy of Science in Jerusalem titled simply 'Formalism 64'.

Alas, neither the transcription of that talk nor the proceedings of the congress seem to be electronically available.

A few years later, Robinson published 'From a formalist's point of view' in 'Dialectica' Vol. 23, No. 1 (1969), pp. 45-49 where he restated and clarified points he had made in that talk. This article is available for those with JSTOR access
https://www.jstor.org/stable/42968450

For a secondary account we may consult Robinson's biographer Joseph Dauben (yes, the same Dauben who wrote the acclaimed biography of Georg Cantor). There is an available article 'Abraham Robinson and Nonstandard Analysis: History, Philosophy, and Foundations of Mathematics'

https://conservancy.umn.edu/bitstream/handle/11299/185659/11_07Dauben.pdf

In the section 'Robinson and "Formalism 64"' (p. 10), Dauben discusses the quote in question.

<quote>
Robinson emphasized two factors in rejecting his earlier Platonism in favor of a formalist position:

(i) Infinite totalities do not exist in any sense of the word (i.e., either really or ideally). More precisely, any mention, or purported mention, of infinite totalities is, literally, meaningless.

(ii) Nevertheless, we should continue the business of Mathematics 'as usual,' i.e. we should act as if infinite totalities really existed.
</quote>

Re: Mathematics, science and Abraham Robinson

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 by: sobriquet - Wed, 18 May 2022 16:24 UTC

On Wednesday, May 18, 2022 at 2:42:22 PM UTC+2, WM wrote:
> sobriquet schrieb am Dienstag, 17. Mai 2022 um 19:52:54 UTC+2:
>
> > I see no conceptual difference between the claim that there is no biggest natural number (or biggest prime number) and the claim that there is no smallest number, since we can always add 1 to a number or divide a number by 2.
> >
> > Infinitely subdividing an area of two is equivalent to the claim that there is no biggest finite number of subdivisions beyond which the area we're subdividing would suddenly no longer sum up to two.
> Both supply potential infinity, i.e., numbers growing larger than any given number and (positive) numbers shrinking smaller than any given positive number. But that is not the infinite of set theory, namely actual infinity.
>
> All numbers you get by induction have ℵo successors before ω:
> ∀n ∈ ℕ_ind: |ℕ \ {1, 2, 3, ..., n}| = ℵo .
> They cannot be exhausted, because they remain always there.

But then what does it mean when we prove something with mathematical induction?
I would assume that if we proof a property P(n) for natural numbers n with induction, that
means that for all natural numbers n, P(n) holds.
But since there is an infinite set of natural numbers, the property has been proven for an actual
infinite number of cases.

So if we prove that we can subdivide a given finite area into an arbitrary natural number of
parts decreasing in size, and the parts sum up to the original area we started
out with, that means we know that the actual infinite sum of subdivided areas decreasing
in size indefinitely yields the original area.

>
> But the set of all natural numbers exhausts also these successors:
> |ℕ \ {1, 2, 3, ...}| = 0
> or
> {0, 1, 2, 3, ..., ω} \ ℕ = {0, ω}.
>
> Regards, WM

Re: Mathematics, science and Abraham Robinson

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Subject: Re: Mathematics, science and Abraham Robinson
From: askaske...@gmail.com (WM)
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 by: WM - Wed, 18 May 2022 18:25 UTC

sobriquet schrieb am Mittwoch, 18. Mai 2022 um 18:25:04 UTC+2:
> On Wednesday, May 18, 2022 at 2:42:22 PM UTC+2, WM wrote:
> > sobriquet schrieb am Dienstag, 17. Mai 2022 um 19:52:54 UTC+2:
> >
> > > I see no conceptual difference between the claim that there is no biggest natural number (or biggest prime number) and the claim that there is no smallest number, since we can always add 1 to a number or divide a number by 2.
> > >
> > > Infinitely subdividing an area of two is equivalent to the claim that there is no biggest finite number of subdivisions beyond which the area we're subdividing would suddenly no longer sum up to two.
> > Both supply potential infinity, i.e., numbers growing larger than any given number and (positive) numbers shrinking smaller than any given positive number. But that is not the infinite of set theory, namely actual infinity.
> >
> > All numbers you get by induction have ℵo successors before ω:
> > ∀n ∈ ℕ_ind: |ℕ \ {1, 2, 3, ..., n}| = ℵo .
> > They cannot be exhausted, because they remain always there.

> But then what does it mean when we prove something with mathematical induction?

We prove it for all definable natural numbers.
> I would assume that if we proof a property P(n) for natural numbers n with induction, that
> means that for all natural numbers n, P(n) holds.

You know that the set of numbers which this is proved for is always finite?
> But since there is an infinite set of natural numbers, the property has been proven for an actual
> infinite number of cases.

By induction we can prove that the numbers reached by induction have ℵo successors before ω.Therefore they belong to a finite set. It is impossible to subdivide ℕ into two consecutive infinite aleph_0-sets.The successor numbers cannot be identified by induction. They can only be used collectively.

The set of all natural numbers contains also these successors:
ℕ \ {1, 2, 3, ...} = { }
>
> So if we prove that we can subdivide a given finite area into an arbitrary natural number of
> parts decreasing in size, and the parts sum up to the original area we started
> out with, that means we know that the actual infinite sum of subdivided areas decreasing
> in size indefinitely yields the original area.

Yes that holds for every definable subdivision into n parts.

Regards, WM

Re: Mathematics, science and Abraham Robinson

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Subject: Re: Mathematics, science and Abraham Robinson
From: askaske...@gmail.com (WM)
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 by: WM - Wed, 18 May 2022 18:32 UTC

FredJeffries schrieb am Mittwoch, 18. Mai 2022 um 18:14:16 UTC+2:
>
> <quote>
> Robinson emphasized two factors in rejecting his earlier Platonism in favor of a formalist position:
>
> (i) Infinite totalities do not exist in any sense of the word (i.e., either really or ideally). More precisely, any mention, or purported mention, of infinite totalities is, literally, meaningless.
>
> (ii) Nevertheless, we should continue the business of Mathematics 'as usual,' i.e. we should act as if infinite totalities really existed.
> </quote>

Let's continue with the origianl source:

"I feel quite unable to grasp the idea of an actual infinite totality. To me there appears to exist an unbridgeable gulf between sets or structures of one, or two, or five elements, on one hand, and infinite structures on the other hand [...] I must regard a theory which refers to an infinite totality as meaningless in the sense that its terms and sentences cannot posses the direct interpretation in an actual structure that we should expect them to have by analogy with concrete (e.g., empirical) situations." [A. Robinson: "Formalism 64", North-Holland, Amsterdam, p. 230f]

Regards, WM

Re: Mathematics, science and Abraham Robinson

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Subject: Re: Mathematics, science and Abraham Robinson
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 by: FromTheRafters - Wed, 18 May 2022 18:36 UTC

WM was thinking very hard :
> sobriquet schrieb am Mittwoch, 18. Mai 2022 um 18:25:04 UTC+2:
>> On Wednesday, May 18, 2022 at 2:42:22 PM UTC+2, WM wrote:
>>> sobriquet schrieb am Dienstag, 17. Mai 2022 um 19:52:54 UTC+2:
>>>
>>>> I see no conceptual difference between the claim that there is no biggest
>>>> natural number (or biggest prime number) and the claim that there is no
>>>> smallest number, since we can always add 1 to a number or divide a number
>>>> by 2.
>>>>
>>>> Infinitely subdividing an area of two is equivalent to the claim that
>>>> there is no biggest finite number of subdivisions beyond which the area
>>>> we're subdividing would suddenly no longer sum up to two.
>>> Both supply potential infinity, i.e., numbers growing larger than any given
>>> number and (positive) numbers shrinking smaller than any given positive
>>> number. But that is not the infinite of set theory, namely actual infinity.
>>>
>>> All numbers you get by induction have ℵo successors before ω:
>>> ∀n ∈ ℕ_ind: |ℕ \ {1, 2, 3, ..., n}| = ℵo .
>>> They cannot be exhausted, because they remain always there.
>
>> But then what does it mean when we prove something with mathematical
>> induction?
>
> We prove it for all definable natural numbers.
>
>> I would assume that if we proof a property P(n) for natural numbers n with
>> induction, that means that for all natural numbers n, P(n) holds.
>
> You know that the set of numbers which this is proved for is always finite?
>
>> But since there is an infinite set of natural numbers, the property has been
>> proven for an actual infinite number of cases.
>
> By induction we can prove that the numbers reached by induction have ℵo
> successors before ω.

Go ahead, we'll wait. While we are waiting, why don't you tell us what
'successors before ω' means here.

Re: Mathematics, science and Abraham Robinson

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Subject: Re: Mathematics, science and Abraham Robinson
From: franz.fr...@gmail.com (Fritz Feldhase)
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 by: Fritz Feldhase - Wed, 18 May 2022 18:39 UTC

On Wednesday, May 18, 2022 at 8:32:16 PM UTC+2, WM wrote:

> Let's continue with the origianl source:
>
> "I feel quite unable to grasp the idea of an actual infinite totality. To me there appears to exist an unbridgeable gulf between sets or structures of one, or two, or five elements, on one hand, and infinite structures on the other hand [...] I must regard a theory which refers to an infinite totality as meaningless in the sense that its terms and sentences cannot posses the direct interpretation in an actual structure that we should expect them to have by analogy with concrete (e.g., empirical) situations." [A. Robinson: "Formalism 64", North-Holland, Amsterdam, p. 230f]

So what?! Do you psychotic asshole full of shit understand the significance of the statement

"(ii) Nevertheless, we should continue the business of Mathematics 'as usual,' i.e. we should act as if infinite totalities really existed." [A. Robinson]

?

Obviously not. After all, your "business" is crankery and psychotic mumbo-jumbo.

Re: Mathematics, science and Abraham Robinson

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Subject: Re: Mathematics, science and Abraham Robinson
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 by: Timothy Golden - Wed, 18 May 2022 18:54 UTC

On Wednesday, May 18, 2022 at 2:32:16 PM UTC-4, WM wrote:
> FredJeffries schrieb am Mittwoch, 18. Mai 2022 um 18:14:16 UTC+2:
> >
> > <quote>
> > Robinson emphasized two factors in rejecting his earlier Platonism in favor of a formalist position:
> >
> > (i) Infinite totalities do not exist in any sense of the word (i.e., either really or ideally). More precisely, any mention, or purported mention, of infinite totalities is, literally, meaningless.
> >
> > (ii) Nevertheless, we should continue the business of Mathematics 'as usual,' i.e. we should act as if infinite totalities really existed.
> > </quote>
> Let's continue with the origianl source:
>
> "I feel quite unable to grasp the idea of an actual infinite totality. To me there appears to exist an unbridgeable gulf between sets or structures of one, or two, or five elements, on one hand, and infinite structures on the other hand [...] I must regard a theory which refers to an infinite totality as meaningless in the sense that its terms and sentences cannot posses the direct interpretation in an actual structure that we should expect them to have by analogy with concrete (e.g., empirical) situations." [A. Robinson: "Formalism 64", North-Holland, Amsterdam, p. 230f]
>
> Regards, WM

I wonder to what degree those concerned with natural valued infinity are possibly entertaining the birth of the continuum?
As you start plopping down positions on a line and they keep going at some large value the discernment of the smaller values seems imperceptible. Particularly thinking in terms of large n a relative position abstraction with arbitrarily fine granularity ensues.

This bears out as we consider the decimal value as a natural value. For instance as 1/3= 0.333... then dropping the decimal point we are dealing in a natural value 333.... Even a value such as 2/5 in perfection will yield 0.4000... which again as a natural value is 4000...
These sorts of infinite precision values are in denial of epsilon/delta theory, whereas 2/5=0.4000 is a finite precision instance. In other words this is a computationally valid instance and if we did wish to work in greater precision we could. Dropping the decimal from this instance we see 2/5 as 4000 and of course its data can be recovered by building up a structure xe where x is the natural value and e is the decimal place. The 'e' portion is arguably natural valued, but it is of a different meaning. It's position is a matter of placing a secondary unity upon the otherwise purely natural value.

In this way the continuum can be built by having a high regard for the natural value; augmenting it with a new sense of unity; and as well keeping a regard for epsilon/delta theory and its adjustable precision. In effect those presuming perfection in their rational values have been working with an infinite form of the natural value and never saw it as ambiguous. Could these be the infinite concerns of the natural valued philosophy? I do credit in part WM's dark number with helping me develop the gray number that is the continuum; the more truly 'real' value. That the interpretation could come back then onto the naturals in this way is an interesting turn. So then are values such as 333... dark values? Hah! I have managed to instantiate them!

Re: Mathematics, science and Abraham Robinson

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Subject: Re: Mathematics, science and Abraham Robinson
From: franz.fr...@gmail.com (Fritz Feldhase)
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 by: Fritz Feldhase - Wed, 18 May 2022 18:56 UTC

On Wednesday, May 18, 2022 at 8:25:52 PM UTC+2, WM wrote:
> sobriquet schrieb am Mittwoch, 18. Mai 2022 um 18:25:04 UTC+2:
> >
> > But then what does it mean when we prove something with mathematical induction?
> >
> We prove it for all [...] natural numbers.

Or all elements in in IN.

Right.

> > I would assume that if we proof a property P(n) for natural numbers n with induction, that
> > means that for all natural numbers n, P(n) holds.

Using symbols: An(n e IN -> P(n)) or An e IN: P(n). Right!

> You know that the set of numbers which this is proved for is always finite?

No, we don't know that IN is finite.

Hint: "Mathematical induction is a mathematical proof technique. It is essentially used to prove that a statement P(n) holds for every natural number n = 0, 1, 2, 3, ... ; that is, the overall statement is a sequence of infinitely many cases P(0), P(1), P(2), P(3), ... ." (Wikipedia)

> > But since there is an infinite set of natural numbers, the property has been proven for an actual infinite number of cases.

Yes. See the quote from Wikipedia.

[Quote from the German page: "Die vollständige Induktion ist eine mathematische Beweismethode, nach der eine Aussage für alle natürlichen Zahlen bewiesen wird, die größer oder gleich einem bestimmten Startwert sind. Da es sich um unendlich viele Zahlen handelt..."]

> [...] the numbers [that] have ℵo successors before ω [...] belong to a finite set.

Sure, each and every "number" n belongs to the finite set {n}. An incredible insight, Mückenheim!

On the other hand, each and every number in IN has ℵo successors "before ω" (< ω), but IN is infinite.

Re: Mathematics, science and Abraham Robinson

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Subject: Re: Mathematics, science and Abraham Robinson
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 by: sergi o - Wed, 18 May 2022 19:07 UTC

On 5/18/2022 1:25 PM, WM wrote:
> sobriquet schrieb am Mittwoch, 18. Mai 2022 um 18:25:04 UTC+2:
>> On Wednesday, May 18, 2022 at 2:42:22 PM UTC+2, WM wrote:
>>> sobriquet schrieb am Dienstag, 17. Mai 2022 um 19:52:54 UTC+2:
>>>
>>>> I see no conceptual difference between the claim that there is no biggest natural number (or biggest prime number) and the claim that there is no smallest number, since we can always add 1 to a number or divide a number by 2.
>>>>
>>>> Infinitely subdividing an area of two is equivalent to the claim that there is no biggest finite number of subdivisions beyond which the area we're subdividing would suddenly no longer sum up to two.
>>> Both supply potential infinity, i.e., numbers growing larger than any given number and (positive) numbers shrinking smaller than any given positive number. But that is not the infinite of set theory, namely actual infinity.
>>>
>>> All numbers you get by induction have ℵo successors before ω:
>>> ∀n ∈ ℕ_ind: |ℕ \ {1, 2, 3, ..., n}| = ℵo .
>>> They cannot be exhausted, because they remain always there.
>
>> But then what does it mean when we prove something with mathematical induction?
>
> We prove it for all definable natural numbers.

wrong, it is provee for all natural numbers.

>
>> I would assume that if we proof a property P(n) for natural numbers n with induction, that
>> means that for all natural numbers n, P(n) holds.
>
> You know that the set of numbers which this is proved for is always finite?

wrong, he said all natural numbers. Did you miss that ?

>
>> But since there is an infinite set of natural numbers, the property has been proven for an actual
>> infinite number of cases.
>
> By induction we can prove that the numbers reached by induction have ℵo successors before ω.

wrong, By induction we can prove that the numbers reached by induction upto ω.

>Therefore they belong to a finite set.

Wrong, it is all natural numbers, the entire set, ℕ

> It is impossible to subdivide ℕ into two consecutive infinite aleph_0-sets.

wrong, odd and even... etc

> The successor numbers cannot be identified by induction.

You stopped at k again. Nobody stopped induction, it goes all the way out, like Chuck Norris.

>They can only be used collectively.

yes, used numbers after collectively harvested can be plowed into the fields like fertilizer.

>
> The set of all natural numbers contains also these successors:
> ℕ \ {1, 2, 3, ...} = { }

intentional diversion, and misdirection.

>>
>> So if we prove that we can subdivide a given finite area into an arbitrary natural number of
>> parts decreasing in size, and the parts sum up to the original area we started
>> out with, that means we know that the actual infinite sum of subdivided areas decreasing
>> in size indefinitely yields the original area.
>
> Yes that holds for every definable subdivision into n parts.

your "definable" is meaningless, with its beeps, flashes, raps, hoofs, giggles

>
> Regards, WM

Re: Mathematics, science and Abraham Robinson

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Subject: Re: Mathematics, science and Abraham Robinson
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 by: Fritz Feldhase - Wed, 18 May 2022 19:28 UTC

On Wednesday, May 18, 2022 at 9:07:55 PM UTC+2, sergi o wrote:
> On 5/18/2022 1:25 PM, WM wrote:
> >
> > It is impossible to subdivide ℕ into two consecutive infinite [....] sets.

That's indeed true! (Big surprise!)

> wrong, odd and even...

Not "consecutive" (though infiite).

What he's talking about here is a partition of IN into two _infinite_ sets {a_1, a_2, a_3 ...} and {b_1, b_2, b_3, ...} such that

a_i < b_j, for all i.j e IN.

(/Partition/ of IN into two sets {a_1, a_2, a_3 ...}, {b_1, b_2, b_3, ...} here means: {a_1, a_2, a_3 ...} =/= {}, {b_1, b_2, b_3, ...} =/= {}, {a_1, a_2, a_3 ...} n {b_1, b_2, b_3, ...} = {} and {a_1, a_2, a_3 ...} u {b_1, b_2, b_3, ...} = IN.)

Need to see a proof? :-P

> > ... that holds for every definable subdivision into n parts.
> >
> your "definable" is meaningless, with its beeps, flashes, raps, hoofs, giggles

Indeed. :-)

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