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Do we really need the concept of infinity?

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Jerry Kraus

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Aug 29, 2017, 11:08:36 AM8/29/17
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Wouldn't very, very large, or very, very numerous be sufficient, as conceptions, for calculus, for example, or for set theory? Why must we resort to the absolute concept of endlessness? Doesn't this cause more problems conceptually, than it is worth?

Peter Percival

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Aug 29, 2017, 11:19:41 AM8/29/17
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Jerry Kraus wrote:
> Wouldn't very, very large, or very, very numerous be sufficient, as
> conceptions, for calculus, for example,

Is 'all subsets of R are Lebesque measurable' consistent (with ZF)?
Yes, if an inaccessible set exists.

> or for set theory?

Set theory is the theory of infinite sets.

> Why must we resort to the absolute concept of
> endlessness? Doesn't this cause more problems conceptually, than it

Yes for the cranks of sci.logic and sci.math.

> is worth?
>


--
Do, as a concession to my poor wits, Lord Darlington, just explain
to me what you really mean.
I think I had better not, Duchess. Nowadays to be intelligible is
to be found out. -- Oscar Wilde, Lady Windermere's Fan

Jerry Kraus

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Aug 29, 2017, 11:38:17 AM8/29/17
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Well, Peter, for all practical purposes, extremely large is quite sufficient for the concept of limits, that forms the basis for Calculus. We can simply substitute "extremely large" for infinity-- as in, the limit of the function as x becomes extremely large -- and make this value precisely as large as we need it to be in context.


Not really sure where you got the idea that all sets are infinite Peter. Again, for most practical purposes, most sets are quite finite, indeed.

You see, Peter, absolutes, like the concept of infinity, are a kind of oversimplification of reality that may make mathematics less effective, rather than more effective. And, also, may result in theoretical periphrasis that is more of an obstacle to scientific progress, rather than the reverse.

Peter Percival

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Aug 29, 2017, 11:59:04 AM8/29/17
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Jerry Kraus wrote:
> On Tuesday, August 29, 2017 at 9:19:41 AM UTC-6, Peter Percival
> wrote:
>> Jerry Kraus wrote:
>>> Wouldn't very, very large, or very, very numerous be sufficient,
>>> as conceptions, for calculus, for example,
>>
>> Is 'all subsets of R are Lebesque measurable' consistent (with ZF)?
>> Yes, if an inaccessible set exists.
>>
>>> or for set theory?
>>
>> Set theory is the theory of infinite sets.
>>
>>> Why must we resort to the absolute concept of endlessness?
>>> Doesn't this cause more problems conceptually, than it
>>
>> Yes for the cranks of sci.logic and sci.math.
>>
>>> is worth?

> Well, Peter, for all practical purposes, extremely large is quite
> sufficient for the concept of limits, that forms the basis for
> Calculus. We can simply substitute "extremely large" for infinity--

How? In the *definition* of "the limit of f(x) as x tends to infinity"
infinitely large numbers do not feature. On the other hand

the limit of f(x) as x tends to infinity

does not mean the same as

the limit of f(x) as x tends to C

where C is some large but finite number. Choose any C you like and a
function f can be easily found that exemplifies the difference.

> as in, the limit of the function as x becomes extremely large -- and
> make this value precisely as large as we need it to be in context.
>
> Not really sure where you got the idea that all sets are infinite

I haven't got any such idea. If finite sets are all you're interested
in, then set theory is redundant. (An case in point is often mentioned
here: ZF-Inf+~Inf is bi-interpretable with first-order PA.)

> Peter. Again, for most practical purposes, most sets are quite
> finite, indeed.

No doubt there are reasonable meanings of "most practical purposes" for
which that is true. But the number of finite sets is less than the
number of sets in general.

> You see, Peter, absolutes, like the concept of infinity, are a kind

What do you mean by the concept of infinity being absolute?

> of oversimplification of reality that may make mathematics less
> effective, rather than more effective. And, also, may result in
> theoretical periphrasis that is more of an obstacle to scientific
> progress, rather than the reverse.

A few years ago Zdislav V. Kovarik made a post listing a dozen or more
meaning of the word "infinity" as used in different branches of
mathematics. I'm hoping that he won't mind me reposting it:

There is a long list of "infinities (with no claim to exhaustiveness):
infinity of the one-point compactification of N,
infinity of the one-point compactification of R,
infinity of the two-point compactification of R,
infinity of the one-point compactification of C,
infinities of the projective extension of the plane,
infinity of Lebesgue-type integration theory,
infinities of the non-standard extension of R,
infinities of the theory of ordinal numbers,
infinities of the theory of cardinal numbers,
infinity adjoined to normed spaces, whose neighborhoods are
complements of relatively compact sets,
infinity adjoined to normed spaces, whose neighborhoods are
complements of bounded sets,
infinity around absolute G-delta non-compact metric spaces,
infinity in the theory of convex optimization,
etc.;

each of these has a clear definition and a set of well-defined rules
for handling it.

And the winner is...
the really, really real infinity imagined by inexperienced debaters of
foundations of mathematics; this one has the advantage that it need
not be defined ("it's just there, don't you see?") and the user can
switch from one set of rules to another, without warning, and without
worrying about consistency, for the purpose of scoring points in idle
and uneducated (at least on one side) debates.

Peter Percival

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Aug 29, 2017, 12:04:59 PM8/29/17
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Jerry Kraus wrote:

> Well, Peter, for all practical purposes, extremely large is quite

For all practical purposes we don't need music, poetry and drama either.
Nor do we need Jews, Negroes or homosexuals. What? Your plumber is a
Jewish homosexual? I need only point out that there are non-Jewish
heterosexual plumbers (there'll even be some in your neighbourhood) to
prove my point.

David Petry

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Aug 29, 2017, 1:25:52 PM8/29/17
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On Tuesday, August 29, 2017 at 8:08:36 AM UTC-7, Jerry Kraus wrote:
> Wouldn't very, very large, or very, very numerous be sufficient, as conceptions, for calculus, for example, or for set theory? Why must we resort to the absolute concept of endlessness? Doesn't this cause more problems conceptually, than it is worth?

I suspect you're just trolling.

For thousands of years (at least since Aristotle), mathematicians have agreed that the notion of an actual infinity is not part of mathematics. But they viewed mathematics as being closely connected to science. But then Cantor and his cohorts came along and said that mathematics is an art form, and not a science. And they tell us that the theory of infinity is a really beautiful theory.

And we must accept the claim that the theory of infinity is beautiful, because the artists who created the theory tell us it is beautiful. And, of course, mathematics can be defined as what mathematicians do, and anyone who asks the wrong questions is just a crackpot. And the story gets weirder and weirder the more you dig into it.

If we agree that the purpose of mathematics is to provide a framework for reasoning about the real world, and we agree that mathematics is defined by its purpose, and not merely by its form, then we are forced to agree that the notion of an actual infinity is not needed in mathematics. This has been argued ad nauseam in this newsgroup. It seems there's no point in continuing the discussion. Not here, anyways.

Bill

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Aug 29, 2017, 1:35:50 PM8/29/17
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David Petry wrote:
>
> If we agree that the purpose of mathematics is to provide a framework for reasoning about the real world, ...


Why should we agree to that? Which real world? Do you mean complex world?

Jerry Kraus

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Aug 29, 2017, 2:12:49 PM8/29/17
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On Tuesday, August 29, 2017 at 10:04:59 AM UTC-6, Peter Percival wrote:
> Jerry Kraus wrote:
>
> > Well, Peter, for all practical purposes, extremely large is quite
>
---------------------------------------------------------------------------
> For all practical purposes we don't need music, poetry and drama either.

Well, just speaking democratically of course, Peter, more people would see a need for music, poetry and drama, than for infinity. We could take a poll, I suppose.


> Nor do we need Jews, Negroes or homosexuals. What? Your plumber is a
> Jewish homosexual? I need only point out that there are non-Jewish
> heterosexual plumbers (there'll even be some in your neighbourhood) to
> prove my point.
---------------------------------------------------------------------------

I see I've struck a nerve here. You see I'm not saying we can't speculate on the notion of infinity "ad infinitum" -- hahaha, little pun there! -- but, I'm wondering if mathematicians might not find something better to do with their time.

Jerry Kraus

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Aug 29, 2017, 2:15:20 PM8/29/17
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On Tuesday, August 29, 2017 at 11:25:52 AM UTC-6, David Petry wrote:
> On Tuesday, August 29, 2017 at 8:08:36 AM UTC-7, Jerry Kraus wrote:
> > Wouldn't very, very large, or very, very numerous be sufficient, as conceptions, for calculus, for example, or for set theory? Why must we resort to the absolute concept of endlessness? Doesn't this cause more problems conceptually, than it is worth?
>
> I suspect you're just trolling.
>
> For thousands of years (at least since Aristotle), mathematicians have agreed that the notion of an actual infinity is not part of mathematics. But they viewed mathematics as being closely connected to science. But then Cantor and his cohorts came along and said that mathematics is an art form, and not a science. And they tell us that the theory of infinity is a really beautiful theory.
>
> And we must accept the claim that the theory of infinity is beautiful, because the artists who created the theory tell us it is beautiful. And, of course, mathematics can be defined as what mathematicians do, and anyone who asks the wrong questions is just a crackpot. And the story gets weirder and weirder the more you dig into it.
>
----------------------------------------------------------------------------
> If we agree that the purpose of mathematics is to provide a framework for reasoning about the real world, and we agree that mathematics is defined by its purpose, and not merely by its form, then we are forced to agree that the notion of an actual infinity is not needed in mathematics. This has been argued ad nauseam in this newsgroup. It seems there's no point in continuing the discussion. Not here, anyways.

OK, David, sorry for stating the obvious then. That's really all I'm saying, that Cantor and much of the speculation regarding the nature of infinity, and large versus small infinities etc. etc. etc. may not be the best use of the average mathematician's time. However elegant the conceptions may be, of course.

Quadibloc

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Aug 29, 2017, 2:17:01 PM8/29/17
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For calculus, one can get by with limits - one doesn't need the actual infinite,
only the idea that things can get as big as you want.

John Savard

David Petry

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Aug 29, 2017, 3:30:11 PM8/29/17
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In order for me to answer your question, you would have to explain to me how I would go about arguing with you about what "should" be.

The problem is that by accepting Cantor's ideas, mathematics becomes deficient and even crippled as a language of science and technology.

It's simply a fact that the mathematicians who do useful work pay no attention to Cantor's ideas.



Dan Christensen

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Aug 29, 2017, 3:43:50 PM8/29/17
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In real analysis, we do need the notion of sets that are finite, infinite, countable or uncountable.


Dan

Jerry Kraus

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Aug 29, 2017, 3:56:44 PM8/29/17
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Possibly. But, couldn't the concept of extremely large, to the point of not being practically countable, be substituted for "infinite" or "uncountable"? And mightn't this concept be somewhat simpler, more practical, more accurate, and less prone to theoretical confusion and periphrasis than the concept of absolute infinity?

FredJeffries

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Aug 29, 2017, 4:14:41 PM8/29/17
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On Tuesday, August 29, 2017 at 12:56:44 PM UTC-7, Jerry Kraus wrote:

> Possibly. But, couldn't the concept of extremely large, to the point of not being practically countable, be substituted for "infinite" or "uncountable"? And mightn't this concept be somewhat simpler, more practical, more accurate, and less prone to theoretical confusion and periphrasis than the concept of absolute infinity?

You've obviously never taken a class in numerical analysis

Jerry Kraus

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Aug 29, 2017, 4:17:51 PM8/29/17
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I realize the mathematical convention of infinity is employed. I'm arguing it's confusing, inaccurate, and non-essential.



FredJeffries

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Aug 29, 2017, 4:21:39 PM8/29/17
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On Tuesday, August 29, 2017 at 8:38:17 AM UTC-7, Jerry Kraus wrote:

> Well, Peter, for all practical purposes, extremely large is quite sufficient for the concept of limits, that forms the basis for Calculus. We can simply substitute "extremely large" for infinity-- as in, the limit of the function as x becomes extremely large -- and make this value precisely as large as we need it to be in context.

What, precisely, IS the difference between the "infinity" used in first-semester calculus and your "extremely large". Is it merely a new name for the same concept?

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

>
> Not really sure where you got the idea that all sets are infinite Peter. Again, for most practical purposes, most sets are quite finite, indeed.
>
> You see, Peter, absolutes, like the concept of infinity, are a kind of oversimplification of reality that may make mathematics less effective, rather than more effective.

Just the opposite, in fact. That's why differentials and integrals, etc are studied. Because they are SIMPLER that numerical approximations.

FredJeffries

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Aug 29, 2017, 4:23:39 PM8/29/17
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So you don't know what the subject of numerical analysis is about.
https://en.wikipedia.org/wiki/Numerical_analysis

> I'm arguing it's confusing, inaccurate, and non-essential.

You are wrong.

Me

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Aug 29, 2017, 4:25:45 PM8/29/17
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On Tuesday, August 29, 2017 at 10:17:51 PM UTC+2, Jerry Kraus wrote:

> I'm arguing it's confusing, inaccurate, and non-essential.

You are an idiot then.

Jerry Kraus

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Aug 29, 2017, 4:28:16 PM8/29/17
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On Tuesday, August 29, 2017 at 2:21:39 PM UTC-6, FredJeffries wrote:
> On Tuesday, August 29, 2017 at 8:38:17 AM UTC-7, Jerry Kraus wrote:
>
> > Well, Peter, for all practical purposes, extremely large is quite sufficient for the concept of limits, that forms the basis for Calculus. We can simply substitute "extremely large" for infinity-- as in, the limit of the function as x becomes extremely large -- and make this value precisely as large as we need it to be in context.
>
-----------------------------------------------------------------------
> What, precisely, IS the difference between the "infinity" used in first-semester calculus and your "extremely large". Is it merely a new name for the same concept?
-----------------------------------------------------------------------

Actually, Fred, that's an excellent question. It's a different, and a much more ordinary conception. Gee, there's a whole lot of stuff here, and, I really don't know if I can figure out exactly how much, there's so much! Actually, it's a rather humbler conception. It's a recognition of ignorance, rather than an assertion of unknowable fact.

Dan Christensen

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Aug 29, 2017, 5:02:44 PM8/29/17
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Whether extremely large or not, any finite set X has the property that, for all functions f: X --> X, if f is injective (1-to-1), then f is also surjective (onto). Infinite sets do not have this property (Dedekind).


Dan

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

Jim Burns

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Aug 29, 2017, 6:05:01 PM8/29/17
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On 8/29/2017 2:12 PM, Jerry Kraus wrote:
> On Tuesday, August 29, 2017 at 10:04:59 AM UTC-6,
> Peter Percival wrote:
>> Jerry Kraus wrote:

>>> Well, Peter, for all practical purposes, extremely
>>> large is quite
>>
>> For all practical purposes we don't need music,
>> poetry and drama either.
>
> Well, just speaking democratically of course, Peter,
> more people would see a need for music, poetry and
> drama, than for infinity. We could take a poll,
> I suppose.

Why would we take a poll? It isn't choice of _either_
infinity _or_ music, poetry and drama.

Suppose you liked sweet onion on your hamburger, but
_most people_ didn't, why would that be a reason for
_you_ to stop putting sweet onion on your hamburger?
(Mmmm. Hamburgers.) It wouldn't even be a reason if
_you_ one some _other_ day did not want sweet onion
on your burger.

In mathematics, infinity barely moves the needle on the
Bizarre-O-Meter(tm).

In physics, you'd have mobs of physicists grabbing
pitchforks and torches if you tried to take their
infinities away, especially their _continuous media_
(metaphorical pitchforks and metaphorical torches,
I hope, but I'm assuming that taking away their
infinities is metaphorical, too.)

If you don't want to use infinity, then don't.
How is someone else using infinities even an issue?
Unless you're like someone that considers it important
that no one _else_ puts sweet onion on their hamburger.
That's the only way I can understand this.

Shobe, Martin

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Aug 29, 2017, 8:14:27 PM8/29/17
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What should they be doing instead?

Martin Shobe

Shobe, Martin

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Aug 29, 2017, 8:18:20 PM8/29/17
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On 8/29/2017 2:56 PM, Jerry Kraus wrote:
> On Tuesday, August 29, 2017 at 1:43:50 PM UTC-6, Dan Christensen wrote:
>> On Tuesday, August 29, 2017 at 2:17:01 PM UTC-4, Quadibloc wrote:
>>> On Tuesday, August 29, 2017 at 9:08:36 AM UTC-6, Jerry Kraus wrote:
>>>> Wouldn't very, very large, or very, very numerous be sufficient, as conceptions,
>>>> for calculus, for example, or for set theory? Why must we resort to the
>>>> absolute concept of endlessness? Doesn't this cause more problems
>>>> conceptually, than it is worth?
>>>
>>> For calculus, one can get by with limits - one doesn't need the actual infinite,
>>> only the idea that things can get as big as you want.
>>>
>>
>> In real analysis, we do need the notion of sets that are finite, infinite, countable or uncountable.
>>
>>
>> Dan
>
> Possibly. But, couldn't the concept of extremely large, to the point of not being practically countable, be substituted for "infinite" or "uncountable"?

Rather then answer this directly, lets take a look at something
concrete. In analysis, limits are an important concept. What would the
definition of a limit be with this change?

> And mightn't this concept be somewhat simpler, more practical, more accurate, and less prone to theoretical confusion and periphrasis than the concept of absolute infinity?
>

People have tried to eliminate infinity. So far, they have been more
complex, less practical, less accurate, and more prone to theoretical
confusion. (Periphrasis doesn't appear to have anything to do with it).

Martin Shobe

John Gabriel

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Aug 29, 2017, 9:28:23 PM8/29/17
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On Tuesday, 29 August 2017 10:08:36 UTC-5, Jerry Kraus wrote:
> Wouldn't very, very large, or very, very numerous be sufficient, as conceptions, for calculus, for example, or for set theory? Why must we resort to the absolute concept of endlessness? Doesn't this cause more problems conceptually, than it is worth?

If you are dealing with limits, then what you describe is exactly what is used. For example, to say 0 = Lim_{x \to \infty} \frac{1}{x} means:

The value that the fraction 1/x can never attain as x becomes very large, is zero.

But limits are not required in calculus. In fact, the reason why calculus works has nothing to do with limits. The New Calculus is evidence of this.

To give you a simple example, consider the concept of area. Before I came along, it was understood to be the product of the sides of a given rectangle. Later the term quadrature was used even with respect to irregular areas, but it was misunderstood as a limit of an infinite sum which is nonsense.

I realised that the academic understanding of area did not apply to all areas in general. So in order to have one definition of area, I discovered that an area is well defined as the product of two arithmetic means. Nothing about infinity here and the concept applies to all areas. Similarly, the product of three arithmetic means describes volume for all types. Cavalieri had a vague idea but he never understood my definitions in terms of arithmetic means.

The mean value theorem is about the arithmetic mean of innumerably many ordinates of a function in a given interval. It is impossible to find this mean through the standard way, that is, to add up all the ordinates and divide by infinity since the sum itself is a super task and infinity is not a number. However, because the sum telescopes when replaced with derivatives, it is possible to find the arithmetic mean.

The following article describes the process using the bogus mainstream calculus and a patch (positional derivative) I provided:

https://drive.google.com/open?id=0B-mOEooW03iLZG1pNlVIX2RTR0E

Positional derivative:

https://drive.google.com/open?id=0B-mOEooW03iLVVg3QWtOdkxUbVk

As you can see, there is nothing about infinity there. In the New Calculus, this is done without limits, infinity, infinitesimals or any other ill-formed concept:

https://drive.google.com/open?id=0B-mOEooW03iLblJNLWJUeGxqV0E

Comments are unwelcome and will be ignored.

Posted on this newsgroup in the interests of public education and to eradicate ignorance and stupidity from mainstream mythmatics.

gils...@gmail.com (MIT)
huiz...@psu.edu (HARVARD)
and...@mit.edu (MIT)
david....@math.okstate.edu (David Ullrich)
djo...@clarku.edu
mar...@gmail.com

John Gabriel

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Aug 29, 2017, 9:36:17 PM8/29/17
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Watch this YT video for more details:

https://www.youtube.com/watch?v=_vvP19UGK70

Dan Christensen

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Aug 29, 2017, 10:44:59 PM8/29/17
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On Tuesday, August 29, 2017 at 9:28:23 PM UTC-4, John Gabriel wrote:

>
> But limits are not required in calculus. In fact, the reason why calculus works has nothing to do with limits. The New Calculus is evidence of this.
>

Maybe if your Wacky New Calclueless could deal with limits, you might finally be able to determine the derivative of such simple functions as y=x in your goofy system. And NO, Troll Boy, it is NOT "undefined." What a moron.


Dan

Ross A. Finlayson

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Aug 29, 2017, 11:20:32 PM8/29/17
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Huzzah, Burns: "Bizarre-o-Meter".

Or, excuse me, "Bizarre-O-Meter(tm)".

The: "... if you tried to
take their infinities away,
their _continuous media_, ...",
the implied tension and the
fanaticism for "the continuous media",
I forward your opinion.

Infinity, though, Jerry, I don't really
think "Do we really need the concept of
infinity" as much as "does the concept
of infinity really need us".

Here that's where there is infinity,
then there basically is whether there
is or not.

It's pretty well established in the
classical laws the continuum mechanics
are pretty well established reasonable
expectation of usually "continuous"
terms and how in the arithmetic the
expression (as of cases of induction)
the discrete is in terms of infinitely-
many terms. It's just not so for where
it's not.


That's of course besides that there's
all the solutions in the middle, here
all finite terms and finite combinatorics
and results in finite groups and usually
the all sorts of finite algebras as altogether
build for usual families all sorts of expressions
finite and infinite, and here for numerical methods
and of terms.

There's certainly "all" the finite, that's no
different than "infinite" to me.

"Than" or "from", "no different from infinite"
or "no difference from infinite".

Then "than" or "from" is an ad-hoc convention
for example for expressions for different and
difference in the value for example, with
approximations as not different (in terms,
always if not eventually with errors, for
for the functional and replete besides the
pragmatic).

That's an example, a convention or accommodation
for "infinitely-many" any why or why not in terms,
while in a first-order system, measurement or
along those lines is built in finite terms,
classical constructions and as above.

It's not that I care about "infinity",
it's certainly all that it is whether I care.

Makes sense to me for anybody else to feel the same way.

Then, the mathematics and finite and discrete
mathematics as a discipline (among for example
the world of established results in continuous
functions, here often and in the canon expressions
as above), as a discipline, quite certainly isn't
lacking in quite most usual applications, for
that there is yet the "extra".

Plainly it's the continuous has infinitely-many
discrete terms or what would be parts or here
some continuous "media" (or usually, "region")
in the terms, it would be a contradiction for
it not to (have infinitely-many parts), the
continuous media. Here "infinitely many parts"
is "each the same" or "the same size". (Clearly
continuous media can be put to finitely many
parts of various sizes, for example a line
on a plane.)


"Finitism" is completely reasonable (finitism
with infinite integers, or any some of them
in order that go on past ever their count).
"Retro-finitism" or figuring to negate
is for crankety trolls. Retro-finitism
is for crankety trolls and they can get bent.

"Finitism" is perfectly reasonable: in terms.

As a concept, though, "Infinity" it seems rather
the point of the integers to have that there.


Peter Percival

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Aug 30, 2017, 2:34:29 AM8/30/17
to
Jerry Kraus wrote:
>
> Possibly. But, couldn't the concept of extremely large, to the
> point of not being practically countable, be substituted for
> "infinite" or "uncountable"?

How? In the *definition* of "the limit of f(x) as x tends to infinity"
infinitely large numbers do not feature. On the other hand

the limit of f(x) as x tends to infinity

does not mean the same as

the limit of f(x) as x tends to C

where C is some large but finite number. Choose any C you like and a
function f can be easily found that exemplifies the difference.

> And mightn't this concept be somewhat
> simpler, more practical, more accurate, and less prone to theoretical
> confusion and periphrasis than the concept of absolute infinity?

How is *just plain wrong* simpler, more practical, more accurate, and
less prone to theoretical confusion and periphrasis?

John Gabriel

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Aug 30, 2017, 8:36:36 AM8/30/17
to
On Wednesday, 30 August 2017 01:34:29 UTC-5, Peter Percival wrote:
> Jerry Kraus wrote:
> >
> > Possibly. But, couldn't the concept of extremely large, to the
> > point of not being practically countable, be substituted for
> > "infinite" or "uncountable"?
>
> How? In the *definition* of "the limit of f(x) as x tends to infinity"
> infinitely large numbers do not feature. On the other hand
>
> the limit of f(x) as x tends to infinity
>
> does not mean the same as
>
> the limit of f(x) as x tends to C

You err because if C were "infinity", then there is no limit. So you are writing irrelevant nonsense.

Dan Christensen

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Aug 30, 2017, 10:33:45 AM8/30/17
to
On Wednesday, August 30, 2017 at 2:34:29 AM UTC-4, Peter Percival wrote:
> Jerry Kraus wrote:
> >
> > Possibly. But, couldn't the concept of extremely large, to the
> > point of not being practically countable, be substituted for
> > "infinite" or "uncountable"?
>
> How? In the *definition* of "the limit of f(x) as x tends to infinity"
> infinitely large numbers do not feature. On the other hand
>
> the limit of f(x) as x tends to infinity
>
> does not mean the same as
>
> the limit of f(x) as x tends to C
>

"Tending to infinity" can be formalized in terms of the reals without a symbol for infinity, e.g.

lim(x --> oo):f(x)=L <=> For all epsilon > 0, there exists y such that...

The infinity symbol in the standard limit notation is just a convenient shorthand. It is essential, however, to distinguish finite, infinite, countable and uncountable sets.


Dan

David Petry

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Aug 30, 2017, 10:57:08 AM8/30/17
to
On Tuesday, August 29, 2017 at 12:56:44 PM UTC-7, Jerry Kraus wrote:
> On Tuesday, August 29, 2017 at 1:43:50 PM UTC-6, Dan Christensen wrote:

> > In real analysis, we do need the notion of sets that are finite, infinite, countable or uncountable.

Yes, of course. That's why Gauss and Poincare and so many others were complete morons when it came to real analysis; they didn't have the sophisticated understanding of infinity that Dan has.


> Possibly. But, couldn't the concept of extremely large, to the point of not being practically countable, be substituted for "infinite" or "uncountable"?


Many truly outstanding mathematicians have agreed with you:

 "Actual infinity does not exist. What we call infinite is only the endless possibility of creating new objects no matter how many exist already". -- Poincare

'Infinity is nothing more than a figure of speech which helps us talk about limits. The notion of a completed infinity doesn't belong in mathematics' -- Gauss (paraphrased)

Simon Roberts

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Aug 30, 2017, 11:04:11 AM8/30/17
to
On Wednesday, August 30, 2017 at 10:33:45 AM UTC-4, Dan Christensen wrote:
> On Wednesday, August 30, 2017 at 2:34:29 AM UTC-4, Peter Percival wrote:
> > Jerry Kraus wrote:
> > >
> > > Possibly. But, couldn't the concept of extremely large, to the
> > > point of not being practically countable, be substituted for
> > > "infinite" or "uncountable"?
> >
> > How? In the *definition* of "the limit of f(x) as x tends to infinity"
> > infinitely large numbers do not feature. On the other hand
> >
> > the limit of f(x) as x tends to infinity
> >
> > does not mean the same as
> >
> > the limit of f(x) as x tends to C
> >
>
> "Tending to infinity" can be formalized in terms of the reals without a symbol for infinity, e.g.
>
> lim(x --> oo):f(x)=L <=> For all epsilon > 0, there exists y such that...

The definition for L includes L. Bothersome. This is [even] not done in "Pocket Oxford Dictionary."

I know, a big part of completing some math(s) is done with an educated guess and then checking if the guess is correct. If so, for example using the definition of the limit then the definition has been then verified...?

We know, always, verification, is seldom proof.

>
> The infinity symbol in the standard limit notation is just a convenient shorthand. It is essential, however, to distinguish finite, infinite, countable and uncountable sets.

Dan, what is oo a symbol for? A symbol represents something. does it not?

BTW, I have not always a firm, unwavering stance. Only in many events, cases or instances a mere cynic.

Peter Percival

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Aug 30, 2017, 11:32:30 AM8/30/17
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Gauss was talking about such things as "x tending to infinity" (where
there is no "infinity" and the definition shows as much). He was not
talking about infinite sets.

Peter Percival

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Aug 30, 2017, 11:37:09 AM8/30/17
to
Simon Roberts wrote:

>
> Dan, what is oo a symbol for? A symbol represents something. does it
> not?

The is a fool-proof way of finding out what oo means in such expressions as

lim(x --> oo) f(x)

and that method is *look the definition up in an analysis text*. If
you'll just do that, you will find that infinity (no matter how
symbolized) does *not* occur in the definition of lim(x --> oo) f(x).

A question for you, Simon: if you want to know what oo means, why
wouldn't you look up the definition is a text book? Isn't it the most
obvious thing to do?

John Gabriel

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Aug 30, 2017, 11:48:15 AM8/30/17
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On Tuesday, 29 August 2017 12:25:52 UTC-5, David Petry wrote:
> On Tuesday, August 29, 2017 at 8:08:36 AM UTC-7, Jerry Kraus wrote:
> > Wouldn't very, very large, or very, very numerous be sufficient, as conceptions, for calculus, for example, or for set theory? Why must we resort to the absolute concept of endlessness? Doesn't this cause more problems conceptually, than it is worth?
>
> I suspect you're just trolling.
>
> For thousands of years (at least since Aristotle), mathematicians have agreed that the notion of an actual infinity is not part of mathematics. But they viewed mathematics as being closely connected to science. But then Cantor and his cohorts came along and said that mathematics is an art form, and not a science. And they tell us that the theory of infinity is a really beautiful theory.
>
> And we must accept the claim that the theory of infinity is beautiful, because the artists who created the theory tell us it is beautiful. And, of course, mathematics can be defined as what mathematicians do, and anyone who asks the wrong questions is just a crackpot. And the story gets weirder and weirder the more you dig into it.
>
> If we agree that the purpose of mathematics is to provide a framework for reasoning about the real world, and we agree that mathematics is defined by its purpose,

In most cases, you don't get to define concepts randomly in mathematics because there is a very good chance that they won't work! Bogus mainstream mythmatics requires much more than just definitions: one has to create many rules in order to maintain theory that remains error free. But even in this light, it is evident that mythmaticians have failed dismally.

There are THREE words that should NEVER be used in STEM. These are:

i. PRINCIPLE

ii. LAW

iii. RULE

STEM is governed by logic and rational thinking. By definition, a law, principle or rule need not be reasonable and is based on belief. Better to say:

"Teach properties"

Not

"Teach principles"

The inference here might be that not all principles are necessarily valid. For example, one can question whether the Newtonian mechanics formulas could have been formulated differently, but one cannot question unchangeable formulas, that is, pi = circumference length / diameter length, Pythagoras, etc.

People make rules or decrees, but logic dictates properties.

Property: an attribute, quality, or characteristic of something.

Principle: a fundamental truth or proposition that serves as the foundation for a system of belief or behavior or for a chain of reasoning.

Law: the system of rules that a particular country or community recognizes as regulating the actions of its members and may enforce by the imposition of penalties.

Rule: a principle that operates within a particular sphere of knowledge, describing or prescribing what is possible or allowable.

Dan Christensen

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Aug 30, 2017, 12:12:54 PM8/30/17
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Fortunately, these silly notions were NOT central to the bulk of their work. Even they might have died in obscurity otherwise. Do you think there might be a lesson there for the modern internet crank?


Dan




David Petry

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Aug 30, 2017, 1:02:12 PM8/30/17
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On Wednesday, August 30, 2017 at 8:32:30 AM UTC-7, Peter Percival wrote:

> > 'Infinity is nothing more than a figure of speech which helps us talk
> > about limits. The notion of a completed infinity doesn't belong in
> > mathematics' -- Gauss (paraphrased)
>
> Gauss was talking about such things as "x tending to infinity" (where
> there is no "infinity" and the definition shows as much). He was not
> talking about infinite sets.


Mueckenheim has shown us a quote from Cantor in which he (Cantor) claims to have proven that the above quote from Gauss is wrong. Thus the quote is relevant.

FredJeffries

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Aug 30, 2017, 1:16:52 PM8/30/17
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On Wednesday, August 30, 2017 at 8:04:11 AM UTC-7, Simon Roberts wrote:
> On Wednesday, August 30, 2017 at 10:33:45 AM UTC-4, Dan Christensen wrote:

> > "Tending to infinity" can be formalized in terms of the reals without a symbol for infinity, e.g.
> >
> > lim(x --> oo):f(x)=L <=> For all epsilon > 0, there exists y such that...
>
> The definition for L includes L.

It's not a definition for "L".
It's a definition for "lim(x --> oo):f(x)"

Jim Burns

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Aug 30, 2017, 1:33:01 PM8/30/17
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On 8/30/2017 10:56 AM, David Petry wrote:
> On Tuesday, August 29, 2017 at 12:56:44 PM UTC-7,
> Jerry Kraus wrote:
>> On Tuesday, August 29, 2017 at 1:43:50 PM UTC-6,
>> Dan Christensen wrote:

>>> In real analysis, we do need the notion of sets
>>> that are finite, infinite, countable or uncountable.
>
> Yes, of course. That's why Gauss and Poincare and
> so many others were complete morons when it came to
> real analysis; they didn't have the sophisticated
> understanding of infinity that Dan has.

It's entirely possible that Dan (or I) have a more
sophisticated understanding of infinity than
Gauss or Poincare.
(I'd have to check to see if that were so, but my
point would be the same if we were talking about some
_other_ geniuses of the past. Archimedes? Aristotle?
<Grunt-Click>, the celebrated inventor of fire?)
This shouldn't be seen as any sort of criticism of
Gauss or Poincare.

If I have seen further than others, it is by standing
upon the shoulders of giants.
-- Isaac Newton

If we understand infinity better than they do, all it
may mean is that we were clever enough to get ourselves
born _after_ the debates that led to the current consensus
around infinity.

----
I stumbled on a post by John Baez, somewhere else, about
something else, that addressed the question of why
there is such consensus in mathematics, much more than
anywhere else, it seems.[1]

I think that, if you ignore the process of reaching
consensus in mathematics, you present an unrealistic
picture of what actually happens. And you do a disservice
to those geniuses of the past, such as Gauss and Poincare,
by insisting (in effect) that they would -- if alive today
-- ignore the debates that led to _today's_ consensus, and
would carry on making the same claims today that you quote
them making in the past.

<Baez>

John Baez 7 October 2011 at 04:01

I think Valeria hit the nail on the head: mathematicians
crave consensus.

If any sort of argument is of the sort that it only
convinces 50% of mathematicians, we'll either say it's
"not mathematics", or discuss, polish and/or demolish
the argument until convinces either 99% of mathematicians
or just 1%. (Example: Cantor's proofs.)

If someone doesn't play the game according to the usual
rules, we'll make up a new game and say they're playing
that game instead, thus eliminating potential controversy.
(Example: intuitionistic mathematics.)

Finally, we reward people who quickly admit their errors,
instead of fighting on endlessly. We say they're smart,
not wimps. (Example: Edward Nelson.) People who fight on
endlessly are labelled crackpots and excluded from the
community. (Examples: too numerous to list here.)

</Baez>
<http://m-phi.blogspot.com/2011/10/inconsistency-of-pa-and-consensus-in.html?showComment=1317956471992#c5147900263397954505>

Note:
"People who fight on endlessly are labelled crackpots
and excluded from the community."

Me

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Aug 30, 2017, 3:01:41 PM8/30/17
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On Wednesday, August 30, 2017 at 5:04:11 PM UTC+2, Simon Roberts wrote:

> Dan, what is "oo" a symbol for? A symbol [denotes] something. Does it not?

No, not in this context. Here "oo" does not denote a mathematical object. It's just shorthand for some slightly more involved state of affairs. See: https://en.wikipedia.org/wiki/Limit_of_a_function#Limits_at_infinity

On the other hand, it *is* possible to use it as a symbol for a certain mathematical object: https://en.wikipedia.org/wiki/Extended_real_number_line#Limits

Peter Percival

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Aug 30, 2017, 3:53:25 PM8/30/17
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David Petry wrote:

> If we agree that the purpose of mathematics is to provide a framework

This "we" being who?

> for reasoning about the real world, and we agree that mathematics is
> defined by its purpose, and not merely by its form, then we are
> forced to agree that the notion of an actual infinity is not needed
> in mathematics. This has been argued ad nauseam in this newsgroup.
> It seems there's no point in continuing the discussion. Not here,
> anyways.
>


Me

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Aug 30, 2017, 4:01:51 PM8/30/17
to
On Wednesday, August 30, 2017 at 4:57:08 PM UTC+2, David Petry wrote:

> Gauss and Poincare [...] they didn't have the sophisticated understanding
> of infinity [we have today].

Right. You know THERE IS progress in mathematics (and logic). :-)

Me

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Aug 30, 2017, 4:10:33 PM8/30/17
to
On Tuesday, August 29, 2017 at 7:25:52 PM UTC+2, David Petry wrote:

> For thousands of years (at least since Aristotle), mathematicians have
> agreed that the notion of an actual infinity is not part of mathematics.

Indeed. But this period ended after "set theory" was developed by Cantor, Dedekind and Frege (after some debate, of course).

"Cantor's work was well received by some of the prominent mathematicians of his day, such as Richard Dedekind [or Gottlob Frege --Me]. But his willingness to regard infinite sets as objects to be treated in much the same way as finite sets was bitterly attacked by others, particularly Kronecker. There was no objection to a 'potential infinity' in the form of an unending process, but an 'actual infinity' in the form of a completed infinite set was harder to accept."

Source: H.B. Enderton: Elements of Set Theory.

You know, there *is* progress in mathematics (and logic).

John Gabriel

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Aug 30, 2017, 4:31:24 PM8/30/17
to
On Wednesday, 30 August 2017 10:32:30 UTC-5, Peter Percival wrote:
> David Petry wrote:
> > On Tuesday, August 29, 2017 at 12:56:44 PM UTC-7, Jerry Kraus wrote:
> >> On Tuesday, August 29, 2017 at 1:43:50 PM UTC-6, Dan Christensen
> >> wrote:
> >
> >>> In real analysis, we do need the notion of sets that are finite,
> >>> infinite, countable or uncountable.
> >
> > Yes, of course. That's why Gauss and Poincare and so many others
> > were complete morons when it came to real analysis; they didn't have
> > the sophisticated understanding of infinity that Dan has.
> >
> >
> >> Possibly. But, couldn't the concept of extremely large, to the
> >> point of not being practically countable, be substituted for
> >> "infinite" or "uncountable"?
> >
> >
> > Many truly outstanding mathematicians have agreed with you:
> >
> > "Actual infinity does not exist. What we call infinite is only the
> > endless possibility of creating new objects no matter how many exist
> > already". -- Poincare
> >
> > 'Infinity is nothing more than a figure of speech which helps us talk
> > about limits. The notion of a completed infinity doesn't belong in
> > mathematics' -- Gauss (paraphrased)
>
> Gauss was talking about such things as "x tending to infinity" (where
> there is no "infinity" and the definition shows as much). He was not
> talking about infinite sets.

Wrong. Gauss was addressing the concept of "infinity" in general, which includes infinite sets.

"x tending to infinity" is no different from "the set has no last member". In fact, the very process described by "x tending to infinity" regards an "infinite set".

But how could you know? Your sense of judgment is so perverted and extraordinary to the point of ridicule. Chuckle.

John Gabriel

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Aug 30, 2017, 4:36:00 PM8/30/17
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On Wednesday, 30 August 2017 15:10:33 UTC-5, Me wrote:
> On Tuesday, August 29, 2017 at 7:25:52 PM UTC+2, David Petry wrote:
>
> > For thousands of years (at least since Aristotle), mathematicians have
> > agreed that the notion of an actual infinity is not part of mathematics.
>
> Indeed. But this period ended after "set theory" was developed by Cantor, Dedekind and Frege (after some debate, of course).

It never ended. All that happened, was that a group of orangutans to which you belong, decided to decree the nonsense to be established knowledge. Mueckenheim has routed the lot of you so badly, you don't know your rear end from your face. All you can do is hurl insults.

You can call Cantor's rot "knowledge", but it doesn't change the fact that it remains rot. Chuckle. There was a very good reason Cantor ended up in a mental asylum. He was a delusional non-mathematician like most of you. If you had any aptitude in math, you would have dismissed Cantor's delusions a long time ago.

7777777

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Aug 30, 2017, 4:54:58 PM8/30/17
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I am still waiting for your answer to my question:
what you have is an endless set of all the natural numbers {1,2,3,...}. So is this an infinite set?

Simon Roberts

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Aug 30, 2017, 4:56:32 PM8/30/17
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On Wednesday, August 30, 2017 at 3:01:41 PM UTC-4, Me wrote:
> On Wednesday, August 30, 2017 at 5:04:11 PM UTC+2, Simon Roberts wrote:
>
> > Dan, what is "oo" a symbol for? A symbol [denotes] something. Does it not?
>
> No, not in this context. Here "oo" does not denote a mathematical object. It's just shorthand for some slightly more involved state of affairs. See: https://en.wikipedia.org/wiki/Limit_of_a_function#Limits_at_infinity

yes very difficult, you're correct. As quoted:

the limit of f as x approaches infinity is L. that's a wrap.
>
> On the other hand, it *is* possible to use it as a symbol for a certain mathematical object: https://en.wikipedia.org/wiki/Extended_real_number_line#Limits

further, and farther and so on, et cetera, etc...stated:

To make things completely formal, the Cauchy sequences definition of {\displaystyle \mathbb {R} } \mathbb {R} allows us to define {\displaystyle +\infty } +\infty as the set of all sequences of rationals which, for any {\displaystyle K>0} K>0, from some point on exceed {\displaystyle K} K. We can define {\displaystyle -\infty } -\infty similarly.

ok, he said it. "we can." good enough for me.


Me

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Aug 30, 2017, 5:01:16 PM8/30/17
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On Wednesday, August 30, 2017 at 10:36:00 PM UTC+2, John Gabriel wrote:
> On Wednesday, 30 August 2017 15:10:33 UTC-5, Me wrote:
> > On Tuesday, August 29, 2017 at 7:25:52 PM UTC+2, David Petry wrote:
> > >
> > > For thousands of years (at least since Aristotle), mathematicians have
> > > agreed that the notion of an actual infinity is not part of mathematics.
> > >
> > Indeed. But this period ended after "set theory" was developed by Cantor,
> > Dedekind and Frege (after some debate, of course).
> >
> It never ended. All that happened, was that a group of orangutans to which
> you belong, decided to decree the nonsense to be established knowledge.
> [...] You can call Cantor's rot "knowledge", but it doesn't change the fact
> that it remains rot.

If you say so. An alternative view:

"So wurde schließlich durch die gigantische Zusammenarbeit von Frege, Dedekind, Cantor das Unendliche auf den Thron gehoben [...]."

["Finally, by the gigantic collaboration between Frege, Dedekind, Cantor the Infinite was lifted to the throne"]

(David Hilbert, Das Unendliche, 1925)

Bolzano; Cantor, Dedekind, Frege; Peano, Russell & Whitehead, Zermelo, ..., Hilbert: a group of orangutans, all of them!
Message has been deleted
Message has been deleted

Me

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Aug 30, 2017, 5:42:41 PM8/30/17
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On Wednesday, August 30, 2017 at 10:54:58 PM UTC+2, 7777777 wrote:

> what you have is an endless set of all the natural numbers {1,2,3,...}.

In set theory, we do not call this set "endless" but /infinite/. Indeed, the set of all natural numbers (defined in the usual way) is infinite (in the context of set theory, of course).

What do we mean by this? There are several DEFINITIONS for the notion of an /infinite/ set (in the context of set theory).

I just stumbled over a very appealing one by Tarski (1924):

A set is A /finite/ iff every non-empty set B of subsets of A contains an element which is minimal in respect to (strict) inclusion, i.e. an element C e B such that for no C' e B: C' c C.

A non-finite set is said to be /infinite/.
____________________

Now, we may derive from these definitions:

A set A is infinite iff there is a non-empty set B of subsets of A, such that no element in B is minimal in respect to (strict) inclusion (i.e. for every element C e B there is an C' e B such that C' c C).

Again:

A set A is infinite iff there is a non-empty set B of subsets of A, such that for every C e B there is an C' e B such that C' c C.

Let's consider the set A = {1, 2, 3, ...}, for example. Then B = {{1, 2, 3, ...}, {2, 3, 4,...}, {3, 4, 5, ...}, ...} would be a non empty set of subsets of A. Now for each and every set C in B there is a set C' in B such that C' c C: if C = {n, n+1, n+2, ...} in B, then C' = {n+1, n+2, n+3, ...} is in B too, and C' c C.

John Gabriel

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Aug 30, 2017, 6:23:52 PM8/30/17
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On Wednesday, 30 August 2017 16:01:16 UTC-5, Me wrote:
> On Wednesday, August 30, 2017 at 10:36:00 PM UTC+2, John Gabriel wrote:
> > On Wednesday, 30 August 2017 15:10:33 UTC-5, Me wrote:
> > > On Tuesday, August 29, 2017 at 7:25:52 PM UTC+2, David Petry wrote:
> > > >
> > > > For thousands of years (at least since Aristotle), mathematicians have
> > > > agreed that the notion of an actual infinity is not part of mathematics.
> > > >
> > > Indeed. But this period ended after "set theory" was developed by Cantor,
> > > Dedekind and Frege (after some debate, of course).
> > >
> > It never ended. All that happened, was that a group of orangutans to which
> > you belong, decided to decree the nonsense to be established knowledge.
> > [...] You can call Cantor's rot "knowledge", but it doesn't change the fact
> > that it remains rot.
>
> If you say so. An alternative view:
>
> "So wurde schließlich durch die gigantische Zusammenarbeit von Frege, Dedekind, Cantor das Unendliche auf den Thron gehoben [...]."

Zu welchem thron? Der imaginäre?

>
> ["Finally, by the gigantic collaboration between Frege, Dedekind, Cantor the Infinite was lifted to the throne"]
>
> (David Hilbert, Das Unendliche, 1925)
>
> Bolzano; Cantor, Dedekind, Frege; Peano, Russell & Whitehead, Zermelo, ..., Hilbert: a group of orangutans, all of them!

Ja. Sie sind die Orang-Utans, auf die ich mich beziehe. Kichern!

John Gabriel

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Aug 30, 2017, 6:27:37 PM8/30/17
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{1,2,3,...} is a pathological concept. Anything with an ellipsis in it is ill-formed nonsense.

Me

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Aug 30, 2017, 6:56:09 PM8/30/17
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On Thursday, August 31, 2017 at 12:23:52 AM UTC+2, John Gabriel wrote:

> > Bolzano; Cantor, Dedekind, Frege; Peano, Russell & Whitehead, Zermelo, ...,
> > Hilbert: a group of orangutans, all of them!
> >
> Ja. Sie sind die Orang-Utans, auf die ich mich beziehe. Kicher!

Good to know! :-)

Message has been deleted

Me

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Aug 30, 2017, 7:13:02 PM8/30/17
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On Thursday, August 31, 2017 at 12:27:37 AM UTC+2, John Gabriel wrote:

> {1,2,3,...} is a pathological concept. Anything with an ellipsis in it is
> ill-formed nonsense.

If you say so. We do not need this notion ("...") in set theory. It's just a convinient way to express certain things.

Instead "{0,1,2,...}" you might as well write "w" (omega) in the context of set theory. /w/ can be defined without referring to "..." etc.

Hint: Since it can be proved that the Peano Axioms hold for w with 0 := {} and s(x) := x U {x}, as well as 0 e w, 1 e w, 2 e w, 3 e w (where 1 := s(0), 2 := s(1), 3 := s(2)), and especially that for any x e w: s(x) e w, "{0,1,2,...}" certainly might NOT be the "ill-formed nonsense" you consider it to be.

Me

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Aug 30, 2017, 7:22:36 PM8/30/17
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On Thursday, August 31, 2017 at 1:13:02 AM UTC+2, Me wrote:

> We do not need this notion ("...") in set theory. It's just a convinient way
> to express certain things.
>
> Instead "{0,1,2,3,...}" you might as well write "w" (omega) in the context of
> set theory. /w/ can be defined without referring to "..." etc.
>
> Hint: Since it can be proved that the Peano Axioms hold for w with 0 := {}
> and s(x) := x U {x}, as well as 0 e w, 1 e w, 2 e w, 3 e w (where 1 := s(0),
> 2 := s(1), 3 := s(2)), and especially that for any x e w: s(x) e w,
> "{0,1,2,...}" certainly might NOT be the "ill-formed nonsense" you consider
> it to be.

Hint:

{1,2,3,...} := w \ {0}

is a proper definition.

Then we can prove 0 !e {1,2,3,...}, 1 e {1,2,3,...}, 2 e {1,2,3,...}, 3 e {1,2,3...} as well as

for all x e {1,2,3,...}: s(x) e {1,2,3,...} .

Since we usually -later- define "+" in a way such that

n+1 = s(n) (for any n e w)

this means: 0 !e {1,2,3,...}, 1 e {1,2,3,...}, 2 e {1,2,3,...}, 3 e {1,2,3,...} as well as

for all n e {1,2,3,...}: n+1 e {1,2,3,...} .

Hence the notion "{1,2,3,...}" seems rather reasonable TO ME (in this case).

Shobe, Martin

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Aug 30, 2017, 8:10:55 PM8/30/17
to
On 8/30/2017 9:56 AM, David Petry wrote:
> On Tuesday, August 29, 2017 at 12:56:44 PM UTC-7, Jerry Kraus wrote:
>> On Tuesday, August 29, 2017 at 1:43:50 PM UTC-6, Dan Christensen wrote:
>
>>> In real analysis, we do need the notion of sets that are finite, infinite, countable or uncountable.
>
> Yes, of course. That's why Gauss and Poincare and so many others were complete morons when it came to real analysis; they didn't have the sophisticated understanding of infinity that Dan has.
>
>
>> Possibly. But, couldn't the concept of extremely large, to the point of not being practically countable, be substituted for "infinite" or "uncountable"?
>
>
> Many truly outstanding mathematicians have agreed with you:
>
> "Actual infinity does not exist. What we call infinite is only the endless possibility of creating new objects no matter how many exist already".. -- Poincare
>
> 'Infinity is nothing more than a figure of speech which helps us talk about limits. The notion of a completed infinity doesn't belong in mathematics' -- Gauss (paraphrased)
>

I've been unable to locate this one. Do you have a source?

Martin Shobe

Me

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Aug 30, 2017, 9:23:32 PM8/30/17
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"I protest firstly against the use of an infinite magnitude as a completed one, which never has been allowed in mathematics. The infinite is only a mode of speaking, when we in principle talk about limits which are approached by certain ratios as closely as desired whereas others are allowed to grow without reservation."

[C.F. Gauß, letter to H.C. Schumacher (12 July 1831)]

(Translation Mückenheim?)


David Petry

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Aug 30, 2017, 9:43:54 PM8/30/17
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On Wednesday, August 30, 2017 at 10:33:01 AM UTC-7, Jim Burns wrote:

> I think that, if you ignore the process of reaching
> consensus in mathematics, you present an unrealistic
> picture of what actually happens.

> "mathematicians crave consensus" -- John Baez


In other words, the mathematicians desperately want to stay in the Matrix; the red pill is not for them.

David Petry

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Aug 30, 2017, 10:01:56 PM8/30/17
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On Wednesday, August 30, 2017 at 1:10:33 PM UTC-7, Me wrote:
> On Tuesday, August 29, 2017 at 7:25:52 PM UTC+2, David Petry wrote:
>
> > For thousands of years (at least since Aristotle), mathematicians have
> > agreed that the notion of an actual infinity is not part of mathematics.
>
> Indeed. But this period ended after "set theory" was developed by Cantor, Dedekind and Frege (after some debate, of course).
[...]
> You know, there *is* progress in mathematics (and logic).

Here's another way of looking at things:

As intelligent, self-aware beings, we can reason about how we think about and understand reality. And then it is eminently reasonable to claim that mathematics has to be consistent with our understanding of how we reason about reality. That is, we have the ability to reason about the distinction between reality and make believe, and mathematics should provide us with a way to formalize that reasoning. But, the reasoning underlying Cantor's ideas about infinite sets is incompatible with our understanding of how we distinguish between reality and make believe, and thus it is eminently reasonable to believe that Cantor's ideas should be, and inevitably will be, expunged from mathematics.

If you don't want to discuss this topic, that is, of course, fine, but I don't know why you (and so many others) feel you have to make silly noises to prevent such a discussion from taking place.

John Gabriel

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Aug 30, 2017, 10:04:30 PM8/30/17
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It is ill-formed nonsense as I consider to be. Peano was a juvenile idiot. Nothing he did was even worth mentioning. His so-called "axioms" are a joke.

Bijective cardinality does not replace "..." and gives no support to the concept of infinity.

https://www.youtube.com/watch?v=C_AIIxG8AFI

David Petry

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Aug 30, 2017, 10:05:35 PM8/30/17
to
On Wednesday, August 30, 2017 at 5:10:55 PM UTC-7, Shobe, Martin wrote:

> > Many truly outstanding mathematicians have agreed with you:
> >
> > "Actual infinity does not exist. What we call infinite is only the endless possibility of creating new objects no matter how many exist already".. -- Poincare
> >
> > 'Infinity is nothing more than a figure of speech which helps us talk about limits. The notion of a completed infinity doesn't belong in mathematics' -- Gauss (paraphrased)
> >
>
> I've been unable to locate this one. Do you have a source?
>
> Martin Shobe


"Me" gave you the source for Gauss' quote. I took the Poincare quote from Morris Kline's book "Mathematics: the loss of certainty". I don't know the original source.

John Gabriel

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Aug 30, 2017, 10:05:48 PM8/30/17
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A bigger idiot than John Baez does not exist today. Well, maybe Stephen Hawking?

John Gabriel

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Aug 30, 2017, 10:08:21 PM8/30/17
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Original quote is:

"I protest against the use of an infinite quantity as something completed, which is never permissible in mathematics."

http://ac.els-cdn.com/0315086079900296/1-s2.0-0315086079900296-main.pdf?_tid=103ea0bc-8df1-11e7-99ae-00000aacb361&acdnat=1504145395_119da349937d82296f8bf5cc8cf28441

John Gabriel

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Aug 30, 2017, 10:11:04 PM8/30/17
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Note especially how Gauss thought a little like me:

. . . 3 is not as close to the true value of pi as
is 3.14, and 3.14159 is still closer. By adding additional
places to the right of the decimal, it is
possible to approximate the true value of A as closely
as one likes. But Gauss insisted that one could not
assume all the terms of the decimal expansion to be
given to determine pi exactly. To do so would involve
an infinite number of terms, and thus comprise an
actually infinite set of numbers, which Gauss refused
to a flow in rigorous mathematics [Dauben 1977, 861.]

I have been telling you all these things but you are brainwashed. Mueckenheim has been telling you these things also but you are hard of hearing.

John Gabriel

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Aug 30, 2017, 10:13:20 PM8/30/17
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To make it simple for you: 3.14159... is a junk concept because there is no such thing as an infinite set.

0.333... is even more absurd because if infinity were possible, then a contradiction of an important number theorem arises.

Jim Burns

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Aug 30, 2017, 10:14:44 PM8/30/17
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David, The Matrix is a movie.
"This is the Matrix and you need a red pill to see that"
is not an argument.

If you have an argument, make it. But this is not
new territory. I don't expect you to find an argument
that hasn't already been hashed out well before you or
I were born.

Your "red pill" argument indicates to me that you think
you don't have a fresh argument, either, just a promise
of enlightenment if we sign over our judgment to you
(ie, take the red pill).

Here's part of what you decided to snip:
<Baez>
If any sort of argument is of the sort that it only
convinces 50% of mathematicians, we'll either say it's
"not mathematics", or discuss, polish and/or demolish
the argument until convinces either 99% of mathematicians
or just 1%. (Example: Cantor's proofs.)
</Baez>

A bit of (highly hypothetical) history:
_Someone_ already had these arguments about infinity.
The argument for potential infinity got demolished.
A consensus formed.

Now, you want us to "red-pill" ourselves, by which
you mean, disregard the arguments that led us to
today's consensus. And _why_ should we disregard
those arguments? Because "The Matrix".

John Gabriel

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Aug 30, 2017, 10:17:44 PM8/30/17
to
On Wednesday, 30 August 2017 21:14:44 UTC-5, Jim Burns wrote:
> On 8/30/2017 9:43 PM, David Petry wrote:
> > On Wednesday, August 30, 2017 at 10:33:01 AM UTC-7,
> > Jim Burns wrote:
>
> >> I think that, if you ignore the process of reaching
> >> consensus in mathematics, you present an unrealistic
> >> picture of what actually happens.
> >
> >> "mathematicians crave consensus" -- John Baez
> >
> > In other words, the mathematicians desperately want
> > to stay in the Matrix; the red pill is not for them.
>
> David, The Matrix is a movie.
> "This is the Matrix and you need a red pill to see that"
> is not an argument.
>
> If you have an argument, make it. But this is not
> new territory. I don't expect you to find an argument
> that hasn't already been hashed out well before you or
> I were born.

Chuckle. That right there is proof that you worship the doctrines of the matrix. Case closed.

Dan Christensen

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Aug 30, 2017, 10:46:08 PM8/30/17
to
On Wednesday, August 30, 2017 at 10:04:30 PM UTC-4, John Gabriel wrote:

>
> It is ill-formed nonsense as I consider to be. Peano was a juvenile idiot. Nothing he did was even worth mentioning.


I think someone here is green with envy here, someone who cannot even prove that 2+2=4 in his goofy system of mathematics. It is a trivial exercise from Peano's original axioms from over a century ago. I guess that makes YOU the juvenile idiot, right, Troll Boy?


Dan

Me

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Aug 30, 2017, 10:56:21 PM8/30/17
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On Thursday, August 31, 2017 at 4:04:30 AM UTC+2, John Gabriel wrote:

> Peano was a juvenile idiot. Nothing he did was even worth mentioning.
> His so-called "axioms" are a joke.

Alternative facts?

The truth is:
https://en.wikipedia.org/wiki/Peano_axioms
https://en.wikipedia.org/wiki/Giuseppe_Peano

Me

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Aug 30, 2017, 11:00:04 PM8/30/17
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On Thursday, August 31, 2017 at 4:05:48 AM UTC+2, John Gabriel wrote:

> A bigger idiot than John Baez does not exist today.

John Gabriel?

Dan Christensen

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Aug 30, 2017, 11:00:23 PM8/30/17
to
On Wednesday, August 30, 2017 at 11:04:11 AM UTC-4, Simon Roberts wrote:
> On Wednesday, August 30, 2017 at 10:33:45 AM UTC-4, Dan Christensen wrote:
> > On Wednesday, August 30, 2017 at 2:34:29 AM UTC-4, Peter Percival wrote:
> > > Jerry Kraus wrote:
> > > >
> > > > Possibly. But, couldn't the concept of extremely large, to the
> > > > point of not being practically countable, be substituted for
> > > > "infinite" or "uncountable"?
> > >
> > > How? In the *definition* of "the limit of f(x) as x tends to infinity"
> > > infinitely large numbers do not feature. On the other hand
> > >
> > > the limit of f(x) as x tends to infinity
> > >
> > > does not mean the same as
> > >
> > > the limit of f(x) as x tends to C
> > >
> >
> > "Tending to infinity" can be formalized in terms of the reals without a symbol for infinity, e.g.
> >
> > lim(x --> oo):f(x)=L <=> For all epsilon > 0, there exists y such that...
>
> The definition for L includes L.

This is NOT the definition for L. It defines this particular notation. If it presents too many strange symbols for you, what with the "-->" and "oo" symbols, think of it as a binary predicate:

LimInf(f,L) <=> ....


Dan
Message has been deleted

Dan Christensen

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Aug 30, 2017, 11:04:36 PM8/30/17
to
How about that delusional idiot who calls himself "the greatest mathematician ever," but after all these years, he STILL cannot even prove that 2+2=4 in his goofy system. Anyone you know, Troll Boy??? (Hee, hee!)


Dan

Me

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Aug 30, 2017, 11:05:15 PM8/30/17
to
On Thursday, August 31, 2017 at 4:46:08 AM UTC+2, Dan Christensen wrote:

> I think someone here is green with envy here ...

Or rather just completely nuts, I guess.

Dan Christensen

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Aug 30, 2017, 11:22:53 PM8/30/17
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We now know that Gauss was occasionally a bit hasty in his judgement, but most of us can only hope to be wrong as often as he was. Unlike you, he did not pursue these silly notions and make them the central focus of his career. Had he done so, he would most likely have died in obscurity. Do you think there might be a lesson here for you, Troll Boy?


Dan

Simon Roberts

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Aug 30, 2017, 11:23:48 PM8/30/17
to
On Wednesday, August 30, 2017 at 10:11:04 PM UTC-4, John Gabriel wrote:
> On Wednesday, 30 August 2017 21:08:21 UTC-5, John Gabriel wrote:
> > On Wednesday, 30 August 2017 21:05:35 UTC-5, David Petry wrote:
> > > On Wednesday, August 30, 2017 at 5:10:55 PM UTC-7, Shobe, Martin wrote:
> > >
> > > > > Many truly outstanding mathematicians have agreed with you:
> > > > >
> > > > > "Actual infinity does not exist. What we call infinite is only the endless possibility of creating new objects no matter how many exist already".. -- Poincare
> > > > >
> > > > > 'Infinity is nothing more than a figure of speech which helps us talk about limits. The notion of a completed infinity doesn't belong in mathematics' -- Gauss (paraphrased)
> > > > >
> > > >
> > > > I've been unable to locate this one. Do you have a source?
> > > >
> > > > Martin Shobe
> > >
> > >
> > > "Me" gave you the source for Gauss' quote. I took the Poincare quote from Morris Kline's book "Mathematics: the loss of certainty". I don't know the original source.
> >
> > Original quote is:
> >
> > "I protest against the use of an infinite quantity as something completed, which is never permissible in mathematics."
> >
> > http://ac.els-cdn.com/0315086079900296/1-s2.0-0315086079900296-main.pdf?_tid=103ea0bc-8df1-11e7-99ae-00000aacb361&acdnat=1504145395_119da349937d82296f8bf5cc8cf28441
>
> Note especially how Gauss thought a little like me:

whow, dude! what?

zelos...@outlook.com

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Aug 31, 2017, 2:03:25 AM8/31/17
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On Tuesday, August 29, 2017 at 5:08:36 PM UTC+2, Jerry Kraus wrote:
> Wouldn't very, very large, or very, very numerous be sufficient, as conceptions, for calculus, for example, or for set theory? Why must we resort to the absolute concept of endlessness? Doesn't this cause more problems conceptually, than it is worth?

Having it "really really really" big does not suffice it in most cases. For example in calculus and integration, any finite amount will result in the integral being 0, even countably infinite is not "big enough" to generate any integral. To have non-zero it needs to be uncountably large.

So infinity is invaluable. To mathematicians it is not an issue, it is only to cranks that cling to their intuition. As long as you follow things logically it doesn't give much problem, counterintuitive it does but not many other problems compared to all the problems that arise if infinity is not used.

zelos...@outlook.com

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Aug 31, 2017, 2:04:33 AM8/31/17
to
Mr Crank! Nice to see you again!

What number theorem would that be? Is it yet another one of your thing where you do not understand the definition or theorem and then argue against a strawman?

Shobe, Martin

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Aug 31, 2017, 8:54:15 AM8/31/17
to
There's the first half. However, the second half doesn't appear in it
(even as a paraphrase).

Martin Shobe

Shobe, Martin

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Aug 31, 2017, 8:54:50 AM8/31/17
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Then the paraphrase isn't accurate.

Martin Shobe

Jerry Kraus

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Aug 31, 2017, 9:33:56 AM8/31/17
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On Wednesday, August 30, 2017 at 7:23:32 PM UTC-6, Me wrote:
> On Thursday, August 31, 2017 at 2:10:55 AM UTC+2, Shobe, Martin wrote:
> > On 8/30/2017 9:56 AM, David Petry wrote:
> > >
> > > 'Infinity is nothing more than a figure of speech which helps us talk
> > > about limits. The notion of a completed infinity doesn't belong in
> > > mathematics' -- Gauss (paraphrased)
> > >
> > I've been unable to locate this one. Do you have a source?
>
-------------------------------------------------------------------------
> "I protest firstly against the use of an infinite magnitude as a completed one, which never has been allowed in mathematics. The infinite is only a mode of speaking, when we in principle talk about limits which are approached by certain ratios as closely as desired whereas others are allowed to grow without reservation."
-----------------------------------------------------------------------
>
------------------------------------------------------------------
> [C.F. Gauß, letter to H.C. Schumacher (12 July 1831)]
--------------------------------------------------------------------
>
> (Translation Mückenheim?)

I think what Gauss may be saying, effectively, is that by its very nature, the concept of infinity is incomprehensible, and represents something of a parallel to the theological concept of eternity. Now, the fact that something is incomprehensible doesn't mean that we can't discuss it, assuming we employ certain mutually agreed upon assumptions regarding it. However, such discussions are unlikely to have any practical significance, at all. Hence the famous medieval scholastic query, "How many angels can fit on the head of a pin?" I suspect Cantor's speculations are somewhat analogous to this.


Peter Percival

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Aug 31, 2017, 11:17:57 AM8/31/17
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Two different notions of infinity are being mixed up here. Gauss is
talking about the infinity in "x tends to infinity" (i.e., no infinity
at all). Cantor was talking about infinite sets, ordered and unordered.
What do you (and the other nutters) gain by taking about both as if
they were the same thing?
>
>


--
Do, as a concession to my poor wits, Lord Darlington, just explain
to me what you really mean.
I think I had better not, Duchess. Nowadays to be intelligible is
to be found out. -- Oscar Wilde, Lady Windermere's Fan

FredJeffries

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Aug 31, 2017, 11:44:16 AM8/31/17
to
On Thursday, August 31, 2017 at 6:33:56 AM UTC-7, Jerry Kraus wrote:
> On Wednesday, August 30, 2017 at 7:23:32 PM UTC-6, Me wrote:
> > On Thursday, August 31, 2017 at 2:10:55 AM UTC+2, Shobe, Martin wrote:
> > > On 8/30/2017 9:56 AM, David Petry wrote:
> > > >
> > > > 'Infinity is nothing more than a figure of speech which helps us talk
> > > > about limits. The notion of a completed infinity doesn't belong in
> > > > mathematics' -- Gauss (paraphrased)
> > > >
> > > I've been unable to locate this one. Do you have a source?
> >
> -------------------------------------------------------------------------
> > "I protest firstly against the use of an infinite magnitude as a completed one, which never has been allowed in mathematics. The infinite is only a mode of speaking, when we in principle talk about limits which are approached by certain ratios as closely as desired whereas others are allowed to grow without reservation."
> -----------------------------------------------------------------------
> >
> ------------------------------------------------------------------
> > [C.F. Gauß, letter to H.C. Schumacher (12 July 1831)]
> --------------------------------------------------------------------
> >
> > (Translation Mückenheim?)
>
> I think what Gauss may be saying, effectively, is that by its very nature, the concept of infinity is incomprehensible, and represents something of a parallel to the theological concept of eternity.

No, he isn't. We don't have to guess at what Gauss was saying. We can actually read it.

The 1831 correspondence with Schumacher
which prompted the oft-quoted passage may be found in Band 8 of his
collected works beginning at page 210

http://www.wilbourhall.org/pdfs/Carl_Friedrich_Gauss_Werke___8.pdf
and go to page 220 of 472
The passage itself appears on page 216 (226 of 472), second paragraph.

They were discussing parallel lines.

See also William C Waterhouse "Gauss on Infinity", Historia Mathematica 6 (1979), 430-436

http://ac.els-cdn.com/0315086079900296/1-s2.0-0315086079900296-main.pdf?_tid=32ca4802-8e5f-11e7-8952-00000aab0f02&acdnat=1504192698_ef87764f250eb492856a4d5c0a8c309d

> Now, the fact that something is incomprehensible doesn't mean that we can't discuss it, assuming we employ certain mutually agreed upon assumptions regarding it. However, such discussions are unlikely to have any practical significance, at all. Hence the famous medieval scholastic query, "How many angels can fit on the head of a pin?" I suspect Cantor's speculations are somewhat analogous to this.

But you don't actually know anything about his "speculations" nor the motivations for them.

He was investigating convergence of trigonometric series which were solutions to equations concerning heat flow, which in an industrial era, had great "practical significance".

FredJeffries

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Aug 31, 2017, 11:47:59 AM8/31/17
to
On Tuesday, August 29, 2017 at 1:28:16 PM UTC-7, Jerry Kraus wrote:
> On Tuesday, August 29, 2017 at 2:21:39 PM UTC-6, FredJeffries wrote:
> > On Tuesday, August 29, 2017 at 8:38:17 AM UTC-7, Jerry Kraus wrote:
> >
> > > Well, Peter, for all practical purposes, extremely large is quite sufficient for the concept of limits, that forms the basis for Calculus. We can simply substitute "extremely large" for infinity-- as in, the limit of the function as x becomes extremely large -- and make this value precisely as large as we need it to be in context.
> >
> -----------------------------------------------------------------------
> > What, precisely, IS the difference between the "infinity" used in first-semester calculus and your "extremely large". Is it merely a new name for the same concept?
> -----------------------------------------------------------------------
>
> Actually, Fred, that's an excellent question.

To which you refuse to give an answer.

> It's a different, and a much more ordinary conception. Gee, there's a whole lot of stuff here, and, I really don't know if I can figure out exactly how much, there's so much! Actually, it's a rather humbler conception. It's a recognition of ignorance, rather than an assertion of unknowable fact.

Who was it that was complaining about periphrasis?

Jerry Kraus

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Aug 31, 2017, 11:48:06 AM8/31/17
to
A valid question, Peter. After all, set theory didn't exist, at least in Western mathematics -- the Chinese had some much earlier versions -- until the late nineteenth century. So, Gauss obviously wasn't talking about set theory when he dismissed the concept of infinity. On the other hand, the concept of infinity as employed in set theory is obviously related to, and derived from, the concept of infinity used in calculus. Now, precisely how they differ, and how they are related, is rather difficult to determine, I would suggest. Precisely because, the concept of infinity itself is essentially incomprehensible, as is the concept of eternity in theology. So, problems will exist in the practical applications of any concept of infinity, whatsoever, whatever distinctions may exist between different types of infinities.

John Gabriel

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Aug 31, 2017, 12:05:15 PM8/31/17
to
On Tuesday, 29 August 2017 10:08:36 UTC-5, Jerry Kraus wrote:
> Wouldn't very, very large, or very, very numerous be sufficient, as conceptions, for calculus, for example, or for set theory? Why must we resort to the absolute concept of endlessness? Doesn't this cause more problems conceptually, than it is worth?

http://www.sciencedirect.com/science/article/pii/0315086079900296

The first paragraph which is only assertion of the author (NOT GAUSS!) states:

"His celebrated statement has no connection to the set theory to which it was later applied."

Actually it has every connection!

Gauss was addressing the concept of "infinity" in general, which includes infinite sets even if they had not been formalised at the time. Sets are indexed by natural numbers which are considered to be part of "infinite sets".

"x tending to infinity" is no different from "the set has no last member". In fact, the very process described by "x tending to infinity" regards an "infinite set".

Peter Percival

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Aug 31, 2017, 12:06:40 PM8/31/17
to
Jerry Kraus wrote:
> [...] On the other hand, the concept of infinity as employed in
> set theory is obviously related to, and derived from, the concept of
> infinity used in calculus. Now, precisely how they differ, and how
> they are related, is rather difficult to determine, I would suggest.

See Jourdain's /Introduction/ to Cantor's /Contributions to the founding
of the theory of transfinite numbers/ (Dover) for the connection between
calculus (convergence of Fourier series specifically) and Cantor's early
work on the transfinite.

> Precisely because, the concept of infinity itself is essentially
> incomprehensible,

*Which* concept of infinity among the various concepts of infinity that
mathematicians have considered are you referring to?

> as is the concept of eternity in theology. So,
> problems will exist in the practical applications of any concept of
> infinity, whatsoever, whatever distinctions may exist between
> different types of infinities.
>


John Gabriel

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Aug 31, 2017, 12:07:16 PM8/31/17
to
No. He was even though set theory hadn't been formally established. The concept applies regardless.

John Gabriel

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Aug 31, 2017, 12:11:36 PM8/31/17
to
Moreover, Gauss states clearly that:

"...one could not assume that all the terms of the decimal expansion to be given to determine pi exactly."

Since such terms are indexed by natural numbers which are thought to be part of an imaginary infinite set, Gauss was *most definitely* including infinite sets or "infinite" anything else!

John Gabriel

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Aug 31, 2017, 12:16:20 PM8/31/17
to
Waterhouse is a modern orangutan academic and on the second page he misguidedly refers to the parallel postulate but there is no such thing and zero relevance:

https://www.linkedin.com/pulse/part-5-axioms-postulates-mathematics-john-gabriel

John Gabriel

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Aug 31, 2017, 12:22:04 PM8/31/17
to
Waterhouse's article not only highlights the ubiquitous ignorance of mainstream academics, but it also highlights the intellectual dishonesty and moral bankruptcy of these academics. Writing such papers has only one agenda - to further the established mainstream ignorance at all costs.

David Petry

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Aug 31, 2017, 12:46:25 PM8/31/17
to
On Wednesday, August 30, 2017 at 7:14:44 PM UTC-7, Jim Burns wrote:

> Now, you want us to "red-pill" ourselves, by which
> you mean, disregard the arguments that led us to
> today's consensus.

My claim is that we live in a very different world from the world in which Cantor's theory arose. Computers, and the imminent rise of artificial intelligence, can impact our view of mathematics. Pure mathematicians are still living in the nineteenth century.

John Gabriel

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Aug 31, 2017, 12:56:37 PM8/31/17
to
Mythmaticians are indeed living in Cantor's Fools Paradise. There is no artificial intelligence (AI) besides humans. Advanced automation is not equal to AI. But let me not stray from the topic...

Red pill and blue pill are stupid ways to make a convincing argument. Many mainstream ideas are problematic because they are ill formed. Such ideas do not exit in the realm of noumena. You cannot convince someone his ideas are ill formed when he insists that you have to use his ridiculous rules to accomplish the same. Rules, decrees, principles and laws have no place in rational thought, only in the palaces of Kings. Beliefs do not belong to mathematics.

You have to show that their ideas are ill formed and lead to many contradictions. This has been done and hopefully will continue to be done. Perhaps we can erase the stupidity of the current generation so that not more generations are lost in the conformity of the current mainstream orangutans.

burs...@gmail.com

unread,
Aug 31, 2017, 12:57:13 PM8/31/17
to
Compared to abacus etc.. did the computer meanwhile
cross the infinity border? I don't think anything has

fundamentally changed. It doesn't matter whether you
can compute 1 mio digits of pi. Its still the same math.

burs...@gmail.com

unread,
Aug 31, 2017, 12:58:24 PM8/31/17
to
bird brains such as John Gabriel couldn't use
internet in the nineteenth century, thats true.

FredJeffries

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Aug 31, 2017, 1:32:37 PM8/31/17
to
On Thursday, August 31, 2017 at 8:48:06 AM UTC-7, Jerry Kraus wrote:

> A valid question, Peter. After all, set theory didn't exist, at least in Western mathematics -- the Chinese had some much earlier versions -- until the late nineteenth century. So, Gauss obviously wasn't talking about set theory when he dismissed the concept of infinity. On the other hand, the concept of infinity as employed in set theory is obviously related to, and derived from, the concept of infinity used in calculus. Now, precisely how they differ, and how they are related, is rather difficult to determine, I would suggest. Precisely because, the concept of infinity itself is essentially incomprehensible, as is the concept of eternity in theology. So, problems will exist in the practical applications of any concept of infinity, whatsoever, whatever distinctions may exist between different types of infinities.

On the contrary, it was the development of precise notions of limits and transfinite numbers and sets in the 19th century which gave clear separation between the mathematical notions of infinite/transfinite and the metaphysical/theological speculations and made of the former a scientific investigation.

I notice that you have made no fewer than 9 posts in this thread and have yet to say anything accurate on the ostensible subject. You might want to actually learn something about a subject before you go off embarrassing yourself by pronouncing it to be nonsense. Unless, of course, you enjoy being a troll (a task you seem to have succeeded in with this thread, as it approaches 100 messages in number).

David Petry

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Aug 31, 2017, 1:34:59 PM8/31/17
to
On Thursday, August 31, 2017 at 9:56:37 AM UTC-7, John Gabriel wrote:
> On Thursday, 31 August 2017 11:46:25 UTC-5, David Petry wrote:
> > On Wednesday, August 30, 2017 at 7:14:44 PM UTC-7, Jim Burns wrote:
> >
> > > Now, you want us to "red-pill" ourselves, by which
> > > you mean, disregard the arguments that led us to
> > > today's consensus.
> >
> > My claim is that we live in a very different world from the world in which Cantor's theory arose. Computers, and the imminent rise of artificial intelligence, can impact our view of mathematics. Pure mathematicians are still living in the nineteenth century.


> Red pill and blue pill are stupid ways to make a convincing argument.

You really have to consider the audience.


> You cannot convince someone his ideas are ill formed when he insists that you have to use his ridiculous rules to accomplish the same.

That's actually a very good point, which mathematicians generally miss.


> You have to show that their ideas are ill formed and lead to many contradictions.


That's problematic. The mathematicians have a different understanding of the word "consistent" than sane people.

David Petry

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Aug 31, 2017, 1:47:36 PM8/31/17
to
On Thursday, August 31, 2017 at 9:57:13 AM UTC-7, burs...@gmail.com wrote:

> Am Donnerstag, 31. August 2017 18:46:25 UTC+2 schrieb David Petry:
> > On Wednesday, August 30, 2017 at 7:14:44 PM UTC-7, Jim Burns wrote:
> >
> > > Now, you want us to "red-pill" ourselves, by which
> > > you mean, disregard the arguments that led us to
> > > today's consensus.
> >
> > My claim is that we live in a very different world from the world in which Cantor's theory arose. Computers, and the imminent rise of artificial intelligence, can impact our view of mathematics. Pure mathematicians are still living in the nineteenth century.


> Compared to abacus etc.. did the computer meanwhile
> cross the infinity border? I don't think anything has
>
> fundamentally changed. It doesn't matter whether you
> can compute 1 mio digits of pi. Its still the same math.


What I have been claiming over the years is that a more "modern" way to view mathematics is to think of the computer as the mathematicians' microscope, and then view mathematics as the science of the computational phenomena we observe when we look through that microscope.

All of the mathematics that is relevant to reasoning about the real world is part of the science of phenomena observable in that microscope. Cantor's theory is incompatible with that view.

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