A fast and loose history of mathematical infinity

[David Petry]

First of all, anyone who claims that Cantor's set theory is in any way essential for the mathematics that is relevant to science and technology simply doesn't know what he's talking about.

Here's a fast and loose history of mathematical infinity:

For thousands of years mathematicians accepted the idea that mathematics is closely related to science, and that since the notion of a completed (or "actual") infinity is not part of science, it is also not part of mathematics.

The mathematicians did, however, accept the idea of a potential infinity. Here's the intuition behind that:

We can think of infinity as a destination that can never be reached. That is, no matter how far we go in the direction of infinity, there's still an infinite distance beyond that to go. Note that this is true in science, and hence it must be true in mathematics if mathematical reasoning is to be compatible with scientific reasoning.

The claim behind the notion of an actual infinity is that even though we cannot in any meaningful sense actually reach infinity, we can talk about infinity as if we could reach it. But that has nothing to do with scientific reasoning, and hence should not be part of mathematical reasoning.

And Gauss, who is arguably the greatest mathematician to have ever lived, summed it up nicely by saying, "the actual infinite has no role to play in mathematics. In mathematics, infinity is nothing more than a figure of speech that mathematicians find to be very useful when reasoning about limits". (that's a paraphrase)

And if it's not obvious what Gauss meant by that, here's a way to think about it that should be intuitively appealing: when we think about infinity, we should think about what happens along the journey towards infinity, but it's meaningless to ask what happens when we actually get there.

So Cantor claimed that Gauss and all the mathematicians before him were wrong about infinity. He claimed that the only requirement that his theory of infinity needs to meet in order to be accepted as part of mathematics is that the theory be formally consistent with the parts of mathematics that had been formalized by the middle of the nineteenth century. And evidently he claimed that there was no need for his theory to be consistent with the parts of mathematical reasoning that was done informally at that time. In particular, the connection between mathematical reasoning and scientific reasoning had not been formalized by the middle of the nineteenth century, and he felt that mathematicians should ignore that part of mathematical reasoning.

It seems that it was mostly Hilbert's endorsement of Cantor's theory that led mathematicians to accept the theory as part of mathematics. And to justify it, they claimed that mathematics is really an art, and not part of science.

What I have been claiming for over 32 years now, is that there have been some major advances in our understanding of scientific reasoning over the past century, and that these advances in our understanding of scientific reasoning could be formalized and integrated into mathematics at the foundational level. And I genuinely believe that eventually mathematicians will accept that, and it will be seen as a major advance in our understanding of mathematics.

The relevant key advance in our understanding of scientific reasoning is what has been called the "simulation hypothesis". This says that everything we regard as part of reality (both the physical world and the world of the mind) can be simulated on a digital computer. So what I claim is that this leads to the idea that we can think of the digital computer as a "microscope" that lets a peer deeply into a world of computation, and then we can define mathematics to be the science that studies the phenomena and patterns we observe when we peer through that microscope. I've chosen to call this idea "observability", and it gives us a way to formalize a link between scientific reasoning and mathematical reasoning. This "science of phenomena observable in the world of computation" includes all of the mathematics that is relevant to science and technology.

There's something actually evil about insisting that mathematics is an art and not a science. First, art always includes an element of deception. And Cantor's theory of infinite set includes an element of deception. That's not a good thing

Furthermore, the mathematicians who insist that mathematics must be viewed as an art, will claim that anyone (such as myself) who wants to build a foundation for mathematics that incorporates the idea of observability (i.e. meaningful mathematical assertions must have observable implications) is not even doing mathematics and is not welcome in the mathematics community. And sometimes they go so far as to claim that what I want to do is evil, and that I must be motivated by a desire to deprive the mathematicians of their intellectual freedom. And what they accuse me of doing is what they are doing.

[FromTheRafters]

David Petry expressed precisely :

I find that adding a number to itself and then dividing the result by
two gives me back the original number. How *must* this relate to
physics in your view?

"Set theory is the mathematical theory of well-determined collections,
called sets, of objects that are called members, or elements, of the
set. Pure set theory deals exclusively with sets, so the only sets
under consideration are those whose members are also sets. The theory
of the hereditarily-finite sets, namely those finite sets whose
elements are also finite sets, the elements of which are also finite,
and so on, is formally equivalent to arithmetic. So, the essence of set
theory is the study of infinite sets, and therefore it can be defined
as the mathematical theory of the actual—as opposed to
potential—infinite."

https://plato.stanford.edu/entries/set-theory/

[sergi o]

On 5/27/2022 6:27 AM, David Petry wrote:
>
> First of all, anyone who claims that Cantor's set theory is in any way essential for the mathematics that is relevant to science and technology simply doesn't know what he's talking about.

<flush>

[Alan Mackenzie]

David Petry <david...@gmail.com> wrote:

[ .... ]

> Furthermore, the mathematicians who insist that mathematics must be
> viewed as an art, will claim that anyone (such as myself) who wants to
> build a foundation for mathematics that incorporates the idea of
> observability (i.e. meaningful mathematical assertions must have
> observable implications) is not even doing mathematics and is not
> welcome in the mathematics community.

"Wants"? You implied you've been wanting this for 32 years. What's been
stopping you actually building this new foundation? Is it perhaps just
too difficult? Or, given the foundation we already have, is it perhaps
just redundant?

> And sometimes they go so far as to claim that what I want to do is
> evil, and that I must be motivated by a desire to deprive the
> mathematicians of their intellectual freedom.

This sounds unlikely. Can you substantiate this allegation? Give a
name, a date, and a publication where some mathematician has used the
word "evil" (or a synonym) to describe your proposed development.

> And what they accuse me of doing is what they are doing.

And what is this thing that you are accused of "doing"? I suspect that
if you actually achieved anything like what you say you want, you'd have
no difficulty getting respected publication for it.

--
Alan Mackenzie (Nuremberg, Germany).

[Timothy Golden]

Even if we forgo the arithmetic infinity, as say a cosmology or universe that works upon a shell model (which carries numerous congruences to observation) there will still be the differential to deal with. Some believe we could bottom out in a quantized basis at say the Plank. I don't believe it. Conservation under rotation denies this interpretation.

What would you make of an infinite value such as:
333...3345 ?
Through a chain of reasoning I find a nearby value is already in use within mathematics:
1/3 = 0.333...
Through the destruction of the rational value we can treat the radix point value as fundamental or at least structural, whereas the rational value is an operation on two values, thus ensuring it is not fundamental. In hinsight the radix point value is a natural value augmented with a little dot that indicates a unital position; a secondary form of unity. Engaging the 1/3 example, deleting the radix point exposes the natural value:
333...
for which an equivalent representation is:
333...33
which can be incremented to:
333...34
and so forth. Nobody thus far has falsified this construction. The usage of ellipses is controversial and always will have to be placed at that untouchable zone that you describe infinity as being at. In this way everything here is consistent.

The choice to allow or not allow this format is perhaps a bifurcated approach. That the choice extends onto the radix point value is relevant. Should we reject the ellipses then all is even better with epsilon/delta answering the problem in the rationals that it answered in the irrationals: now epsilon/delta can apply to all values universally. The gray number is born. This is more the nature of the continuum as we deal with it in reality. Engineers and physicists ought not be entitled to a different interpretation of their floating point value versus the mathematician. These splits are false. This is a strong crux.

[Jim Burns]

On 5/27/2022 7:27 AM, David Petry wrote:

> First of all, anyone who claims that
> Cantor's set theory is in any way essential for
> the mathematics that is relevant to science and
> technology simply doesn't know what he's
> talking about.

I have a different claim.

Science-and-technologists choose to use
mathematics based on Cantor's set theory,
and,
as best we can tell, there is no reason for
them to choose differently.

> For thousands of years mathematicians accepted
> the idea that mathematics is closely related to science,
> and that since the notion of a completed (or "actual")
> infinity is not part of science, it is also not
> part of mathematics.
>
> The mathematicians did, however, accept the idea of
> a potential infinity.

The terms "actual infinity" and "potential infinity"
are going to cause problems in this discussion.
One poster (Can you guess who?) has taken them for
his own, and uses them in ways others do not.

> We can think of infinity as a destination that
> can never be reached. That is, no matter how far we go
> in the direction of infinity, there's still an infinite
> distance beyond that to go. Note that this is true in
> science, and hence it must be true in mathematics
> if mathematical reasoning is to be compatible with
> scientific reasoning.

There doesn't seem to be room for the real numbers
in this description of infinity. Speaking as
a sometime-dabbler in scientific reasoning,
I want my real numbers.

I want my description of a continuum.
Essential or not, describing things with a continuum
can be useful -- if in no other way, then useful
to my imagination. I choose to use real numbers,
if I am not prevented from doing so.

I think that what makes a continuum continuum-y
is that there are enough points in it that a
function which is _continuous at each point_
cannot have the leaps that we associate with
discontinuities.

Consider _only_ the positive rationals.
The function f
f(x) =
{ 0 for x^2 < 2
( 1 for x^2 > 2

is continuous at every point --
at every _rational_ point.
And yet, there is this leap at ...
well, we can't say where, really.
That seems to be the root of the problem.

So, we include sqrt(2) in our continuum.
f(sqrt(2)) must have a value, but, no matter what
it is, f(x) can't be continuous at sqrt(2).
This is the correct answer.
When we consider continuous functions,
we want to exclude f(x).

Generally,
when we partition the continuum into LEFT and
RIGHT, each point in LEFT to the left of each
point in RIGHT,
we want a point between LEFT and RIGHT at which
functions can be continuous, thus restricting
"continuous functions" to _what we mean_ by
"continuous functions".

In other words,
being continuum-y requires Dedekind completeness.

However,
Dedekind completeness requires us to lay aside
the notion of infinity as endless sequence.

No endless sequence of points can contain
all of the points of the continuum.
From an endless sequence of points, we can partition
the continuum into LEFT, RIGHT, and BETWEEN,
with no sequence-point in BETWEEN.

If there are no points in BETWEEN, we can define
some g(x) continuous everywhere in LEFT and in RIGHT
but which makes some leap in going from LEFT to
RIGHT.

We don't want that, our continuum wouldn't be
continuum-y. But points in BETWEEN aren't
in the endless sequence. Therefore, the endless
sequence doesn't contain all points.
And that's true of any endless sequence of points.

This is what I mean by saying
there isn't enough room in the endless-sequence-style
infinity for real numbers.
A reminder: I want continuum-y real numbers.

> The claim behind the notion of an actual infinity is
> that even though we cannot in any meaningful sense
> actually reach infinity, we can talk about infinity
> as if we could reach it.

I am going to excuse myself from discussion of what
should or should not be called "actual infinity".

It seems to me that ________ infinity can be
discussed by describing _one of_ some collection
which happens to be infinite, and then using
only truth-preserving inferences to proceed from
that description to further claims about that
collection.

For example,
we can describe _one of_ the collection of cuts
LEFT and RIGHT of the continuum as having a
point-between. That makes the continuum
sufficiently continuum-y for our purposes.
And we go on from there, truth-preserving-ly.

> What I have been claiming for over 32 years now,
> is that there have been some major advances in
> our understanding of scientific reasoning over
> the past century, and that these advances in our
> understanding of scientific reasoning could be
> formalized and integrated into mathematics at
> the foundational level.

32 years is a long time.

It seems to me that foundations are intended
to support whatever we choose to build on them.
Mixing metaphors, a good foundation is a Swiss Army
knife. We won't necessarily find a use for the
spork. That's not a reason to remove the spork.

It sounds to me as though you want to remove
all "unnecessary" tools from our foundations.
Like removing the spork from a Swiss Army knife.
I don't see any good argument for doing so.

> And sometimes they go so far as to claim that
> what I want to do is evil,

I don't think formalizing falsification in
mathematics is evil. I question how useful it
would be. The philosophy of science has moved on
from falsification. But no one needs to pay any
attention to my questions.

I think that Lysenkoism is at least anti-truth
and could be very evil indeed, depending upon
how its regulations are enforced.

At various times, you have sounded as though
you were advocating some form of Lysenkoism.

> and that I must be motivated by a desire to deprive
> the mathematicians of their intellectual freedom.
> And what they accuse me of doing is what they are doing.

32 years is a long time.

What if you had spent 32 years working on whatever
you wanted to work on, such as on formalizing
falsification, and then whoever was interested in
your work joined in?

Is there something wrong with the picture I paint?

[Earle Jones]

On Fri May 27 04:27:09 2022 David Petry wrote:
>
> First of all, anyone who claims that Cantor's set theory is in any way essential for the mathematics that is relevant to science and technology simply doesn't know what he's talking about.
>
> Here's a fast and loose history of mathematical infinity:
>
> For thousands of years mathematicians accepted the idea that mathematics is closely related to science, and that since the notion of a completed (or "actual") infinity is not part of science, it is also not part of mathematics.
>
> The mathematicians did, however, accept the idea of a potential infinity. Here's the intuition behind that:
>
> We can think of infinity as a destination that can never be reached. That is, no matter how far we go in the direction of infinity, there's still an infinite distance beyond that to go. Note that this is true in science, and hence it must be true in mathematics if mathematical reasoning is to be compatible with scientific reasoning.
>
> The claim behind the notion of an actual infinity is that even though we cannot in any meaningful sense actually reach infinity, we can talk about infinity as if we could reach it. But that has nothing to do with scientific reasoning, and hence should not be part of mathematical reasoning.
>
> And Gauss, who is arguably the greatest mathematician to have ever lived, summed it up nicely by saying, "the actual infinite has no role to play in mathematics. In mathematics, infinity is nothing more than a figure of speech that mathematicians find to be very useful when reasoning about limits". (that's a paraphrase)
>
> And if it's not obvious what Gauss meant by that, here's a way to think about it that should be intuitively appealing: when we think about infinity, we should think about what happens along the journey towards infinity, but it's meaningless to ask what happens when we actually get there.
>
> So Cantor claimed that Gauss and all the mathematicians before him were wrong about infinity. He claimed that the only requirement that his theory of infinity needs to meet in order to be accepted as part of mathematics is that the theory be formally consistent with the parts of mathematics that had been formalized by the middle of the nineteenth century.

[Clip some here]

Can you give us a reference where Cantor claimed that Gauss was wrong about infinity?

Thanks,

earle
*

[WM]

Cantor condemned Gauß' rejection of the actual infinite in several places:
"The erroneous in Gauss' letter consists in his sentence that the finished infinite could not become an object of mathematical consideration. [...] The finished infinite can be found, in a sense, in the numbers ,  + 1, ..., , ..." [G. Cantor, letter to R. Lipschitz (19 Nov 1883)]
"My opposition to Gauss consists in the fact that Gauss rejects as inconsistent (I mean he does so unconsciously, i.e., without knowing this notion) all multitudes with exception of the finite and therefore categorically and basically discards the actual infinite which I call transfinitum, and together with this he declares the transfinite numbers as impossible, the existence of which I have established." [G. Cantor, letter to D. Hilbert (27 Jan 1900)]
"it seems that the ancients haven't had any clue of the transfinite, the possibility of which is even strongly rejected by Aristotle and his school like in newer times by d'Alembert, Lagrange, Gauss, Cauchy, and their adherents." [G. Cantor, letter to G. Peano (21 Sep 1895)]
From https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf

Regards, WM

[WM]

Jim Burns schrieb am Freitag, 27. Mai 2022 um 20:39:04 UTC+2:
> On 5/27/2022 7:27 AM, David Petry wrote:
>
> > First of all, anyone who claims that
> > Cantor's set theory is in any way essential for
> > the mathematics that is relevant to science and
> > technology simply doesn't know what he's
> > talking about.
> I have a different claim.
>
> Science-and-technologists choose to use
> mathematics based on Cantor's set theory,
> and,
> as best we can tell, there is no reason for
> them to choose differently.

That is a ridiculous claim. Mathematics has no connection with set theory other than some artificial constructs by set theorists which in spite of the completly wrong set theory obtain some correct mathematics accidentltally.

Regards, WM

[sergi o]

On 5/27/2022 2:12 PM, WM wrote:
> Jim Burns schrieb am Freitag, 27. Mai 2022 um 20:39:04 UTC+2:
>> On 5/27/2022 7:27 AM, David Petry wrote:
>>
>>> First of all, anyone who claims that
>>> Cantor's set theory is in any way essential for
>>> the mathematics that is relevant to science and
>>> technology simply doesn't know what he's
>>> talking about.
>> I have a different claim.
>>
>> Science-and-technologists choose to use
>> mathematics based on Cantor's set theory,
>> and,
>> as best we can tell, there is no reason for
>> them to choose differently.
>

<snip crap>

>
> Regards, WM

[sergi o]

On 5/27/2022 2:08 PM, WM wrote:
> Earle Jones schrieb am Freitag, 27. Mai 2022 um 21:01:55 UTC+2:
>> On Fri May 27 04:27:09 2022 David Petry wrote:
>
>>> So Cantor claimed that Gauss and all the mathematicians before him were wrong about infinity. He claimed that the only requirement that his theory of infinity needs to meet in order to be accepted as part of mathematics is that the theory be formally consistent with the parts of mathematics that had been formalized by the middle of the nineteenth century.
>> [Clip some here]
>>
>> Can you give us a reference where Cantor claimed that Gauss was wrong about infinity?
>>

> Cantor

<snip troll book stuff>

> From https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf
>
> Regards, WM

"references" from a troll book


mmernoch keine Definition, wie zu erwarten war, wie schon seit Jahren.

Was stellen Sie sich vor unter dem Durchschnitt einer potentiell unendlichen Menge potentiell unendlicher Mengen?

Was soll denn in einer "potentiell unendlichen Menge" enthalten sein
und was nicht? SIE haben noch nicht einmal definiert, was denn eine
"potentiell unendliche Menge" eigentlich sein soll, und ebensowenig,
was denn "in einer pootentiell unendlichen Menge enthalten" sein soll.
Alles nur nichtssagendes Geschwurbel, aber nichts wirklich substanti-
elles ...

Wir reden hier von unendlichen Mengen, nicht von Taschenrechnern. Sie behaupten, der mathematisch saubere Begriff sei die 'potentiell unendliche Menge'
und der Cantor'sche Begriff der 'aktual unendlichen Menge' führe zu Widersprüchen.
Sie sind sehr weit davon entfernt, diese Behauptung beweisen zu können; u.a. weil Ihre Begriffsbildungen verschwommen sind.

Das ist also die Mückmeatik - in gewissem Sinne verschwommen, d.h. dummes Geschwätz.

Es ist doch unfair Rindviecher mit diesem Vergleich zu beleidigen, die
koennen ssich doch kaum wehren ...

Touche'. Ich möchte mich hiermit bei allen vierbeinigen Rindviechern von Herzen entschuldigen. Sie mit WM zu vergleichen, war für die vierbeinigen
Rindviecher höchst beleidigend. So eine Entgleisung wird hoffentlich nicht wieder vorkommen.

[FredJeffries]

On Friday, May 27, 2022 at 4:27:15 AM UTC-7, david...@gmail.com wrote:
>
> We can think of infinity as a destination that can never be reached.

We CAN think of 'infinity' that way, yes. But we don't have to. If all one can see is nails, one thinks that the only tool needed is a hammer.

Your being around for 32 years means that you have many times seen the repost of Zdislav V. Kovarik's from August 8, 1999

https://groups.google.com/g/sci.math/c/wM6drf_B0Po/m/QO6yQP2avs8J

[Earle Jones]

On Fri May 27 04:27:09 2022 David Petry wrote:
>

> First of all, anyone who claims that Cantor's set theory is in any way essential for the mathematics that is relevant to science and technology simply doesn't know what he's talking about.
>
> Here's a fast and loose history of mathematical infinity:

*
Your "fast and loose" history of mathematical infinity is not very "fast", but it certainly is "loose."


You seem to be saying that if it isn't relative to science, then it isn't mathematics. Let me give an example. I was born in the year 1931. That is a prime number. My friend Barry was born in 1933. That is also a prime number. Together, our birth years form a prime pair. Prime pairs are a fascinating subject in mathematics. Is there an infinite number of prime pairs? If you know, please publish it and become famous. The number of primes is infinite (a high-school proof), but the number of prime pairs is still unknown. What does that have to do with Science, which is a process for discovery of knowledge?

What Cantor gave us is a platform for discussion about infinite quantities. Which is larger, the number of integers or the number of even integers?

earle
*

[Jim Burns]

On 5/27/2022 3:12 PM, WM wrote:
> Jim Burns schrieb
> am Freitag, 27. Mai 2022 um 20:39:04 UTC+2:
>> On 5/27/2022 7:27 AM, David Petry wrote:

>>> First of all, anyone who claims that
>>> Cantor's set theory is in any way essential for
>>> the mathematics that is relevant to science and
>>> technology simply doesn't know what he's
>>> talking about.
>>
>> I have a different claim.
>>
>> Science-and-technologists choose to use
>> mathematics based on Cantor's set theory,
>> and,
>> as best we can tell, there is no reason for
>> them to choose differently.
>
> That is a ridiculous claim.

I phrased my claim the way I did to emphasize
the difference between my claim and the claims
of whoever David Petry is concerned with.

It should be clear to anyone familiar with
science-and-technology mathematics that its various
parts _can be_ treated within a Cantor-compliant
set theory such as ZFC.

On the other hand,
these parts can also be considered apart from
set theory foundations. We can prove many
important theorems about the real numbers
without having proved the existence of them
from the axioms of ZFC.

But this relationship to Cantor's set theory seems
relevant to the current discussion.
It is not required, but it is not forbidden.


If we take a wider view of science, technology,
engineering, mathematics, and their ilk,
it is typical for such folk to leave debates within
a neighboring discipline to the experts within
that neighboring discipline. Once the experts
have sorted things out to some sort of consensus,
STEM-types will be very happy to use the consensus.
But chemists don't often consent to becoming
physicists in order to use physics in their chemistry.
Biologists don't often consent to becoming chemists
in order to use chemistry in their biology.

It looks to me exactly backwards to suggest that
science-and-technologists long to offer advice to
mathematicians on how to do mathematics. Apart
from the practical difficulties, it looks to me
as though they'd run a mile to avoid needing to
do that. They've got other things to do,
science-and-technology things.

> Mathematics has no connection with set theory other than
> some artificial constructs by set theorists which
> in spite of the completly wrong set theory obtain
> some correct mathematics accidentltally.

...which is to say,
mathematics has no connection with set theory
other than the connection I mentioned.
Fair enough.

[David Petry]

On Friday, May 27, 2022 at 8:43:42 AM UTC-7, Alan Mackenzie wrote:

Give a
> name, a date, and a publication where some mathematician has used the
> word "evil" (or a synonym) to describe your proposed development.

It happened a number of times in this newsgroup, about 25 years ago or so, that someone with a PhD in mathematics would insinuate that the only conceivable reason someone would be opposed to Cantor's theory being in mathematics is antisemitism.

> if you actually achieved anything like what you say you want, you'd have
> no difficulty getting respected publication for it.

You may honestly believe that.

[David Petry]

On Friday, May 27, 2022 at 11:39:04 AM UTC-7, Jim Burns wrote:
> On 5/27/2022 7:27 AM, David Petry wrote:

> > We can think of infinity as a destination that
> > can never be reached. That is, no matter how far we go
> > in the direction of infinity, there's still an infinite
> > distance beyond that to go. Note that this is true in
> > science, and hence it must be true in mathematics
> > if mathematical reasoning is to be compatible with
> > scientific reasoning.


> There doesn't seem to be room for the real numbers
> in this description of infinity. Speaking as
> a sometime-dabbler in scientific reasoning,
> I want my real numbers.

The "real" numbers used in science are finite precision real numbers. What I have been advocating is that the "real" numbers used in science should be finite precision real numbers along with an error estimate.

I've been saying that one of the advances in scientific reasoning made in the past century is the recognition that we must always take uncertainty into consideration, both in practice and theory.

> It sounds to me as though you want to remove
> all "unnecessary" tools from our foundations.

No, you really are missing the point. Consistency is the essential point. I want to formalize scientific reasoning, and then I suggest that mathematics should be formally consistent with that formalization of scientific reasoning. And I argue that Cantor's ideas about infinity are not consistent with scientific reasoning.

> At various times, you have sounded as though
> you were advocating some form of Lysenkoism.

Ok, so you are proving that you are not really understanding my arguments. No surprise there.

[David Petry]

Thank you, Wolfgang, for posting that.

Maybe my memory is just playing tricks on me, but I think I remember you posting something Cantor wrote where he specifically addressed Gauss' quote where he (Gauss) said that infinity is merely a figure of speech (facon de parler).

[David Petry]

On Friday, May 27, 2022 at 12:21:01 PM UTC-7, FredJeffries wrote:


> Your being around for 32 years means that you have many times seen the repost of Zdislav V. Kovarik's from August 8, 1999
>
> https://groups.google.com/g/sci.math/c/wM6drf_B0Po/m/QO6yQP2avs8J

Actually I do remember that.

If the topic here were Poincare's assertion that Cantor's theory of infinite sets is a disease from which mathematics will one day be cured, then that article is merely a distraction.

[David Petry]

On Friday, May 27, 2022 at 3:03:36 PM UTC-7, Earle Jones wrote:


> Which is larger, the number of integers or the number of even integers?

You're in over your head.

[David Petry]

On Friday, May 27, 2022 at 3:52:15 PM UTC-7, Jim Burns wrote:

> It should be clear to anyone familiar with
> science-and-technology mathematics that its various
> parts _can be_ treated within a Cantor-compliant
> set theory such as ZFC.

That's not actually true. Scientists and technologists include a natural language narrative when they use mathematics, and if they use set theoretic concepts at all, it's merely window dressing.

[Alan Mackenzie]

David Petry <david...@gmail.com> wrote:
> On Friday, May 27, 2022 at 8:43:42 AM UTC-7, Alan Mackenzie wrote:

>> Give a name, a date, and a publication where some mathematician has
>> used the word "evil" (or a synonym) to describe your proposed
>> development.

> It happened a number of times in this newsgroup, about 25 years ago or
> so, that someone with a PhD in mathematics would insinuate that the
> only conceivable reason someone would be opposed to Cantor's theory
> being in mathematics is antisemitism.

All sorts of foul things get said in this newsgroup, and I'm sure that
was the case 25 years ago too, if perhaps to a lesser degree. But has
any mathematician said "evil" about it in polite discussion, say in an
academic journal?

>> if you actually achieved anything like what you say you want, you'd have
>> no difficulty getting respected publication for it.

> You may honestly believe that.

I do. Do you not? The key thing is actually having something to
publish, some results. After these 32 years, do you have some
alternative foundation of mathematics, or at least the rudiments of one?
If not, why not? I suspect the real reason is that it is too difficult.
Maybe there is no such alternative foundation (which comes to the same
thing in the end). Set theory wasn't developed complete and finished by
one person in a couple of years.

So if you have any concrete ideas, why not publish them here for review
and criticism?

[FromTheRafters]

No answer? One could say that there are more integers than there are
even integers since, as sets, one is a proper subset of the other.

The "size" or "cardinality" of these inductive sets is the same though
because the infinite sets' cardinalities (type of infinity) are not the
same as the finite sets' cardinalities (number of elements) with regard
to size.

[WM]

Alan Mackenzie schrieb am Samstag, 28. Mai 2022 um 10:57:15 UTC+2:

> All sorts of foul things get said in this newsgroup, and I'm sure that
> was the case 25 years ago too, if perhaps to a lesser degree. But has
> any mathematician said "evil" about it in polite discussion, say in an
> academic journal?

Presently journals are censored by the majority of matheologians. But there are books.

"those of us who work in probability theory or any other area of applied mathematics have a right to demand that this disease, for which we are not responsible, be quarantined and kept out of our field. In this view, too, we are not alone; and indeed have the support of many non-Bourbakist mathematicians. [E.T. Jaynes: "Probability theory: The logic of science", edited by G.L. Bretthorst, Cambridge Univ. Press (2003) pp. XXII & XXVII & 672f]

"Pure mathematics and science are finally being reunited and, mercifully, the Bourbaki plague is dying out. [M. Gell-Mann: "Nature conformable to herself", Bulletin of the Santa Fe Institute 7 (1992) p. 7]

"The ordinary diagonal Verfahren I believe to involve a patent confusion of the program and object aspects of the decimal fraction, which must be apparent to any who imagines himself actually carrying out the operations demanded in the proof. In fact, I find it difficult to understand how such a situation should have been capable of persisting in mathematics." [P.W. Bridgman: "A physicist's second reaction to Mengenlehre", Scripta Mathematica 2 (1934) p. 225ff]

"Examples of (according to Brouwer) meaningless word play are the second number class and the higher power sets. [D. van Dalen: "Mystic, geometer, and intuitionist: The life of L.E.J. Brouwer", Oxford Univ. Press (2002)]

"We cannot admit the assumption of an infinite number, of beings or of objects coexisting, without being trapped by manifest contradictions. [A. Cauchy "Sept lecons de physique générale", Gauthier-Villars, Paris (1868) p. 23]

"Feferman shows in his article "Why a little bit goes a long way" on the basis of a number of case studies that the mathematics currently required for scientific applications can all be carried out in an axiomatic system whose basic justification does not require the actual infinite." [S. Feferman: private communication]

"The result of the preceding discussion is that our axioms, if interpreted as meaningful statements, necessarily presuppose a kind of Platonism, which cannot satisfy any critical mind and which does not even produce the conviction that they are consistent. [K. Gödel: "The present situation in the foundations of mathematics" (1933) in S. Feferman et al. (eds.): "Kurt Gödel, collected works, Vol. III, unpublished essays and lectures", Oxford Univ. Press, (1995) p. 50]

"Kronecker, who visited me at the beginning of July, declared with the friendliest smile that he had much correspondence about my last paper with Hermite in order to demonstrate to him that all that was only 'humbug'." (Cantor)

And finally even a journal! "Cantor's diagonal proof engages us in an endless, potentially infinite and quite senseless paradoxical "game of two honest tricksters". [A.A. Zenkin: "Logic of actual infinity and G. Cantor's diagonal proof of the uncountability of the continuum", Review of Modern Logic 9 (2004) p. 28]

Regards, WM

[Timothy Golden]

Odd infinite number: 333...33
Even infinite number: 333...34

[WM]

david...@gmail.com schrieb am Samstag, 28. Mai 2022 um 07:02:53 UTC+2:

> Maybe my memory is just playing tricks on me, but I think I remember you posting something Cantor wrote where he specifically addressed Gauss' quote where he (Gauss) said that infinity is merely a figure of speech (facon de parler).

Cantor addressed it directly in the quoted letter to Lipschitz of 19 Nov 1883:
Sie machen mich in Ihrem Schreiben auf eine Stelle bei Gauss in einem Briefe an Schumacher v. 12' Juli 1831 aufmerksam, worin er "protestirt gegen den Gebrauch einer unendlichen Grösse als einer Vollendeten, welcher in der Mathematik niemals erlaubt ist. Das Unendliche ist nur eine facon de parler, indem man eigentlich von Grenzen spricht, denen gewisse Verhältnisse so nahe kommen als man will, während anderen ohne Einschränkung zu wachsen verstattet ist."
In diesen beiden Sätzen ist Falsches mit Wahrem durcheinander gemengt;

Regards, WM

[Jim Burns]

On 5/28/2022 1:15 AM, David Petry wrote:
> On Friday, May 27, 2022 at 3:52:15 PM UTC-7,
> Jim Burns wrote:

>> It should be clear to anyone familiar with
>> science-and-technology mathematics that its various
>> parts _can be_ treated within a Cantor-compliant
>> set theory such as ZFC.
>
> That's not actually true.

No, that's actually true.

Axiom of infinity
--> some inductive set

Minimal inductive set
--> natural numbers

Partition of pairs ⟨i,j⟩ of naturals by i₁+j₂ = i₂+j₁
--> integers

Partition of pairs ⟨p,q⟩ of integers by p₁*q₂ = p₂*q₁
--> rationals

Partition of Cauchy sequences ⟨ r₁, r₂, r₃, ... ⟩
of rationals by
𝑙𝑖𝑚 ⟨ r₁₁-r₂₁, r₁₂-r₂₂, r₁₃-r₂₃, ... ⟩ = 0
--> reals

> Scientists and technologists include
> a natural language narrative when they use mathematics,
> and if they use set theoretic concepts at all,
> it's merely window dressing.

That addresses some other claim, which I haven't made.

You have confused my claim that we *CAN*
base science-and-technology mathematics on
Cantor-compliant set theory
with a straw-man claim that we *MUST*
base science-and-technology mathematics on
Cantor-compliant set theory.

Your edited version of my claim doesn't fit well
with my next paragraph.

>> On the other hand,
>> these parts can also be considered apart from
>> set theory foundations. We can prove many
>> important theorems about the real numbers
>> without having proved the existence of them
>> from the axioms of ZFC.

There are reasons that are enough for many people
to use ZFC as a foundation. That's not the same as
these people using ZFC a lot.

Suppose you live or work in a building with a
totally non-metaphorical foundation.
You could spend your whole life in that building
having close to no thoughts about its foundation.
That's actually the most probable outcome,
if the foundation continues to work as it should.

[WM]

Jim Burns schrieb am Samstag, 28. Mai 2022 um 19:25:16 UTC+2:
> On 5/28/2022 1:15 AM, David Petry wrote:
> > On Friday, May 27, 2022 at 3:52:15 PM UTC-7,
> > Jim Burns wrote:
>
> >> It should be clear to anyone familiar with
> >> science-and-technology mathematics that its various
> >> parts _can be_ treated within a Cantor-compliant
> >> set theory such as ZFC.
> >
> > That's not actually true.
> No, that's actually true.
>
> Axiom of infinity
> --> some inductive set
>
> Minimal inductive set
> --> natural numbers

No. Every member of the inductive set has ℵo successors before ω:
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo

But all natural numbers have no successors before ω:
{0, 1, 2, 3, ..., ω} \ ℕ = {0, ω}.

Therefore the potentially infinite inductive collection ℕ_def is not the same as ℕ.

> There are reasons that are enough for many people
> to use ZFC as a foundation.

They have to hypnotize themselves, like you, that by purely exchanging X's and O's in
XOOO...
XOOO...
XOOO...
XOOO...
...
the whole matrix can be emptied of all O's.

Regards, WM

[sergi o]

On 5/27/2022 11:44 PM, David Petry wrote:
> On Friday, May 27, 2022 at 8:43:42 AM UTC-7, Alan Mackenzie wrote:
>
> Give a
>> name, a date, and a publication where some mathematician has used the
>> word "evil" (or a synonym) to describe your proposed development.
>
> It happened a number of times in this newsgroup, about 25 years ago or so, that someone with a PhD in mathematics would insinuate that the only conceivable reason someone would be opposed to Cantor's theory being in mathematics is antisemitism.

=> name, a date, and a publication where some mathematician has used the word "evil" (or a synonym) to describe your proposed development.

>
>> if you actually achieved anything like what you say you want, you'd have
>> no difficulty getting respected publication for it.
>
> You may honestly believe that.
>

so you are just making stuff up.

[sergi o]

wtf ? kisses and hugs..... ?

[sergi o]

On 5/27/2022 11:58 PM, David Petry wrote:
> On Friday, May 27, 2022 at 11:39:04 AM UTC-7, Jim Burns wrote:
>> On 5/27/2022 7:27 AM, David Petry wrote:
>
>>> We can think of infinity as a destination that
>>> can never be reached. That is, no matter how far we go
>>> in the direction of infinity, there's still an infinite
>>> distance beyond that to go. Note that this is true in
>>> science, and hence it must be true in mathematics
>>> if mathematical reasoning is to be compatible with
>>> scientific reasoning.
>
>
>> There doesn't seem to be room for the real numbers
>> in this description of infinity. Speaking as
>> a sometime-dabbler in scientific reasoning,
>> I want my real numbers.
>
>
> The "real" numbers used in science are finite precision real numbers.

That is a generalization, which is wrong.

> What I have been advocating is that the "real" numbers used in science should be finite precision real numbers along with an error estimate.

another generalization, and a wrong one again.

>
> I've been saying that one of the advances in scientific reasoning made in the past century is the recognition that we must always take uncertainty into consideration, both in practice and theory.

another vague general statement.

>
>
>> It sounds to me as though you want to remove
>> all "unnecessary" tools from our foundations.
>
>
> No, you really are missing the point. Consistency is the essential point. I want to formalize scientific reasoning, and then I suggest that mathematics should be formally consistent with that formalization of scientific reasoning. And I argue that Cantor's ideas about infinity are not consistent with scientific reasoning.

you have not argued anything at all, just make unfounded statements and generalizations. useless verbage.

>
>
>> At various times, you have sounded as though
>> you were advocating some form of Lysenkoism.
>
>
> Ok, so you are proving that you are not really understanding my arguments. No surprise there.
>

yes, it is Lysenkoism or worser Michurianism Petry is peddeling

[sergi o]

is the average of an odd infinite number and even infinite number odd or even ?

[Jim Burns]

On 5/28/2022 12:58 AM, David Petry wrote:
> On Friday, May 27, 2022 at 11:39:04 AM UTC-7,
> Jim Burns wrote:
>> On 5/27/2022 7:27 AM, David Petry wrote:

>>> We can think of infinity as a destination that
>>> can never be reached. That is, no matter how far we go
>>> in the direction of infinity, there's still an infinite
>>> distance beyond that to go. Note that this is true in
>>> science, and hence it must be true in mathematics
>>> if mathematical reasoning is to be compatible with
>>> scientific reasoning.
>
>> There doesn't seem to be room for the real numbers
>> in this description of infinity. Speaking as
>> a sometime-dabbler in scientific reasoning,
>> I want my real numbers.
>
> The "real" numbers used in science are finite precision
> real numbers.

By "I want my real numbers", I mean that
I want the things we use to derive the Balmer series
for the hydrogen spectrum from the Hamiltonian
𝐇 = 𝐩²/2m - e²/𝐫

These things I refer to are not finite-precision.
I want _them_

> What I have been advocating is that
> the "real" numbers used in science should be
> finite precision real numbers along with
> an error estimate.

We already have real numbers with errors.
For example, from January 8, 2021, I see a report
of the age of the universe as 13.77 ± .04 billion
years.
https://www.space.com/universe-age-14-billion-years-old

What could you be saying, then?
That I _should not_ derive the Balmer series
for the hydrogen spectrum from the Hamiltonian
𝐇 = 𝐩²/2m - e²/𝐫 ?

Why not?

> I've been saying that one of the advances in
> scientific reasoning made in the past century is
> the recognition that we must always take
> uncertainty into consideration, both in practice
> and theory.

"Uncertainty" is a word with a few not-always-
-compatible uses.

The finite-precision use of "uncertainty" is
much older than the 20th century. I couldn't say
how old. As long as we've measured things?

The fairly new 20th-century use of "uncertainty" is
closely related to deriving the Balmer series
for the hydrogen spectrum from the Hamiltonian
𝐇 = 𝐩²/2m - e²/𝐫

There is a _minimum uncertainty_ ∆𝐱⋅∆𝐩 >= ℏ/2
in measurements of displacement 𝐱 and momentum 𝐩
This can be derived from their commutator relation
[𝐱,𝐩] = 𝑖ℏ

You know what else can be derived from their
commutator relation? The Balmer series.

I don't know what you're going on about.

If you decide to chuckle about how clueless
I am for not knowing what you're going on
about, more power to you. It'd be a curious thing,
though, if you didn't take the opportunity to
at least try to explain better what you're
going on about. Maybe 32 years doesn't seem
very long to you, and you're hoping to do
this for 32 more years.

>> It sounds to me as though you want to remove
>> all "unnecessary" tools from our foundations.
>
> No, you really are missing the point.
> Consistency is the essential point.

"Consistent" v1.

> I want to formalize scientific reasoning,
> I want to formalize scientific reasoning,
> and then I suggest that mathematics should be
> formally consistent with that formalization of
> scientific reasoning. And I argue that
> Cantor's ideas about infinity are not consistent

"Consistent" v2.

> with scientific reasoning.

Consistency v1 (self-consistency) is valuable
in itself.
"Those who can mae you believe absurdities
can make you commit attrocities." (Voltaire?)

Consistency v2 (co-consistency) is only valuable
if your intention is to reject what you don't
_already know_ is useful to scientists.

Why do that?
Why do that, when we've been surprised
again and again by what has turned out to be
valuable?

>> At various times, you have sounded as though
>> you were advocating some form of Lysenkoism.
>
> Ok, so you are proving that you are not really
> understanding my arguments. No surprise there.

Lucy for you that there are all these other people,
beating on your door, who _don't_ need more explanation
from you about what your argument is.

[mitchr...@gmail.com]

The only way to deal with infinity is the Continuum Hypothesis
It is containing an infinity instead as sizes made of the infinitely small.
Counting forever will not complete to any infinity.

Mitchell Raemsch

[Earle Jones]

*
Mitchell:

Don't forget that forever is a long time -- especially the last part.

(Thanks to Woody Allen.)

[mitchr...@gmail.com]

I count on it. It never concludes.
Does the multiverse die if our universe does.
Religion knows better than the science death cult....

[Ross A. Finlayson]

On Friday, May 27, 2022 at 8:57:27 AM UTC-7, timba...@gmail.com wrote:

> On Friday, May 27, 2022 at 7:27:15 AM UTC-4, david...@gmail.com wrote:
> > First of all, anyone who claims that Cantor's set theory is in any way essential for the mathematics that is relevant to science and technology simply doesn't know what he's talking about.
> >

> > Here's a fast and loose history of mathematical infinity:
> >

> > For thousands of years mathematicians accepted the idea that mathematics is closely related to science, and that since the notion of a completed (or "actual") infinity is not part of science, it is also not part of mathematics.
> >
> > The mathematicians did, however, accept the idea of a potential infinity. Here's the intuition behind that:

> >
> > We can think of infinity as a destination that can never be reached. That is, no matter how far we go in the direction of infinity, there's still an infinite distance beyond that to go. Note that this is true in science, and hence it must be true in mathematics if mathematical reasoning is to be compatible with scientific reasoning.
> >

> > The claim behind the notion of an actual infinity is that even though we cannot in any meaningful sense actually reach infinity, we can talk about infinity as if we could reach it. But that has nothing to do with scientific reasoning, and hence should not be part of mathematical reasoning.
> >
> > And Gauss, who is arguably the greatest mathematician to have ever lived, summed it up nicely by saying, "the actual infinite has no role to play in mathematics. In mathematics, infinity is nothing more than a figure of speech that mathematicians find to be very useful when reasoning about limits". (that's a paraphrase)
> >
> > And if it's not obvious what Gauss meant by that, here's a way to think about it that should be intuitively appealing: when we think about infinity, we should think about what happens along the journey towards infinity, but it's meaningless to ask what happens when we actually get there.
> >
> > So Cantor claimed that Gauss and all the mathematicians before him were wrong about infinity. He claimed that the only requirement that his theory of infinity needs to meet in order to be accepted as part of mathematics is that the theory be formally consistent with the parts of mathematics that had been formalized by the middle of the nineteenth century. And evidently he claimed that there was no need for his theory to be consistent with the parts of mathematical reasoning that was done informally at that time. In particular, the connection between mathematical reasoning and scientific reasoning had not been formalized by the middle of the nineteenth century, and he felt that mathematicians should ignore that part of mathematical reasoning.
> >
> > It seems that it was mostly Hilbert's endorsement of Cantor's theory that led mathematicians to accept the theory as part of mathematics. And to justify it, they claimed that mathematics is really an art, and not part of science.
> >
> > What I have been claiming for over 32 years now, is that there have been some major advances in our understanding of scientific reasoning over the past century, and that these advances in our understanding of scientific reasoning could be formalized and integrated into mathematics at the foundational level. And I genuinely believe that eventually mathematicians will accept that, and it will be seen as a major advance in our understanding of mathematics.
> >
> > The relevant key advance in our understanding of scientific reasoning is what has been called the "simulation hypothesis". This says that everything we regard as part of reality (both the physical world and the world of the mind) can be simulated on a digital computer. So what I claim is that this leads to the idea that we can think of the digital computer as a "microscope" that lets a peer deeply into a world of computation, and then we can define mathematics to be the science that studies the phenomena and patterns we observe when we peer through that microscope. I've chosen to call this idea "observability", and it gives us a way to formalize a link between scientific reasoning and mathematical reasoning. This "science of phenomena observable in the world of computation" includes all of the mathematics that is relevant to science and technology.
> >
> > There's something actually evil about insisting that mathematics is an art and not a science. First, art always includes an element of deception. And Cantor's theory of infinite set includes an element of deception. That's not a good thing
> >
> > Furthermore, the mathematicians who insist that mathematics must be viewed as an art, will claim that anyone (such as myself) who wants to build a foundation for mathematics that incorporates the idea of observability (i.e. meaningful mathematical assertions must have observable implications) is not even doing mathematics and is not welcome in the mathematics community. And sometimes they go so far as to claim that what I want to do is evil, and that I must be motivated by a desire to deprive the mathematicians of their intellectual freedom. And what they accuse me of doing is what they are doing.
> Even if we forgo the arithmetic infinity, as say a cosmology or universe that works upon a shell model (which carries numerous congruences to observation) there will still be the differential to deal with. Some believe we could bottom out in a quantized basis at say the Plank. I don't believe it. Conservation under rotation denies this interpretation.
>
> What would you make of an infinite value such as:
> 333...3345 ?
> Through a chain of reasoning I find a nearby value is already in use within mathematics:
> 1/3 = 0.333...
> Through the destruction of the rational value we can treat the radix point value as fundamental or at least structural, whereas the rational value is an operation on two values, thus ensuring it is not fundamental. In hinsight the radix point value is a natural value augmented with a little dot that indicates a unital position; a secondary form of unity. Engaging the 1/3 example, deleting the radix point exposes the natural value:
> 333...
> for which an equivalent representation is:
> 333...33
> which can be incremented to:
> 333...34
> and so forth. Nobody thus far has falsified this construction. The usage of ellipses is controversial and always will have to be placed at that untouchable zone that you describe infinity as being at. In this way everything here is consistent.
>
> The choice to allow or not allow this format is perhaps a bifurcated approach. That the choice extends onto the radix point value is relevant. Should we reject the ellipses then all is even better with epsilon/delta answering the problem in the rationals that it answered in the irrationals: now epsilon/delta can apply to all values universally. The gray number is born. This is more the nature of the continuum as we deal with it in reality. Engineers and physicists ought not be entitled to a different interpretation of their floating point value versus the mathematician. These splits are false. This is a strong crux.

Quantization and the Planck is a really great deal: wells, strings, atoms.

(Units.)

I think of it as a hologram, then, its state is its projection.

This way "rational dissipation" is still all tidal (toroidal, ...).

That the hologram or projection is Euclidean, ....

The interval arithmetic you'll find addresses methods.

(Floating point arithmetic.)

If you think of real number as integer part, and, non-integr part,
in the integer lattice, it fulfills analysis neatly, full piece-wise.

Only the "real" part, the real number: ....

[Ross A. Finlayson]

That is just regress that goes to impasse.

"For any large number, there is always a number much "larger", ...".

"If infinity is largest I am one."

"If infinity is largest I am one: or zero, or a tiny infinitesimal."

People's most usual notion of infinitesimal,
is, "for any infinity, its infinitesimal is its reciprocal".

Then, in large bounded spaces of computer arrays, is the idea
to fill the memory, data, then run out the single instruction
multiple data, that, it would be large and free in usual CPU terms.

For that "this being an infinity of these the reciprocal is infinitesimal".

Then for floating point, it seems like "for interval arithmetic,
modulate epsilon or the infinitesimals, keeping under bounds
1/epsilon or infinity, ... 'infinity computer', ...".

Or where that's so it's a thing, usual bounds, ....

Anyways Mitch here is an "intuitionist", for when
constructivism just won't do.

For us, ....

Anyways it's simply acknowledged the truism then
"so what, and how so??". Not "wrong".

Generally, not wrong.

[Ross A. Finlayson]

Gauss just knew that according to limits he would never be proven wrong, ....

Or, you know, figure that other people would "know" that.

[mitchr...@gmail.com]

Sizes of infinity are made by their inverse or an infinity
of the infinitesimal. That also builds all integers the same.

[Jim Burns]

On 5/28/2022 1:38 PM, WM wrote:
> Jim Burns schrieb
> am Samstag, 28. Mai 2022 um 19:25:16 UTC+2:

>> There are reasons that are enough for many people
>> to use ZFC as a foundation.
>
> They have to hypnotize themselves, like you,
> that by purely exchanging X's and O's in

...like this...

X:X O:X O:X O:X ...
X:X O:X O:X O:X ...
X:X O:X O:X O:X ...
X:X O:X O:X O:X ...
X:X O:X O:X O:X ...
X:X O:X O:X O:X ...
X:X O:X O:X O:X ...
X:X O:X O:X O:X ...
X:X O:X O:X O:X ...
... ... ... ...

> the whole matrix can be emptied of all O's.

Hold my beer.

1/1:1/1 1/2:2/1 1/3:4/1 1/4:7/1 ...
2/1:3/1 2/2:5/1 2/3:8/1 2/4:12/1 ...
3/1:6/1 3/2:9/1 3/3:13/1 3/4:18/1 ...
4/1:10/1 4/2:14/1 4/3:19/1 4/4:25/1 ...
5/1:15/1 5/2:20/1 5/3:26/1 5/4:33/1 ...
6/1:21/1 6/2:27/1 6/3:34/1 6/4:42/1 ...
7/1:28/1 7/2:35/1 7/3:43/1 7/4:52/1 ...
8/1:36/1 8/2:44/1 8/3:53/1 8/4:63/1 ...
9/1:45/1 9/2:54/1 9/3:64/1 9/4:75/1 ...
... ... ... ...

[Ross A. Finlayson]

I'm very curious Birkhoff, Ramsey, Balmer, there are
some very great numbers in the series: what surprise
to show up under estimation, what is a usual guess.

So, if you could tell me about your interest in the "atomic
spectrum of hydrogen" it's interesting all the terms.

Mostly the spectrum analysis is here: when I think of physics
and "there is an algorithm and it is quant", it's that the channels
or signal, of the spectrum, is about key components in spectrum
as what are also the values in the field.

[David Petry]

On Saturday, May 28, 2022 at 5:08:42 AM UTC-7, WM wrote:

> "The result of the preceding discussion is that our axioms, if interpreted as meaningful statements, necessarily presuppose a kind of Platonism, which cannot satisfy any critical mind and which does not even produce the conviction that they are consistent. [K. Gödel: "The present situation in the foundations of mathematics" (1933) in S. Feferman et al. (eds.): "Kurt Gödel, collected works, Vol. III, unpublished essays and lectures", Oxford Univ. Press, (1995) p. 50]

Do you know who wrote that? Was it Godel, or Feferman? It sounds to me like something Feferman would say, and not like something Godel would say, but if I'm wrong, I'd like to know.

[David Petry]

On Saturday, May 28, 2022 at 1:57:15 AM UTC-7, Alan Mackenzie wrote:

> David Petry <david...@gmail.com> wrote:


> > It happened a number of times in this newsgroup, about 25 years ago or
> > so, that someone with a PhD in mathematics would insinuate that the
> > only conceivable reason someone would be opposed to Cantor's theory
> > being in mathematics is antisemitism.

> All sorts of foul things get said in this newsgroup, and I'm sure that
> was the case 25 years ago too, if perhaps to a lesser degree. But has
> any mathematician said "evil" about it in polite discussion, say in an
> academic journal?

The people who said things like that were not people who were in the habit of saying foul things.

I've never seen the ideas I promote discussed in a journal.

> So if you have any concrete ideas, why not publish them here for review
> and criticism?

I'm tempted to say, "you've got to be kidding", but you're not, are you? The people here seem unable to grasp the very simplest of ideas that I'm promoting.

[sergi o]

On 5/28/2022 11:40 PM, David Petry wrote:
> On Saturday, May 28, 2022 at 1:57:15 AM UTC-7, Alan Mackenzie wrote:
>> David Petry <david...@gmail.com> wrote:
>
>
>>> It happened a number of times in this newsgroup, about 25 years ago or
>>> so, that someone with a PhD in mathematics would insinuate that the
>>> only conceivable reason someone would be opposed to Cantor's theory
>>> being in mathematics is antisemitism.
>
>> All sorts of foul things get said in this newsgroup, and I'm sure that
>> was the case 25 years ago too, if perhaps to a lesser degree. But has
>> any mathematician said "evil" about it in polite discussion, say in an
>> academic journal?
>
> The people who said things like that were not people who were in the habit of saying foul things.
>
> I've never seen the ideas I promote discussed in a journal.

of course.

>
>
>> So if you have any concrete ideas, why not publish them here for review
>> and criticism?
>
> I'm tempted to say, "you've got to be kidding", but you're not, are you? The people here seem unable to grasp the very simplest of ideas that I'm promoting.
>
>
>

what ideas are you promoting ? and can you be specific ?

[David Petry]

On Saturday, May 28, 2022 at 10:25:16 AM UTC-7, Jim Burns wrote:
> On 5/28/2022 1:15 AM, David Petry wrote:
> > On Friday, May 27, 2022 at 3:52:15 PM UTC-7,
> > Jim Burns wrote:
>
> >> It should be clear to anyone familiar with
> >> science-and-technology mathematics that its various
> >> parts _can be_ treated within a Cantor-compliant
> >> set theory such as ZFC.
> >
> > That's not actually true.

> No, that's actually true.

I think I do understand what you are saying, and I think you do not understand what I'm saying. And I think that what you are saying is trivial to the point of being silly, and I think that I am presenting a new idea that would be of great value to society if I could get society to accept it.

Here's what I think you are saying:

1) Mathematics is by definition something that is consistent with ZFC.
2) Hence, when scientists use mathematics, they are using something that is consistent with ZFC.
3) If and when scientists use reasoning that cannot be formalized within ZFC, they are not using mathematics.
4) There's absolutely no possible reason to change anything.

Here's what I am saying:

1) Scientific reasoning can be formalized in such a way that every abstract principle used in scientific reasoning would be part of that formalization.
2) The key to doing this is the principle of "observability", as I have described it in the article that started this thread.
3) A full formalization of scientific reasoning would not be consistent with ZFC.
4) It would be eminently reasonable to equate "mathematics" with the formalization of scientific reasoning.
5) Mathematics as the formalization of scientific reasoning is something that a great many people can apply in their professions, whereas "mathematics" defined to be the implications of ZFC is of virtually no use to anyone who does productive work.
6) ZFC is an impediment to progress in science and technology, especially AI.

If you (JB) can't prove to me that you understand what I am saying, there will be no reason for me to continue engaging with you.

[David Petry]

On Saturday, May 28, 2022 at 12:51:02 PM UTC-7, Jim Burns wrote:


> We already have real numbers with errors.
> For example, from January 8, 2021, I see a report
> of the age of the universe as 13.77 ± .04 billion
> years.

Exactly. And I'm saying that keeping track of those errors is so important that it would be reasonable to say that finite precision reals along with error estimates should be the kind of "real" numbers upon which analysis is built, and the foundational logic of mathematics should be able to deal the inherent uncertainties.

[FromTheRafters]

David Petry used his keyboard to write :

This might help:

https://ebrary.net/48217/mathematics/godel_incompleteness_metaphysics_arithmetic

[WM]

Jim Burns schrieb am Sonntag, 29. Mai 2022 um 00:48:12 UTC+2:
> On 5/28/2022 1:38 PM, WM wrote:
> > Jim Burns schrieb
> > am Samstag, 28. Mai 2022 um 19:25:16 UTC+2:
> >> There are reasons that are enough for many people
> >> to use ZFC as a foundation.
> >
> > They have to hypnotize themselves, like you,
> > that by purely exchanging X's and O's in
> ...like this...
>
> X:X O:X O:X O:X ...
> X:X O:X O:X O:X ...
> X:X O:X O:X O:X ...
> X:X O:X O:X O:X ...
> X:X O:X O:X O:X ...
> X:X O:X O:X O:X ...
> X:X O:X O:X O:X ...
> X:X O:X O:X O:X ...
> X:X O:X O:X O:X ...
> ... ... ... ...
> > the whole matrix can be emptied of all O's.
> Hold my beer.

Drink it. Maybe you get clear.

>
> 1/1:1/1 1/2:2/1 1/3:4/1 1/4:7/1 ...
> 2/1:3/1 2/2:5/1 2/3:8/1 2/4:12/1 ...
> 3/1:6/1 3/2:9/1 3/3:13/1 3/4:18/1 ...
> 4/1:10/1 4/2:14/1 4/3:19/1 4/4:25/1 ...
> 5/1:15/1 5/2:20/1 5/3:26/1 5/4:33/1 ...
> 6/1:21/1 6/2:27/1 6/3:34/1 6/4:42/1 ...
> 7/1:28/1 7/2:35/1 7/3:43/1 7/4:52/1 ...
> 8/1:36/1 8/2:44/1 8/3:53/1 8/4:63/1 ...
> 9/1:45/1 9/2:54/1 9/3:64/1 9/4:75/1 ...
> ... ... ... ...

Exchanging the first few terms may feign such a matrix. But you cannot complete it. Mathematics however *proves* that the O's will never disappear - how far you may extend to attempts.

Regards, WM

[WM]

According to Feferman it was Gödel in 1933, but I have not checked the original. I am sure though that Feferman did not lie.

Regards, WM

[Alan Mackenzie]

OK, thanks for saying something definite.

Doesn't the maths which keeps track of these errors exist already?

I think you are proposing that our current body of knowledge known as
real analysis should be swept away and replaced by something new which in
place of exact values has approximations.

If I'm right here, why would this new notion of real numbers have to
replace the standard one, rather than existing alongside of it? Would
there, for example, be irreconcilable contradictions between the two
versions of real number?

--
Alan Mackenzie (Nuremberg, Germany).

[Python]

Petry's idiotic claim is especially pathetic because construtivist
real analysis exists:

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

David is not even remotely interested. He has no interest in
math, actually. He is just frustated to have been drop out of
academia and is trolling this place for decades with stupid,
nonsensical rant.

[sergi o]

you are out of math.

> Mathematics however *proves* that the O's will never disappear

Your Mathematics cannot *prove* anything, it is fake math.



>
> Regards, WM

[sergi o]

On 5/29/2022 12:03 AM, David Petry wrote:
> On Saturday, May 28, 2022 at 10:25:16 AM UTC-7, Jim Burns wrote:
>> On 5/28/2022 1:15 AM, David Petry wrote:
>>> On Friday, May 27, 2022 at 3:52:15 PM UTC-7,
>>> Jim Burns wrote:
>>
>>>> It should be clear to anyone familiar with
>>>> science-and-technology mathematics that its various
>>>> parts _can be_ treated within a Cantor-compliant
>>>> set theory such as ZFC.
>>>
>>> That's not actually true.
>
>> No, that's actually true.
>
>
> I think I do understand what you are saying, and I think you do not understand what I'm saying. And I think that what you are saying is trivial to the point of being silly, and I think that I am presenting a new idea that would be of great value to society if I could get society to accept it.
>
> Here's what I think you are saying:

no.

>
> 1) Mathematics is by definition something that is consistent with ZFC.

you generalize too much, "by definition" is bogus

> 2) Hence, when scientists use mathematics, they are using something that is consistent with ZFC.

you generalize too much

> 3) If and when scientists use reasoning that cannot be formalized within ZFC, they are not using mathematics.

wrong.

> 4) There's absolutely no possible reason to change anything.

your weird conclusion.

>
> Here's what I am saying:
>
> 1) Scientific reasoning can be formalized in such a way that every abstract principle used in scientific reasoning would be part of that formalization.

wrong, too general.

> 2) The key to doing this is the principle of "observability", as I have described it in the article that started this thread.

that is physics, not math.

> 3) A full formalization of scientific reasoning would not be consistent with ZFC.

too general, show me evidence.

> 4) It would be eminently reasonable to equate "mathematics" with the formalization of scientific reasoning.

wrong. math is a language

> 5) Mathematics as the formalization of scientific reasoning is something that a great many people can apply in their professions, whereas "mathematics" defined to be the implications of ZFC is of virtually no use to anyone who does productive work.

wrong.

> 6) ZFC is an impediment to progress in science and technology, especially AI.

wrong, another general statement. Show me how all of AI depends on ZFC....

>
> If you (JB) can't prove to me that you understand what I am saying, there will be no reason for me to continue engaging with you.

you dont understand your own subject.

>
>
>

[sergi o]

agree => stupid, nonsensical rants

[Alan Mackenzie]

David Petry <david...@gmail.com> wrote:
> On Saturday, May 28, 2022 at 1:57:15 AM UTC-7, Alan Mackenzie wrote:
>> David Petry <david...@gmail.com> wrote:


>> > It happened a number of times in this newsgroup, about 25 years ago
>> > or so, that someone with a PhD in mathematics would insinuate that
>> > the only conceivable reason someone would be opposed to Cantor's
>> > theory being in mathematics is antisemitism.

>> All sorts of foul things get said in this newsgroup, and I'm sure that
>> was the case 25 years ago too, if perhaps to a lesser degree. But has
>> any mathematician said "evil" about it in polite discussion, say in an
>> academic journal?

> The people who said things like that were not people who were in the
> habit of saying foul things.

Without seeing the context from 25 years ago, it's difficult to judge why
they might have said such things - whether in jest, or exasperation, or
whatever.

> I've never seen the ideas I promote discussed in a journal.

I'm trying to get you to discuss them here. ;-) If you find set theory,
or part of it, as the basic mathematical foundation in some way bad, why
not try (?again) to say why, and to suggest what might replace it.

Up till now, I think you've just said that you find it unaesthetic; that
it's not useful for science. I think you'll need some justification that
applies purely to mathematics; some reason that set theory just doesn't
work very well for mathematics, and some inkling of what might work
better.

>> So if you have any concrete ideas, why not publish them here for
>> review and criticism?

> I'm tempted to say, "you've got to be kidding", but you're not, are you?

Definitely not.

> The people here seem unable to grasp the very simplest of ideas that
> I'm promoting.

My feeling, having read quite a lot of your posts, is that you haven't
been very specific on what you're promoting. In another post on this
thread, though, you have stated you want to replace real analysis with
some new form of real numbers which are approximate rather than exact.
But real numbers are quite a way up the stack of classes of number
currently constructed from set theory. Do you have ideas how we might
formulate natural numbers, integers, rationals, ....., complex numbers,
.... without using set theory?

That would surely be worth enduring the flack that anybody who actually
does anything here has to endure.

[WM]

Exchanging an X and an O from X.....O to O.....X will not make any of them disappear and will not make any of them multiply. It will simply maintain an X and an O. I would never have believed before that matheologians are dishonest enough (because not even an earthworm can be stupid enough) to contradict this simple fact.

Regards, WM

[Ross A. Finlayson]

Somehow it didn't surprise me that I had completely trampled my side of the fence, ....

It being light on that side though my marks were soon gone, ....

Of course something like "Foundations of the Continuous and Discrete" for
mathematics includes any all this infinite and infinitesimals, numbers,
these days it's scattered all over reference from antiquity, for what
some few "axioms" like inverse and otherwise for laws make examples
of logic in the continuous and discrete that are simple and free of otherwise
conclusion, while at the same time framed all modern mathematics.

This is letters, ....

[JVR]

An earthworm can hardly be dumber than McMuck, but it can certainly be longer. Suppose, for the sake of argument,
that there is an earthworm exactly as long as the Hotel in Hilbertsheim. Could such an earthworm be persuaded that
it is exactly as long as two other equally long earthworms?
Definitely not. Therefore, McMuck is right: Anybody who thinks he can turn 1, 2, 3, .... or infinitely many X's into O's
is playing 3-card-monty.

[sergi o]

typically, your references are bad, and do not apply. They are red herrings

You even admit you did not read the above reference.

[WM]

david...@gmail.com schrieb am Sonntag, 29. Mai 2022 um 07:03:13 UTC+2:

> 1) Scientific reasoning can be formalized in such a way that every abstract principle used in scientific reasoning would be part of that formalization.
> 2) The key to doing this is the principle of "observability", as I have described it in the article that started this thread.
> 3) A full formalization of scientific reasoning would not be consistent with ZFC.
> 4) It would be eminently reasonable to equate "mathematics" with the formalization of scientific reasoning.
> 5) Mathematics as the formalization of scientific reasoning is something that a great many people can apply in their professions, whereas "mathematics" defined to be the implications of ZFC is of virtually no use to anyone who does productive work.
> 6) ZFC is an impediment to progress in science and technology, especially AI.
>

7) Return to meaning and truth as the essence of mathematics. (From https://ebrary.net/48217/mathematics/godel_incompleteness_metaphysics_arithmetic.)

Regards, WM

[Ross A. Finlayson]

AMS document is what is called an AMS document,
it's American Mathematical Society, very usual: most
all documents are templates off AMS documents.

Articles, ....

[Timothy Golden]

On Saturday, May 28, 2022 at 6:29:05 PM UTC-4, Ross A. Finlayson wrote:
> On Friday, May 27, 2022 at 8:57:27 AM UTC-7, timba...@gmail.com wrote:
> > On Friday, May 27, 2022 at 7:27:15 AM UTC-4, david...@gmail.com wrote:
> > > First of all, anyone who claims that Cantor's set theory is in any way essential for the mathematics that is relevant to science and technology simply doesn't know what he's talking about.
> > >
> > > Here's a fast and loose history of mathematical infinity:
> > >
> > > For thousands of years mathematicians accepted the idea that mathematics is closely related to science, and that since the notion of a completed (or "actual") infinity is not part of science, it is also not part of mathematics.
> > >
> > > The mathematicians did, however, accept the idea of a potential infinity. Here's the intuition behind that:
> > >

> > > We can think of infinity as a destination that can never be reached. That is, no matter how far we go in the direction of infinity, there's still an infinite distance beyond that to go. Note that this is true in science, and hence it must be true in mathematics if mathematical reasoning is to be compatible with scientific reasoning.
> > >

Thanks Ross for at least responding, though you didn't touch the infinite instances.
Quantization of space fails to me when we simply ponder rotation.
Take a mug on a table, rotate is around a few times, and ponder the positions of its parts (atoms) as engaged within a quantized space. There is no way to account for the rearrangement of the atoms of the solid object. The mug should not have conserved its geometry; even under a partial turn. Entertaining this method seriously for a moment the work done in rearranging those molecules should come out as work, so maybe through some turns of calculus you might wind up with a form of momentum, but really it won't work out on face value. The mug should just drag to a halt under this work theory, so even this fails to hold with physical correspondence.

The existence of solids is somewhat built into all of our understanding. Without solids we would have very little sensibility to gain. The ruler turned to fluid is no ruler at all. The straight edge will never be. Even a taught line takes no being. Those who wish to dismiss physical correspondence don't care about such arguments apparently, yet the foundations of counting and accountability rest upon a stable basis that allows us to progress as far as we've gotten, anyways. As to how far we have gotten; so long as arguments like this thread are still alive it seems that our primitive status is upheld.

The biggest fail is the lack of correspondence to the physical basis. This includes unidirectional time; not bidirectional time as Einstein formulated it. The real value as fundamental is a losing proposition. The two-signed number has swamped the flow. Time was a one-signed number. So what comes next? How about trying out the three-signed number? It works perfectly and you'll have geometry and the complex numbers in one fell swoop and there is no need to stop at three. Meanwhile the breakpoint makes itself known even as the usual algebraic properties hold. That a form of emergent spacetime arises here in the generalization of sign is the mark that makes polysign worthy. The complex value from the same laws that get the real value are a sure sign too.

I am trying to get to some version of quantized space-time-matter that would resolve the problem laid out above to do with rotation. Topology seems the nearest topic within mathematics, but I never found much convincing there. Dive into topology and pick it apart next?

[Graham Cooper]

The problem is using only known digits in the countable list.

0. 0 0 0
0. 1 1 1
0. 1 0 1

To define ANTI_DIAGONAL PROPERLY use UNKNOWN DIGIT [?]


0. [0] 0 0
0. 1 [1] 1
0. 1 0 [1]
0. 1 0 0 [?] <------------- ANTI-DIAGONAL

Just let AD have an unknown digit

------------------------------------------------------------------------------------------------


What is a LIST OF REALS# ?
What is TRI-STATE-COMPUTABILITY?
What reals are missing?

A REAL# is a sequence beginning 0.abcd...
where abcd e { 0 , 1 , ? }

Thats right! We've changed the definition of computable
to include non-terminating algorithms.

Given a LIST OF normal reals abcd e {0 ,1}

R (standard)
0.0010...
0.1100...
0.0101...
...

The DIAGONAL D = 0.010...

Let a LIST OF TURING MACHINES calculate each row

TM1 = 0.0010...
TM2 = 0.1100...
TM3 = 0.0101...
...

Can a TURING MACHINE calculate the ANTIDIAGONAL AD ?

AD = 0.101...

Assume TM4 is the ANTI-DIAGONAL

TM1 = 0.0010...
TM2 = 0.1100...
TM3 = 0.0101...
TM4 = 0.101[?]

What is the VALUE of R_4_4 [?]

If R_4_4 is 0 then TM4_4 should output 1
If R_4_4 is 1 then TM4_4 should output 0

By the poor definition of AD its impossible for any row to equal it
or any computer program to compute it?

But that's ABSURD, its simple for a Turing Machine to emulate all the
other turing machines and flip the digits 0-->1 and 1-->0

The error is in the DEFINITION of ANTI-DIAGONAL, its like "who shaves the barber"? an impossible question.

A TM could output 0.1010...
and another could output 0.1011...

so the AD is computable as a PAIR.

-----------------------------------------------------

What is TRI-STATE computability ?

A TM outputs 1 for TRUE
and any other output for FALSE

That is 2-STATE computability.

But what about WEB SERVERS that never terminate?
They are computing too, so TRI-STATE-COMPUTABILITY
includes a non-halting TM as a "computed" result.

Consider computations running for INFINITE TIME and you have
a theoretical computer result.

Considering there are NON COMPUTABLE REALS (chaitans omega, busy beaver) these actually have results at INFINITE TIME

In fact if we GUESSED the value of BUSY BEAVER for the size of the turing machine we could calculate those values

What is TRI-STATE-COMPUTABLE useful for?

1 you can program a HALT FUNCTION.

LOOPS(X)
if HALT(X X) LOOPS(X)

HALT(loops,loops)
always gets it wrong, if loops(loops) halts it goes into a loop vice versa

BUT using TRI-STATE-COMPUTABILITY HALT(loops,loops) FAILS TO HALT ITSELF

The small set of self contradiction programs (halt, busy beaver, omega)
run forever, this is a nearly detectable result state.


So how do you list ANTI-DIAGONAL, CHAITANS OMEGA and the BUSY BEAVER FUNCTION
(non computable) on a COMPUTABLE LIST OF REALS?

Use TRI-STATE REALS#

R#
0.0010..
0.1100..
0.0011..
0.100?..
...


NOTICE THE ?
Its like a REAL DIGIT using TRI-STATE-REALS

But if no TM can compute CHAITANS OMEGA, how can you list it with
a list of TURING MACHINES?

This is the EUREKA PART! (8-D


A TURING MACHINE that is allowed to NOT HALT on certain values
can list all the other HALT VALUES (and in infinite time only the self
contradictory non-halting values are left over)


TM5 0. 1 0 1 0 ? 0 ? 0 0 0 ? 1 1 ? .... <-------- CHAITANS' OMEGA

TM5 is not CHAITANS OMEGA per se
but by giving NON HALTING VALUES [?]
it is a _SPECIFICATION_ OF THE NUMBER.


So TRI-STATE-COMPUTABLES can list all DEFINITIONS of Reals

What is the ANTI-DIAGONAL you ask? ITS A NON FULLY DEFINED REAL
so it just has 1 GAP

TM4 0. 1 0 1 ? 1 0 1 0 1 1 ... <---------- ANTI-DIAGONAL


CONCLUSION

TRI-STATE REALS can list all non computable functions (as Turing Machine Specifications that run for infinite time) they will have GAPS [?]

thinking of real numbers with just peano digits in a computable paradigm
leads to contradiction, computers sometimes dont halt as part of
the programs definition, so no wonder you get |R|>oo nonsense thinking
of reals as only having digits.

R_n_digit = 0 | 1 | ?

[Jim Burns]

On 5/29/2022 1:03 AM, David Petry wrote:

> Here's what I think you [JB] are saying:

>
> 1) Mathematics is by definition something that is
> consistent with ZFC.

No.
I am saying that
the mathematics which you consider to be
all the mathematics
are in fact all theorems of ZFC.

The proofs of these theorems are the work of
various mathematicians. The existence of the proofs
is not mystical.

( I don't think how you're using "consistent"
( is how I'm using it. For clarity's sake,
( I'm avoiding using "consistent".

> 2) Hence, when scientists use mathematics,
> they are using something that is consistent with ZFC.

No.
When scientists use mathematics, we know that their
conclusions are justifiable, assuming the axioms of
ZFC.

It didn't have to be ZFC that got assumed.

I think that the point of one foundation for all of
that is to minimize the assumptions from which we
work. Less is more, in this case.

I think that scientists use which mathematics
they use because
(i) they find it useful, for science-y reasons.
(ii) it's been okayed by Those People Over There
for whom mathematics is a life's-work, not merely
a convenient-to-the-hand wrench or soldering iron.

> 3) If and when scientists use reasoning that
> cannot be formalized within ZFC,
> they are not using mathematics.

No.

There are more things in heaven and earth, Horatio,
than are dreamt of in your mathematics.

( Weird.
( Isn't "That's not mathematics" _your_ shtick?

> 4) There's absolutely no possible reason to
> change anything.

No.
You (DP) have given no reason to change anything.

What counts as a reason to change anything
is a judgment I get to make when I do scientific
reasoning. It is a judgment which others get to
make when they do scientific reasoning.

This arrangement has some nice features.
Those who make these judgments
(i) know the circumstances
(ii) will bear the consequences, good or bad.

I would want any "improvement" on the current
arrangement to have these features.
Or a really good argument why the "improvement"
is even better without them.

> Here's what I am saying:
>
> 1) Scientific reasoning can be formalized in
> such a way that every abstract principle used in
> scientific reasoning would be part of that
> formalization.

I read "can be" here as you saying "but not yet".
Feel free to clarify that.

I don't think all of scientific reasoning can be
formalized.
I think that it's good that it can't be.

Formal systems have some very nice strengths.
Those strengths are not all the strengths.
I don't see why we can't have it all.

I'm certain you (DP) haven't told me why we can't.
Until then, no.

> 2) The key to doing this is the principle of
> "observability", as I have described it in the article
> that started this thread.

Observables are very important in quantum mechanics.
A lot of interesting experimental work has been done in
the last 32 years on the foundations of quantum mechanics.

For example, we have learned how to crack open
the box holding Schroedinger's cat (rubidium atom)
_just the tiniest bit_ -- just enough for a _partial_
collapse into definite states.

If you're thinking of something quantum, in addition to
the other objections I have, you're way too early
on putting this developing field into foundations.


There's another principle I would like to set on the
table next to yours: the principle of _uniformity_
For some collections, for some claims, that claim is
_uniformly_ true of each individual in the collection.

Consider the collection of things finitely-many '+1'
from 0. We finite beings can't _observe_ all of that
collection. However, we can _reason about_ all of
that collection, starting with the claim about j
(which is in the collection) that j is finitely-many
'+1' from 0.

We derive further claims which are also uniformly
true of that collection. We know they are because
we only use inferences which preserve that uniform
truth when we derive them.

> 3) A full formalization of scientific reasoning
> would not be consistent with ZFC.

Since the status of a full formalization of scientific
reasoning is "some day, I hope, but not yet",
this is (Harry Frankfurt's) bullshit ==
worse than a lie.

> 4) It would be eminently reasonable to equate
> "mathematics" with the formalization of
> scientific reasoning.

Since what "mathematics" means is currently something
you want to change, it strikes me as less than useful
to use "mathematics" as though everyone knows what's
meant by it.

I suggest that you equate "snarfblargle" with the
formalization of scientific reasoning. That would
reduce the chance of someone thinking _incorrectly_
that they knew what you were saying.

> 5) Mathematics as the formalization of scientific
> reasoning is something that a great many people
> can apply in their professions,

Scientists reason about science.
Philosophers of science reason about
scientific reasoning.

What I expect is that a vanishing few beyond
philosophers of science will care at all.
And that much assumes this formalization will exist
sometime in the future.

> whereas "mathematics" defined to be the implications
> of ZFC is of virtually no use to anyone who does
> productive work.
>
> 6) ZFC is an impediment to progress in science and
> technology, especially AI.

If AI is to progress (and I expect it will),
it will need to deal with unformalized knowledge.

Inch by inch, workers in AI are learning to do this.
Consider machine learning.

> If you (JB) can't prove to me that you understand
> what I am saying, there will be no reason for me
> to continue engaging with you.

My hope is that I have set a good example for you
to follow. You have clearly not understood what
I've been saying. Did I cease engaging with you?
No. I tried to explain what I've been saying.

The technical term for this is "conversation".

[Jim Burns]

For each m/n in the matrix,
k/1 is in the first column of the matrix.

k = (m+n-1)*(m+n-2)/2+m

What is in the matrix are all and only m/n,
for each BEFORE and AFTER =< m
and each BEFORE and AFTER =< n,
some j ends BEFORE and j+1 begins AFTER.

For each BEFORE and AFTER =< k,
some j ends BEFORE and j+1 begins AFTER,
thus k/1 is in the first column.

> Mathematics however *proves* that the O's
> will never disappear - how far you may
> extend to attempts.

Infinity is not a reallyreallyreallyreallyreallyreally
large number. It is a different kind of thing.

k = (m+n-1)*(m+n-2)/2+m
proves it is a different kind of thing.

[WM]

Jim Burns schrieb am Sonntag, 29. Mai 2022 um 21:44:13 UTC+2:
> On 5/29/2022 1:03 AM, David Petry wrote:

> > 4) There's absolutely no possible reason to
> > change anything.
> No.
> You (DP) have given no reason to change anything.

Nobody could give more obvious evidence than this: Exchanging an X and an O from X.....O to O.....X will not make any of them disappear and will not make any of them multiply. It will simply maintain an X and an O.

But you are too dishonest to confess what you must know already. Therefore your words in this matter are hollow words.

Regards, WM

[WM]

Jim Burns schrieb am Sonntag, 29. Mai 2022 um 22:28:00 UTC+2:

> For each m/n in the matrix,
> k/1 is in the first column of the matrix.
>
> k = (m+n-1)*(m+n-2)/2+m

Only for such which have aleph_0 successors. These successors can be handle collectively
|ℕ \ {1, 2, 3, ...}| = 0 .
but not individually
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.

> Infinity is not a reallyreallyreallyreallyreallyreally
> large number. It is a different kind of thing.

Even in infinity exchanging an X and an O from X.....O to O.....X will not make any of them disappear and will not make any of them multiply. It will simply maintain an X and an O. Without this simple logic all other rules concerning the infinity would vanish in dust.

Regards, WM

[Ross A. Finlayson]

Reverse them, same?

Just find where they're reversible and where they do, it's the middle.

[Ross A. Finlayson]

When I went to college it was basically that the field was topology.

Though I really found the quest notion, as much as for "why topology"
as "that topology".

Then of course it's "that topology", ..., is agreeable.

It's suitably general for analysis, ....

[Jim Burns]

On 5/29/2022 4:55 PM, WM wrote:
> Jim Burns schrieb
> am Sonntag, 29. Mai 2022 um 22:28:00 UTC+2:

>> For each m/n in the matrix,
>> k/1 is in the first column of the matrix.
>>
>> k = (m+n-1)*(m+n-2)/2+m
>
> Only for such which have aleph_0 successors.

What is in the matrix are all and only m/n,
for each BEFORE and AFTER =< m
and each BEFORE and AFTER =< n,
some j ends BEFORE and j+1 begins AFTER.

For each BEFORE and AFTER =< k,
some j ends BEFORE and j+1 begins AFTER,

thus k/1 is in the first column.

> These successors can be handle collectively
> |ℕ \ {1, 2, 3, ...}| = 0 .
> but not individually
> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
>
>> Infinity is not a reallyreallyreallyreallyreallyreally
>> large number. It is a different kind of thing.
>
> Even in infinity exchanging an X and an O from
> X.....O to O.....X will not make any of them disappear
> and will not make any of them multiply. It will simply
> maintain an X and an O. Without this simple logic
> all other rules concerning the infinity would
> vanish in dust.

No swap disappears an O.

No swap moves an O out of the matrix.

After all the swaps, no O is in the matrix.

Infinity is not a reallyreallyreallyreallyreallyreally
large number. It is a different kind of thing.

That's what we're proving here.

[zelos...@gmail.com]

fredag 27 maj 2022 kl. 13:27:15 UTC+2 skrev david...@gmail.com:
> First of all, anyone who claims that Cantor's set theory is in any way essential for the mathematics that is relevant to science and technology simply doesn't know what he's talking about.
>
> Here's a fast and loose history of mathematical infinity:
>
> For thousands of years mathematicians accepted the idea that mathematics is closely related to science, and that since the notion of a completed (or "actual") infinity is not part of science, it is also not part of mathematics.
>
> The mathematicians did, however, accept the idea of a potential infinity. Here's the intuition behind that:
>
> We can think of infinity as a destination that can never be reached. That is, no matter how far we go in the direction of infinity, there's still an infinite distance beyond that to go. Note that this is true in science, and hence it must be true in mathematics if mathematical reasoning is to be compatible with scientific reasoning.
>
> The claim behind the notion of an actual infinity is that even though we cannot in any meaningful sense actually reach infinity, we can talk about infinity as if we could reach it. But that has nothing to do with scientific reasoning, and hence should not be part of mathematical reasoning.
>
> And Gauss, who is arguably the greatest mathematician to have ever lived, summed it up nicely by saying, "the actual infinite has no role to play in mathematics. In mathematics, infinity is nothing more than a figure of speech that mathematicians find to be very useful when reasoning about limits". (that's a paraphrase)
>
> And if it's not obvious what Gauss meant by that, here's a way to think about it that should be intuitively appealing: when we think about infinity, we should think about what happens along the journey towards infinity, but it's meaningless to ask what happens when we actually get there.
>
> So Cantor claimed that Gauss and all the mathematicians before him were wrong about infinity. He claimed that the only requirement that his theory of infinity needs to meet in order to be accepted as part of mathematics is that the theory be formally consistent with the parts of mathematics that had been formalized by the middle of the nineteenth century. And evidently he claimed that there was no need for his theory to be consistent with the parts of mathematical reasoning that was done informally at that time. In particular, the connection between mathematical reasoning and scientific reasoning had not been formalized by the middle of the nineteenth century, and he felt that mathematicians should ignore that part of mathematical reasoning.
>
> It seems that it was mostly Hilbert's endorsement of Cantor's theory that led mathematicians to accept the theory as part of mathematics. And to justify it, they claimed that mathematics is really an art, and not part of science.
>
> What I have been claiming for over 32 years now, is that there have been some major advances in our understanding of scientific reasoning over the past century, and that these advances in our understanding of scientific reasoning could be formalized and integrated into mathematics at the foundational level. And I genuinely believe that eventually mathematicians will accept that, and it will be seen as a major advance in our understanding of mathematics.
>
> The relevant key advance in our understanding of scientific reasoning is what has been called the "simulation hypothesis". This says that everything we regard as part of reality (both the physical world and the world of the mind) can be simulated on a digital computer. So what I claim is that this leads to the idea that we can think of the digital computer as a "microscope" that lets a peer deeply into a world of computation, and then we can define mathematics to be the science that studies the phenomena and patterns we observe when we peer through that microscope. I've chosen to call this idea "observability", and it gives us a way to formalize a link between scientific reasoning and mathematical reasoning. This "science of phenomena observable in the world of computation" includes all of the mathematics that is relevant to science and technology.
>
> There's something actually evil about insisting that mathematics is an art and not a science. First, art always includes an element of deception. And Cantor's theory of infinite set includes an element of deception. That's not a good thing
>
> Furthermore, the mathematicians who insist that mathematics must be viewed as an art, will claim that anyone (such as myself) who wants to build a foundation for mathematics that incorporates the idea of observability (i.e. meaningful mathematical assertions must have observable implications) is not even doing mathematics and is not welcome in the mathematics community. And sometimes they go so far as to claim that what I want to do is evil, and that I must be motivated by a desire to deprive the mathematicians of their intellectual freedom. And what they accuse me of doing is what they are doing.

EVerything you wrote here is wrong.

Get over yourself, mathematics is better today than it has ever been.

[Fritz Feldhase]

On Sunday, May 29, 2022 at 11:57:04 PM UTC+2, Jim Burns wrote:
>
> After all the swaps, no O is in the matrix.

Actually, there is no matrix "after all the swaps", you silly crank.

WM is just considering the matrices which are terms of a certain series.

There is no other relevant matrix in this context. (Note that supertasks aren't parts of mathematical/set theoretic arguments/proofs.)

That's why HE is right and you are talking nonsens.

WM> Exchanging an X and an O from
WM> X.....O to O.....X will not make any of them disappear
WM> and will not make any of them multiply. It will simply
WM> maintain an X and an O.

Indeed.

> After all swaps

Nonsense.

[Jim Burns]

On 5/30/2022 1:57 AM, Fritz Feldhase wrote:
> On Sunday, May 29, 2022 at 11:57:04 PM UTC+2,
> Jim Burns wrote:

>> After all the swaps, no O is in the matrix.
>
> Actually, there is no matrix "after all the swaps",
> you silly crank.

after all the swaps, P(k) ==
∀i ∈ ⟨1,2,3,...⟩, ∃j>=i, ∀k>=j, P(k)

all the swaps ==
⟨ swap(1), swap(2), swap(3), ... ⟩

matrix(i+1) = swap(i)(matrix(i))

∀i ∈ ⟨1,2,3,...⟩, ⟨1,...,i⟩ is finite

⟨1,...,i⟩ is finite ==
for each BEFORE and AFTER in ⟨1,...,i⟩

some j ends BEFORE and j+1 begins AFTER

> WM is just considering the matrices which are
> terms of a certain series.

So am I. That's my point.

WM is considering (vaguely) a certain series of
matrices in order to wave his hands and declare them
"contradictory", thus "proving" dark numbers.

I am de-vaguifying that series, that same series,
with no dark numbers, in order to show that the
counter-intuitive pseudo-contradictory claims
are properties of the "good" finitely-distanced
swaps, matrices, numbers.

Infinitely-many finitely-distanced things.
The pseudo-contradictions are between
things up to a thing (finite) and
all the things (infinite).

And you are...
I have no idea what you're doing.
Did you get bored with mathematics?

> There is no other relevant matrix in this context.
> (Note that supertasks aren't parts of
> mathematical/set theoretic arguments/proofs.)
>
> That's why HE is right and you are talking nonsens.
>
> WM> Exchanging an X and an O from
> WM> X.....O to O.....X will not make any of them

> WM> disappear and will not make any of them multiply.
> WM> It will simply maintain an X and an O.

>
> Indeed.
>
>> After all swaps
>
> Nonsense.

Up to each FINITELY-DISTANCED swap,
each O is in the matrix.

For each O, for each place in the matrix,
for each of INFINITELY-MANY swaps,
there is a FINITELY-DISTANCED swap after it
after which that O is never in that place.

...because
for _all_ swaps,
any place with a swap-into has a swap-out-of,

but,
for swaps _up to_ some swap,
that's not so,
not all places with a swap-into
have a swap-out-of-and-up-to-that-swap.

[David Petry]

On Sunday, May 29, 2022 at 4:28:58 AM UTC-7, Alan Mackenzie wrote:
> David Petry <david...@gmail.com> wrote:
> > On Saturday, May 28, 2022 at 12:51:02 PM UTC-7, Jim Burns wrote:
>
>
> >> We already have real numbers with errors.
> >> For example, from January 8, 2021, I see a report
> >> of the age of the universe as 13.77 ± .04 billion
> >> years.
>
> > Exactly. And I'm saying that keeping track of those errors is so
> > important that it would be reasonable to say that finite precision
> > reals along with error estimates should be the kind of "real" numbers
> > upon which analysis is built, and the foundational logic of mathematics
> > should be able to deal the inherent uncertainties.

> I think you are proposing that our current body of knowledge known as
> real analysis should be swept away and replaced by something new which in
> place of exact values has approximations.

Not really. The theoretical development of the subject would be different. The idea is that real analysis would be developed in a way that is useful in science--i.e. science must always deal with uncertainty. But then, as an abstraction, we could consider the limit as the uncertainty goes to zero, and recover the parts of current real analysis that have the possibility of being relevant to science. The parts of real analysis that rely on set theoretic principles in an essential way would be lost.

[David Petry]

On Sunday, May 29, 2022 at 4:40:49 AM UTC-7, Python wrote:

> Petry's idiotic claim is especially pathetic because construtivist
> real analysis exists:
>
> https://en.wikipedia.org/wiki/Constructive_analysis
>
> David is not even remotely interested. He has no interest in
> math, actually. He is just frustated to have been drop out of
> academia and is trolling this place for decades with stupid,
> nonsensical rant.

Is it just me, or does Jean Pierre Messager (Python) come across as being a little bit haughty?

FWIW, everything he says about me is nonsense.

[David Petry]

Yes, that's absolutely right. I've written about that in a few of my essays.

(From https://ebrary.net/48217/mathematics/godel_incompleteness_metaphysics_arithmetic.)

Thank you for that link. It's cleared up a few things for me.

[FromTheRafters]

David Petry explained :

Since he posts as Python, why do you feel the need to name-drop? Are
you a stalker?

[David Petry]

On Sunday, May 29, 2022 at 12:44:13 PM UTC-7, Jim Burns wrote:
> My hope is that I have set a good example for you
> to follow.

Wow! That's all I can say. Wow!

[David Petry]

On Sunday, May 29, 2022 at 9:49:38 PM UTC-7, zelos...@gmail.com wrote:


> EVerything you wrote here is wrong.

Your witty one-liners always crack me up.

[David Petry]

On Monday, May 30, 2022 at 5:50:31 AM UTC-7, FromTheRafters wrote:

> Since he posts as Python, why do you feel the need to name-drop? Are
> you a stalker?

I believe that if you pay close attention, you will conclude that he is the stalker.

[Timothy Golden]

At some level isn't the diagonalization a play on dimensional reasoning?
If you have a 1D<->2D mapping then you as well have a 1D<->3D, 1D<->4D, and so forth.
As to which you actually are dealing in: this is known as the domain of discourse right?
At some level you have to explicitly state which of these things is the one that you are working on, which will entail as well the structure by which they got organized.

I guess in effect, as mesmerizing as it is to do the mapping, to claim its purity is false because the 'other' side is not free-standing. It had to be constructed up front. Otherwise you can't say which of the many, and some of those many have many means of organization. Still the number splitting is cool. It is mesmerizing. There are simpler ways to split numbers. Still, I think as we admit in more types into the value; for instance the decimal point and the sign, then we see better correspondence to the physical world. That these more complicated forms still sit on the back of the natural value I think is satisfying to those of the natural valued persuasion. That the radix point itself at some level is another natural value... hmmmm... I wonder would happen if you code that into your splitter? It's generally a small value relative to the magnitude. Sort of monkeying around, but realizing the radix point to be engaging in the same modulo principals as the magnitude, then pushing farther to an arbitrary unital you wind up with a new interpretation of the rational value. It is of course the dirty reradixer all over again; maybe just not quite so dirty. The interval is after all not in a fraction anymore. It's more like this:
"Let there be a secondary form of unity such that 1,234 base units are one secondary unit."
It's a mixed modulo system really. The rational value is nearby though it is down in the subunits. That's why this construction is slightly cleaner: it avoids the usage of division.

Modulo principals have already exposed support for spacetime through polysign analysis. This includes the unidirectional nature of time along with its confusing zero dimensional aspect. Try moving a mug around in time. Any luck? If not, that is zero dimensions of freedom, sir. And what exactly gets you three dimensional space? It's the same method. Unfortunately though we used points in the analysis. If we use mugs we do actually find more degrees of freedom. That our zero dimensional point required three coordinated values makes our zero dimensional point three dimensional, doesn't it? Ambiguity ensues once again. That these procedures occur in time as well we must not deny. That the mug analysis gets more then three degrees to specify it's relative position is discomforting. Here the solid awareness should have been invoked earlier. However, we still practice deference to Euclid. No doubt he would have been quicker on the draw here than we are. The accumulation as we are trained under threat of failure upon it does not add up. Some sort of general dimensional analysis or so forth has to be gotten to. Even the paper from which most have done their work must be accounted for and the symbols upon it. How we as elements of spacetime are dealing with these problems: mathematicians have never taken on the burden it seems. Yet if physics if born upon a basis then isn't that basis mathematical?

Now this can wrap back onto your dimensional creation, for all are pulling three dimensions out of a hat via empirical correspondence, and that for what they claim to be a zero dimensional point. None have troubled over time. And yet as prisoners of spacetime we are all caught within these limitations. I will hold to the belief that a semi-classical resolution can still exist. A place where philosophy, mathematics, and physics are not foreign to each other. Obviously the early steps are attempts. We need not bow so low to history. We must not bow so low. Every ambiguity is an opening.

[Alan Mackenzie]

David Petry <david...@gmail.com> wrote:
> On Sunday, May 29, 2022 at 4:28:58 AM UTC-7, Alan Mackenzie wrote:
>> David Petry <david...@gmail.com> wrote:
>> > On Saturday, May 28, 2022 at 12:51:02 PM UTC-7, Jim Burns wrote:

>> >> We already have real numbers with errors. For example, from
>> >> January 8, 2021, I see a report of the age of the universe as 13.77
>> >> ± .04 billion years.

>> > Exactly. And I'm saying that keeping track of those errors is so
>> > important that it would be reasonable to say that finite precision
>> > reals along with error estimates should be the kind of "real"
>> > numbers upon which analysis is built, and the foundational logic of
>> > mathematics should be able to deal the inherent uncertainties.

>> I think you are proposing that our current body of knowledge known as
>> real analysis should be swept away and replaced by something new which
>> in place of exact values has approximations.

> Not really. The theoretical development of the subject would be
> different. The idea is that real analysis would be developed in a way
> that is useful in science--i.e. science must always deal with
> uncertainty.

But science must deal with the ideal cases, from which there are
uncertain variations, too. Jim can reasonably expect to be able to
derive the spectrum of the hydrogen atom from its Hamiltonian, and if I
understood him correctly, that process depends on exactitude, rather than

uncertainty.

> But then, as an abstraction, we could consider the limit as the
> uncertainty goes to zero, and recover the parts of current real
> analysis that have the possibility of being relevant to science.

That begs the question whether these parts might constitute the whole of
real analysis.

> The parts of real analysis that rely on set theoretic principles in an
> essential way would be lost.

Is real analysis not a coherent whole? Could it be so separated into two
parts, leaving a "useful" part and the rest, and still be coherent?

Could you give an example of a part of real analysis which relies on set
theory "in an essential way", and which would be thus lost?

[sergi o]

On 5/29/2022 9:58 AM, WM wrote:
> sergi o schrieb am Sonntag, 29. Mai 2022 um 16:01:28 UTC+2:
>> On 5/29/2022 4:41 AM, WM wrote:
>

>>> Mathematics however *proves* that the O's will never disappear

>> Your Mathematics cannot *prove* anything, it is fake math.
>
> Exchanging an X and an O from X.....O to O.....X will not make any of them disappear and will not make any of them multiply. It will simply maintain an X and an O. I would never have believed before that matheologians are dishonest enough (because not even an earthworm can be stupid enough) to contradict this simple fact.
>
> Regards, WM
>

so now you are saying your "swaps" does nothing.

agree, it is slight of hand to mislead people.

you wont use math, equations to fully describe it, because your failure is obvious.

[FromTheRafters]

Timothy Golden laid this down on his screen :

It seems to me to be close to reasoning about space-filling curves too.

[WM]

Jim Burns schrieb am Sonntag, 29. Mai 2022 um 23:57:04 UTC+2:

> No swap disappears an O.
>
> No swap moves an O out of the matrix.
>
> After all the swaps, no O is in the matrix.

This is a contradiction, not even tolerable in fairy tales, let alone in mathematics. Infinity is not an excuse.

> Infinity is not a reallyreallyreallyreallyreallyreally
> large number. It is a different kind of thing.

Then the diagonal argument would not prove uncountable sets.

No line contains the antidiagonal number.
After all lines the antidiagonal number is contained.

Abrkadabra - Simsalabim

> That's what we're proving here.

Yes!

Regards, WM

[WM]

Fritz Feldhase schrieb am Montag, 30. Mai 2022 um 07:57:17 UTC+2:
> On Sunday, May 29, 2022 at 11:57:04 PM UTC+2, Jim Burns wrote:
> >
> > After all the swaps, no O is in the matrix.
> Actually, there is no matrix "after all the swaps", you silly crank.
>
> WM is just considering the matrices which are terms of a certain series.
>
> There is no other relevant matrix in this context. (Note that supertasks aren't parts of mathematical/set theoretic arguments/proofs.)

Not of mathematical but of set theoretical arguments.

The belief in the possibility to finish infinite bijections raises the paradoxical result that Adolf A. Fraenkel explained by Laurence Sterne's novel "The life and opinions of Tristram Shandy, gentleman": "Well known is the story of Tristram Shandy who undertakes to write his biography, in fact so pedantically, that the description of each day takes him a full year. Of course he will never get ready if continuing that way. But if he lived infinitely long (for instance a 'countable infinity' of years [...]), then his biography would get 'ready', because, expressed more precisely, every day of his life, how late ever, finally would get its description because the year scheduled for this work would some time appear in his life." [A. Fraenkel: "Einleitung in die Mengenlehre", 3rd ed., Springer, Berlin (1928) p. 24] "If he is mortal he can never terminate; but did he live forever then no part of his biography would remain unwritten, for to each day of his life a year devoted to that day's description would correspond." [A.A. Fraenkel, A. Levy: "Abstract set theory", 4th ed., North Holland, Amsterdam (1976) p. 30]

To have an example with a simpler ratio consider Scrooge McDuck who per day earns 10 $ and spends 1 $. The dollar bills are enumerated by the natural numbers. McDuck receives and spends them in natural order. If he lived forever he would go bankrupt by the same argument. (Using coins he would get rich.)

>
> That's why HE is right and you are talking nonsens.
>
> WM> Exchanging an X and an O from
> WM> X.....O to O.....X will not make any of them disappear
> WM> and will not make any of them multiply. It will simply
> WM> maintain an X and an O.
>
> Indeed.
>
> > After all swaps
>
> Nonsense.

In fact! But unfortuantely necessary to assume for a statement about the state after all swaps, i.e., the complete Abzählung.

Regards, WM

[WM]

Jim Burns schrieb am Montag, 30. Mai 2022 um 13:29:59 UTC+2:

> Up to each FINITELY-DISTANCED swap,
> each O is in the matrix.

And there are no other swapy.

>
> For each O, for each place in the matrix,
> for each of INFINITELY-MANY swaps,
> there is a FINITELY-DISTANCED swap after it
> after which that O is never in that place.
>
> ...because
> for _all_ swaps,
> any place with a swap-into has a swap-out-of,
>
> but,
> for swaps _up to_ some swap,
> that's not so,
> not all places with a swap-into
> have a swap-out-of-and-up-to-that-swap.

And all definable places belong to a finite domain. Thereafter the O's disappear from sight but not from the matrix. What you call disappearing O's is in fact O's getting dark.

Regards, WM

[FromTheRafters]

David Petry explained on 5/30/2022 :

He does seem overly antagonistic at times.

[sergi o]

look, just identify each fraction with the row and column indexes, m and n

now, generate the equation that maps these into a one dimensional array (a set) (m,n => k)

"swap in and swap out" is not math, it is WMs way to stay confused.

Sets, like the matrix of rationals, are not variables.

Use precision, write the equations.

lack of them indicate an aversion to math.

[Ross A. Finlayson]

Here, for, "reversing the swaps, their reversibility",
is "what is the hedge".

[FromTheRafters]

WM formulated the question :

> Jim Burns schrieb am Montag, 30. Mai 2022 um 13:29:59 UTC+2:
>
>> Up to each FINITELY-DISTANCED swap,
>> each O is in the matrix.
>
> And there are no other swapy.
>>
>> For each O, for each place in the matrix,
>> for each of INFINITELY-MANY swaps,
>> there is a FINITELY-DISTANCED swap after it
>> after which that O is never in that place.
>>
>> ...because
>> for _all_ swaps,
>> any place with a swap-into has a swap-out-of,
>>
>> but,
>> for swaps _up to_ some swap,
>> that's not so,
>> not all places with a swap-into
>> have a swap-out-of-and-up-to-that-swap.
>
> And all definable places belong to a finite domain.

Define "Definable places" and show how their domain must be finite.

Aren't your Xs and Os placed on a countably infinite by countably
infinite (N x N) table? Are you thinking that a subset of *all*
permutations of the table must be countable?

[FromTheRafters]

sergi o explained :

I remember reading somewhere that when writing a mathematical proof one
should use words rather than symbols where possible because one should
be able to explain the concept, through the proof, to the average
'Joe'.

Then again, the squareroot of two is an acceptable number now, things
change over time. WM just doesn't change with them and holds on to old
discarded notions.

[Ross A. Finlayson]

This idea about the Achilles and tortoise, and,
the _round_ track, and, the _straight_, track,
helps to establish the track in terms of ratios and pi.

I.e., to stopwatch Achilles and the tortoise at all,
only when they arrive around the round track in the same time,
is so contrived as wherever they meet, pass, or start together,
then there is an inner track for the tortoise, that the
Achilles and the tortoise round around the round track, in
the same time. (Then waiting for it to finish and both times.)

Then, in terms of pi, for the straight track, it was so many paving
stones each the Achilles and the tortoises' circles, that
straightened out on the straight track, is a function of
the pavers, in pi.

Because, to stopwatch the beginning and ending of the straight
track, the starter would have to go faster than both Achilles and
the tortoise.

The point here is "Zeno motion, and foot-races: length of circles".

The pavers or path elements the perimeter: for the
"spiral, ..., or round space-filling curve", then is for the
path element of the line integral that is the straight track.

The "line integral" or "path integral", here is the point of
"measuring Achilles and the tortoise in a line". Summing
the curve elements from that "these length bits are non-zero,
though they are the path elements and not, stretched, these
relaxed unstressed path elements, as a line, would have the
length of this line ..., its line integral", that's probably
the course's first "space-filling curve, of lines with no width".

[Ross A. Finlayson]

On Monday, May 30, 2022 at 7:48:59 AM UTC-7, FromTheRafters wrote:

Think of diagonalization as "first is built a structure the space,
that supports all the values in the function,
then in the infinite case antidiaognalization exists to tear it down".


I.e. usually it's presented as for cases "this antidiagonalizes,
that antidiagonalizes" when it's actually "these ALL, ...,
antidiagonalize", at the very end.

This way it's easier to leave it out when the usual
cases are "these all" and "those all", besides
"none of them".


As usual I'm banging my shoe here but at least by now
the point, ..., is for taking.

Then, the first space-filling curve, is-a the line,
but, lines have no width so no area so no space.

Or a point isn't a "space-filling curve", ...,
simply of course as by its space.

As "integrable elements" though it usually is.
(Points, space filling, contiguous/continuous, curve, space.)

Ever since I let out Euclidean geometry to point-and-space
geometry with lines, planes, ..., making Euclidean geometry,
this way it still is.

Indeed after the uncountability and paradox slates,
geometry is a wide open space.

[WM]

FromTheRafters schrieb am Montag, 30. Mai 2022 um 17:02:53 UTC+2:

> > And all definable places belong to a finite domain.
> Define "Definable places" and show how their domain must be finite.

Try to define a natural number (such that another one knows what you mean) that is not finite. The maximum of all attempts over time, even over the lifetime of the universe, will be finite.

>
> Aren't your Xs and Os placed on a countably infinite by countably
> infinite (N x N) table? Are you thinking that a subset of *all*
> permutations of the table must be countable?

We consider here only those permutations which are prescribed by Cantor's

k = (m + n - 1)(m + n - 2)/2 + m

in ascending order of k:

1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2, 5/1, 1/6, 2/5, 3/4, 4/3, 5/2, 6/1, ...

or in the laguage of my matrices:

XOOO...
XOOO...
XOOO...
XOOO...
...

XXOO...
OOOO...
XOOO...
XOOO...
...

XXOO...
XOOO...
OOOO...
XOOO...
...

XXXO...
XOOO...
OOOO...
OOOO...
...

and so on.

All those are countable of course. The result however shows that "countable" is an unsound notion.

Regards, WM

[Jim Burns]

On 5/30/2022 8:50 AM, David Petry wrote:
> On Sunday, May 29, 2022 at 12:44:13 PM UTC-7,
> Jim Burns wrote:

>>> If you (JB) can't prove to me that you understand
>>> what I am saying, there will be no reason for me
>>> to continue engaging with you.

>>
>> My hope is that I have set a good example for you

>> to follow. You have clearly not understood what
>> I've been saying. Did I cease engaging with you?
>> No. I tried to explain what I've been saying.

>> My hope is that I have set a good example for you
>> to follow.
>
> Wow! That's all I can say. Wow!

You're welcome.

>> The technical term for this is "conversation".

[sergi o]

Nope, math proofs are quite formal, and have a format to follow. moat all higher math books are stuffed with formal proofs. Equations are used to be
specific, and they are unchanging with time, understood by all. Math proofs are for other math professionals.

The average joe writing proof, typically is diluted and does not cover all cases, and depends upon words/sentances which can be interpreted differently
over time, and to different people. I have a excellent book that discusses the drawbacks and limitations of language.

WM offers spoofs, his intent is to misslead people.

https://monks.scranton.edu/files/courses/Math299/math-299-lecture.pdf

[sergi o]

On 5/29/2022 3:50 PM, WM wrote:
> Jim Burns schrieb am Sonntag, 29. Mai 2022 um 21:44:13 UTC+2:
>> On 5/29/2022 1:03 AM, David Petry wrote:
>
>>> 4) There's absolutely no possible reason to
>>> change anything.
>> No.
>> You (DP) have given no reason to change anything.
>
> Nobody could give more obvious evidence than this: Exchanging an X and an O from X.....O to O.....X will not make any of them disappear and will not make any of them multiply. It will simply maintain an X and an O.
>
> But you are too dishonest

Why do you state he is dishonest ? You have not provided clarity into your swaparoofest.


> Regards, WM

[FromTheRafters]

Yes, formal proofs must comply to the input characteristics of the
algorithmic provers. Natural language won't do for this.

[Jim Burns]

On 5/30/2022 8:40 AM, David Petry wrote:
> On Sunday, May 29, 2022 at 4:28:58 AM UTC-7,
> Alan Mackenzie wrote:
>> David Petry <david...@gmail.com> wrote:

>>> On Saturday, May 28, 2022 at 12:51:02 PM UTC-7,

>>> Jim Burns wrote:

>>>> We already have real numbers with errors.
>>>> For example, from January 8, 2021, I see a report
>>>> of the age of the universe as 13.77 ± .04 billion
>>>> years.
>>
>>> Exactly. And I'm saying that keeping track of those
>>> errors is so important that it would be reasonable to
>>> say that finite precision reals along with error
>>> estimates should be the kind of "real" numbers upon
>>> which analysis is built, and the foundational logic of
>>> mathematics should be able to deal the inherent
>>> uncertainties.
>
>> I think you are proposing that our current body of
>> knowledge known as real analysis should be swept away
>> and replaced by something new which in place of
>> exact values has approximations.
>
> Not really.
> The theoretical development of the subject would be
> different. The idea is that real analysis would be
> developed in a way that is useful in science--
> i.e. science must always deal with uncertainty.

> But then, as an abstraction, we could consider the
> limit as the uncertainty goes to zero, and recover
> the parts of current real analysis that have the

> possibility of being relevant to science. The parts of

> real analysis that rely on set theoretic principles
> in an essential way would be lost.

Suppose that you had a theory of real numbers based on
geometry, instead of on set theory. Would that resolve
the concerns you have?

----
Assuming it would resolve your concerns...

We want to geometrically define

(i)
Addition and subtraction, linear order

Left as an exercise for the reader.

(ii)
Multiplication and division

Assume that a unit segment [0,1] is defined.

Construct similar triangles with side-lengths
1 : x : ... and y : p : ...
Similar triangles, thus p = x*y

Construct similar triangles with side-lengths
y : x : ... and 1 : q : ...
Similar triangles, thus q = x/y

(iii)
No infinite and no infinitesimal elements.

This is the Archimedean property.
For any two distances x and y,
there is some finite number n of replications of x
such that n*x > y

One can see this as the formalization of limiting
geometric operations to finitely-many (though
unbounded-many) steps.

(iv)
All least upper bounds of bounded non-empty sets.

This is Dedekind completeness.

It is equivalent to the intermediate value theorem,
which we could state as
if two continuous curves cross,then they intersect.

This is much younger than the Archimedean property.
It seems to me that this is not because Dedekind
completeness is so weird, but because it is so banal
that it took that long to realize it needed stating.


If (i),(ii),(iii),(iv) give you what you want,
I suggest that you continue to use everyone else's
real numbers. What I described is everyone else's
real numbers.

( Yes, (i),(ii),(iii),(iv) imply uncountably-many
( points, especially (iv).
( It's a geometric line.
( What reasons are there,
( beyond "This guy wants them to be countable",
( do we have for thinking they are countable?