On Friday, 26 August 2022 at 16:32:18 UTC+2, malcolm.ar...@gmail.com wrote:
> On Friday, 26 August 2022 at 14:44:29 UTC+1, skepd...@gmail.com wrote:
> > On Friday, 26 August 2022 at 15:39:16 UTC+2, malcolm.ar...@gmail.com wrote:
> > > acos(sqrt(2)) = PI/4
> > > You can prove that geometrically (draw a right angled triangle with unit small sides. It's an isosceles triangle because the
> > > small sides are of equal length, and the angle between them is a right angle. So the other two angles must be PI/4).
> > >
> > > The numbers are irrational (the sides of the triangle are not ratios of each other, and the angle isn't a ratio of the angle an
> > > arc subtends with two radii of its own length). But we can specify them exactly.
> > Which has what to do with arithmetic on real numbers?
> >
> You said that no "operations" could be performed on irrationals of infinite precision.
> So I gave an example of an operation. We've got one irrational (root 2) and one operation
> (arc cosine) and, if we're working in radians, we obtain another irrational (PI/4). These are
> exact.

You have given no example of an operation.
You have merely produced a tautological relation.

Please familiarise yourself with the notion of an "operator". Thanks.

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

On Friday, 26 August 2022 at 16:01:21 UTC+2, richar...@gmail.com wrote:
> Nope.
>
> Nothing says that operations on the Reals need to give the full results
> in finite time/steps.
Shit-for-brains fucking idiot. You are literally claiming/implying that you can perform operations on the Reals.

So PERFORM the "addition" operation! What's e + π + √2 equal to?

On 8/28/22 5:01 PM, Skep Dick wrote:
> On Friday, 26 August 2022 at 16:01:21 UTC+2, richar...@gmail.com wrote:
>> Nope.
>>
>> Nothing says that operations on the Reals need to give the full results
>> in finite time/steps.
> Shit-for-brains fucking idiot. You are literally claiming/implying that you can perform operations on the Reals.
>
> So PERFORM the "addition" operation! What's e + π + √2 equal to?
>
>
>

e + π + √2

Didn't claim I could REPRESENT the value.

Perhaps you don't understand the normal definition of operations on The
Reals.

Reals tend to be represented by limits, so an operation on The Reals
results in a Limit, so given any finite error you can compute the result
to at least that close in finite time.

If I wanted to, I could convert the above problem into an limit form
with the various limits that allow us to compute each of the terms.

But, you don't really want that, you just want to be a dick that shows
how little he knows about what he is talking about.

Skep Dick <skepdick22@gmail.com> writes:

> On Saturday, 27 August 2022 at 01:59:19 UTC+2, Ben Bacarisse wrote:

>>If you want to know what I mean the information is readily available
>>on the web.
>
> That is soooo peculiar to me, the location of YOUR meaning is NOT in
> your own head, but on the web?!?

>> I could, but I won't, because it's standard and available in any
>> course on real analysis.

> OH! You mean the coursework that has been bullshiting us about
> epsilon-delta calculus for centuries?

and...

>>It's a troll move to get me to waste time telling you things you
>>should have learnt before commenting on this topic.
>
> I did learn them.

So you do know what I meant. I thought so. Why did you pretend
otherwise?

Do you have a point to make about this standard definition of an
infinite sum that seems to have upset you so much?

--
Ben.

On Monday, 29 August 2022 at 04:17:25 UTC+2, Ben Bacarisse wrote:
> Do you have a point to make about this standard definition of an
> infinite sum that seems to have upset you so much?
You mean other than the point I have been making all along?!? That for somebody who insists on "clarity" you sure are a highly skilled obcurantist!

You simply refuse to tackle the issue head on, and instead just keep deflecting with irrelevant nonsense like infinite sums!

The "∞" above the Σ is just as problematic as the "∞" in lim(x -> ∞)!!!

You are binding "∞" to a parameter in lim(x -> ∞)! What is the domain of lim such that "∞" is in it?
You are binding "∞" to a parameter in Σ(n=0, ∞)! What is the domain of Σ such that "∞" is in it?

For somebody so uptight about "usual definitions" the standard notion of "parametric polymorphism" seems entirely above your paygrade!

https://ncatlab.org/nlab/show/polymorphism

On Monday, 29 August 2022 at 04:17:25 UTC+2, Ben Bacarisse wrote:
> So you do know what I meant. I thought so. Why did you pretend
> otherwise?
No, actually! I still have absolutely NO fucking clue what you mean by Σ(x=0, ∞)f(x) either! And for the exact same reason!

Why is "∞" in the domain of Σ when x ∈ ℝ !?!?!

But you can keep on pretending that you care about "clarity"; and you can keep on playing victim to "intentional misunderstanding".

On 8/29/22 3:15 AM, Skep Dick wrote:
> On Monday, 29 August 2022 at 04:17:25 UTC+2, Ben Bacarisse wrote:
>> Do you have a point to make about this standard definition of an
>> infinite sum that seems to have upset you so much?
> You mean other than the point I have been making all along?!? That for somebody who insists on "clarity" you sure are a highly skilled obcurantist!
>
> You simply refuse to tackle the issue head on, and instead just keep deflecting with irrelevant nonsense like infinite sums!
>
> The "∞" above the Σ is just as problematic as the "∞" in lim(x -> ∞)!!!
>
> You are binding "∞" to a parameter in lim(x -> ∞)! What is the domain of lim such that "∞" is in it?
> You are binding "∞" to a parameter in Σ(n=0, ∞)! What is the domain of Σ such that "∞" is in it?
>
> For somebody so uptight about "usual definitions" the standard notion of "parametric polymorphism" seems entirely above your paygrade!
>
> https://ncatlab.org/nlab/show/polymorphism
>

The Domain of the limit parameter of a sumation operator is
Union(Intergers, Infinity)

The Domain of the limit parameter of the lim operatior is Union(Real,
Infinity).

Those operators are "overloaded" on the type of that parameter, and use
one definition for the finite value, and a different definition for the
infinite value.

Using the definition on the page you quoted, it is basically "ad-hoc"
polymorphism. Though, maybe, depending on the expresiveness of the
language, you could use a parameteric polymorphism with perhaps an
ad-hoc rule deep in.

On 8/29/22 3:30 AM, Skep Dick wrote:
> On Monday, 29 August 2022 at 04:17:25 UTC+2, Ben Bacarisse wrote:
>> So you do know what I meant. I thought so. Why did you pretend
>> otherwise?
> No, actually! I still have absolutely NO fucking clue what you mean by Σ(x=0, ∞)f(x) either! And for the exact same reason!
>
> Why is "∞" in the domain of Σ when x ∈ ℝ !?!?!
>
> But you can keep on pretending that you care about "clarity"; and you can keep on playing victim to "intentional misunderstanding".

Because they are different parameters, and each parameter might have a
different domain.

Lots of things have different domains for their different parameters.

On Monday, 29 August 2022 at 13:21:59 UTC+2, richar...@gmail.com wrote:
> The Domain of the limit parameter of a sumation operator is
> Union(Intergers, Infinity)
>
> The Domain of the limit parameter of the lim operatior is Union(Real,
> Infinity).
So you are outright admitting that the domain is discontinuous ?!?!?!

Ooops!

> Those operators are "overloaded" on the type of that parameter, and use
> one definition for the finite value, and a different definition for the
> infinite value.
🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣

So there is a discontinuity on [0, ∞) where the value of x magically jumps from being "finite" to being "infinite" ?!?!?!?!?!

> Using the definition on the page you quoted, it is basically "ad-hoc"
> polymorphism. Though, maybe, depending on the expresiveness of the
> language, you could use a parameteric polymorphism with perhaps an
> ad-hoc rule deep in.
I've told you a dozen times that I am talking about parametric polymorphism..
I've also told you a dozen times that I am not interested in the polymorphism itself, but the **parametricity** of the polymorphism!

IF you are parametrising the limit; or Σ function on [0, ∞) ∪ ∞ THEN you are parametrising the limit or Σ functions on [0,∞]

You are treating ∞ as part of the continuum despite your claims to the contrary!

On Monday, 29 August 2022 at 13:23:29 UTC+2, richar...@gmail.com wrote:
> On 8/29/22 3:30 AM, Skep Dick wrote:
> > On Monday, 29 August 2022 at 04:17:25 UTC+2, Ben Bacarisse wrote:
> >> So you do know what I meant. I thought so. Why did you pretend
> >> otherwise?
> > No, actually! I still have absolutely NO fucking clue what you mean by Σ(x=0, ∞)f(x) either! And for the exact same reason!
> >
> > Why is "∞" in the domain of Σ when x ∈ ℝ !?!?!
> >
> > But you can keep on pretending that you care about "clarity"; and you can keep on playing victim to "intentional misunderstanding".
> Because they are different parameters, and each parameter might have a
> different domain.
>
> Lots of things have different domains for their different parameters.
Yeah, but you are supposed to be doing calculus! The study of **continuous** functions!

But then you can't even explain the discontinuity on [0, ∞) where x jumps from finite to infinite.

On 8/29/22 8:03 AM, Skep Dick wrote:
> On Monday, 29 August 2022 at 13:21:59 UTC+2, richar...@gmail.com wrote:
>> The Domain of the limit parameter of a sumation operator is
>> Union(Intergers, Infinity)
>>
>> The Domain of the limit parameter of the lim operatior is Union(Real,
>> Infinity).
> So you are outright admitting that the domain is discontinuous ?!?!?!
>
> Ooops!
>
>> Those operators are "overloaded" on the type of that parameter, and use
>> one definition for the finite value, and a different definition for the
>> infinite value.
> 🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣
>
> So there is a discontinuity on [0, ∞) where the value of x magically jumps from being "finite" to being "infinite" ?!?!?!?!?!
>
>> Using the definition on the page you quoted, it is basically "ad-hoc"
>> polymorphism. Though, maybe, depending on the expresiveness of the
>> language, you could use a parameteric polymorphism with perhaps an
>> ad-hoc rule deep in.
> I've told you a dozen times that I am talking about parametric polymorphism.
> I've also told you a dozen times that I am not interested in the polymorphism itself, but the **parametricity** of the polymorphism!
>
> IF you are parametrising the limit; or Σ function on [0, ∞) ∪ ∞ THEN you are parametrising the limit or Σ functions on [0,∞]
>
> You are treating ∞ as part of the continuum despite your claims to the contrary!
>

So, since they are not defined as parameterized, maybe you system can't
handle it.

So the problem is YOU are treating it as part of a continuum that it
isn't part of.

This is the problem of trying to re-frame a system, YOU need to resolve
the issue, not just say the old system was flawed because it can't adapt
to your new system.

On 8/29/22 8:07 AM, Skep Dick wrote:
> On Monday, 29 August 2022 at 13:23:29 UTC+2, richar...@gmail.com wrote:
>> On 8/29/22 3:30 AM, Skep Dick wrote:
>>> On Monday, 29 August 2022 at 04:17:25 UTC+2, Ben Bacarisse wrote:
>>>> So you do know what I meant. I thought so. Why did you pretend
>>>> otherwise?
>>> No, actually! I still have absolutely NO fucking clue what you mean by Σ(x=0, ∞)f(x) either! And for the exact same reason!
>>>
>>> Why is "∞" in the domain of Σ when x ∈ ℝ !?!?!
>>>
>>> But you can keep on pretending that you care about "clarity"; and you can keep on playing victim to "intentional misunderstanding".
>> Because they are different parameters, and each parameter might have a
>> different domain.
>>
>> Lots of things have different domains for their different parameters.
> Yeah, but you are supposed to be doing calculus! The study of **continuous** functions!
>
> But then you can't even explain the discontinuity on [0, ∞) where x jumps from finite to infinite.
>

Who said I was doing calculus.

Note, the definition of continuous doesn't apply at infinity, as
infinity is functions in calculus are only defined over The Reals, and
infinity isn't a member of The Real. It is only present in the system as
a possible limit with a special definition for those limits.

Note, Limit Theory is NOT part of Calculus, but a base that Calculus is
built on.

On Monday, 29 August 2022 at 14:12:35 UTC+2, richar...@gmail.com wrote:
> So, since they are not defined as parameterized, maybe you system can't
> handle it.
Shame! Are you tripping up over syntax?

sigma(start, end, f)

In [1]: def sigma(start, end, f):
...: sum = 0
...: for x in range(start, end + 1):
...: sum = sum + f(x)
...: return sum

In [2]: sigma(0,2, (lambda x:x))
Out[2]: 3

In [3]: sigma(0,2, (lambda x: 2**x))
Out[3]: 7
> So the problem is YOU are treating it as part of a continuum that it
> isn't part of.
What? Yes YOU are treating it as part of the continuum! Not me!

[0, ∞) UNION ∞ is [0,∞]

> This is the problem of trying to re-frame a system, YOU need to resolve
> the issue, not just say the old system was flawed because it can't adapt
> to your new system.
How many fucking times do I have to explain this to you?!?!?

I am not talking about a NEW system. I am talking about YOUR system.

On Monday, 29 August 2022 at 14:16:42 UTC+2, richar...@gmail.com wrote:
> Who said I was doing calculus.
You are doing REAL analysis. Are you not ?!?!

Real analysis is a branch of mathematical analysis that analyses the behaviour of real numbers, sequences and series, and real functions.
Convergence, limits, **continuity**, **smoothness**, **differentiability**,

> Note, the definition of continuous doesn't apply at infinity
Obviously it doesn't apply AT infinity. But it applies ON THE WAY TO infinity.

> infinity is functions in calculus are only defined over The Reals, and
> infinity isn't a member of The Real. It is only present in the system as
> a possible limit with a special definition for those limits.
Which doesn't address DISCONTINUITY between finite and infinite values of x ∊ R.

Approaching infinity is a meaningless notion IF ∀x, x ∊ R Finite(x)

> Note, Limit Theory is NOT part of Calculus, but a base that Calculus is
> built on.
I know! Your "base" is nonsense!

How you say? If you have no foundations - your system collapses!

Skep Dick <skepdick22@gmail.com> writes:

> On Monday, 29 August 2022 at 04:17:25 UTC+2, Ben Bacarisse wrote:
>> Do you have a point to make about this standard definition of an
>> infinite sum that seems to have upset you so much?

> You mean other than the point I have been making all along?!? That for
> somebody who insists on "clarity" you sure are a highly skilled
> obcurantist!
>
> You simply refuse to tackle the issue head on, and instead just keep
> deflecting with irrelevant nonsense like infinite sums!
>
> The "∞" above the Σ is just as problematic as the "∞" in lim(x ->
> ∞)!!!

I have no problem with either, and you won't say what you think the
problem is.

> You are binding "∞" to a parameter in lim(x -> ∞)!

No I'm, not. Wij thinks I am because he does not know that there are
two distinct definitions of limit, and so he thinks that the asymptotic
limit is just the usual one (lim(x->x) f(x)) with ∞ "bound to c". I
thought maybe you /did/ know that there are two definitions, but that
would mean you are pretending not to know. Are you pretending not to
know how lim(x->∞) f(x) is defined? Seems an odd thing to do.

> What is the domain
> of lim such that "∞" is in it?

The asymptotic limit is a function of real-valued functions. ∞ is not
in the domain of the limit functor nor in the domain or co-domain of any
function to which it can be applied. It does not even appear in the
definition of the limit. You could find the answers to these trivial
questions for yourself from any good book on real analysis.

> You are binding "∞" to a parameter in Σ(n=0, ∞)!

No, I'm not.

> What is the domain of Σ such that "∞" is in it?

Likewise. The domain of Σ (in this case) is the set of functions from N
to R.

It seems that you think "∞" is problematic because you don't know how
such limits are defined. (The sums are also defined in terms of
limits.) This is a guess, of course, because all you say is that "it's
problematic".

--
Ben.
"What's the 0th integer greater than 4? It's 4!" (Skep Dick, Aug 2022)

Skep Dick <skepdick22@gmail.com> writes:

> On Monday, 29 August 2022 at 04:17:25 UTC+2, Ben Bacarisse wrote:
>> So you do know what I meant. I thought so. Why did you pretend
>> otherwise?
>
> No, actually!

I wonder what it was you leaned then. When I said I would not "waste
time telling you things you should have learnt before commenting on this
topic" you said:

| I did learn them.

> I still have absolutely NO fucking clue what you mean by Σ(x=0, ∞)f(x)
> either! And for the exact same reason!

The only reason is that you won't read a book on real analysis to find
out. (Actually, I think you did learn -- as you said you did -- and are
just pretending not to know for fun.)

> Why is "∞" in the domain of Σ when x ∈ ℝ !?!?!

∞ is not "in the domain of Σ" (Σ operates on functions) and ∞ is not in
the domain of the function to which it is applied either. And x is
taken to range over N. When x is in R, we usually write an integral,
and the definitions (there's more than one, as you know) are more
complicated.

> But you can keep on pretending that you care about "clarity"; and you
> can keep on playing victim to "intentional misunderstanding".

Since I claim nothing but common-all-garden limits, your not knowing
what I mean must be simply a pretence or a reluctance to pick up a
book.

--
Ben.
"What's the 0th integer greater than 4? It's 4!" (Skep Dick, Aug 2022)

On Monday, 29 August 2022 at 16:19:01 UTC+2, Ben Bacarisse wrote:
> No I'm, not. Wij thinks I am because he does not know that there are
> two distinct definitions of limit
So you are outright admitting to equivocation! I thought so...

Given the TWO DISTINCT definitions of limit what's your decision-procedure for selecting which definition to use.

Because at this point every comp sci person on this forum knows you are talking about dynamic dispatch.

>and so he thinks that the asymptotic
> limit is just the usual one (lim(x->x) f(x)) with ∞ "bound to c".
Well, shouldn't you use DIFFERENT NAMES to distinguish the TWO DISTINCT FUNCTIONS and AVOID CONFUSION.

You know, because you said you care about claity. Or... did that rule fall away now?

> thought maybe you /did/ know that there are two definitions, but that
> would mean you are pretending not to know.
>Are you pretending not to know how lim(x->∞) f(x) is defined? Seems an odd thing to do.
Well, if pointing out THAT lim() is parametrically polymorphic is what you mean by "pretending" then yes - I am "pretending".

I've been "pretending" all along that you are obscuring clarity by using the SAME NAME for TWO DIFFERENT DEFINITIONS!

> The asymptotic limit is a function of real-valued functions. ∞ is not
> in the domain of the limit functor nor in the domain or co-domain of any
> function to which it can be applied.
Ah! And what is your decision-procedure for applicability/non-applicability?

>It does not even appear in the definition of the limit. You could find the answers to these trivial
> questions for yourself from any good book on real analysis.
Any "good" book on real analysis would NOT use the SAME NAME for TWO DIFFERENT DEFINITIONS! Unless the theory was founded upon a foundation which handles polymorphism.

And none of your foundations seem to hit that mark.

> > You are binding "∞" to a parameter in Σ(n=0, ∞)!
> No, I'm not.
Really? So the usual Σ syntax can't be re-written in functional notation as sigma(start, end, f) ?!?!

Maybe you can give us the type-signature of Σ? That would help!

> > What is the domain of Σ such that "∞" is in it?
> Likewise. The domain of Σ (in this case) is the set of functions from N
> to R.
Wait a minute?!? Which parameter to Σ has the type N -> R ?!?!

> It seems that you think "∞" is problematic because you don't know how
> such limits are defined.
That's an outright lie. ∞ is problematic because you are passing it as a parameter to an iterator like Σ.

>(The sums are also defined in terms of
> limits.)
But wait a minute! Aren't the limits defined in terms of sums?

It sure seems to me you have a circular dependency?

> Ben.
> "What's the 0th integer greater than 4? It's 4!" (Skep Dick, Aug 2022)
"I don't understand the difference between Nat -> Nat -> Bool and Nat -> Nat -> Nat" (Ben Bacarisse, Aug 2022)

Skep Dick <skepdick22@gmail.com> writes:

> On Monday, 29 August 2022 at 16:19:01 UTC+2, Ben Bacarisse wrote:
>> No I'm, not. Wij thinks I am because he does not know that there are
>> two distinct definitions of limit
> So you are outright admitting to equivocation! I thought so...

I am struggling to accept that you are honestly this badly informed.
Surely this is just saying nonsense for the sake of saying anything to
get off the hook?

> Given the TWO DISTINCT definitions of limit what's your
> decision-procedure for selecting which definition to use.

You just look at the lim. lim(x->oo) is the asymptotic limit and can't
be defined in the same way as lim(x->c) where c is in R. Two utterly
distinct situations need distinct definitions.

Please tell me you've know this all along? Why would you post anything
on this topic if you did not know these basic facts?

>> and so he thinks that the asymptotic
>> limit is just the usual one (lim(x->x) f(x)) with ∞ "bound to c".

> Well, shouldn't you use DIFFERENT NAMES to distinguish the TWO
> DISTINCT FUNCTIONS and AVOID CONFUSION.

Unfortunately I did not invent the notation and, in fact, it causes
absolutely no confusion to any students I have every come across.
Except for you, but then I think you've know this all along, no? Were
you really confused?

Anyway, now you know what I mean when I say that 0.999... = 1. That
took a long time. You "disagree" so presumably you have some other
definition in mind. Maybe one day you will give it.

>> thought maybe you /did/ know that there are two definitions, but that
>> would mean you are pretending not to know. Are you pretending not to
>> know how lim(x->∞) f(x) is defined? Seems an odd thing to do.
>
> Well, if pointing out THAT lim() is parametrically polymorphic is what
> you mean by "pretending" then yes - I am "pretending".

No, that's no what I meant. I meant pretending not to know in the plain
sense of knowing all along what I meant by an infinite sum while asking
me what I meant as if you didn't know -- ordinary pretending.

Maybe you were not simply pretending not to know. Maybe you really did
not know. But then why would you post on the topic?

> I've been "pretending" all along that you are obscuring clarity by
> using the SAME NAME for TWO DIFFERENT DEFINITIONS!

You really just have to get used to that. Both are limits (so writing
lim seems reasonable) and I've never seen anyone else (well, excpt maybe
Wij) get confused by the two forms since the symbol oo is obviously
special since it's not in R.

>> The asymptotic limit is a function of real-valued functions. ∞ is not
>> in the domain of the limit functor nor in the domain or co-domain of any
>> function to which it can be applied.
>
> Ah! And what is your decision-procedure for
> applicability/non-applicability?

Ah! I see you cut the context because my answer points out that your
question was daft.

As to your new question, see any good book on real analysis will explain
how and when the lim operator can be applied.

>> It does not even appear in the definition of the limit. You could find
>> the answers to these trivial questions for yourself from any good book
>> on real analysis.
>
> Any "good" book on real analysis would NOT use the SAME NAME for TWO
> DIFFERENT DEFINITIONS! Unless the theory was founded upon a foundation
> which handles polymorphism.

I think mathematics is not for you. All the texts will assume a level
of sophistication and flexibility about notation that you find
confusing. Since it does not seem to cause problems for other students,
I don't think anyone will write a text "good enough" for you.

> And none of your foundations seem to hit that mark.

Well, I am sorry about that. Is that why you up gave up on maths? It's
use of notation is not sufficiently like programming abstractions?

There is a great book called "The Mathematical Experience" that
documents what mathematics really is by looking at what mathematicians
actually do. In a case study it gives the Chinese remainder theorem in
a whole bunch of formulations from ancient texts to modern books. The
last example is a hyper-abstract (but hyper precise) definition from a
CS text. The authors remark that "computer science in it's theoretical
formulation is dominated by a spirit of abstraction which defers to no
other branch of mathematics in its zealotry".

Most other mathematicians are quite relaxed about notation.

>> > You are binding "∞" to a parameter in Σ(n=0, ∞)!
>> No, I'm not.
> Really? So the usual Σ syntax can't be re-written in functional
> notation as sigma(start, end, f) ?!?!

Not in any useful way for real analysis, no. In the finite version,
'start' and 'end' will appear in the definition, but in the infinite
case infinite case 'end' (to which oo would appear to be bound) will not
appear.

Topology (right up there in the abstraction stakes) is different, Wij
was not taking about that, so neither am I.

> Maybe you can give us the type-signature of Σ? That would help!

Or you could learn about infinite sums before pontificating about them.
(I know... neither of these will happen.)

I'll note that the vast majority of mathematicians would not care about
the type signature of the summation operator. That's your CS
abstraction zealotry coming out. "Sum" must be abstracted into one
polymorphic functor!

>> > What is the domain of Σ such that "∞" is in it?
>> Likewise. The domain of Σ (in this case) is the set of functions from N
>> to R.
>
> Wait a minute?!? Which parameter to Σ has the type N -> R ?!?!

Whatever "parameter" (I wouldn't use that term) denotes the sequence
being summed. But I don't insist on the severe all-encompassing
abstractions that you appear to prefer -- this is just the infinite sum
operator. Remember, for us relaxed mathematicians, there are two such
operators, even though they look so very alike.

>> It seems that you think "∞" is problematic because you don't know how
>> such limits are defined.
>
> That's an outright lie. ∞ is problematic because you are passing it as
> a parameter to an iterator like Σ.

No, no parameters. No passing. This is us hippy mathematicians happily
using a notation that is not a parameterised, polymorphic abstraction.
What's problematic is trying to shoehorn everything into your view of
such abstractions.

>>(The sums are also defined in terms of
>> limits.)
> But wait a minute! Aren't the limits defined in terms of sums?
>
> It sure seems to me you have a circular dependency?

Only because you cut all the context. "The sums" are the sums in
question, infinite sums, not all sums. They are defined in terms of
limits of sums that are not the ones in question being simple finite
sums. I don't think you are posting in good faith.

> "I don't understand the difference between Nat -> Nat -> Bool and Nat
> -> Nat -> Nat" (Ben Bacarisse, Aug 2022)

Now I know you probably don't aspire to the highest moral standards, but
you really should not pretend that that is a quote. You are free to
think that that is what my words meant, but you should make it clear
that it's your words and your opinion.

--
Ben.
"What's the 0th integer greater than 4? It's 4!" (Skep Dick, Aug 2022)

On Monday, 29 August 2022 at 18:36:27 UTC+2, Ben Bacarisse wrote:
> Skep Dick <skepd...@gmail.com> writes:
>
> > On Monday, 29 August 2022 at 16:19:01 UTC+2, Ben Bacarisse wrote:
> >> No I'm, not. Wij thinks I am because he does not know that there are
> >> two distinct definitions of limit
> > So you are outright admitting to equivocation! I thought so...
> I am struggling to accept that you are honestly this badly informed.
I am struggling to accept that you are trying to pin this on me! Mr "Usual Definitions".

You ARE equivocating. This is not my opinion. It's a fact! In accordance with the definition of "equivocation".

"In logic, equivocation ("calling two different things by the same name") is an informal fallacy resulting from the use of a particular word/expression in multiple senses within an argument."
> Surely this is just saying nonsense for the sake of saying anything to
> get off the hook?
Get fucked! I am not the one equivocating. If you don't care about clarity and avoiding logical fallacies (which actively harms understanding) just say so!

People who can't (self)reflect are the worst!

> > Given the TWO DISTINCT definitions of limit what's your
> > decision-procedure for selecting which definition to use.
> You just look at the lim. lim(x->oo) is the asymptotic limit and can't
> be defined in the same way as lim(x->c) where c is in R. Two utterly
> distinct situations need distinct definitions.
So why don't you have a notation for asymptotic and non-asymptotic limits?

That's so peculiar! The two functions even behave semantically differently!

> Please tell me you've know this all along? Why would you post anything
> on this topic if you did not know these basic facts?
Because the medium of communication (books!) is NOT conducive to encoding the polymorphic aspect of the lim() operator.

On the other hand source code IS conducive to encoding the polymorphic aspect of functions. I am sure you've heard of generics in programming language theory?

> Unfortunately I did not invent the notation and, in fact, it causes
> absolutely no confusion to any students I have every come across.
So the fact that every student of yours abandons the epsilon-delta method the moment they cross ocer into a Physics lecture was no signal of importance to you?

The fact that Physicists (to this day) use calculus with infinitesimals is of no importance to you?

> Except for you, but then I think you've know this all along, no? Were
> you really confused?
I am very confused. Just not about the topic you think I am confused about.

> Anyway, now you know what I mean when I say that 0.999... = 1. That
> took a long time.
No, I still don't. But hey! Keep lying to yourself.

>You "disagree" so presumably you have some other definition in mind.
Yes. I have told you my definition/theorem multiple times. (.999... = 1) ↔ False

>Maybe one day you will give it.
Maybe one day you'l acknowledge it.

> > Well, if pointing out THAT lim() is parametrically polymorphic is what
> > you mean by "pretending" then yes - I am "pretending".
> No, that's no what I meant. I meant pretending not to know in the plain
> sense of knowing all along what I meant by an infinite sum while asking
> me what I meant as if you didn't know -- ordinary pretending.
I didn't know what you meant by an infinite sum. That's why I asked.

In fact - you even admitted that your definitions (often) mean more than one thing! I know that. Which is precisely why I ask. So I can establish WHICH meaning you had in mind.

> Maybe you were not simply pretending not to know. Maybe you really did
> not know. But then why would you post on the topic?
To find out what you mean? Unequivocally.

> > I've been "pretending" all along that you are obscuring clarity by
> > using the SAME NAME for TWO DIFFERENT DEFINITIONS!
> You really just have to get used to that. Both are limits (so writing
> lim seems reasonable)
Both 1 and 2 are numbers, yet for some strange reason we use different symbols for them.
Both * and / are operators, yet for some strange reason we use different symbols for them.

>and I've never seen anyone else (well, excpt maybe
> Wij) get confused by the two forms since the symbol oo is obviously
> special since it's not in R.
It is obvious that ∞ is not in R. You've told us. But then you say stuff like "Element of R approaches ∞"

Indeed, I still struggle with the discontinuity from Finite -> Infinite values.

> >> The asymptotic limit is a function of real-valued functions. ∞ is not
> >> in the domain of the limit functor nor in the domain or co-domain of any
> >> function to which it can be applied.
> >
> > Ah! And what is your decision-procedure for
> > applicability/non-applicability?
> Ah! I see you cut the context because my answer points out that your
> question was daft.
Nonsense. My question is explicitly asking you for the source code of your dispatch procedure.

> As to your new question, see any good book on real analysis will explain
> how and when the lim operator can be applied.
That's very unfortunate - I can't explain English to a computer.

They are very.... particular... about equivocation.

> > Any "good" book on real analysis would NOT use the SAME NAME for TWO
> > DIFFERENT DEFINITIONS! Unless the theory was founded upon a foundation
> > which handles polymorphism.
> I think mathematics is not for you.
I think that's another fallacy. The No True Scottsman fallacy.

>All the texts will assume a level
> of sophistication and flexibility about notation that you find
> confusing.
ALL the texts? The source code of every proof assistant which implements Real analysis seems to be quite explicit about the polymorphism at play.

Maybe your encoding+medium is shit?

>Since it does not seem to cause problems for other students,
> I don't think anyone will write a text "good enough" for you.
Well, have you actually pointed out the logical issue to them? Or are you intentionally equivocating to avoid a student revolt?

> > And none of your foundations seem to hit that mark.
> Well, I am sorry about that. Is that why you up gave up on maths? It's
> use of notation is not sufficiently like programming abstractions?
That sounds like a perfectly good reason! Who doesn't want more+better abstractions?!?

Isn't that the entire point of Mathematics? Abstract reasoning!

> There is a great book called "The Mathematical Experience" that
> documents what mathematics really is by looking at what mathematicians
> actually do.
That sounds like a bit of a narrowly-focused text. They probably didn't look at what (meta)mathematicians (who are still mathematicians, by the way) actually do.

>In a case study it gives the Chinese remainder theorem in
> a whole bunch of formulations from ancient texts to modern books. The
> last example is a hyper-abstract (but hyper precise) definition from a
> CS text. The authors remark that "computer science in it's theoretical
> formulation is dominated by a spirit of abstraction which defers to no
> other branch of mathematics in its zealotry".
Ah. But that's just a matter of opinion!

Metamathematics is the Mathematics of Mathematics

> Most other mathematicians are quite relaxed about notation.
You call yourself "relaxed" ?!?!?! Hahahahahah!

You got super uptight about the "greater_than" and "nth_number_greater_than" notation!
And you are still avoiding the notational question about the "nth_successor_of_x"

> >> > You are binding "∞" to a parameter in Σ(n=0, ∞)!
> >> No, I'm not.
> > Really? So the usual Σ syntax can't be re-written in functional
> > notation as sigma(start, end, f) ?!?!
> Not in any useful way for real analysis, no. In the finite version,
> 'start' and 'end' will appear in the definition, but in the infinite
> case infinite case 'end' (to which oo would appear to be bound) will not
> appear.
That's utter nonsense. The behaviour of Σ literally changes depending on whether you put a 5 or a ∞ on its head!!!

> Topology (right up there in the abstraction stakes) is different, Wij
> was not taking about that, so neither am I.
Oh well. At least one of us is talking about topology and homotopy types.

> > Maybe you can give us the type-signature of Σ? That would help!
> Or you could learn about infinite sums before pontificating about them.
> (I know... neither of these will happen.)
Yeah, look - infinities are tricky things. You seem to have landed on the wrong side...

> I'll note that the vast majority of mathematicians would not care about
> the type signature of the summation operator. That's your CS
> abstraction zealotry coming out. "Sum" must be abstracted into one
> polymorphic functor!
That's absolute nonsense! The behaviour of Σ changes depending on whether you put a 5 or a ∞ on its head!
YOU have already abstracted it into one polymorphic functor!!!
The WAY you do algebra presently is polymorphic BY DEFAULT. You are manipulating reals together with integers together with rationals without a second thought!

On Monday, August 29, 2022 at 10:44:47 AM UTC-7, skepd...@gmail.com wrote:
> On Monday, 29 August 2022 at 18:36:27 UTC+2, Ben Bacarisse wrote:
> > Skep Dick <skepd...@gmail.com> writes:
> >
> > > On Monday, 29 August 2022 at 16:19:01 UTC+2, Ben Bacarisse wrote:
> > >> No I'm, not. Wij thinks I am because he does not know that there are
> > >> two distinct definitions of limit
> > > So you are outright admitting to equivocation! I thought so...
> > I am struggling to accept that you are honestly this badly informed.
> I am struggling to accept that you are trying to pin this on me! Mr "Usual Definitions".
>
> ...
>
> Oh, and don't worry about the notation. Apparently, you Mathematicians, are relaxed
>
> about such stuff.
>
A Mathematicians idea of notation may relaxed but it can be stiffened.

We need a theory of notation that allows an identifier to have the form
"lim("+x+"->"+y+") f(x)"
Call that pattern "lim(""->"")" or lim(@->@)
Or the form
"lim("x"->∞)" f(x)
called "lim(""->∞)" or lim(@->∞)
So there two different forms whose description's first terms are the
same but differ thereafter.

Most mathematicians feel this is not worth mentioning.

Skep Dick <skepdick22@gmail.com> writes:

> On Monday, 29 August 2022 at 18:36:27 UTC+2, Ben Bacarisse wrote:
>> Skep Dick <skepd...@gmail.com> writes:
>>
>> > On Monday, 29 August 2022 at 16:19:01 UTC+2, Ben Bacarisse wrote:
>> >> No I'm, not. Wij thinks I am because he does not know that there are
>> >> two distinct definitions of limit
>> > So you are outright admitting to equivocation! I thought so...
>> I am struggling to accept that you are honestly this badly informed.
>
> I am struggling to accept that you are trying to pin this on me! Mr
> "Usual Definitions".

Whose fault is it that you won't look up how limits are defined? Not
mine.

> You ARE equivocating. This is not my opinion. It's a fact! In
> accordance with the definition of "equivocation".
>
> "In logic, equivocation ("calling two different things by the same
> name") is an informal fallacy resulting from the use of a particular
> word/expression in multiple senses within an argument."

The expressions are clearly distinct. There is no equivocating.

>> > Given the TWO DISTINCT definitions of limit what's your
>> > decision-procedure for selecting which definition to use.
>>
>> You just look at the lim. lim(x->oo) is the asymptotic limit and can't
>> be defined in the same way as lim(x->c) where c is in R. Two utterly
>> distinct situations need distinct definitions.
>
> So why don't you have a notation for asymptotic and non-asymptotic
> limits?

It's right there. For some reason you claim you can't see the
difference.

>> Please tell me you've know this all along? Why would you post anything
>> on this topic if you did not know these basic facts?
>
> Because the medium of communication (books!) is NOT conducive to
> encoding the polymorphic aspect of the lim() operator.

Ah, I thought you might have trouble with book learning. Unfortunately
authors are not going to accommodate your desire to have material
presented in one pointlessly uniform way.

> On the other hand source code IS conducive to encoding the polymorphic
> aspect of functions. I am sure you've heard of generics in programming
> language theory?
>
>> Unfortunately I did not invent the notation and, in fact, it causes
>> absolutely no confusion to any students I have every come across.
>
> So the fact that every student of yours abandons the epsilon-delta
> method the moment they cross ocer into a Physics lecture was no signal
> of importance to you?

If they did (and I have no idea if they did) why would I care?

> The fact that Physicists (to this day) use calculus with
> infinitesimals is of no importance to you?

I'm interested in infinitesimals. I like infinitesimals. I wish the
0.999... = 1 deniers knew how to use them.

>> Anyway, now you know what I mean when I say that 0.999... = 1. That
>> took a long time.
>
> No, I still don't. But hey! Keep lying to yourself.

Well I think you are stuck then. Do you always join in discussions even
when you don't know what's being talked about?

>>You "disagree" so presumably you have some other definition in mind.
> Yes. I have told you my definition/theorem multiple times. (.999... =
> 1) ↔ False

You think that's a definition? Well at least we can be sure you won't
be writing any books about the topic!

But please, don't feel you have to say anything more. If you've said as
much as you can, let's leave it at that.

>>Maybe one day you will give it.
> Maybe one day you'l acknowledge it.

Aw, now that's disappointing. It's right there on the Usenet crank
bingo card.

>> ... I meant pretending not to know in the plain
>> sense of knowing all along what I meant by an infinite sum while asking
>> me what I meant as if you didn't know -- ordinary pretending.
>
> I didn't know what you meant by an infinite sum. That's why I asked.

OK. And now we know that you don't like maths books, so we are at an
impasse. Try to keep your objections to what I wrote in proportion to
the degree to with you don't know what I mean.

> In fact - you even admitted that your definitions (often) mean more
> than one thing!

No. The definitions mean only one thing.

>> and I've never seen anyone else (well, excpt maybe
>> Wij) get confused by the two forms since the symbol oo is obviously
>> special since it's not in R.
>
> It is obvious that ∞ is not in R. You've told us. But then you say
> stuff like "Element of R approaches ∞"

Gosh, where? Let me correct that right away. I can't find the post
where I said that so can you give me the message ID? (Please don't let
this be another dishonest non quote.)

>> >> The asymptotic limit is a function of real-valued functions. ∞ is not
>> >> in the domain of the limit functor nor in the domain or co-domain of any
>> >> function to which it can be applied.
>> >
>> > Ah! And what is your decision-procedure for
>> > applicability/non-applicability?
>> Ah! I see you cut the context because my answer points out that your
>> question was daft.
> Nonsense. My question is explicitly asking you for the source code of
> your dispatch procedure.

No, the question I replied to with the text above was "What is the
domain of lim such that "∞" is in it?". You presumably know (now) what a daft
question that was so you are pretending you were asking for a
"decision-procedure" all along. You weren't. You asked a silly question.

>> > Any "good" book on real analysis would NOT use the SAME NAME for TWO
>> > DIFFERENT DEFINITIONS! Unless the theory was founded upon a foundation
>> > which handles polymorphism.
>> I think mathematics is not for you.
>
> I think that's another fallacy. The No True Scottsman fallacy.

Yes, it would be if I were making a logical argument. But I am just
making the observation that you don't appear to be a true Scotsman --
that mathematics is not for you. You can agree or disagree since it's
opinion and not a logical deduction.

>> All the texts will assume a level
>> of sophistication and flexibility about notation that you find
>> confusing.
>
> ALL the texts? The source code of every proof assistant which
> implements Real analysis seems to be quite explicit about the
> polymorphism at play.

I am happy if you have found the rigorous "polymorphic" lim operator you
wanted. Do you still not know what I mean by 0.999... = 1? If not,
then what good was this suitably abstract text?

>> Most other mathematicians are quite relaxed about notation.
>
> You call yourself "relaxed" ?!?!?! Hahahahahah!
>
> You got super uptight about the "greater_than" and
> "nth_number_greater_than" notation!

Uptight? No, quite relaxed about that too.

> And you are still avoiding the notational question about the
> "nth_successor_of_x"

Yes, still just loving the fact that you can answer "What's the 0th
integer greater than 4?" with "It's 4!". Classic!

>
>> >> > You are binding "∞" to a parameter in Σ(n=0, ∞)!
>> >> No, I'm not.
>> > Really? So the usual Σ syntax can't be re-written in functional
>> > notation as sigma(start, end, f) ?!?!
>> Not in any useful way for real analysis, no. In the finite version,
>> 'start' and 'end' will appear in the definition, but in the infinite
>> case infinite case 'end' (to which oo would appear to be bound) will not
>> appear.
>
> That's utter nonsense. The behaviour of Σ literally changes depending
> on whether you put a 5 or a ∞ on its head!!!

Hmm... That's what I said only I gave more detail. Oh well, I think
there will be little understanding between us on any topic now.

>> Topology (right up there in the abstraction stakes) is different, Wij
>> was not taking about that, so neither am I.
> Oh well. At least one of us is talking about topology and homotopy types.
>
>> > Maybe you can give us the type-signature of Σ? That would help!
>> Or you could learn about infinite sums before pontificating about them.
>> (I know... neither of these will happen.)
> Yeah, look - infinities are tricky things. You seem to have landed on the wrong side...
>
>> I'll note that the vast majority of mathematicians would not care about
>> the type signature of the summation operator. That's your CS
>> abstraction zealotry coming out. "Sum" must be abstracted into one
>> polymorphic functor!
>
> That's absolute nonsense! The behaviour of Σ changes depending on
> whether you put a 5 or a ∞ on its head! YOU have already abstracted it
> into one polymorphic functor!!! The WAY you do algebra presently is
> polymorphic BY DEFAULT. You are manipulating reals together with
> integers together with rationals without a second thought!
>
> And equality.... DAMN! Equality is basically one giant polymorphism
> because Univalence axiom!

On 8/29/22 8:21 AM, Skep Dick wrote:
> On Monday, 29 August 2022 at 14:12:35 UTC+2, richar...@gmail.com wrote:
>> So, since they are not defined as parameterized, maybe you system can't
>> handle it.
> Shame! Are you tripping up over syntax?
>
> sigma(start, end, f)
>
> In [1]: def sigma(start, end, f):
> ...: sum = 0
> ...: for x in range(start, end + 1):
> ...: sum = sum + f(x)
> ...: return sum
>
> In [2]: sigma(0,2, (lambda x:x))
> Out[2]: 3
>
> In [3]: sigma(0,2, (lambda x: 2**x))
> Out[3]: 7

yes, but that program won't work with end = ∞, as it will never finish.

And ∞ + 1 isn't a defined value. The problem being that if you try to
treat ∞ just like any other number, you get into all sorts of contradictins.

>
>> So the problem is YOU are treating it as part of a continuum that it
>> isn't part of.
> What? Yes YOU are treating it as part of the continuum! Not me!
>
> [0, ∞) UNION ∞ is [0,∞]

except that [0, ∞) and ∞ are different sorts of things.
>
>> This is the problem of trying to re-frame a system, YOU need to resolve
>> the issue, not just say the old system was flawed because it can't adapt
>> to your new system.
> How many fucking times do I have to explain this to you?!?!?
>
> I am not talking about a NEW system. I am talking about YOUR system.
>
>

But MY system doesn't have a contiuum of [0, ∞] or a parametric overload
of the limit operatior, so it can't be my system.

All that is an effet of you trying to redefine things into a different
fundamental framework.

Any problem there-of, are YOUR problems, not mine.

I know how to take a limit, and how to define it. The fact that you have
issues converting from a system based on limits to one where you are
bound by computability is your own problem, not mine.

On 30/08/2022 01:05, Ben Bacarisse wrote:
> Skep Dick <skepdick22@gmail.com> writes:
>> So the fact that every student of yours abandons the epsilon-delta
>> method the moment they cross ocer into a Physics lecture was no signal
>> of importance to you?
> If they did (and I have no idea if they did) why would I care?

Probably unlike Ben, I spent a significant part of my career
teaching maths to physicists. I never tried to teach them delta-
epsilon analysis. In the UK, basic calculus is done at school, so
they came to university with a decent basic knowledge already of
how to differentiate and integrate. It was more important to teach
them the more advanced things they needed /for physics/. Even more
important to teach them the limits of what they knew -- "If /this/
happens, you need to consult a mathematician." Much the same with
engineers, and others who needed post-school mathematics.

>> The fact that Physicists (to this day) use calculus with
>> infinitesimals is of no importance to you?

Not in my modules they didn't.

> I'm interested in infinitesimals. I like infinitesimals. I wish the
> 0.999... = 1 deniers knew how to use them.

Likewise. But they're not a good way to do calculus. You
either run into all the problems that Newton, Euler, ... ran into in
the 18thC and which led to rigorous analysis in the 19thC, or you
have to work with hyperreals or surreals or similar, and that's an
awful lot of extra baggage to carry around for something so simple
[for most physics/engineering purposes]. It's more fruitful to
teach limits informally, and emphasise that this is for "well-
behaved functions" *only*. Good and bad behaviour illustrated by
examples rather than by formal definitions.

--
Andy Walker, Nottingham.
Andy's music pages: www.cuboid.me.uk/andy/Music
Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Handel

On Tuesday, 30 August 2022 at 02:28:32 UTC+2, richar...@gmail.com wrote:
> On 8/29/22 8:21 AM, Skep Dick wrote:
> > On Monday, 29 August 2022 at 14:12:35 UTC+2, richar...@gmail.com wrote:
> >> So, since they are not defined as parameterized, maybe you system can't
> >> handle it.
> > Shame! Are you tripping up over syntax?
> >
> > sigma(start, end, f)
> >
> > In [1]: def sigma(start, end, f):
> > ...: sum = 0
> > ...: for x in range(start, end + 1):
> > ...: sum = sum + f(x)
> > ...: return sum
> >
> > In [2]: sigma(0,2, (lambda x:x))
> > Out[2]: 3
> >
> > In [3]: sigma(0,2, (lambda x: 2**x))
> > Out[3]: 7
> yes, but that program won't work with end = ∞, as it will never finish.
I think you are lying. The program will work just fine, but it will never finish.
It is just an infinite loop.

The value of “sum” inside the program will continue to accumulate.

> And ∞ + 1 isn't a defined value. The problem being that if you try to
> treat ∞ just like any other number, you get into all sorts of contradictins.
That is just a straw man. I am not treating it as a number any more than you are treating it as a number when you put it on top of Σ.

We can just treat the range() function as polymorphic. It can range over [0, end] for finite values, or [0, ∞) for infinite values.

> >> So the problem is YOU are treating it as part of a continuum that it
> >> isn't part of.
> > What? Yes YOU are treating it as part of the continuum! Not me!
> >
> > [0, ∞) UNION ∞ is [0,∞]
> except that [0, ∞) and ∞ are different sorts of things.
Yes, and? Your UNION operator doesn’t seem to mind the type mismatch?

So what is the result of the operation UNiON([0, ∞), ∞) ?

> >
> But MY system doesn't have a contiuum of [0, ∞]
Then what do you mean by [0, ∞) UNION ∞ ?

>or a parametric overload of the limit operatior, so it can't be my system.
Ohhh! So you are using a different system to Ben? You guys should have a chat!

He says lim() is overloaded for asymptotic and non-asymptotic limits.

> All that is an effet of you trying to redefine things into a different
> fundamental framework.
I am not redefining anything. I am UNIFYING your mess into a framework that deals with polymorphism.

Generic types.

> Any problem there-of, are YOUR problems, not mine.
Your mess is everyone’s problem.

> I know how to take a limit, and how to define it.
You don’t know shit. You can’t even define “define”, but you think you can define a limit.

>The fact that you have
> issues converting from a system based on limits to one where you are
> bound by computability is your own problem, not mine.
It is difficult to explain to an idiot that he doesn’t understand.

I am no more bound by computability than you are bound by your “foundation”.

I am merely demonstrating that my foundation is more foundational.

Andy Walker <anw@cuboid.co.uk> writes:

> On 30/08/2022 01:05, Ben Bacarisse wrote:
<cut>
>> I'm interested in infinitesimals. I like infinitesimals. I wish the
>> 0.999... = 1 deniers knew how to use them.
>
> Likewise. But they're not a good way to do calculus. You
> either run into all the problems that Newton, Euler, ... ran into in
> the 18thC and which led to rigorous analysis in the 19thC, or you
> have to work with hyperreals or surreals or similar, and that's an
> awful lot of extra baggage to carry around for something so simple
> [for most physics/engineering purposes].

Agreed. And where they have been studied, it's been to show that the
somewhat informal arguments made by Euler, Newton and others could, in
fact, be formalised. And, except for some pathologically peculiar
functions, these methods give the same results.

There's a vague idea floating around this group that if only the boring
hide-bound traditionalists would accept infinitesimals, then
0.999... would not be numerically 1. But the sequence of partial sums
denoted by an infinite digit sequence is about as smooth and boringly
convergent as they come. Infinite digits string need to mean something
else altogether before you get any new and fun results, but none of the
0.999... = 1 deniers will ever say what, despite there being ready-made
examples in the literature. But then cranks will always prefer to be
uniquely wrong, than right about some well-known alternative.

--
Ben.