On Friday, March 25, 2022 at 7:16:38 AM UTC-7, sergio wrote:
> On 3/25/2022 6:08 AM, WM wrote:
> > Eram semper recta schrieb am Freitag, 25. März 2022 um 06:58:00 UTC+1:
> >
> >> While you are correct in saying that one can only realise a repeated pattern from a process (formula or method), you are wrong to say that without it you don't have a number.
> >
> > No. A decimal representation that cannot be described does not bring a number to your mind. But that is the task of a decimal representation. As you know there is no infinite decimal representation without a formula prescribing every digit.
> Wrong, you're out of math.
>
>
> >
> > Regards, WM

But, are you _into_ math?

Why do you guys keep playing with learn-nothing trolls?

On 3/25/2022 11:21 AM, Ross A. Finlayson wrote:
> On Friday, March 25, 2022 at 7:16:38 AM UTC-7, sergio wrote:
>> On 3/25/2022 6:08 AM, WM wrote:
>>> Eram semper recta schrieb am Freitag, 25. März 2022 um 06:58:00 UTC+1:
>>>
>>>> While you are correct in saying that one can only realise a repeated pattern from a process (formula or method), you are wrong to say that without it you don't have a number.
>>>
>>> No. A decimal representation that cannot be described does not bring a number to your mind. But that is the task of a decimal representation. As you know there is no infinite decimal representation without a formula prescribing every digit.
>> Wrong, you're out of math.
>>
>>
>>>
>>> Regards, WM
>
> But, are you _into_ math?
>
> Why do you guys keep playing with learn-nothing trolls?
>

some other forums at universities copy this one directly, so bad math must be flagged as bad with the reason why.

On Friday, 25 March 2022 at 13:08:34 UTC+2, WM wrote:
> Eram semper recta schrieb am Freitag, 25. März 2022 um 06:58:00 UTC+1:
>
> > While you are correct in saying that one can only realise a repeated pattern from a process (formula or method), you are wrong to say that without it you don't have a number.

> No. A decimal representation that cannot be described does not bring a number to your mind.

Agreed. I don't know why you said "No" ??

> But that is the task of a decimal representation. As you know there is no infinite decimal representation without a formula prescribing every digit.

Agreed. However, you are WRONG to say that without a decimal representation that one does not have a number. Here are just a few counter-examples: 1/2, 1/3, 5/6, etc. :-)

>
> Regards, WM

On Saturday, 26 March 2022 at 09:57:05 UTC+2, Eram semper recta wrote:
> On Friday, 25 March 2022 at 13:08:34 UTC+2, WM wrote:
> > Eram semper recta schrieb am Freitag, 25. März 2022 um 06:58:00 UTC+1:
> >
> > > While you are correct in saying that one can only realise a repeated pattern from a process (formula or method), you are wrong to say that without it you don't have a number.
>
>
> > No. A decimal representation that cannot be described does not bring a number to your mind.
> Agreed. I don't know why you said "No" ??
> > But that is the task of a decimal representation. As you know there is no infinite decimal representation without a formula prescribing every digit.
> Agreed. However, you are WRONG to say that without a decimal representation that one does not have a number. Here are just a few counter-examples: 1/2, 1/3, 5/6, etc. :-)

Decimal representation was adopted primarily for the following reasons:

1. No need to learn the theory of numbers (fractions) in order to perform the operations of arithmetic. One could say that this system which was advocated by Stevin was the first dumbing down process of teaching arithmetic. Calculators made things a lot worse.

2. It is easy to understand the architecture of the system, that is, an integral and fractional part separated only by a period (radix).

3. The ability to represent numbers *uniquely*.

Main Drawback: Not all numbers can be represented in a given radix system. For example, 1/3 has no representation in decimal, only an approximation. Remember that rewriting a number in positional or radix format entails MEASURE using the exponents of the radix.

>
>
> >
> > Regards, WM

On Saturday, 26 March 2022 at 09:57:05 UTC+2, Eram semper recta wrote:
> On Friday, 25 March 2022 at 13:08:34 UTC+2, WM wrote:
> > Eram semper recta schrieb am Freitag, 25. März 2022 um 06:58:00 UTC+1:
> >
> > > While you are correct in saying that one can only realise a repeated pattern from a process (formula or method), you are wrong to say that without it you don't have a number.
>
>
> > No. A decimal representation that cannot be described does not bring a number to your mind.
> Agreed. I don't know why you said "No" ??
> > But that is the task of a decimal representation. As you know there is no infinite decimal representation without a formula prescribing every digit.

---> Important: A formula DOES NOT prescribe every digit, only those which you can calculate.

> Agreed. However, you are WRONG to say that without a decimal representation that one does not have a number. Here are just a few counter-examples: 1/2, 1/3, 5/6, etc. :-)
>
>
> >
> > Regards, WM

On Saturday, 26 March 2022 at 10:05:31 UTC+2, Eram semper recta wrote:
> On Saturday, 26 March 2022 at 09:57:05 UTC+2, Eram semper recta wrote:
> > On Friday, 25 March 2022 at 13:08:34 UTC+2, WM wrote:
> > > Eram semper recta schrieb am Freitag, 25. März 2022 um 06:58:00 UTC+1:
> > >
> > > > While you are correct in saying that one can only realise a repeated pattern from a process (formula or method), you are wrong to say that without it you don't have a number.
> >
> >
> > > No. A decimal representation that cannot be described does not bring a number to your mind.
> > Agreed. I don't know why you said "No" ??
> > > But that is the task of a decimal representation. As you know there is no infinite decimal representation without a formula prescribing every digit.
> > Agreed. However, you are WRONG to say that without a decimal representation that one does not have a number. Here are just a few counter-examples: 1/2, 1/3, 5/6, etc. :-)
> Decimal representation was adopted primarily for the following reasons:
>
> 1. No need to learn the theory of numbers (fractions) in order to perform the operations of arithmetic. One could say that this system which was advocated by Stevin was the first dumbing down process of teaching arithmetic. Calculators made things a lot worse.
>
> 2. It is easy to understand the architecture of the system, that is, an integral and fractional part separated only by a period (radix).
>
> 3. The ability to represent numbers *uniquely*.
>
> Main Drawback: Not all numbers can be represented in a given radix system.. For example, 1/3 has no representation in decimal, only an approximation. Remember that rewriting a number in positional or radix format entails MEASURE using the exponents of the radix.
>
>
> >
> >
> > >
> > > Regards, WM

What does it mean to MEASURE a number in a given base?

Example 1: Measure 1/4 in decimal.

Solution: 1/4 = 2/10 + 5/100 = 0.25

Example 2: Try to measure 1/3 in decimal.

Nolution: 1/3 = 3/10 + 3/100 + 3/1000 + ?

Adding 3 dots to 0.333 does not complete the measure. Moreover, the most important number theorem in mathematics proves that in order for any number p/q to have a measure in base b, it must be the case that b contains all the prime factors of q.

https://www.academia.edu/39981684/Proof_of_the_most_important_Number_theorem_that_mainstream_mathematics_academics_never_learned

FromTheRafters schrieb am Freitag, 25. März 2022 um 14:47:50 UTC+1:
> It happens that WM formulated :

> > Every written sequence necessarily ends somewhere. But those who believe that
> > infinite sequences could exist without generating formula should know how to
> > continue. What is the next digit following upon 0.322322322322322322322 in an
> > infinite sequence?
> 0.322322322322322322321(9)...
>
> or
>
> 0.322322322322322322322(0)...
>
> You probably meant
>
> 0.(322)...

You don't know how to continue. That is what you should learn.
>
> but it is hard to tell what you are thinking.

It is easy to tell what you are thinking. You claim that an infinite sequence can be given without formula. But you see that it is impossible. Nevertheless you will continue to claim that it is possible.

Regards, WM

sergio schrieb am Freitag, 25. März 2022 um 15:04:02 UTC+1:
> On 3/25/2022 6:11 AM, WM wrote:

> >>> What is the next digit following upon 0.322322322322322322322?

> FACT. All fractions have repeating decimal patterns.

But is the above sequence as fraction?

Regards, WM

Eram semper recta schrieb am Samstag, 26. März 2022 um 08:57:05 UTC+1:
> On Friday, 25 March 2022 at 13:08:34 UTC+2, WM wrote:
> > Eram semper recta schrieb am Freitag, 25. März 2022 um 06:58:00 UTC+1:
> >
> > > While you are correct in saying that one can only realise a repeated pattern from a process (formula or method), you are wrong to say that without it you don't have a number.
>
>
> > No. A decimal representation that cannot be described does not bring a number to your mind.
> Agreed. I don't know why you said "No" ??
> > But that is the task of a decimal representation. As you know there is no infinite decimal representation without a formula prescribing every digit.
> Agreed. However, you are WRONG to say that without a decimal representation that one does not have a number. Here are just a few counter-examples: 1/2, 1/3, 5/6, etc. :-)

I did not say so. I said that there is no infinite decimal representation without a formula prescribing every digit. 1/4 and 1/3 are such formulas.

Regards, WM

horand....@gmail.com schrieb am Freitag, 25. März 2022 um 13:35:59 UTC+1:
> You asked how one knows that fractions have an eventually repeating decimal expansion.

No. I asked how you could know that the sequence will be continued like a fraction.

Regards, WM

WM <askasker48@gmail.com> wrote:
> zelos...@gmail.com schrieb am Donnerstag, 24. März 2022 um 14:49:21 UTC+1:
>> torsdag 24 mars 2022 kl. 11:54:37 UTC+1 skrev WM:
>> > sergio schrieb am Mittwoch, 23. März 2022 um 15:13:57 UTC+1:
>> > > All fractions have repeating decimal patterns.
>> > How would you know that without a formula?
>> > What is the next digit following upon 0.322322322322322322322?

>> None, it ended

> Every written sequence necessarily ends somewhere. But those who
> believe that infinite sequences could exist without generating formula
> should know how to continue. What is the next digit following upon
> 0.322322322322322322322 in an infinite sequence?

Which infinite sequence were you thinking of? We can't read your mind
(thankfully).

If you are thinking about that sequence being the decimal expansion of a
real number N, then the next digit is the integer part of ((N -
0.322322322322322322322) * 10^22).

> Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).

WM explained on 3/26/2022 :
> FromTheRafters schrieb am Freitag, 25. März 2022 um 14:47:50 UTC+1:
>> It happens that WM formulated :
>
>>> Every written sequence necessarily ends somewhere. But those who believe
>>> that infinite sequences could exist without generating formula should know
>>> how to continue. What is the next digit following upon
>>> 0.322322322322322322322 in an infinite sequence?
>> 0.322322322322322322321(9)...
>>
>> or
>>
>> 0.322322322322322322322(0)...
>>
>> You probably meant
>>
>> 0.(322)...
>
> You don't know how to continue. That is what you should learn.

You should learn notation. The question mark is not a numeral.

You are the one clearly missing things. I don't need to convince you of
this, as you have already convinced everyone else.

>> but it is hard to tell what you are thinking.
>
> It is easy to tell what you are thinking. You claim that an infinite sequence
> can be given without formula.

Not at all, I'm saying such a formula describes a particular number,
but its CDE representation does not define the reals. There is a home
for every real, not only just the ones you can think of or jot down.

WM <askasker48@gmail.com> wrote:

> I said that there is no infinite decimal representation without a
> formula prescribing every digit. 1/4 and 1/3 are such formulas.

How can you be serious? You don't understand, any more than John
Gabriel, what it means for a mathematical entity not to exist, do you?

Perhaps you might like to expand on what you think you mean by the
non-existence of such infinite decimal representations.

> Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).

On Saturday, 26 March 2022 at 11:48:39 UTC+2, WM wrote:
> Eram semper recta schrieb am Samstag, 26. März 2022 um 08:57:05 UTC+1:
> > On Friday, 25 March 2022 at 13:08:34 UTC+2, WM wrote:
> > > Eram semper recta schrieb am Freitag, 25. März 2022 um 06:58:00 UTC+1:
> > >
> > > > While you are correct in saying that one can only realise a repeated pattern from a process (formula or method), you are wrong to say that without it you don't have a number.
> >
> >
> > > No. A decimal representation that cannot be described does not bring a number to your mind.
> > Agreed. I don't know why you said "No" ??
> > > But that is the task of a decimal representation. As you know there is no infinite decimal representation without a formula prescribing every digit.
> > Agreed. However, you are WRONG to say that without a decimal representation that one does not have a number. Here are just a few counter-examples: 1/2, 1/3, 5/6, etc. :-)
> I did not say so. I said that there is no infinite decimal representation without a formula prescribing every digit. 1/4 and 1/3 are such formulas.

Neither 1/4 nor 1/3 are formulas. They are complete well-formed numbers. They do not have to be represented in any particular radix system.

Again, I am not sure what you are thinking - you appear to be muddled.

>
> Regards, WM

On Saturday, 26 March 2022 at 11:56:16 UTC+2, WM wrote:
> horand....@gmail.com schrieb am Freitag, 25. März 2022 um 13:35:59 UTC+1:
> > You asked how one knows that fractions have an eventually repeating decimal expansion.
> No. I asked how you could know that the sequence will be continued like a fraction.

The answer to this question is obvious:

1. Any number (no need to say rational number because number implies rational) that can be represented (measured) in a given radix system will have a fixed number of digits.
2. Any number that cannot be represented in a given radix system always has a repeating pattern.
3. Any ratio (such as pi) that has no number describing it, has ONLY an approximation in any given radix system.

>
> Regards, WM

On Saturday, 26 March 2022 at 14:13:02 UTC+2, Eram semper recta wrote:
> On Saturday, 26 March 2022 at 11:56:16 UTC+2, WM wrote:
> > horand....@gmail.com schrieb am Freitag, 25. März 2022 um 13:35:59 UTC+1:
> > > You asked how one knows that fractions have an eventually repeating decimal expansion.
> > No. I asked how you could know that the sequence will be continued like a fraction.
> The answer to this question is obvious:
>
> 1. Any number (no need to say rational number because number implies rational) that can be represented (measured) in a given radix system will have a fixed number of digits.
> 2. Any number that cannot be represented in a given radix system always has a repeating pattern.

3. Any ratio (such as pi) that has no number describing it, has ONLY an approximation in any given radix system.

.... and does not repeat.

>
> >
> > Regards, WM

On Saturday, 26 March 2022 at 06:56:16 UTC-3, WM wrote:
> horand....@gmail.com schrieb am Freitag, 25. März 2022 um 13:35:59 UTC+1:
> > You asked how one knows that fractions have an eventually repeating decimal expansion.
>
> No. I asked how you could know that the sequence will be continued like a fraction.

You fucking, lying piece of shit! Here is the exchange I responded to, again:

On Thursday, 24 March 2022 at 07:54:37 UTC-3, WM wrote:
> sergio schrieb am Mittwoch, 23. März 2022 um 15:13:57 UTC+1:
> > All fractions have repeating decimal patterns.
> How would you know that without a formula?
> What is the next digit following upon 0.322322322322322322322?

You asked two questions that have nothing to do with each other, and you used a bait and switch. In the first question you feign/display ignorance of basic mathematical facts ("How would you know that [all fractions have repeating decimal patterns] without a formula"), and then you have the audacity to invite your readers to guess your sick demented mind on, as I said, a completely unrelated digit sequence.

THAT is why I invited you to fuck off, twice. I herewith repeat the invitation.

WM wrote:

> horand....@gmail.com schrieb am Freitag, 25. März 2022 um 13:35:59
> UTC+1: You asked how one knows that fractions have an eventually
>> repeating decimal expansion.
>
> No. I asked how you could know that the sequence will be continued like
> a fraction. Regards, WM

now watch this amerecan imbecile, laughing about own pericarditis,
damaging his heart. By taking all the corrupt government *mandated*
"covid" lethal injections, called "vaccines". His capitalist government
*wants_him_dead*, and he is laughing. You fucking braindead idiots.

My New Pericarditis Gear
https://153news.net/watch_video.php?v=9MXBO5S9NN7U

then this ugly satanist khazar bitch arriving in hell.

*_"500,000 dead children is worth the lie"_*

Madeleine Albright 60 Mintues - 500,000 DEAD IRAQI KIDS IS WORTH IT
MAY.12.1996 https://www.bitchute.com/video/DorRgjfabjv4/

then the puppet Zelenske, actor "president", media script propaganda

Realize What Your Watching Is Scripted Propaganda _ Become Aware People
https://www.bitchute.com/video/Kys7NRLeyyAS

Irish Deputy PM - The 95% Vaxxed Are Getting Sick From The 5% Unvaxxed
https://www.bitchute.com/video/ueREz2YSnlul

then the corrupt nato, as different from naze_ukraine, bombing bridges, TV
stations, infrastructure, water and power plants etc, daily for three (3)
months.

NATO Attacks Yugoslavia in 1999
https://www.bitchute.com/video/2esL7rg6A3uk

On 3/26/2022 4:56 AM, WM wrote:
> horand....@gmail.com schrieb am Freitag, 25. März 2022 um 13:35:59 UTC+1:
>> You asked how one knows that fractions have an eventually repeating decimal expansion.
>
> No. I asked how you could know that the sequence will be continued like a fraction.
>
> Regards, WM

No, you did not.

On 3/26/2022 4:40 AM, WM wrote:
> sergio schrieb am Freitag, 25. März 2022 um 15:04:02 UTC+1:
>> On 3/25/2022 6:11 AM, WM wrote:
>
>>>>> What is the next digit following upon 0.322322322322322322322?
>
>> FACT. All fractions have repeating decimal patterns.
>
> But is the above sequence as fraction?
>
> Regards, WM

You have failed to identify the repeating decimals.

what does your "?" specifically mean ?

If left as is, in Math it is interpreted as 0.322322322322322322322000000000...

or 0.32232232232232232232200000000(0)

Is that what you mean ?

Or you saying it is irrational ?

On 3/26/2022 4:48 AM, WM wrote:
> Eram semper recta schrieb am Samstag, 26. März 2022 um 08:57:05 UTC+1:
>> On Friday, 25 March 2022 at 13:08:34 UTC+2, WM wrote:
>>> Eram semper recta schrieb am Freitag, 25. März 2022 um 06:58:00 UTC+1:
>>>
>>>> While you are correct in saying that one can only realise a repeated pattern from a process (formula or method), you are wrong to say that without it you don't have a number.
>>
>>
>>> No. A decimal representation that cannot be described does not bring a number to your mind.
>> Agreed. I don't know why you said "No" ??
>>> But that is the task of a decimal representation. As you know there is no infinite decimal representation without a formula prescribing every digit.
>> Agreed. However, you are WRONG to say that without a decimal representation that one does not have a number. Here are just a few counter-examples: 1/2, 1/3, 5/6, etc. :-)
>
> I did not say so. I said that there is no infinite decimal representation without a formula prescribing every digit. 1/4 and 1/3 are such formulas.
>
> Regards, WM

more troll food from the Deceiver

On 3/25/2022 7:19 AM, WM wrote:
> Jim Burns schrieb
> am Donnerstag, 24. März 2022 um 21:20:48 UTC+1:

>> We know enough, there, that we know that there is
>> one and only one point in all the intervals represented
>> by all the finite initial segments of that infinite
>> digit sequence.
>
> Again you forget the dark realm.

We describe the objects we intend to study.

Here, we intend to study a geometric line.
We describe it as having the _Archimedean property_
and _Dedekind completeness_

If your dark realm does not satisfy the description,
we haven't forgotten it,
it's not part of what we intend to study.

> At definable step number n there is always
> an interval described.
> The limit would be a point.

It can't be two points.

Only a a point in _all_ the intervals is represented by
_all_ the intervals.

The line has the _Archimedean property_ which is that,
for any two distances x and y,
there is a "definable" number n of replications of y
such that n*y > x

However, for each "definable" number n
there are intervals shorter than 1/n

_Two_ points are a distance y apart and
for some "definable" n, n*y > 1
The two can't both be in an interval shorter than 1/n
Two can't both be in _all_ intervals.
Thus, two can't both be represented by _all_ intervals.

> But you cannot know the limit from the sequence alone.

*Some of* our knowledge of infinite digit sequences
does not cease.

*Some of* our knowledge of geometric lines
does not cease.

We know that two point can't both be in _all_
the intervals.

>>> See my above example.
>>
>> I guess you mean this:
>>> What is the next digit in 0.448448448448?
>>
>> 0.448448448448 isn't an infinite digit sequence.
>
> It is, but I could not write all its digits.
> So I ceased, hoping you could continue without formula
> (or you would understand that this is impossible).

I don't know what the next digit is.
_If you're telling the truth_ about it being
an infinite digit sequence,
then it _has_ a next digit.
And a next after that. And a next. And so on.
I don't know what those are, either.

We can know a thing and
at the same time not-know a different thing.
We can know d is in {0,1,2,3,4,5,6,7,8,9} and
at the same time not-know what d is.
We can know d < 10 and
at the same time not-know if d is even.

We know enough, there, that we know that there is
one and only one point in all the intervals represented
by all the finite initial segments of that infinite
digit sequence.

On Saturday, March 19, 2022 at 9:35:13 AM UTC-7, WM wrote:
> There is no permutation of the Xs in
>
> XOOO...
> XOOO...
> XOOO...
> XOOO...
> ...
>
> such that all positions are covered by Xs.

Would you agree that there is permutation of the X's and 0's such that all the 0's end up in the first column and all the other columns are "covered" by X's?

> If it were, it would imply, that in the sequence
>
> XOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
>
> all Os could be covered by the X.

I don't see the logic to that.

> Therefore Cantor's notion of countability is trash.

I don't disagree with your conclusion, but I simply don't understand the argument you use to arrive at that conclusion. Do you think you have a better argument than the one Poincare gave us?

Let me remind everybody about the argument that Poincare (and others) made:

1) Above all else, mathematics is a language. That is, it's a system of symbols and rules for manipulating strings of symbols.

2) The purpose of the language is to give us a way to encode and communicate "ideas".

3) The things that can be said to "exist" in mathematics are ideas! Or in other words, the things that "exist" in the language of mathematics are things we can actually talk about, and identify, and describe.

4) Emphatically, ideas can be completely defined in a finite number of words of the language.

5) If we take Cantor's ideas about infinite sets seriously, we would conclude that there cannot be more than "countably" many ideas, since there are not more than "countably" many finite strings of symbols.

6) But then Cantor goes on to "prove" that there "exist" sets (collections of objects) that are bigger than any "countable" collection. And he claims that this "proves" that there "exist" objects in mathematics that are not "ideas" (and hence don't actually "exist")

7) That last conclusion is so absurd that the whole idea of "countable" should be thrown away. It cannot be relevant to the important mathematics that has applications in reasoning about the real world.

On 3/27/2022 11:48 AM, David Petry wrote:
> On Saturday, March 19, 2022 at 9:35:13 AM UTC-7, WM wrote:
>> There is no permutation of the Xs in
>>
>> XOOO...
>> XOOO...
>> XOOO...
>> XOOO...
>> ...
>>
>> such that all positions are covered by Xs.
>
> Would you agree that there is permutation of the X's and 0's such that all the 0's end up in the first column and all the other columns are "covered" by X's?
>
>
>
>> If it were, it would imply, that in the sequence
>>
>> XOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
>>
>> all Os could be covered by the X.
>
> I don't see the logic to that.
>
>
>> Therefore Cantor's notion of countability is trash.
>
> I don't disagree with your conclusion, but I simply don't understand the argument you use to arrive at that conclusion. Do you think you have a better argument than the one Poincare gave us?
>
>
> Let me remind everybody about the argument that Poincare (and others) made:
>
> 1) Above all else, mathematics is a language. That is, it's a system of symbols and rules for manipulating strings of symbols.
>
> 2) The purpose of the language is to give us a way to encode and communicate "ideas".
>
> 3) The things that can be said to "exist" in mathematics are ideas! Or in other words, the things that "exist" in the language of mathematics are things we can actually talk about, and identify, and describe.
>
> 4) Emphatically, ideas can be completely defined in a finite number of words of the language.
>
> 5) If we take Cantor's ideas about infinite sets seriously, we would conclude that there cannot be more than "countably" many ideas, since there are not more than "countably" many finite strings of symbols.
>
> 6) But then Cantor goes on to "prove" that there "exist" sets (collections of objects) that are bigger than any "countable" collection. And he claims that this "proves" that there "exist" objects in mathematics that are not "ideas" (and hence don't actually "exist")
>
> 7) That last conclusion is so absurd that the whole idea of "countable" should be thrown away. It cannot be relevant to the important mathematics that has applications in reasoning about the real world.
>

na, anyone can count sheep, or rocks, they are "countable". Even you (can) count!

On 3/27/2022 12:48 PM, David Petry wrote:

> Let me remind everybody about the argument that
> Poincare (and others) made:
>
> 1) Above all else, mathematics is a language.
> That is, it's a system of symbols and rules for
> manipulating strings of symbols.

I would answer Poincaré and others that a language
is more than symbols and rules. The strings of symbols
or whatnot have meanings.

It seems to me that the purpose of our symbols and rules
is to set up something that mimics ... _something_
Call it the meaning of the symbols and rules, if you permit.

It seems to me that the correctness of
the mimicry of the intended _something_
is more important than being finite or infinite
or uncountable or inaccessible.

Let's say that I describe something.
The power set 𝒫(ℕ) of the natural numbers.
From my description, it follows that 𝒫(ℕ) is uncountable.
But I am philosophically opposed to uncountability.
Let us say.

It seems to me that, if my philosophical objections
force me to describe 𝒫(ℕ) incorrectly, then I should
either reconsider my philosophical objections or I should
reconsider reasoning about 𝒫(ℕ) I should go do something
else, maybe play solitaire, something that is _only_
symbols and rules for manipulating strings of symbols.