Doctor Hachel said :
Log i = 0
i is a second root of f(x)=Log i
N.B. Log is ln.
R.H.
Op 24/05/2025 om 21:26 schreef Richard Hachel:
Doctor Hachel said :
Log i = 0
i is a second root of f(x)=Log i
N.B. Log is ln.
R.H.
Somehow I have more confidence in wolfram alpha than a random usenet crackpot.
https://www.wolframalpha.com/input?i=ln%28i%29
Le 25/05/2025 à 05:07, sobriquet a écrit :
Op 24/05/2025 om 21:26 schreef Richard Hachel:
Doctor Hachel said :
Log i = 0
i is a second root of f(x)=Log i
N.B. Log is ln.
R.H.
Somehow I have more confidence in wolfram alpha than a random usenet
crackpot.
https://www.wolframalpha.com/input?i=ln%28i%29
You have every right to your certainties.
R.H.
On 05/24/2025 12:26 PM, Richard Hachel wrote:
Doctor Hachel said :
Log i = 0
i is a second root of f(x)=Log i
N.B. Log is ln.
R.H.
Take a look at Lambert's W(0) and W(1), ....
(-i) ^ 2 = -1
There is no certainty in math. But when you encounter conflicting claims
Le 25/05/2025 à 15:02, sobriquet a écrit :
There is no certainty in math. But when you encounter conflicting claims
Well, that's exactly the opposite :)
There is nothing but certainties in math, since everything is proved in
a non-discutable way.
And there are open problems and conjectures, that are not been proven
yet. But of course nothing at the low student level of the poor dumb
french man.
Op 25/05/2025 om 16:58 schreef efji:
Le 25/05/2025 à 15:02, sobriquet a écrit :
There is no certainty in math. But when you encounter conflicting claims
Well, that's exactly the opposite :)
There is nothing but certainties in math, since everything is proved
in a non-discutable way.
Not really.. for instance, there are people who reject proofs by contradiction, so some proofs might be acceptable to some while being rejected by others.
Also, things that used to be considered obviously true, like the
shortest distance between two points being a straight line have later
become uncertain (with the potential curvature of geometry as opposed to
flat geometry).
And there is evidence that we can't even have a completely reliable
system where we can prove everything that is true and nothing that is
false, since Gödel has shown that any formal system that includes basic arithmetic must necessarily be incomplete.
Le 25/05/2025 à 17:50, sobriquet a écrit :
Op 25/05/2025 om 16:58 schreef efji:
Le 25/05/2025 à 15:02, sobriquet a écrit :
There is no certainty in math. But when you encounter conflicting
claims
Well, that's exactly the opposite :)
There is nothing but certainties in math, since everything is proved
in a non-discutable way.
Not really.. for instance, there are people who reject proofs by
contradiction, so some proofs might be acceptable to some while being
rejected by others.
Also, things that used to be considered obviously true, like the
shortest distance between two points being a straight line have later
become uncertain (with the potential curvature of geometry as opposed
to flat geometry).
And there is evidence that we can't even have a completely reliable
system where we can prove everything that is true and nothing that is
false, since Gödel has shown that any formal system that includes
basic arithmetic must necessarily be incomplete.
Well, that's true about Gödel's theorem: it says that there exists some propositions that cannot be proven true or false. It does not imply that
what is proved can be "uncertain"!
In practice, what we call "maths" in 2025 is totally proved in a
rigorous way, with the proper hypothesis clearly given. The famous
example of the "straight line" is historical and is related to a period
of time where the logical basis of maths where not strongly established.
We are not in the XIX's century any more!
99.9999% of mathematicians accept the proofs by contradiction,
and I really wonder what is the point of the remaining 0.0001% :)
Op 25/05/2025 om 18:03 schreef efji:
Le 25/05/2025 à 17:50, sobriquet a écrit :
Op 25/05/2025 om 16:58 schreef efji:
Le 25/05/2025 à 15:02, sobriquet a écrit :
There is no certainty in math. But when you encounter conflicting
claims
Well, that's exactly the opposite :)
There is nothing but certainties in math, since everything is proved
in a non-discutable way.
Not really.. for instance, there are people who reject proofs by
contradiction, so some proofs might be acceptable to some while being
rejected by others.
Also, things that used to be considered obviously true, like the
shortest distance between two points being a straight line have later
become uncertain (with the potential curvature of geometry as opposed
to flat geometry).
And there is evidence that we can't even have a completely reliable
system where we can prove everything that is true and nothing that is
false, since Gödel has shown that any formal system that includes
basic arithmetic must necessarily be incomplete.
Well, that's true about Gödel's theorem: it says that there exists
some propositions that cannot be proven true or false. It does not
imply that what is proved can be "uncertain"!
In practice, what we call "maths" in 2025 is totally proved in a
rigorous way, with the proper hypothesis clearly given. The famous
example of the "straight line" is historical and is related to a
period of time where the logical basis of maths where not strongly
established. We are not in the XIX's century any more!
99.9999% of mathematicians accept the proofs by contradiction,
and I really wonder what is the point of the remaining 0.0001% :)
But the same holds for science. It's reasonable to assume that theories
like evolution are proven beyond reasonable doubt, but philosophically speaking, there is no certainty in science.
Le 25/05/2025 à 18:17, sobriquet a écrit :
Op 25/05/2025 om 18:03 schreef efji:
Le 25/05/2025 à 17:50, sobriquet a écrit :
Op 25/05/2025 om 16:58 schreef efji:
Le 25/05/2025 à 15:02, sobriquet a écrit :
There is no certainty in math. But when you encounter conflicting
claims
Well, that's exactly the opposite :)
There is nothing but certainties in math, since everything is
proved in a non-discutable way.
Not really.. for instance, there are people who reject proofs by
contradiction, so some proofs might be acceptable to some while
being rejected by others.
Also, things that used to be considered obviously true, like the
shortest distance between two points being a straight line have
later become uncertain (with the potential curvature of geometry as
opposed to flat geometry).
And there is evidence that we can't even have a completely reliable
system where we can prove everything that is true and nothing that
is false, since Gödel has shown that any formal system that includes
basic arithmetic must necessarily be incomplete.
Well, that's true about Gödel's theorem: it says that there exists
some propositions that cannot be proven true or false. It does not
imply that what is proved can be "uncertain"!
In practice, what we call "maths" in 2025 is totally proved in a
rigorous way, with the proper hypothesis clearly given. The famous
example of the "straight line" is historical and is related to a
period of time where the logical basis of maths where not strongly
established. We are not in the XIX's century any more!
99.9999% of mathematicians accept the proofs by contradiction,
and I really wonder what is the point of the remaining 0.0001% :)
But the same holds for science. It's reasonable to assume that theories
like evolution are proven beyond reasonable doubt, but philosophically
speaking, there is no certainty in science.
Yes, there is no certainty in science, EXCEPT in maths !
Take any maths book or article, take any proposition entitled "Theorem",
you know that it is true forever, without any doubt and without any
chance that somebody in 1000 years in the future could disprove it,
whatever "science" will be at this time.
Humans are fallible creatures. Math communities consist of humans. Conclusion, there is no certainty in math. Though of course it's
extremely unlikely for something to turn out to be false if its proof
has been verified and accepted by the entire community of mathematicians
and has stood the test of time. But there are also other factors
involved. For instance the proof could consist of terabytes of data, so
in that case we might increase our confidence level if we formalize the
proof so it gets checked independently by a computer.
Le 25/05/2025 à 19:17, sobriquet a écrit :
Humans are fallible creatures. Math communities consist of humans.
Conclusion, there is no certainty in math. Though of course it's
extremely unlikely for something to turn out to be false if its proof
has been verified and accepted by the entire community of
mathematicians and has stood the test of time. But there are also
other factors involved. For instance the proof could consist of
terabytes of data, so in that case we might increase our confidence
level if we formalize the
proof so it gets checked independently by a computer.
It seems that your level in mathematics may not be very advanced, and
you appear to be repeating, somewhat awkwardly, what you’ve read in mainstream media.
No, proofs of theorems generally do not involve "terabytes of data" :)
ChatGPT gives the following rough evaluations:
* 3 to 5 millions of theorems proved since the beginning of humanity,
some of them with multiple proofs (e.g. Pythagorean Theorem: more than
400 independent proofs).
* 250000 to 350000 theorems published last year (between 100000 and
120000 maths publications in peer reviews).
Among these millions of theorems, only a few involve a computer to help
the proof. The first one was the "4 colors Theorem" in 1976 that used a computer to check 1936 identified configurations, too long to check
manually.
Each year, a few theorems use computers to be proved, either using
"proof assistants" that formalize and check the logic of hundreds of
pages of inductions, or, like in the case of the 4 colors Theorem, check
a finite number of remaining cases (possibly big) while the main human
proof says something like "for n>N, blablabla".
But although the mainstream media talk a lot about them, they are
totally marginal in the crowd of new theorems.
I'm just interested in math and science at an abstract level from a historical perspective and how technology (AI in particular) has the potential to transform education and the dissemination/accessibility of knowledge and understanding.
https://www.quantamagazine.org/mathematical-beauty-truth-and-proof-in- the-age-of-ai-20250430/
Le 25/05/2025 à 15:02, sobriquet a écrit :
There is no certainty in math. But when you encounter conflicting claims
Well, that's exactly the opposite :)
There is nothing but certainties in math, since everything is proved in
a non-discutable way.
Le 25/05/2025 à 18:17, sobriquet a écrit :
Op 25/05/2025 om 18:03 schreef efji:
Le 25/05/2025 à 17:50, sobriquet a écrit :
Op 25/05/2025 om 16:58 schreef efji:
Le 25/05/2025 à 15:02, sobriquet a écrit :
There is no certainty in math. But when you encounter conflicting
claims
Well, that's exactly the opposite :)
There is nothing but certainties in math, since everything is
proved in a non-discutable way.
Not really.. for instance, there are people who reject proofs by
contradiction, so some proofs might be acceptable to some while
being rejected by others.
Also, things that used to be considered obviously true, like the
shortest distance between two points being a straight line have
later become uncertain (with the potential curvature of geometry as
opposed to flat geometry).
And there is evidence that we can't even have a completely reliable
system where we can prove everything that is true and nothing that
is false, since Gödel has shown that any formal system that includes
basic arithmetic must necessarily be incomplete.
Well, that's true about Gödel's theorem: it says that there exists
some propositions that cannot be proven true or false. It does not
imply that what is proved can be "uncertain"!
In practice, what we call "maths" in 2025 is totally proved in a
rigorous way, with the proper hypothesis clearly given. The famous
example of the "straight line" is historical and is related to a
period of time where the logical basis of maths where not strongly
established. We are not in the XIX's century any more!
99.9999% of mathematicians accept the proofs by contradiction,
and I really wonder what is the point of the remaining 0.0001% :)
But the same holds for science. It's reasonable to assume that theories
like evolution are proven beyond reasonable doubt, but philosophically
speaking, there is no certainty in science.
Yes, there is no certainty in science, EXCEPT in maths !
Take any maths book or article, take any proposition entitled "Theorem",
you know that it is true forever, without any doubt and without any
chance that somebody in 1000 years in the future could disprove it,
whatever "science" will be at this time.
on 5/26/2025, WM supposed :
On 25.05.2025 19:03, efji wrote:
Yes, there is no certainty in science, EXCEPT in maths !
Take any maths book or article, take any proposition entitled
"Theorem", you know that it is true forever, without any doubt and
without any chance that somebody in 1000 years in the future could
disprove it, whatever "science" will be at this time.
That is wrong. Present mathematics simply assumes that all natural
numbers can be used for counting. But that is wrong.
No, you simply misunderstand what countability means.
That is wrong. Present mathematics simply assumes that all natural
numbers can be used for counting. But that is wrong.
Le 26/05/2025 à 16:36, WM a écrit :
That is wrong. Present mathematics simply assumes that all natural
numbers can be used for counting. But that is wrong.
What's the point ?
It is the DEFINITION of "counting". A countable infinite set IS a set equipped with a bijection onto \N.
On 25.05.2025 16:58, efji wrote:
Le 25/05/2025 à 15:02, sobriquet a écrit :
There is no certainty in math. But when you encounter conflicting claims
Well, that's exactly the opposite :)
There is nothing but certainties in math, since everything is proved
in a non-discutable way.
Is this a non-discutable proof?
{1} has infinitely many (ℵo) successors.
If {1, 2, 3, ..., n} has infinitely many (ℵo) successors, then {1, 2,
3, ..., n, n+1} has infinitely many (ℵo) successors, for every
accessible natural number.
Le 26/05/2025 à 16:36, WM a écrit :
That is wrong. Present mathematics simply assumes that all natural
numbers can be used for counting. But that is wrong.
What's the point ?
It is the DEFINITION of "counting". A countable infinite set IS a set equipped with a bijection onto \N.
WM wrote :set equipped with a bijection onto \N.
On 26.05.2025 22:25, efji wrote:
Le 26/05/2025 à 16:36, WM a écrit :
That is wrong. Present mathematics simply assumes that all natural numbers can be used for counting. But that is wrong.
What's the point ?
It is the DEFINITION of "counting". A countable infinite set IS a
bijection.This bijection does not exist because most natural numbers cannot be distinguished as a simple argument shows.
Bijected elements need not be distinguished, it is enough to show a
WM wrote :
On 26.05.2025 22:25, efji wrote:
Le 26/05/2025 à 16:36, WM a écrit :
That is wrong. Present mathematics simply assumes that all natural
numbers can be used for counting. But that is wrong.
What's the point ?
It is the DEFINITION of "counting". A countable infinite set IS a set
equipped with a bijection onto \N.
This bijection does not exist because most natural numbers cannot be
distinguished as a simple argument shows.
Bijected elements need not be distinguished, it is enough to show a bijection.
WM wrote :
On 26.05.2025 22:25, efji wrote:
Le 26/05/2025 à 16:36, WM a écrit :This bijection does not exist because most natural numbers cannot be
That is wrong. Present mathematics simply assumes that all natural
numbers can be used for counting. But that is wrong.
What's the point ?
It is the DEFINITION of "counting". A countable infinite set IS a set
equipped with a bijection onto \N.
distinguished as a simple argument shows.
Bijected elements need not be distinguished, it is enough to show a bijection.
Why are you working together with WM to hijack this thread from another crank?
WM's wierd ideas have been thoroughly discredited on this forum.
Why
cooperate with him to create another monster thread which goes nowhere?
On 5/27/2025 12:34 PM, WM wrote:
On 27.05.2025 18:05, FromTheRafters wrote:[...]
WM wrote :
On 26.05.2025 22:25, efji wrote:
Le 26/05/2025 à 16:36, WM a écrit :This bijection does not exist because most natural numbers cannot be
That is wrong. Present mathematics simply assumes that all natural >>>>>> numbers can be used for counting. But that is wrong.
What's the point ?
It is the DEFINITION of "counting". A countable infinite set IS a
set equipped with a bijection onto \N.
distinguished as a simple argument shows.
Bijected elements need not be distinguished, it is enough to show a
bijection.
You mean it is enough to believe in a bijection?
Either the bijection works or it doesn't. For instance, Cantor Pairing
works with any unsigned integer.
Huh? Show me one unsigned integer that Cantor Pairing does not work
with?
On 5/27/2025 1:13 PM, WM wrote:
On 27.05.2025 22:07, Chris M. Thomasson wrote:
On 5/27/2025 12:34 PM, WM wrote:
On 27.05.2025 18:05, FromTheRafters wrote:[...]
WM wrote :
On 26.05.2025 22:25, efji wrote:
Le 26/05/2025 à 16:36, WM a écrit :This bijection does not exist because most natural numbers cannot
That is wrong. Present mathematics simply assumes that all
natural numbers can be used for counting. But that is wrong.
What's the point ?
It is the DEFINITION of "counting". A countable infinite set IS a >>>>>>> set equipped with a bijection onto \N.
be distinguished as a simple argument shows.
Bijected elements need not be distinguished, it is enough to show a
bijection.
You mean it is enough to believe in a bijection?
Either the bijection works or it doesn't. For instance, Cantor
Pairing works with any unsigned integer.
It does not.
All positive fractions
1/1, 1/2, 1/3, 1/4, ...
2/1, 2/2, 2/3, 2/4, ...
3/1, 3/2, 3/3, 3/4, ...
4/1, 4/2, 4/3, 4/4, ...
...
can be indexed by the Cantor function k = (m + n - 1)(m + n - 2)/2 + m
which attaches the index k to the fraction m/n in Cantor's sequence
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, ... .
Its terms can be represented by matrices. When we attach all indeXes k
= 1, 2, 3, ..., for clarity represented by X, to the integer fractions
m/1 and indicate missing indexes by hOles O, then we get the matrix
M(0) as starting position:
XOOO... XXOO... XXOO... XXXO...
XOOO... OOOO... XOOO... XOOO...
XOOO... XOOO... OOOO... OOOO...
XOOO... XOOO... XOOO... OOOO...
M(0) M(2) M(3) M(4) ...
M(1) is the same as M(0) because index 1 remains at 1/1. In M(2) index
2 from 2/1 has been attached to 1/2. In M(3) index 3 from 3/1 has been
attached to 2/1. In M(4) index 4 from 4/1 has been attached to 1/3.
Successively all fractions of the sequence get indexed. In the limit
we see no fraction without index remaining. Note that the only
difference to Cantor's enumeration is that Cantor does not render
account for the source of the indices.
Every X, representing the index k, when taken from its present
fraction m/n, is replaced by the O taken from the fraction to be
indexed by this k. Its last carrier m/n will be indexed later by
another index. Important is that, when continuing, no O can leave the
matrix as long as any index X blocks the only possible drain, i.e.,
the first column. And if leaving, where should it settle?
As long as indexes are in the drain, no O has left. The presence of
all O indicates that almost all fractions are not indexed. And after
all indexes have been issued and the drain has become free, no indexes
are available which could index the remaining matrix elements, yet
covered by O.
It should go without saying that by rearranging the X of M(0) never a
complete covering can be realized.
It sounds like you are trying to say
, even the following map does not work:
map_to(x) return x + 1
map_from(x) return x - 1
On 05/28/2025 04:22 PM, Chris M. Thomasson wrote:
On 5/27/2025 1:13 PM, WM wrote:
All positive fractions
1/1, 1/2, 1/3, 1/4, ...
2/1, 2/2, 2/3, 2/4, ...
3/1, 3/2, 3/3, 3/4, ...
4/1, 4/2, 4/3, 4/4, ...
...
can be indexed by the Cantor function k = (m + n - 1)(m + n - 2)/2 + m
which attaches the index k to the fraction m/n in Cantor's sequence
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, ... .
Its terms can be represented by matrices. When we attach all indeXes k
= 1, 2, 3, ..., for clarity represented by X, to the integer fractions
m/1 and indicate missing indexes by hOles O, then we get the matrix
M(0) as starting position:
XOOO... XXOO... XXOO... XXXO...
XOOO... OOOO... XOOO... XOOO...
XOOO... XOOO... OOOO... OOOO...
XOOO... XOOO... XOOO... OOOO...
M(0) M(2) M(3) M(4) ...
M(1) is the same as M(0) because index 1 remains at 1/1. In M(2) index
2 from 2/1 has been attached to 1/2. In M(3) index 3 from 3/1 has been
attached to 2/1. In M(4) index 4 from 4/1 has been attached to 1/3.
Successively all fractions of the sequence get indexed. In the limit
we see no fraction without index remaining. Note that the only
difference to Cantor's enumeration is that Cantor does not render
account for the source of the indices.
Every X, representing the index k, when taken from its present
fraction m/n, is replaced by the O taken from the fraction to be
indexed by this k. Its last carrier m/n will be indexed later by
another index. Important is that, when continuing, no O can leave the
matrix as long as any index X blocks the only possible drain, i.e.,
the first column. And if leaving, where should it settle?
As long as indexes are in the drain, no O has left. The presence of
all O indicates that almost all fractions are not indexed. And after
all indexes have been issued and the drain has become free, no indexes
are available which could index the remaining matrix elements, yet
covered by O.
It should go without saying that by rearranging the X of M(0) never a
complete covering can be realized.
It sounds like you are trying to say, even the following map does notDidn't you already have this conversation,
work:
map_to(x) return x + 1
map_from(x) return x - 1
map_to(0) = 1
map_to(1) = 2
...
map_from(1) = 0
map_from(2) = 1
...
?
and why are you having it here?
It seems you're describing a simple book-keeping of an integer continuum
in areal terms.
Also called a geometrization sometimes.
On 05/29/2025 11:25 AM, WM wrote:<snip>
On 29.05.2025 17:37, Ross Finlayson wrote:
It seems you're describing a simple book-keeping of an integer continuum >>> in areal terms.
Also called a geometrization sometimes.
Right. Do you understand the result?
You can build it in Katz' OUTPACING simply enough,
that more is larger.
That depends on a particular structure though, like geometry,
or often enough the integer lattice.
Then that something is recursively self-similar, like the
"infinite balanced binary tree", any node of which is a copy
of the root, or "square Cantor space" where all the sequences
of 0's and 1's are in lexicographic order in the language of 2^w,
that's mostly defined by square Cantor space being an arithmetization
of a geometrization of a line-drawing the interval [0,1].
The only way I'd suggest you're making sense at all is to
agree with everything I say, and retroactively.
On 06/06/2025 06:03 AM, Julio Di Egidio wrote:
Yep, except that "Cantor" arithmetizes [0,1] into [0,1).
Yet, arithmetization is a mere attempt of geometrization,
FromTheRafters <FTR@nomail.afraid.org> wrote:
Why are you working together with WM to hijack this thread from another crank?
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