- #106
phoenixthoth
- 1,605
- 2
seems like a fractal. at least it seems self-similar.
A new point of view on Cantor's diagonalization arguments
- Thread starter Organic
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- Diagonalization Point
In summary, the conversation is discussing a new perspective on Cantor's diagonalization arguments and thanking individuals for their contributions. The conversation also delves into the topic of alephs and the differences between conventional mathematics and the speaker's own system. The speaker's system claims to be more expansive than Cantor's transfinite universes, with different relationships between aleph0 and 2^aleph0. The conversation also touches on the concept of magnitude in relation to the binary tree representation.
- #107
Organic
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If each notation represents more then one value then there is a difference between those trees (finite of infinte).
Again you counting the number of the notations and not the magnitude that each one of them representing.
If you don't remember then my answer to you was:
Your calculation is not right because:
1) aleph0 in my point of view stands for a general notation for any collection of infinitely many elements, and its value is flexible.
2) for example: By saying that a=(aleph0-aleph0) < b=aleph0 we mean that there are aleph0 elements in b that are not covered by a.
Also by a=(aleph0-2^aleph0) < b=aleph0 we mean that there are 2^aleph0 elements in b that are not covered by a.
There are no absolute magnitudes when we deal with collections of infinitely many elements, and no arithmetical operation (finite or infinite) can change their property of being infinitely many elements.
Shortly speaking, no arithmetical operation (finite or infinite) can change the “–“ or “+” sign in a collocation of infinitely many elements.
3) in my Binary tree the aleph0 width magnitude and the 2^aleph0 length magnitude, depends on each other, therefore their relative proportion (notated as width=aleph0 < length=2^aleph0) was not changed by your operation.
You still ignore the inner structure of infinitely many elements, because after your operation we have this list:
Code:
{...,3,2,1}=N
2 2 2
^ ^ ^
| | |
v v v
[b]{[/b]...,1,1,1[b]}[/b]<--> 1
...,1,1,1
...,1,1,0 <--> 2
...,1,1,0
...,1,0,1 <--> 3
...,1,0,1
...,1,0,0 <--> 4
...,1,0,0
...,0,1,1 <--> 5
...,0,1,1
...,0,1,0 <--> 6
...,0,1,0
...,0,0,1 <--> 7
...,0,0,1
...,0,0,0 <--> 8
...,0,0,0
...So, as you see aleph0-1 < aleph0
By the way, the result of your oparation is ((aleph0)-1) < ((2^aleph0)-aleph0)and the reason that it is jusut -aleph0 and not -2^aleph0, can be clearly shown here:
Code:
<---Arithmetic magnitude
{...,3,2,1,0} = Z*
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
{...,[b]1-1-1-1[/b]} <--> 1 Geometric magnitude(based on the
...,1,1,1,[b]0[/b] <--> 2 | thin notations)
...,1,1,[b]0[/b]/ |
...,1,1/0, |
...,1,[b]0[/b], , |
...,1/0, , |
...,1|0, , |
...,1|0, , |
...,[b]0[/b]/ , , |
...,0, , , |
...,0, , , |
...,0, , , |
...,0, , , |
...,0, , , |
...,0, , , |
...,0, , , |
... VAfter your operation we have:
( ((aleph0)-1) < ((2^aleph0)-aleph0) ) < ( aleph0 < 2^aleph0 )
Maybe this can help:
Code:
/[b]1[/b]_1
[b]1[/b]_2
/ \[b]0[/b]_1
[b]1[/b]_4
/\ /[b]1[/b]_1
/ [b]0[/b]_2
/ \[b]0[/b]_1
... [b]1[/b]_8
\ /[b]1[/b]_1
\ [b]1[/b]_2
\/ \[b]0[/b]_1
[b]0[/b]_4
\ /[b]1[/b]_1
[b]0[/b]_2
\[b]0[/b]_1
/[b]1[/b]_1
[b]1[/b]_2
/ \[b]0[/b]_1
[b]1[/b]_4
/\ /[b]1[/b]_1
/ [b]0[/b]_2
/ \[b]0[/b]_1
... [b]0[/b]_8
\ /[b]1[/b]_1
\ [b]1[/b]_2
\/ \[b]0[/b]_1
[b]0[/b]_4
\ /[b]1[/b]_1
[b]0[/b]_2
\[b]0[/b]_1
...
Code:
/[b]1[/b]_2
[b]1[/b]_4
/ \[b]0[/b]_2
[b]1[/b]_8
/\ /[b]1[/b]_2
/ [b]0[/b]_4
/ \[b]0[/b]_2
... [b]1[/b]_16
\ /[b]1[/b]_2
\ [b]1[/b]_4
\/ \[b]0[/b]_2
[b]0[/b]_8
\ /[b]1[/b]_2
[b]0[/b]_4
\[b]0[/b]_2
/[b]1[/b]_2
[b]1[/b]_4
/ \[b]0[/b]_2
[b]1[/b]_8
/\ /[b]1[/b]_2
/ [b]0[/b]_4
/ \[b]0[/b]_2
... [b]0[/b]_16
\ /[b]1[/b]_2
\ [b]1[/b]_4
\/ \[b]0[/b]_2
[b]0[/b]_8
\ /[b]1[/b]_2
[b]0[/b]_4
\[b]0[/b]_2
...
Last edited:
- #108
moshek
- 265
- 0
Dear Organic
What is the relation betwen your theoy to Frege fundaumanetal work fron 1878 that establish the foundation of mathematical Logic ?
Thank you
Moshe
What is the relation betwen your theoy to Frege fundaumanetal work fron 1878 that establish the foundation of mathematical Logic ?
Thank you
Moshe
- #109
Organic
- 1,224
- 0
Hi Moshek,
Please look here:
https://www.physicsforums.com/showthread.php?s=&threadid=16502&perpage=15&pagenumber=1
Please look here:
https://www.physicsforums.com/showthread.php?s=&threadid=16502&perpage=15&pagenumber=1
- #110
moshek
- 265
- 0
Thank you Organic !
There is allot of similarity to your concept of a a number to the way Prege develop first order logic but without the redundancy that you look on. So in some sense Prege work may be consider as a one example to your theory.
What about relativity theory of Ablert Einstein, since Prege was before Him ?
Thank you
Moshek
There is allot of similarity to your concept of a a number to the way Prege develop first order logic but without the redundancy that you look on. So in some sense Prege work may be consider as a one example to your theory.
What about relativity theory of Ablert Einstein, since Prege was before Him ?
Thank you
Moshek
- #111
Organic
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Moshek,
All I have now is Frege's basic information structures + uncertainty and redundancy as bulit-in properties of it.
Through my point of view these proprties have to be taken as basics properties of any modern theoretical research of any information system, like Math language for example.
All I have now is Frege's basic information structures + uncertainty and redundancy as bulit-in properties of it.
Through my point of view these proprties have to be taken as basics properties of any modern theoretical research of any information system, like Math language for example.
- #112
Organic
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After Albert Einstein there is no meaning to talk about space without time and vise versa.
Through my point of view there is no meaning to talk about quantity without structure and vise versa.
General conclusion:
The internal structure of any given quantity (finite or infinite) cannot be ignored.
Through my point of view there is no meaning to talk about quantity without structure and vise versa.
General conclusion:
The internal structure of any given quantity (finite or infinite) cannot be ignored.
Last edited:
- #113
moshek
- 265
- 0
Dear Organic :
The similarity and also the deferent
of your theory to Prege theory
is really amassing me !
Do you mean that Einstein develop new way
to look on the world but he still
was using Newton mathematics
and you suggest us and
alternativ mathematics ?
Moshek
www.icm2006.org
The similarity and also the deferent
of your theory to Prege theory
is really amassing me !
Do you mean that Einstein develop new way
to look on the world but he still
was using Newton mathematics
and you suggest us and
alternativ mathematics ?
Moshek
www.icm2006.org
- #114
Organic
- 1,224
- 0
Hi Moshek,
Newton mathematics was a real break through that gave us the ability to deal with momentum in the real world.
Now it is about time to deal with complexity in the real world, but in my opinion it cannot be done if structural property of any mathematical product is not examined together with its quantitative property.
Shortly speaking any mathematical product is at least structural/quantitative product.
Ferge started to develop the connection between information's structure and logic, but its colleagues did not understand its attitude, ignored the information structure and developed only the private case of information structure of no_redundancy_no_uncertainty form.
Newton mathematics was a real break through that gave us the ability to deal with momentum in the real world.
Now it is about time to deal with complexity in the real world, but in my opinion it cannot be done if structural property of any mathematical product is not examined together with its quantitative property.
Shortly speaking any mathematical product is at least structural/quantitative product.
Ferge started to develop the connection between information's structure and logic, but its colleagues did not understand its attitude, ignored the information structure and developed only the private case of information structure of no_redundancy_no_uncertainty form.
Last edited:
- #115
moshek
- 265
- 0
Organic:
I think that Your are talking about mathematics without any
modeling or equations !
How all this is relate
to how Wittgenstein see mathematics?
Thank you
Moshek
I think that Your are talking about mathematics without any
modeling or equations !
How all this is relate
to how Wittgenstein see mathematics?
Thank you
Moshek
- #116
Hurkyl
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I forgot I had aborted my last reply.
If you'll allow me to use your analogy:
Einstein's great idea was that we should stop pretending we know the answer to "What IS the universe?" and focus on the question "What do we know ABOUT the universe?"
Modern mathematics uses the same idea; we don't care what things are, we only care about what we can do with them.
From a purely logical perspective, this is a phenomenal trick; if all of our theorems are based only on "What can we do with these things?", and are completely independant of "What are these things?", then all of our theorems are valid even if we have the wrong answer to "What are these things?"
This is why I think your approach is fundamentally flawed. You are focusing so hard on answering the question "What are these things?", but the question is irrelevant!
(And, for the record, I don't think your answer is even a valid one to this question)
If you'll allow me to use your analogy:
Einstein's great idea was that we should stop pretending we know the answer to "What IS the universe?" and focus on the question "What do we know ABOUT the universe?"
Modern mathematics uses the same idea; we don't care what things are, we only care about what we can do with them.
From a purely logical perspective, this is a phenomenal trick; if all of our theorems are based only on "What can we do with these things?", and are completely independant of "What are these things?", then all of our theorems are valid even if we have the wrong answer to "What are these things?"
This is why I think your approach is fundamentally flawed. You are focusing so hard on answering the question "What are these things?", but the question is irrelevant!
(And, for the record, I don't think your answer is even a valid one to this question)
- #117
Organic
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What is the connection between not ignoring the structural property
of the natural numbers and the question "What are these things?"
The inner structure of the natural numbers can be shown here:
http://www.geocities.com/complementarytheory/ETtable.pdf
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- #118
Organic
- 1,224
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Dear Hurkyl (this time please answer to this post),
My basic approach about the infinity concept is that redundancy and uncertainty are naturally involved, no more no less.
Therefore aleph0 is a notation that stands for general and flexible "cloud like" thing.
Cantor's thing is a frozen one, mine is not.
I think that my approach is much more interesting and fruitful than Cantor's approach.
Please let me put these two different approaches "on the table" and I would like to examine them together with you.
I am going to write my point of view on Cantor’s approach in a very simple way that (I hope) can be understood by you.
Please read it, and open my eyes to important things that I omit, don’t understand, distorting and so on.
So here it is:
1) Let set C be a complete non-empty binary tree where complete non-empty binary tree exists iff both root AND all its leafs are in C.
2) Let RT be the root , let LF be the all leafs,
therefore RT AND LF are in C --> [RT , LF].
( In my point of view RT XOR LF are in C --> ( (RT… , …LF) OR (LT… , LF] OR
[RT ,…LF) ) AND NOT [RT , LF] where “…” means unreachable. )
Now, from my point of view I see these basic problems when RT AND LF are in C:
1) If both RT AND LF are in C, then C must be a finite set.
2) If C is a non-finite set (through my point of view C is forced to be a non-finite set) then the base value 2 (which is the fundamental structural property of the non-empty binary-tree) cannot exist. Also RT value is unknown.
We must realize that if RT AND LF are in C AND C is a non-finite set, then the structural property of our information (in this case we are talking about the binary tree structure) collapsed into itself (we have no infinitely many elements anymore) and cannot be used as an input by Math language .
Therefore the expression 2^aleph0 cannot exist and we cannot construct
the transfinite universes.
Shortly speaking, what is called uncountable is not uncountable but simply does not exist in any form of input that can be used by Math language.
Through my point of view base 2 can exit
iff (C is finite) OR (RT XOR LF are in C)
Please show me how C is a non-finite set where
(RT AND LF are in C) AND (base 2 exists --> binary tree exists).
Also please tell me what is the value of RT when RT AND LF are in C AND C is a non-finite set.
My basic approach about the infinity concept is that redundancy and uncertainty are naturally involved, no more no less.
Therefore aleph0 is a notation that stands for general and flexible "cloud like" thing.
Cantor's thing is a frozen one, mine is not.
I think that my approach is much more interesting and fruitful than Cantor's approach.
Please let me put these two different approaches "on the table" and I would like to examine them together with you.
I am going to write my point of view on Cantor’s approach in a very simple way that (I hope) can be understood by you.
Please read it, and open my eyes to important things that I omit, don’t understand, distorting and so on.
So here it is:
1) Let set C be a complete non-empty binary tree where complete non-empty binary tree exists iff both root AND all its leafs are in C.
2) Let RT be the root , let LF be the all leafs,
therefore RT AND LF are in C --> [RT , LF].
( In my point of view RT XOR LF are in C --> ( (RT… , …LF) OR (LT… , LF] OR
[RT ,…LF) ) AND NOT [RT , LF] where “…” means unreachable. )
Now, from my point of view I see these basic problems when RT AND LF are in C:
1) If both RT AND LF are in C, then C must be a finite set.
2) If C is a non-finite set (through my point of view C is forced to be a non-finite set) then the base value 2 (which is the fundamental structural property of the non-empty binary-tree) cannot exist. Also RT value is unknown.
We must realize that if RT AND LF are in C AND C is a non-finite set, then the structural property of our information (in this case we are talking about the binary tree structure) collapsed into itself (we have no infinitely many elements anymore) and cannot be used as an input by Math language .
Therefore the expression 2^aleph0 cannot exist and we cannot construct
the transfinite universes.
Shortly speaking, what is called uncountable is not uncountable but simply does not exist in any form of input that can be used by Math language.
Through my point of view base 2 can exit
iff (C is finite) OR (RT XOR LF are in C)
Please show me how C is a non-finite set where
(RT AND LF are in C) AND (base 2 exists --> binary tree exists).
Also please tell me what is the value of RT when RT AND LF are in C AND C is a non-finite set.
- #119
Hurkyl
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Exercise: try defining "structural property" without making any reference to what a natural number "is".
We math people call that a contradiction. And it CAN be used in "Math language" through the Law of contradiction:
"If P implies a contradiction, then P is false"
Or, more formally:
[tex]
P \rightarrow (Q \wedge \neg Q) \vdash \neg P
[/tex]
or equivalently
[tex]
P \rightarrow Q, \neg Q \vdash \neg P
[/tex]
Or, informally, it goes like this:
Let's make an assumption (call it P).
We derive a contradiction.
We conclude our assumption was wrong.
Or, if I may try and translate into Organic-speak:
If P causes our information to collapse, then P cannot be true.
And how does 2^aleph0 relate to anything above this statement in your post?
- #120
Organic
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I did more then thet, it can be found here:
http://www.geocities.com/complementarytheory/count.pdf
You ignore the inner information structure of the natural numbers,
I don't.
Therefore your natural numbers are private case of my natural numbers
as can be clearly shown here:
http://www.geocities.com/complementarytheory/ETtable.pdf
You cannot translate my ideas by your mathematical tools, because you ignore the information structure of the Binary tree as irrelevant to you, and looking only on its quantitative "shadow" that falling on the "real line".
So you did not understand me then:
1) On what basis you translate me?
2) After you read all what I wrote can you answer to this?:
Please show me how C is a non-finite set where
(RT AND LF are in C) AND (base 2 exists --> binary tree exists).
Also please tell me what is the value of RT when RT AND LF are in C AND C is a non-finite set.
Last edited:
- #121
Hurkyl
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The reason for this is because you do not convey your ideas effectively, and you are self-contradictory.
(e.g. admitting that the set of real numbers satisfies the definition of "uncountable", yet in the same breath you assert that the set of real numbers is not "uncountable")
Can you translate it into math-speak?
- #122
moshek
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Hurkyl :
I there is in mathematics
a definition to "definition" ?
Thank you
Moshek
I there is in mathematics
a definition to "definition" ?
Thank you
Moshek
- #123
Hurkyl
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I'm sure there are mathematical theories which have a class of objects called definitions, but in general the term "definition" is meta-mathematical, not mathematical, so to answer your question literally, the answer (in general) is no.
- #124
moshek
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Dear Hurkl!
I was glad for your kind answer to me !
Since you are a matematition
but you can't defind "definition"
Way you ask it from Organic ?
Thank you
Moshek
I was glad for your kind answer to me !
Since you are a matematition
but you can't defind "definition"
Way you ask it from Organic ?
Thank you
Moshek
- #125
Hurkyl
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I'm not asking him to define definition, I'm asking him to provide one. 
I'm probably somewhat more abstract than "mainstream" mathematics, but I would define "definition" metamathematically as simply one or more (precise) mathematical statements initially taken to be true.
The point, then, is that we can carry out logical deductions from the definitions to derive theorems.
I'm probably somewhat more abstract than "mainstream" mathematics, but I would define "definition" metamathematically as simply one or more (precise) mathematical statements initially taken to be true.
The point, then, is that we can carry out logical deductions from the definitions to derive theorems.
- #126
moshek
- 265
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No Hurkl !
The point is only a point !
Let me ask you my question in anothr way:
Do you believe that what happened to physice at 1905
chould Hapend also to mathematics in one day?
Thank you
Moshek
The point is only a point !
Let me ask you my question in anothr way:
Do you believe that what happened to physice at 1905
chould Hapend also to mathematics in one day?
Thank you
Moshek
- #127
Hurkyl
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It happened to mathematics first.
- #128
moshek
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when ?
- #129
Organic
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If you speak about non-Euclidian geometry then you right but what about QM and Bohr's complementary attitude?
No, N and R are enumerable by my definitions because my aleph0 is not your aleph0, no more no less.
Also I see that you don't understand my post about the difference between Cantor's aleph0 and my aleph0.
So, here it is again, but please this time stop on each part of it and please ask me about it, if you think that you don't understand it, thank you:
Dear Hurkyl (this time please answer to this post),
My basic approach about the infinity concept is that redundancy and uncertainty are naturally involved, no more no less.
Therefore aleph0 is a notation that stands for general and flexible "cloud like" thing.
Cantor's thing is a frozen one, mine is not.
I think that my approach is much more interesting and fruitful than Cantor's approach.
Please let me put these two different approaches "on the table" and I would like to examine them together with you.
I am going to write my point of view on Cantor’s approach in a very simple way that (I hope) can be understood by you.
Please read it, and open my eyes to important things that I omit, don’t understand, distorting and so on.
So here it is:
1) Let set C be a complete non-empty binary tree where complete non-empty binary tree exists iff both root AND all its leafs are in C.
2) Let RT be the root , let LF be the all leafs,
therefore RT AND LF are in C --> [RT , LF].
( In my point of view RT XOR LF are in C --> ( (RT… , …LF) OR (LT… , LF] OR
[RT ,…LF) ) AND NOT [RT , LF] where “…” means unreachable. )
Now, from my point of view I see these basic problems when RT AND LF are in C:
1) If both RT AND LF are in C, then C must be a finite set.
2) If C is a non-finite set (through my point of view C is forced to be a non-finite set) then the base value 2 (which is the fundamental structural property of the non-empty binary-tree) cannot exist. Also RT value is unknown.
We must realize that if RT AND LF are in C AND C is a non-finite set, then the structural property of our information (in this case we are talking about the binary tree structure) collapsed into itself (we have no infinitely many elements anymore) and cannot be used as an input by Math language .
Therefore the expression 2^aleph0 cannot exist and we cannot construct
the transfinite universes.
Shortly speaking, what is called uncountable is not uncountable but simply does not exist in any form of input that can be used by Math language.
Through my point of view base 2 can exit
iff (C is finite) OR (RT XOR LF are in C)
Please show me how C is a non-finite set where
(RT AND LF are in C) AND (base 2 exists --> binary tree exists).
Also please tell me what is the value of RT when RT AND LF are in C AND C is a non-finite set.
- #130
Hurkyl
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I'm not talking about non-Euclidean geometry.
I'm talking about the idea of eliminating unnecessary assumptions.
Remember what SR was all about; it was proven that the speed of light is measured to be the same in all reference frames. Physicists of the time were trying to add new and mysterious things to physics to "save" their ideas of how the universe should work. Einstein said to heck with it and said that we might as well study what our measurements say.
My memory of timelines sucks, but I think mathematical formalism has several decades on Einstein. If not, it's in full swing now. *shrug* For instance, look at category theory; it cares nothing about the objects themselves, just the fact that they are objects, and that there are functions between objects.
I'm talking about the idea of eliminating unnecessary assumptions.
Remember what SR was all about; it was proven that the speed of light is measured to be the same in all reference frames. Physicists of the time were trying to add new and mysterious things to physics to "save" their ideas of how the universe should work. Einstein said to heck with it and said that we might as well study what our measurements say.
My memory of timelines sucks, but I think mathematical formalism has several decades on Einstein. If not, it's in full swing now. *shrug* For instance, look at category theory; it cares nothing about the objects themselves, just the fact that they are objects, and that there are functions between objects.
- #131
Organic
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What measurements?
Again, what about QM and Bohr's complementary attitude?
- #132
Hurkyl
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What complementary attitude, and what about it?
- #133
Organic
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http://plato.stanford.edu/entries/qm-copenhagen/#4
Through my point of view Natural numbers are complementary elements, based on discreteness(particle-like)-continuum(wave-like) associations.
The information structure of the standard Natural numbers, is only a private case of these associations, for example:
http://www.geocities.com/complementarytheory/ETtable.pdf
More details can be found here:
http://www.geocities.com/complementarytheory/POV.pdf
Through my point of view Natural numbers are complementary elements, based on discreteness(particle-like)-continuum(wave-like) associations.
The information structure of the standard Natural numbers, is only a private case of these associations, for example:
http://www.geocities.com/complementarytheory/ETtable.pdf
More details can be found here:
http://www.geocities.com/complementarytheory/POV.pdf
Last edited:
- #134
moshek
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Hurkyl :
I am sorry, but what happend
to physics at 1905 by Einstein
Did not happened yet
to mathematics!
Moshek
I am sorry, but what happend
to physics at 1905 by Einstein
Did not happened yet
to mathematics!
Moshek
- #135
matt grime
Science Advisor
Homework Helper
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I can't quite decide *exactly* what happened in 1905 that's so important, but, Moshek, and please don't take this the wrong way, on what basis are you qualified to say what has happened to mathematics at any point?
- #136
suyver
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1905 is Einsteins "magic year": it is the year in which he published special relativity, the resolution of the photoelectric effect (for which he would get the Nobel prize later on), and the theoretical description of Brownian motion (that almost proved beyond doubt the existence of atoms). These three theoretical discoveries gave a profound paradigm shift in physics.
I have no idea if such coincedences have also been seen in the field of mathematics (I am not an expert), but I guess they probably have (Euler, Gauss, other giants?)...
- #137
moshek
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Dear Matt
Thank you for your nice question to me !
I will share with you today here very clear evidents that mathematics is standing today in front of changing paradigms like Einstein did in 1905 to Physics.
Dear Organic:
You may be interested in the following conference that stat today about the futher of Cardinals.
www.as.huji.ac.il/schools/math8/mathsprog.shtml[/URL]
Best
Moshek
Thank you for your nice question to me !
I will share with you today here very clear evidents that mathematics is standing today in front of changing paradigms like Einstein did in 1905 to Physics.
Dear Organic:
You may be interested in the following conference that stat today about the futher of Cardinals.
www.as.huji.ac.il/schools/math8/mathsprog.shtml[/URL]
Best
Moshek
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- #138
matt grime
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Arguably these are partly mathematical discoveries, at least I was taught about two of the three of them in my mathematics degree.
I suppose Russell would have to qualify, as would the category theory 'paradigm shift' (notice that, organic, shift, not change, shift) and then there was the bourbaki school, and the current shift away from that line of thinking. The mathematics of today is vastly different from that in Gauss's day - a lot of the proofs of that period don't stand up to scrutiny now. Then science was seen as a branch of philosophy, natural philosophy, and had according standards of proof. The shift in mathematics to today's view was gradual, and didn't have this alleged sudden epiphany, but it is arguable that saying physics fundamentally changed in 1905 is missing the years of research that went on up to that point that allowed these new discoveries to be accepted.
And we haven't even mentioned Cantor, Godel, Zermelo-Frankel, Nash, chaos theory, Mandelbrot, and various others to a lesser degree (Brylinski's loop space, Tsygan's simultaneous and independent discovery of Cyclic homology with Connes, Keller and Rickard's independent and simultaneous work on derived equivalences, Brauer's ground breaking work on representation theory...). What standards do you want to use to compare them?
I suppose Russell would have to qualify, as would the category theory 'paradigm shift' (notice that, organic, shift, not change, shift) and then there was the bourbaki school, and the current shift away from that line of thinking. The mathematics of today is vastly different from that in Gauss's day - a lot of the proofs of that period don't stand up to scrutiny now. Then science was seen as a branch of philosophy, natural philosophy, and had according standards of proof. The shift in mathematics to today's view was gradual, and didn't have this alleged sudden epiphany, but it is arguable that saying physics fundamentally changed in 1905 is missing the years of research that went on up to that point that allowed these new discoveries to be accepted.
And we haven't even mentioned Cantor, Godel, Zermelo-Frankel, Nash, chaos theory, Mandelbrot, and various others to a lesser degree (Brylinski's loop space, Tsygan's simultaneous and independent discovery of Cyclic homology with Connes, Keller and Rickard's independent and simultaneous work on derived equivalences, Brauer's ground breaking work on representation theory...). What standards do you want to use to compare them?
- #139
matt grime
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- #140
moshek
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Matt :
Did you ever read the 4 last lines
of Hilbert lecture at Paris 1900?
Organic already share with us that he dedicate almost 20 years from his life in trying the develop new perspective about mathematics and he share with us his discoveries very gently.
That fact that he is not famiyar with ordinary mathematics ( he admit that already ) is not necessarily relevant to the question if him material have the potential to make an Organic sift in mathematics.
Take care
Moshek
Did you ever read the 4 last lines
of Hilbert lecture at Paris 1900?
Organic already share with us that he dedicate almost 20 years from his life in trying the develop new perspective about mathematics and he share with us his discoveries very gently.
That fact that he is not famiyar with ordinary mathematics ( he admit that already ) is not necessarily relevant to the question if him material have the potential to make an Organic sift in mathematics.
Take care
Moshek