Paradoxes that remain paradoxical even when you understand the underlying theory
Clash Royale CLAN TAG#URR8PPP
up vote
5
down vote
favorite
It strikes me that the Banach-Tarski paradox (rearranging ball partitions)
is not dispelled even when you understand the underlying mathematics.
Perhaps Parrondo's paradox in Game Theory (sawtooth losing → winning) is analogous, in that it retains for me a sense of magic
even after studying simulations.
Perhaps Simpson's paradox in statistics (two plusses's→minus) is borderline, in that, even though it is easy to fall into it, it is
also easy to see why one's intuition was incorrect.
Q.Which mathematical "paradoxes" remain paradoxical even when you understand them
thoroughly?
I would judge Braess' paradox
(adding a shortcut to a road network impedes traffic) as the type of paradox that
is dispelled upon understanding it. Whereas the Banach-Tarski paradox remains
(for me) paradoxical even though I think I understand the mathematics
behind it.
A defensible answer to Q is: No paradoxes remain paradoxical when thoroughly understood—that's what it means to "understand"!
foundations paradoxes
asked Jan 16 '16 at 2:17
Joseph O'Rourke
17.2k248104
add a comment |Â
up vote
5
down vote
favorite
It strikes me that the Banach-Tarski paradox (rearranging ball partitions)
is not dispelled even when you understand the underlying mathematics.
Perhaps Parrondo's paradox in Game Theory (sawtooth losing → winning) is analogous, in that it retains for me a sense of magic
even after studying simulations.
Perhaps Simpson's paradox in statistics (two plusses's→minus) is borderline, in that, even though it is easy to fall into it, it is
also easy to see why one's intuition was incorrect.
Q.Which mathematical "paradoxes" remain paradoxical even when you understand them
thoroughly?
I would judge Braess' paradox
(adding a shortcut to a road network impedes traffic) as the type of paradox that
is dispelled upon understanding it. Whereas the Banach-Tarski paradox remains
(for me) paradoxical even though I think I understand the mathematics
behind it.
A defensible answer to Q is: No paradoxes remain paradoxical when thoroughly understood—that's what it means to "understand"!
foundations paradoxes
asked Jan 16 '16 at 2:17
Joseph O'Rourke
17.2k248104
1
Maybe it's just me but I find something very simple that everyone understands paradoxical: voting with millions of voters...why does it matter if I vote? This is really the Sorites paradox in disguise, but the pile of sand version doesn't do it for me.
– Matt Samuel
Jan 16 '16 at 2:39
1
I think "paradox" here means "counter to the understanding we derive from the world around us in everyday life". Mathematics is (in some sense) a model of this real world. We make up definitions of abstract objects that mimic the behaviour of the real world, and then we reason about these objects, hoping that our reasoning will tell us something interesting about the real objects they represent. So, when the mathematics gives us results that are counter-intuitive, my feeling is that the mathematical definitions might not be suitable models of reality.
– bubba
Jan 16 '16 at 2:43
My other answer is the one you anticipated in your last paragraph :-)
– bubba
Jan 16 '16 at 2:45
1
Perhaps : en.wikipedia.org/wiki/Burali-Forti_paradox
– DanielV
Jan 16 '16 at 2:59
add a comment |Â
up vote
5
down vote
favorite
up vote
5
down vote
favorite
It strikes me that the Banach-Tarski paradox (rearranging ball partitions)
is not dispelled even when you understand the underlying mathematics.
Perhaps Parrondo's paradox in Game Theory (sawtooth losing → winning) is analogous, in that it retains for me a sense of magic
even after studying simulations.
Perhaps Simpson's paradox in statistics (two plusses's→minus) is borderline, in that, even though it is easy to fall into it, it is
also easy to see why one's intuition was incorrect.
Q.Which mathematical "paradoxes" remain paradoxical even when you understand them
thoroughly?
I would judge Braess' paradox
(adding a shortcut to a road network impedes traffic) as the type of paradox that
is dispelled upon understanding it. Whereas the Banach-Tarski paradox remains
(for me) paradoxical even though I think I understand the mathematics
behind it.
A defensible answer to Q is: No paradoxes remain paradoxical when thoroughly understood—that's what it means to "understand"!
foundations paradoxes
asked Jan 16 '16 at 2:17
Joseph O'Rourke
17.2k248104
It strikes me that the Banach-Tarski paradox (rearranging ball partitions)
is not dispelled even when you understand the underlying mathematics.
Perhaps Parrondo's paradox in Game Theory (sawtooth losing → winning) is analogous, in that it retains for me a sense of magic
even after studying simulations.
Perhaps Simpson's paradox in statistics (two plusses's→minus) is borderline, in that, even though it is easy to fall into it, it is
also easy to see why one's intuition was incorrect.
Q.Which mathematical "paradoxes" remain paradoxical even when you understand them
thoroughly?
I would judge Braess' paradox
(adding a shortcut to a road network impedes traffic) as the type of paradox that
is dispelled upon understanding it. Whereas the Banach-Tarski paradox remains
(for me) paradoxical even though I think I understand the mathematics
behind it.
A defensible answer to Q is: No paradoxes remain paradoxical when thoroughly understood—that's what it means to "understand"!
foundations paradoxes
asked Jan 16 '16 at 2:17
Joseph O'Rourke
17.2k248104
edited Jan 16 '16 at 2:37
asked Jan 16 '16 at 2:17
Joseph O'Rourke
17.2k248104
asked Jan 16 '16 at 2:17
Joseph O'Rourke
17.2k248104
asked Jan 16 '16 at 2:17
Joseph O'Rourke
17.2k248104
17.2k248104
1
Maybe it's just me but I find something very simple that everyone understands paradoxical: voting with millions of voters...why does it matter if I vote? This is really the Sorites paradox in disguise, but the pile of sand version doesn't do it for me.
– Matt Samuel
Jan 16 '16 at 2:39
1
I think "paradox" here means "counter to the understanding we derive from the world around us in everyday life". Mathematics is (in some sense) a model of this real world. We make up definitions of abstract objects that mimic the behaviour of the real world, and then we reason about these objects, hoping that our reasoning will tell us something interesting about the real objects they represent. So, when the mathematics gives us results that are counter-intuitive, my feeling is that the mathematical definitions might not be suitable models of reality.
– bubba
Jan 16 '16 at 2:43
My other answer is the one you anticipated in your last paragraph :-)
– bubba
Jan 16 '16 at 2:45
1
Perhaps : en.wikipedia.org/wiki/Burali-Forti_paradox
– DanielV
Jan 16 '16 at 2:59
add a comment |Â
1
Maybe it's just me but I find something very simple that everyone understands paradoxical: voting with millions of voters...why does it matter if I vote? This is really the Sorites paradox in disguise, but the pile of sand version doesn't do it for me.
– Matt Samuel
Jan 16 '16 at 2:39
1
I think "paradox" here means "counter to the understanding we derive from the world around us in everyday life". Mathematics is (in some sense) a model of this real world. We make up definitions of abstract objects that mimic the behaviour of the real world, and then we reason about these objects, hoping that our reasoning will tell us something interesting about the real objects they represent. So, when the mathematics gives us results that are counter-intuitive, my feeling is that the mathematical definitions might not be suitable models of reality.
– bubba
Jan 16 '16 at 2:43
My other answer is the one you anticipated in your last paragraph :-)
– bubba
Jan 16 '16 at 2:45
1
Perhaps : en.wikipedia.org/wiki/Burali-Forti_paradox
– DanielV
Jan 16 '16 at 2:59
1
1
Maybe it's just me but I find something very simple that everyone understands paradoxical: voting with millions of voters...why does it matter if I vote? This is really the Sorites paradox in disguise, but the pile of sand version doesn't do it for me.
– Matt Samuel
Jan 16 '16 at 2:39
Maybe it's just me but I find something very simple that everyone understands paradoxical: voting with millions of voters...why does it matter if I vote? This is really the Sorites paradox in disguise, but the pile of sand version doesn't do it for me.
– Matt Samuel
Jan 16 '16 at 2:39
1
1
I think "paradox" here means "counter to the understanding we derive from the world around us in everyday life". Mathematics is (in some sense) a model of this real world. We make up definitions of abstract objects that mimic the behaviour of the real world, and then we reason about these objects, hoping that our reasoning will tell us something interesting about the real objects they represent. So, when the mathematics gives us results that are counter-intuitive, my feeling is that the mathematical definitions might not be suitable models of reality.
– bubba
Jan 16 '16 at 2:43
I think "paradox" here means "counter to the understanding we derive from the world around us in everyday life". Mathematics is (in some sense) a model of this real world. We make up definitions of abstract objects that mimic the behaviour of the real world, and then we reason about these objects, hoping that our reasoning will tell us something interesting about the real objects they represent. So, when the mathematics gives us results that are counter-intuitive, my feeling is that the mathematical definitions might not be suitable models of reality.
– bubba
Jan 16 '16 at 2:43
My other answer is the one you anticipated in your last paragraph :-)
– bubba
Jan 16 '16 at 2:45
My other answer is the one you anticipated in your last paragraph :-)
– bubba
Jan 16 '16 at 2:45
1
1
Perhaps : en.wikipedia.org/wiki/Burali-Forti_paradox
– DanielV
Jan 16 '16 at 2:59
Perhaps : en.wikipedia.org/wiki/Burali-Forti_paradox
– DanielV
Jan 16 '16 at 2:59
add a comment |Â
3 Answers
3
active
oldest
votes
up vote
3
down vote
accepted
One of my favorites: The real numbers are a vector space over the rationals. Therefore there is a basis for this vector space (a consequence of the Axiom of Choice), and such one basis must lie in the unit interval, since you can replace any basis element by a multiple between 0 and 1.
A lot of paradoxes come from the Axiom of Choice, which nevertheless strikes me as intuitive and a lot of paradoxes come from our failure to understand that infinite sets don't have to correspond to our expectations for finite sets.
answered Jan 16 '16 at 3:10
Syd Henderson
1,55868
Nice example! And that basis must be uncountable, and presumably cannot be described---We just know it exists.
– Joseph O'Rourke
Jan 16 '16 at 15:24
@Joseph: Unless you add more axioms to ZFC, it is consistent that there is no "reasonable" description at all. And unless you add the axiom of choice, it is possible that there is no such basis to begin with.
– Asaf Karagila♦
Jan 20 '16 at 21:47
But in all seriousness, how is this paradoxical?
– Asaf Karagila♦
Jan 20 '16 at 21:50
add a comment |Â
up vote
1
down vote
From the title it is clear that the issue is psychological.
Purely personal example: Though this is not a paradox, existence of countable dense and sigma additivity leads to the notion that there exist an open dense set in the real line of arbitrarily small measure.
Another thing is we accept the standard three axioms a metric has to satisfy. This leads to $p$-adic metric spaces where two open sets can intersect only one is contained in another.
I am reminded of the saying, " A man convinced against his will is of the same opinion still".
Let me quote more.
Einstein about statistical/quantum mechanics,"God does not play dice".
Gordon on Hilbert's finiteness theorem on invariants
"This is not mathematics, this is theology".
I think Kronecker also had problems with Cantor's theory of infinity.
answered Jan 16 '16 at 2:54
P Vanchinathan
14k12035
add a comment |Â
up vote
0
down vote
The axiom of choice states that for every set of nonempty sets, there exists a function that assigns to each of those sets one element of that set. I find it very weird that the axiom of choice might not be true. It's not weird that it is not provable in ZF. Rather since it's not provable in ZF, I'm not sure that it's true and I feel like if it turns out not to be true, then that's very wierd.
answered Aug 18 at 1:03
Timothy
262211
add a comment |Â
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
One of my favorites: The real numbers are a vector space over the rationals. Therefore there is a basis for this vector space (a consequence of the Axiom of Choice), and such one basis must lie in the unit interval, since you can replace any basis element by a multiple between 0 and 1.
A lot of paradoxes come from the Axiom of Choice, which nevertheless strikes me as intuitive and a lot of paradoxes come from our failure to understand that infinite sets don't have to correspond to our expectations for finite sets.
answered Jan 16 '16 at 3:10
Syd Henderson
1,55868
Nice example! And that basis must be uncountable, and presumably cannot be described---We just know it exists.
– Joseph O'Rourke
Jan 16 '16 at 15:24
@Joseph: Unless you add more axioms to ZFC, it is consistent that there is no "reasonable" description at all. And unless you add the axiom of choice, it is possible that there is no such basis to begin with.
– Asaf Karagila♦
Jan 20 '16 at 21:47
But in all seriousness, how is this paradoxical?
– Asaf Karagila♦
Jan 20 '16 at 21:50
add a comment |Â
up vote
3
down vote
accepted
One of my favorites: The real numbers are a vector space over the rationals. Therefore there is a basis for this vector space (a consequence of the Axiom of Choice), and such one basis must lie in the unit interval, since you can replace any basis element by a multiple between 0 and 1.
A lot of paradoxes come from the Axiom of Choice, which nevertheless strikes me as intuitive and a lot of paradoxes come from our failure to understand that infinite sets don't have to correspond to our expectations for finite sets.
answered Jan 16 '16 at 3:10
Syd Henderson
1,55868
Nice example! And that basis must be uncountable, and presumably cannot be described---We just know it exists.
– Joseph O'Rourke
Jan 16 '16 at 15:24
@Joseph: Unless you add more axioms to ZFC, it is consistent that there is no "reasonable" description at all. And unless you add the axiom of choice, it is possible that there is no such basis to begin with.
– Asaf Karagila♦
Jan 20 '16 at 21:47
But in all seriousness, how is this paradoxical?
– Asaf Karagila♦
Jan 20 '16 at 21:50
add a comment |Â
up vote
3
down vote
accepted
up vote
3
down vote
accepted
One of my favorites: The real numbers are a vector space over the rationals. Therefore there is a basis for this vector space (a consequence of the Axiom of Choice), and such one basis must lie in the unit interval, since you can replace any basis element by a multiple between 0 and 1.
A lot of paradoxes come from the Axiom of Choice, which nevertheless strikes me as intuitive and a lot of paradoxes come from our failure to understand that infinite sets don't have to correspond to our expectations for finite sets.
answered Jan 16 '16 at 3:10
Syd Henderson
1,55868
One of my favorites: The real numbers are a vector space over the rationals. Therefore there is a basis for this vector space (a consequence of the Axiom of Choice), and such one basis must lie in the unit interval, since you can replace any basis element by a multiple between 0 and 1.
A lot of paradoxes come from the Axiom of Choice, which nevertheless strikes me as intuitive and a lot of paradoxes come from our failure to understand that infinite sets don't have to correspond to our expectations for finite sets.
answered Jan 16 '16 at 3:10
Syd Henderson
1,55868
answered Jan 16 '16 at 3:10
Syd Henderson
1,55868
answered Jan 16 '16 at 3:10
Syd Henderson
1,55868
answered Jan 16 '16 at 3:10
Syd Henderson
1,55868
1,55868
Nice example! And that basis must be uncountable, and presumably cannot be described---We just know it exists.
– Joseph O'Rourke
Jan 16 '16 at 15:24
@Joseph: Unless you add more axioms to ZFC, it is consistent that there is no "reasonable" description at all. And unless you add the axiom of choice, it is possible that there is no such basis to begin with.
– Asaf Karagila♦
Jan 20 '16 at 21:47
But in all seriousness, how is this paradoxical?
– Asaf Karagila♦
Jan 20 '16 at 21:50
add a comment |Â
Nice example! And that basis must be uncountable, and presumably cannot be described---We just know it exists.
– Joseph O'Rourke
Jan 16 '16 at 15:24
@Joseph: Unless you add more axioms to ZFC, it is consistent that there is no "reasonable" description at all. And unless you add the axiom of choice, it is possible that there is no such basis to begin with.
– Asaf Karagila♦
Jan 20 '16 at 21:47
But in all seriousness, how is this paradoxical?
– Asaf Karagila♦
Jan 20 '16 at 21:50
Nice example! And that basis must be uncountable, and presumably cannot be described---We just know it exists.
– Joseph O'Rourke
Jan 16 '16 at 15:24
Nice example! And that basis must be uncountable, and presumably cannot be described---We just know it exists.
– Joseph O'Rourke
Jan 16 '16 at 15:24
@Joseph: Unless you add more axioms to ZFC, it is consistent that there is no "reasonable" description at all. And unless you add the axiom of choice, it is possible that there is no such basis to begin with.
– Asaf Karagila♦
Jan 20 '16 at 21:47
@Joseph: Unless you add more axioms to ZFC, it is consistent that there is no "reasonable" description at all. And unless you add the axiom of choice, it is possible that there is no such basis to begin with.
– Asaf Karagila♦
Jan 20 '16 at 21:47
But in all seriousness, how is this paradoxical?
– Asaf Karagila♦
Jan 20 '16 at 21:50
But in all seriousness, how is this paradoxical?
– Asaf Karagila♦
Jan 20 '16 at 21:50
add a comment |Â
up vote
1
down vote
From the title it is clear that the issue is psychological.
Purely personal example: Though this is not a paradox, existence of countable dense and sigma additivity leads to the notion that there exist an open dense set in the real line of arbitrarily small measure.
Another thing is we accept the standard three axioms a metric has to satisfy. This leads to $p$-adic metric spaces where two open sets can intersect only one is contained in another.
I am reminded of the saying, " A man convinced against his will is of the same opinion still".
Let me quote more.
Einstein about statistical/quantum mechanics,"God does not play dice".
Gordon on Hilbert's finiteness theorem on invariants
"This is not mathematics, this is theology".
I think Kronecker also had problems with Cantor's theory of infinity.
answered Jan 16 '16 at 2:54
P Vanchinathan
14k12035
add a comment |Â
up vote
1
down vote
From the title it is clear that the issue is psychological.
Purely personal example: Though this is not a paradox, existence of countable dense and sigma additivity leads to the notion that there exist an open dense set in the real line of arbitrarily small measure.
Another thing is we accept the standard three axioms a metric has to satisfy. This leads to $p$-adic metric spaces where two open sets can intersect only one is contained in another.
I am reminded of the saying, " A man convinced against his will is of the same opinion still".
Let me quote more.
Einstein about statistical/quantum mechanics,"God does not play dice".
Gordon on Hilbert's finiteness theorem on invariants
"This is not mathematics, this is theology".
I think Kronecker also had problems with Cantor's theory of infinity.
answered Jan 16 '16 at 2:54
P Vanchinathan
14k12035
add a comment |Â
up vote
1
down vote
up vote
1
down vote
From the title it is clear that the issue is psychological.
Purely personal example: Though this is not a paradox, existence of countable dense and sigma additivity leads to the notion that there exist an open dense set in the real line of arbitrarily small measure.
Another thing is we accept the standard three axioms a metric has to satisfy. This leads to $p$-adic metric spaces where two open sets can intersect only one is contained in another.
I am reminded of the saying, " A man convinced against his will is of the same opinion still".
Let me quote more.
Einstein about statistical/quantum mechanics,"God does not play dice".
Gordon on Hilbert's finiteness theorem on invariants
"This is not mathematics, this is theology".
I think Kronecker also had problems with Cantor's theory of infinity.
answered Jan 16 '16 at 2:54
P Vanchinathan
14k12035
From the title it is clear that the issue is psychological.
Purely personal example: Though this is not a paradox, existence of countable dense and sigma additivity leads to the notion that there exist an open dense set in the real line of arbitrarily small measure.
Another thing is we accept the standard three axioms a metric has to satisfy. This leads to $p$-adic metric spaces where two open sets can intersect only one is contained in another.
I am reminded of the saying, " A man convinced against his will is of the same opinion still".
Let me quote more.
Einstein about statistical/quantum mechanics,"God does not play dice".
Gordon on Hilbert's finiteness theorem on invariants
"This is not mathematics, this is theology".
I think Kronecker also had problems with Cantor's theory of infinity.
answered Jan 16 '16 at 2:54
P Vanchinathan
14k12035
answered Jan 16 '16 at 2:54
P Vanchinathan
14k12035
answered Jan 16 '16 at 2:54
P Vanchinathan
14k12035
answered Jan 16 '16 at 2:54
P Vanchinathan
14k12035
14k12035
add a comment |Â
add a comment |Â
up vote
0
down vote
The axiom of choice states that for every set of nonempty sets, there exists a function that assigns to each of those sets one element of that set. I find it very weird that the axiom of choice might not be true. It's not weird that it is not provable in ZF. Rather since it's not provable in ZF, I'm not sure that it's true and I feel like if it turns out not to be true, then that's very wierd.
answered Aug 18 at 1:03
Timothy
262211
add a comment |Â
up vote
0
down vote
The axiom of choice states that for every set of nonempty sets, there exists a function that assigns to each of those sets one element of that set. I find it very weird that the axiom of choice might not be true. It's not weird that it is not provable in ZF. Rather since it's not provable in ZF, I'm not sure that it's true and I feel like if it turns out not to be true, then that's very wierd.
answered Aug 18 at 1:03
Timothy
262211
add a comment |Â
up vote
0
down vote
up vote
0
down vote
The axiom of choice states that for every set of nonempty sets, there exists a function that assigns to each of those sets one element of that set. I find it very weird that the axiom of choice might not be true. It's not weird that it is not provable in ZF. Rather since it's not provable in ZF, I'm not sure that it's true and I feel like if it turns out not to be true, then that's very wierd.
answered Aug 18 at 1:03
Timothy
262211
The axiom of choice states that for every set of nonempty sets, there exists a function that assigns to each of those sets one element of that set. I find it very weird that the axiom of choice might not be true. It's not weird that it is not provable in ZF. Rather since it's not provable in ZF, I'm not sure that it's true and I feel like if it turns out not to be true, then that's very wierd.
answered Aug 18 at 1:03
Timothy
262211
answered Aug 18 at 1:03
Timothy
262211
answered Aug 18 at 1:03
Timothy
262211
answered Aug 18 at 1:03
Timothy
262211
262211
add a comment |Â
add a comment |Â
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1
Maybe it's just me but I find something very simple that everyone understands paradoxical: voting with millions of voters...why does it matter if I vote? This is really the Sorites paradox in disguise, but the pile of sand version doesn't do it for me.
– Matt Samuel
Jan 16 '16 at 2:39
1
I think "paradox" here means "counter to the understanding we derive from the world around us in everyday life". Mathematics is (in some sense) a model of this real world. We make up definitions of abstract objects that mimic the behaviour of the real world, and then we reason about these objects, hoping that our reasoning will tell us something interesting about the real objects they represent. So, when the mathematics gives us results that are counter-intuitive, my feeling is that the mathematical definitions might not be suitable models of reality.
– bubba
Jan 16 '16 at 2:43
My other answer is the one you anticipated in your last paragraph :-)
– bubba
Jan 16 '16 at 2:45
1
Perhaps : en.wikipedia.org/wiki/Burali-Forti_paradox
– DanielV
Jan 16 '16 at 2:59