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Vladimir Arnold on formal thinking
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In an interview, Vladimir Arnold talks about his teaching experience in France and condemns the formal thinking of the students. In the end he concludes that although their reasoning is logically correct, they understand nothing. If their deduction is correct (as he says), what is wrong with the given solution? What do the students not understand (or where does exactly their mathematical understanding fail?)? How can oneself make sure to "understand properly"? He says:
For example, at a written examination in
dynamical systems for fourth-year students at
Paris-Dauphine, one problem was to find the
limit of the solution of a system of Hamiltonian
equations on the phase plane starting with some
given initial point when time goes to infinity. The
idea was to choose the initial point on a separatrix
of a saddle, with the limit being the saddle
point.
Preparing the examination problem, I made
an arithmetical error, and the phase curve (the
energy-level curve containing the initial point)
was a closed oval instead of the separatrix. The
students discovered this and concluded that
there exists a finite time $T$ at which the solution
returns to the initial point. Using the unicity
theorem, they were able to deduce that for any
integer $n$ the value of the solution at time $nT$ is
still the initial point. Then came the conclusion:
since the limit at infinite time coincides with the
limit for any subsequence of times going to infinity,
the limit is equal to the initial point! This solution was invented independently by several
good students sitting at different places in the
examination hall. In all this reasoning, there are
no logical mistakes. It is a correct deduction
which one may also generate by a computer. It
is apparent that the authors understood nothing.
soft-question dynamical-systems education math-history
edited Aug 12 at 7:16
wchargin
1,0981024
asked Aug 11 at 16:47
Breaking M_a_t
599315
add a comment |Â
up vote
10
down vote
favorite
In an interview, Vladimir Arnold talks about his teaching experience in France and condemns the formal thinking of the students. In the end he concludes that although their reasoning is logically correct, they understand nothing. If their deduction is correct (as he says), what is wrong with the given solution? What do the students not understand (or where does exactly their mathematical understanding fail?)? How can oneself make sure to "understand properly"? He says:
For example, at a written examination in
dynamical systems for fourth-year students at
Paris-Dauphine, one problem was to find the
limit of the solution of a system of Hamiltonian
equations on the phase plane starting with some
given initial point when time goes to infinity. The
idea was to choose the initial point on a separatrix
of a saddle, with the limit being the saddle
point.
Preparing the examination problem, I made
an arithmetical error, and the phase curve (the
energy-level curve containing the initial point)
was a closed oval instead of the separatrix. The
students discovered this and concluded that
there exists a finite time $T$ at which the solution
returns to the initial point. Using the unicity
theorem, they were able to deduce that for any
integer $n$ the value of the solution at time $nT$ is
still the initial point. Then came the conclusion:
since the limit at infinite time coincides with the
limit for any subsequence of times going to infinity,
the limit is equal to the initial point! This solution was invented independently by several
good students sitting at different places in the
examination hall. In all this reasoning, there are
no logical mistakes. It is a correct deduction
which one may also generate by a computer. It
is apparent that the authors understood nothing.
soft-question dynamical-systems education math-history
edited Aug 12 at 7:16
wchargin
1,0981024
asked Aug 11 at 16:47
Breaking M_a_t
599315
5
Arnold is saying, that while the French think they're clever, he (Arnold) actually is clever.
– Lord Shark the Unknown
Aug 11 at 16:57
18
Arnold blames his students for Arnold having concocted a faulty examination text.
– Did
Aug 11 at 17:01
3
Isn't the conclusion wrong? In this case the solution to the ODE is periodic. It can't converge unless it's constant.
– Jack M
Aug 11 at 17:10
2
@JackM The reasoning is correct given the assumption that the solution approaches a limit. These students failed to recognize that this assumption was false, despite being perfectly aware that the phase curve was a closed oval.
– littleO
Aug 11 at 18:18
add a comment |Â
up vote
10
down vote
favorite
up vote
10
down vote
favorite
In an interview, Vladimir Arnold talks about his teaching experience in France and condemns the formal thinking of the students. In the end he concludes that although their reasoning is logically correct, they understand nothing. If their deduction is correct (as he says), what is wrong with the given solution? What do the students not understand (or where does exactly their mathematical understanding fail?)? How can oneself make sure to "understand properly"? He says:
For example, at a written examination in
dynamical systems for fourth-year students at
Paris-Dauphine, one problem was to find the
limit of the solution of a system of Hamiltonian
equations on the phase plane starting with some
given initial point when time goes to infinity. The
idea was to choose the initial point on a separatrix
of a saddle, with the limit being the saddle
point.
Preparing the examination problem, I made
an arithmetical error, and the phase curve (the
energy-level curve containing the initial point)
was a closed oval instead of the separatrix. The
students discovered this and concluded that
there exists a finite time $T$ at which the solution
returns to the initial point. Using the unicity
theorem, they were able to deduce that for any
integer $n$ the value of the solution at time $nT$ is
still the initial point. Then came the conclusion:
since the limit at infinite time coincides with the
limit for any subsequence of times going to infinity,
the limit is equal to the initial point! This solution was invented independently by several
good students sitting at different places in the
examination hall. In all this reasoning, there are
no logical mistakes. It is a correct deduction
which one may also generate by a computer. It
is apparent that the authors understood nothing.
soft-question dynamical-systems education math-history
edited Aug 12 at 7:16
wchargin
1,0981024
asked Aug 11 at 16:47
Breaking M_a_t
599315
In an interview, Vladimir Arnold talks about his teaching experience in France and condemns the formal thinking of the students. In the end he concludes that although their reasoning is logically correct, they understand nothing. If their deduction is correct (as he says), what is wrong with the given solution? What do the students not understand (or where does exactly their mathematical understanding fail?)? How can oneself make sure to "understand properly"? He says:
For example, at a written examination in
dynamical systems for fourth-year students at
Paris-Dauphine, one problem was to find the
limit of the solution of a system of Hamiltonian
equations on the phase plane starting with some
given initial point when time goes to infinity. The
idea was to choose the initial point on a separatrix
of a saddle, with the limit being the saddle
point.
Preparing the examination problem, I made
an arithmetical error, and the phase curve (the
energy-level curve containing the initial point)
was a closed oval instead of the separatrix. The
students discovered this and concluded that
there exists a finite time $T$ at which the solution
returns to the initial point. Using the unicity
theorem, they were able to deduce that for any
integer $n$ the value of the solution at time $nT$ is
still the initial point. Then came the conclusion:
since the limit at infinite time coincides with the
limit for any subsequence of times going to infinity,
the limit is equal to the initial point! This solution was invented independently by several
good students sitting at different places in the
examination hall. In all this reasoning, there are
no logical mistakes. It is a correct deduction
which one may also generate by a computer. It
is apparent that the authors understood nothing.
soft-question dynamical-systems education math-history
edited Aug 12 at 7:16
wchargin
1,0981024
asked Aug 11 at 16:47
Breaking M_a_t
599315
edited Aug 12 at 7:16
wchargin
1,0981024
edited Aug 12 at 7:16
wchargin
1,0981024
edited Aug 12 at 7:16
wchargin
1,0981024
1,0981024
asked Aug 11 at 16:47
Breaking M_a_t
599315
asked Aug 11 at 16:47
Breaking M_a_t
599315
asked Aug 11 at 16:47
Breaking M_a_t
599315
599315
5
Arnold is saying, that while the French think they're clever, he (Arnold) actually is clever.
– Lord Shark the Unknown
Aug 11 at 16:57
18
Arnold blames his students for Arnold having concocted a faulty examination text.
– Did
Aug 11 at 17:01
3
Isn't the conclusion wrong? In this case the solution to the ODE is periodic. It can't converge unless it's constant.
– Jack M
Aug 11 at 17:10
2
@JackM The reasoning is correct given the assumption that the solution approaches a limit. These students failed to recognize that this assumption was false, despite being perfectly aware that the phase curve was a closed oval.
– littleO
Aug 11 at 18:18
add a comment |Â
5
Arnold is saying, that while the French think they're clever, he (Arnold) actually is clever.
– Lord Shark the Unknown
Aug 11 at 16:57
18
Arnold blames his students for Arnold having concocted a faulty examination text.
– Did
Aug 11 at 17:01
3
Isn't the conclusion wrong? In this case the solution to the ODE is periodic. It can't converge unless it's constant.
– Jack M
Aug 11 at 17:10
2
@JackM The reasoning is correct given the assumption that the solution approaches a limit. These students failed to recognize that this assumption was false, despite being perfectly aware that the phase curve was a closed oval.
– littleO
Aug 11 at 18:18
5
5
Arnold is saying, that while the French think they're clever, he (Arnold) actually is clever.
– Lord Shark the Unknown
Aug 11 at 16:57
Arnold is saying, that while the French think they're clever, he (Arnold) actually is clever.
– Lord Shark the Unknown
Aug 11 at 16:57
18
18
Arnold blames his students for Arnold having concocted a faulty examination text.
– Did
Aug 11 at 17:01
Arnold blames his students for Arnold having concocted a faulty examination text.
– Did
Aug 11 at 17:01
3
3
Isn't the conclusion wrong? In this case the solution to the ODE is periodic. It can't converge unless it's constant.
– Jack M
Aug 11 at 17:10
Isn't the conclusion wrong? In this case the solution to the ODE is periodic. It can't converge unless it's constant.
– Jack M
Aug 11 at 17:10
2
2
@JackM The reasoning is correct given the assumption that the solution approaches a limit. These students failed to recognize that this assumption was false, despite being perfectly aware that the phase curve was a closed oval.
– littleO
Aug 11 at 18:18
@JackM The reasoning is correct given the assumption that the solution approaches a limit. These students failed to recognize that this assumption was false, despite being perfectly aware that the phase curve was a closed oval.
– littleO
Aug 11 at 18:18
add a comment |Â
4 Answers
4
active
oldest
votes
up vote
17
down vote
accepted
The solution has no limit, of course, because the phase curve is a closed oval.
However, the statement of the question assumed (erroneously) that the solution does have a limit. If it were true that the solution approaches a limit, then the argument given by these students would be completely correct.
The students should have recognized that the solution does not approach a limit, but rather is periodic.
answered Aug 11 at 18:08
littleO
26.3k540102
3
I went to university in France... I find it quite shocking that fourth-year students would make a mistake like this. I think Arnold's point about lacking a conceptual understanding is that if you were for instance visualizing the problem as you went along, you would spot the problem instantly.
– Jack M
Aug 11 at 18:25
5
I'd put this down to examination stress rather than a lack of understanding. Only an unusually confident student is going to put down an answer that contradicts the premise of the question. And if the only other answer you can find seems wrong, well, better to put that down and move on than to not answer at all, or even worse, to waste too much time trying to find a solution that doesn't exist. In a real-world situation those same students might very well have solved the problem correctly.
– Harry Johnston
Aug 12 at 1:37
add a comment |Â
up vote
2
down vote
Not myself being a professional mathematician, I have had a lot of professors express similar frustrated sentiments to my lectures.
Notwithstanding the fact that in the anecdote Arnold provides his students are merely responding to his own error in an examination setting (what else would he expect them to do but try to solve the problem given with the tools they'd been taught?), it sounds like he's making a distinction between a person understanding the formal components that make up a proof, and a person having a coherent intuition of the subject as a whole that would guide them to the formal components that make up a proof.
If you subscribe to Church's Thesis then you could explore Arnold's sentiments by the following extreme example:
Consider the programming language Malbolge, which was invented to be pathological. While programming is possible, and you could create a running program on your computer by copying someone else's code, or even by stringing together working functions that people have created, you could not afterwards claim to understand Malbolge like you might claim to understand Python. Nor could someone who modifies a working 'Hello World' program in Malbolge to print 'Goodbye World' claim to understand how Malgolbe interprets the function as a whole.
That's my interpretation of the passage anyway. Soft answer for a soft question, hope it was helpful.
answered Aug 11 at 17:30
Leland Reardon
588
add a comment |Â
up vote
1
down vote
He says, in my opinion, that a theory should be studied from its main conclusions and problems it allows to solve backward to the axioms, that is, by making abstraction, that is, by remaking mathematics. People should get their hands dirty with problems and only by banging their head against all problem difficulties they will understand the real appropriateness of axioms of a theory and at least naive mistakes can be avoided. The problem is that the main conclusions can be very numerous (infinity many) or at least much more numerous than the axioms and the theorems that the axioms follows and so it takes a great deal of time and force of will in order to examine them all. So he tries to incite seriuous students through this kind of provoking remarks.
answered Aug 11 at 17:56
trying
4,2811722
add a comment |Â
up vote
1
down vote
Surely,the students' answers were logically valid,but not logically sound? More experience with similar faulty questions would help them develop the habit of checking all initial assumptions in any given questions, and through this 'personal empiricism' gain intuition. Simply put, trust nothing, check everything.
answered Aug 11 at 19:10
Zeno
211
add a comment |Â
4 Answers
4
active
oldest
votes
4 Answers
4
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
17
down vote
accepted
The solution has no limit, of course, because the phase curve is a closed oval.
However, the statement of the question assumed (erroneously) that the solution does have a limit. If it were true that the solution approaches a limit, then the argument given by these students would be completely correct.
The students should have recognized that the solution does not approach a limit, but rather is periodic.
answered Aug 11 at 18:08
littleO
26.3k540102
3
I went to university in France... I find it quite shocking that fourth-year students would make a mistake like this. I think Arnold's point about lacking a conceptual understanding is that if you were for instance visualizing the problem as you went along, you would spot the problem instantly.
– Jack M
Aug 11 at 18:25
5
I'd put this down to examination stress rather than a lack of understanding. Only an unusually confident student is going to put down an answer that contradicts the premise of the question. And if the only other answer you can find seems wrong, well, better to put that down and move on than to not answer at all, or even worse, to waste too much time trying to find a solution that doesn't exist. In a real-world situation those same students might very well have solved the problem correctly.
– Harry Johnston
Aug 12 at 1:37
add a comment |Â
up vote
17
down vote
accepted
The solution has no limit, of course, because the phase curve is a closed oval.
However, the statement of the question assumed (erroneously) that the solution does have a limit. If it were true that the solution approaches a limit, then the argument given by these students would be completely correct.
The students should have recognized that the solution does not approach a limit, but rather is periodic.
answered Aug 11 at 18:08
littleO
26.3k540102
3
I went to university in France... I find it quite shocking that fourth-year students would make a mistake like this. I think Arnold's point about lacking a conceptual understanding is that if you were for instance visualizing the problem as you went along, you would spot the problem instantly.
– Jack M
Aug 11 at 18:25
5
I'd put this down to examination stress rather than a lack of understanding. Only an unusually confident student is going to put down an answer that contradicts the premise of the question. And if the only other answer you can find seems wrong, well, better to put that down and move on than to not answer at all, or even worse, to waste too much time trying to find a solution that doesn't exist. In a real-world situation those same students might very well have solved the problem correctly.
– Harry Johnston
Aug 12 at 1:37
add a comment |Â
up vote
17
down vote
accepted
up vote
17
down vote
accepted
The solution has no limit, of course, because the phase curve is a closed oval.
However, the statement of the question assumed (erroneously) that the solution does have a limit. If it were true that the solution approaches a limit, then the argument given by these students would be completely correct.
The students should have recognized that the solution does not approach a limit, but rather is periodic.
answered Aug 11 at 18:08
littleO
26.3k540102
The solution has no limit, of course, because the phase curve is a closed oval.
However, the statement of the question assumed (erroneously) that the solution does have a limit. If it were true that the solution approaches a limit, then the argument given by these students would be completely correct.
The students should have recognized that the solution does not approach a limit, but rather is periodic.
answered Aug 11 at 18:08
littleO
26.3k540102
edited Aug 11 at 18:16
answered Aug 11 at 18:08
littleO
26.3k540102
answered Aug 11 at 18:08
littleO
26.3k540102
answered Aug 11 at 18:08
littleO
26.3k540102
26.3k540102
3
I went to university in France... I find it quite shocking that fourth-year students would make a mistake like this. I think Arnold's point about lacking a conceptual understanding is that if you were for instance visualizing the problem as you went along, you would spot the problem instantly.
– Jack M
Aug 11 at 18:25
5
I'd put this down to examination stress rather than a lack of understanding. Only an unusually confident student is going to put down an answer that contradicts the premise of the question. And if the only other answer you can find seems wrong, well, better to put that down and move on than to not answer at all, or even worse, to waste too much time trying to find a solution that doesn't exist. In a real-world situation those same students might very well have solved the problem correctly.
– Harry Johnston
Aug 12 at 1:37
add a comment |Â
3
I went to university in France... I find it quite shocking that fourth-year students would make a mistake like this. I think Arnold's point about lacking a conceptual understanding is that if you were for instance visualizing the problem as you went along, you would spot the problem instantly.
– Jack M
Aug 11 at 18:25
5
I'd put this down to examination stress rather than a lack of understanding. Only an unusually confident student is going to put down an answer that contradicts the premise of the question. And if the only other answer you can find seems wrong, well, better to put that down and move on than to not answer at all, or even worse, to waste too much time trying to find a solution that doesn't exist. In a real-world situation those same students might very well have solved the problem correctly.
– Harry Johnston
Aug 12 at 1:37
3
3
I went to university in France... I find it quite shocking that fourth-year students would make a mistake like this. I think Arnold's point about lacking a conceptual understanding is that if you were for instance visualizing the problem as you went along, you would spot the problem instantly.
– Jack M
Aug 11 at 18:25
I went to university in France... I find it quite shocking that fourth-year students would make a mistake like this. I think Arnold's point about lacking a conceptual understanding is that if you were for instance visualizing the problem as you went along, you would spot the problem instantly.
– Jack M
Aug 11 at 18:25
5
5
I'd put this down to examination stress rather than a lack of understanding. Only an unusually confident student is going to put down an answer that contradicts the premise of the question. And if the only other answer you can find seems wrong, well, better to put that down and move on than to not answer at all, or even worse, to waste too much time trying to find a solution that doesn't exist. In a real-world situation those same students might very well have solved the problem correctly.
– Harry Johnston
Aug 12 at 1:37
I'd put this down to examination stress rather than a lack of understanding. Only an unusually confident student is going to put down an answer that contradicts the premise of the question. And if the only other answer you can find seems wrong, well, better to put that down and move on than to not answer at all, or even worse, to waste too much time trying to find a solution that doesn't exist. In a real-world situation those same students might very well have solved the problem correctly.
– Harry Johnston
Aug 12 at 1:37
add a comment |Â
up vote
2
down vote
Not myself being a professional mathematician, I have had a lot of professors express similar frustrated sentiments to my lectures.
Notwithstanding the fact that in the anecdote Arnold provides his students are merely responding to his own error in an examination setting (what else would he expect them to do but try to solve the problem given with the tools they'd been taught?), it sounds like he's making a distinction between a person understanding the formal components that make up a proof, and a person having a coherent intuition of the subject as a whole that would guide them to the formal components that make up a proof.
If you subscribe to Church's Thesis then you could explore Arnold's sentiments by the following extreme example:
Consider the programming language Malbolge, which was invented to be pathological. While programming is possible, and you could create a running program on your computer by copying someone else's code, or even by stringing together working functions that people have created, you could not afterwards claim to understand Malbolge like you might claim to understand Python. Nor could someone who modifies a working 'Hello World' program in Malbolge to print 'Goodbye World' claim to understand how Malgolbe interprets the function as a whole.
That's my interpretation of the passage anyway. Soft answer for a soft question, hope it was helpful.
answered Aug 11 at 17:30
Leland Reardon
588
add a comment |Â
up vote
2
down vote
Not myself being a professional mathematician, I have had a lot of professors express similar frustrated sentiments to my lectures.
Notwithstanding the fact that in the anecdote Arnold provides his students are merely responding to his own error in an examination setting (what else would he expect them to do but try to solve the problem given with the tools they'd been taught?), it sounds like he's making a distinction between a person understanding the formal components that make up a proof, and a person having a coherent intuition of the subject as a whole that would guide them to the formal components that make up a proof.
If you subscribe to Church's Thesis then you could explore Arnold's sentiments by the following extreme example:
Consider the programming language Malbolge, which was invented to be pathological. While programming is possible, and you could create a running program on your computer by copying someone else's code, or even by stringing together working functions that people have created, you could not afterwards claim to understand Malbolge like you might claim to understand Python. Nor could someone who modifies a working 'Hello World' program in Malbolge to print 'Goodbye World' claim to understand how Malgolbe interprets the function as a whole.
That's my interpretation of the passage anyway. Soft answer for a soft question, hope it was helpful.
answered Aug 11 at 17:30
Leland Reardon
588
add a comment |Â
up vote
2
down vote
up vote
2
down vote
Not myself being a professional mathematician, I have had a lot of professors express similar frustrated sentiments to my lectures.
Notwithstanding the fact that in the anecdote Arnold provides his students are merely responding to his own error in an examination setting (what else would he expect them to do but try to solve the problem given with the tools they'd been taught?), it sounds like he's making a distinction between a person understanding the formal components that make up a proof, and a person having a coherent intuition of the subject as a whole that would guide them to the formal components that make up a proof.
If you subscribe to Church's Thesis then you could explore Arnold's sentiments by the following extreme example:
Consider the programming language Malbolge, which was invented to be pathological. While programming is possible, and you could create a running program on your computer by copying someone else's code, or even by stringing together working functions that people have created, you could not afterwards claim to understand Malbolge like you might claim to understand Python. Nor could someone who modifies a working 'Hello World' program in Malbolge to print 'Goodbye World' claim to understand how Malgolbe interprets the function as a whole.
That's my interpretation of the passage anyway. Soft answer for a soft question, hope it was helpful.
answered Aug 11 at 17:30
Leland Reardon
588
Not myself being a professional mathematician, I have had a lot of professors express similar frustrated sentiments to my lectures.
Notwithstanding the fact that in the anecdote Arnold provides his students are merely responding to his own error in an examination setting (what else would he expect them to do but try to solve the problem given with the tools they'd been taught?), it sounds like he's making a distinction between a person understanding the formal components that make up a proof, and a person having a coherent intuition of the subject as a whole that would guide them to the formal components that make up a proof.
If you subscribe to Church's Thesis then you could explore Arnold's sentiments by the following extreme example:
Consider the programming language Malbolge, which was invented to be pathological. While programming is possible, and you could create a running program on your computer by copying someone else's code, or even by stringing together working functions that people have created, you could not afterwards claim to understand Malbolge like you might claim to understand Python. Nor could someone who modifies a working 'Hello World' program in Malbolge to print 'Goodbye World' claim to understand how Malgolbe interprets the function as a whole.
That's my interpretation of the passage anyway. Soft answer for a soft question, hope it was helpful.
answered Aug 11 at 17:30
Leland Reardon
588
answered Aug 11 at 17:30
Leland Reardon
588
answered Aug 11 at 17:30
Leland Reardon
588
answered Aug 11 at 17:30
Leland Reardon
588
588
add a comment |Â
add a comment |Â
up vote
1
down vote
He says, in my opinion, that a theory should be studied from its main conclusions and problems it allows to solve backward to the axioms, that is, by making abstraction, that is, by remaking mathematics. People should get their hands dirty with problems and only by banging their head against all problem difficulties they will understand the real appropriateness of axioms of a theory and at least naive mistakes can be avoided. The problem is that the main conclusions can be very numerous (infinity many) or at least much more numerous than the axioms and the theorems that the axioms follows and so it takes a great deal of time and force of will in order to examine them all. So he tries to incite seriuous students through this kind of provoking remarks.
answered Aug 11 at 17:56
trying
4,2811722
add a comment |Â
up vote
1
down vote
He says, in my opinion, that a theory should be studied from its main conclusions and problems it allows to solve backward to the axioms, that is, by making abstraction, that is, by remaking mathematics. People should get their hands dirty with problems and only by banging their head against all problem difficulties they will understand the real appropriateness of axioms of a theory and at least naive mistakes can be avoided. The problem is that the main conclusions can be very numerous (infinity many) or at least much more numerous than the axioms and the theorems that the axioms follows and so it takes a great deal of time and force of will in order to examine them all. So he tries to incite seriuous students through this kind of provoking remarks.
answered Aug 11 at 17:56
trying
4,2811722
add a comment |Â
up vote
1
down vote
up vote
1
down vote
He says, in my opinion, that a theory should be studied from its main conclusions and problems it allows to solve backward to the axioms, that is, by making abstraction, that is, by remaking mathematics. People should get their hands dirty with problems and only by banging their head against all problem difficulties they will understand the real appropriateness of axioms of a theory and at least naive mistakes can be avoided. The problem is that the main conclusions can be very numerous (infinity many) or at least much more numerous than the axioms and the theorems that the axioms follows and so it takes a great deal of time and force of will in order to examine them all. So he tries to incite seriuous students through this kind of provoking remarks.
answered Aug 11 at 17:56
trying
4,2811722
He says, in my opinion, that a theory should be studied from its main conclusions and problems it allows to solve backward to the axioms, that is, by making abstraction, that is, by remaking mathematics. People should get their hands dirty with problems and only by banging their head against all problem difficulties they will understand the real appropriateness of axioms of a theory and at least naive mistakes can be avoided. The problem is that the main conclusions can be very numerous (infinity many) or at least much more numerous than the axioms and the theorems that the axioms follows and so it takes a great deal of time and force of will in order to examine them all. So he tries to incite seriuous students through this kind of provoking remarks.
answered Aug 11 at 17:56
trying
4,2811722
answered Aug 11 at 17:56
trying
4,2811722
answered Aug 11 at 17:56
trying
4,2811722
answered Aug 11 at 17:56
trying
4,2811722
4,2811722
add a comment |Â
add a comment |Â
up vote
1
down vote
Surely,the students' answers were logically valid,but not logically sound? More experience with similar faulty questions would help them develop the habit of checking all initial assumptions in any given questions, and through this 'personal empiricism' gain intuition. Simply put, trust nothing, check everything.
answered Aug 11 at 19:10
Zeno
211
add a comment |Â
up vote
1
down vote
Surely,the students' answers were logically valid,but not logically sound? More experience with similar faulty questions would help them develop the habit of checking all initial assumptions in any given questions, and through this 'personal empiricism' gain intuition. Simply put, trust nothing, check everything.
answered Aug 11 at 19:10
Zeno
211
add a comment |Â
up vote
1
down vote
up vote
1
down vote
Surely,the students' answers were logically valid,but not logically sound? More experience with similar faulty questions would help them develop the habit of checking all initial assumptions in any given questions, and through this 'personal empiricism' gain intuition. Simply put, trust nothing, check everything.
answered Aug 11 at 19:10
Zeno
211
Surely,the students' answers were logically valid,but not logically sound? More experience with similar faulty questions would help them develop the habit of checking all initial assumptions in any given questions, and through this 'personal empiricism' gain intuition. Simply put, trust nothing, check everything.
answered Aug 11 at 19:10
Zeno
211
answered Aug 11 at 19:10
Zeno
211
answered Aug 11 at 19:10
Zeno
211
answered Aug 11 at 19:10
Zeno
211
211
add a comment |Â
add a comment |Â
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5
Arnold is saying, that while the French think they're clever, he (Arnold) actually is clever.
– Lord Shark the Unknown
Aug 11 at 16:57
18
Arnold blames his students for Arnold having concocted a faulty examination text.
– Did
Aug 11 at 17:01
3
Isn't the conclusion wrong? In this case the solution to the ODE is periodic. It can't converge unless it's constant.
– Jack M
Aug 11 at 17:10
2
@JackM The reasoning is correct given the assumption that the solution approaches a limit. These students failed to recognize that this assumption was false, despite being perfectly aware that the phase curve was a closed oval.
– littleO
Aug 11 at 18:18