Mixing
Relative weight-age of the Mathematical concepts and ideas.
$begingroup$
My question has its roots in the following question that I had asked earlier:
Prove that the sum of digits of $(999...9)^{3}$ (cube of integer with $n$ digits $9$) is $18n$
Now while going through some classical texts on Number Theory, I had come across this statement that such results as the sum of digits of a number are of trivial nature and are not studied too rigorously.
Similarly, books on trigonometry often suggest that versine and coversine as ratios of much less importance which may be "skipped".
I often wonder what leads to such statements in mathematics wherein one attributes certain level of relative importance to some topics, theorems or identities while lending just a cursory remark upon the others. Is there a formal way to identify or measure as to what is of significant importance and worth pursuing in mathematics and what may be not?
A counter example could be Fermat's last theorem and its proof. The statement itself may appear trivial and lacking any practical applications but it still has been pursued rigorously by mathematicians centuries over.
Simply put how do we assess the relative "worth" of the countless mathematical ideas?Further adding to it do we have any hierarchy whereby we can quantify the relative weight-age of various concepts.
soft-question
asked Jan 7 at 4:43
naveen dankalnaveen dankal
4,55921348
$endgroup$
add a comment |
$begingroup$
My question has its roots in the following question that I had asked earlier:
Prove that the sum of digits of $(999...9)^{3}$ (cube of integer with $n$ digits $9$) is $18n$
Now while going through some classical texts on Number Theory, I had come across this statement that such results as the sum of digits of a number are of trivial nature and are not studied too rigorously.
Similarly, books on trigonometry often suggest that versine and coversine as ratios of much less importance which may be "skipped".
I often wonder what leads to such statements in mathematics wherein one attributes certain level of relative importance to some topics, theorems or identities while lending just a cursory remark upon the others. Is there a formal way to identify or measure as to what is of significant importance and worth pursuing in mathematics and what may be not?
A counter example could be Fermat's last theorem and its proof. The statement itself may appear trivial and lacking any practical applications but it still has been pursued rigorously by mathematicians centuries over.
Simply put how do we assess the relative "worth" of the countless mathematical ideas?Further adding to it do we have any hierarchy whereby we can quantify the relative weight-age of various concepts.
soft-question
asked Jan 7 at 4:43
naveen dankalnaveen dankal
4,55921348
$endgroup$
5
$begingroup$
Anything about digit sums, even if it is non-trivial, is typically considered recreational mathematics because it deals with writing numbers down (and who does that?) and only in a specific non-natural way (using as many digits as we happen to have fingers). -- Of course, "worth" is subject to time and fashion, e.g., which trigonometric functions are considered most important may depend on whether you work in Fourier analysis or in maritime navigation (and even in the latter case it depends on whether you want to work with tables or have a pocket calculator available)
$endgroup$
– Hagen von Eitzen
Jan 7 at 5:19
add a comment |
$begingroup$
My question has its roots in the following question that I had asked earlier:
Prove that the sum of digits of $(999...9)^{3}$ (cube of integer with $n$ digits $9$) is $18n$
Now while going through some classical texts on Number Theory, I had come across this statement that such results as the sum of digits of a number are of trivial nature and are not studied too rigorously.
Similarly, books on trigonometry often suggest that versine and coversine as ratios of much less importance which may be "skipped".
I often wonder what leads to such statements in mathematics wherein one attributes certain level of relative importance to some topics, theorems or identities while lending just a cursory remark upon the others. Is there a formal way to identify or measure as to what is of significant importance and worth pursuing in mathematics and what may be not?
A counter example could be Fermat's last theorem and its proof. The statement itself may appear trivial and lacking any practical applications but it still has been pursued rigorously by mathematicians centuries over.
Simply put how do we assess the relative "worth" of the countless mathematical ideas?Further adding to it do we have any hierarchy whereby we can quantify the relative weight-age of various concepts.
soft-question
asked Jan 7 at 4:43
naveen dankalnaveen dankal
4,55921348
$endgroup$
My question has its roots in the following question that I had asked earlier:
Prove that the sum of digits of $(999...9)^{3}$ (cube of integer with $n$ digits $9$) is $18n$
Now while going through some classical texts on Number Theory, I had come across this statement that such results as the sum of digits of a number are of trivial nature and are not studied too rigorously.
Similarly, books on trigonometry often suggest that versine and coversine as ratios of much less importance which may be "skipped".
I often wonder what leads to such statements in mathematics wherein one attributes certain level of relative importance to some topics, theorems or identities while lending just a cursory remark upon the others. Is there a formal way to identify or measure as to what is of significant importance and worth pursuing in mathematics and what may be not?
A counter example could be Fermat's last theorem and its proof. The statement itself may appear trivial and lacking any practical applications but it still has been pursued rigorously by mathematicians centuries over.
Simply put how do we assess the relative "worth" of the countless mathematical ideas?Further adding to it do we have any hierarchy whereby we can quantify the relative weight-age of various concepts.
soft-question
soft-question
asked Jan 7 at 4:43
naveen dankalnaveen dankal
4,55921348
asked Jan 7 at 4:43
naveen dankalnaveen dankal
4,55921348
edited Jan 7 at 8:31
naveen dankal
asked Jan 7 at 4:43
naveen dankalnaveen dankal
4,55921348
asked Jan 7 at 4:43
naveen dankalnaveen dankal
4,55921348
asked Jan 7 at 4:43
naveen dankalnaveen dankal
4,55921348
4,55921348
5
$begingroup$
Anything about digit sums, even if it is non-trivial, is typically considered recreational mathematics because it deals with writing numbers down (and who does that?) and only in a specific non-natural way (using as many digits as we happen to have fingers). -- Of course, "worth" is subject to time and fashion, e.g., which trigonometric functions are considered most important may depend on whether you work in Fourier analysis or in maritime navigation (and even in the latter case it depends on whether you want to work with tables or have a pocket calculator available)
$endgroup$
– Hagen von Eitzen
Jan 7 at 5:19
add a comment |
5
$begingroup$
Anything about digit sums, even if it is non-trivial, is typically considered recreational mathematics because it deals with writing numbers down (and who does that?) and only in a specific non-natural way (using as many digits as we happen to have fingers). -- Of course, "worth" is subject to time and fashion, e.g., which trigonometric functions are considered most important may depend on whether you work in Fourier analysis or in maritime navigation (and even in the latter case it depends on whether you want to work with tables or have a pocket calculator available)
$endgroup$
– Hagen von Eitzen
Jan 7 at 5:19
5
5
$begingroup$
Anything about digit sums, even if it is non-trivial, is typically considered recreational mathematics because it deals with writing numbers down (and who does that?) and only in a specific non-natural way (using as many digits as we happen to have fingers). -- Of course, "worth" is subject to time and fashion, e.g., which trigonometric functions are considered most important may depend on whether you work in Fourier analysis or in maritime navigation (and even in the latter case it depends on whether you want to work with tables or have a pocket calculator available)
$endgroup$
– Hagen von Eitzen
Jan 7 at 5:19
$begingroup$
Anything about digit sums, even if it is non-trivial, is typically considered recreational mathematics because it deals with writing numbers down (and who does that?) and only in a specific non-natural way (using as many digits as we happen to have fingers). -- Of course, "worth" is subject to time and fashion, e.g., which trigonometric functions are considered most important may depend on whether you work in Fourier analysis or in maritime navigation (and even in the latter case it depends on whether you want to work with tables or have a pocket calculator available)
$endgroup$
– Hagen von Eitzen
Jan 7 at 5:19
add a comment |
1 Answer
1
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oldest
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$begingroup$
It may be impossible to assess the potential future practical applications of a mathematical result. I base this on some examples:
Brouwer's Fixed-Point Theorem to explain the cause of some instances of heart fibrillation (a medical condition, quickly fatal if not treated immediately).
Some results about closest-packing of n-spheres (the Kepler Problem in higher dimensions) have applications in design of efficient data-transmission systems that have error-detection/correction in them.
A theorem in Knot Theory that has an interpretation in statistical thermodynamics. (See the book "On Knots" by Kaufmann.)
The theory of Hilbert spaces is indispensable in quantum mechanics. The physicist Erwin Schrodinger re-discovered quite a bit of it with his "matrix mechanics", initially unaware that he was finding a new application for some previously-developed math .
Basic number theory as a foundation for modern encryption. The great number-theorist G. H.Hardy (in A Marhematician's Apology) wrote (proudly, I think) "Nothing I have ever done is of any use." I think he meant this in the sense that a sonata has no "use".
answered Jan 7 at 20:51
DanielWainfleetDanielWainfleet
34.8k31648
$endgroup$
1
$begingroup$
BTW it's Erwin Schrödinger with an "E" :)
$endgroup$
– 0x539
Jan 7 at 21:27
$begingroup$
@0x539 . . Noted & corrected
$endgroup$
– DanielWainfleet
Jan 9 at 3:56
add a comment |
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$begingroup$
It may be impossible to assess the potential future practical applications of a mathematical result. I base this on some examples:
Brouwer's Fixed-Point Theorem to explain the cause of some instances of heart fibrillation (a medical condition, quickly fatal if not treated immediately).
Some results about closest-packing of n-spheres (the Kepler Problem in higher dimensions) have applications in design of efficient data-transmission systems that have error-detection/correction in them.
A theorem in Knot Theory that has an interpretation in statistical thermodynamics. (See the book "On Knots" by Kaufmann.)
The theory of Hilbert spaces is indispensable in quantum mechanics. The physicist Erwin Schrodinger re-discovered quite a bit of it with his "matrix mechanics", initially unaware that he was finding a new application for some previously-developed math .
Basic number theory as a foundation for modern encryption. The great number-theorist G. H.Hardy (in A Marhematician's Apology) wrote (proudly, I think) "Nothing I have ever done is of any use." I think he meant this in the sense that a sonata has no "use".
answered Jan 7 at 20:51
DanielWainfleetDanielWainfleet
34.8k31648
$endgroup$
1
$begingroup$
BTW it's Erwin Schrödinger with an "E" :)
$endgroup$
– 0x539
Jan 7 at 21:27
$begingroup$
@0x539 . . Noted & corrected
$endgroup$
– DanielWainfleet
Jan 9 at 3:56
add a comment |
$begingroup$
It may be impossible to assess the potential future practical applications of a mathematical result. I base this on some examples:
Brouwer's Fixed-Point Theorem to explain the cause of some instances of heart fibrillation (a medical condition, quickly fatal if not treated immediately).
Some results about closest-packing of n-spheres (the Kepler Problem in higher dimensions) have applications in design of efficient data-transmission systems that have error-detection/correction in them.
A theorem in Knot Theory that has an interpretation in statistical thermodynamics. (See the book "On Knots" by Kaufmann.)
The theory of Hilbert spaces is indispensable in quantum mechanics. The physicist Erwin Schrodinger re-discovered quite a bit of it with his "matrix mechanics", initially unaware that he was finding a new application for some previously-developed math .
Basic number theory as a foundation for modern encryption. The great number-theorist G. H.Hardy (in A Marhematician's Apology) wrote (proudly, I think) "Nothing I have ever done is of any use." I think he meant this in the sense that a sonata has no "use".
answered Jan 7 at 20:51
DanielWainfleetDanielWainfleet
34.8k31648
$endgroup$
1
$begingroup$
BTW it's Erwin Schrödinger with an "E" :)
$endgroup$
– 0x539
Jan 7 at 21:27
$begingroup$
@0x539 . . Noted & corrected
$endgroup$
– DanielWainfleet
Jan 9 at 3:56
add a comment |
$begingroup$
It may be impossible to assess the potential future practical applications of a mathematical result. I base this on some examples:
Brouwer's Fixed-Point Theorem to explain the cause of some instances of heart fibrillation (a medical condition, quickly fatal if not treated immediately).
Some results about closest-packing of n-spheres (the Kepler Problem in higher dimensions) have applications in design of efficient data-transmission systems that have error-detection/correction in them.
A theorem in Knot Theory that has an interpretation in statistical thermodynamics. (See the book "On Knots" by Kaufmann.)
The theory of Hilbert spaces is indispensable in quantum mechanics. The physicist Erwin Schrodinger re-discovered quite a bit of it with his "matrix mechanics", initially unaware that he was finding a new application for some previously-developed math .
Basic number theory as a foundation for modern encryption. The great number-theorist G. H.Hardy (in A Marhematician's Apology) wrote (proudly, I think) "Nothing I have ever done is of any use." I think he meant this in the sense that a sonata has no "use".
answered Jan 7 at 20:51
DanielWainfleetDanielWainfleet
34.8k31648
$endgroup$
It may be impossible to assess the potential future practical applications of a mathematical result. I base this on some examples:
Brouwer's Fixed-Point Theorem to explain the cause of some instances of heart fibrillation (a medical condition, quickly fatal if not treated immediately).
Some results about closest-packing of n-spheres (the Kepler Problem in higher dimensions) have applications in design of efficient data-transmission systems that have error-detection/correction in them.
A theorem in Knot Theory that has an interpretation in statistical thermodynamics. (See the book "On Knots" by Kaufmann.)
The theory of Hilbert spaces is indispensable in quantum mechanics. The physicist Erwin Schrodinger re-discovered quite a bit of it with his "matrix mechanics", initially unaware that he was finding a new application for some previously-developed math .
Basic number theory as a foundation for modern encryption. The great number-theorist G. H.Hardy (in A Marhematician's Apology) wrote (proudly, I think) "Nothing I have ever done is of any use." I think he meant this in the sense that a sonata has no "use".
answered Jan 7 at 20:51
DanielWainfleetDanielWainfleet
34.8k31648
edited Jan 9 at 3:55
answered Jan 7 at 20:51
DanielWainfleetDanielWainfleet
34.8k31648
answered Jan 7 at 20:51
DanielWainfleetDanielWainfleet
34.8k31648
answered Jan 7 at 20:51
DanielWainfleetDanielWainfleet
34.8k31648
34.8k31648
1
$begingroup$
BTW it's Erwin Schrödinger with an "E" :)
$endgroup$
– 0x539
Jan 7 at 21:27
$begingroup$
@0x539 . . Noted & corrected
$endgroup$
– DanielWainfleet
Jan 9 at 3:56
add a comment |
1
$begingroup$
BTW it's Erwin Schrödinger with an "E" :)
$endgroup$
– 0x539
Jan 7 at 21:27
$begingroup$
@0x539 . . Noted & corrected
$endgroup$
– DanielWainfleet
Jan 9 at 3:56
1
1
$begingroup$
BTW it's Erwin Schrödinger with an "E" :)
$endgroup$
– 0x539
Jan 7 at 21:27
$begingroup$
BTW it's Erwin Schrödinger with an "E" :)
$endgroup$
– 0x539
Jan 7 at 21:27
$begingroup$
@0x539 . . Noted & corrected
$endgroup$
– DanielWainfleet
Jan 9 at 3:56
$begingroup$
@0x539 . . Noted & corrected
$endgroup$
– DanielWainfleet
Jan 9 at 3:56
add a comment |
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$begingroup$
Anything about digit sums, even if it is non-trivial, is typically considered recreational mathematics because it deals with writing numbers down (and who does that?) and only in a specific non-natural way (using as many digits as we happen to have fingers). -- Of course, "worth" is subject to time and fashion, e.g., which trigonometric functions are considered most important may depend on whether you work in Fourier analysis or in maritime navigation (and even in the latter case it depends on whether you want to work with tables or have a pocket calculator available)
$endgroup$
– Hagen von Eitzen
Jan 7 at 5:19