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What does this Jacques Hadamard quote mean?

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What does this Jacques Hadamard quote mean?



Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)
Which kinds of Philosophy.SE questions should be taken from (or tolerated in)…What is the difference between a statement and a proposition?What are some introductory books about the philosophy of mathematics?Does relating objects implies in the search of a common unity?What is a straight line?What does mathematical constructivism gain us philosophically?Intuitionism and physicswhat is the ontology-ideology distinction in phil of mathIs anything truly continuous?Why do mathematical Axioms work so well in science?Distinguishing between procedure-like and collection-like mathematical objects










5















What does this Jacques Hadamard quote mean?




The shortest path between two truths in the real domain passes through the complex domain.




Is this a philosophical statement?
what is its mathematical background?










share|improve this question



















  • 1





    See Jacques Hadamard, The Psychology of Invention in the Mathematical Field (or.ed.1945), page 122 : it is about the role of intuition in the process of discovery (with some examples).

    – Mauro ALLEGRANZA
    13 hours ago






  • 2





    More specifically, H refers to use by Cardan of imaginary quantities in his calculations with roots of equations, in a time when imaginary numbers were not yet rigorously defined.

    – Mauro ALLEGRANZA
    12 hours ago











  • Neither "real" nor "complex" are used in the colloquial sense, they refer to real and complex numbers. What he means is that many problems of real analysis (computation of integrals, summing of series, solving differential equations) are easier solved by passing to the complex domain and using its methods. It raises issues in philosophy of mathematics in justifying such "transition through the imaginary" considered by Husserl, see On Husserl's Mathematical Apprenticeship, pp.20-23.

    – Conifold
    6 hours ago















5















What does this Jacques Hadamard quote mean?




The shortest path between two truths in the real domain passes through the complex domain.




Is this a philosophical statement?
what is its mathematical background?










share|improve this question



















  • 1





    See Jacques Hadamard, The Psychology of Invention in the Mathematical Field (or.ed.1945), page 122 : it is about the role of intuition in the process of discovery (with some examples).

    – Mauro ALLEGRANZA
    13 hours ago






  • 2





    More specifically, H refers to use by Cardan of imaginary quantities in his calculations with roots of equations, in a time when imaginary numbers were not yet rigorously defined.

    – Mauro ALLEGRANZA
    12 hours ago











  • Neither "real" nor "complex" are used in the colloquial sense, they refer to real and complex numbers. What he means is that many problems of real analysis (computation of integrals, summing of series, solving differential equations) are easier solved by passing to the complex domain and using its methods. It raises issues in philosophy of mathematics in justifying such "transition through the imaginary" considered by Husserl, see On Husserl's Mathematical Apprenticeship, pp.20-23.

    – Conifold
    6 hours ago













5












5








5








What does this Jacques Hadamard quote mean?




The shortest path between two truths in the real domain passes through the complex domain.




Is this a philosophical statement?
what is its mathematical background?










share|improve this question
















What does this Jacques Hadamard quote mean?




The shortest path between two truths in the real domain passes through the complex domain.




Is this a philosophical statement?
what is its mathematical background?







philosophy-of-science philosophy-of-mathematics






share|improve this question















share|improve this question













share|improve this question




share|improve this question








edited 7 hours ago









Eliran

4,90131433




4,90131433










asked 13 hours ago









The Last JediThe Last Jedi

985




985







  • 1





    See Jacques Hadamard, The Psychology of Invention in the Mathematical Field (or.ed.1945), page 122 : it is about the role of intuition in the process of discovery (with some examples).

    – Mauro ALLEGRANZA
    13 hours ago






  • 2





    More specifically, H refers to use by Cardan of imaginary quantities in his calculations with roots of equations, in a time when imaginary numbers were not yet rigorously defined.

    – Mauro ALLEGRANZA
    12 hours ago











  • Neither "real" nor "complex" are used in the colloquial sense, they refer to real and complex numbers. What he means is that many problems of real analysis (computation of integrals, summing of series, solving differential equations) are easier solved by passing to the complex domain and using its methods. It raises issues in philosophy of mathematics in justifying such "transition through the imaginary" considered by Husserl, see On Husserl's Mathematical Apprenticeship, pp.20-23.

    – Conifold
    6 hours ago












  • 1





    See Jacques Hadamard, The Psychology of Invention in the Mathematical Field (or.ed.1945), page 122 : it is about the role of intuition in the process of discovery (with some examples).

    – Mauro ALLEGRANZA
    13 hours ago






  • 2





    More specifically, H refers to use by Cardan of imaginary quantities in his calculations with roots of equations, in a time when imaginary numbers were not yet rigorously defined.

    – Mauro ALLEGRANZA
    12 hours ago











  • Neither "real" nor "complex" are used in the colloquial sense, they refer to real and complex numbers. What he means is that many problems of real analysis (computation of integrals, summing of series, solving differential equations) are easier solved by passing to the complex domain and using its methods. It raises issues in philosophy of mathematics in justifying such "transition through the imaginary" considered by Husserl, see On Husserl's Mathematical Apprenticeship, pp.20-23.

    – Conifold
    6 hours ago







1




1





See Jacques Hadamard, The Psychology of Invention in the Mathematical Field (or.ed.1945), page 122 : it is about the role of intuition in the process of discovery (with some examples).

– Mauro ALLEGRANZA
13 hours ago





See Jacques Hadamard, The Psychology of Invention in the Mathematical Field (or.ed.1945), page 122 : it is about the role of intuition in the process of discovery (with some examples).

– Mauro ALLEGRANZA
13 hours ago




2




2





More specifically, H refers to use by Cardan of imaginary quantities in his calculations with roots of equations, in a time when imaginary numbers were not yet rigorously defined.

– Mauro ALLEGRANZA
12 hours ago





More specifically, H refers to use by Cardan of imaginary quantities in his calculations with roots of equations, in a time when imaginary numbers were not yet rigorously defined.

– Mauro ALLEGRANZA
12 hours ago













Neither "real" nor "complex" are used in the colloquial sense, they refer to real and complex numbers. What he means is that many problems of real analysis (computation of integrals, summing of series, solving differential equations) are easier solved by passing to the complex domain and using its methods. It raises issues in philosophy of mathematics in justifying such "transition through the imaginary" considered by Husserl, see On Husserl's Mathematical Apprenticeship, pp.20-23.

– Conifold
6 hours ago





Neither "real" nor "complex" are used in the colloquial sense, they refer to real and complex numbers. What he means is that many problems of real analysis (computation of integrals, summing of series, solving differential equations) are easier solved by passing to the complex domain and using its methods. It raises issues in philosophy of mathematics in justifying such "transition through the imaginary" considered by Husserl, see On Husserl's Mathematical Apprenticeship, pp.20-23.

– Conifold
6 hours ago










3 Answers
3






active

oldest

votes


















5














Considering




An important paper concerning the distribution of prime numbers was Riemann's 1859 memoir "On the Number of Primes Less Than a Given Magnitude", the only paper he ever wrote on the subject. Riemann introduced new ideas into the subject, the chief of them being that the distribution of prime numbers is intimately connected with the zeros of the analytically extended Riemann zeta function of a complex variable. In particular, it is in this paper of Riemann that the idea to apply methods of complex analysis to the study of the real function π(x) originates. Extending the ideas of Riemann, two proofs of the asymptotic law of the distribution of prime numbers were obtained independently by Jacques Hadamard and Charles Jean de la Vallée Poussin and appeared in the same year (1896). Both proofs used methods from complex analysis, [...]




https://en.wikipedia.org/wiki/Prime_number_theorem#History_of_the_proof_of_the_asymptotic_law_of_prime_numbers



it seems very likely this quote means something in the spirit of:




Because complex numbers differ in certain ways from real numbers—their structure is simpler in some respects and richer in others—there are differences in detail between real and complex analysis.




https://www.britannica.com/science/analysis-mathematics/Complex-analysis




The precise definition of what it means for a function defined on the real line to be differentiable or integrable will be given in the Real Analysis course. In this course, we will look at what it means for functions defined on the complex plane to be differentiable or integrable and look at ways in which one can integrate complex-valued functions. Surprisingly, the theory turns out to be considerably easier than the real case! Thus the word ‘complex’ in the title refers to the presence of complex numbers, and not that the analysis is harder!




https://personalpages.manchester.ac.uk/staff/charles.walkden/complex-analysis/complex_analysis.pdf




And in fact, complex numbers are not more complicated than reals: in some ways, they are simpler. For instance, polynomials always have roots. Likewise, complex analysis is often simpler than real analysis: for example, every differentiable function is differentiable as often as we please, and has a power series expansion.




Stewart, I., & Tall, D. (2018). Complex analysis. Cambridge University Press.






share|improve this answer








New contributor




alkchf is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.



























    5














    I don't think there is much philosophical significance in what he said. Basically, he is saying that the complex field is a nice field to work with---and indeed it is. For example, every n degree polynomial in C[x] has exactly n roots in C, while R[x] does not enjoy this property. Of course, there any many reasons why C is nice. Another one is that the general linear group GLn(C) is a connected topological space (path connected, in fact), while GLn(R) is disconnected.






    share|improve this answer























    • Yes. Another example: The "shortest path" to calculating many real-valued definite integrals is use contour integration and the residue theorem in the complex plane.

      – olooney
      9 hours ago






    • 5





      Oh, and electrical engineers routinely work in the complex plane even though they only care about the real component, because $e^it$ is easier to work with than $sin(t)$ and $cos(t)$. They pass "through" the complex plane on their way to a real valued solution for simplicity sake.

      – olooney
      9 hours ago











    • @olooney Ah, indeed! That is a very nice illustration. My answer obviously shows my bias for theoretical maths.

      – Eli Bashwinger
      9 hours ago



















    5














    It's actually misquoted. From:
    http://homepage.divms.uiowa.edu/~jorgen/hadamardquotesource.html




    A longer and more nuanced formulation appears (in English) in Hadamard's An Essay on the Psychology of Invention in the Mathematical Field (Princeton U. Press, 1945; Dover, 1954; Princeton U. Press, as The Mathematician's Mind, 1996), page 123: "It has been written that the shortest and best way between two truths of the real domain often passes through the imaginary one." Note the use of "way" rather than "path", "of" rather than "in", "imaginary" rather than "complex", the added characterization "and best", the qualification "often", and the mysterious introduction "It has been written...".




    Now you may be wondering...if he didn't actually write this quote, who did?. It comes from Paul Painlevé's Analyse des travaux scientifiques (Gauthier-Villars, 1900; reprinted in Librairie Scientifique et Technique, Albert Blanchard, Paris, 1967, pp. 1-2; reproduced in Oeuvres de Paul Painlevé, Éditions du CNRS, Paris, 1972-1975, vol. 1, pp. 72-73)




    (TRANSLATED) The natural development of this work soon led the geometers in their studies to embrace imaginary as well as real values of the variable. The theory of Taylor series, that of elliptic functions, the vast field of Cauchy analysis, caused a burst of productivity derived from this generalization. It came to appear that, between two truths of the real domain, the easiest and shortest path quite often passes through the complex domain.




    So...it was a literal statement about mathematics which was nicely morphed into sounding philosophical.






    share|improve this answer








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    Rob Bird is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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      3 Answers
      3






      active

      oldest

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      3 Answers
      3






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      5














      Considering




      An important paper concerning the distribution of prime numbers was Riemann's 1859 memoir "On the Number of Primes Less Than a Given Magnitude", the only paper he ever wrote on the subject. Riemann introduced new ideas into the subject, the chief of them being that the distribution of prime numbers is intimately connected with the zeros of the analytically extended Riemann zeta function of a complex variable. In particular, it is in this paper of Riemann that the idea to apply methods of complex analysis to the study of the real function π(x) originates. Extending the ideas of Riemann, two proofs of the asymptotic law of the distribution of prime numbers were obtained independently by Jacques Hadamard and Charles Jean de la Vallée Poussin and appeared in the same year (1896). Both proofs used methods from complex analysis, [...]




      https://en.wikipedia.org/wiki/Prime_number_theorem#History_of_the_proof_of_the_asymptotic_law_of_prime_numbers



      it seems very likely this quote means something in the spirit of:




      Because complex numbers differ in certain ways from real numbers—their structure is simpler in some respects and richer in others—there are differences in detail between real and complex analysis.




      https://www.britannica.com/science/analysis-mathematics/Complex-analysis




      The precise definition of what it means for a function defined on the real line to be differentiable or integrable will be given in the Real Analysis course. In this course, we will look at what it means for functions defined on the complex plane to be differentiable or integrable and look at ways in which one can integrate complex-valued functions. Surprisingly, the theory turns out to be considerably easier than the real case! Thus the word ‘complex’ in the title refers to the presence of complex numbers, and not that the analysis is harder!




      https://personalpages.manchester.ac.uk/staff/charles.walkden/complex-analysis/complex_analysis.pdf




      And in fact, complex numbers are not more complicated than reals: in some ways, they are simpler. For instance, polynomials always have roots. Likewise, complex analysis is often simpler than real analysis: for example, every differentiable function is differentiable as often as we please, and has a power series expansion.




      Stewart, I., & Tall, D. (2018). Complex analysis. Cambridge University Press.






      share|improve this answer








      New contributor




      alkchf is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.
























        5














        Considering




        An important paper concerning the distribution of prime numbers was Riemann's 1859 memoir "On the Number of Primes Less Than a Given Magnitude", the only paper he ever wrote on the subject. Riemann introduced new ideas into the subject, the chief of them being that the distribution of prime numbers is intimately connected with the zeros of the analytically extended Riemann zeta function of a complex variable. In particular, it is in this paper of Riemann that the idea to apply methods of complex analysis to the study of the real function π(x) originates. Extending the ideas of Riemann, two proofs of the asymptotic law of the distribution of prime numbers were obtained independently by Jacques Hadamard and Charles Jean de la Vallée Poussin and appeared in the same year (1896). Both proofs used methods from complex analysis, [...]




        https://en.wikipedia.org/wiki/Prime_number_theorem#History_of_the_proof_of_the_asymptotic_law_of_prime_numbers



        it seems very likely this quote means something in the spirit of:




        Because complex numbers differ in certain ways from real numbers—their structure is simpler in some respects and richer in others—there are differences in detail between real and complex analysis.




        https://www.britannica.com/science/analysis-mathematics/Complex-analysis




        The precise definition of what it means for a function defined on the real line to be differentiable or integrable will be given in the Real Analysis course. In this course, we will look at what it means for functions defined on the complex plane to be differentiable or integrable and look at ways in which one can integrate complex-valued functions. Surprisingly, the theory turns out to be considerably easier than the real case! Thus the word ‘complex’ in the title refers to the presence of complex numbers, and not that the analysis is harder!




        https://personalpages.manchester.ac.uk/staff/charles.walkden/complex-analysis/complex_analysis.pdf




        And in fact, complex numbers are not more complicated than reals: in some ways, they are simpler. For instance, polynomials always have roots. Likewise, complex analysis is often simpler than real analysis: for example, every differentiable function is differentiable as often as we please, and has a power series expansion.




        Stewart, I., & Tall, D. (2018). Complex analysis. Cambridge University Press.






        share|improve this answer








        New contributor




        alkchf is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
        Check out our Code of Conduct.






















          5












          5








          5







          Considering




          An important paper concerning the distribution of prime numbers was Riemann's 1859 memoir "On the Number of Primes Less Than a Given Magnitude", the only paper he ever wrote on the subject. Riemann introduced new ideas into the subject, the chief of them being that the distribution of prime numbers is intimately connected with the zeros of the analytically extended Riemann zeta function of a complex variable. In particular, it is in this paper of Riemann that the idea to apply methods of complex analysis to the study of the real function π(x) originates. Extending the ideas of Riemann, two proofs of the asymptotic law of the distribution of prime numbers were obtained independently by Jacques Hadamard and Charles Jean de la Vallée Poussin and appeared in the same year (1896). Both proofs used methods from complex analysis, [...]




          https://en.wikipedia.org/wiki/Prime_number_theorem#History_of_the_proof_of_the_asymptotic_law_of_prime_numbers



          it seems very likely this quote means something in the spirit of:




          Because complex numbers differ in certain ways from real numbers—their structure is simpler in some respects and richer in others—there are differences in detail between real and complex analysis.




          https://www.britannica.com/science/analysis-mathematics/Complex-analysis




          The precise definition of what it means for a function defined on the real line to be differentiable or integrable will be given in the Real Analysis course. In this course, we will look at what it means for functions defined on the complex plane to be differentiable or integrable and look at ways in which one can integrate complex-valued functions. Surprisingly, the theory turns out to be considerably easier than the real case! Thus the word ‘complex’ in the title refers to the presence of complex numbers, and not that the analysis is harder!




          https://personalpages.manchester.ac.uk/staff/charles.walkden/complex-analysis/complex_analysis.pdf




          And in fact, complex numbers are not more complicated than reals: in some ways, they are simpler. For instance, polynomials always have roots. Likewise, complex analysis is often simpler than real analysis: for example, every differentiable function is differentiable as often as we please, and has a power series expansion.




          Stewart, I., & Tall, D. (2018). Complex analysis. Cambridge University Press.






          share|improve this answer








          New contributor




          alkchf is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
          Check out our Code of Conduct.










          Considering




          An important paper concerning the distribution of prime numbers was Riemann's 1859 memoir "On the Number of Primes Less Than a Given Magnitude", the only paper he ever wrote on the subject. Riemann introduced new ideas into the subject, the chief of them being that the distribution of prime numbers is intimately connected with the zeros of the analytically extended Riemann zeta function of a complex variable. In particular, it is in this paper of Riemann that the idea to apply methods of complex analysis to the study of the real function π(x) originates. Extending the ideas of Riemann, two proofs of the asymptotic law of the distribution of prime numbers were obtained independently by Jacques Hadamard and Charles Jean de la Vallée Poussin and appeared in the same year (1896). Both proofs used methods from complex analysis, [...]




          https://en.wikipedia.org/wiki/Prime_number_theorem#History_of_the_proof_of_the_asymptotic_law_of_prime_numbers



          it seems very likely this quote means something in the spirit of:




          Because complex numbers differ in certain ways from real numbers—their structure is simpler in some respects and richer in others—there are differences in detail between real and complex analysis.




          https://www.britannica.com/science/analysis-mathematics/Complex-analysis




          The precise definition of what it means for a function defined on the real line to be differentiable or integrable will be given in the Real Analysis course. In this course, we will look at what it means for functions defined on the complex plane to be differentiable or integrable and look at ways in which one can integrate complex-valued functions. Surprisingly, the theory turns out to be considerably easier than the real case! Thus the word ‘complex’ in the title refers to the presence of complex numbers, and not that the analysis is harder!




          https://personalpages.manchester.ac.uk/staff/charles.walkden/complex-analysis/complex_analysis.pdf




          And in fact, complex numbers are not more complicated than reals: in some ways, they are simpler. For instance, polynomials always have roots. Likewise, complex analysis is often simpler than real analysis: for example, every differentiable function is differentiable as often as we please, and has a power series expansion.




          Stewart, I., & Tall, D. (2018). Complex analysis. Cambridge University Press.







          share|improve this answer








          New contributor




          alkchf is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
          Check out our Code of Conduct.









          share|improve this answer



          share|improve this answer






          New contributor




          alkchf is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
          Check out our Code of Conduct.









          answered 12 hours ago









          alkchfalkchf

          2613




          2613




          New contributor




          alkchf is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
          Check out our Code of Conduct.





          New contributor





          alkchf is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
          Check out our Code of Conduct.






          alkchf is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
          Check out our Code of Conduct.





















              5














              I don't think there is much philosophical significance in what he said. Basically, he is saying that the complex field is a nice field to work with---and indeed it is. For example, every n degree polynomial in C[x] has exactly n roots in C, while R[x] does not enjoy this property. Of course, there any many reasons why C is nice. Another one is that the general linear group GLn(C) is a connected topological space (path connected, in fact), while GLn(R) is disconnected.






              share|improve this answer























              • Yes. Another example: The "shortest path" to calculating many real-valued definite integrals is use contour integration and the residue theorem in the complex plane.

                – olooney
                9 hours ago






              • 5





                Oh, and electrical engineers routinely work in the complex plane even though they only care about the real component, because $e^it$ is easier to work with than $sin(t)$ and $cos(t)$. They pass "through" the complex plane on their way to a real valued solution for simplicity sake.

                – olooney
                9 hours ago











              • @olooney Ah, indeed! That is a very nice illustration. My answer obviously shows my bias for theoretical maths.

                – Eli Bashwinger
                9 hours ago
















              5














              I don't think there is much philosophical significance in what he said. Basically, he is saying that the complex field is a nice field to work with---and indeed it is. For example, every n degree polynomial in C[x] has exactly n roots in C, while R[x] does not enjoy this property. Of course, there any many reasons why C is nice. Another one is that the general linear group GLn(C) is a connected topological space (path connected, in fact), while GLn(R) is disconnected.






              share|improve this answer























              • Yes. Another example: The "shortest path" to calculating many real-valued definite integrals is use contour integration and the residue theorem in the complex plane.

                – olooney
                9 hours ago






              • 5





                Oh, and electrical engineers routinely work in the complex plane even though they only care about the real component, because $e^it$ is easier to work with than $sin(t)$ and $cos(t)$. They pass "through" the complex plane on their way to a real valued solution for simplicity sake.

                – olooney
                9 hours ago











              • @olooney Ah, indeed! That is a very nice illustration. My answer obviously shows my bias for theoretical maths.

                – Eli Bashwinger
                9 hours ago














              5












              5








              5







              I don't think there is much philosophical significance in what he said. Basically, he is saying that the complex field is a nice field to work with---and indeed it is. For example, every n degree polynomial in C[x] has exactly n roots in C, while R[x] does not enjoy this property. Of course, there any many reasons why C is nice. Another one is that the general linear group GLn(C) is a connected topological space (path connected, in fact), while GLn(R) is disconnected.






              share|improve this answer













              I don't think there is much philosophical significance in what he said. Basically, he is saying that the complex field is a nice field to work with---and indeed it is. For example, every n degree polynomial in C[x] has exactly n roots in C, while R[x] does not enjoy this property. Of course, there any many reasons why C is nice. Another one is that the general linear group GLn(C) is a connected topological space (path connected, in fact), while GLn(R) is disconnected.







              share|improve this answer












              share|improve this answer



              share|improve this answer










              answered 11 hours ago









              Eli BashwingerEli Bashwinger

              508413




              508413












              • Yes. Another example: The "shortest path" to calculating many real-valued definite integrals is use contour integration and the residue theorem in the complex plane.

                – olooney
                9 hours ago






              • 5





                Oh, and electrical engineers routinely work in the complex plane even though they only care about the real component, because $e^it$ is easier to work with than $sin(t)$ and $cos(t)$. They pass "through" the complex plane on their way to a real valued solution for simplicity sake.

                – olooney
                9 hours ago











              • @olooney Ah, indeed! That is a very nice illustration. My answer obviously shows my bias for theoretical maths.

                – Eli Bashwinger
                9 hours ago


















              • Yes. Another example: The "shortest path" to calculating many real-valued definite integrals is use contour integration and the residue theorem in the complex plane.

                – olooney
                9 hours ago






              • 5





                Oh, and electrical engineers routinely work in the complex plane even though they only care about the real component, because $e^it$ is easier to work with than $sin(t)$ and $cos(t)$. They pass "through" the complex plane on their way to a real valued solution for simplicity sake.

                – olooney
                9 hours ago











              • @olooney Ah, indeed! That is a very nice illustration. My answer obviously shows my bias for theoretical maths.

                – Eli Bashwinger
                9 hours ago

















              Yes. Another example: The "shortest path" to calculating many real-valued definite integrals is use contour integration and the residue theorem in the complex plane.

              – olooney
              9 hours ago





              Yes. Another example: The "shortest path" to calculating many real-valued definite integrals is use contour integration and the residue theorem in the complex plane.

              – olooney
              9 hours ago




              5




              5





              Oh, and electrical engineers routinely work in the complex plane even though they only care about the real component, because $e^it$ is easier to work with than $sin(t)$ and $cos(t)$. They pass "through" the complex plane on their way to a real valued solution for simplicity sake.

              – olooney
              9 hours ago





              Oh, and electrical engineers routinely work in the complex plane even though they only care about the real component, because $e^it$ is easier to work with than $sin(t)$ and $cos(t)$. They pass "through" the complex plane on their way to a real valued solution for simplicity sake.

              – olooney
              9 hours ago













              @olooney Ah, indeed! That is a very nice illustration. My answer obviously shows my bias for theoretical maths.

              – Eli Bashwinger
              9 hours ago






              @olooney Ah, indeed! That is a very nice illustration. My answer obviously shows my bias for theoretical maths.

              – Eli Bashwinger
              9 hours ago












              5














              It's actually misquoted. From:
              http://homepage.divms.uiowa.edu/~jorgen/hadamardquotesource.html




              A longer and more nuanced formulation appears (in English) in Hadamard's An Essay on the Psychology of Invention in the Mathematical Field (Princeton U. Press, 1945; Dover, 1954; Princeton U. Press, as The Mathematician's Mind, 1996), page 123: "It has been written that the shortest and best way between two truths of the real domain often passes through the imaginary one." Note the use of "way" rather than "path", "of" rather than "in", "imaginary" rather than "complex", the added characterization "and best", the qualification "often", and the mysterious introduction "It has been written...".




              Now you may be wondering...if he didn't actually write this quote, who did?. It comes from Paul Painlevé's Analyse des travaux scientifiques (Gauthier-Villars, 1900; reprinted in Librairie Scientifique et Technique, Albert Blanchard, Paris, 1967, pp. 1-2; reproduced in Oeuvres de Paul Painlevé, Éditions du CNRS, Paris, 1972-1975, vol. 1, pp. 72-73)




              (TRANSLATED) The natural development of this work soon led the geometers in their studies to embrace imaginary as well as real values of the variable. The theory of Taylor series, that of elliptic functions, the vast field of Cauchy analysis, caused a burst of productivity derived from this generalization. It came to appear that, between two truths of the real domain, the easiest and shortest path quite often passes through the complex domain.




              So...it was a literal statement about mathematics which was nicely morphed into sounding philosophical.






              share|improve this answer








              New contributor




              Rob Bird is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
              Check out our Code of Conduct.
























                5














                It's actually misquoted. From:
                http://homepage.divms.uiowa.edu/~jorgen/hadamardquotesource.html




                A longer and more nuanced formulation appears (in English) in Hadamard's An Essay on the Psychology of Invention in the Mathematical Field (Princeton U. Press, 1945; Dover, 1954; Princeton U. Press, as The Mathematician's Mind, 1996), page 123: "It has been written that the shortest and best way between two truths of the real domain often passes through the imaginary one." Note the use of "way" rather than "path", "of" rather than "in", "imaginary" rather than "complex", the added characterization "and best", the qualification "often", and the mysterious introduction "It has been written...".




                Now you may be wondering...if he didn't actually write this quote, who did?. It comes from Paul Painlevé's Analyse des travaux scientifiques (Gauthier-Villars, 1900; reprinted in Librairie Scientifique et Technique, Albert Blanchard, Paris, 1967, pp. 1-2; reproduced in Oeuvres de Paul Painlevé, Éditions du CNRS, Paris, 1972-1975, vol. 1, pp. 72-73)




                (TRANSLATED) The natural development of this work soon led the geometers in their studies to embrace imaginary as well as real values of the variable. The theory of Taylor series, that of elliptic functions, the vast field of Cauchy analysis, caused a burst of productivity derived from this generalization. It came to appear that, between two truths of the real domain, the easiest and shortest path quite often passes through the complex domain.




                So...it was a literal statement about mathematics which was nicely morphed into sounding philosophical.






                share|improve this answer








                New contributor




                Rob Bird is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                Check out our Code of Conduct.






















                  5












                  5








                  5







                  It's actually misquoted. From:
                  http://homepage.divms.uiowa.edu/~jorgen/hadamardquotesource.html




                  A longer and more nuanced formulation appears (in English) in Hadamard's An Essay on the Psychology of Invention in the Mathematical Field (Princeton U. Press, 1945; Dover, 1954; Princeton U. Press, as The Mathematician's Mind, 1996), page 123: "It has been written that the shortest and best way between two truths of the real domain often passes through the imaginary one." Note the use of "way" rather than "path", "of" rather than "in", "imaginary" rather than "complex", the added characterization "and best", the qualification "often", and the mysterious introduction "It has been written...".




                  Now you may be wondering...if he didn't actually write this quote, who did?. It comes from Paul Painlevé's Analyse des travaux scientifiques (Gauthier-Villars, 1900; reprinted in Librairie Scientifique et Technique, Albert Blanchard, Paris, 1967, pp. 1-2; reproduced in Oeuvres de Paul Painlevé, Éditions du CNRS, Paris, 1972-1975, vol. 1, pp. 72-73)




                  (TRANSLATED) The natural development of this work soon led the geometers in their studies to embrace imaginary as well as real values of the variable. The theory of Taylor series, that of elliptic functions, the vast field of Cauchy analysis, caused a burst of productivity derived from this generalization. It came to appear that, between two truths of the real domain, the easiest and shortest path quite often passes through the complex domain.




                  So...it was a literal statement about mathematics which was nicely morphed into sounding philosophical.






                  share|improve this answer








                  New contributor




                  Rob Bird is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                  Check out our Code of Conduct.










                  It's actually misquoted. From:
                  http://homepage.divms.uiowa.edu/~jorgen/hadamardquotesource.html




                  A longer and more nuanced formulation appears (in English) in Hadamard's An Essay on the Psychology of Invention in the Mathematical Field (Princeton U. Press, 1945; Dover, 1954; Princeton U. Press, as The Mathematician's Mind, 1996), page 123: "It has been written that the shortest and best way between two truths of the real domain often passes through the imaginary one." Note the use of "way" rather than "path", "of" rather than "in", "imaginary" rather than "complex", the added characterization "and best", the qualification "often", and the mysterious introduction "It has been written...".




                  Now you may be wondering...if he didn't actually write this quote, who did?. It comes from Paul Painlevé's Analyse des travaux scientifiques (Gauthier-Villars, 1900; reprinted in Librairie Scientifique et Technique, Albert Blanchard, Paris, 1967, pp. 1-2; reproduced in Oeuvres de Paul Painlevé, Éditions du CNRS, Paris, 1972-1975, vol. 1, pp. 72-73)




                  (TRANSLATED) The natural development of this work soon led the geometers in their studies to embrace imaginary as well as real values of the variable. The theory of Taylor series, that of elliptic functions, the vast field of Cauchy analysis, caused a burst of productivity derived from this generalization. It came to appear that, between two truths of the real domain, the easiest and shortest path quite often passes through the complex domain.




                  So...it was a literal statement about mathematics which was nicely morphed into sounding philosophical.







                  share|improve this answer








                  New contributor




                  Rob Bird is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                  Check out our Code of Conduct.









                  share|improve this answer



                  share|improve this answer






                  New contributor




                  Rob Bird is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                  Check out our Code of Conduct.









                  answered 9 hours ago









                  Rob BirdRob Bird

                  512




                  512




                  New contributor




                  Rob Bird is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                  Check out our Code of Conduct.





                  New contributor





                  Rob Bird is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                  Check out our Code of Conduct.






                  Rob Bird is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                  Check out our Code of Conduct.



























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