The Mathematician's Mind: The Psychology of Invention in the Mathematical Field by Jacques Hadamard - ISBN 0486201074 - Princeton University Press 1945

note that this was finished much earlier but without recall, recall starts a month before the current date to already start with very spaced recalls


Being interested in creativity and convinced that mathematics, as a discipline using abstraction to solve problems, does require such a skill, unlike the nearly automated techniques taught during school. Also previously discovered some of the work of Hadamard in the history of mathematics.

Pre-reading model

Draw a schema (using PmGraphViz or another solution) of the situation of the area in the studied domain before having read the book.


  • Introduction
    • distinction between discovery and invention (p.xi)
    • "As my master, Hermite, told me: <<We are rather servants than master in Mathematics.>>" (p.xii)
  • 1 General Views and Inquiries
    • The Mathematics "Bump" (p3)
    • Psychologists' Views on the Subject (p6)
    • Mathematical Enquiries (p7)
    • Some Criticisms (p9)
    • Pointcarre's Statements (p11)
    • Looking at One's Own Unconsciousness (p14)
    • Instances in Other Fields (p15)
      • enunciation a potential process starting by "incubation" then "illumination" following Graham Wallas' Art of Thought (p16)
    • The Chance Hypothesis (p18)
      • "explanation by pure chance is equivalent to no explanation at all and to asserting that there are effects without causes" (p19)
  • 2 Discussions on Unconsciousness
    • The Manifold Character of Unconsciousness (p23)
    • Frindge-Consciousness (p24)
    • Successive Layers in the Unconscious (p26)
  • 3 The Unconscious and Discovery
    • Combination of Ideas (p29)
      • "it is obvious that invention or discovery, be it in mathematics or anywhere else, takes place by combining ideas." (p29)
        • footnote on the etymology of cogito as "to shake together"
        • an intuitive principle that is coherent with evolutionary epistemology and the necessity to handle modularity
    • The Following Steps (p30)
      • "this first process, this building up of numerous combinations, is only the beginning of creation, even, as we should say, preliminary to it" (p30)
    • To Invent is to Chose (p30)
    • Coming Back to the Unconscious (p31)
      • mention of "mathematical beauty" or "geometric elegance" probably close to Dijkstra sensibility on "mathematical elegance"
      • "it is clear that no significant discovery or invention can take place without the will of finding" (p31)
      • "that invention is choice, that this choice is imperatively governed by the sense of scientific beauty" (p31)
    • Other Views on Incubation (p32)
      • offering 2 chief hypothesis (p33)
        • freshness or rest-hypothesis
        • absence of interference or forgetting-hypothesis
    • Discussion of These Ideas (p34)
    • Other Views on Illuminations. An Intination Stage (p38)
    • Further Theories on the Unconscious (p40)
  • 4 The Preparation Stage. Logic and Chance
    • Throughout Conscious Work (p43)
    • Conscious Work as Preparatory (p45)
    • Pointcarre's View on the Mode of Action of Preparatory Work (p46)
      • Pointcarre's metaphor on ideas as combinations "liked the hooked atoms of Epicurus" sounds close to Seedea:Research.Research#CreativityIsPhysical view on "natural creativity" (information is physical, creativity is physical)
    • Logic and Chance (p47)
    • Personal Instances (p50)
    • The Case of Pascal (p53)
    • Attempts to Govern our Unconscious (p54)
  • 5 The Later Conscious Work
    • The Fourth Stage (p56)
      • "preparation, incubation and illumination"
      • "a rather mechanical part of the work [...] verification"
    • A Statement by Paul Valery (p57)
    • Numerical Calculators (p58)
    • Appreciation of One's Own Work (p60)
      • "every stage of the research, has to be, so to speak, articulated to the folowing one by a result in a precise form, which I should propoe to call a relay-result" (p62)
    • Incubation and Relay-results (p63)
  • 6 Discovery as a Synthesis. The Help of Signs
    • Synthesis in Discovery (p64)
    • The Use of Signs (p66)
    • Words and Wordless Thought (p66)
      • Boileau's famous verses "
        Ce qui se conceoit bien s'enonce clairement,
        Et les mots pour le dire arrivent aisement,
        " (p70)
    • Mental Pictures in Usual Thought (p71)
    • Mental Pictures in Tense Thought (p73)
    • Personal Observations (p75)
      • "I have been told by some friends that I have a special way of looking when indulging in mathematical research. I hardly doubt that this especially accompanies the construction of the schema in question." (p78)
        • see also Confirmation holism on Wikipedia, also known as Theory Laden or Theory Dependance
          • a bias that seems to let some who developed their own model to "see what other can not see", yet begging for experimental confirmation
        • "Quand on demande a Michel Foucault ce qu'un livre est pour lui, il repond: c'est une boite a outils. Proust.... disait que son livre etait comme des lunettes: voyez si elles vous conviennent, si vous percevez grace a elles ce que vous n'auriez pas pu saisir autrement; sinon laissez mon livre" page 73 of Mille Plateaux also published in La Guerre des Idees and Literary philosophers
    • Respective Roles of Full Consciousness and Frindge-Consciousness (p80)
    • Other Stages of Research (p81)
    • Another Conception (p83)
    • An Inquiry among Mathematicians (p83)
    • Some Ideas of Descartes (p86)
    • Other Thinkers (p89)
    • Is Thought in words without inconvenience (p92)
    • A Valuable Description (p96)
    • Comparison with Another Question Concerning Imagery (p97)
    • Can Imagery be Educated? (p98)
      • "Such an auto-education of mental processes [Titchener ability to use auditory and visuals images] seems to me to be one of the most remarkable achievement in psychology." (p98-99)
    • Using Relay-results (p99)
    • General Remarks (p99)
    • Addendum (p99)
  • 7 Different Kinds of Mathematical Minds
    • The Case of Common Sense (p100)
    • Second Step: The Student in Mathematics (p103)
    • Logic and Intuitive Minds: A Political Aspect of the Question (p106)
    • Pointcarre's View of the Distinction (p108)
    • Application of Our Previous Data (p112)
    • (A) More or Less Depths in the Unconscious (p112)
      • "It is quite natural to speak of an intuitive mind if the zone where ideas are combined is deeper, and of a logical if that zone is rather superficial." (p113)
      • "If that zone is deeper, there will be more difficulty in bringing the result to the knowledge of consciousness and it is likely to happen for what is strictly necessary." (p113)
    • (B) More or Less Narrowly Directed Thought (p114)
    • (C) Different Auxiliary Representations (p114)
    • Other Differences in Mathematical Minds (p115)
  • 8 Paradoxical Cases of Intuition
    • Fermat 1601-1661 (p116)
    • Riemann 1826-1866 (p117)
    • Galois 1811-1831 (p118)
    • A Case in the Work of Pointcarre (p121)
    • Historical Comparisons (p122)
      • "we must admit that some part of the mental process develop so deeply in the unconscious, even important ones, remain hidden from our conscious self." (p122)
  • 9 The General Direction of Research
    • Two Conceptions of Invention (p124)
      • "One could say that application's constant relation to theory is the same as that of the leaf to the tree : one supports the other, but the former feeds the latter." (p125)
    • The Choice of Subjects (p126)
      • "Then, how are we to select the subject of research? This delicate choice is one of the most important things in research; according to it we form, generally in a reliable manner, our judgement of the value of a scientist." (p126)
    • Direction of Inventive Work and Desire of Originality (p131)
  • Final Remarks
    • (p133)
  • Appendix 1 An inquiry into the working methods of mathematicians
    • (p137)
  • Appendix 2 A testimonial from professor Einstein
    • (p142)
  • Appendix 3 The invention of infinitesimal calculus
    • (p144)

Overall remarks and questions

  • appreciating and enjoying beauty is also a training
    • like training an evolutionary filter to remove and increase quality again and again.
  • if reading is an act of not simply acquiring words but "executing" the thoughts of the author, how can one allows himself to meta-think without having competing executions during the reading?
    • is there a most efficient way to handle that "battle for consciousness" of each executing series of thoughts
  • instead of focusing on the quality of thoughts, one could think in term of "cognitive drag"

See also


Point A, B and C are debatable because of e, f and j.


(:new_vocabulary_start:) scantily a sieve (:new_vocabulary_end:)

Post-reading model

Draw a schema (using PmGraphViz or another solution) of the situation of the area in the studied domain after having read the book. Link it to the pre-reading model and align the two to help easy comparison.

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Other read books linking to the The Mathematician's Mind page :

The Tinkerer's Accomplice
The Princeton Companion to Mathematics

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