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notes








1


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2


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3


     :   ,  , , ,  . ..




4


   : Jillian E. Lauer andStella F. Lourenco, Spatial Processing inInfancy Predicts Both Spatial andMathematical Aptitude inChildhood, Psychological Science 27, no. 10 (2016): 129198.




5


   : Margalit Fox, Katherine Johnson Dies at101; Mathematician Broke Barriers atNASA, NewYork Times, Feb. 24, 2020,  2010 Fayetteville (NC) Observer.




6


 .. ..




7


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8


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9


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10


  :   26 1791       (Letters ofthe Wordsworth Family From 1787 to1855, ed. William Knight, vol.1 (Cambridge: Ginn andCompany, 1907), 28),  ,      ,      .   , , , ,  ,     .




11


,  : Joan Baum, Onthe Importance ofMathematics toWordsworth, Modern Language Quarterly 46, no. 4 (1985):392.




12


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13


  :  ,         Dublin University Magazine 19 (1842): 95,    ,     .78   Life ofSir William Rowan Hamilton ( ).        ,  ,    ,    .       1820  ,   ,   ,     ,   ;   ,  ,        ,      ,    .     ?




14


   : Zerah Colburn, AMemoir ofZerah Colburn: Written byHimself. Containing anAccount ofthe First Discovery ofHis Remarkable Powers; His Travels inAmerica andResidence inEurope; aHistory ofthe Various Plans Devised forHis Patronage; His Return tothis Country, andthe Causes which Led Him toHis Present Profession; with His Peculiar Methods ofCalculation (Springfield, MA: G. andC. Merriam, 1833),72.




15


    : Graves, Life ofSir William Rowan Hamilton, 7879.




16


 :     16 1827   Graves, Life ofSir William Rowan Hamilton,261.




17


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18


ʠ1922,    ,     :   (-   )    ,   ( )   -,    3 (#g3).  ,       ,   - ,         .




19


 , : Mr. Lincolns Early Life: How He Educated Himself, NewYork Times, Sep. 4, 1864: 5. ,            .




20


  demonstrate, demonstration         . ..




21


   (American Tract Society)  ,    1825     . ..




22


 , :     Jesse William Weik, TheReal Lincoln; aPortrait (Boston: Houghton Mifflin, 1922), 240.        : Tales ofImpossibility (Princeton: Princeton University Press, 2019),   ,      ,    .




23


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24


 , ,   .  .. ..




25


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26


  . ..




27


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28


 ,  ,          7     .




29


,    :  Geometry without Axioms, Quarterly Journal ofEducation XIII (1833):105.




30


  (proposition):   蠖 A. Kucharski, Euclid asFounding Father, Nautilus, Oct. 13, 2016; http://dev.nautil.us/issue/41/selection/euclid-as-founding-father (http://dev.nautil.us/issue/41/selection/euclid-as-founding-father).




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32


        ;     :      .         ࠫ,        .




33


  : Thomas Jefferson, TheEssential Jefferson, ed. Jean M. Yarbrough (Indianapolis: Hackett Publishing, 2006),193.




34


  : Thomas Jefferson, ThePapers ofThomas Jefferson, Retirement Series, ed. J. Jefferson Looney, vol.4 (Princeton: Princeton University Press, 2008), 429; https://press.princeton.edu/books/ebook/9780691184623/the-papers-of-thomas-jefferson-retirement-series-volume-4 (https://press.princeton.edu/books/ebook/9780691184623/the-papers-of-thomas-jefferson-retirement-series-volume-4).




35


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36


  :    ,      : Quarterly Journal ofEducation, vol.XIII (1833): 105.      !




37


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38


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. .   . . ..




39


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40


    :   ,   ,  . ,        ,    ,  ,    ,    ,       . . ..




41


   ,    1930,    :   170; 170-    :   36,     , ..,    ,   :  !      .




42


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43


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44


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45


   : James J. Sylvester, APlea forthe Mathematician, Nature 1 (1870): 26163.




46


   1950: Kenneth E. Brown, Why Teach Geometry? Mathematics Teacher 43, no. 3 (1950): 1036; https://www.jstor.org/stable/27953519 (https://www.jstor.org/stable/27953519).




47


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48


  : Whitney, Lincoln theCitizen,178.




49


   ࠖ  . ..




50


 :      16 2013: Two-Column Proofs That Two-Column Proofs Are Terrible, Math with Bad Drawings (blog), mathwithbaddrawings.com/2013/10/16/two-column-proofs-that-two-column-proofs-are-terrible/ (http://mathwithbaddrawings.com/2013/10/16/two-column-proofs-that-two-column-proofs-are-terrible/).




51


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52


            .




53


       ,   ,  :                  . ...   .  (1990) 6. . ..




54


    ,   ,   .  :    1           1.   ,   :    ( ࠖ ).      ,       .    . , ,       . ..




55


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56


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57


       II,       .




58


  ,              ,     ;   ,  ,    ,     .




59


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60


, , : Henri Poincar, TheValue ofScience, trans. G. B. Halsted (NewYork: TheScience Press, 1907),23.




61


  ,  ,    ,       .[ : Ƞ ,   ,      ,   ,         ,     . ,  I.  .-. . .]




62


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63


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64


   Quod Erat Demonstrandum,    .




65


     Common Core,           K-12,    .    ,     .  , ,   Common Core ,    ,   .




66


   : M. J. Nathan, etal., Actions Speak Louder with Words: TheRoles ofAction andPedagogical Language forGrounding Mathematical Proof, Learning andInstruction 33 (2014): 18293.




67


        !




68


  : Jeremy Gray, Henri Poincar: AScientific Biography (Princeton: Princeton University Press, 2012),26.




69


  1970: David Lewis andStephanie Lewis, Holes, Australasian Journal ofPhilosophy 48, no. 2 (1970): 20612.




70


    2014:https://forum.bodybuilding.com/showthread.php?t162056763&page=1 (https://forum.bodybuilding.com/showthread.php?t162056763%26page%3d1).




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72


 , :www.youtube.com/watch?v=W0tYRVQvKbM (https://www.youtube.com/watch?v%3dW0tYRVQvKbM).




73


,  :  ,     ,      , ,       .          .




74


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76


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77


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78


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79


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80


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81


,   : Dantzig, Henri Poincar,3.




82


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87


   : Diese Tendenz scheint heute selbstverstandlich; sie war es voracht Jahren nicht; es bedurfte derEnergie unddes Temperamentes vonEmmy Noether, um sie zumAllgemeingut derTopologen zumachen undsie inder Topologie, ihren Fragestellungen undihren Methoden, diejenige Rolle spielen zulassen, diesie heute spielt. Paul Alexandroff andHeinz Hopf, Topologie I: Erster Band. Grundbegriffe derMengentheoretischen Topologie Topologie derKomplexe Topologische Invarianzs?tze undAnschliessende Begriffsbildungen Verschlingungen imn-Dimensionalen Euklidischen Raum Stetige Abbildungen vonPolyedern (Berlin: Springer-Verlag, 1935), ix.       .




88


 :      Ernst Breitenberger, Johann Benedikt Listing, History ofTopology, ed. I. M. James (Amsterdam: North-Holland, 1999), 90924.




89


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