Search Results for Peano

Biographies

1. Peano biography
   
   • Giuseppe Peano .
     
   • Giuseppe Peano's parents worked on a farm and Giuseppe was born in the farmhouse 'Tetto Galant' about 5 km from Cuneo.
     
   • Among Peano's teachers in his first year at the University of Turin was D'Ovidio who taught him analytic geometry and algebra.
     
   • Peano continued to study pure mathematics in his third year and found that he was the only student to do so.
     
   • The others had continued their studies at the Engineering School which Peano himself had originally intended to do.
     
   • On 29 September 1880 Peano graduated as doctor of mathematics.
     
   • Peano joined the staff at the University of Turin in 1880, being appointed as assistant to D'Ovidio.
     
   • Peano was appointed assistant to Genocchi for 1881-82 and it was in 1882 that Peano made a discovery that would be typical of his style for many years, he discovered an error in a standard definition.
     
   • Genocchi was by this time quite old and in relatively poor health and Peano took over some of his teaching.
     
   • Peano was about to teach the students about the area of a curved surface when he realised that the definition in Serret's book, which was the standard text for the course, was incorrect.
     
   • Peano immediately told Genocchi of his discovery to be told that Genocchi already knew.
     
   • This book Course in Infinitesimal Calculus although based on Genocchi's lectures was edited by Peano and indeed it has much in it written by Peano himself.
     
   • published with additions by Dr Giuseppe Peano.
     
   • So that nothing will be attributed to me which is not mine, I must declare that I have had no part in the compilation of the aforementioned book and that everything is due to that outstanding young man Dr Giuseppe Peano ..
     
   • Peano received his qualification to be a university professor in December 1884 and he continued to teach further courses, some for Genocchi whose health had not recovered sufficiently to allow him to return to the University.
     
   • In 1886 Peano proved that if f (x, y) is continuous then the first order differential equation dy/dx = f (x, y) has a solution.
     
   • Four years later Peano showed that the solutions were not unique, giving as an example the differential equation dy/dx = 3y2/3 , with y(0) = 0.
     
   • In addition to his teaching at the University of Turin, Peano began lecturing at the Military Academy in Turin in 1886.
     
   • In 1888 Peano published the book Geometrical Calculus which begins with a chapter on mathematical logic.
     
   • A more significant feature of the book is that in it Peano sets out with great clarity the ideas of Grassmann which certainly were set out in a rather obscure way by Grassmann himself.
     
   • This book contains the first definition of a vector space given with a remarkably modern notation and style and, although it was not appreciated by many at the time, this is surely a quite remarkable achievement by Peano.
     
   • In 1889 Peano published his famous axioms, called Peano axioms, which defined the natural numbers in terms of sets.
     
   • These were published in a pamphlet Arithmetices principia, nova methodo exposita which, according to [Peano : Life and Works of Giuseppe Peano (Dordrecht, 1980).',5)">5] were:- .
     
   • The pamphlet was written in Latin and nobody has been able to give a good reason for this, other than [Peano : Life and Works of Giuseppe Peano (Dordrecht, 1980).',5)">5]:- .
     
   • Genocchi died in 1889 and Peano expected to be appointed to fill his chair.
     
   • Before the appointment could be made Peano published another stunning result.
     
   • Peano's continuous space-filling curves cannot be 1-1 of course, otherwise Netto's theorem would be contradicted.
     
   • Hausdorff wrote of Peano's result in Grundzuge der Mengenlehre in 1914:- .
     
   • In December 1890 Peano's wait to be appointed to Genocchi's chair was over when, after the usual competition, Peano was offered the post.
     
   • In 1891 Peano founded Rivista di matematica, a journal devoted mainly to logic and the foundations of mathematics.
     
   • The first paper in the first part is a ten page article by Peano summarising his work on mathematical logic up to that time.
     
   • Peano had a great skill in seeing that theorems were incorrect by spotting exceptions.
     
   • When Corrado Segre submitted an article to Rivista di matematica Peano pointed out that some of the theorems in the article had exceptions.
     
   • Of course this was so against Peano's rigorous approach to mathematics that he argued strongly:- .
     
   • It was not only Corrado Segre who suffered from Peano's outstanding ability to spot lack of rigour.
     
   • Of course it was the precision of his thinking, using the exactness of his mathematical logic, that gave Peano this clarity of thought.
     
   • Peano pointed out an error in a proof by Hermann Laurent in 1892 and, in the same year, reviewed a book by Veronese ending the review with the comment:- .
     
   • From around 1892, Peano embarked on a new and extremely ambitious project, namely the Formulario Mathematico.
     
   • In many ways this grand idea marks the end of Peano's extraordinary creative work.
     
   • Peano began trying to convert all those around him to believe in the importance of this project and this had the effect of annoying them.
     
   • However Peano and his close associates, including his assistants, Vailati, Burali-Forti, Pieri and Fano soon became deeply involved with the work.
     
   • When describing a new edition of the Formulario Mathematico in 1896 Peano writes:- .
     
   • When the calculus volume of the Formulario was published Peano, as he had indicated, began to use it for his teaching.
     
   • Peano, who was a good teacher when he began his lecturing career, became unacceptable to both his students and his colleagues by the style of his teaching.
     
   • One of his students, who was actually a great admirer of Peano, wrote:- .
     
   • The professor was a law unto himself in his own subject and Peano was not prepared to listen to his colleagues when they tried to encourage him to return to his old style of teaching.
     
   • The Formulario Mathematico project was completed in 1908 and one has to admire what Peano achieved but although the work contained a mine of information it was little used.
     
   • However, perhaps Peano's greatest triumph came in 1900.
     
   • It was a triumph for Peano and Russell, who attended the Congress, wrote in his autobiography:- .
     
   • The Congress was the turning point of my intellectual life, because there I met Peano.
     
   • Peano remained in Paris for this Congress and listened to Hilbert's talk setting out ten of the 23 problems which appeared in his paper aimed at giving the agenda for the next century.
     
   • Peano was particularly interested in the second problem which asked if the axioms of arithmetic could be proved consistent.
     
   • Even before the Formulario Mathematico project was completed Peano was putting in place the next major project of his life.
     
   • In 1903 Peano expressed interest in finding a universal, or international, language and proposed an artificial language "Latino sine flexione" based on Latin but stripped of all grammar.
     
   • Peano's career was therefore rather strangely divided into two periods.
     
   • Of his personality Kennedy writes in [Peano : Life and Works of Giuseppe Peano (Dordrecht, 1980).',5)">5]:- .
     
   • Peano may not only be classified as a 19th century mathematician and logician, but because of his originality and influence, must be judged one of the great scientists of that century.
     
   • Although Peano is a founder of mathematical logic, the German mathematical philosopher Gottlob Frege is today considered the father of mathematical logic.
     
   • Extract from Rivista di matematica (1895) showing the original Peano axioms.
     
   • Honours awarded to Giuseppe Peano .
     
   • http://www-history.mcs.st-andrews.ac.uk/Biographies/Peano.html .
     

2. Genocchi biography
   
   • However, as Kennedy explains in [Peano : Life and Works of Giuseppe Peano (Dordrecht, 1980).',1)">1], he was not ideally suited to the practice of law:- .
     
   • However, he was not entirely successful as a teacher either [Peano : Life and Works of Giuseppe Peano (Dordrecht, 1980).',1)">1]:- .
     
   • During the year 1881-82 Peano served as his assistant.
     
   • Kennedy writes in [Peano : Life and Works of Giuseppe Peano (Dordrecht, 1980).',1)">1]:- .
     
   • Genocchi's style as a teacher is also described in [Peano : Life and Works of Giuseppe Peano (Dordrecht, 1980).',1)">1]:- .
     
   • In 1882 Genocchi broke his kneecap and Peano took over his teaching.
     
   • This book was based on Genocchi's lectures but was largely the work of Peano.
     
   • Universita di Torino (1889-90), 195-202.',2)">2] Peano explained how the publication arose:- .
     
   • The book was important and widely aclaimed despite some unease over whether Peano had Genocchi's full agreement to publish the text with the additions he made.
     

3. Vailati biography
   
   • Peano was appointed as assistant to D'Ovidio at Turin in 1880 and Vailati was among the first group of students for whom he had to care.
     
   • Clearly Peano had a major influence on Vailati who would become interested in topics on which Peano worked.
     
   • However, probably mainly due to Peano, Vailati realised that mathematics was the subject for him so he continued to study at Turin for his mathematics degree which he was awarded in 1888.
     
   • Then in 1892 he was appointed as Peano's second assistant, a position he held until 1895.
     
   • After completing his years as Peano's assistant, Vailati became assistant in projective grometry and then one year later he became Volterra's assistant at Turin.
     
   • Vailati worked on mathematical logic, working closely with Peano on this topic, and also on the history and methodology of science.
     
   • Kennedy writes in [Peano : Life and Works of Giuseppe Peano (Dordrecht, 1980).',1)">1]:- .
     
   • di matematica 8 (1909), 206-7.',2)">2] Peano describes Vailati:- .
     

4. Vacca biography
   
   • He moved to Turin where he became Peano's assistant in November 1897.
     
   • Our work on the logic of Leibniz was almost completed (at least we thought so) when we had the pleasure, at the International Congress of Philosophy (August, 1900), of making the acquaintance of Mr Giovanni Vacca, at that time mathematical assistant at the University of Turin, who had examined, the year before, the manuscripts of Leibniz preserved in Hannover, and had extracted from them several formulae of logic inserted in the "Formulaire de Mathematiques" of Mr Peano.
     
   • However before this, in 1902, Vacca's position as Peano's assistant came to an end and he had returned to Genoa.
     
   • In 1904 Vacca returned to Turin and was assistant to Peano for one further year.
     
   • For example in 1928 Peano presented a paper by Vacca on Fermat's method of descent to the Academy of Sciences of Turin.
     

5. Faa di Bruno biography
   
   • Faa di Bruno travelled to Paris where he studied at the Sorbonne under Cauchy who [Peano : Life and Works of Giuseppe Peano (Dordrecht, 1980).',1)">1]:- .
     
   • In 1898 the printing press was purchased by Peano for 407 lire and he printed the Rivista di Matematica on it for several years.
     
   • In [Peano : Life and Works of Giuseppe Peano (Dordrecht, 1980).',1)">1] Faa di Bruno is described as follows:- .
     

6. Burali-Forti biography
   
   • At the start of the 1894-95 academic session, Burali-Forti became Peano's assistant at the University of Turin.
     
   • Burali-Forti was a close friend of Peano's but his closest friend and mathematical collaborator was Roberto Marcolongo.
     
   • Kennedy writes in [Peano : Life and Works of Giuseppe Peano (Dordrecht, 1980).',2)">2]:- .
     

7. Boggio biography
   
   • A text which Boggio wrote on the differential calculus with geometrical applications, published in 1921, was reviewed by his colleague Peano who says the books use of vector methods:- .
     
   • Kennedy writes in [Peano : Life and Works of Giuseppe Peano (Dordrecht, 1980).',1)">1]:- .
     
   • Boggio was mainly interested in mathematical physics and potential theory after his work with Peano when he held the assistant position in Turin.
     

8. Pieri biography
   
   • However, after he moved to Turin, Pieri became influenced by Peano at the University and Burali-Forti who was a colleague at the Military Academy.
     
   • He improved on results of Pasch and Peano and then, in 1905, Pieri gave the first axiomatic definition of complex projective geometry which does not build on real projective geometry.
     
   • In the field of the philosophy of sciences the Italian phalanx was supreme: Peano, Burali-Forti, Padoa, Pieri absolutely dominated the discussion.
     
   • Peano in [Academia pro Interlingua, Discussiones 4 (1913), 31-35.',8)">8] writes:- .
     

9. D'Ovidio biography
   
   • He had two famous assistants, Peano (1880-83) and Corrado Segre (1883-84).
     
   • Kennedy writes in [Peano : Life and Works of Giuseppe Peano (Dordrecht, 1980).',1)">1]:- .
     

10. Bruno biography
    
    • His lifestyle is described in [Peano : Life and Works of Giuseppe Peano (Dordrecht, 1980).',1)">1]:- .
      

11. Siacci biography
    
    • In [Peano : Life and Works of Giuseppe Peano (Dordrecht, 1980).',1)">1] his mathematical contributions are given:- .
      

12. Whitehead biography
    
    • In fact they had attended the International Congress of Mathematicians in Paris in 1900 and there they had learnt about Peano's work on the foundations of mathematics.
      
    • This led to them study Peano's papers and this must have been a major factor in getting their collaboration started.
      

13. Frege biography
    
    • He must have hoped that this first volume of what he viewed would be his greatest achievement would be well received, but except for one review by Peano, it was ignored by his contemporaries.
      
    • Frege's influence in the short term came through the work of Peano, Wittgenstein, Husserl, Carnap and Russell.
      

14. Grassmann biography
    
    • Other who were directly influenced included Hankel, Peano, Whitehead, and Klein.
      
    • Much of Peano's contributions were, as he acknowledges himself, based on the ideas of Grassmann.
      

15. Milne Archibald biography
    
    • He read papers at meetings of the Society such as Notes on the equation of the parabolic cylinder on Friday 9 January 1914, The Conformal Representation of the Quotient of two Bessel Functions on 24 January 1916, and Note on the Peano-Baker method of solving linear differential equations on 11 February 1916.
      

16. Padoa biography
    
    • Padua belonged to Peano's school of mathematical logic, popularising this type of work.
      

17. Lipschitz biography
    
    • Peano gave an existence theorem for this differential equation, giving conditions which guarantee at least one solution.
      

18. Veronese biography
    
    • However Peano strongly criticised the notion due to the lack of rigour of Veronese's description and also for the fact that he did not justify his use of infinitesimal and infinite segments.
      

19. Wilder biography
    
    • an exposition of the basic theories of modern mathematics: the theory of sets, the real number system (on the basis of the Peano axioms) and the theory of groups (including some of its applications in algebra and geometry).
      

20. Hahn biography
    
    • In a paper introducing local connectedness he characterised the sets which a point can traverse in a continuous motion; this is, the continuous images of a time interval or a segment (now often called Peano continua).
      

21. Veblen biography
    
    • His often quoted dissertation under Moore, on a system of axioms of Euclidean geometry, followed the trend of development of Pasch (1882) and Peano (1889, 1894) rather than that of Hilbert (1899) and Pieri (1899).
      

22. Segre Beniamino biography
    
    • Beniamino Segre's teachers at Turin University included Peano, Fano, Fubini and Corrado Segre (a not too close relative).
      

23. Poretsky biography
    
    • These include: various documents related to Poretsky's lectures on mathematical logic for mathematics department students at Kazan University which were intended to be given for three semesters in the autumn of 1887 and all of 1888 but were delivered only during the 1888 Spring semester, a complete mathematical logic program compiled by Poretsky, materials related to Poretsky's father and family, Poretsky's Magister's (master's) dissertation and the decision of the physics-mathematics faculty council to award him the doctorate in astronomy rather than the Magister, a complete list of the sources he used (including Boole, Jevons, Schroder, and Peano), biographical data and materials regarding his illness and subsequent dismissal from Kazan University.
      

24. Bottasso biography
    
    • He was a lecturer at the 'Conferenze matematiche' organised by Giuseppe Peano at the Turin University between 1915 and 1916, speaking to the high school teachers about numerical calculus.
      

25. Hildebrant biography
    
    • The Peano Axioms; 3.
      

26. Spencer Tony biography
    
    • The basis for the orthogonal invariants of a larger number of matrices is then deduced by using Peano's theorem.
      

27. Couturat biography
    
    • The topics covered in the correspondence include: the foundations of geometry, extension versus intension in logic, the Russell paradox, the axiom of choice, the controversies with Poincare, logic, Leibniz, Peano, Kant, arithmetical induction, mathematical existence, politics, international language, and some personal matters.
      

28. Mandelbrot biography
    
    • Odd indeed they were, there were curves - one dimensional lines in effect - which filled two dimensional spaces, there were curves which were well behaved, that is nice and continuous but which had no slope at any point (not just some points, ANY points) and they went by strange names, the Peano Space filling curve, the Sierpinski gasket, the Koch curve, the Cantor Ternary set.
      

History Topics

1. Abstract linear spaces
   
   • The first to give an axiomatic definition of a real linear space was Peano in a book published in Torino in 1888.
     Go directly to this paragraph
     
   • Peano's 1888 book Calcolo geometrico secondo l'Ausdehnungslehre di H.
     
   • It was many years before this notation was to become accepted, in fact Peano's book seems to have had very little influence for many years.
     
   • In Chapter IX of the book Peano gives axioms for a linear space.
     
   • It is hard to believe that Peano writes the following in 1888.
     
   • Peano goes on to state the existence of a zero object 0 and says that 0a = 0, that a - b means a + (-b) and states it is easy to show that a - a = 0 and 0 + a = a.
     
   • Peano defines a linear system to be any system of objects satisfying his four conditions.
     
   • Peano considers entire functions f(x) of a variable x, defines the sum of f1(x) and f2(x) and the product of f(x) by a real number m.
     
   • Peano defines linear operators on a linear space, shows that by using coordinates one obtains a matrix.
     
   • However Pincherle did not base his work on that of Peano, rather on the abstract operator theory of Leibniz and d'Alembert.
     Go directly to this paragraph
     
   • Although never attaining the level of abstraction which Peano had achieved, Hilbert and his student Schmidt looked at infinite dimensional spaces of functions in 1904.
     Go directly to this paragraph
     

2. Set theory references
   
   • G Heinzmann (ed.), Poincare, Russell, Zermelo et Peano : textes de la discussion (1906-1912) sur les fondements des mathematiques : des antinomies a la predicativite (Paris, 1986).
     
   • G Heinzmann (ed.), Poincare, Russell, Zermelo et Peano : textes de la discussion (1906-1912) sur les fondements des mathematiques : des antinomies a la predicativite (Paris, 1986).
     

3. Set theory references
   
   • G Heinzmann (ed.), Poincare, Russell, Zermelo et Peano : textes de la discussion (1906-1912) sur les fondements des mathematiques : des antinomies a la predicativite (Paris, 1986).
     
   • G Heinzmann (ed.), Poincare, Russell, Zermelo et Peano : textes de la discussion (1906-1912) sur les fondements des mathematiques : des antinomies a la predicativite (Paris, 1986).
     

4. Set theory
   
   • Although not of major importance in the development of set theory it is worth noting that Peano introduced the symbol belongs for 'is an element of' in 1889.
     Go directly to this paragraph
     
   • The first person to explicitly note that he was using such an axiom seems to have been Peano in 1890 in dealing with an existence proof for solutions to a system of differential equations.
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Famous Curves

Societies etc

1. International Congress Speakers
   
   • Giuseppe Peano, Logica matematica.
     

References

1. References for Peano
   
   • References for Giuseppe Peano .
     
   • D A Gillies, Frege, Dedekind, and Peano on the foundations of arithmetic (Assen, 1982).
     
   • H C Kennedy, Giuseppe Peano (Basel, 1974).
     
   • H C Kennedy, Peano : Life and Works of Giuseppe Peano (Dordrecht, 1980).
     
   • R Murawski, Giuseppe Peano- pioneer and promoter of symbolic logic, Uniwersytet im.
     
   • G Peano, Selected works of Giuseppe Peano, with a biographical sketch and bibliography by H C Kennedy (London, 1973).
     
   • H C Kennedy, Peano's concept of number, Historia Mathematica 1 (1974), 387-408.
     
   • G Lolli, On the 50th anniversary of Peano (1858-1932), Scientia (Milano) 117 (5-8) (1982), 361-367.
     
   • F Palladino, The letters of Giuseppe Peano in the correspondence of Ernesto Cesaro (Italian), Nuncius Ann.
     
   • W V O Quine, Peano as logician, Hist.
     
   • M Segre, Peano's axioms in their historical context, Archive for History of Exact Science 48 (3-4) (1994), 201-342.
     
   • E A Zaitsev, An interpretation of Peano's logic, Archive for History of Exact Science 46 (4) (1994), 367-383.
     
   • http://www-history.mcs.st-andrews.ac.uk/References/Peano.html .
     

2. References for Genocchi
   
   • H C Kennedy, Peano : Life and Works of Giuseppe Peano (Dordrecht, 1980).
     
   • G Peano, Angelo Genocchi, Annuario R.
     

3. References for Vailati
   
   • H C Kennedy, Peano : Life and Works of Giuseppe Peano (Dordrecht, 1980).
     
   • G Peano, In Memoriam di Giovanni Vailati, Boll.
     

4. References for Burali-Forti
   
   • H C Kennedy, Peano : Life and Works of Giuseppe Peano (Dordrecht, 1980).
     
   • R Feys, Peano et Burali-Forti, precurseurs de la logique combinatoire, Actes du XIeme Congres International de Philosophie, Bruxelles V (Louvain, 1953), 70-72.
     

5. References for Pieri
   
   • H C Kennedy, Peano : Life and Works of Giuseppe Peano (Dordrecht, 1980).
     
   • G Peano, Mario Pieri, Academia pro Interlingua, Discussiones 4 (1913), 31-35.
     

6. References for Boggio
   
   • H C Kennedy, Peano : Life and Works of Giuseppe Peano (Dordrecht, 1980).
     

7. References for Poincare
   
   • G Heinzmann, Poincare, Russell, Zermelo et Peano (Paris, 1986).
     
   • A Drago, Poincare versus Peano and Hilbert about the mathematical principle of induction, in Henri Poincare : science et philosophie, Nancy, 1994 (Berlin, 1996), 513-527; 586.
     

8. References for Vacca
   
   • H C Kennedy, Peano : Life and Works of Giuseppe Peano (Dordrecht, 1980).
     

9. References for Faa di Bruno
   
   • H C Kennedy, Peano : Life and Works of Giuseppe Peano (Dordrecht, 1980).
     

10. References for Bruno
    
    • H C Kennedy, Peano : Life and Works of Giuseppe Peano (Dordrecht, 1980).
      

11. References for Siacci
    
    • H C Kennedy, Peano : Life and Works of Giuseppe Peano (Dordrecht, 1980).
      

12. References for D'Ovidio
    
    • H C Kennedy, Peano : Life and Works of Giuseppe Peano (Dordrecht, 1980).
      

13. References for Bhaskara II
    
    • V Madhukar Mallayya and K Jha, Bhaskara's concept of numeration in decuple proportions - earliest reference in Vedas with Yaska's 'Nirukta' throwing light on the notion of succession in enumeration : an anticipation of Peano's axioms, Ganita-Bharati 17 (1-4) (1995), 85-91.
      

14. References for Zermelo
    
    • G Heinzmann (ed.), Poincare, Russell, Zermelo et Peano : textes de la discussion (1906-1912) sur les fondements des mathematiques : des antinomies a la predicativite (Paris, 1986).
      

15. References for Schroder
    
    • V Peckhaus, Ernst Schroder und die 'pasigraphischen Systeme' von Peano und Peirce, Modern Logic 1 (2-3) (1990/91), 174-205.
      

16. References for Dedekind
    
    • D A Gillies, Frege, Dedekind, and Peano on the foundations of arithmetic (Assen, 1982).
      

17. References for Bhaskara
    
    • V Madhukar Mallayya and K Jha, Bhaskara's concept of numeration in decuple proportions - earliest reference in Vedas with Yaska's 'Nirukta' throwing light on the notion of succession in enumeration : an anticipation of Peano's axioms, Ganita-Bharati 17 (1-4) (1995), 85-91.
      

18. References for Volterra
    
    • A Guerraggio, Memoirs of Volterra and Peano on the motion of the poles (Italian), Archive for History of Exact Science 31 (2) (1984), 97-126.
      

19. References for Frege
    
    • D A Gillies, Frege, Dedekind, and Peano on the foundations of arithmetic (Assen, 1982).
      

20. References for Poretsky
    
    • N I Styazhkin, History of Mathematical Logic from Leibniz to Peano (Cambridge, Mass., 1969).
      

21. References for Pasch
    
    • H C Kennedy, The origins of modern axiomatics: Pasch to Peano, Amer.
      

Additional material

1. Edmund Landau: 'Foundations of Analysis' Prefaces
   
   • I do not, to be sure, prove, the consistency of the five Peano axioms (because that can not be done), but each of them is obviously independent of the preceding ones.
     
   • In the Foundations of Analysis course I begin with the Peano axioms for the natural numbers and get through the theory of the real and of the complex numbers.
     
   • He returned my manuscript to me with the remark that he had found it necessary to add further axioms to Peano's in the course of the development, because the standard procedure, which I had followed, had proved to be incomplete at a certain point.
     
   • Grandjot's axioms can all be proved (as we could have learned from Dedekind), so that everything remains based on Peano's axioms (cf.
     
   • On the basis of his five axioms, Peano defines x + y for fixed x and all y as follows: .
     
   • All would be well if - and this is not done in Peano's method because order is introduced only after addition - one had the concept "numbers ≤ y" and could speak of the set of y's for which there is an f(z), defined for z ≤ y, with the properties .
     

2. Publications of Alessandro Padoa
   
   • A Padoa, Sul teorema di Cantor-Bernstein-Peano, PM (1907), 23.
     
   • Peano all'ideografia logica, PM (1933), 3; 15.
     

3. A N Whitehead addresses the British Association in 1916, Part 2
   
   • The final impulse to modern logic comes from the independent discovery of the importance of the logical variable by Frege and Peano.
     
   • Frege went further than Peano, but by an unfortunate symbolism rendered his work so obscure that no one fully recognised his meaning who had not found it out for himself.
     

4. Gian-Carlo Rota: Alonzo Church
   
   • It cannot be a complete coincidence that several outstanding logicians of the twentieth century found shelter in asylums at some time in their lives: Cantor, Zermelo, Godel, Peano, and Post are some.
     

5. Publications of Corrado Segre
   
   • C Segre, Su alcuni indirizzi nelle investigazioni geometriche, Rivista di matematica diretta da G Peano I (1891), 42-66.
     

6. Ernest Hobson addresses the British Association in 1910, Part 3
   
   • I am very far from wishing to minimise the high philosophic interest of the attempt made by the Peano-Russell school to exhibit Mathematics as a scheme of deductive logic.
     

7. Hans Hahn: 'The crisis in intuition
   
   • Peano ..
     

8. Oskar Bolza: 'Calculus of Variations
   
   • In order, however, to make the book accessible to a larger circle of readers, I have systematically given references to the following standard works: Encyclopaedie der mathematischen Wissenschaften, especially the articles on "Allgemeine Functionslehre" (Pringsheim) and "Differential- und Integralrechnung" (Voss); Jordan, Cours d'Analyse, second edition); Genocchi-Peano, Differentialrechnung und Grundzuge der Integralrechnung, translated by Bohlmann and Schepp; occasionally also to Dini, Theorie der Functionen einer veraaderlichen reelen Grosse, translated by Lutroth and Schepp; Stolz, Grundzuge der Differential- und Integralrechnung.
     

9. Bertrand Russell on Euclid
   
   • Pasch, Vorlesungen uber neuere Geometrie, Leipzig, 1882; Peano, I Principii di Geometria, Turin, 1889.] .
     

10. Gian-Carlo Rota: Alonzo Church
    
    • It cannot be a complete coincidence that several outstanding logicians of the twentieth century found shelter in asylums at some time in their lives: Cantor, Zermelo, Godel, Peano, and Post are some.
      

11. Felix Klein on intuition
    
    • This idea of building up science purely on the basis of axioms has since been carried still further by Peano, in his logical calculus ..
      

12. R L Wilder: 'Cultural Basis of Mathematics II
    
    • In particular, Bell devotes at least 25 pages to the development of what he calls "mathematical logic." Can there be any possible doubt that this subject, not considered part of mathematics in our culture in 1900, despite the pioneering work of Peano and his colleagues, is now in such "good standing" that any impartial definition of mathematics must be broad enough to include it? .
      

13. Publications of Corrado Segre
    
    • C Segre, Su alcuni indirizzi nelle investigazioni geometriche, Rivista di matematica diretta da G Peano I (1891), 42-66.
      

Quotations

1. A quotation by Peano
   
   • A quotation by Giuseppe Peano .
     
   • http://www-history.mcs.st-andrews.ac.uk/Quotations/Peano.html .
     

Chronology

1. Chronology for 1880 to 1890
   
   • Peano proves that if f(x, y) is continuous then the first order differential equation dy/dx = f(x, y) has a solution.
     
   • He puts arithmetic on a rigorous foundation giving what were later known as the "Peano axioms".
     
   • Peano publishes Arithmetices principia, nova methodo exposita (The Principles of Arithmetic) which gives the Peano axioms defining the natural numbers in terms of sets.
     
   • Peano discovers a space filling curve.
     

2. Mathematical Chronology
   
   • Peano proves that if f(x, y) is continuous then the first order differential equation dy/dx = f(x, y) has a solution.
     
   • He puts arithmetic on a rigorous foundation giving what were later known as the "Peano axioms".
     
   • Peano publishes Arithmetices principia, nova methodo exposita (The Principles of Arithmetic) which gives the Peano axioms defining the natural numbers in terms of sets.
     
   • Peano discovers a space filling curve.
     

3. Chronology for 1890 to 1900
   
   • Peano discovers a space filling curve.