Piccola storia della matematica, 1: a book of mine published in 2012 dealing with history of mathematics from the beginnings till Reinassance.
Matemática escolar desde un punto de vista superior, I: a book I wrote in collaboration with Carlo Madonna and Alberto Barcia (Universidad Autonoma de Madrid) published in 2011, which deals with fundamental aspects of arithmetics and set theory, with an historical-didactic approach.
Piccola storia della matematica, 2: a book of mine published in 2010 dealing with history of mathematics in XVII and XVIII centuries.
Other publications: expository or research.
Notes...: some of my texts, notes and books about various mathematical subjects matematica. I subdivide them into: didactic, which practically require no mathematical knowledge, graduate. thus at a PhD and research level.
Books: some classic mathematical book, at various levels, which I suggest.
Links: Institutions: departments, institutes, etc., Electronic Repositories specialized in mathematics, History of mathematics, Texts on line about mathematics.
Subjects: Differential Geometry a subject I studied for several years.
Some classic texts...: classic texts of mathematics to download from my site.

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Piccola storia della matematica, 1

Negli ultimi anni mi sono interessato di storia della matematica: l'editore Alphatest mi ha commissionato un "bignamino" relativo al periodo che va dalla Preistoria alla fine del Cinquecento: Piccola storia della matematica, vol.1. Si tratta di una sintetica ma esauriente esposizione degli sviluppi della matematica dalle origini fino al Rinascimento, che cerca di esporre anche i principali sviluppi della matematica nei paesi extra-europei (Egitto, India, Cina, Maya, etc.), cercando di fornire un rapido inquadramento storico: oltre a una descrizione dei principali risultati dell'epoca, di una discussione delle fonti e anche di biografie, vere o presunte, di matematici del passato, comprende diverse schede concepite per stimolare il lettore in percorsi interdisciplinari.
    Indice
  1. Storia e preistoria della matematica: Cosa intendiamo per matematica?; Preistoria dei numeri; La notazione numerica e la scrittura; La misurazione del tempo e i cicli naturali; La misurazione di aree e volumi; Matematica, astronomia e religione.
  2. La matematica in Mesopotamia e nell'Antico Egitto: Gli albori della storia della matematica; Il sistema sessagesimale, la notazione posizionale e "l'antenato dello zero"; Il periodo paleo-babilonese; Geometria, aritmetica e "algebra" babilonesi; Matematica lungo il corso del Nilo; Esempi di aritmetica e geometria egizia.
  3. La matematica dei popoli indoeuropei: India e Grecia: Gli Indoeuropei e gli albori della matematica Indiana; Gli Indoeuropei nella Grecia arcaica e le origini del pensiero speculativo; Le scuole ionica e pitagorica e gli inizi della geometria greca; La mistica pitagorica dei numeri; La scoperta degli incommensurabili, il punto geometrico e l'infinito in matematica; Le costruzioni con riga e compasso nella matematica del v secolo.
  4. L'età classica della matematica greca: Problemi e idee della matematica al tempo di Platone; L'influenza di Platone sulla matematica greca; La geometria solida; L'opera perduta di Eudosso di Cnido; Aristotele e la transizione verso la fioritura scientifica ellenistica; I frammenti della scienza ellenistica.
  5. Apogeo ellenistico: Euclide, Archimede e Apollonio: Euclide e i suoi Elementi; Il più grande matematico dell'Antichità; Il calcolo di lunghezze, aree, volumi e... granelli di sabbia; Archimede e il metodo di Eudosso; Il metodo meccanico di Archimede; Apollonio di Perga e la teoria delle coniche.
  6. La matematica nell'Impero Romano: Roma, la tradizione ellenica e l'Ellenismo; Alcuni campi di applicazione della matematica presso i Romani; Erone, Diofanto e la tradizione aritmetica babilonese; Astronomia e trigonometria nell'alto Impero; La geometria nel basso Impero; Il tramonto della matematica in Occidente.
  7. Splendori della matematica asiatica: Origine e caratteri della matematica cinese; I Nove capitoli: gli Elementi della matematica cinese; Geometria e aritmetica nella Cina antica; Matematica e astronomia nell'India classica; L'apice della matematica indiana.
  8. La matematica nel mondo islamico: La cultura islamica al crocevia fra oriente e occidente; Al-Khwarizmi e la creazione dell'algebra; La geometria islamica fra teoria e pratica; Alcuni elementi innovativi della matematica islamica; La trigonometria; La diffusione del sapere matematico dell'Islam nell'Europa medievale.
  9. La matematica nel Rinascimento: La cultura europea fra Medioevo e Rinascimento; Geometria, architettura e pittura nel Quattrocento; Il Rinascimento algebrico; Tartaglia, Cardano e Ferrari; I numeri complessi e l'inizio del simbolismo algebrico; L'alba della rivoluzione scientifica.
Buona lettura!

Matemática escolar desde un punto de vista superior, I

Ho collaborato nel 2011 alla stesura del volume Alberto Barcia, Paolo Caressa y Carlo G. Madonna Matemática escolar desde un punto de vista superior, I: conjuntos y números, Ediciones de la Universidad Autónoma de Madrid, in spagnolo: si tratta di una introduzione all'aritmetica che utilizza il linguaggio della teoria degli insiemi e che affronta questi argomenti classici ma elementari dal punto di vista della matematica universitaria. Di questo libro ho scritto le note storiche e collaborato alla stesura della parte insiemistica e assiomatica, oltre che alla redazione degli esercizi.
L'immagine della copertina è una mia foto scattata a Perugia e che ritrae un particolare della Fontana maggiore capolavoro di Nicola e Giovanni Pisano, in particolare la nicchia dove la Aritmetica e Retorica sono allegoricamente ritratte: mi sembrava una scelta appropriata dato che nel libro si trattano la logica e l'aritmetica.
    Índice general
  1. Introdución
  2. El lenguaje de la matemática: La estructura lógica de la matemática; ¿Que es una teoría matemática?; Conjuntos y operaciones con los conjuntos; Producto cartesiano y relaciones de equivalencias; Aplicaciones; Grupos y estructuras algebraicas; Notas históricas; Ejercicios.
  3. Números y sus operaciones: Axiomática para los números naturales; Definición recursiva de suma y relación de orden; Divisibilidad entre números naturales; El sistema posicional; mcd y mcm de dos números naturales; Criterios de divisibilidad; Números primo y factorización prima; Notas históricas; Ejercicios.
  4. Números enteros, racionales y decimales: Los números enteros; Los números racionales; Los números decimales; Notas históricas; Ejercicios.
  5. Soluciones.

Piccola storia della matematica, 2

Negli ultimi anni mi sono interessato di storia della matematica: l'editore Alphatest mi ha commissionato un "bignamino" relativo al periodo che va dalla fine del Cinquecento all'inizio dell'Ottocento: Piccola storia della matematica, vol.2. Si tratta di una sintetica ma esauriente esposizione degli sviluppi della matematica in questa epoca, che comprende anche un rapido inquadramento storico: vi sono descrite le principali idee matematiche fiorite in quel periodo, senza ovviamente entrare nei dettagli, e racchiude una cinquantina di biografie di matematici, più alcune schede concepite per stimolare il lettore in percorsi interdisciplinari.
    Indice
  1. La matematica nei secoli xvii e xviii: Aspetti della matematica dei secoli xvii e xviii; Galileo e la nascita della scienza moderna; Elementi di novità nella scienza e nella matematica seicentesche; La scoperta dell'utilità della matematica; La circolazione del sapere scientifico; Il contesto storico e culturale.
  2. La rinascita della geometria: Algebra e geometria agli inizi del Seicento; Fermat: teoria dei numeri e geometria analitica; La geometria algebrica di Cartesio; La nuova prospettiva di Desargues; Pascal: geometria e teologia; La parabola della geometria nel xvii secolo.
  3. Le origini del calcolo infinitesimale: L'influenza di Galileo; I precursori di Newton e Leibniz in Francia; L'aritmetica dell'infinito e il teorema fondamentale del calcolo; La scoperta newtoniana del calcolo infinitesimale; Leibniz: logica, algebra e analisi; La querelle Newton vs Leibniz.
  4. Sviluppi del calcolo infinitesimale nel xviii secolo: I problemi sollevati dal calcolo infinitesimale; I Bernoulli: una dinastia di matematici; L'enigma delle serie infinite; Il calcolo delle probabilità; Le equazioni differenziali; Lo stato dell'analisi alla metà del Settecento.
  5. L'irripetibile opera di Eulero: Il più prolifico matematico di tutti i tempi; I metodi di Eulero e la loro efficacia; I contributi al calcolo infinitesimale; Gli integrali ellittici: da Senigallia a Berlino; Il gusto per la teoria dei numeri; Applicazioni, problemi e soluzioni.
  6. La matematica fra lumi e Rivoluzione: Un matematico enciclopedico; Matematica e politica; Geometria e rivoluzione; Probabilità e determinismo in un sol uomo; Lo scandalo della geometria e la passione per l'aritmetica; La teoria di Fourier e gli sviluppi del concetto di funzione.
  7. Lagrange: un filosofo stoico nell'epoca dei lum: Il meglio di Italia e Francia alla corte di Berlino; La formalizzazione analitica del calcolo delle variazioni; I contributi alla meccanica celeste; La grande sintesi della meccanica analitica; Le funzioni analitiche e la teoria dei numeri; Facile come l'algebra?
Spero che leggere questo libriccino sia divertente almeno quanto lo è stato scriverlo...

Other publications

txt Grattacieli e crisi economiche, Xla Tangente 42, dicembre 2013, pp.24-27.
Una semplice e divulgativa discussione del concetto di correlazione in statistica viene seguita da un bizzarro esempio di applicazione, vale a dire la teoria per la quale le crisi economiche globali e la costruzione di nuovi grattacieli da record siano collegate in qualche modo. Applicando un po' di sano scetticismo e qualche conto banale cerchiamo di ridimensionare la questione...

txt Le tre vite di Giuseppe Luigi Lagrange, Xla Tangente 40, agosto 2013, pp.27-30.
Un breve articolo nel quale si riassume la carriera scientifica di Lagrange nel bicentenario della sua morte, mostrando l'indole cosmopolita di questo saggio uomo di scienza, che ha vissuto i primi trent'anni della sua vita a Torino, per poi spostarsi nella prestigiosa accademia della Berlino di Federico II e terminare la sua parabola umana e scientifica, carico di onori e gloria, nella Parigi del periodo rivoluzionario e napoleonico.

txt La matematica degli antichi Romani II, Xla Tangente 39, giugno 2013, pp.17-19.
La seconda parte di un articolo divulgativo nel quale si illustrano i principali esempi di applicazione della matematica presso gli antichi Romani, utilizzando alcune fonti classiche della letteratura latina e alcune idee di moderni studiosi di scienza del mondo antico.

txt La matematica degli antichi Romani, Xla Tangente 38, aprile 2013, pp.17-20.
La prima parte di un articolo divulgativo nel quale si collezionano i principali esempi di applicazione della matematica presso gli antichi Romani, mostrando come l'elaborazione di concetti matematici algoritmici fosse comune nel mondo tardo repubblicano e imperiale, contrariamente a quanto comunemente si crede.

txt Storia e didattica della scienza: concetti ed esempi, Revista de Didácticas Específicas 7, dicembre 2012, pp.23-44.
ABSTRACT: In this note we discuss, by means of examples, the relation between history and didactics of science: after recalling the classical arguments which advocates the importance of the historical approach for the didactics of science, a geometrical metaphor is introduced to provide a conceptual framework, which we call conceptual triangle. By means of this metaphor we analyse some examples of scientists who used history both when performing their research and as a didactic tool. Next we introduce the motivations which, in our opinion, are more relevant to the integration of history with didactics, quoting several examples.

txt Il gruppo dei quattro: da Klein a Piaget!, Xla Tangente 36, dicembre 2012, pp.27-30.
Un articolo divulgativo che tratta di alcune applicazioni della teoria dei gruppi a domini non prettamente scientifici, come la teoria della percezione e della conoscenza di Poincaré e la psicologia cognitiva di Piaget.

txt La formula del serial killer, Xla Tangente 32, aprile 2012, pp.29-32.
Un articolo divulgativo che introduce alcune tecniche matematiche che sono utilizzate per definire il profilo geografico di un serial killer e, sperabilmente, trovare il suo nascondiglio in modo da poterlo catturare: il tutto è stato di ispirazione per l'episodio pilota della serie televisiva Numb3rs.

txt Spread, bond e derivati: parole e numeri per i nostri soldi, Xla Tangente 31, febbraio 2012, pp.19-22.
Un articolo divulgativo che spiega in modo (spero) semplice alcuni concetti di base della finanza entrati nel linguaggio e nell'immaginario comune da un po' di tempo a questa parte. Non mancate di visitare il sito della rivista XlaTangente, che contiene molto materiale interessante di matematica divulgativa.

txt Complex Foliated Tori and their Moduli Spaces (with Adriano Tomassini), Communications in Contemporary Mathematics, 7 no.3 (2005), 1-15.   [versione txt dvi]
ABSTRACT: We consider complex foliated tori, their periods and polarizations on them. We define the moduli space of polarized complex foliated tori and show that it is a normal analytic complex space. Finally we discuss some examples.

txt Examples of Poisson Modules, II, Rendiconti del Circolo Matematico di Palermo(2) 53 (2004), 23-60.   [versione txt dvi]
ABSTRACT: We give a geometric description of different classes of Poisson modules as introduced in [1]: we start with tensors tangent to leaves on a Poisson manifold, consider Poisson structures on bundles and also an example of Poisson module on a manifold which does not come from any vector bundle; finally we use this language to sketch some integral calculus on Poisson manifolds: we suggest how to introduce integration, homology and cohomology in our setting.

txt Examples of Poisson Modules, I, Rendiconti del Circolo Matematico di Palermo(2) 52 (2003), 419-452.   [versione txt dvi]
ABSTRACT: We sketch some differential calculus on Poisson algebras and introduce a concept of module and representation on a Poisson algebras; we give examples and consider cohomologies connecting these constructions to the algebra of Poisson brackets.

txt On Poisson-Kähler structures (with Adriano Tomassini), preprint Università di Parma, 2001   [versione txt dvi]
ABSTRACT: Regular Poisson structures and foliated (almost) complex structures are considered on manifolds, as a generalization of the Kähler case. We discuss the existence of compatible Poisson and (almost) complex structures in general and we give examples of such structures on product of spheres, Poisson manifolds equipped with Dirac brackets induced by contact forms, Iwasawa manifold, etc.


Notes...

I collect here some unpublished mathematical writing which are intelligible (my personal archive is wider): I have classified them according to three levels of difficulty, since mathematics is an initiatory science, thus one cannot expect to understand immediately a subject, but has to return on it time and again, and each time he advances by a step in the stair of knowledge: in the top it seems to run around a circle, but actually you are ascending...
"Didactic" writings should be understood by any reader of average education; "undergraduate" writings are addressed to motivated reader and correspond more or less to what is teached in the first years of college studies; "graduate" writings correspond to higher level and are addressed to mathematicians.
Texts are provided in HTML or PDF, the latter being also given in the DVI format (compressed according to the ZIP format) commonly used in scientific communications: to read them free software is available: Acrobat Reader for PDF files and TeX for DVI files.
Currently there are few text available in English: if you read Italian then check the italian version of this page...

Popular science notes

txt Quasi per caso (2016)
Le slide di una conferenza sul concetto di successione casuale e sulla generazione di numeri pseudo-casuali, con molte animazioni Javascript: le pagine con le animazioni contengono delle brevi istruzioni per l'uso passando col mouse sopra i pulsanti e le caselle di testo.

txt Funzioni viste in diagonale (2007)
Si tratta di una breve nota divulgativa nella quale tento di spiegare in modo semplice l'argomento diagonale di Cantor applicato alla enumerazione di tutte le funzioni possibili.

txt Idiozie non euclidee (2004)
È una nota che ho scritto per rabbia, in seguito ad un delirante articolo apparso sul quotidiano La Repubblica così pieno di idiozie in fatto di matematica da far venire il sangue alla testa... Ringrazio Domenico Fiorenza per avermi segnalato un errore nel testo, dovuto al suddetto afflusso ematico alla cervice.

txt Una lettera sulla riforma dell'Università (1995)
Si tratta di un intervento che ho scritto anni fa, quando si vagheggiava una ipotesi di riforma universitaria (che purtroppo ha poi dato luogo all'attuale sfacelo) su sollecitazione del prof. Claudio Procesi che stava raccogliendo pareri in materia.

txt Geometria e Topologia (1995)
Una breve introduzione alla geometria e alla topologia scritta per una pagina WEB pubblicata qualche tempo fa dall'Università di Firenze per un progetto didattico sulle scuole superiori.

txt Matematica in versi (1993)
Una breve nota sulle sfide matematiche in voga nel Rinascimento, ed in particolare sulla storia dell'equazione di terzo grado, la cui prima soluzione fu data in versi da Nicola Tartaglia.


Research notes

I collect here some notes, preprints and other stuff of more specialistic taste (this does not mean "less clear") and some papers on the research themes I worked on.
txt La matemática de los Romanos: una vindicacíon (2012)
Slides of a talk I gave on April 26, 2012 at the department of Didáctica Específicas of Autonomous University of Madrid. It is a discussion of the mathematics of ancient Romans, which tries to debukn some myths about Roman science using recent studies about mathematics in the roman world and the concept of science by the Romans (compare books by Cuomo and Lehoux).

txt Historia y didáctica: ¿Quién sirve a quién? (2012)
Slides of a seminar I gave for the Máster Universitario en Didácticas Específicas en el Aula, Museos y Espacios Naturales, held on April 25, 2012 at the department of Didáctica Específicas of Autonomous University of Madrid. It contains a discussion on the relationships between history and didactic of science, which propose a couple of metaphors along with some examples, and some original ideas about the contributions that didactics may give to the study of the history of science.

txt Alcuni cenni di geometria simplettica (2002)   [versione txt dvi]
Testo che contiene le note di un seminario dato nell'inverno 2002 a Firenze nell'àmbito del seminario di geometria simplettica.

txt Strutture di Poisson e strutture complesse (2001)   [versione txt dvi]
Nota che che riassume le motivazioni e gli esempi di alcune ricerche recenti sul connubio fra strutture di Poisson e strutture simplettiche: questi risultati sono stati presentati al congresso Proprietà geometriche delle varietà reali e complesse, III tenutosi a Mondello dal 1 settembre all'8 settembre 2002.

txt Omologia di Hochschild e di Poisson a confronto (2001)   [versione txt dvi]
Testo di una conferenza da me tenuta a Tor Vergata, maggio 2001, nell'àmbito del workshop Gruppi Quantici e dintorni organizzato presso il Dipartimento di Matematica dell'Università di Roma "Tor Vergata": si tratta di alcune osservazioni sui legami fra queste due omologie sia dal punto di vista algebrico che geometrico.

txt Formalità e varietà simplettiche (2000)   [versione txt dvi]
Testo di un seminario tenuto il 15 giugno 2000 presso il dipartimento di Matematica dell'Università di Firenze U.Dini: si tratta di una disamina di un teorema di Merkulov.

txt The Algebra of Poisson Brackets (2000)   [versione txt dvi]
Text written for the Young Algebra Seminar (YAS), Tor Vergata January 17, 2000: it is an exposition of the algebraic formalism behind Poisson geometry, plus a rapid introduction to the concept of Poisson module I tried to exploit at the time.

txt Osservazioni sul formalismo hamiltoniano (2000)   [versione txt dvi]
Testo di una conferenza da me tenuta nell'aprile 2000 a Perugia nell'àmbito del workshop Gruppi Quantici organizzato da Nicola Ciccoli presso il Dipartimento di Matematica dell'Università di Perugia: si tratta di alcune mie riflessioni scaturite dalla lettura del classico Mathematical Foundations of Quantum Mechanics di von Neumann ed ampliate a fornire una breve introduzione all'algebra del formalismo hamiltoniano (secondo la mia interpretazione).

txt Un seminario sul teorema di Borel-Weil (1998)   [versione txt dvi]
Note del seminario tenuto nell'estate del 1998 a Roma I (seminari degli studenti). Richiede un po' di conoscenze, almeno un bienno di matematica universitaria, per essere compreso.

txt Strutture di Poisson (1995)   [versione txt dvi]
Note di alcuni seminari che ho dato a Firenze nel 1995, e che introducono alle strutture di Poisson: algebre e varietà di Poisson, con qualche riguardo per i gruppi di Poisson-Lie.


Books

Here it follows a list of some mathematical books (on fundamental arguments) which I consider to be beautiful books in themselves, and not only as handbooks or merely technical texts: I think that, whatever kind of mathematician one can be, a professional mathematician could, or better should, read them (and hopefully, she/he would do it).
Differential geometry books are quoted in this section of the site.
R. Abraham, J. Marsden, Foundations of Mechanics.
An complete introduction to Mechanics (both Lagrangian and Hamiltonian), which not only is written in a clear and fresh way, but also contains practically all fundamental results in the area, stated and proved with rigor, and with enlightening examples and exercises. It is also worth in its preliminaries, mainly about geometry, which make it suitable for self-study and a reference work not yet surpassed.

V. Arnold, Mathematical methods of classical mechanics.
Lighter than the previous one, it is written in a wonderfully brilliant style, it comes in short at the hearth of the subject and contains a lot of examples and exercises very stimulating. More technical or advanced arguments are treated in appendices at the end of the book.

E. Artin, Galois Theory.
The classic introduction to Galois Theory, short but which contemplates the fundamental parts of the theory, as based upon linear algebra.

R. Bott, L.W. Tu, Differential Forms in Algebraic Topology.
A complete text in algebraic topology, which deals with cohomology, homology, homotopy and spectral sequences: indeed it starts with differentiable manifolds, thus by de Rham cohomology, and touches many other areas in algebraic topology, in a concrete but elegant and rigorous way. A chief-work.

N. Bourbaki, Algèbre.
Still the more complete and rigorous text about linear algebra and its ramifications. Rich in exercises, to me the best of the series.

C. Chevalley, Theory of Lie Groups.
The first modern text about Lie groups: agile and elegant it is still a graceful reading, with several insights.

A. Church, Mathematical Logic.
The best text ever written about mathematical logic, but unfortunately incomplete: indeed the second volume was never written, so that Gödel theorems, computability and set theories does not figure in, but the deepness and erudition of the volume make it unique even in this half form. The introduction alone is a chief work in scholarship and rigor.

P. Cohen, Set Theory and Continuum Hypothesis.
Classical text on set theory, oriented toward the exposition of Cohen results on independence of axioms of set theory: actually it contains a rapid but very clear introduction to the subject.

H.-D. Ebbinghaus et al., Zahlen.
This book is an example of how one can write both about mathematics and its history at the same time, in such a way that both historians and mathematicians can use it as a reference: it deals with the concept of number from its origins, both (pre)historical and logical, until the most recent conceptions about the subject. It is written by a pool of high level mathematicians, which made the exposition heterogeneous, pleasant and erudite. (There's an English translation, but the original German version is much cheaper!),

F.R. Gantmacher, The Theory of Matrices.
In spite of its title, it is a linear algebra book at the highest level, which deals completely with the structure theory of linear operators in vector spaces (and much more). It is full of examples, applications and insights (say the mechanical interpretation of Gauss algorithm!).

K. Gödel, On Undecidable Propositions of Formal Mathematical Systems.
These are the notes from a series of lectures given by Gödel about his most famous theorem: it is a brilliant exposition, not difficult to read, because the author works in the second order logic, which makes his genial argumentations much more simpler, especially for mathematicians.

F. Hirzebruch, Topological Methods in Algebraic Geometry.
A text in the middle between complex geometry and algebraic topology, which essentially it is a diversion about Riemann-Roch theorem: with this in mind, the author has written one of the most brilliant and clear introductions to the geometry of complex manifolds I know.

K. Knopp, Theory and Applications of Infinite Series.
This is an old book, but still delightful: it provides a complete exposition of series theory, starting from scratch, thus from the definition of real numbers (a discussion with many interesting historical notes) and goes into the deepest details of numerical, power and function series theory, ending with results on asymptotic and divergent series.

A.N. Kolmogorov, S. Fomin, Elements of real and functional analysis.
A classic text of higher analysis which contains all basic notions needed to study functional analysis, and which includes also preliminaries about sets, topology, integration and Fourier analysis with examples and explanations very clear and simple. It would be silly not to start studying functional analysis from here.

A.A. Kostrikin, Introduction à l'algèbre.
This book (translated from the Russian and which is also available in the English translation) is the most suitable introduction to modern algebra I know: on the one hand it deals with many questions no longer found in contemporary books (for example symmetric functions), on the other hand it provides, in its second half, a complete introduction to abstract algebra and to its applications in maths. The style is rigorous, yet elementary, and exercises and examples are very involving: one could define it as "a concrete introduction to abstract algebra".

S. Lang, Linear Algebra.
The best introductory treatise on linear algebra, with examples from analysis and geometry, which deals with all the main arguments in an elementary and clear way. Suggested to everyone.

J. Milnor, Morse Theory.
Notes from Milnor lectures on calculus of variations and the topology of manifolds: it contains the shortest yet readable introduction to Riemannian geometry I know (exotic notation aside) and it is such a pleasure to read it that it makes one want to work in Morse theory.

J. Milnor, Topology from the differentiable viewpoint.
It is a precious booklet which, in a handful of pages, collects all fundamental result about topology of manifolds in a clear, short and brilliant way.

D. Mumford, Abelian Varieties.
A non-introductory book to abelian varieties which is complete and full of brilliant insights: on reading it, it is evident how the author has clear ideas about the subject and how he is able to communicate them to the reader. The theory is exposed both in its classic and in its modern disguise: as pleasure for minds.

J. von Neumann, Mathematical Foundations of Quantum Mechanics.
This text should fill the gap of mathematical rigor in quantum mechanics of the 30s of XX century (the author complains about Dirac's theory since Dirac could not justify some key step in his reasonings: these justifications will be given later by L. Schwartz with his theory of distributions). To do that, von Neumann invents a new approach to quantum mechanics, by using spectral theory of unbounded operators in Hilbert spaces, which he himself invent to this aim. In spite of the fact that the language is somewhat changed and the theory now is placed inside operator algebras theory (also due to von Neumann) whis book is still a pleasant and valuable source, especially in showing how von Neumann had clear the mysterious relationships betwenn abstract mathematics and physical phaenomena which it can describe.

L.S. Pontriagin, Topological Groups.
An absolute chief-work: it introduces the reader into the realm of topological groups, Lie groups and their algebras, starting practically from scratch. The exposition is very clear, examples are carefully selected and argumentations are absolutely elegant. Some parts are less renowned, as the chapter on topological rings, but not less intriguing.

C. Shannon, A mathematical theory of communication.
Classical text of applied mathematics: herein Shannon introduces information theory, and binds it to the theory of code transmission (hot spots at that time); the mathematical treatment is very elegant, and some results are really deep.

H. Weyl, Space, Time, Matter.
A text about general relativity theory, written short after the theory was first published: it contains an examination (with an historical and philosophical taste) of the problem of space-time in relativistic mechanics, and its relationships with linear algebra and riemannian geometry. Today it can be difficult to read because of notation (modern tensor notation was not yet developed) but nevertheless it keeps its brilliance and elegance.

H. Weyl, The Theory of Groups and Quantum Mechanics.
For a half it is an algebra book, and for another half it is a physic book: this old text shows how a great mind can have a unitarian view of different disciplines and theories. Here you'll find also representation theory of groups with many a concrete examples.

H. Weyl, The Classical Groups.
Another great book which may be difficult to read because of the tensor notation(or better because of its absence): but it is, in my opinion, the most elegant work by Weyl, and the subject is developed at such a deep level that it can still constitute an introduction to the algebraic part of invariant theory. Great scholarship.

Links:


Departments and other mathematical institutions

AMS: American Mathematical Society
European Differential Geometry Endeavour
Ministero dell'Università e della Ricerca Scientifica e Tecnologica
Istituto Nazionale di Alta Matematica "F.Severi", Roma
Dipartimento di Matematica "G.Castelnuovo", Roma I.
Dipartimento di Matematica, Roma II.
Dipartimento di Matematica, Roma III
Dipartimento di Matematica "U.Dini", Firenze
Dipartimento di Matematica Applicata "G.Sansone", Firenze
Dipartimento di Matematica, Parma
Scuola Normale Superiore, Pisa
Ecole Normal Supérieur, Paris
Max Planck Institute, Bonn
Erwin Schrödinger Institut, Wien
Institute for the Advanced Studies, Princeton
MSRI, Berkeley
IHES, Bures sur Yvette
MIT, Boston
CIRM (Luminy)
EMS
TMR - Europe
AMS Calendar
Oberwolfach
ACM-France
IMPR and its origin

Electronic Repositories

Math Archives contains good material: texts, programs and much more.
Preprint lanl: most mathematicians post their papers here.
Hilberts Mathematische Probleme contain resources about the famous Hilbert list of mathematical problems, dating back to 1900.
MathWeb Journals
Zentralblatt MATH
Journal of Lie Theory
Journal of Algebraic Geometry
MathSearch
UPenn

History of Mathematics

There's a definitive site on the subject, which is the one whom many others borrow informations, and it is the first in the following list of links:
The MacTutor History of Mathematics archive A huge site with as many informations as you like on the subject, with more than 1000 biographies with major mathematicians listed alphabetically and chronologically, an index of topics, and a birthplace map.
Euclid's Elements complete English edition.
Gauss Society whose aim is to preserve the memory of the scholar and the person Carl Friedrich Gauß, in co-operation with the Georg-August University of Göttingen, the Academy of Sciences of Göttingen and the city of Göttingen. They organize lectures, publish article, etc.
The Evariste Galois Archive A resource of biographical material in various languages about this great genius, written by Bernard Bychan.
Die Mathematischen Probleme von David Hilbert.
Muslim Scientists and Islamic Civilization: a site devoted to Muslim golden age of science (in particular to great Muslim thinkers); it's a well-done and interesting site, with a lot of informations and useful links on Islamic civilization.
Los Matemáticos y su Historia Un sitio en español que se halla en la Universidad de Santiago de Chile.
W. W. Rouse Ball's A Short Account of the History of Mathematics: seventeenth and early eighteenth century mathematicians, under David Wilkins's History of Mathematics pages.
History of Mathematics A site with informations not bounded to Western thought, but spanning to arabic, indian and Japanese mathematicians.
Catalogue of the Scientific Community, a collection of 631 detailed biographies on members of the scientific community during the 16th and 17th centuries, compiled by Richard S. Westfall in the department of History and Philosophy of Science at Indiana University. The Catalog is part of the Galileo Project at Rice University, a hypermedia exhibit with pages on Galileo's Villa, Maps of Galileo's World, Timelines of Galileo's Life & Era, and Galileo Project Resources: it deserves a visit!.
Mesopotamian Mathematics
Mathematics: Ancient Science and Its Modern Fates at the Library of Congress Vatican Exhibit

Texts on-line

[see also on-line texts on computer science, on-line texts on literature and on-line texts on philosophy].

Links collections to texts scattered on the WEB

Gallica, bibliothèque numérique de la Bibliothèque nationale de France is a delicious collection of tens of thousands of books, mainly scanned in pdf (not in text format) from the national French library: there are several hundreds of mathematical books (collected works, journals, monographies, etc.) in many idioms (French, English, German, Italian, ...) up to the beginning of XX century.
Goettingen State and University Library Server has more than 4000 documents and 1800000 images on line! It has a huge mathematical section especially journals (as Inventiones Mathematicae, Numeriche Mathematik, Geometric and Functional Analysis, Mathematische Zeitschrift and more).
Numérisation de documents anciens mathématiques collects French mathematical journals: Annales de l'institut Fourier (1949-1998), Annales de l'institut Henri Poincaré (1930-1964), Annales mathématiques Blaise Pascal (1994-2000), Annales scientifiques de l'École normale supérieure (1864-1998), Annales de l'université de Grenoble (1945-1948), Bulletin de la SMF (1872-1993), Journées Équations aux dérivées partielles (1974-2003), Mémoires de la SMF (1964-1993), Publications mathématiques de l'IHÉS (1959-1998) [possibly updated to more recent dates].
Kolekcja matematyczno-fizyczna at Biblioteka Wirtualna Nauki in Poland leaves on line the following collections (pdf files): Acta Arithmetica (1935-1965), Annales Polonici Mathematici (1955-1961), Banach Center Publications (1976-1983), Colloquium Mathematicum (1947-1961), Fundamenta Mathematicae (1920-1993), International Journal of Applied Mathematics and Computer Science (2001-2004), Prace matematyczno-fizyczne (1888-1952), Pisma M. Smoluchowskiego (1924-1928), Rozprawy Matematyczne (1952-1955), Studia Mathematica (1929-1982). Moreover, one also finds here several volumes of the Monografie matematyczne tomy 1-33, 37, 39, 40, 42, 43, 47, with classic books by Banach, Kuratowski, Saks, and others (in French, English and Polish).
A comprehensive list of free e-books on mathematics, with many classic ones: a must!
A comprehensive list of math e-books at UPENN.
Astrophysics Data System (ADS) Virtual Library is a collection of some classic texts in celestial mechanics.
Proceedings of the National Academy of Sciences of the United States of America on line!!! (the complete collection!!!).
AMS bookstore offers some classics of modern mathematics, just to quote a few: Dynamical Systems by Birkhoff, Geometric Asymptoticsby Guillemin and Sternbergs, and more!
Bullettin of the American Mathematical Society on line.
On line books on Celestial Mechanics all classic: also contains links toward archives with more Celestial Mechanics texts (and also on Mathematics and Physics).
MSRI a list of e-books on mathematics (higher level) even very recent.
MathWeb: Books still from AMS: a list of titles with links and references.
Online Mathematics Textbooks is a very useful list of free on-line books about mathematics, maintained by George Cain.
Decio Cocolicchio maintains a huge list of on-line books, spanning from algebra, to numeric calculus, to quantum physics.
Un meta-link di appunti universitari.
Hereinafter I leave some links toward single on-line texts (books, notes, exercises, etc.).

Algebra and Linear Algebra

Robert Ash home page contains some fine books on line about: algebra, number theory, commutative algebra, complex variables.
Andries E. Brouwer home page contains some lecture notes about algebra, arithmetics, logic and topology.
John A. Beachy pages contain many of the definitions and theorems from the area of mathematics generally called abstract algebra. They are intended for undergraduate students taking an abstract algebra class at the junior/senior level, as well as for students taking their first graduate algebra course. It is based on the books Abstract Algebra, by John A. Beachy and William D. Blair, and Abstract Algebra II, by John A. Beachy.
Algebra Lineare.
esercizi ed appunti di geometria, presso l'università di Roma II Tor Vergata.
A primer of Linear Algebra di Stéphane Marchand-Maillet.
A course on Linear Algebra di Harold V. McIntosh.
Linear Algebra textbook home page.

Calculus & Analysis

Alain Connes WEB pages: the web site of one of the most outstanding contemporary mathematicians, with many downloadable papers and books!!!
Terence Tao's WEB site with many interesting notes on variuos arguments about combinatorics, Fourier, PDE, number theory, etc. Great!!!
Timothy Gowers WEB pages, with many challenging exercises, lectures and math material.
Elementary Calculus: An Approach Using Infinitesimals On-line Edition, by H. Jerome Keisler. This is a calculus textbook at the college Freshman level based on Abraham Robinson's infinitesimals, which date from 1960. Robinson's modern infinitesimal approach puts the intuitive ideas of the founders of the calculus on a mathematically sound footing, and is easier for beginners to understand than the more common approach via limits.
Calculus Lecture Notes.
More Calculus notes
Esercizi di Analisi (di livello medio-alto).
Appunti di analisi matematica per l'università (.html)
esercizi ed appunti di analisi, presso l'università di Roma II Tor Vergata.

Geometry

Euclid's Elements with notes, animations and more on David E. Joyce page, which also contains many interesting notes about the history of mathematics.
The Manifolds Page.
Books by Jean-Pierre Demailly and most notably his comprehensive treatise on line about algebraic and analytic geometry: a must!
An Introduction to Riemannian Geometry lecture notes at Lund University, good for an introduction to the subject.
Elementary Differential Geometry.
Notes on Differential Geometry by B. Csikós at Budapest University.
Differential Geometry and General Relativity - some notes about Riemann Geometry and its application to Relativity.
Natural operations in differential geometry a book by I.Kolár, P.W.Michor and J.Slovák, whose first three chapters contain an introduction to modern differential geometry (not for beginners).
Differential Geometry - Dynamical Systems.
The Mathematical Papers of Georg Friedrich Bernhard Riemann, classical seminal papers which gave to Differential Geometry its shape.
On Alan Weinstein pages you'll find his book on line about the geometry of non commutative algebras, and many survey articles on differential geometry, on various subjects (symplectic, riemannian, Poisson, etc.).

Lie algebras, Lie groups and their representations

Lie Groups and Representations by Peter Woit at Columbia.
Peter Cameron's classical groups page
Sulla pagina di Claudio Procesi ci sono gli appunti dei corsi su gruppi di Lie, algebra commutativa e topologia algebrica, ma soprattutto i suoi Appunti di Teoria degli Invarianti e di teoria delle rappresentazioni.
Introduzione a gruppi ed algebre in meccanica quantistica di M.Toller

Logic and Set Theory

Fundamenta Mathematicae on line (1920-1993) in the Kolekcja matematyczno-fizyczna at Biblioteka Wirtualna Nauki in Poland.
Bullettin of Symbolic Logic on line: each paper is in postcript format.
Paul Bernays: Philosopher of Mathematics is project aimed to prepare and publish a volume of English translations of his papers on the philosophy of mathematics. The originals were written in either German or French, with Bernays himself supplying an English translation in one case. Some of these papers have been collected and published in German under the title Abhandlungen zur Philosophie der Mathematik.
Practical Foundations of Mathematics by Paul Taylor.

Mechanics

Isaac Newton on line project: a beautiful collection of Newton's works in pdf format, among them the Philosophiae Naturalis Principia Mathematica and his Opticks.
Books on line about statistical mechanics and mathematical physics by various scientists from Rome I University (Gallavotti, Gentile, Antoci, Bonetto, Benfatto).
Quantization as an Eigenvalue Problem by Harold V. McIntosh.
Appunti di Fisica Matematica.
Introduzione alla teoria dei campi quantistici di M.Toller
Appunti di chimica teorica di F. Tarantelli con molto materiale di fisica matematica.
Stringy Black Holes by Martijn Derix and Jan Pieter van der Schaar.

Number Theory

A lot of online number theory lecture notes at www.numbertheory.org. Compare also their list of Online Theses in Number Theory.
Bruce Ikenaga's notes on elementary number theory.
Micheal Filaseta's page contains a lot of resources and texts on line about number theory.
Franz Lemmermeyer offers a huge collection of notes, lectures and more on algebraic geometry, elliptic curves and number theory.
http://www.maths.tcd.ie/pub/Maths/Courseware/ contains a lot of lecture notes given at Dublin's Trinity College School of Mathematics, about number theory, elliptic curves and representation theory of finite groups.

Other Interesting Topics

The Electronic Journal of Combinatorics, a reviewed journal of combinatorics that is free on the www.
Graph Theory by Reinhard Diestel, available online.
Mikhail Gromov WEB pages contains few notes to read, but worth of a visit: he is one of the great mathematicians of our times.
David Mumford's Modeling the World with Mathematics: clocks, waves, chaos and chance: really exciting!
Doron Zeilberger Home Page, with many papers, reflections and texts on maths.
Herbert Wilf's Home Page contains several books on line about computational mathematics, namely:
  1. generatingfunctionology (publisher: Academic Press/Harcourt Brace)
  2. "A=B", by Marko Petkovsek, Doron Zeilberger and myself. (publisher: A K Peters, Ltd.)
  3. Algorithms and Complexity. Copyright 1985.
  4. East Side, West Side (Lecture notes on combinatorial objects and Maple programs for generating them. Copyright 1999.)
  5. Lectures on Integer Partitions (From PIMS lectures given in summer 2000 at U. of Victoria. Copyright 2000.)
  6. Lecture Notes on Numerical Analysis (By Dennis Deturck and myself; For a junior-senior level course; covers numerical solution of ODE's and numerical linear algebra. Copyright 2002.)
They are all free to download!!! And here you'll find also other interesting notes in mathematics.
Some Lecture notes for the mathematical courses of the London School of Economics and Political Sciences.
Numerical Recipes Software's Books On-Line hosted by the Universal Library, contains free books on Numerical Calculus with examples written in C, Fortran 77, Fortran 90.
Introduction to Statistics of Financial Markets by Jürgen Franke, Wolfgang Härdle, Christian Hafner.

Subjects:


Differential Geometry

The term Differential Geometry was used for the first time by Luigi Bianchi in his classical books Lezioni di geometria differenziale (1894), which, along with the Leçons sur la théorie générale des surfaces (1887-96) by Gaston Darboux, has been the standard treatise in the subject for many years.
Actually Differential Geometry was born as the natural application of the analytical techniques developed in XVII and XVIII cent. to analytic geometry, founded by René Descartes. The idea of Descartes is to identify the plane and the space resp. with the sets of pairs (x,y) and triples (x,y,z) of numbers: then curves and surfaces are described by sets of points corresponding to solutions of equations:
f(x,y)=0               g(x,y,z)=0
When, with the creation of calculus, the notion of differentiable function was introduced, curves whose defining functions are differentiable where investigated, for example by Christian Huygens, Leonard Euler and Gaspard Monge. Euler introduced a concept of curvature for a surface described by an equation g(x,y,z)=0 in his Recherches sur la courbure des surfaces (1760), where he gave a method to compute the curvature of a curve displayed on a surface, while Monge laid down the main results on plane and space curves in his Applications de l'Analyse à la Géometrie (1807). Many pupils of Monge continued his researches in the filed, just to quote a few ones: Charles Dupin, Jean Baptiste-Marie Charles Meusnier, Jean-Victoire Poncelet and Olindo Rodrigues.
But the cornerstone of classical Differential Geometry is the paper of C. F. Gauss Disquisitiones generales circa superficies curvas (1823-26) where all the main definitions and theorems regarding pre-Riemannian Differential Geometry may be found: Gauss attaches to a surface two bilinear forms which encode all its metrical properties and begins the study of the metrical properties of the surface in themselves, rather than merely study the surface as a subset of the euclidean space.
This idea was expanded and generalized by Bernhard Riemann who, in 1854, discussed a lecture to get the title of Privattdozent which was published as Über die Hypothesen, welche der Geometrie zu Grunde liegen, where Riemann states the concept of a metric on a manifold.
Since then, Differential Geometry is the study of structures on smooth manifolds, namely sections of bundles over manifolds which are invariant w.r.t. some transformations group. The classical example is that of Riemannian metrics, thus bilinear, symmetric and non degenerate 2-forms on the tangent bundle of a manifold.
A manifold may be thought as a piecewise object whose pieces look like small regions in the euclidean plane; this means that around a point in a manifold geometry looks like usual euclidean geometry, while in the large (given that pieces are patched together in a smooth way) its geometry may be pretty complicated. For example, the spherical surface of earth is described in atlases by charts, which are plane regions: charts overlap in a smooth way.
The concept of manifold was introduced in its present generality at the beginning of XX century, as far as I know in Hermann Weyl's book Die Idee der Riemannschen Fläche (1913), but the concept of smooth manifold was surely clear to Henri Poincaré (end of XIX century).
Manifolds are widely used in Mechanics: for example both General Relativity and Hamiltonian Mechanics can't be properly formulated without them, and indeed many developments of Differential Geometry were stimulated by Mechanics and Physics (even today).
Among the on-line texts on mathematics I've left some links toward informations and lecture notes (mainly introductory) in Differential Geometry and its applications.
In my pages you'll find some classic text in Differential Geometry by great masters of the past.
There are thousands of books in Differential Geometry and its branches: here I quote my favorite ones.
M. Do Carmo, Differential Geometry of Curves and Surfaces Prentice Hall, 1976.
An excellent introduction to the classical theory, in a modern language; very well written.

W.M. Boothby: An introduction to differentiable manifolds and Riemannian geometry, Academic Press, 1986.
This is an introduction to differential and riemannian geometry in a clear and elementary style, though stating all notions with precision and at a satisfactory level of generality.

M. Spivak, A comprehensive introduction to differential geometry.
The huge books of this series are written in a style so pleasant and penetrating which can be read far more easily than many small booklets on the subject. Spivak introduces in details all main themes in modern differential geometry, with emphasis, in the last three volumes, on global analysis on manifolds. He does not give a complete picture, according to contemporary parameters, since both complex and symplectic geometry (nowadays very studied) are only touched briefly, but nevertheless it remains, along with do Carmo books, the best gate through the realm of differential geometry. Exercises are the chosen and exhibited excellently.

M.M. Postnikov: Riemannian Geometry, Springer, 2001
This is a complete and up to date exposition of modern Riemannian Geometry and Differential Geometry in general (structures on vector bundles); a pleasant reading.

A.T. Fomenko: Differential Geometry and Topology, Consultant Bureau, 1987
A second course on Differential Geometry covering homology, Morse theory, 3-manifolds, symmetric spaces and symplectic geometry.

Kobayashi-Nomizu: Foundations of Differential Geometry (2 vols), Wiley, 1963-69
This is the standard reference, concise and fairly complete, even though it was written some 40 years ago!

Some classic texts...

Classics of mathematics

To read classical works, which is usually done and required in the fields of philosophy and humanities, is instead disregarded in scientific areas: the reasons are essentially two: 1) many texts become "obsolescent" sine theories they explain are incomplete or become surpassed; 2) notations used are different from ours, and often impossible to decipher, like hieroglyphics. Just to make an example (maybe a bit extreme), Frege's Begriffsschrift, a milestone in mathematical logic, is difficult to read not because it is written in German, but because of the weird notation with which it expresses quantifiers.
Anyway, I believe that reading classics is well-spent time: if it is true that mathematics should not be reread but only rewritten, in the sense that the same notions can be expressed in different ways, simpler and easier to understand with a suitable notation (and modern one is almost always better than ancient one) it is also true that ideas of great mathematicians are still there, among those pages and those symbols and, it is well known, great mathematical intuitions are quite rare. Moreover, mathematics has an undoubted luck with respect to other sciences: its notions are, when correct, definitive. there are no wrong theories in mathematics. A theory may be old-fashioned, it can have parallel theories or also concurrent theories, but, if some consequences follow from some axioms, this is true ab eterno: the most fulgid example is the one of Euclidean geometry. Euclid's' theorems are still true, and they'll be forever, indeed it makes no sense to tie them to temporality: hence Euclid's' Elements can still be read not only as a historical curiosity, but as an actual handbook: instead, if you try to read Aristotle's Physics the result is not the same...
Of course it is true that the Dialogo sui massimi sistemi by Galileo is still a valid source for classic kinematics, but in any case the Weltanschauung has changed since then, and the nature we observe nowadays it is not the one observed by Galileo: the Euclidean plane, instead, is still exactly the same!
Having said that, I would like to add that this collection of fragments is but the infinitesimal part of any reasonable collection of sources of the history of mathematics: rather, it is a list of a few texts I transcribed and sometimes translated just because they intrigued me by their beauty and deepness, and which are scarcely to be found (this is very irritating: it's easy to retrieve in a bookshop or on the Net the Aristotle's Physics, but not Archimedes' Arenarium, which is much more modern, astounding and genial).
txt Carl Friedrich Gauss, Disquisitiones generales circa superficies curvas (1828)
This work is the corner stone of the vast building of Differential Geometry: Gauss was the first to use in a systematic way differential and integral calculus, and also linear algebra which he was also founding, to study the geometry of surfaces in space adopting an intrinsic approach. In particular in this work the fundamental forms of a surface are introduced, and the so-called theorema egregium is proved, which states the isometric invariance of these forms. The transcription follows the classic Werke edition.

txt Bernhard Riemann, Ueber die Hypothesen, welche der Geometrie zu Grunde liegen (1854)
This brief note was written by Riemann to get the position of Privattdozent at the G\"ottingen University, Gauss was in the audience: in these few pages Riemann has changed geometry, by introducing concepts which would manifest themselves in their entirety only in the XX century, and which would pervade physical applications of mathematics in a definitive way (from relativity to quantum mechanics). Therein is it introduced the concept of a metric on a manifold of any dimension. The transcription follows the third edition, as it appears in the collected works.

txt Eugenio Beltrami, Saggio di interpetrazione della geometria non euclidea (1868)
This is the work in which Beltrami shows that Lobachevsky's non-Euclidean Geometries may be realized inside Euclidean spaces: namely, by using the tools from differential geometry of his times, Beltrami builds a surface in the space whose geometry follows Lobachevsky axioms. In this way the non-Euclidean axiomatic system is incarnated (although partially) in a model, and this discovery can be restated as a first step in proving that non-Euclidean geometries is not contradictory, as far as Euclidean geometry is. It will be David Hilbert, after some thirty years, to establish that it is not possible to build a complete surface of the Beltrami type in the Euclidean space. The transcription has been conducted on the first volume of complete works.

txt Giuseppe Peano, Intégration par séries des équation différentielles linéaires (1888)
In this short note Peano proves the formula for the series integration of a linear system of ordinary differential equations: the importance of this work lies in that to formulate and to prove the theorem, Peano uses in a conscious and efficient way concepts from Linear Algebra, namely the concept of a vector space and of a linear transformation, which he discloses axiomatically, borrowing from Grassmann works, with incredibly modern spirit and approach. The first part is a crash course on Linear Algebra which even today could substitute several lecture notes by our teachers (but for some terminology: Peano adopts the term complex instead of vector, and substitution instead of transformation, following the habits of his epoch). So, even if it is always mentioned for natural numbers axioms, Peano is also the founder of the axiomatics of Linear Algebra as far as we still understand it.