What textbooks had the biggest impact on you as a mathematician?

My answer to a question in Quora: What textbooks had the biggest impact on you as a mathematician?

Of course, books which I read as a schoolboy, in the last 3 years of secondary school (Years 8–10 in the education system of my home country, that is, I was 14 to 16 years old). The list is bizarre and reflects a rather steep learning curve which was quite common in my generation.

Year 8

  • Vasilyev and Gutenmacher, Lines and Curves.
  • Gelfand, Glagoleva, Schnol, Functions and Graphs.
  • Kirillov, Limits.

(These three small books were among little cute booklets which were mailed to me from Mathematics Correspondence School; true masterpieces of mathematical didactic.)

  • Courant and Robbins, What is Mathematics? (Russian translation).
  • R. Hartshorn, Foundations of Projective Geometry (Russian translation). A little gem of abstraction.

Year 9

  • O. Yu. Schmidt, Abstract Group Theory (1933, republished with minor changes from its first edition of 1916).
  • R.Baire, Lecons sur les Fonctions Discontinues (1932, Russian translation of the French original of 1904). This was perhaps the first book of descriptive set theory. After reading it, I had never had trouble with functions of real variables.

Only later I realised that Schmidt and Baire helped me to put mathematics into a historic perspective and see it as something dynamic, developing.

Years 9 and 10

  • Birkhoff and MacLane, A Survey of Modern Algebra (in English)
  • Gorenstein, Finite Groups (in English) (a hardcore research monograph, it was the Bible of finite group theory for a couple of decades)

From Gorenstein on, my reading switched to research mode, highly selective, with tight focus on key ideas and arguments.

I was privileged to meet, in later years, Gelfand, Gorenstein, and Hartshorn in person and thank them for their books. Gelfand looked at me with suspicion and asked: ”And what have you learnt from “Functions and Graphs?” My answer: “The general principle: always look at the simplest possible example” delighted him. He immediately started to explain that this was his most important contribution to pedagogy of mathematics, and that in his famous Seminar he was always able to find a example simpler than the one given by the speaker.

The question about textbooks was asked to me by Michael Kheifetz; he also asked me:

What’s the most amazing conversation you have had about mathematics?

I can answer it now: quite a number of conversations with Israel Moiseevich Gelfand that followed this first encounter.

An interested reader can find comparative analysis of Schmidt and Gorenstein in my paper Mathematics discovered, invented, and inherited (Section 3).