Theoretical Books on Mathematics

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Discussion Overview

The discussion revolves around recommendations for rigorous theoretical books on mathematics across various branches, including algebra, geometry, calculus, linear algebra, and real analysis. Participants are invited to contribute their own suggestions and opinions on the theoretical nature of the listed texts.

Discussion Character

  • Exploratory
  • Debate/contested
  • Technical explanation

Main Points Raised

  • One participant proposes a list of theoretical mathematics books across several branches, emphasizing the importance of focusing on the "Why?" rather than the "How?".
  • Another participant challenges the inclusion of Simmons' Calculus, arguing that it is not rigorous and is "hand wavy".
  • A different participant expresses surprise at the absence of Walter Rudin's real analysis book from the list, implying it should be included for its rigor.
  • Another suggestion is made for Coddington's "Introduction to Ordinary Differential Equations", which is described as theoretical at an elementary level, with all concepts proved from the start.
  • One participant recommends Tom Apostol's "Mathematical Analysis" and Paul Halmos' "Finite Dimensional Vector Space" as additional theoretical texts.

Areas of Agreement / Disagreement

Participants do not appear to reach consensus on the rigor of certain books, with differing opinions on Simmons' Calculus and the inclusion of specific titles like Rudin's work. Multiple competing views remain regarding what constitutes a rigorous theoretical text.

Contextual Notes

Some participants express differing criteria for what qualifies as a theoretical book, leading to varying opinions on the rigor of suggested texts. There is also a lack of consensus on the definitions of "theoretical" versus "practical" approaches in mathematics.

Who May Find This Useful

This discussion may be useful for students, educators, and mathematics enthusiasts seeking rigorous theoretical texts for deeper understanding in various branches of mathematics.

Kalvino
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What are some rigorous theoretical books on mathematics for each branch of it? I have devised a fantastic list of my own and would like to hear your sentiments too.

Elementary Algebra:

Gelfand's Algebra
Gelfand's Functions & Graphs
Burnside's Theory of Equations
Euler's Analysis of the Infinite
Bellman's Introduction to Inequalities
Umbarger's Logarithms

Elementary Geometry:

Kiselev's Geometry
Lang's Geometry
Gelfand's Trigonometry
Gelfand's Method of Coordinates
Gutenmacher's Lines & Curves

Overview: Serge Lang's Basic Mathematics

Calculus:

Spivak's Calculus
Apostol's Calculus
Courant's Introduction to Calculus & Analysis
Simmons' Calculus with Analytic Geometry
Hubbard's Vector Calculus

Linear Algebra:

Lang's Introduction to Linear Algebra
Axler's Linear Algebra Done Right
Friedberg's Linear Algebra
Hoffman-Kunze's Linear Algebra
Roman's Advanced Linear Algebra

Real Analysis:

Binmore's Mathematical Analysis
Pugh's Real Mathematical Analysis
Folland's Real Analysis
McDonald's A Course in Real Analysis

You may make additions to my list or add more branches like Topology, Complex Analysis and Differential Geometry if you like, but remember; the books should focus on the "Why?" rather than the "How?" or in other words; should be highly theoretical. Books like Stewart's Calculus don't classify as being theoretical.
 
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I wouldn't say Simmons is a rigorous book. Yes, it is a good book, however, it is very hand wavy.
 
I am surprised that you do not have Walter Rudin's real analysis book listed.
 
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introduction to ordinary differential equations by coddington , i would consider theoretical at the elementary level. Everything is proved, starts with complex numbers 1st page!
 
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How about Tom Apostol's Mathematical Analysis and Paul Halmos' Finite Dimensional Vector Space"?
 

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