Literate Scientist

This is one of the first books I bought more than 5 years ago when I started reading math and physics books after more than 10 year break… I wanted to refresh my math knowledge and especially to learn discrete math that I mostly missed during my Chemistry education. I stumbled upon this book in a bookshop and liked its binding, paper quality and layout inside. I found the book very didactic and now looking back with all knowledge I gained afterwards from other books I would definitely say it is a very good textbook to start learning computer science.

Discrete Mathematics

- Dmitry Vostokov @ LiterateScientist.com -

If you ask me now what book I recommend for a broad overview of mathematics I would not hesitate to point to this latest book that I just started reading:

The Princeton Companion to Mathematics (Hardcover)

Although it is 1000 page book with two columns of text it is actually intended to be read from cover to cover! This book is now on top of my math overview recommendations which previously included these books:

• The Road to Reality
• Comprehensive Mathematics for Computer Scientists
• Mathematics: Form and Function
• All the Mathematics You Missed
• The Nature and Growth of Modern Mathematics

- Dmitry Vostokov @ LiterateScientist.com -

This is out-of-print book that is hard to find now. I was very keen and fortunate to buy it from a 3rd-party seller a few years ago and immediately read it then. It gives great overview of modern mathematics from the perspective of the founder of category theory. The similar overview of modern mathematics can be found in The Road to Reality, 2 volumes of Comprehensive Mathematics for Computer Scientists and the recent 1000 page “The Princeton Companion to Mathematics”. One note that I didn’t like is the following passage from page 439:

“Not all outside influences are really fruitful. For example, one engineer came up with the notion of a fuzzy set…”

Mathematics: Form and Function

Here is the table of contents showing the breadth of material:

CHAPTER I
Origins of Formal Structure
The Natural Numbers
Infinite Sets
Permutations
Time and Order
Space and Motion
Symmetry
Transformation Groups
Boolean Algebra
Calculus, Continuity and Topology
Human Activity and Ideas
Mathematical Activities
Axiomatic Structure

Chapter II From Whole Numbers to Rational Numbers
Properties of Natural Numbers
The Peano Postulates
Natural Numbers Described by Recursion
Number Theory
Integers
Rational Numbers
Congruence
Cardinal Numbers
Ordinal Numbers
What Are Numbers?

Chapter III Geometry
Spatial Activities
Proofs without Figures
The Parallel Axiom
Hyperbolic Geometry
Elliptic Geometry
Geometric Magnitude
Geometry by Motion
Orientation
Groups in Geometry
Geometry by Groups
Solid Geometry
Is geometry a Science?

Chapter IV Real Numbers
Measures of Magnitude
Magnitude as a Geometric Measure
Manipulations of Magnitudes
Comparison of Magnitudes
Axioms for the Reals
Vector Geometry
Analytic Geometry
Trigonometry
Complex Numbers
Stereographic Projection and Infinity
Are Imaginary Numbers Real?
Abstract Algebra Revealed
The Quaternions - and Beyond
Summary

Chapter V Functions, Transformations, and Groups
Types of Functions
Maps
Whats is a Function?
Functions as Sets of Pairs
Transformation Groups
Groups
Galois Theory
Construction of Groups
Simple Groups
Summary: Ideas of Image and Composition

Chapter VI Concepts of Calculus
Origins
Integration
Derivatives
The Fundamental Theorem of the Integral Calculus
Kepler’s Laws and Newton’s Laws
Differential Equations
Foundations of Calculus
Approximations and Taylor’s Series
Partial Derivatives
Differential Forms
Calculus Becomes Analysis
Interconnections of the Concepts

Chapter VII Linear Algebra
Sources of Linearity
Transformations versus Matrices
Eigenvalues
Dual Spaces
Inner Product Spaces
Orthogonal Matrices
Adjoints
The Principal Axis Theorem
Bilinearity and Tenso Products
Collapse by Quotients
Exterior Algebra and Differential Forms
Similarity and Sums
Summary

Chapter VIII Forms of Space
Curvature
Gaussian Curvature for Surfaces
Arc Length and Intrinsic geometry
Many-Valued Functions and Riemann Surfaces
Examples of Manifolds
Intrinsic Surfaces and Topological Spaces
Manifolds
Smooth Manifolds
Paths and Quantities
Riemann Metrics
Sheaves
What Is Geometry?

Chapter IX Mechanics
Kepler’s Laws
Momentum, Work, and Energy
Lagrange’s Equations
Velocities and Tangent Bundles
Mechanics in Mathematics
Hamilton’s Principle
Hamilton’s Equations
Tricks versus Ideas
The Principal Function
The Hamilton-Jacobi Equation
The Spinning Top
The Form of Mechanics
Quantum Mechanics

Chapter X Complex Analysis and Topology
Functions of a Complex Variable
Pathological Functions
Complex Derivatives
Complex Integration
Paths in the Plane
The Cauchy Theorem
Uniform Convergence
Power Series
The Cauchy Integral Formula
Singularities
Riemann Surfaces
Germs and Sheaves
Analysis, Geometry, and Topology

Chapter XI Sets, Logic, and Categories
The Hierarchy of Sets
Axiomatic Set Theory
The Propositional Calculus
First Order Language
The Predicate Calculus
Precision and Understanding
Godel Incompleteness Theorems
Independence Results
Categories and Functions
Natural Transformations
Universals
Axioms on Functions
Intuitionistic Logic
Independence by Means of Sheaves
Foundation or Organization?

Chapter XII The Mathematical Network
The Formal
Ideas
The Network
Subjects, Specialties, and Subdivisions
Problems
Understanding Mathematics
Generalization and Abstraction
Novelty
Is Mathematics True?
Platonism
Preferred Directions for Research
Summary

- Dmitry Vostokov @ LiterateScientist.com -

Reading general science books especially those bordering on philosophy requires understanding basic principles of philosophy and its logic as well as possessing critical thinking skills. The following short book which I’m half way through seems to be good:

The Philosophers Toolkit: A Compendium of Philosophical Concepts and Methods

- Dmitry Vostokov @ LiterateScientist.com -

I’ve just finished reading this book and despite some bad reviews on Amazon I wasn’t disappointed. The author’s writing style is a bit unusual with some unknown words that you need a dictionary but after a chapter I became used to it and it wasn’t a problem anymore. If you heard about Gödel theorems before but cannot repeat precisely what they are about then this book is for you and you will find detail-free sketch of the proof very clear. I really liked the author’s attack on positivism and postmodernism especially in the light of previously read Fashionable Nonsense. I also liked the conclusion at the end of the book that the life of Gödel was “incomplete” too. The book discusses Vienna Circle and Wittgenstein, relationship between Einstein and Gödel and even some political issues in the Institute for Advanced Study related to Gödel’s life (this is why I included this book under Politics category too). One remark about bad review from the professor who participated in publishing edited works of Kurt Gödel: I can publish works of Aristotle nowadays without much efforts. Will it give me the right to judge other works and proclaim without sound justification that they don’t know philosophy?

Incompleteness: The Proof and Paradox of Kurt Godel (Great Discoveries)

- Dmitry Vostokov @ LiterateScientist.com -

This one I discovered last year and just finished reading. Interesting collection of articles written in the late 70’s and early 80’s about application of mathematics. My favourite were two articles about statistics and two articles about Husserl. The article about Kant biography was nice as well.

Discrete Thoughts

- Dmitry Vostokov @ LiterateScientist.com -

Very good book to learn about mathematical logic, distinction between syntax and semantics, different interpretations of formal languages and how this leads to various different non-classical logics. It will deepen your understanding of mathematics if you studied or encountered only classical propositional and predicate logic and want to learn more about fuzzy logic, for example, among many others. All necessary prerequisites are covered in the first 230 pages of this 500 page book including informal set theory and topology. I read most of this book couple of years ago and want to re-read it soon.

Classical and Nonclassical Logics: An Introduction to the Mathematics of Propositions

- Dmitry Vostokov @ LiterateScientist.com -

Just finished reading this book written by Steven Sloman. Very readable explanation about a causal modeling framework involving probability distributions and causal graphs, explanation of counterfactuals, logic of intervention, Markov equivalence, explanation discounting, causal structures and language, reasoning and decision making, the difference between observation and action. I became interested in causality from software troubleshooting and debugging perspective where we need to diagnose causes (defects) from symptoms (failures) and this book was a good introduction to me.

Causal Models: How People Think about the World and Its Alternatives

- Dmitry Vostokov @ LiterateScientist.com -