Literate Scientist

This is a book I bought a few years ago and started reading immediately but put aside and only this summer read it fully from cover to cover. In order to appreciate its content you need some degree of mathematical and computer science maturity. For example, if you have never heard of his theorems and only read Incompleteness: The Proof and Paradox of Kurt Godel or similar popular book then you would have difficulty going through the book and it would appear boring. It is not an entertaining or bedside reading. This is why I put it aside on the first reading although I knew about this theorem since I read “Mathematics: The Loss of Certainty” more than 25 years ago being a schoolboy (in Russian translation). Just before writing this review I ordered “There’s Something About Godel: The Complete Guide to the Incompleteness Theorem” and the latter looks like less heavy reading judged from excerpts from its publisher website. Putting all these reminiscences aside I really enjoyed second reading of “Godel’s Theorem”. It really clarified some points from ¬B->¬A or PA & ¬Con(PA) perspectives and made me curious about fixpoints. I even borrowed the latter term and introduced them for crash dump analysis and debugging: “a dereference fixpoint”. I also liked chapters 4 and 6 about using Godel’s theorems outside mathematics and clarifying misconceptions in Rucker’s and Penrose’s books. However, after a few months I cannot recall anything definite what I read from that book although I felt good that I understood everything while reading so perhaps the book requires the 3rd reading for me I’m going to give it another try after “There’s Something About Godel” and update this review.

Godel’s Theorem: An Incomplete Guide to Its Use and Abuse

- Dmitry Vostokov @ LiterateScientist.com -

I read this book from cover to cover while flying on a plane from Dublin to St. Petersburg and back. That was so wonderful reading experience - I couldn’t put the book down during those flights. I recall that I visited the Department of Mathematics a few times when I studied Chemistry in Moscow State University although at that time I knew next to nothing about Russian mathematicians. The book touched me so deeply that I bought the main work of Florensky: The Pillar and Ground of the Truth, the history of Russian philosophy and several books explaining Orthodox Church. This is the best mathematics history book I have ever read, my feelings perhaps comparable to those that I experienced when I finished reading Mathematics: The Loss of Certainty by Morris Kline but that was more than 20 years ago.

Naming Infinity: A True Story of Religious Mysticism and Mathematical Creativity

- Dmitry Vostokov @ LiterateScientist.com -

This is a book that I noticed in a bookshop 6 years ago. I was curious by its title and front cover because at school I was interested in foundations of mathematics and abstract algebra ideas. I bought this book and from it I first heard of and learnt about category theory.

Conceptual Mathematics: A First Introduction to Categories

Very accessible and highly recommended as the first introduction but it requires probably the second reading if you are not used to mathematical abstractions. Fortunately there is the second edition coming after almost 15 years that seems have extra 50 pages added and I’m looking forward to reading it too.

Conceptual Mathematics: A First Introduction to Categories (Second Edition)

- Dmitry Vostokov @ LiterateScientist.com -

This book I bought a few years ago but only started reading 4 months ago and just finished:

Understanding the Infinite

I must say that it was not a light read and it requires certain mathematical maturity beyond undergraduate courses. The first part deals with Cantor and Zermelo set theories and axioms. It is very dry sometimes and chapters are long which was not good for me because I was only reading 10 - 12 pages per week while commuting. In many places the author assumes that a reader already knows a lot about logic and set theory, for example, at the end, he devotes a page or two about Putman modal logic and uses freely its quantifiers without explaining them. Some glossary at the end would have greatly benefited this book. What I found clarifying is the fact that there are two foundations of set theory: the notions of logical and combinatorial collections. For the latter the Axiom of Choice is self-evident and is no longer controversial. The second part starting from chapter VI is more philosophical and concerns with epistemology and ontology of the infinite. At least at the beginning it clarifies the difference between potential and actual infinity. In the middle we see the use of schemas to avoid quantifiers. At the end of the book the author discusses the theory of indefinite large and small, its extrapolations to infinite and provides examples from mathematical analysis. The main theme of the book, as I understand it, is that our intuition about infinity arises from intuitive understanding of indefinitely large sets, their hierarchies and extrapolations.

- Dmitry Vostokov @ LiterateScientist.com -

These two volumes I bought a few years ago, started reading the first chapter and then other books got reading priority, for example, Rosen’s “Life Itself”. A few weeks ago I picked up the first volume again and started reading from the beginning. I’m was really amazed how I understand it better after reading Rosen’s books. These volumes are highly recommended to learn about models of reality and mathematical modeling itself. The first chapter that discusses the relationship of models to observation is awesome. The book requires an undergraduate engineering level of mathematics: linear algebra, calculus and a bit of mathematical analysis. You will also learn about catastrophe-theoretic models, chaos, cellular automata, geometry of human affairs, patterns, fractals, and many other things. There is even a discussion about controversies in catastrophe theory involving Rene Thom. I think the first volume of this book set is a prerequisite reading before starting with classic Structural Stability And Morphogenesis.

Reality Rules, 2 Volume Set

- Dmitry Vostokov @ LiterateScientist.com -

OpenTask plans to publish the extended and edited version of this blog as a book:

Literate Scientists and Their Books: An Independent Guide to Understanding Reality (ISBN: 978-1906717520)

- Dmitry Vostokov @ LiterateScientist.com -

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 -