Zettelkastens and Literate Programs

I've been re-reading John Harrison's Handbook of Practical Logic and Automated Reasoning (a wonderful book), with an eye towards writing notes for my Zettelkasten.

What I have done, out of habit, is I've reworked the code in Standard ML (as opposed to OCaml). I've added a "register" (mathematicians would recognize them as "theorem environments" in LaTeX) for source code, and the template for writing such notes on a 5.5 inch by 4.25 inch (A6) slip of paper looks like:

[id number]		Code [name of chunk]
[English summary of the code, or the algorithm, or...]
⟨[name of chunk]⟩≡
[Standard ML code here]

Just replace the bracketed italicized text (including the brackets) with your own text. I do write a horizontal line separating the "code block" from the "text block", and underline the "Code" part of the title.

I follow the general heuristics Knuth suggests for literate programming, even using the names of code chunks in angled braces as if they were statements. Following some of the early handwritten documents on Standard ML, I underline keywords like "if", "then", "else", "case", "fun", etc., in the code block.

While reading Harrison's book, I add unit tests, write up examples, etc. This isn't done "in the abstract", I actually type the code up on a computer (e.g., here is my implementation of most of the propositional logic code).

One advantage to this is that we can prove properties about the code snippet in branches off the code zettel. Another advantage, giving a good human-level summary of the code snippet helps greatly with Harrison's book, since the techniques from chapter 3 make another appearance in chapter 6 (i.e., 300 pages later) and I forget some of the nuances surrounding the implementation ("Is skolem called before or after nnf?"), I can just glance at the snippets for guidance.

Another useful aspect of using a Zettelkasten for studying theorem provers is that it facilitates blending notes from multiple books seamlessly. I noticed this accidentally, when I was unsatisfied with Harrison's description of Herbrand universes, then started reading Chang and Lee's Symbolic Logic and Mechanical Theorem Proving (1973) description of Herbrand universes.

There are other aspects of Zettelkasten which seems uniquely suitable for studying theorem provers, but I digress.

I'm experimenting with using a Zettelkasten for literate programming "by hand" more generally, which is a bizarre but fascinating experience, especially when exploring monads in Standard ML.

Anyways, I just wanted to make note of this quirky note-taking system for literate programming.

(Ramblings on) Writing Notes on Quantum Field Theory

So, in the long run, my aim is to write great notes on quantum field theory and quantum gravity. Since quantum gravity depends on quantum field theory, it makes sense to begin there!

For those uninterested in my rambling thought process, here's the punch line: just as integrals and derivatives are first covered symbolically in calculus, then rigorously in analysis...we likewise believe that a naive and symbolic approach first ought to be covered, then a rigorous and axiomatic approach second.

Greiner and Reinhardt's Field Quantization provides a good level for detail, at least for the naive/symbolic treatment of field theory.

What Other People Do

The basic approach other books take is "Well, here's Feynman diagrams. Quantum Field Theory just plays with these...here's how you get Feynman rules...and here's renormalization, the end."

This is not terrible. But it is lacking a certain je ne sais quoi.

So instead, perhaps I should look at it from the mathematical perspective. This has its own problems.

Depends on...

The problem I have is with dependencies! It doesn't make sense to write about quantum field theory without first writing about classical field theory, quantum mechanics, and a bit about functional analysis.

I have written a note about Relativistic Quantum Mechanics [pdf] (which may make more sense after reading my notes on Lie Groups, Lie Algebras, and their Representations [pdf]).

However, there is still more to do with quantum mechanics. Particularly, the subject of scattering theory is lacking. (Erik Koelink has some great Lecture Notes [tudelft.nl] too)

Despite my notes on Functional Techniques in Path Integral Quantization [pdf], I still feel lacking in the "path integral" department.

Perhaps I should write notes on measure theory, functional analysis, then tackle Glimm and Jaffe's Quantum physics: a functional integral point of view?

With classical field theory, the subject quickly becomes a can of worms (sadly enough).

Gauge theory, as Derek Wise notes in his blog post "The geometric role of symmetry breaking in gravity", is intimately connected to Cartan Geometry.

There are dozens of exercises/examples to consider in gauge theory: Yang-Mills Theory, Born-Infield Action, Non-linear Sigma-Model, Non-linear Electrodynamics, Chern-Simons Theory, etc.

What Outline

So far, I've been considering my obstacles...but what about an outline?

The model I am following is the treatment of integration and differentiation in mathematics. First we have the naive symbolic manipulations (as done in calculus), then later we have the formal and rigorous proof based approach (as done in analysis).

Perhaps we should begin with naive field theory, where we obtain classical field theory "naively" from a "many body" problem.

This has merit from modelling fields as densities on the intuitive level.

Canonical quantization of this scheme becomes a triviality.

The problem with this approach is: what about the treatment of gauge theories, and their quantization?

After a few miracles, I expect to end up working with path integral quantization and formal calculus.

Naive treatment on quantizing gauge systems ought to be considered a bit more closely...

So that concludes the "naive" approach, and we begin the Axiomatic Approach. We should clarify the term "Axiom" means specification (not "God given truth", as dictionaries insist!).

The axiomatic approach would be done in a "guess-and-check" manner, modifying the axioms as necessary.

We naturally begin with Wightman axioms for the canonical approach, and the Osterwalder-Schrader axioms for the path integral approach. (Quickly, we ought to prove these two are equivalent!)

Kac's Vertex Algebras for Beginners takes the Wightman axioms, then extends them to conformal field theory through some magic. Perhaps this would be a worth-while example to consider?

Puzzles

Recently I've been more interested in puzzles.

Project Euler is the classic example of puzzles which require either higher math or computational skill (or both!).

Facebook has a collection of puzzles too, motivated from the engineering perspective.

But note: Facebook uses these puzzles for hiring people. Plus, the puzzles are not always mathematically oriented.

I suppose a good mathematician should always set up puzzles for themselves. As Socrates remarked:

SOCRATES: Indeed, Lysimachus, I should be very wrong in refusing to aid in the improvement of anybody. And if I had shown in this conversation that I had a knowledge which Nicias and Laches have not, then I admit that you would be right in inviting me to perform this duty; but as we are all in the same perplexity, why should one of us be preferred to another? I certainly think that no one should; and under these circumstances, let me offer you a piece of advice (and this need not go further than ourselves). I maintain, my friends, that every one of us should seek out the best teacher whom he can find, first for ourselves, who are greatly in need of one, and then for the youth, regardless of expense or anything. But I cannot advise that we remain as we are. And if any one laughs at us for going to school at our age, I would quote to them the authority of Homer, who says, that

'Modesty is not good for a needy man.'

Let us then, regardless of what may be said of us, make the education of the youths our own education. (Emphasis added, from Plato's Laches)

For example, I know a little bit about representations of Lie groups and Lie algebras (one can always learn more!)...but what about the representation of the quaternion group induced from the irreducible representations of SU(2)? How does it decompose into irreps? Etc.

Knuth remarked somewhere what helped him understand the representation theory for the symmetric group was writing a program which generated the permutation matrix representations.

I suspect writing a program which does these sorts of computations is a great puzzle for any mathematician that's savvy with programming.

Reading Material

And now, for something completely different.

A few papers I want to read:
When physics helps mathematics: calculation of the sophisticated multiple integral, 13 pages;
Some algebraic properties of differential operators, 15 pages;
Introduction to supergravity, 152 pages;
Fermionic impurities in Chern-Simons-matter theories, 31 pages;
Spinors and Twistors in Loop Gravity and Spin Foams, 16 pages;
Quaternionic Analysis, Representation Theory and Physics, 60 pages.

Problem Notebooks

I've been looking for the infamous Kourovka notebook, but there are apparently others.

Recall the Kourovka notebook is a collection of open problems in group theory. There are other notebooks with open problems in other fields.

For example, the Dniester Notebook [usask.ca] (pdf, 56 pages) has problems in ring theory.

The notebook stopped being published years ago. Now it's been open sourced.

The Sverdlovsk notebook stopped publishing back in 1989; it was a collection of open problems for semigroups, but it cannot be found.

Other problem notebooks might exist too, but I am unaware of them...

People don't read anymore :(

I have been wandering around book stores recently, and stumbled upon the Landmark Herodotus. If you recall, I wrote a few posts on Herodotus.

People ought to be reading the text, and produce their own notes, producing a similar result (as Landmark) which is personalized.

Don't mistake me: I think Landmark is a wonderful resource! It has helped with maps, and so on...but people should be doing this on their own.

The problem Landmark posed (to me) parallels Cliff Notes. The books are produced as an example of how to take notes while reading books...not a replacement for reading!

Extemporaneously producing maps is an invaluable skill. The map doesn't have to be precise, e.g. drawing Asia minor as a rectangle, or Greece as three rectangles.

There are some elements to, e.g., the Landmark Herodotus that are pleasant, and makes it a great resource to borrow from. For example: what did Herodotus get wrong?

It appears that he got a lot of Egyptian history wrong, for example.

But my point is: reading is studying! You have to make a book your own through notes, maps, references to other books, etc.

How you study is entirely up to you; how you take notes, well, that is entirely up to you. But reading without taking notes is like dancing: nobody does it unless drunk or insane.

People just confuse the mechanics of reading (parsing letters into words) with the procedure called "reading" (obtaining information from books, and evaluating it). *sigh*

P.S. Happy New Year!

Writing a History Book?

So I have been thinking about writing a book on history, but a different sort of book. The style is slightly inspired from mathematical writing and Herodotus, where each chapter is divided into "chunks".

A "chunk" discusses one subject, which is summarized in a single sentence --- the sentence is highlighted in bold, at the beginning of the chunk.

The idea is that there would be an overview, summarizing the chapter, which consists of just the "summary sentences". This way, the reader can get a cursory understanding of history, while looking for details in the appropriate section.

It would be nice to make it a website in the form of Real World Haskell, The Django Book, or Zend's Comment System. I think the Zend system is freely available via GIT.

Each paragraph ends with its own comments. However, with history, the author must be well prepared to deal with editing obscene comments. History is quite a sensitive subject.

After carefully examining the tools available, ucomment is the best choice.

It allows the author to dynamically update the document, while keeping the comments. Additionally, it uses the Sphinx markup language (which allows output to LaTeX).

Outline of Commutative Geometry

So, preparing for my discussion of noncommutative geometry, I need to discuss "commutative geometry".

What's going on here? Well, lets begin with the simplest notion of a space: a topological space.

I have discussed in my notebk [googlecode.com] the notion of a topological space and continuous functions.

However, the algebra of continuous real-valued (or, more generally, complex-valued) functions encode the topology.

So I need to write notes reconstructing the topological data for $X$ from the ring structure and properties which $C(X)$ satisfy.

Vector Bundles

The next sort of space we can work with is a vector bundle. What's this guy?

Well, it's really a fibre bundle whose fibre forms a vector space. What's a fibre bundle?

It's a generalization of the product space where we fix one of the spaces.

Where does this occur? In vector calculus!

We are working with $\mathbb{R}^{3}$. A vector field assigns to each point in $\mathbb{R}^{3}$ a vector. But vectors live in "linear spaces" (or vector spaces).

So secretly we have $X=\mathbb{R}^{3}$ be the underlying space, and the total space be $E=F\times X$ consisting of "tangent vectors" (an ordered pair consisting of the vector assigned by the vector field, and its base point).

The fibre here is a vector space. Moreover, it is $F=\mathbb{R}^{3}$ as a vector space.

This is the simplest example of a vector bundle. So what?

Well, vector fields can be represented through ordered triples. That is, three smooth functions represent each component of the vector field (the x part, y part, and z part).

So algebraically we have $C^{\infty}(\mathbb{R}^{3})\times C^{\infty}(\mathbb{R}^{3})\times C^{\infty}(\mathbb{R}^{3})$ represent all possible vector fields on our space.

This is a free module over $C^{\infty}(\mathbb{R}^{3})$. So are vector bundles represented by free modules?

Not really, we use projective modules (which is more general).

There are a few other things to discuss on this matter, e.g., global sections, and so forth.

Spinor Bundles

This should be discussed in some detail, as there are few good references on the subject.

Even nLab's entry on Spinor bundles is lacking, alas!

However, we need to encode this data in a spectral triple. See the nLab's entry, it is quite good.

See also Alain Connes' "On the spectral characterization of manifolds" (arXiv:0810.2088) for details.

This would take some time to write up.

Noncommutative Rejoinder

I suspect that by taking an arbitrary ring, instead of the ring of smooth functions (or continuous functions, or...), we begin working with noncommutative geometry.

Is this algebraic geometry? No, not really. Algebraic geometers use polynomials to encode their geometric objects.

On the other hand, what we are doing here is considering the structure of rings and modules over our rings to encode geometric properties and data.

Of course, I may be misinformed on what algebraic geometers do...I frankly never understood it well enough to satisfy myself.

Math to think about

There are several interesting directions I'd like to investigate. So interesting, I have decided to let you in on it too!

Moonshine

No living man cannot deny interest in moonshine. Terry Gannon's "Monstrous moonshine and the classification of CFT" (arXiv:math/9906167) provides a great review.

The basic idea is that we have a way to associate to "algebraic stuff" (e.g., groups) some "modular stuff".

What's great is, the word "stuff" is used in the technical sense of the word.

This would require reviewing group theory, finite groups, as well as some ring theory (for the "algebraic stuff"). To discuss "modular stuff", we'd need to review complex analysis, functional analysis. The connection here would require studying conformal field theory, to some degree.

Robert Wilson's Finite Simple Groups is a wonderful reference for finite simple groups; and as always SPLAG is a good reference too.

Noncommutative Geometry

People mean many things by "Noncommutative Geometry", here I mean Connes' approach.

I suppose this first requires us to consider what "commutative geometry" is!

This requires knowledge of commutative algebra and differential Geometry. The idea is to model "differential calculus over a commutative ring" (as Wikipedia calls it), i.e., consider the algebraic "grammar" underlying differential geometry.

Commutative algebra describes this algebraic "grammar". Studying this model in commutative algebra is precisely what I mean by "commutative geometry" (where projective modules correspond to vector bundles, and so on).

Noncommutative geometry, on the other hand, generalizes this model to the noncommutative setting!

I still need to write up my notes on differential geometry, but there are no good references for "commutative geometry"! There are a few books on commutative algebra, though...

I should type up my notes on algebraic topology too, since spin bundles are a "principal Spin bundle". Although I have some notes written on the Spin group (see my Lie groups notes), I should review it some more. Michelson and Lawson's Spin Geometry is a wonderful book to consider...

Operator algebras need to be reviewed for considering spectral triples. The algebra we typically work with are von Neumann algebras which are related to C* algebras.

Some references for operator algebras:

1. Kadison and Ringrose, Fundamentals of the theory of operator algebras vol. I and II
2. Blackadar. Operator Algebras: Theory of C*-Algebras and von Neumann Algebras. Encyclopaedia of Mathematical Sciences. Springer-Verlag, 2005.
3. Yasuyuki Kawahigashi, "Conformal Field Theory and Operator Algebras" arXiv:0704.0097 (18 pages)
4. Meghna Mittal, Vern Paulsen, "Operator Algebras of Functions." arXiv:0907.5184
5. John M. Erdman Lecture Notes on Operator Algebras (129 pages)
6. J. A. Erdos, C*-Algebras (51 pages).
7. N.P. Landsman, "Lecture notes on C*-algebras, Hilbert C*-modules, and quantum mechanics" arXiv:math-ph/9807030 (89 pages).
8. Jacob Lurie's Course Notes on Von Neumann Algebras, quite comprehensive!
9. Wassermann, Operators on Hilbert space [ps] (70 pages)
10. VFR Jones, von Neumann algebras [pdf] (150 pages)
11. NP Landsman's Lecture Notes on Operator Algebras [pdf] (64 pages).
12. John Hunter and Bruno Nachtergaele, Applied Analysis (free, legal ebook!)

And, of course, there is Connes' Noncommutative Geometry [pdf], as well as Connes and Marcolli's Noncommutative Geometry, Quantum Fields and Motives [pdf]

Basic Physics Macros

Continuing from my post LaTeX Macros for Personal Notes, I'd like to discuss some macros for physics.

I am using the "ISEE" approach to tackling examples, where we have four major steps:

1. "Identify" what do we have and what are we looking for?
2. "Set Up" what are the relevant concepts and equations? Set up the equations.
3. "Execute" Carry out the scratch work
4. "Evaluate" Look back, reflect, what were the key points and key ideas?

We are working with a lot of examples, and the examples are long (compared to math!). So we need to indicate when the examples are done.

Following Euclid, we introduce a \qefsymbol which is used at the end of examples and constructions. This is done just as QED is used at the end of proofs.

I will use the amsthm package.

\usepackage{amsthm}

\theoremstyle{plain}
\newtheorem{thm}{Theorem}[chapter]
\newtheorem{prop}[thm]{Proposition}

\theoremstyle{definition}
\newtheorem{defn}[thm]{Definition}
\newtheorem{ex}[thm]{Example}
\newtheorem{fact}{Experimental Fact}
\newtheorem{prob}[thm]{Problem}
\newtheorem{construction}[thm]{Construction}
\newtheorem{con}[thm]{Conjecture}
\newtheorem*{notation}{Notation}
\newtheorem*{assume}{Assumption}
\newtheorem*{quest}{Question}

\theoremstyle{remark}
\newtheorem{rmk}[thm]{Remark}
\newtheorem{sch}[thm]{Scholium}

\newcommand\qefsymbol{\ensuremath\blacksquare}
%{\ensuremath\triangle} % perhaps \ensuremath\triangle if one prefers...

\makeatletter
\newenvironment{example}{\begin{ex} %
  \let\qedsymbol\qefsymbol % this is a temporary "let"
  \pushQED{\qed}}%
  {\popQED\@endpefalse\end{ex}}

\newenvironment{construct}%
  {\begin{construction}\pushQED{\qed}}%
  {\popQED\end{construction}}
\makeatother

Now, to keep track of which step of ISEE we are at, I'd like to introduce the following code:

\font\manual=manfnt

\newcommand\identify{\noindent\llap{\manual\char'170\rm\kern.5em}\textbf{Identify:}}
\newcommand\setup{\noindent\llap{\manual\char'170\rm\kern.5em}\textbf{Set up:}}
\newcommand\execute{\noindent\llap{\manual\char'170\rm\kern.5em}\textbf{Execute:}}
\newcommand\evaluate{\noindent\llap{\manual\char'170\rm\kern.5em}\textbf{Evaluate:}}

There are other matters to discuss, like units and so forth, which I'll tackle next time...

MetaPost, Plotting, and numerical precision

So, to write up diagrams in LaTeX, you need to use Metapost. But Metapost doesn't use floating point arithmetic.

As Claudio Beccari's "Floating point numbers and METAFONT, METAPOST, TEX, and PostScript Type 1 fonts" (TUGboat [pdf]) notes, 32 bit integers are used to represent real numbers. The first 16 bits form the fractional part, 14 bits the integer part, 1 bit for the sign, and 1 bit for special purposes.

So, that means we have 16 log(2)/log(10) digits of precision, or about 4 digits. Now lets remember:

1 PS point = 1.00375 points
1 pica = 12 PS points
1 inch = 72 PS points = 72.27 points = 6 pica
1 cm = 28.3464567 PS points = 2.36220472 pica

We have precision of 2^-16 points, or about 0.000868055556 inches, or 0.00220486111 centimeters.

That's decent for output but not for intermediate computations. For example, if we were to plot x^x, we may lose a lot of precision.

Plots in Metapost

Lets consider a simple plot of $f(x)=x^{2}$.

numeric u;
u := 1pc; % units

vardef f(expr x) = x*x enddef;

beginfig(0)
  % draw the axes
  drawdblarrow (-3u-ahlength,0)--(3u+ahlength,0);
  drawdblarrow (0,0-2ahlength)--(0,9u+ahlength);

  % plot the function
  draw (-3u,f(-3)*u)
    for i=-3+0.05 step 0.05 until 3:
    ..(i*u,f(i)*u)
  endfor;
endfig;
end;

Remember that ahlength is the length of the arrow head.

This basic scheme can be generalized if we add numerics x0 and x1 which control where the plot begins and ends (respectively), as well as the step size dx which is taken to be "small enough".

Revising our code:

numeric u;
numeric dx;
u := 1pc; % units
dx := 0.05; 

vardef f(expr x) = x*x enddef;

beginfig(0)
  numeric x[];

  x0 := -3; % start plotting at x=-3
  x1 := 3; % stop plotting at x=+3

  % draw the axes
  drawdblarrow (x0*u-ahlength,0)--(x1*u+ahlength,0); % x-axis
  drawdblarrow (0,0-2ahlength)--(0,f(x1)*u+ahlength); % y-axis

  % plot the function
  draw (x0*u,f(x0)*u)
    for i=x0+dx step dx until 3:
    ..(i*u,f(i)*u)
  endfor;
endfig;
end;

This makes things a little complicated. What we are doing is computing the pairs (x,y) and then scaling them, then plotting.

The dx is the change in x before scaling. The points plotted have a change in x that amounts to dx*u=0.6pt approximately.

But we can do more! If we specify how big we want this plot to be, i.e. it has to fit within X inches, then we can determine the scale u by this.

Specifically, u := X/(x[1]-x[0]) is the scale factor definition.

If we demand that dx*u=0.6pt hold, which is "sufficiently good" for practical purposes, then we also define dx := (3pt)/(5*u).

The interested reader may want to read Learning Metapost by Doing.

LaTeX Macros for Personal Notes

So last time, I discussed the notion of personal mathematical notes (as opposed to expository mathematical notes) and would like to discuss some LaTeX macros which enable writing personal notes.

The basic scheme is to write in "chunks" (to borrow a term from literate programming). We've all seen examples of this, Bagchi and Wells refer to it as "labeled style" in their paper Varieties of Mathematical Prose

But each "chunk" is a self-contained concept, example, discussion, etc.

For a good example of this writing style, see On Euler's Footsteps.

LaTeX Code

I am taking CWEB's style. So, the code listing I have is as follows:

% chunk.sty
\ProvidesPackage{chunk}[2011/12/15 Cunking commands for personal notes]
\makeatletter

\@ifundefined{@addpunct}{
  \def\@addpunct#1{\ifnum\spacefactor>\@m \else#1\fi}
  }{}

\newcounter{chunk@ctr}

\newcommand\M{\medbreak\noindent%
  \refstepcounter{chunk@ctr}%
  \textbf{\thechunk@ctr\@addpunct{.}}\quad\ignorespaces}

% deprecated macro:
% \newcommand\N[1]{\M\textbf{#1\@addpunct{.}}\quad\ignorespaces}

% superior implementation:

\def\N{\@ifstar
        \NStar%
        \NNoStar%
}
\def\NStar#1{\medbreak\noindent\textbf{#1\@addpunct{.}\quad}\ignorespaces}
\def\NNoStar#1{\M\textbf{#1\@addpunct{.}\quad}\ignorespaces}

% permits writing \N*{Un-numbered chunk} for a chunk
% without a numeric label!

\makeatother
% end of cunk.sty

Note that it is completely self-contained code, and you do not need amsgen package. If you already loaded it, then no worries!

Each chunk is numbered. We use \M for unlabeled chunks, and \N{My favorite chunk!} for labeled chunks (which is labeled "My favorite chunk!").

So lets write up some example usage:

\documentclass{article}
\usepackage{chunk} % make it in the same directory
% or put it in ~/texmf/tex/latex/ and run "sudo texhash"
\title{Example Notes}
\author{Alex Nelson}
\date{\today}
\begin{document}
\maketitle

\N{Introduction}
Today we will solve all the problems in the universe. 

\M Lorem ipsum dolor sit amet, consectetur adipisicing elit, 
sed do eiusmod tempor incididunt ut labore et dolore magna 
aliqua. Ut enim ad minim veniam, quis nostrud exercitation 
ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis 
aute irure dolor in reprehenderit in voluptate velit esse 
cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat 
cupidatat non proident, sunt in culpa qui officia deserunt 
mollit anim id est laborum.

\N{Conclusion?} Nobel Prize please!

\end{document}

As far as bugs, I don't think there are any...it's too minimalistic!

To Do

The chunk counter is rather minimalistic, and doesn't count within any section. This has to be changed by hand if the user wants to use these macros and number chunks within each chapter...

In the time honored tradition of mathematicians, this exercise is left for the reader!

Notes on "How to Read a Book"

These are just some quick notes from Adler's How to Read a Book (Touchstone, 1972).

Four Levels of Reading

The first level is "Elementary Reading". In mastering this level, one learns the rudiments of the art of reading, receives basic training in reading, and acquires initial reading skills.

The reader is merely concerned with language as it is employed by the writer. At this level of reading, the question asked of the reader is "What does the sentence say?"

The second level of reading is called "Inspectional Reading". Its aim is to get the most out of a book within a given amount of time (e.g., an hour before going to bed).

(I have heard that humans work best at 90 minute time intervals, although I do not know if this is factual or not.)

Inspectional reading is the art of skimming systematically.

Whereas the question that is asked at the first level is "What does the sentence say?" the question typically asked at this level is "What is the book about?" That is a surface question; others of a similar nature are "What is the structure of the book?" or "What are its parts?"

Upon completing an inspectional reading of a book, no matter how short the time you had to do it in, you should also be able to answer the question, "What kind of book is it-a novel, a history, a scientific treatise?"

The third level of reading we will call "Analytical Reading".

The analytical reader must ask many, and organized, questions of what one is reading.

Analytical reading is preeminently for the sake of understanding. Adler claims it isn't needed if one is reading for entertainment, although I disagree with him.

The fourth and highest level of reading we will call "Syntopical Reading".

When reading syntopically, the reader reads many books, not just one, and places them in relation to one another and to a subject about which they all revolve. But mere comparison of texts is not enough. Syntopical reading involves more. With the help of the books read, the syntopical reader is able to construct an analysis of the subject that may not be in any of the books. It is obvious, therefore, that syntopical reading is the most active and effortful kind of reading.

[...] Let it suffice for the moment to say that syntopical reading is not an easy art, and that the rules for it are not widely known. Nevertheless, syntopical reading is probably the most rewarding of all reading activities. The benefits are so great that it is well worth the trouble of learning how to do it.

I think that this is what most (good) mathematicians do when studying a subject.

Also note that each stage is contained in the next higher stage...so elementary reading is contained in inspectional reading; inspectional reading is contained in analytical reading; analytical reading is contained in syntopical reading.

Inspectional Reading

There are two types of inspectional reading that are related to each other. Adler has a list of suggestions on systematic skimming:

1. Look at the Title page and, if the book has one, at its Preface.
2. Study the table of contents to help get an idea of the books' structure. Sometimes this helps a lot (e.g. when reading "The Cambridge Medieval History" or any other "Cambridge ______ History"), other times it doesn't help that much. Other books it helps with: Gibbon's Decline and Fall, Karl Marx's Das Kapital, Milton's Paradise Lost.
3. Check the index to get a gist of the topics covered.
4. If the book has a jacket, read the publisher's blurb.
5. Look at the chapters that seem pivotal.
6. Read a paragraph or two on each page, or perhaps several sequential pages. But not more than that. You are skimming, after all! Every couple of pages, read a couple of paragraphs.

The other aspect to inspectional reading is summed up in this single rule: In tackling a difficult book for the first time, read it through without ever stopping to look up or ponder the things you do not understand right away.

Instead of speed reading, a better idea is that Every book should be read no more slowly than it deserves, and no more quickly than you can read it with satisfaction and comprehension.

Finally, do not try to understand every word or page of a difficult book the first time through. This is the most important rule of all; it is the essence of inspectional reading. Do not be afraid to be, or to seem to be, superficial. Race through even the hardest book. You will then be prepared to read it well the second time.

[...]

[...] The first stage of inspectional reading-the stage we have called systematic skimming-serves to prepare the analytical reader to answer the questions that must be asked during the first stage of that level. Systematic skimming, in other words, anticipates the comprehension of a book's structure.

Active Reading

Adler writes, If your aim in reading is to profit from it-to grow somehow in mind or spirit-you have to keep awake. That means reading as actively as possible.

How do we "actively read"?

Ask questions while you read: questions that you yourself must try to answer in the course of reading. Adler suggests specifically four questions:

1. What is the book about on the whole?
2. What is being said in detail, and how? That is, what are the main ideas, assertions, and arguments?
3. Is the book true, in part or in whole?
4. So what?

Adler argues that this is the reader's obligation.

Inspectional reading will answer the first two questions, but not the second two. Analytical reading will answer the last two; and the last question is the most important for syntopical reading.

Good books are over your head; they would not be good for you if they were not. After all: no pain, no gain.

So to reiterate, there are three things to look at:

1. Studying the structure of the work.
2. Studying the logical propositions made and organized into chains of inference.
3. Evaluation of the merits of the arguments and conclusions.

Mark up your books!

I do not know if I agree with this practice, but Adler suggests it to become more actively involved in reading: marking up your book in the margins.

Although I must admit I do this when reading technical papers, and at times I have printed out (legally) downloaded chapters from Springer--Verlag books.

Adler provides a list of possible markups:

1. Underline major points, important statements, etc.
2. Vertical line in the margins for passages too long to be underlined, or already underlined but really important.
3. Write a star, asterisk, or other symbol in the margin for really, really important statements. Pretend you can only use 10--12 per book, so make them count!
4. Write numbers in the margin to indicate a sequence of points in an argument the author is making.
5. In the margins, write "Cf. [pg] xxx" to refer to other page xxx in the book. Sometimes, in math books, I also write "See so-and-so's title, pg xx for more on some subject." This helps a lot in mathematics.
6. Circle key words or phrases.
7. Write notes in the margins, at the top of the page, and/or at the bottom of the page. Also, Adler remarks The endpapers at the back of the book can be used to make a personal index of the author's points in the order of their appearance.

On this last point, Adler remarks The front endpapers are better reserved for a record of your thinking. After finishing the book and making your personal index on the back endpapers, tum to the front and try to outline the book [...] as an integrated structure, with a basic outline and an order of parts.

Bert Webb has suggested Twelve Ways To Mark Up A Book [typepad.com] although note that a post-it note is horrible for a book: it will make the paper brittle over time, and eventually (shockingly) it will shatter when used.

What about Library Books?

You cannot (well, should not) do this with library books. What to do?

Well, back in the day they had these things called a "Commonplace book [wikipedia.org]" where one would write down a passage from a book, and then some comments on it.

This is a viable alternative, but it leads us to our next segment: note making.

Note Making

There are three types of note making.

"Structural note-making" are notes primarily concerning the book's struture, and not its substance-at least not in detail.

"Conceptual note-making" are notes answering questions on the book's truth and significance. They concern the author's concepts, and also your own, as they have been deepened or broadened by your reading of the book.

"Dialectical note-making" are notes about the shape of the discussion- the discussion that is engaged in by all of the authors, even if unbeknownst to them. This is syntopical reading based notes, and requires several books.

Adler notes that But in order to forget them as separate acts, you have to learn them first as separate acts (55).

Analytical Reading

There are several rules Adler gives to read analytically.

RULE 1. "You must know what kind of book you are reading, and you should know this as early in the process as possible, preferably before you begin" (60). Is it fiction (a novel, a play, a poem, an epic) or non-fiction?

We can infer some information from the title. Adler quips Again, however, to group books as being of the same kind is not enough; to follow this first rule of reading you must know what that kind is (64).

The title gives different information for fiction books, compared to non-fiction books. And in non-fiction, the title is different for mathematics and science, compared to more liberal arts subjects.

We can also note that there is a difference between "theoretical" and "practical" books: it is the distinction between knowledge and action.

Theoretical books teach you that something is the case. Practical books teach you how to do something you want to do or think you should do (66).

RULE 2. State the unity of the whole book in a single sentence, or at most a short paragraph. (75--76).

In other words, the "unity" of the book is what you would tell your friend or family over dinner.

This should involve, what Dr MacElroy calls, Detail (with a capital "D"): any fact, figure, proper nouns, number, statistics, ratios, or capitalized words.

Compare these two statements summarizing a hypothetical book:

Statement 1: Beatles attack many Southern crops.

Statement 2: Each Summer, Japanese Beatles attack over 300 different kinds of flowers, foliage, and fruit in North Carolina, South Carolina, Georgia, Tennessee, and Alabama.

See the difference? I hope so...

RULE 3. Set forth the major parts of the book, and show how these are organized into a whole, by being ordered to one another and to the unity of the whole (76).

In a sense, this is a self-seimilar aspect to note taking.

Adler remarks Hence the third rule involves more than just an enumeration of the parts. It means outlining them, that is, treating the parts as if they were subordinate wholes, each with a unity and complexity of its own (84).

When you look at my notes on Herodotus' Histories, Books I and II, you can see that my notes for book I are written up in such a way that the first outline I wrote is:
first logos: the story of Croesus (1.1-94)
second logos: the rise of Cyrus the Great (1.95-140)
third logos: affairs in Babylonia and Persia (1.141-216)

I then went back, and then expanded on each of these.

I. The story of Croesus.
7–25. Lydian History.
26–56. Croesus of Lydia.
57–64. History of Athens.
65–68. History of Sparta.
II. Rise of Cyrus.
69–84. Croesus attempts (and fails) at conquering Assyrians; Cyrus conquers Sardis.
85–92. Croesus as Cyrus' slave.
93–94. Lydian culture.
95–130. Medes history, rise of Deioces, Phraortes.
106–125. Background of Cyrus' birth, upbringing, etc.
126–130. Cyrus overthrowing Cyaxeres by using the Persians, becomes ruler of the Persians.
131–140. Culture of the Persians.
III. Affairs in Babylonia and Persia.
141–176. Persian conquest of the Ionians.
178–200. Babylon, its History; battle of Babylon; customs of Babylonians.
201–216. Death of Cyrus.

Each of these can be expanded, in turn, to write more Details. As Adler writes:

A good book, like a good house, is an orderly arrangement of parts. Each major part has a certain amount of independence. As we will see, it may have an interior structure of its own, and it may be decorated in a different way from other parts. But it must also be connected with the other parts-that is, related to them functionally-for otherwise it would not contribute its share to the intelligibility of the whole.

[...]

Let us return now to the second rule, which requires you to state the unity of a book. A few illustrations of the rule in operation may guide you in putting it into practice.

Let us begin with a famous case. You probably read Homer's Odyssey in school. If not, you must know the story of Odysseus, or Ulysses, as the Romans call him, the man who took ten years to return from the siege of Troy only to find his faithful wife Penelope herself besieged by suitors. It is an elaborate story as Homer tells it, full of exciting adventures on land and sea, replete with episodes of all sorts and many complications of plot. But it also has a single unity of action, a main thread of plot that ties everything together.

Aristotle, in his Poetics, insists that this is the mark of every good story, novel, or play. To support his point, he shows how the unity of the Odyssey can be summarized in a few sentences.

A certain man is absent from home for many years; he is jealously watched by Poseidon, and left desolate. Meanwhile his home is in a wretched plight; suitors are wasting his substance and plotting against his son. At length, tempest-tossed, he himself arrives; he makes certain persons acquainted with him; he attacks the suitors with his own hand, and is himself preserved while he destroys them.

"This," says Aristotle, "is the essence of the plot; the rest is episode." (77--79)

Adler gives two warnings:

(1) a good author will help you summarize the book in a single sentence [usually in the preface],

(2) there is no single correct "single-sentence summary" for a book...there may be many different such summaries.

Regarding the self-similarity to note-taking, Adler remarks:

Hence the third rule involves more than just an enumeration of the parts. It means outlining them, that is, treating the parts as if they were subordinate wholes, each with a unity and complexity of its own.

[...] According to the second rule, we had to say : The whole book is about so and so and such and such. That done, we might obey the third rule by proceeding as follows: (1) The author accomplished this plan in five major parts, of which the first part is about so and so, the second part is about such and such, the third part is about this, the fourth part about that, and the fifth part about still another thing. (2) The first of these major parts is divided into three sections, of which the first considers X, the second considers Y, and the third considers Z. (3) In the first section of the first part, the author makes four points, of which· the first is A, the second B, the third C, and the fourth D. And so on and so forth. (84)

This may seem like too much work, but it is done habitually. It does not have to be written down, it may be stored mentally in one's memory.

How much outlining should one do? Adler quips No book deserves a perfect outline because no book is perfect (85). Remember: the outline is of the book, not the subject.

Sometimes the outline is longer than the book (e.g. Medieval commentaries on Aristotle is typically longer than the original, since it includes more than an outline...it also includes examples, etc.).

When reading, e.g., Das Kapital (vol. I) I essentially had to rewrite, sentence by sentence, the first four chapters. But everything after that was simple to understand.

RULE 4. Find out what the Author's problems were (92). The author is trying to answer some question, supposedly the book has the [a?] solution.

Not all questions were explicitly stated. When we have a list of the questions, we should ask ourselves: Which are primary and which secondary? Which questions must be answered first, if others are to be answered later? (93)

We can see that, like the previous rules, this applies to the "self-similar" parts of the book.

What sort of questions can we ask? Well...

If you know the kinds of questions anyone can ask about anything, you will become adept in detecting an author's problems. They can be formulated briefly : Does something exist? What kind of thing is it? What caused it to exist, or under what conditions can it exist, or why does it exist? What purpose does it serve? What are the consequences of its existence? What are its characteristic properties, its typical traits? What are its relations to other things of a similar sort, or of a different sort? How does it behave? These are all theoretical questions. What ends should be sought? What means should be chosen to a given end? What things must one do to gain a certain objective, and in what order? Under these conditions, what is the right thing to do, or the better rather than the worse? Under what conditions would it be better to do this rather than that? These are all practical questions. (93--94)

So, these are the four rules of reading which cover up to (and including) analytical reading. To reiterate, these rules are:

1. Classify the book according t o kind and subject matter.
2. State what the whole book is about with the utmost brevity.
3. Enumerate its major parts in their order and relation, and outline these parts as you have outlined the whole.
4. Define the problem or problems the author is trying to solve.

Coming to Terms with the Author

Sometimes the author uses special terms in a particular way (mathematicians know this best of all people).

For example, if one is reading economic works from the 18th to mid-19th centuries, the word "value" has many different but related meanings depending on the author.

In fact, terms are so important, Adler adds a fifth rule:

RULE 5. FIND THE IMPORTANT WORDS AND THROUGH THEM COME TO TERMS WITH THE AUTHOR. Note that the rule has two parts. The first part is to locate the important words, the words that make a difference. The second part is to determine the meaning of these words, as used, with precision. (98)

These two steps can be thought of slightly differently:

As we have pointed out, each of the rules of interpretive reading involves two steps. To get technical for a moment, we may say that these rules have a grammatical and a logical aspect. The grammatical aspect is the one that deals with words. The logical step deals with their meanings or, more precisely, with terms. (99)

These two steps are reciprocal, though. We identify the grammatical aspect by locating the passages with key terms, and we determine the meaning of the key terms by understanding their meaning with respect to that passage.

Sometimes there are typographical indicators of introducing key terms. In mathematics, this is done explicitly in a block definition that looks like:

Definition. A "term" is ...

Other times, it can be inferred from the table of contents (e.g., in economic texts one can immediately see that price, value, labour, output, productivity, etc., are key terms).

With regards to determining their meaning, the general rule is you have to discover the meaning of a word you do not understand by using the meanings of all the other words in the context that you do understand (107).

Sadly, we must come to acknowledge that (in general) There is no rule of thumb for doing this. The process is something like the trial-and-error method of putting a jigsaw puzzle together (108).

The Author's Intent

The author's propositions are nothing but expressions of personal opinion unless they are supported by reasons (115).

We want to know not merely what the author's propositions are, but also why the author thinks we ought to be persuaded to accept them.

So, we have some additional rules:

RULE 5. Find the important words and come to terms.

RULE 6. Mark the most important sentences in a book and discover the propositions they contain.

RULE 7. Locate or construct the basic arguments in the book by finding them in the connection of sentences.

How do we find the key sentences? The heart of the author's communication lies in the major affirmations and denials the author is making, and the reasons the author gives for so doing.

Perhaps you are beginning to see how essential a part of reading it is to be perplexed and know it. Wonder is the beginning of wisdom in learning from books as well as from nature. If you never ask yourself any questions about the meaning of a passage, you cannot expect the book to give you any insight you do not already possess (123).

Adler goes on to give another indicator:

This suggests one further clue to the location of the principal propositions. They must belong to the main argument of the book. They must be either premises or conclusions. Hence, if you can detect those sentences that seem to form a sequence, a sequence in which there is a beginning and an end, you probably have put your finger on the sentences that are important. (123)

Adler urges us to read and re-read the sentences which puzzle us rather than interest us.

How do we construct the basic arguments of the text? Adler remarks:

The translation of one English sentence into another, however, is not merely verbal. The new sentence you have formed is not a verbal replica of the original. If accurate, it is faithful to the thought alone. That is why making such translations is the best test you can apply to yourself, if you want to be sure you have digested the proposition, not merely swallowed the words. If you fail the test, you have uncovered a failure of understanding. If you say that you know what the author means, but can only repeat the author's sentence to show that you do, then you would not be able to recognize the author's proposition if it were presented to you in other words. (126)

So rewrite the argument in your own words.

Another good test is to exemplify the proposition:

There is one other test of whether you understand the proposition in a sentence you have read. Can you point to some experience you have had that the proposition describes or to which the proposition is in any way relevant? Can you exemplify the general truth that has been enunciated by referring to a particular instance of it? To imagine a possible case is often as good as citing an actual one. If you cannot do anything at all to exemplify or illustrate the proposition, either imaginatively or by reference to actual experiences, you should suspect that you do not know what is being said. (127)

If we fail to obtain a translation, then as a fall-back we should attempt an example.

We will, in fact, note that many sentences do not contain an argument at all. So let us reformulate rule 7:

RULE 7'. Find if you can the paragraphs in a book that state its important arguments; but if the arguments are not thus expressed, your task is to construct them, by taking a sentence from this paragraph, and one from that, until you have gathered together the sequence of sentences that state the propositions that compose the argument.

A good book should summarize its arguments as it goes along, though.

If the book contains arguments at all, then you must know what they are, and be able to summarize them.

Several tips:

(1) arguments consist of sentences. If you can spot the conclusion, the arguments must be nearby...and if you have the arguments, where is it heading?

(2) discriminate between the kind of argument that points to one or more particular facts as evidence for some generalization and the kind that offers a series of general statements to prove some further generalizations (132). In other words, is the argument inductive or deductive (respectively)?

(3) observe what things the author says we must assume, which of the author's statements can be proved or otherwise evidenced, and what need not be proved because it is self-evident. In other words, what is the logical status of each statement: assumption, provable, or axiom?

So, knowing the terms, propositions, and arguments leads us to the next rule of reading:

RULE 8. Find out what the author's solutions are.

So, we really want to know What is being said in detail, and how? To answer that, we use rules 5 through 8. Recall that these rules are:

RULE 5. Come to terms with the author by interpreting his key words.

RULE 6. Grasp the author's leading propositions by dealing with his most important sentences.

RULE 7. Know the author's arguments, by finding them in, or constructing them out of, sequences of sentences.

RULE 8. Determine which of his problems the author has solved, and which the author has not; and of the latter, decide which the author knew he had failed to solve.

Answering the Question: "So what?"

We should remember that as intellectuals, when we criticize we don't do it...in the conventional "You're an idiot"-sense of the word "criticize". In other words: no ad-hominems.

Concentrate on the core points the pieces make - words/phrases, references, examples, quotes, statistics -- and ask yourself if these can be criticized because they are vague, ambiguous, unhelpful, misguided, etc.

Discuss what is "missing" from a given text: Think about missing links in terms of what articles give and what they gloss over (i.e., never make clear, never substantiate, etc. etc. etc.).

Don't come out and say stuff, you slowly give parcel stuff out. Be careful with the words you use.

Criticism is working through the piece, showing differences with other pieces, discuss weaknesses and strengths.

Intellectuals are not interested in the person or their background, it's what they say and to be fair even if you loathe them.

We are very cautious. Instead think about 1 sentence statements, get involved with "This «statement said» is wrong/weird/exaggerated."

A huge red flag comes from statements like "All Americans know..."

Think about what we are given within what we are given. That is, given the piece, use only the piece...don't wander off and obtain statistics. Sometimes that is good, especially in the sciences.

Also, if they just say something without citation — e.g., make a claim without evidence, that is valid criticism.

We can summarize these points are the following rules:

RULE 9. Do not begin criticism until you have completed your outline and your interpretation of the book. (Do not say you agree, disagree, or suspend judgment, until you can say "I understand.")

RULE 10. Do not disagree disputatiously or contentiously.

RULE 11. Demonstrate that you recognize the difference between knowledge and mere personal opinion by presenting good reasons for any critical judgment you make.

Moreover, where we may critique an author may be specified in the last batch of rules:

RULE 12. Show wherein the author is uninformed.

To support the claim "the author is uninformed", you must:

1. be able to state the knowledge that the author lacks and
2. show how
   1. it is relevant, and how
   2. it makes a difference to the author's conclusions.

RULE 13. Show wherein the author is misinformed.

The author is making assertions contrary to fact, i.e. proposing as true or more probable what is in fact false or less probable. The writer is claiming to have knowledge which the writing does not possess. This kind of defect should be pointed out only if it is relevant to the author's conclusions. And to support the remark you must be able to argue the truth or greater probability of a position contrary to the author's.

RULE 14. Show wherein the author is illogical.

The reader must be able to show (respectfully) how the author's argument lacks cogency. We are concerned with this defect only to the extent that the major conclusions are affected by it.

RULE 15. Show wherein the author's analysis or account is incomplete.

It is not enough to say that a book is incomplete. Anyone can say that of any book. There is no point in making this remark, unless the reader can define the inadequacy precisely, either by his own efforts as a "knower" or through the help of other books.

There is nothing wrong with controversy, but (as Adler remarks) Good controversy should not be a quarrel about assumptions (155).

Of course, these rules need to be modified for various genres of writing.

Genres of Writing

Literature and Poetry

We do not look for "truth" in literature or poetry. We look for its effects on us.

Also literature typically consists of a number of "episodes", which can be thought of as a short story. These are composed together to form a "macroscale" story.

For example, Mark Twain's Adventures of Tom Sawyer has each chapter be a short story, but they are related to each other and weave several "macroscale" stories (e.g. Tom and Becky's relationship, Tom and Huck witnessing the murder, which leads them to become "pirates" along with another child, etc. etc. etc.).

First observe: the elements of fiction are its episodes and incidents, its characters, and their thoughts, speeches, feelings, and actions.

Second: we said terms are connected with propositions; analogously, the elements of fiction are connected by the total scene or background against which they stand out in relief.

Adler remarks:

You will recall that the first three questions are: first, What is the book about as a whole?; second, What is being said in detail, and how?; and third, Is the book true, in whole or part? The application of these three questions to imaginative literature was covered in the last chapter. The first question is answered when you are able to describe the unity of the plot of a story, play, or poem-"plot" being construed broadly to include the action or movement of a lyric poem as well as of a story. The second question is answered when you are able to discern the role that the various characters play, and recount, in your own words, the key incidents and events in which they are involved. And the third question is answered when you are able to give a reasoned judgment about the poetical truth of the work. Is it a likely story? Does the work satisfy your heart and your mind? Do you appreciate the beauty of the work? In each case, can you say why? (emphasis added, 215--16)

Plays

When reading plays, Adler suggests imagining we have the play going on inside our "inner theater" (cf. "inner monologue").

The only complete way to read a play is to see it performed, just as the only complete way to read music is to hear it performed.

Lyrical Poetry

I combine the first two rules of reading lyrical poetry together: to read it through without stopping, whether you think you understand it or not, and simultaneously read it out loud.

What questions can we ask of lyrical poetry? Usually they are rhetorical, though they may also be syntactical.

Why do certain words pop out of the poem and stare you in the face? Is it because the rhythm marks them? Or the rhyme? Or are the words repeated? Do several stanzas seem to be about the same ideas; if so, do these ideas form any kind of sequence? Anything of this sort that you can discover will help your understanding (230).

To be understood, the poem must be read aloud. But also, after some period of time we should return to it. Reading lyrical poetry is a lifetime job.

History

One might want to refer to Theodore Roosevelt's "History as Literature".

Actually, a lot of history may be viewed as a novel. One could legitimately read Herodotus as a sequence of episodes which describe the interaction between the Greeks and the Persians.

Actually, I have been wondering how to take notes on history in a way that is effective. I feel that recording the events as episodes is a legitimate way to do it.

History discusses events, persons, or institutions. There are two types of propositions:

(1) those statements regarding events, persons, or institutions;

(2) how the story is told, i.e., who is the hero, where the author places the climax, how the author develops the aftermath.

NB: when taking notes on history, we could be inspired by Jaegwon Kim's theory of structured events. An event is an ordered triple (x, P, t) where:
x is/are Object(s), i.e., what persons or institutions or locations or...;
P is a property; and
t is a date or temporal ordering.
Example: (Lincoln is assassinated, 1865). Index cards become cute and handy.

Adler gives two rules for reading history: The first is: if you can, read more than one history of an event or period that interests you. The second is: read a history not only to learn what really happened at a particular time and place in the past, but also to learn the way men act in all times and places, especially now (241).

What questions may be asked while reading history? We may note that

...the historian tells a story, and that story, of course, occurred in time. Its general outlines are thus determined, and we do not have to search for them. But there is more than one way to tell a story, and we must know how the historian has chosen to tell his. Does he divide his work into chapters that correspond to years or decades or generations? Or does he divide it according to other rubrics of his own choosing? Does he discuss, in one chapter, the economic history of his period, and cover its wars and religious movements and literary productions in others? Which of these is most important to him? If we discover that, if we can say which aspect of the story he is telling seems to him most fundamental, we can understand him better. (242)

When asking "What of it?" Adler quips History, which tells us of the actions of men of the past, often does lead us to make changes, to try to better our lot (243).

I know this sounds like fiction, but once upon a time politicians were extraordinarily well read in history...but modern politicians are barely literate.

Future Directions

It is kind of open what I am going to do with this notebook/blog. I don't really know myself. Personally, I prefer TeX as my markup language as opposed to html, which is why I don't write as much as I should.

(Rant: TeX is just so much more convenient! It actually allows me to change notation with the flip of the wrist! For example, consider \let\propersubset=\subsetneq, then I simply use A\propersubset B and if I hate the notation...well, one line of code changed! With html, I have to change everything by hand. Or counters...html has no counter macros, grr...)

It would be nice to "categorify" Bourbaki's work. Take that with a grain of salt! What I mean by this is to be as comprehensive as Bourbaki (I'm reading through his Algebra right now), presenting definitions as e.g. a group object instead of a group, a magma object instead of a magma, etc.

That is to say, present the same material "internalized" in an arbitrary category. So for an example of this, consider the following definition:

Definition 1. A Magma Object consists of an object M in a monoidal category C equipped with a morphism μ:M⊗M→M.

I must confess that such an endeavor is appealing to me, but I might do it in LaTeX (taking advantage of its flexibility).

Short Summary

I think I will probably end up writing up a cohesive collection of notes — in the spirit of Bourbaki — that is self contained covering all of mathematics (from Set Theory and Category Theory foundations to...whatever!). However, I think I'll TeX it up, and post it online.

(As an aside, there is an interesting project called LuaTeX (there is also LuaLaTeX). There is no memory limits for it, and it has embedded Lua code. Perhaps it would be interesting to use this when writing notes involving numerical calculations?)

A Remark on Reading Bourbaki

A very brief and small remark that may be helpful to those that are studying Bourbaki. The first volume of the Elements of Mathematics (The Theory of Sets) is more or less useless.

The only useful (and in my humble opinion, coherent) parts of that book is the "Summary of Results".

Although, if one were really "hardcore", one would have a collection of e.g. composition notebooks to write notes on the series. By giving actual explanation and examples, it should expand the size of the text several fold.

Also, it helps to create a "cheat sheet" of notation. Bourbaki used bizarre notation since, I assume, they had to work with typewriters.

Unfortunately, no one uses their notation. So, we are forced to come up with a "Rosetta stone" to translate their notation into modern notation.

Some guidelines for your "Rosetta Stone for Bourbaki" might be:

1. Include the book and page numbers where it is first introduced or defined.
2. Include definitions, since those too are "Bourbaki-dependent".
3. Have a separate "Stone" for each book.

I just thought that it may be helpful to someone trying to read through this ancient tome…

Addendum Tuesday August 16, 2011 at 01:10:11PM (PDT)

After some more research, I found out that the bizarre system Bourbaki uses in The Theory of Sets is really something called "Epsilon Calculus."

Math Overflow had a discussion on Bourbaki's epsilon calculus which is useful, and the Stanford Encyclopedia of Philosophy's page is instructive.

I doubt that this system was intended to be fully used, since (as some pointed out in the math overflow discussion) "even trivial proofs require an astonishing number of steps directly from axioms. Existence of the empty set can be proved with 11,225,997 steps and transfinite recursion can be proved with 11,777,866,897,976 steps."

The internet encyclopedia of philosophy states:

The growing awareness of the larger meaning and significance of epsilon calculi has only come in stages. Hilbert and Bernays introduced epsilon terms for several meta-mathematical purposes, as above, but the extended presentation of an epsilon calculus, as a formal logic of interest in its own right, in fact only first appeared in Bourbaki's Elements de Mathematique (although see also Ackermann 1937-8). Bourbaki's epsilon calculus with identity (Bourbaki, 1954, Book 1) is axiomatic, with Modus Ponens as the only primitive inference or derivation rule. Thus, in effect, we get:

(X ∨ X) → X,
X → (X ∨ Y),
(X ∨ Y) → (Y ∨ X),
(X ∨ Y) → ((Z ∨ X) → (Z ∨ Y)),
Fy → FεxFx,
x = y → (Fx ↔ Fy),
(x)(Fx ↔ Gx) → εxFx = εxGx.

This adds to a basis for the propositional calculus an epsilon axiom schema, then Leibniz' Law, and a second epsilon axiom schema, which is a further law of identity. Bourbaki, though, used the Greek letter tau rather than epsilon to form what are now called "epsilon terms"; nevertheless, he defined the quantifiers in terms of his tau symbol in the manner of Hilbert and Bernays, namely:

(∃x)Fx ↔ FεxFx,
(x)Fx ↔ Fεx¬Fx;

and note that, in his system the other usual law of identity, "x = x", is derivable.

The principle purpose Bourbaki found for his system of logic was in his theory of sets, although through that, in the modern manner, it thereby came to be the foundation for the rest of mathematics. Bourbaki's theory of sets discriminates amongst predicates those which determine sets: thus some, but only some, predicates determine sets, i.e. are "collectivisantes". All the main axioms of classical Set Theory are incorporated in his theory, but he does not have an Axiom of Choice as a separate axiom, since its functions are taken over by his tau symbol. The same point holds in Bernays' epsilon version of his set theory (Bernays 1958, Ch VIII).

Source: Internet Encyclopedia of Philosophy.

Over at the nLab, the entry on Choice Operators really helps explain what that pesky τ operator is in Bourbaki's Theory of Sets.

LaTeX Macros...

So here are some LaTeX3 macros I've found or written that are pretty useful. This post covers the following packages I've written/patched together:

• Bourbaki inspired dangerous bend environments
• Macros for underbrackets and overbrackets, similar to underbrace and overbrace
• Misner, Thorne, and Wheeler type equations
• Exercises and Answers in the style of the TeXbook
• Style like the TeXbook in LaTeX macros

Addendum (9:30 AM (PST) 15 December 2011): I have modified the danger.sty code, and it is now more robust. For more macros, see my Notebk's wiki page for others and documentation.

Dangerous Bends!

Bourbaki used Dangerous Bend symbols to indicate some tricky reasoning. Knuth used this too in his TeX book, among other places.

I too use it in my notes...but I use it as an environment. It's safer this way ;) At any rate the code is contained a file danger.sty, reproduced below:

\NeedsTeXFormat{LaTeX2e}
\ProvidesPackage{danger}[2009/08/06 Danger and Double Danger Environments]

\usepackage{manfnt}
% or if manfnt is unavailable, uncomment the next two lines
%\font\manual=manfnt
%\def\dbend{{\manual\char127}} % dangerous bend sign

%%
% This macro header is what controls the ``dangerous bend''
% paragraph
%%

% Danger, Will Robinson!
\newenvironment{danger}{\medbreak\noindent\hangindent=2pc\hangafter=-2%
  \clubpenalty=10000%
  \hbox to0pt{\hskip-\hangindent\dbend\hfill}\small\ignorespaces}%
  {\medbreak\par}

% Danger! Danger!
\newenvironment{ddanger}{\medbreak\noindent\hangindent=3pc\hangafter=-2%
  \clubpenalty=10000%
  \hbox to0pt{\hskip-\hangindent\dbend\kern2pt\dbend\hfill}\small\ignorespaces}%
  {\medbreak\par}

The above code is just literally cut/paste from Knuth's TeXbook macros. On the other hand, in LaTeX a better implementation might be:

\font\manual=manfnt
\def\dbend{{\manual\char127}} % dangerous bend sign

% Danger, Will Robinson!
\newenvironment{danger}{\medbreak\noindent\hangindent=2pc\hangafter=-2%
  \clubpenalty=10000%
  \hbox to0pt{\hskip-\hangindent\dbend\hfill}\small\ignorespaces}%
  {\medbreak\par}

% Danger! Danger!
\newenvironment{ddanger}{\medbreak\noindent\hangindent=3pc\hangafter=-2%
  \clubpenalty=10000%
  \hbox to0pt{\hskip-\hangindent\dbend\kern2pt\dbend\hfill}\small\ignorespaces}%
  {\medbreak\par}

Underbrackets and Overbrackets

In math mode, you can use underbrace and overbrace, but there are no corresponding brackets macros. This is sad, because I'm more fond of brackets than I am of braces.

So I fiddled around and pieced together the brackets.sty file:

\NeedsTeXFormat{LaTeX2e}
\ProvidesPackage{brackets}[2009/08/06 Overbracket and Underbracket racket]

\makeatletter
\def\overbracket{\@ifnextchar [ {\@overbracket} {\@overbracket
[\@bracketheight]}}
\def\@overbracket[#1]{\@ifnextchar [ {\@over@bracket[#1]}
{\@over@bracket[#1][0.3em]}}
\def\@over@bracket[#1][#2]#3{%\message {Overbracket: #1,#2,#3}
\mathop {\vbox {\m@th \ialign {##\crcr \noalign {\kern 3\p@
\nointerlineskip }\downbracketfill {#1}{#2}
                              \crcr \noalign {\kern 3\p@ }
                              \crcr  $!\hfil$ \displaystyle {#3}\hfil $%
                              \crcr} }}\limits}
\def\downbracketfill#1#2{$!\m@th$ \setbox \z@ \hbox {$!\braceld$$}
                  \edef\@bracketheight{\the\ht\z@}\downbracketend{#1}{#2}
                  \leaders \vrule \@height #1 \@depth \z@ \hfill
                  \leaders \vrule \@height #1 \@depth \z@ \hfill
\downbracketend{#1}{#2}$}
\def\downbracketend#1#2{\vrule depth #2 width #1\relax}


\def\underbracket{%
  \@ifnextchar[{\@underbracket}{\@underbracket [\@bracketheight]}%
}
\def\@underbracket[#1]{%
  \@ifnextchar[{\@under@bracket[#1]}{\@under@bracket[#1][0.4em]}%
}
\def\@under@bracket[#1][#2]#3{%\message {Underbracket: #1,#2,#3}
 \mathop{\vtop{\m@th \ialign {##\crcr $!\hfil$ \displaystyle {#3}\hfil $!$%
 \crcr \noalign {\kern 3\p@ \nointerlineskip }\upbracketfill {#1}{#2}
       \crcr \noalign {\kern 3\p@ }}}}\limits}
\def\upbracketfill#1#2{$!\m@th$ \setbox \z@ \hbox {$!$\braceld$!$}
                    \edef\@bracketheight{\the\ht\z@}\bracketend{#1}{#2}
                    \leaders \vrule \@height #1 \@depth \z@ \hfill
                    \leaders \vrule \@height #1 \@depth \z@ \hfill \bracketend
               {#1}{#2}$}
\def\bracketend#1#2{\vrule height #2 width #1\relax}
\makeatother


% Makes limits on sums and integrals pretty
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}

The other part of the code, which % Makes limits pretty, allows stuff to be written on top of the sum's limits. It's taken from Voss's math mode notes.

Equations with Misner, Thorne and Wheeler type Arrows

This is going to be a bit harder to explain (it's one of those "You have to have seen it to understand what I'm talking about" type situations). So look at page 139, equation 5.15a for example. Note the arrows pointing to the terms in the equation? Yeah, I'd like to do that in LaTeX, but how?

Unfortunately you have to use the picture environment (or, at least, that's the only way I know how to do it!). Consider the following example document:

\documentclass{amsart}
\usepackage{brackets}

\newlength\textwidthcm
\textwidthcm=.03514598035146\textwidth

\begin{document}
{\catcode`p=12 \catcode`t=12 \gdef\cm#1pt{#1cm}}
{\catcode`p=12 \catcode`t=12 \gdef\dimensionless#1pt{#1}}

\begin{equation}\label{eq:one}
E=mc^{2}
\end{equation}


\setlength{\unitlength}{1cm}
\begin{picture}(\expandafter\dimensionless\the\textwidthcm, 2.5)(0,0)
  \linethickness{0.5pt}
  \put(5,1.5){$\displaystyle Z[0]=\underbracket[0.25pt]{\int ~\mathcal{D}\phi\;\;\; }
               \!\!\!\exp(\int\mathcal{L}d^{4}x)$}
  \put(-.4,1.5){\refstepcounter{equation}{(\arabic{equation})\label{eq:four}}}
  \put(6.75,0.55){\vector(0,1){0.4}}
  \put(6.75,0.55){\line(1,0){1.4}}
  \put(8.15,0.5){\makebox{$\begin{pmatrix}
                        $!sum$$\ 
                        $!over$$\\ 
                        $!histories$$
                        \end{pmatrix}$}}
\end{picture}
Woah what is eq \eqref{eq:four} again? Don't forget the text width is \the\textwidth ~or 
equivalently \expandafter\cm\the\textwidthcm ~or 
\expandafter\dimensionless\the\textwidthcm
\end{document}

This produces the following text:

The only disadvantage is that for each equation you want to do, you have to do this by hand.

Exercises and Answers Macros From The TeXbook

\NeedsTeXFormat{LaTeX2e}
\ProvidesPackage{exercises}[2009/08/06 LaTeX version of exercise macros from the TeXbook]
%%%%%%
% Options: number within either the chapter or the part
%          default is to number the exercises/answers via sections
%%%%%%%%
\newif\if@dump
\@dumpfalse
\newif\if@section
\newif\if@ch@pter
\newif\if@p@rt
\@sectiontrue\@ch@pterfalse\@p@rtfalse
\DeclareOption{chapter}{\@sectionfalse\@ch@pterfalse\@p@rtfalse}
\DeclareOption{part}{}
\def\dump@nswer{0}
\DeclareOption{dump}{\@dumptrue}
\ProcessOptions\relax

\usepackage{manfnt}


\newcounter{sectionCtr}
\newcounter{exno}
\setcounter{exno}{0}


\refstepcounter{sectionCtr}
\if@section
\newcommand{\upd@teCtr}{\ifnum \value{sectionCtr}=\value{section}%
\refstepcounter{exno}\else\setcounter{exno}{1}\setcounter{sectionCtr}{\value{section}}\fi}
\else\if@ch@pter%
\newcommand{\upd@teCtr}{\ifnum \value{sectionCtr}=\value{chapter}
\refstepcounter{exno}\else\setcounter{exno}{1}\setcounter{sectionCtr}{\value{chapter}}\fi}
\else\if@p@rt%
\newcommand{\upd@teCtr}{\ifnum \value{sectionCtr}=\value{part}%
\refstepcounter{exno}\else\setcounter{exno}{1}\setcounter{sectionCtr}{\value{part}}\fi}
\else\@sectiontrue\setcounter{exno}{1}\setcounter{sectionCtr}{\value{section}}
\newcommand{\upd@teCtr}{\ifnum \value{sectionCtr}=\value{section}%
\refstepcounter{exno}\else\setcounter{exno}{1}\setcounter{sectionCtr}{\value{section}}\fi}
\fi\fi\fi

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Exercise Environment
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% well, to make it an environment, one would instead do the following:
% \newenviornment{exercise}{\medbreak\upd@teCtr%
%  \noindent\llap{\mantriangleright\kern.15em}% triangle in margin
%  \small{\textbf{EXERCISE \thesection.\arabic{exno}}}\\%
%  \noindent}{}
% that is, stuff the command into an environment
\newcommand{\exercise}{\medbreak\upd@teCtr%
  \noindent\llap{\mantriangleright\kern.15em}% triangle in margin
  \small{\textbf{EXERCISE \thesection.\arabic{exno}}}\\%
  \noindent}
\newcommand{\dexercise}{\medbreak\upd@teCtr%
  \noindent\llap{\mantriangleright\kern.15em}% triangle in margin
  \small{\textbf{EXERCISE \arabic{sectionCtr}.\arabic{exno}}}\\%
  \noindent}
\newcommand{\dangerexercise}{\dbend \dexercise}
\newcommand{\ddangerexercise}{\dbend\dbend \dexercise}


% formatting macro
\if@dump
  \def\ansno#1.#2:{\medbreak\noindent%
    \hbox to\parindent{\bf\hss(Answer to #1.#2)\enspace}\ignorespaces}
\else
  \def\ansno#1.#2:{\medbreak\noindent%
    \hbox to \parindent{\bf\hss #1.#2.\enspace}\ignorespaces}
\fi

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Answers Command
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\if@dump % if we are dumping the answers directly where they're placed
  \def\ans{} % ans is defined as nothing
  \newcommand{\answer}[1]{\par\medbreak % \answer simply
      \ansno\arabic{sectionCtr}.\arabic{exno}: \\% prints directly
      #1} % out when it's called 
  \newcommand{\dumpanswers}{} % \dumpanswers is empty
\else % else we are dumping it into a file
  \newwrite\ans%
  \immediate\openout\ans=answers % file for answers to exercises
  \newcommand{\answer}[1]{\par\medbreak
    \immediate\write\ans{}%
    \immediate\write\ans{\string\ansno\arabic{sectionCtr}.\arabic{exno}:}%
    \immediate\write\ans{ \detokenize{#1}}}
  \newcommand{\dumpanswers}{\immediate\closeout\ans\input{answers}}
\fi

One could easily turn the exercises command into an environment, but it'd be trickier to turn the \answer command into an environment. The \answer spits out all the answers to a file answers.tex which can be included at the end of the main document.

To write the answers, one should call the \dumpanswers command in its own section/chapter. It does everything necessary to include the answers in the document.

The Poor Man's TeXbook Style

\NeedsTeXFormat{LaTeX2e}
\ProvidesPackage{TeXbook}[2009/08/06 Poor man's \TeX{}book style for \LaTeX]

% the following is the TeXbook's exact specifications
%\usepackage[textheight=38pc,headsep=10pc,top=16pc,%
% textwidth=30pc,inner=6pc,marginparwidth=8pc,marginparsep=1cm]{geometry}
% the following is OUR preferred specifications!
\usepackage[top=6pc,textwidth=30pc,inner=6pc,%
marginparwidth=10pc,marginparsep=1cm]{geometry}
\normalbaselineskip=12pt
\baselineskip=12pt
\abovedisplayskip=6pt plus 3pt minus 1pt
\belowdisplayskip=6pt plus 3pt minus 1pt
\abovedisplayshortskip=0pt plus 3pt
\belowdisplayshortskip=4pt plus 3pt

\usepackage{fancyhdr,marginnote}

\pagestyle{fancy}
\if@twoside
\fancyhead[LE,RO]{\thepage}
\fancyheadoffset[OR,EL]{8pc}
\else
\fancyhead[L]{\S\thesection~\nouppercase{\rightmark}}
\fancyhead[R]{\thepage}
\fancyheadoffset[R]{8pc}
\renewcommand{\sectionmark}[1]{\markboth{}{#1}}
\fi
\cfoot{}
\renewcommand{\headrulewidth}{0.4pt}

\if@twoside
\renewcommand\marginpar[1]{\-\marginnote{\footnotesize{\emph{#1}}}}
\else
\renewcommand\marginpar[1]{\-\marginnote{\raggedright\footnotesize{\emph{#1}}}}%
\fi

This has a decent sized margin, the header is also slightly extended. It's really just a poor man's TeXbook style file, using the exact specifications from the manmac.tex macros file that is freely available from CTAN.

Note that the style file is not exactly as the TeXbook specifies, I thought the \headsep was too large. The textwidth and placement from the inner margin are the same, the marginparwidth is slightly larger so one can write annotations in the margins (I love using \marginpar).

I'm writing up some notes on slice and comma categories, as well as adjoint functors, so sit tight while I polish them up in the next couple of days...