Thursday, December 29, 2011

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).

Saturday, December 24, 2011

My Differential Geometry Images

So I am writing some notes on differential geometry, and using metapost for tricky diagrams. Here are a few of the tricky diagrams I have.

The first three images are:

The third image is a tad bit bigger:

Note the fonts in this picture are messed up, since I had to use postscript fonts. It looks far more beautiful in TeX using metapost, trust me!

numeric u;
color yellow;
u := 1pc;
yellow = red+green;

verbatimtex \input amssym.tex etex;

% sphere
beginfig(0)
  draw fullcircle scaled 4u;
  draw (-2u,0)..(0,-.7u)..(2u,0);
  draw (-2u,0)..(0,.7u)..(2u,0) dashed evenly;
endfig;

% plane + cylinder
beginfig(1)
  draw (0,-u)--(0,3u)--(u,4u)--(u,0)--cycle;

  draw (3u,0)..(4u,-.5u)..(5u,0);
  draw (3u,0)..(4u,.5u)..(5u,0) dashed evenly;

  draw (3u,4u)..(4u,4.5u)..(5u,4u);
  draw (5u,4u)..(4u,3.5u)..(3u,4u);

  draw (3u,0)--(3u,4u);
  draw (5u,0)--(5u,4u);
endfig;

% saddle
beginfig(2)
  path p[];
  z[0] = (2u,4u);
  z[1] = (2.5u,3u);
  z[2] = (1.5u,2.5u);
  z[3] = (4u,0u);
  z[4] = (6.5u,u);
  z[5] = (7.u,1.75u);
  z[6] = (6.75u,0.5u);

  p[0] = z[0]..z[1]..z[2];
  p[1] = z[1]..z[3]..z[4];
  p[2] = z[5]..z[4]..z[6];

  p[3] = z[0]--(point 0.75*length(p[1]) of p[0]);
  p[4] = z[2]--(1.75u,-2u);
  p[5] = z[6]--(6.5u,-2u);
  p[6] = z[5]--(point 0.9*length(p[1]) of p[2]);
  for i=0 upto 6:
    draw p[i];
  endfor;

  draw (4u,-u){up}..(point 0.6*length(p1) of p1);
  draw (point 0.6*length(p1) of p1)..{down}(5.125u,-.75u) dashed evenly;
  draw (1.75u,-2u)..(4u,-u)..(6.5u,-2u);

  p7 = ((0,0)--(2u*unitvector(direction 0 of p[6]))) shifted point (length p6) of p[6];
  draw p[7] dashed evenly;

  p8 = ((0,0)--(3.5u*unitvector(direction 0 of p3))) shifted point (length p3) of p3;
  draw p8 dashed evenly;

  p9 = (point (length p8) of p8)..(5.125u,-.75u)..(point (length p7) of p7);
  draw p9 dashed evenly;

endfig;

beginfig(3)
  picture Rn;
  picture sphere;
  picture torus;
  % R^n
  Rn = image(
      for i=1 upto 3:
      draw (-.5u,i*u)--(3.5u,i*u) withcolor 0.75white;
      draw (i*u,-.5u)--(i*u,3.5u) withcolor 0.75white;
    endfor;
    drawdblarrow (-.5u,0)--(3.5u,0);
    drawdblarrow (0,-.5u)--(0,3.5u);
    );
  % sphere  
  sphere = image( 
      draw fullcircle scaled 4u;
    draw ((-2u,0)..(0,-.7u)..(2u,0));
    draw ((-2u,0)..(0,.7u)..(2u,0)) dashed evenly;
    );
  % torus
  torus = image(
      path hole;
    path uhole;
      draw
      (0,2u)..(-2u,0)..(0,-2u)..(2u,-u)..(4u,-2u)..(6u,0)..(4u,2u)..(2u,u)..cycle;
    hole = halfcircle rotated 180 scaled 2u shifted (4.5u,0);
    draw hole;
    uhole = (point 0.1*length(hole) of hole)
     ..(0.5[(point 0.1*length(hole) of hole),(point 0.9*length(hole) of hole)]+(0,.5u))
     ..(point 0.9*length(hole) of hole);
    draw uhole; 
    );
  
  draw Rn;
  draw Rn shifted (20u,-5u);

  draw torus shifted (8u,4u);

  path localPatch;
  path imageOfF;
  path preimageOfG;
  path imageOfG;
  path imageOfPsi;
  path intersection;
  localPatch = (fullcircle scaled 2u shifted (1.5u,1.5u));
  fill localPatch withcolor 0.75[blue,white];
  draw localPatch dashed evenly;

  z[0] = (((u,0)--(u,2u)) intersectionpoint localPatch);
  draw (((0,u)--(u,u)) intersectionpoint localPatch)
  --(((2u,u)--(4u,u)) intersectionpoint localPatch) withcolor 0.75[black,blue];
  draw (((0,2u)--(2u,2u)) intersectionpoint localPatch)
  --(((2u,2u)--(4u,2u)) intersectionpoint localPatch) withcolor 0.75[black,blue];
  draw (((u,0)--(u,2u)) intersectionpoint localPatch)
  --(((u,2u)--(u,4u)) intersectionpoint localPatch) withcolor 0.75[black,blue];
  draw (((2u,0)--(2u,2u)) intersectionpoint localPatch)
  --(((2u,2u)--(2u,4u)) intersectionpoint localPatch) withcolor 0.75[black,blue];

  imageOfF = fullcircle xscaled 3u yscaled 1.5u shifted (8u,4u);
  fill imageOfF withcolor 0.75[blue,white];
  draw imageOfF dashed evenly;

  picture preG;

  preG = image(
  
  preimageOfG = unitsquare scaled 2u rotated 45 shifted (22u,-5u);

    fill preimageOfG withcolor 0.5[yellow,white];
    fill fullcircle xscaled u yscaled 5u rotated -23 shifted (22u,-3u)
      withcolor 0.5[green,white];
      draw (((21u,-4u)--(21u,-5u)) intersectionpoint preimageOfG)
    --(((21u,-5u)--(21u,-3u)) intersectionpoint preimageOfG)
    withcolor 0.5[black,yellow];
  draw (((22u,-2u)--(22u,-3u)) intersectionpoint preimageOfG)
    --(((22u,-4u)--(22u,-8u)) intersectionpoint preimageOfG)
    withcolor 0.5[black,yellow];
  draw (((23u,-3u)--(23u,-3.5u)) intersectionpoint preimageOfG)
    --(((23u,-3.5u)--(23u,-8u)) intersectionpoint preimageOfG)
    withcolor 0.5[black,yellow];
  draw (((21u,-4u)--(22u,-4u)) intersectionpoint preimageOfG)
    --(((22u,-4u)--(25u,-4u)) intersectionpoint preimageOfG)
    withcolor 0.5[black,yellow];
  draw (((21u,-3u)--(22u,-3u)) intersectionpoint preimageOfG)
    --(((22u,-3u)--(25u,-3u)) intersectionpoint preimageOfG)
    withcolor 0.5[black,yellow];
  draw preimageOfG dashed evenly;

  clip currentpicture to preimageOfG;
    );
  draw preG;

  imageOfG = (19u,2.5u)--(20u,3u)--(21.5u,3u)--(21u,2.5u)--cycle;


  fill imageOfG withcolor 0.5[white,yellow];

  imageOfPsi = fullcircle xscaled 2u yscaled u rotated -45
    shifted (20u,3u);
  fill imageOfPsi withcolor 0.75[blue,white];


  numeric t[];
  (t[0],whatever) = imageOfPsi intersectiontimes ((19u,2.5u)--(20u,3u));
  (t[1],whatever) = imageOfPsi intersectiontimes ((21u,2.5u)--(19u,2.5u));
  (t[2],whatever) = imageOfPsi intersectiontimes ((20u,3u)--(23u,3u));  
  (t[3],whatever) = imageOfPsi intersectiontimes ((19u,2.5u)--(20u,2.5u));
  
  intersection = (subpath(t0,t3) of imageOfPsi)--%(20u,3u)--
    (subpath(t1,t2) of imageOfPsi)--(20u,3u)--cycle;
  
  fill intersection withcolor 0.5[green,white];
  draw imageOfPsi dashed evenly;
  draw imageOfG dashed evenly;
  draw sphere shifted (20u,4u);

  p[0] := (point 0.25*length(localPatch) of localPatch)..(4u,4u)
  ..(point 0.5*length(imageOfF) of imageOfF);
  p[1] := (point 0 of imageOfF)..(12u,5u)..(19.5u,3.5u);
  p[2] := (21.5u,3u)..(23u,0)..(23u,-3u);
  p[3] := (point .75*length(localPatch) of localPatch)..(1.5u,-u)
  ..(10u,-4u)..(21.5u,-4.5u);
  drawarrow p[0];
  
  drawarrow p1;

  drawarrow p2;

  drawarrow p3 dashed evenly;

  label.top(btex $x$ etex, point 0.5*length(p0) of p0);
  label.top(btex $\varphi$ etex, point 0.75*length(p1) of p1);
  label.rt(btex $y^{-1}$ etex, point 0.5*length(p2) of p2);
  label.bot(btex $y^{-1}\circ\varphi\circ x$ etex, point 0.6*length(p3) of p3);

  label(btex $U$ etex, (1.5u,1.5u)) withcolor 0.5[black,blue];

  draw (8u,4u) withpen pencircle scaled 3;
  label.lft(btex $p$ etex, (8u,4u)) withcolor 0.75[blue,black];
  label.bot(btex $M$ etex, (12u,2u));

  label.rt(btex $N$ etex, (22u,4u));

  label.lft(btex $\Bbb{R}^{n}$ etex, (20u,-2u));

  label.lft(btex $\Bbb{R}^{m}$ etex, (0,3u));
endfig;


end;

Thursday, December 22, 2011

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.

Tuesday, December 20, 2011

Ancient Geography

Reading Herodotus requires a greater knowledge of geography than I previously implied.

Fortunately, all the geography you need to know is encoded in a free map. Ironically it comes from a video game titled "Rome: Total Realism".

So, without further ado, here is the map:

Does it seem small? Well, it's a link, so click on it for the huge map.

Sunday, December 18, 2011

Experiments as "registers" in notes

Mathematics is about theorems, science is about experiments. But theorems are expressed rigorously in formal logic...what about experiments?

The "old fashioned" way would be to write up a lab report and carry out the experiment! However, nowadays, there are some programs to simulate experiments.

For example, the KDE Education Project has a few programs for physics, astronomy, and chemistry. Sadly, no biology (although I wouldn't know what that would look like...).

Additionally, if one is writing notes on history, there is a beautiful globe program one could use...

At any rate, for classical physics, this is perhaps the ideal set up. It allows the reader to play with the experiment, which is the entire idea of an experiment(!), and if so possessed one could set it up in real life.

Plus this is free, a small perk.

Physics

There is the Step program which is useful for classical mechanics.

Better, the physical systems being studied can be represented as a program.

And when you get to numerical analysis, you can perform a "How accurate was this?" series of exercises ;)

It is multiplatform, so anyone on Windows, Mac OS X, or *nix can use the KDE Education suite. It may be worth while to consider :)

However, I wonder if there is not a better program out there? The qt framework is always a wee bit heavy duty.

It would be nice if someone invented a scripting language that produced flash videos, or something along those lines...since qt runs slow on my 733MHz Pentium 3 processor :'(

Chemistry

I have not thought about writing up chemistry notes, although I feel if I write biology notes then it would become necessary to study chemistry.

Personally, chemistry seems to be a sibling to physics, it just is concerned with different scales. So experiment is very important in chemistry, and it is just as repeatable as in physics.

The Kalzium suite simulates chemistry to a degree. Everything is always "to a degree"!

Biology?

I am unaware of any similar such program for biology...but there are few experiments which are reproducible in biology!

Perhaps something like "virtual fruit flies" or "virtual peas" would be nice to study genetics, but I think it would be best to do it the "old fashioned way" (i.e., actually carry out the experiment!).

Take Home Message

If you are a mathematician writing notes on science, and are unable to perform the experiments: write a program.

Friday, December 16, 2011

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]

Thursday, December 15, 2011

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 xx, 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!

Tuesday, December 13, 2011

Personal v. Expository notes

Personal Notes Are For One's Self

I recently stumbled across Thinking Mathematically by J. Mason, L. Burton, K. Stacey (Amazon) which has an interesting approach to writing personal mathematical notes.

I say "personal" as opposed to "expository" because they are really personal scratchwork rather than explanations.

What's really cute is Mason, et al., espouse a sort of "markup language" approach (they call it a "rubric"). Let me review this a little.

When "entering" a mathematics problem, there are three things to ask ourselves, which constitute the entry phase:

  1. What am I know? (What is given?)
  2. What do I want?
  3. What can I introduce?
Know, want, introduce. These three things we should always, always, always ask ourselves! Even in long proofs, we should ask ourselves these questions! We should ask ourselves when we get stuck!

With introducing stuff. . . we can do several types of introduction.

Notation:
Assigning values and meanings to variables.
Organization:
Recording and arranging what you know.
Representation:
Choose particular representatives which are easier to manipulate.
Bear in mind that rephrasing the question is particularly useful.

Now, reviewing your work is also critical. There are several different ways of doing this:

Check:
the resolution;
Reflect:
on the key ideas and key moments;
Extend:
to a wider context.
The best way to get the most out of reviewing is to write up your resolution for someone else, in such a way that one can follow what you have done and why.

We can check several things:

  1. Check calculations;
  2. Check arguments to ensure computations are appropriate;
  3. Consequences of conclusions to see if they are reasonable;
  4. Check that the resolution fits the question.
This is sort of subconsciously done.

We also reflect in finitely many ways:

  1. What are the key ideas and key moments?
  2. What are the implications of conjectures and arguments?
  3. Can the resolution be made clearer?
It helps one see things that otherwise would have been missed.

As far as extending, one should really be generalizing the problem. For example: how many squares are on a 3 × 3 chess board? There are 9 instances of 1 × 1 squares, 4 instances of 2 × 2 squares, and a single 3 × 3 square. Thus there are 14 squares altogether. Now, to extend:

  1. How many squares are on an n × n board?
  2. How many rectangles are on a 3 × 3 board? Extend this to n × n boards.
  3. What if we start with an m × n board? How many squares are there in it?
  4. Why work only in two dimensions?
  5. Why count squares with edges parallel to the original?

When stuck, try re-entering the entry phase. This can be done through:

  1. Summarize everything known and wanted;
  2. Rephrase the question in a more appealing way;
  3. Re-read or re-digest the problem.
This is useful sometimes.

Conjecturing is a cyclic four-step procedure:

  1. Articulate a conjecture (and while making it, believe it);
  2. Check the conjecture covers all known cases and examples;
  3. Distrust the conjecture. Try to refute it by finding a nasty case or example; use it to make predictions which can be checked;
  4. Get a sense of why the conjecture is right, or how to modify it, on new examples (go back to step 1).
Note you can start anywhere in this procedure.

It's not too long until one gets to a state where one says "I don't believe it's possible" which leads to the questions

  1. Why can it not be done?
  2. All right, what can be done?
Asking "What can be done" is a critical step in conjecturing.

Now, critical mathematical thinking should be nurtured by thinking three things while doing or reading a proof:

  1. Every statement made should be treated as a conjecture.
  2. Try to defeat and prove conjectures simultaneously.
  3. Look critically at other people's proofs.
There are some mathematical registers that some authors suggest using while writing notes. The collection is sometimes called a rubric:
I Know:
What is given? What is known?
I Want:
What do we want to prove?
Introduce:
Try contributing some:
Notation:
Assigning values and meanings to variables.
Organization:
Recording and arranging what you know.
Representation:
Choose particular representatives which are easier to manipulate.
Stuck!:
"I do not understand...", "I do not know what to do about...", "I cannot see how to...", "I cannot see why..." — try going back to the entry portion "want/know/introduce", or make a conjecture.
AHA!:
Whenever you have a good idea, write it down. Usually, they are of the form: "AHA! Try...", "AHA! Maybe...", or "AHA! But why...".
Check:
the mathematics. This means:
  1. Check calculations;
  2. Check arguments to ensure computations are appropriate;
  3. Consequences of conclusions to see if they are reasonable;
  4. Check that the resolution fits the question, i.e., our answer is the answer to the question asked.
Reflect:
meditate on:
  1. What are the key ideas and key moments?
  2. What are the implications of conjectures and arguments?
  3. Can the resolution be made clearer?
Extend:
generalize to other settings.
The real trick is to change "I'm stuck, panic!" to "I'm stuck, okay, so what can be done about it?"

Monday, December 12, 2011

MathJax, Updates

Updates

I've just updated the design of the blog. I think it looks negligibly better.

I'll also try to write more here rather than on my notebk's wiki.

I posted some exercises for Sean [pdf] which are fun calculus problems.

I suspect what I'll do next is write up my notes on differential geometry from Osserman's course I audited a couple years ago, as well as my notes on algebraic topology (from Dr Schwarz's courses). Then I'll work on spin geometry, "commutative geometry", analysis, and so on.

By "commutative geometry", I really mean spectral triples using commutative rings (Here I am sloppy, but meh I am a sloppy person!). It is also called Differential Calculus over Commutative Algebras, although there are no real texts on the subject...

I'll have to review the prime spectrum of the commutative ring $C(M)$ of continuous functions on a topological space $M$ and how it relates to the topology of $M$.

If we let $M$ be a smooth manifold, then we work with $C^{\infty}(M)$ — I am told there is a theorem due to Shields which says if $M$ and $N$ are smooth manifolds and $C^{\infty}(M)$ is isomorphic to $C^{\infty}(N)$ then $M$ and $N$ are diffeomorphic. How interesting! But I cannot find this theorem...

At any rate, vector bundles over $M$ may be considered by looking at the projective modules over $C^{\infty}(M)$.

We consider algebraic analogs for sections, vector fields, covector fields, and so on. It is really quite cute.

Noncommutative geometry is similar in setting up a dictionary between "algebraic stuff" and "geometric stuff", at least how Connes approaches it. It's just that the "geometric stuff" we work with is a smooth Riemannian manifold $M$ equipped with a spin structure, we consider spin bundles over it, and so on.

MathJax

I am experimenting with MathJax on blogger, so bear with me people.

My reference for this subject is the thread at stackexchange on it.

Consider the "Harmonic Series" \[ \sum^{\infty}_{n=1}\frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\cdots \] which diverges famously.

MathJax uses the \(...\) or $...$ for "inline mathematics" and \[...\] or $$...$$ for "display math", e.g., the mathematics produced above.

I don't know whether to keep it or not, because MathJax is sluggish on some computers. But it is the "way of the future", like blimps and autogyros.

Saturday, November 12, 2011

Some more References

I found a number of free references on various subjects in mathematical physics, available free, legally, online.

Mathematics

Partial Differential Equations

  1. Alexander Komech, and Andrew Komech,
    "Book of Practical PDEs."
    Eprint Lecture note 33/2007, 125 pages.
  2. Alexander Komech,
    "Lectures on elliptic partial differential equations (Pseudodifferential operator approach)".
    Eprint Lecture note 32/2007, 44 pages.
  3. Norbert Ortner, and Peter Wagner,
    "Distribution-Valued Analytic Functions - Theory and Applications".
    Eprint Lecture note 37/2008, 133 pages.
  4. A.D.R. Choudary, Saima Parveen, Constantin Varsan,
    "Partial Differential Equations An Introduction".
    Eprint arXiv:1004.2134v1 [math.AP], 204 pages.
  5. Robert Geroch,
    "Partial Differential Equations of Physics".
    Eprint arXiv:gr-qc/9602055v1, 57 pages.

Differential Geometry

  1. Jürgen Jost,
    "The principles and concepts of geometric analysis".
    Eprint Lecture note 12/2001, 15 pages.
    Note: "geometric analysis" studies things like Morse functions, etc.

Homological Algebra

  1. Volker Runde,
    "Abstract harmonic analysis, homological algebra, and operator spaces".
    Eprint arXiv:math/0206041v6 [math.FA], 12 pages.
  2. Mohamed Barakat, Markus Lange-Hegermann,
    "An Axiomatic Setup for Algorithmic Homological Algebra and an Alternative Approach to Localization".
    Eprint arXiv:1003.1943v3 [math.AC], 30 pages.
  3. Joseph Krasil'shchik, Alexander Verbovetsky,
    "Homological Methods in Equations of Mathematical Physics".
    Eprint arXiv:math/9808130v2 [math.DG], 150 pages.
  4. R. P. Thomas,
    "Derived categories for the working mathematician".
    Eprint arXiv:math/0001045v2 [math.AG], 13 pages.
  5. Semen Podkorytov,
    "On homology of map spaces".
    Eprint arXiv:1102.1645v1 [math.AT], 9 pages.

K-Theory

  1. Max Karoubi,
    "K-theory. An elementary introduction".
    Eprint arXiv:math/0602082v1 [math.KT], 22 pages
  2. Ioannis P. Zois,
    "18 Lectures on K-Theory".
    Eprint arXiv:1008.1346v1 [math.KT], 137 pages.

Mathematical Physics

This is kind of physics stuff, kind of mathematics stuff, and doesn't really fit cleanly into either category.
  1. Andreas Knauf,
    "Number theory, dynamical systems and statistical mechanics".
    Eprint: Lecture note 3/1998, 41 pages.
  2. Bruce Hunt,
    "Geometry of super Yang-Mills and supergravity".
    Eprint: Lecture note 4/1999, 59 pages.
  3. Bruce Hunt,
    "Conformal and gauge symmetry in D = 2 QFT".
    Eprint: Lecture note 5/1999, 115 pages.
  4. Friedemann Brandt,
    "Lectures on Supergravity".
    Eprint: Lecture note 13/2002, 50 pages.
  5. Alexander Komech,
    "Lectures on Quantum Mechanics (nonlinear PDEs point of view)".
    Eprint: Lecture note 25/2005, 231 pages.

Physics

  1. Christopher J. Fewster,
    "Lectures on quantum field theory in curved spacetime".
    Eprint Lecture note 39/2008, 62 pages.

Tuesday, November 1, 2011

Quantifiers

I have been thinking about quantifiers, and the notation around them...since most books use differing notation that is not easy to read.

Since I am trying to be consistent throughout all of mathematics, it seems natural to suggest that the colon ":" should be read as "such that".

In this case, one should write "∃x : P(x)" since one usually writes in natural language "There is some x such that P(x)".

Likewise we often find expressions "If xX, then P(x)"...so it would be natural to use the notation "∀x∈X, P(x)" or "∀ x, x∈X and P(x)". After all, "x∈ X" is really a predicate "isIn(x,X)"...

I'm currently revising my notes on logic in Fascicles 0 of my Elements of Mathematics, which is why I bring this notational problem up! Hopefully, I will be done with revising and improving my logic chapter soon.

Sunday, October 30, 2011

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.

Friday, October 28, 2011

Epsilon Calculus

So, this is a post consisting of notes for myself on Hilbert's ε-calculus...specifically with regards to Bourbaki's use of it.

What is ε Calculus?

We work with first order languages. So, we have predicates, terms, etc.

Let A be a predicate. Then we interpret εx A as "some x that satisfies A".

We could look at it as returning a term t which satisfies A, or if no such term exists it just returns any term for which A is false.

This seems like a pain to program up, since "return any term which satisfies such-and-such, non-deterministically" is...weird!

How did Bourbaki use it?

Well, formally, Bourbaki used a slightly different notation, and the Internet Encylopedia of Philosophy notes [utm.edu] that "Bourbaki's epsilon calculus with identity (Bourbaki, 1954, Book 1) is axiomatic, with Modus Ponens as the only primitive inference or derivation rule."

This reminds me of the simple propositional calculus Mendelson introduces in chapter 1 of Introduction to Mathematical Logic (fourth ed.). If you are unfamiliar with it, wikipedia [wikipedia.org] has a review of the system. It has one rule of inference (modus ponens) and 3 axioms.

Effectively, the ε operator acts as a choice operator, which means that the set theory framework set up by Bourbaki automatically has the axiom of choice induced by this ε operator.

Moreover, this is a global choice operator, so Bourbaki's axioms are equivalent to something like Zermelo set theory + Axiom of Global Choice.

Aside: Bourbaki Set Theory

I want to discuss in some detail the axioms of Bourbaki's set theory...because it is not ever discussed anywhere.

There is (II §1.4, pg 67 of Theory of Sets) the first axiom, "the axiom of extent":

A1. (∀ x)(∀ y)((x⊂y and y⊂x) implies (x=y)).

This is the axiom of extensionality we all know and love that defines set equality as "two sets are equal if and only if they have the same elements".

The next axiom is what we would call the axiom of pairing, and Bourbaki calls it the "the axiom of the set of two elements" (Theory of Sets §1.5, pg 69):

A2. (∀ x)(∀ y) Collz(z=x or z=y)

"This axiom says that if x and y are objects, then there is a set whose only elements are x and y" (Theory of Sets II §1.5, pg 69).

In modern set theory, this is the axiom of pairing which says for all x and y there exists a z such that z={x,y}. This is a little bit sloppy, but that's the content of the axiom.

Next, we have the axiom of the ordered pair (Theory of Sets II §2.1, pg 72):

(∀ x)(∀ x')(∀ y) (∀ y') (((x,y) = (x',y')) implies (x=x' and y=y'))

Nothing controversial here, just the definition of an ordered pair.

There are two axioms left. There is the axiom of the set of subsets (Theory of Sets II §5.1, pg 101):

(∀ X) CollY (Y⊂X)

This amounts to specifying that the power set of any set X exists.

Now the last axiom for Bourbaki's set theory is the axiom of infinity (Theory of Sets III §6.1, pg 183):

A5. There exists an infinite set.

That's it! That's everything! As homework exercise, prove it's formally equivalent to Zermelo's axioms...

Now, what about the axiom of choice? Well, actually, Bourbaki has it covered:

Theorem 1 (Zermelo). Every set E may be well-ordered.

The proof (pp. 153--54) uses, yep you guessed it, the ε choice operator. Basically, take the power set, then throw away the element E in the power set. For each element in this collection of proper subsets, choose an element using ε-calculus.

This induces an ordering of elements because the power set is a complete lattice (with respect to the ordering given by inclusion). Thus we obtain an ordering on the set, and it follows any (nonempty?) set can be well-ordered.

This is equivalent to the axiom of choice. If we allow the scheme of ε extensionality:

∀x ((A(x) if and only if B(x)) implies (εx A = εx B))

What happens? As a consequence to this, we get the Axiom of Global Choice [wikipedia.org].

In Automated Theorem Proving

Martin Giese , Wolfgang Ahrendt's "Hilbert's epsilon-Terms in Automated Theorem Proving" (eprint [citeseerx.ist.psu.edu]) discusses the role the epsilon calculus has had in automated theorem proving.

Tuesday, October 18, 2011

Proclus on Euclid

Proclus wrote some commentaries on Euclid's Elements and found a pattern to the presentation of material:

Every Problem and every Theorem that is furnished with all its parts should contain the following elements: an enunciation, an exposition, a specification, a construction, a proof, and a conclusion. Of these enunciation states what is given and what is being sought from it, a perfect enunciation consists of both these parts. The exposition takes separately what is given and prepares it in advance for use in the investigation. The specification takes separately the thing that is sought and makes clear precisely what it is. The construction adds what is lacking in the given for finding what is sought. The proof draws the proposed inference by reasoning scientifically from the propositions that have been admitted. The conclusion reverts to the enunciation, confirming what has been proved.

Lincoln used this approach in his rhetoric, as Hirsch & Van Haften notes in their book Abraham Lincoln and the Structure of Reason. In an interview, they explained each step.

For the enunciation, think in terms of: Why are we here. It contains short, indisputable facts. They are part of the given. It also includes a sought. This is a high level statement of the general issue being discussed.

For the exposition, think in terms of: What do we need to know relating to what is given. These are additional facts, generally fairly simple, and indisputable. These facts take what was in the enunciation’s given, and prepare for use in the investigation (in the construction).

For the specification, think: What are we trying to prove. The specification is a more direct restatement of the enunciation’s sought. While the sought is frequently neutrally stated, the specification is a direct statement of the proposition to be proved.

For the construction, think: How do the facts lead to what is sought. The construction adds what is lacking in the given for finding what is sought.

For the proof, think in terms of: How does the admitted truth confirm the proposed inference. The proof draws the proposed inference by reasoning scientifically from the propositions that have been admitted.

For the conclusion, think: What has been proved. The conclusion reverts back to the enunciation confirming what has been proved. The conclusion should be straightforward, forceful, and generally short.

It might be useful if anyone ever goes into mathematics...or Law...

Addendum: Modern Mathematics

It seems that this format can be used in modern mathematics. Andrei Rodin's "Doing and Showing" (arXiv:1109.4298 [math.HO]) notes on page 25 how a modern theorem/proof can be formulated in the Euclidean tradition:

Theorem 3:
Any closed subset of a compact space is compact

Proof:
Let F be a closed subset of compact space T and {Fα} be an arbitrary centered system of closed subsets of subspace FT. Then every Fα is also closed in T, and hence {Fα} is a centered system of closed sets in T. Therefore ∩Fα = ∅. By Theorem 1 it follows that F is compact.

Although the above theorem is presented in the usual for today's mathematics form "proposition-proof", its Euclidean structure can be made explicit without re-interpretations and paraphrasing:

[enunciation:]
Any closed subset of a compact space is compact

[exposition:]
Let F be a closed subset of compact space T

[specification: absent].

[construction:]
[Let] {Fα} [be] an arbitrary centered system of closed subsets of subspace FT.

[proof :] [E]very Fα is also closed in T, and hence {Fα} is a centered system of closed sets in T. Therefore ∩Fα=∅. By Theorem 1 it follows that F is compact.

[conclusion: absent ].

The absent specification can be formulated as follows:

"I say that F is a compact space"

while the absent conclusion is supposed to be a literal repetition of the enunciation of this theorem.

Sunday, October 16, 2011

Role of Examples

I have noticed authors use examples in math texts in two ways: demonstrating how to perform an algorithm, or silently giving an exercise to the reader.

An instance of the "demonstration" type example would be found in any calculus textbook that goes through a calculation step-by-step detailing what is happening, etc.

The "exercise" type of example is found in most upper division and graduate texts. It gives enough information to say "Well, this gadget is an example of a group/ring/field/measure/space/etc." but fails to prove it.

So which type of example is appropriate for mathematical writing?

Well, it depends on the writing. I'm writing so I can remind myself of some field or concept, so I can learn it all over again in 5 minutes.

With this sort of writing, the "demonstrations" are more appropriate.

However, in monographs the latter seems conventional. Regarding monographs I'd like to quote Serge Lang:

I have not written this course in the style I would use for an advanced monograph, on sophisticated topics. One writes an advanced monograph for oneself, because one wants to give permanent form to one's vision of some beautiful part of mathematics, not otherwise accessible, somewhat in the manner of a composer setting down his symphony in musical notation. [Emphasis added]

From the Preface to A First Course in Calculus by Lang.

Maybe there is a happy medium, I don't know...

Friday, October 14, 2011

Index Cards

I have been using index cards when studying math, at least now.

It's really useful to write down definitions and theorems on one side (one definition/theorem per card) and explanations of why it's useful on the other side (possibly sketching the proof).

It's been really useful while reading through Kirillov's Lectures on tensor categories and modular functor (eprint [math.sunysb.edu]).

Gel'fand's Generalized Functions reads like a dinner table conversation between mathematicians, without formal definitions in the grocery-list Bourbakist manner. Index cards help out a lot here, enabling you to write down where references are (which was, coincidentally, why they were invented in the first place!).

It's also wonderful when you begin to write a book, you can just collate the index cards of theorems and definitions. Then writing is just a matter of inserting literary "glue" between already existing material.

One can form a wiki [http://takingnotenow.blogspot.com] using index cards.

But instead I prefer to organize notes [studygs.net] slightly differently. It also enables me to "automatically" cite by keeping track of the source in the upper right corner, and label it as a definition or a theorem in the upper left corner.

Plus I am not married to any collation in this scheme. On the other hand, I have to look through all my notes to get to various definitions --- there is no organization to it!

Just more stuff to ponder...

Monday, September 19, 2011

Bourbaki and Sets

Don't worry too much about Bourbaki and Set Theory...Bourbaki didn't really use first order logic, anyways.

At any rate, there was an excellent discussion of the formalism discussed in Bourbaki's Theory of Sets over at the nForum a while back (I don't know how I missed it!).

Someone noted that Bourbaki's set theory is "equivalent" to ZGC (Zarmelo with the Axiom of Global Choice).

Addendum: for more details on Bourbaki's set theory, see the entry on Hilbert's Epsilon Calculus.

Wikipedia claims that Zarmelo set theory is "strong enough to carry out almost all ordinary mathematics not directly connected with set theory or logic" (Wikipedia).

Although, it should be noted, later editions of Bourbaki included the axiom of replacement...making it resemble ZFC more than ZGC (The Bourbaki View, fn 15).

Implications for Automated Theorem Provers

I got my hands ahold of a wonderful book:

  • John Harrison, Handbook of Practical Logic and Automated Reasoning. Cambridge University Press (2009).

Why is it so wonderful? Well, in chapter 6 it discusses Mizar-like declarative proof systems. In fact, the code accompanying the book actually implements a simplified version of it.

It wouldn't be terribly difficult to change the wording to use the Mathematical Vernacular.

Additionally, since the programming language is OCaml, one could take advantage of Camlp4 to extend the grammar to make the syntax prettier.

But I digress.

If one were to implement mathematics formally, one couldn't use ZGC without having problems covering topology (or category theory — large categories are defined how?).

However, if one were to take the conservative extension to ZFC, one would still run into problems (again, large categories?). What one would have to do is use BNG Set Theory.

So for any budding cyber-Bourbaki groups out there: you can't use ZGC, ZFC, but you must use BNG (which is not too bad!).

Although, the more I think about it, the more it doesn't matter as long as the behaviour of sets are "sufficiently nice". The implementation details don't matter for, e.g., the ordered pair...just the fact that (x,y)=(a,b) if and only if x=a and y=b. But now I'm just rambling!

Aside on Axioms and Specifications

One thing I always wondered about: why not use naive specifications instead of constructions (e.g., the ordered pair (x,y) as {{x,∅},{y}} or a million other different ways) or rigid axioms?

The answer is that a specification is axiomatic. One specifies the axiomatic behaviour through specifications, it would be logically equivalent to giving an axiomatic framework.

Addendum

To any budding cyber-Bourbakites out there, you will be relieved to find out that there exists an automated proof checker, the "Zermelo Proof Checker", which is based on the BNG axioms for set theory and some minor type theory.

If, like me, you use older hardware...you will run into problems when using zprover. Namely, eprover will always time out, saying something like "CPU time limit exceeded".

Well, zprover is a script that calls eprover and has a parameter that can be (and should be) changed on line 10:

TIME_LIMIT=5 # in seconds

Change it to 60, or something greater.