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...