Various Math Notes

Alice laughed. "There's no use trying," she said: "one can't believe impossible things."

"I daresay you haven't had much practice," said the Queen. "When I was your age, I always did it for half-an-hour a day. Why, sometimes I've believed as many as six impossible things before breakfast."

Through the Looking-Glass, and What Alice Found There (1871)
by Lewis Carroll (nom de plume for mathematician and logician Charles Lutwidge Dodgson).

"If I had some ham I could have ham and eggs if I had some eggs."

Anonymous Logician

From a Student of Set Theory

Might not a mouse
in iron grip of owl, review
his forest world
in wonder 'midst his fear?

And see his meadow home below,
and tree and stream as new,
and think
"How beautiful from here?"

What songs the Sirens sang, or what name Achilles used when he hid himself among women, though puzzling questions, are not beyond all conjecture.

Sir Thomas Browne, 1658

I have written some notes, placed them in PDF format and collected links to them from this page.

Integration and the Riesz Representation Theorem. Last modified on 10/5/20 at 16:49.

This is a book I have written (maybe 90% finished) on, more or less, Lebesgue Integration.

Actually more than less. It includes a bunch of appendices on interesting but ancillary topics. I guess I just didn't know when to stop. Some of these appendices have been excised and listed as independent pdfs below. The first 150 pages are about measure and integration and the presentation is quite compact, with a lot of material contained therein. Chapter 7 remains to be finished. Chapter 7 is one of those chapters where applications are collected, and it may end at some point but it can never be truly finished. The last 350 pages represent a smattering of interesting facts mentioned here and there in the text. These are standard facts from topology, algebra, set theory and so on, which are referenced in the integration material. A final (short) appendix wrapping up the basic facts about spectral theory for essentially-self-adjoint (unbounded) operators remains to be included.

Except for early material on ordinals and cardinals in Appendix A the appendices do not contain results essential to the purpose of the book, and could be included in greatly "cut-down" form, or even eliminated, without much harm. To give an example, the appendix on Abstract Algebra runs 70 pages or so. The result I really wanted to include here was Stone's Theorem, that every Boolean ring is isomorphic to a ring which consists of a collection of subsets of some set with intersection and symmetric difference as the ring operations. This is a nice result because it says you can always think about Boolean rings in this simple way. About 65 pages of algebra could be cut by simply quoting standard results from a beginning Abstract Algebra course. And though it is a great result I never actually use Stone's Theorem in the text. Possibly I have just made a good case arguing for a large amount of judicious pruning. But there you are, and here it is.

Anyway, I hope this book turns out to be useful for somebody. Enjoy!

Ruminations Concerning Foundational Issues in Modern Mathematics. Last modified on 04/30/10 at 12:34.
Notes on Differential Geometry, Part I. Last modified on 10/8/20 at 14:08.
In these notes I define differentiable manifolds, record necessary facts from Multi-Variable Calculus and talk about many of the topics that can be dealt with using first derivatives and the "locally Euclidean" nature of these manifolds. The exterior derivative, Lie derivative and other entitities which measure how a manifold deviates from "flat" will be discussed in Part II.
A Brief Intro to Number Theory, including the Law of Quadratic Reciprocity and a gesture in the direction of cryptosystems. Last modified on 09/19/23 at 11:27.
Could 1-1+1-1+1-1+...... equal 1/2? Notes on Unconventional Convergence Last modified on 05/1/17 at 07:24.
Some of the Easy Parts of Complex Analysis Last modified on 08/9/16 at 17:07.
Here is an Introduction to the Hyperreal Numbers. Last modified on 06/27/18 at 19:41.
Point-Set Topology. Last modified on 11/20/15 at 07:43.
A Bit of Basic Set Theory. Last modified on 05/30/13 at 21:06.
Notes on Hilbert Spaces and their Operators, and Discussion of More General Linear Spaces Too... Last modified on 04/22/20 at 10:16.
Introductory Topics From Abstract Algebra. Last modified on 04/22/20 at 10:37.
Various Topics Including Metrization, Topological Groups and Uniformities. Last modified on 05/26/13 at 17:16.
A Book Containing Some Basic Facts About Vectors, Including Calculus Using Vectors Last modified on 08/28/05 at 20:21.
A proof of the usual partial fraction decomposition result from Calculus. Last modified on 11/13/15 at 08:03.
Notes on solving 2 by 2 linear DE systems Last modified on 05/11/08 at 21:07.
A few notes on ordinary differential equations (linearization, some nonlinear DEs, Laplace transforms) Last modified on 03/6/06 at 20:46.
Some (unfinished) notes about the shape of cables hanging between anchor posts ... Last modified on 11/25/05 at 20:19.

Here is a link to a complete course on Scientific Computation with daily scripts, tests and so on.

I developed this class and taught it for five or six years for (mainly) undergraduate science and engineering majors. It was generally regarded as a very hard but very useful class. It is organized around MATLAB, which seems to be the software of choice for many Engineering programs. It was really fun (but really hard) to teach after I disabused myself of a few silly notions. It is not a math class, with proofs of all facts presented. And it is not a physics class either. There is wide-ranging math content here, but as a collection of tools; there is plenty of physics here, but physical theory must be developed elsewhere. The point of this class is approximation and modeling ... in many different practical, physical situations. The course content puts phenomenal power in the hands of the students who take it. Pretty great stuff. Last modified on 11/9/20 at 18:55.

Instructions that will help students of a first course in "Classical" Statistics:
Keystrokes for some Stat functions on the TI-83 and TI-84. Last modified on 08/19/20 at 14:17.
How to guess a good sample size for a specified margin of error. Last modified on 08/13/20 at 14:35.
The Monty Hall Problem. Last modified on 08/13/20 at 16:21.
A simple but useful version of Bayes' Theorem. Last modified on 08/13/20 at 16:21.
WORLD MEDICAL ASSOCIATION DECLARATION OF HELSINKI Ethical Principles for Medical Research Involving Human Subjects Last modified on 08/13/20 at 16:23.

In fact the notion of truth a la Tarski avoids complete triviality by the use of the magical expression "meta." We presuppose the existence of a meta-world, in which logical operations already make sense. The world of discourse can therefore be interpreted in the meta-world. The truth value of "A" is determined by "meta-A," and we can in turn explain "meta-A" by "meta-meta-A" ...

We are facing a transcendental explanation of logic: "The rules of logic have been given to us by Tarski, who in turn got them from Mr. Meta-tarski." This is similar to asserting that "Physical particles act this way because they must obey the laws of physics. "

paraphrased from a note of Jean-Yves Girard

I was looking something up in Ronald Brown's "Topology and Groupoids" and he quoted the character Theseus in Shakespeare's "A Midsummer Night's Dream" commenting on the role of Poets:

The Poet's Eye in a Fine Frenzy Rolling
Doth Glance From Heaven to Earth, From Earth to Heaven,
And as Imagination Bodies Forth the Forms of Things Unknown
The Poet's Pen Turns Them to Shapes, and Gives to Airy Nothing
A Local Habitation and a Name.

Brown remarks that the verse could apply to mathematicians and their creations as well as to poets.

And now, of course, I'm quoting Brown.

The Axiom of Choice (google it, if you wish) is a foundational mathematical assumption used explicitly or implicitly by most modern mathematicians. Many of the most intricate triumphs of recent mathematics depend upon it: in some cases the standard proofs fail without the axiom. In (many) other cases the result is known to be unprovable without AC. It has numerous surprising consequences and alternative formulations. The Well-Ordering Principle and Zorn's Lemma are two of these. The following is a joke, and most mathematicians think it is really funny. The "dry humor" type of funny, but still quite good. It is a quote from mathematician Jerry Boma.

The Axiom of Choice is obviously true, the Well-Ordering Principle obviously false, and who can tell about Zorn's Lemma?

Here is another joke on more or less the same topic containing a hint of irony. Or perhaps that's not quite right. Anyway Frechet, Tarski and Lebesgue are titans of early 20th century mathematics. Polish-American mathematician Jan Mycielski relates this anecdote in a 2006 article in the Notices of the AMS.

Tarski tried to publish his theorem (the equivalence between AC and "every infinite set A has the same cardinality as AxA") in Comptes Rendus, but Frechet and Lebesgue refused to present it. Frechet wrote that an implication between two well known (true) propositions is not a new result, and Lebesgue wrote that an implication between two false propositions is of no interest.

Meditating upon the ancient Latin truism "Audentis Fortuna Iuvat 1," and the excerpt from a Jack Vance novel found below, I have decided to set up a bucket at the back of class each day. After five minutes I will examine the contents of this bucket, and proceed with the remainder of the discussion as prompted by the relevant muse, and the goddess Fortuna.
"What are your fees?" inquired Guyal cautiously.

"I respond to three questions," stated the augur.

"For twenty terces I phrase the answer in clear and actionable language;

for ten I use the language of cant, which occasionally admits of ambiguity;

for five, I speak a parable which you must interpret as you will;

and for one terce, I babble in an unknown tongue."

             Jack Vance, Tales of the Dying Earth

1. [Virgil Aeneid X.284]   Back

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This page was last modified on 06/12/22 at 09:03.