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?"



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

Ruminations Concerning Foundational Issues in Modern Mathematics. Last modified on 04/30/10 at 12:34.
Introductory Topics From Abstract Algebra. Last modified on 10/16/17 at 13:42.
Here is an Introduction to Tensors. Last modified on 03/19/17 at 12:50.
Notes on Differential Geometry, Part I. Last modified on 02/26/17 at 13:34.
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 the "locally flat" 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.
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.
Notes on the Discrete Fourier Transform Last modified on 02/4/13 at 19:39.
Quaternions and Rotations Last modified on 10/11/17 at 11:08.
A Very Brief Intro to Number Theory Last modified on 02/3/11 at 16:24.
A proof of the usual partial fraction decomposition result from Calculus. Last modified on 11/13/15 at 08:03.
Overview of (Mostly) Real Linear Algebra Last modified on 11/2/17 at 13:44.
A Book Containing Some Basic Facts About Vectors, Including Calculus Using Vectors Last modified on 08/28/05 at 20:21.
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.
Abbreviated Version of the Vector Notes for Trigonometry Last modified on 08/28/05 at 20:22.
Some (unfinished) notes about the shape of cables hanging between anchor posts ... Last modified on 11/25/05 at 20:19.
Here is an Introduction to the Hyperreal Numbers. Last modified on 01/6/14 at 12:13.
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.
Various Topics Including Metrization, Topological Groups and Uniformities. Last modified on 05/26/13 at 17:16.


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 "Audentes fortuna adiuvat," 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



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