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.

Synopsis of discussions in a Linear Algebra class (Winter 07) Last modified on 01/27/07 at 06:12.
Review problems with solutions for the final for a Linear Algebra class (Winter 06) Last modified on 03/30/06 at 10:51.
A Book Containing Some Basic Facts About Vectors, Including Calculus Using Vectors, in PDF format Last modified on 08/28/05 at 08:21.
Abbreviated Version of the Vector Notes for Trigonometry Last modified on 08/28/05 at 08:22.
Some notes about the shape of cables hanging between anchor posts ... Last modified on 11/25/05 at 08:19.
Here is an Introduction, in PDF Format, to Tensors. Last modified on 10/15/07 at 04:56.
Here is an Introduction, in PDF Format, to the Hyperreal Numbers. Last modified on 07/8/07 at 02:10.
Here are some notes in PDF format from my last Differential Equations class. Last modified on 03/6/06 at 08:46.
More DE class notes. These are on linear systems in two variables. Last modified on 06/7/08 at 09:14.
Ruminations Concerning Foundational Issues in Modern Mathematics. Last modified on 05/30/09 at 05:05.
Introductory Topics From Abstract Algebra. Last modified on 08/7/07 at 02:46.
Point-Set Topology. Last modified on 05/30/09 at 05:05.
A Bit of Basic Set Theory. Last modified on 08/7/07 at 02:46.
Various Topics Including Metrization, Topological Groups and Uniformities. Last modified on 08/7/07 at 02:46.

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.

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This page was last modified on 11/3/09 at 08:32.
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