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Topic: What does one call vector geometry without a coordinate system?
Replies: 32   Last Post: Sep 9, 2013 4:45 PM

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 fom Posts: 1,968 Registered: 12/4/12
Re: What does one call vector geometry without a coordinate system?
Posted: Sep 3, 2013 9:24 PM

On 9/3/2013 8:15 AM, Shmuel (Seymour J.) Metz wrote:
> In <T6CdnWc136RgA7nPnZ2dnUVZ_gCdnZ2d@giganews.com>, on 09/02/2013
> at 09:21 AM, fom <fomJUNK@nyms.net> said:
>

>> My general background had been in pure mathematics
>> and the role for the classical groups in relation
>> to geometric situations had never been explained
>> clearly.

>
> I'm surprised, given <http://en.wikipedia.org/wiki/Erlangen_program>.
> Surely if yoou studied fibre bundles you've encountered [S]O(n) and
> [S]U(n) in a geometrical context.
>

Beyond calculus, my formal mathematical education
consists of

10 wks Linear Algebra (textbook by Gilbert Strang)

30 wks Abstract Algebra (textbook by I. N. Herstein)

30 wks Real Analysis in R^n (textbook by Jerrold Marsden)

10 wks Complex Analysis (textbook by William Derrick)

10 wks Numerical Analysis (textbook by S. D. Conte and Carl de Boor)

10 wks Mathematical Logic (textbook by Herbert Enderton)

I had started courses in set theory, differential equations,
and topology. It had been at that time that I became ill, and,
simultaneously, it had been at that time when I came to have
difficulties because of a fascination with the continuum
question.

All Enderton really says about identity is that it is a
"logical symbol". To see that identity is syntactically
eliminable with respect to the axiom of extensionality
introduces Leibniz' identity of indiscernibles, although
that principle is not part of the standard account. It
is just unfortunate that I could not have reconciled the
facts of the matter in easier ways.

As for any mathematics other than that above, it is what
I have been able to acquire through personal reading from
books bought, primarily, from used book stores. Because
more recent needs have been directed at specific subjects,
I have had to make specific purchases. Lesniewski's
collected works was probably the worst. I checked prices
for four years before finally conceding to the fact that
it would always be expensive.

Ten years ago I found these newsgroups and hoped to get
a little input and help. Unfortunately, George Greene
saw my posts and immediately started a flame. It had
my mathematical knowledge and work on my set theoretic
ideas, I knew nothing of the philosophical views from
which those attacks had been launched.

Well, that deficiency, at least, is resolved.

It is difficult to teach yourself mathematics when
one has no one to help guide one's efforts or to
whom one may turn with questions. And, without companions
or colleagues with whom to speak of these matters, it
is difficult to gain a fluency and a confidence with the
subject.

Oh well. I do my best.