-------- Original Message --------

On Wed, 1 Nov 2023 13:13:12 -0000 (UTC), Don <g@crcomp.net> wrote:

pete wrote:

Robert Carnegie wrote:

Dimensional Traveler wrote:

On 10/31/2023 9:39 AM, Don wrote:

Michael wrote:

Joel Polowin wrote:

Ted wrote:

That's not the way the Liaden old universe worked. For instance, the >>>>>>>>> value of Pi was not the same, setting aside a completely different set

of stars & planets.

Given that pi can be calculated in multiple ways as the sum of mathematical
converging infinite series, it's hard to see how that could be... Archie
Plutonium's "theories" notwithstanding. Of course, that's the mathematical
constant, which can differ from the _physical_ value (e.g. the ratio between
the circumference and diameter of a circle) depending on the curvature of
space.

Of course, in anything other than flat, Euclidean space, the ratio >>>>>>> of the circumference of a circle to its diameter is not constant, but >>>>>>> depends on its diameter (at least) and position (in a negatively >>>>>>> curved space, I think).

Allow me to use the groups as an adhoc classroom. Let me know if my pi >>>>>> pertinent philosophy shown below doesn't make sense.
The ratio of a circle's circumference to its diameter is an
observable fact. And accordingly associated with Aristotlean thought. >>>>>> OTOH, notions of non-Euclidean space are Platonic. And non-
Euclidean space Platonically pulls pi apart to the breaking point? >>>>>>

Purely thought experiment until such time as we can observe
non-Euclidean space to conduct actual experiments.

Since Einstein, we're living in non-Euclidean
space. I think it goes as far as planets
moving in "straight lines" that just happen to
become ellipses because spacetime is bendy.

This is correct.

From a purely Platonic perspective perhaps?

quantum mechanics would appear to be in the strange position of
agreeing with all observations made, while disputing that any
observations can actually be made at all.

_Alice in Quantumland_ (Gilmore)

It's insane isn't it?

Even the explanations in /Science News/, particularly of "quantum
pairs" (for some reason), often don't make sense to me. Granted I
never studied quantum physics -- but the articles are supposed to be
written so an educated but not in the topic of the article person can understand the concepts.

Well, for quantum physics, that's "understand the concepts in a
cartoony way", of course.

Richard Feynman said, "Anyone who says they understand quantum physics, doesn't."


--
.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

rick wrote:

Don wrote:

Paul wrote:

pete wrote:

Robert wrote:

Dimensional wrote:

Don wrote:

Michael wrote:

Joel wrote:

Ted wrote:

That's not the way the Liaden old universe worked. For instance, the >>>>>>>>>> value of Pi was not the same, setting aside a completely different se
of stars & planets.

Given that pi can be calculated in multiple ways as the sum of mathema
converging infinite series, it's hard to see how that could be... Arch
Plutonium's "theories" notwithstanding. Of course, that's the mathemat
constant, which can differ from the _physical_ value (e.g. the ratio b
the circumference and diameter of a circle) depending on the curvature
space.

Of course, in anything other than flat, Euclidean space, the ratio >>>>>>>> of the circumference of a circle to its diameter is not constant, but >>>>>>>> depends on its diameter (at least) and position (in a negatively >>>>>>>> curved space, I think).

Allow me to use the groups as an adhoc classroom. Let me know if my pi >>>>>>> pertinent philosophy shown below doesn't make sense.
The ratio of a circle's circumference to its diameter is an
observable fact. And accordingly associated with Aristotlean thought. >>>>>>> OTOH, notions of non-Euclidean space are Platonic. And non-
Euclidean space Platonically pulls pi apart to the breaking point? >>>>>>>

Purely thought experiment until such time as we can observe
non-Euclidean space to conduct actual experiments.

Since Einstein, we're living in non-Euclidean
space. I think it goes as far as planets
moving in "straight lines" that just happen to
become ellipses because spacetime is bendy.

This is correct.

From a purely Platonic perspective perhaps?

quantum mechanics would appear to be in the strange position of
agreeing with all observations made, while disputing that any
observations can actually be made at all.

_Alice in Quantumland_ (Gilmore)

It's insane isn't it?

Even the explanations in /Science News/, particularly of "quantum
pairs" (for some reason), often don't make sense to me. Granted I
never studied quantum physics -- but the articles are supposed to be
written so an educated but not in the topic of the article person can
understand the concepts.

Well, for quantum physics, that's "understand the concepts in a
cartoony way", of course.

Richard Feynman said, "Anyone who says they understand quantum physics, doesn't."

Did Feynman proclaim an uncertainty principled precept permutation?

"Anyone who is not shocked by quantum theory has not understood it."
- Niels Bohr

Danke,

--
Don.......My cat's )\._.,--....,'``. https://crcomp.net/reviews.php telltale tall tail /, _.. \ _\ (`._ ,. Walk humbly with thy God.
tells tall tales.. `._.-(,_..'--(,_..'`-.;.' Make 1984 fiction again.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On Wed, 20 Dec 2023 23:35:11 -0800 (PST), Andrew McDowell
<mcdowell_ag@sky.com> wrote:

<snippo>

I believe that school geometry is educationally very useful, because it allows people to learn and practice with mathematical proof. I was entranced by proof, perhaps because almost every other form of argument I had seen before was, at best, occupying

a position that might need to be abandoned in the light of further evidence or the other side of the story - and few enough paid even that much attention to the facts of the case.

The teacher of the Geometry course I took in High School (long, long
ago ... but not far, far away) /explicitly stated/ that the purpose of
the course was to teach Deductive Reasoning.

We had to list and justify each and every step in our proofs. A
skipped step or a wrong justification produced a failure.

I suppose you had to do as well.

When I eventually read Euclid, as part of the set known as The Great
Books of the Western World, I found that we had covered only a part of
the material. But that's not as bad as it seems -- a lot of what we
didn't cover was covered in algebra and what we got of number theory.

Still, his proof that you can multiply two irrational numbers and get
a rational result (that is, that a rectangle with two irrational sides
can have a rational area) was interesting.
--
"Here lies the Tuscan poet Aretino,
Who evil spoke of everyone but God,
Giving as his excuse, 'I never knew him.'"

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

Andrew McDowell <mcdowell_ag@sky.com> wrote:

I believe that school geometry is educationally very useful, because it all= >ows people to learn and practice with mathematical proof. I was entranced b= >y proof, perhaps because almost every other form of argument I had seen bef= >ore was, at best, occupying a position that might need to be abandoned in t= >he light of further evidence or the other side of the story - and few enoug= >h paid even that much attention to the facts of the case.

Right, and this is what is never explained to students. The reason why we teach geometry is to teach proofs, because it is a convenient and small
system in which we can explain method of proof.

We could teach proofs in some other way, but geometry is a convenient one
that is traditional for the purpose.

This is why promoting "geometry with proofs" classes as are increasingly
common is so insidious. Because it teaches only the useless part without
the part that is actually important.

When I was a student I was very upset that Euclidian geometry did not reflect the real world and that there were many things in it that did not make sense
in the context of reality. Nobody ever told me that the whole thing is just
a game to teach proofs and that part of why it is so useful is that it does
NOT reflect reality. I did not actually figure this out until I was in grad school.
--scott

--
"C'est un Nagra. C'est suisse, et tres, tres precis."

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On Thu, 21 Dec 2023 08:59:12 -0800, Paul S Person
<psperson@old.netcom.invalid> wrote:

On Wed, 20 Dec 2023 23:35:11 -0800 (PST), Andrew McDowell ><mcdowell_ag@sky.com> wrote:

<snippo>

I believe that school geometry is educationally very useful, because it allows people to learn and practice with mathematical proof. I was entranced by proof, perhaps because almost every other form of argument I had seen before was, at best, occupying

a position that might need to be abandoned in the light of further evidence or the other side of the story - and few enough paid even that much attention to the facts of the case.

The teacher of the Geometry course I took in High School (long, long
ago ... but not far, far away) /explicitly stated/ that the purpose of
the course was to teach Deductive Reasoning.

We had to list and justify each and every step in our proofs. A
skipped step or a wrong justification produced a failure.

I suppose you had to do as well.

When I eventually read Euclid, as part of the set known as The Great
Books of the Western World, I found that we had covered only a part of
the material. But that's not as bad as it seems -- a lot of what we
didn't cover was covered in algebra and what we got of number theory.

Still, his proof that you can multiply two irrational numbers and get
a rational result (that is, that a rectangle with two irrational sides
can have a rational area) was interesting.

Why? (pi + 1) is an irrational number which is obviously exactly one
more than pi and the proof that e^*(i*pi)+1 = 0 is a 2nd year vector
calculus problem.

(My junior high math teacher put it on the blackboard one day without
proof in response to a question from a student about "what's the best
part of mathematics?" and said "come back and discuss it with me in
ten years when you've done university level math". Since after
university graduation I signed up for teacher's training I did just
that "Hey Mr Peters, you remember the day in grade 9 when you
said...")

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On Sat, 23 Dec 2023 04:04:36 -0800 (PST), Robert Carnegie <rja.carnegie@excite.com> wrote:

On Thursday 21 December 2023 at 16:59:20 UTC, Paul S Person wrote:

On Wed, 20 Dec 2023 23:35:11 -0800 (PST), Andrew McDowell
<mcdow...@sky.com> wrote:

<snippo>

I believe that school geometry is educationally very useful, because it allows people to learn and practice with mathematical proof. I was entranced by proof, perhaps because almost every other form of argument I had seen before was, at best,

occupying a position that might need to be abandoned in the light of further evidence or the other side of the story - and few enough paid even that much attention to the facts of the case.

The teacher of the Geometry course I took in High School (long, long
ago ... but not far, far away) /explicitly stated/ that the purpose of
the course was to teach Deductive Reasoning.

We had to list and justify each and every step in our proofs. A
skipped step or a wrong justification produced a failure.

I suppose you had to do as well.

When I eventually read Euclid, as part of the set known as The Great
Books of the Western World, I found that we had covered only a part of
the material. But that's not as bad as it seems -- a lot of what we
didn't cover was covered in algebra and what we got of number theory.

Still, his proof that you can multiply two irrational numbers and get
a rational result (that is, that a rectangle with two irrational sides
can have a rational area) was interesting.

Is that Euclid or your high school teacher?

Euclid.

As I said, I ran across this when I read his work later. It is the
next-to-last theorem in the chapter on Numbers, which are actually
line lengths measured by a line arbitrarily chosen to have length "1".

I'm not sure if the clever bit is simply casting
it in geometry or... Let me see, a square of
unit side has a diagonal of length square root
of 2. A square whose side is sqrt(2) has...
diagonal length 2 and area 2 (?), which if it's
right may be a coincidence. Then I suppose
we want the proof that sqrt(2) is irrational. ><https://en.wikipedia.org/wiki/Irrational_number>
which considers the matter more generally,
records that (long before Euclid) that may have been
fighting talk amongst followers of Pythagoras.

Well, you are close, but no cigar.

It is true that Euclid defines sqrt(2) as "rational in square" because
a square of side sqrt(2) would have a rational area. This does
complicate matters and may be one reason this part of Euclid was not
used in High School Geometry, since many of those taking it were also
taking Algebra and may have found juggling two different views of "rational/irrational" in their minds a bit much.

But the proof is not about something so simple as a square root. It is
about numbers of the form
a + sqrt(b) [1]
and
a - sqrt(b)
The assertion (proved geometrically) is that the area of the rectangle
will be
a*a + sqrt(b)*sqrt(b)
which, of course, is (in algebra) simply
(c + d)*(c - d) = c*c + d*d

Note that, in geometry, a and b will always be > 0. They are, after
all, line lengths. In algebra, of course, no such restriction occurs.

Also note that I never said that /any/ two irrational numbers could do
this, only that some pairs of them could.

Also note that, whatever the algebraic form may be called, Euclid
proved it [2] long before algebra existed.

[1] Note that I am using the algebraic notation to avoid having to
look up and copy what Euclid used.

[2] Well, recorded the proof, anyway. My understanding is that Euclid
is very much a school textbook: it combines material from many other geometricians into one convenient package and maybe adds a few
additional things as well, who can say?
--
"Here lies the Tuscan poet Aretino,
Who evil spoke of everyone but God,
Giving as his excuse, 'I never knew him.'"

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On Fri, 22 Dec 2023 20:22:13 -0800, The Horny Goat <lcraver@home.ca>
wrote:

On Thu, 21 Dec 2023 08:59:12 -0800, Paul S Person ><psperson@old.netcom.invalid> wrote:

On Wed, 20 Dec 2023 23:35:11 -0800 (PST), Andrew McDowell >><mcdowell_ag@sky.com> wrote:

<snippo>

I believe that school geometry is educationally very useful, because it allows people to learn and practice with mathematical proof. I was entranced by proof, perhaps because almost every other form of argument I had seen before was, at best,

occupying a position that might need to be abandoned in the light of further evidence or the other side of the story - and few enough paid even that much attention to the facts of the case.

The teacher of the Geometry course I took in High School (long, long
ago ... but not far, far away) /explicitly stated/ that the purpose of
the course was to teach Deductive Reasoning.

We had to list and justify each and every step in our proofs. A
skipped step or a wrong justification produced a failure.

I suppose you had to do as well.

When I eventually read Euclid, as part of the set known as The Great
Books of the Western World, I found that we had covered only a part of
the material. But that's not as bad as it seems -- a lot of what we
didn't cover was covered in algebra and what we got of number theory.

Still, his proof that you can multiply two irrational numbers and get
a rational result (that is, that a rectangle with two irrational sides
can have a rational area) was interesting.

Why? (pi + 1) is an irrational number which is obviously exactly one
more than pi and the proof that e^*(i*pi)+1 = 0 is a 2nd year vector
calculus problem.

That's not multiplying two irrational number together.

And let's see the /Euclidean geometrical/ proof of your monstrosity.

Noting that there is no line in Euclidean geometry with a length of
"0". Among other difficulties.

As to your question: because it never occurred to me that that could
be the case.

And I said it was "interesting", not that it revealed the secrets of
the Universe to me.
--
"Here lies the Tuscan poet Aretino,
Who evil spoke of everyone but God,
Giving as his excuse, 'I never knew him.'"

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On Sat, 23 Dec 2023 06:55:49 -0500, Tony Nance <tnusenet17@gmail.com>
wrote:

UH - > is Tony, >> is me

(My junior high math teacher put it on the blackboard one day without
proof in response to a question from a student about "what's the best
part of mathematics?" and said "come back and discuss it with me in
ten years when you've done university level math". Since after
university graduation I signed up for teacher's training I did just
that "Hey Mr Peters, you remember the day in grade 9 when you
said...")

Heh - over many years, I've worn out two t-shirts with that equation on
it. Hm...now that I think about it, I should start looking for #3.

Main thing I remember from the day in 2nd year calculus when the
professor provided that e^(i * pi) + 1 = 0 was the reaction from the
prof when I said "I first saw that formula in grade 9!"

He was a bit relieved when I told him that our 9th grade teacher had
said "those of you who go on in mathematics will see that formula
again - remember it and be impressed when you see the proof!"

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)