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 Zero And Infinity, Neither are defined.
GG-Doz
Posted: Mar 29 2017, 02:54 AM


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QUOTE (Double sharp @ Mar 29 2017, 02:20 AM)
QUOTE (Kodegadulo @ Mar 28 2017, 10:25 PM)
QUOTE (GG-Doz @ Mar 28 2017, 09:36 PM)
QUOTE (Kodegadulo @ Mar 28 2017, 06:46 PM)
A time when "pink" did not mean a rosy light red but a putrid greyish-yellow

How'd that happen?

See Pink Used to Be Yellow (No Really).

Word meanings and usages evolve with the times. They blow this way and that with the latest fashions. It is best to work with current definitions. It is folly to be like Wendy, and seek some "original" meaning of a word. No matter how far back you go, there was always some previous generation who defined it differently, and who thought those young upstarts (who themselves were biquennia in our past) were spouting a lot of radical slang.

Indeed, my favourite example of this is that the ancestor of the word speed did not mean what we currently mean by speed.

That makes "Need for speed" sounds like some sort of rags to riches thing
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Shaun
Posted: Mar 29 2017, 07:52 AM


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QUOTE (Kodegadulo @ Mar 28 2017, 04:35 PM)


Wendy brings ill-repute to this site, and by association, to all of us, with every post.

I think we are going overboard with these vindictive attacks.

Perhaps all concerned should voluntarily take a few days off to cool off and get their world back into perspective.

I am therefore also closing this thread
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icarus
Posted: Mar 29 2017, 10:44 PM


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I have reopened the topic. Please do not discuss the dead and let them lie in peace.
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Double sharp
Posted: Mar 31 2017, 03:45 AM


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Regarding the hyperbolic-geometry part of it, I think I should note incidentally that as you increase the size of a circle, it does eventually tend to becoming a straight line, just like in Euclidean geometry. But it has to tend to becoming a horocycle first, and then expand "past infinity" and become a series of hypercycles equidistant to the same line, until it becomes that limiting line.
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GG-Doz
Posted: Mar 31 2017, 07:17 PM


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Let's call infinity H for Hyperreal
The problem with infinity is that H±x=H, which is better described as ∀H ±x ∈H, or, in English, Any Member of H, added or subtracted by any number, is a member of H
What if we treat H as a specific hyperreal? This means H-H=0, 2H = 2H, and so on.
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Double sharp
Posted: Apr 1 2017, 05:26 AM


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When dealing with infinity in the complex numbers, it is probably best to think of it as CP1, the Riemann sphere. You can actually think of hypercycles as "circles of imaginary radius"; the distance from the "centre" of the Poincaré disc model to a point at Euclidean distance r is f(r) = 2 artanh r = ln [(1+r)/(1-r)], and this is only real and nonnegative for r in [0, 1). As r approaches 1, clearly f(r) approaches infinity. So r goes up around the real axis from zero to infinity on the Riemann sphere:

user posted image

and then as we increase r further, (1+r)/(1-r) goes up from negative infinity to approach -1, so these "ultra-ideal points" have imaginary hyperbolic distance from the centre of the hyperbolic plane (taking the principal value of the logarithm) going from ∞ to approaching πi as the point approaches infinite Euclidean distance and the hypercycle approaches a straight line.
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jrus
Posted: Apr 19 2017, 09:55 PM


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Cantor’s diagonal argument, doodling around with point-set topology and real analysis, worrying about the axiom of choice and the Banach–Tarski paradox, etc. are pretty much irrelevant to anyone who isn’t a professional pure mathematician.

From the perspective of scientists, engineers, artists, applied mathematicians, computer programmers, etc., the whole Cantorian world amounts to amusing navel gazing whose only practical value is getting pedantic mathematicians off your back when you want to use (“naive”) calculus or Fourier analysis &c. Feel free to accept or reject Cantor’s arguments, consider them to be insightful gems or pure nonsense promulgated by a cult; it doesn’t affect anything in the physical world one way or another.

As for “infinity”, the problem is primarily that people have a very limited idea of what a “number” or a “proportion” or a “geometrical transformation” can mean. If we taught about stereographic projection and more generally about inversive geometry in high school, along with some projective geometry and homogeneous and barycentric coordinate models thereof, we’d end up with much less confusion about the usefulness of proportions with 0 in either numerator or denominator.
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Kodegadulo
Posted: Apr 19 2017, 10:02 PM


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Mathematicians get a bad rap because their business is investigating abstractions, and it's not at all their business to find useful applications for their abstractions. The whole field of mathematics might be dismissed as irrelevant navel-gazing with no possible application to anything practical in life ... until some physicist or engineer or other suddenly discovers an application for it, as has happened so many times.

Good luck trying to do without the math, when you do wind up needing it...
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jrus
Posted: Apr 19 2017, 10:06 PM


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Sure, fine. But anything useful in the world that can be done in a Cantorian model can also be done just as well without it. In practice, our best physical measurements have ~15 digits of precision. Worrying about whether there are an uncountable number of possible points in some “continuum” is a fun amusement for a rainy day, but is practically irrelevant.

There’s all kinds of useful mathematics out there, and only a tiny fraction of it depends in any critical way on the current dominant model of the “real numbers”.

In a general mathematics paper, when someone says “consider x a real number”, what they really mean is “go bug your college analysis teacher for the precise definitions, let me get on with my argument”.

One sad side effect is that the invention of the “real number” has side-lined a lot of rich and beautiful structure.

http://www.math.nyu.edu/faculty/edwardsd/carnegie.pdf

http://www.math.nyu.edu/faculty/edwardsd/stjohn.pdf
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Kodegadulo
Posted: Apr 19 2017, 10:08 PM


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Something like

\(\displaystyle\sum_{n=1}^{\infty} n = -\frac{1}{10_{\mathrm{z}}}\)

a result attributed to Ramanujan, seems at first a paradoxical, and hence utterly useless. Except for the fact that string theorists are all over it.
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jrus
Posted: Apr 19 2017, 10:14 PM


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Now you’re talking about something almost entirely unrelated, in a way that the mainstream mathematicians would mostly thrash you for. Meh. YMMV.
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Kodegadulo
Posted: Apr 19 2017, 10:43 PM


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@jrus: Well I don't hang out with many mathematicians, so I have no experience with what they would or would not trash. Seems like you've got a beef about this specific mathematical issue, due to some bad interactions with somebody.

I tend to think that even if a particular mathematical subject seems entirely without practical use, the fact that at least some humans on this planet are capable of grokking it raises our collective IQ. If studying one "useless" idea possibly leads someone to study something related that does turn out to reap dividends, then the "useless" idea served a purpose: it honed that mind.
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Kanonier
Posted: Apr 20 2017, 07:53 AM


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QUOTE (Kodegadulo @ Apr 19 2017, 10:43 PM)
I tend to think that even if a particular mathematical subject seems entirely without practical use, the fact that at least some humans on this planet are capable of grokking it raises our collective IQ.



and your earlier point:
"The whole field of mathematics might be dismissed as irrelevant navel-gazing with no possible application to anything practical in life ... until some physicist or engineer or other suddenly discovers an application for it, as has happened so many times."

It is unfortunate, though, that so many mathematical ideas which seemed useless found applications when applied to warfare ...

Grokking? Thankyou for this word; but I had to look it up! I had not seen it anywhere before.

grok
v.
"to understand empathically," 1961, arbitrary formation by U.S. science fiction writer Robert A. Heinlein (1907-1988) in his book "Stranger in a Strange Land." In popular use 1960s; perhaps obsolete now except in internet technology circles.


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Kodegadulo
Posted: Apr 20 2017, 02:09 PM


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Back in the 1950sd=1170sz experts were predicting that we were perhaps a century away from landing men on the moon. I'm beginning to think they were approximately correct. I mean, we're unquennia away from doing it now. But because of the Cold War and Sputnik, NASA pulled off the stunt of doing it nearly a century early, with hardly more combined computing power than most of us today carry around in our pockets. As it stands, my parents let me stay up late and I got to watch Neil Armstrong walk on the moon, on grainy black-and-white TV, in my lifetime. Funny how an existential threat can so focus so many minds.
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Kanonier
Posted: Apr 20 2017, 03:58 PM


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QUOTE (Kodegadulo @ Apr 20 2017, 02:09 PM)
Funny how an existential threat can so focus so many minds.

True.

But your words have nothing to do with zero or infinity.

Perhaps you need a thread for reminiscing about your childhood memories and rambling stories of the war?
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Kodegadulo
Posted: Apr 20 2017, 06:10 PM


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@Kanonier,. Hmm you're right. I've been going randomly off topic . Let me see about splitting out the offending posts. What's the best cutoff point? Opinions? I mean, was jrus' rant about real numbers on 4/19/2017 10:06 PM still on topic? It was in reaction to that that i started getting philosophical...
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Shaun
Posted: Apr 21 2017, 07:08 AM


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QUOTE (Kodegadulo @ Apr 20 2017, 07:10 PM)
What's the best cutoff point?

Maybe we should simply abandon the thread?
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Kanonier
Posted: Apr 22 2017, 06:51 AM


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QUOTE (Shaun @ Apr 21 2017, 07:08 AM)

Maybe we should simply abandon the thread?

Yes - it goes nowhere.
Admins should check members often, stop them wandering offtopic!
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Shaun
Posted: May 23 2017, 05:08 PM


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QUOTE (Kanonier @ Apr 22 2017, 07:51 AM)
QUOTE (Shaun @ Apr 21 2017, 07:08 AM)

Maybe we should simply abandon the thread?

Yes - it goes nowhere.
Admins should check members often, stop them wandering offtopic!

OK. Closed until such time as someone insists they want it opened again.
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