
Re: NonEuclidean Arithmetic
Posted:
Sep 13, 2012 12:07 PM


On Wed, Sep 12, 2012 at 2:13 PM, Paul Tanner <upprho@gmail.com> wrote:
>>> You think that a thing exists only if you name it or know its nature? >>> Hubris, anyone? >>> >> >> Did the game of chess exist before it was invented? Of course not. > > Does an omniscient being exist? Then chess and anything else that > humans could possibly "cook up" exists in the mind of said being prior > (and I mean logically prior) to their cooking. >
I don't have a meaning of "omniscient" handy, sorry. To "know everything" still need not imply anticipating all possibilities. A person who is a "walking encyclopedia" is nevertheless not expected to describe the computer games that will be popular in 2025.
I don't think humans have any idea what "omniscient" means, but it's an impressivesounding word and they like to bandy it about.
> To say that "it does not exist unless and until *I* think of it" is > where the hubris lies. >
Right, for "an institution" to remain viable, for it to have a halflife, it needs multiple players and participants. Single individuals do not make games.
I do think of J.H. Conway as the inventor of surreals, not their discoverer, as if they'd been there all along. But he needed Knuth and others to provide more of an institution, so they could become a page on MathWorld.
Likewise, some friends and I created a "quadray coordinate system" (an invention) for which we have a writeup on Wikipedia. It's an invention. It didn't exists before the 1990s as far as I know, though I could imagine stumbling across a writeup in some old journal.
> And as long as you try to escape the charge of hubris by allowing for > other beings to think of it and so giving "existence" to it  even if > you don't know about it, then these possible "other beings" cannot be > arbitrarily limited to less than a possible omniscient being. >
You sound prosecutorial, using this phrase "escape the charge" as if now I'm somehow engaged in running away from something.
The real numbers are a social construct invented over the years and they continue to be reinvented, as the meaning of "real number" changes as new types of number are coined and infused into the shared vista.
>> "In 1960, Abraham Robinson provided an answer following the first >> approach. The extended set is called the hyperreals and contains >> numbers less in absolute value than any positive real number." [ >> http://en.wikipedia.org/wiki/Infinitesimal ] > > I have written about these hyperreals many times at Math Forum. My > first time has been cited here: >
Yes, good for you. I'm not suggesting you were unaware or "not omniscient" visavis said esoteric language games. No need to prove your awareness of these topics.
> ... > > Rucker has a Ph.D in symbolic logic, the language of set theory, and > so infinite sets and transfinite (ordinal and cardinal) numbers are > familiar to him. He relates this stuff to the general public in a > masterful way. He relates the theologian Anselm's description of God > (no mater how high I can conceive, I attain not to God, but only to > what is beneath God) to the modern set theoretic reflection principle > and how it is used to talk about Omega. I can't recommend his book > enough for regular folk." >
I am rather completely unimpressed.
>> Yeah, so what. Awe yourself with Cantorism if that floats your boat. >> Many gifted at math will have no need for such arcana. >> > > "Such arcana." Again: Hubris, anyone? >
Given how you use the word "Hubris" it's starting to come across as a badge of honor. I like to create distance between myself and some brands of BS artists / game player, no question, especially the ones into "omniscience" (ten foot pole time, when they come on the scene).
>> >> If I hand you a pencil and say it has length pi in the unit system I'm >> using, and then I put put five such pencils end to end and say that's >> 5 * length of pi, then I have told you no lie. > > Straw person. I keep telling you that what I'm talking about is when > both factors are irrationals. >
OK, so repeated addition is OK up to the point where instead of 1/2 of something, you want 1/pi of it, and that something happens to be irrational already. That's when repeated addition breaks down, but scaling has no problem, as even if there's nothing "computable" about this problem, we can still imagine a line getting longer or shorter and if we can do that, we know what multiplication means.
>> You have claimed I am wrong or lying for wanting to provide this kind >> of continuity from N through R, and therefore I have no choice but to >> cross you off my list as a compatible teacher. We just wouldn't get >> along in staff meetings. I'll have to get my pedagogical ideas from >> thinkers I better understand. >> > > Again, straw person. I never said you can't scale an irrational by a > rational number. I proved above that you can. Stick to what I actually > say again and again. Prove me wrong when I say you can't model > repeated addition when both factors are irrational. >
An omniscient being would likely have no trouble modeling any two real numbers multiplying, as a repeated addition problem. I'm not omniscient and yet I've given a number of examples already.
The scaling cartoon shows a volume, length or area changing size and I've been talking about how this could be a case of repeatedly adding:
2/pi < e(1/pi) < 3/pi since 2 < e < 3.
I can compute e using (1+n)^(1/n) where n > 0. 1/pi is in the neighborhood of 1/3  we teach estimation all the time, so students have a sense of the ballpark, can recognize a probably wrong answer. I'm adding a number close to 1/3 to itself less than 3 times, so I shouldn't be getting 1. It's like I divided a pie into a little more than 3 slices but then only took a little more than two of them. 2/3 or 1/3 +1/3 would be an estimate.
>>> n = 1e8 >>> (1+n)**(1/n) 2.7182817983473577 >>> from math import pi >>> (1+n)**(1/n) * (1/pi) 0.8652559698474173
About 86% of the pie has been taken (e is closer to 3 than 2). Of course n could be smaller and 1/pi could be hooked to a Ramanujan engine to get more digits. We'd still be in the ballpark of 86% and the "repeated addition" model helped us conceptualize what was happening and gave us some initial estimates.
Thanks to my learning about multiplication as repeated addition with whole numbers, and then with fractions, this extending to real numbers is not conceptually a problem. When the teacher sides with Devlin and tries to censor my thoughts, take away my repeated addition model, I fight back, as it helps me and I see no reason to surrender this tool of thought to these believers in omniscient beings who accuse me of hubris. They sound incoherent to me, and boastful of their supposedly superior model, which involves a pencil getting longer or something  that's supposed to be "the better way". I'd rather just have multiple models and not denigrate the repeated addition one, which still serves.
> Again, see the surreals. > > They have already dealt with all this. Read about it. >
"They"  you're sounding so awed by authority.
"They" invented some new language games (they're cast in terms of games  hackenbush in particular).
I'm supposed to be awed?
I can invent games too, with others, and have. I guess that makes me one of "them"? Or does one need a PhD in "symbolic logic"?
> Just as having different levels of infinity does not falsify the idea > that the lowest level of infinity is truly an infinity, having > different levels of continua does not falsify the idea that the lowest > level of continuum is truly a continuum. >
You can have more angels on the head of a pin if not all of them are real. By definition (no need for proof at first as axioms are the magic wand).
> You dump on me because you think that I don't have an open mind. But > with your ridiculing of that which do not understand, the mathematics > of the infinite and the continuous, I think that the saying "practice > what you preach" would be good to remember. > >  End of Forwarded Message >
I think you're overly impressed by finite mortals who concoct new language games and institutionalize them. Because they strut and puff in a manner you accept (they have the right degrees), you see questioning their approach to omniscience as "hubris" and/or as science denying behavior.
By their own rules, I can wave my magic wand and make a lot of their stuff go away in my sandcastle. I have that right, within mathematics, by its own rules. I don't have to play every game that's ever been invented. I don't have to agree to every set of axioms and definitions that comes along. Let other people play those games. Life is short.
On the other hand, they shouldn't take away my games by the same token. We're using quadrays, more whole number volumes (where cubes give irrational volumes), and a sphere packing matrix. We don't define "height, width, depth" as "three linearly independent dimensions" (a definition we don't need). We have a different way of organizing concepts. And I say "we" because there's published literature, bibliographies, the whole nine yards. Another way of doing math. Nothing and no one to stop me, even by the rules of math itself. It's a different ethnicity one could say. I'll even say "different race" since I know the term "race" has no scientific meaning and is therefore up for grabs to mean "whatever ethnic group" e.g. "the race of elementary school math teachers" (which I regard as an ethnicity, a subculture, though Litvin disagrees on maththinkingl).
Kirby
 End of Forwarded Message
Message was edited by: kirby urner

