
Re: NonEuclidean Arithmetic
Posted:
Sep 13, 2012 3:39 PM


On Thu, Sep 13, 2012 at 12:07 PM, kirby urner <kirby.urner@gmail.com> wrote: > 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 proprietorial, using this phrase "escape the charge" as if > now I'm somehow now 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. >
I avoided the term "God"  a being as we all know according to many who believe in various types of scholastic theologies knows our futures and any and all futures of any and all universes, but maybe it seems I should have not to make my point.
My point is that the idea that a thing does not exist unless and until it exists in the mind of some human being is clearly and unavoidably a rejection of that philosophy of mathematics called mathematical realism as well as a rejection of major forms of theism.
The belief that mathematical objects can have some sort of existence outside of the human mind is mathematical realism:
http://en.wikipedia.org/wiki/Philosophy_of_mathematics#Mathematical_realism
http://en.wikipedia.org/wiki/Philosophy_of_mathematics#Platonism
http://en.wikipedia.org/wiki/Philosophy_of_mathematics#Empiricism
http://en.wikipedia.org/wiki/Philosophy_of_mathematics#Mathematical_monism
If you wish to dump on the belief that mathematical objects can have some sort of existence outside of the human mind and promote that they cannot have some sort of existence outside of the human mind, then OK, but note that in doing so you are dumping on many people's belief in God, since many people's notion of God is that anything we could possibly "invent" or "discover" (or even our futures or any future of any universe in any multiverse) is already known to this God. Perhaps then you could be more respectful of the beliefs of those who believe differently than you.
No, mathematical realism does not imply the existence of such a God, but the existence of such a God does imply mathematical realism, and so negating mathematical realism yields negating the existence of such a God. (Notice I said "such a God" since I'm allowing for the possibility that some might believe in an ignorant God.)
Specifically on reals: If you or anyone else wishes to reject the existence of any being outside of humanity in whose mind real numbers are an object of thought, then be my guest, but claiming that it's not possible that some such being exists really and clearly is a form of that human hubris that says, "Not only am I the center of all that is, not only am I the measure of all things, I am all that could know of real numbers."
Side note 1: If you wish to believe that the sum total of all existence, including any multiverse http://en.wikipedia.org/wiki/Multiverse#Multiverse_hypotheses_in_physics (that has been recently proposed in theoretical physics) and anything possibly beyond all that  and this includes the set of all past events in the space and time of this sum totally of all existence taken together as a well ordered set  is merely finite, then be my guest.
Side note 2: In the main philosophy of science journals in the US and the UK there has been an ongoing debate starting in the later 20th century as to whether the past can be infinite. The general line is that of those who have written the articles, those who argue that it can be infinite happen to be atheists and those who argue that is cannot be infinite happen to be theists who have tried using that argument as the basis for a proof of the existence of a timeless and infinite God (that would be the causal ground to a claimed logically necessary beginning of time).
But note that the prior posited multiverse has been suggested to be infinite, and that claim and the claim that the past is infinite means that infinite being of some sort exists, which opens all kinds of doors to all kinds of possibilities, including Infinite Being, if you know what I mean.
> >>> >>> 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.
OK, but make that "repeated addition repeatedly redefined" while recalling that scaling has no problem and requires no redefinition from day one.
> >>> 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.
Yet another straw person. You have not given an example modeling any two *irrational* numbers multiplying, as a (finite) repeated addition problem, where the product is exactly obtained on the real number line without redefining things yet again with this "roughly" stuff. (Again, as I said before, I say "finite" since any positive real can be represented not only as an infinite sum but also as an infinite product.)
You have not because it cannot be done.
But generally speaking for any two real number numbers (including any two irrational numbers) a and b I showed how we can obtain exactly where product ab is on the real number line given points 1, a, and b on the line  in fact, it's a proof of the following theorem using "without loss of generality": Given real number (point) 1 and any two positive real numbers (points) a and b, we can construct the exact location of real number (point) ab on the real number line:
Connect 1 on the xaxis to b on the yaxis, and, parallel to that drawn line segment, connect a on the xaxis to the yaxis, and this point on the yaxis is ab. That is, in terms of distance from 0 or magnitude or absolute value: ab is to b as a is to 1  that is, written in terms of ratios or proportions: ab:b as a:1.
And via commutativity we have the other way as an alternative:
Connect point 1 on the xaxis to point a on the y axis, and, parallel to that drawn line segment, connect b on the xaxis to the yaxis, and this point on the yaxis is ab. That is, in terms of distance from 0 or magnitude or absolute value: ab is to a as b is to 1  that is, written in terms of ratios or proportions: ab:a as b:1.
> > 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. >
With repeated redefinition of "repeated addition" you have gotten to the point of giving up what the function actually is, which is a,b > f(a,b)  you have redefined things so much that we now just have a,b  > *approximately* f(a,b).
> 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
Keep it all you want. Just be up front about the fact that you have to redefine things every step of the way and then get to the point where you even have to replace what the function actually is, which is a,b  > f(a,b), with a,b > *approximately* f(a,b).
[In reply to what some of the greatest mathematicians in history have done on the topics of the infinite and the continuum, Kirby writes:]
> "They"  you're sound 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"? ... > 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). ... > I think you're overly impressed by finite mortals who concoct > new language games and institutionalize them. >
Like I said before, you can ridicule what you do not understand and the accomplishments of all those who are way beyond your capabilities all you want. It speaks for itself.
> On the other hand, they shouldn't take away my games by > the same token.
They are not trying to.
It's just that you ought to be upfront with your repeated redefinitions of what "repeated addition" means all the way up to your replacing the actual binary operation a,b > f(a,b), with something else entirely, a,b > *approximately* f(a,b). Again, do this all you want, but just be up front about what I just said, that's all.
One final point based on your "you can have more angels on the head of a pin if not all of them are real": If you want to proceed along the line that, say, a positive integer is "real" but an irrational number is not, then be my guest, but do you realize how amateurish this sounds?
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