On Wed, Sep 12, 2012 at 11:13 AM, kirby urner <firstname.lastname@example.org> wrote: > On Tue, Sep 11, 2012 at 9:21 PM, Paul Tanner <email@example.com> wrote: > >>> >>> The so-called "real numbers" are a relatively recent cultural >>> invention, and multiplication was going on a long time before they >>> were promulgated. >>> >> >> They just didn't understand the continuum even though they were using >> it, and so it is not recent. >> Just because you don't give it a name or know its nature does not mean >> that it therefore does not exist. > > That's a telling statement. You consider the "real numbers" to > "exist" and "to have been discovered". They're an empirical > phenomenon in your mental geography, like Mt. Fuji. >
You think that a thing exists only if you name it or know its nature? Hubris, anyone?
> Those who don't > use them, don't play the same games, are "science deniers" because > they deny the "fact" or these "real numbers". >
Nonsense. I merely point out the complete inconsistency of denying the existence of that which one uses. Everyone - and I mean everyone who takes Algebra I and beyond - is taught to use mathematics based on the binary function on the reals R that we name "multiplication", R x R -> R such that the domain of argument x happens to be R or an uncountable subset of R, this domain of x being uncountable guaranteeing that almost all of the elements of the domain of x are not computable real numbers, which explains why my talking about the cardinality of the continuum being larger than the cardinality of the natural numbers is so utterly relevant.
No, we don't tell them this difference in cardinality and its utterly profound set of implications until later in more advanced treatments if they take these courses or if they educate themselves, but so what? Are we supposed to actually lie to them by implying to them the lie that all the real numbers on which the math they are taught to use are computable, which is what we do when we tell them that repeated addition as computation of the product from two given factors can model the multiplying of any two real numbers when the truth is almost the complete opposite of that? What kind of good math education is that?
By the way, let's again look at that that model I gave of "repeated addition" that shows that any element b in any field with characteristic 0 and that contains the natural numbers can be written as the sum of n elements for any natural number n: b = n(b/n) = (1_n + ... + 1_n)(b/n) = (b/n)_1 + ... + (b/n)_n. Note that since we have surjective multiplication, that means that the product of any two elements in such a field can be written as the sum of n elements.
But does this salvage repeated addition for all of R? Not if repeated addition means being able to compute the product from the two originally given factors, since what I just did, simply derived from some of the algebraic properties, does not actually compute the product from those two given field elements as original factors.
He shows why we cannot compute the product of two non-computable real numbers, which shows that repeated addition as the computation of the product of two numbers cannot be performed on two non-computable real numbers, noting that almost all real numbers are not computable.
> > I hope you're aware that not even every distinguished professor of > mathematics with peer reviewed published papers believes as you do > about the reals. What you have stated is one among many points of > view, by no means universally accepted, even among scientists. > > To say the real numbers are "a cultural artifact", "an institution", > "a set of conventions", "a set of practices", "a set of language > games" is not unheard of or "fringe" in any way. Many anthropologists > will discuss numeracy and alpha-numeracy in this manner. All math is > ethno-math, for them and for me. >
Then I hope you're aware that the notion of a continuum has been around for thousands of years, even if as "a cultural artifact", "an institution", > "a set of conventions", "a set of practices", "a set of language > games", and so, even if you hold to the idea that a thing exists only if someone as a notion of it, then the continuum has existed for thousands of years.
> Anyway, it's becoming clear to me that you're more of a dogmatist than > I'd realized
You confuse being a dogmatist with wanting to avoid sloppiness and wanting to have consistency.