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Topic: Non-Euclidean Arithmetic
Replies: 108   Last Post: Sep 13, 2012 3:39 PM

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 Paul A. Tanner III Posts: 5,920 Registered: 12/6/04
Re: Non-Euclidean Arithmetic
Posted: Sep 12, 2012 5:13 PM

On Wed, Sep 12, 2012 at 3:41 PM, kirby urner <kirby.urner@gmail.com> wrote:
> On Wed, Sep 12, 2012 at 9:26 AM, Paul Tanner <upprho@gmail.com> wrote:
>

>>> 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?
>>

>
> 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.

To say that "it does not exist unless and until *I* think of it" is where the hubris lies.

And as long as one tries to escape the charge of hubris by allowing for other beings to think of it and so giving "existence" to it even if one does not think of it, then not only must these possible "other beings" be included, they also cannot be arbitrarily limited to less than a possible omniscient being.

> People have gone on to invent the set of hyperreals which set is
> defined to contain infinitesimals that are smaller than any real
> number.
>
> I think many students feel ripped off when they find out about the
> hyperreals as the reals were already billed as containing all those
> distances on the number line, yet here we have a number > 0 that's not
> real only because it's too small to be real. Since when did reals
> kick out the really small numbers? What gives?
>
> "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:

http://users.soe.ucsc.edu/~karplus/abe/math_education.html

In the section, "Book reviews and lists of books" is this citation right after a citation of many of what "mattsmom" wrote, who is one who used to post here and elsewhere throughout Math Forum:

"Paul A. Tanner III recommended

* Infinity and the Mind by Rudy Rucker. Tanner says "In Rucker's book, you'll find much about the tension between the historical 'potential infinity vs. actual infinity' debate. Many before and some after Cantor, including some noted mathematicians, rejected the actually infinite and accepted instead only the potentially infinite. Rejecting infinitesimals, the actually infinitely small, led to the theory of limits, 200 years after Newton and Leibniz."

* The Mathematical Experience by Phillip J. Davis and Reuben Hersh. Tanner says "In the book by Davis and Hersh, there is a great write-up about Abraham Robinson's accomplishment in the 1960's in making infinitesimals, the actually infinitely small, logically mathematically sound for the first time in history."

However:

Forget about the hyperreals - go all the way to the surreal numbers.

I have talked about these before also, most recently here:

"Re: Who is the greatest published mathematician in history? If you
http://mathforum.org/kb/message.jspa?messageID=7759023

Partial quote from my post that I link to that is cited further above:

"The "love-knot" denotation of infinity, which we see in calculus, etc. associated with limits, is not the actual infinity, the simplest of which is denoted by lower case omega, the ordinal number of the natural numbers, nor is it what Rucker calls the Absolute Infinity, what he describes as the sort of imaginary ordinal number of the class of all sets. A rather nice book by John H. Conway, published in the 1970's, ON NUMBERS AND GAMES, in the context of his surreal numbers, relates these three types of infinity in what is possibly the most philosophically mind-blowing equation ever conceived. It sets oo, (the love-knot, sometimes called potential infinity) equal to omega (lower case omega, the simplest actual infinity) rooted to V, the class of all sets, which Rucker calls the Absolute Infinity, the infinity of the mystics, denoted by him as Omega, upper case omega.

...

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."

On this number field, here is one recent paper published this year:

"THE ABSOLUTE ARITHMETIC CONTINUUM AND THE UNIFICATION OF ALL NUMBERS GREAT AND SMALL"
http://www.ohio.edu/people/ehrlich/Unification.pdf

Partial quote:

"Abstract.

In his monograph On Numbers and Games, J. H. Conway introduced a real closed field containing the reals and the ordinals as well as a great many less familiar numbers including - [omega], [omega]/2, 1/[omega], sqrt[omega] and [omega]-n to name only a few. Indeed, this particular real closed field, which Conway calls No, is so remarkably inclusive that, subject to the proviso that numbers-construed here as members of ordered fields-be individually definable in terms of sets of NBG (von Neumann-Bernays Godel set theory with global choice), it may be said to contain "All Numbers Great and Small." In this respect, No bears much the same relation to ordered fields that the system R of real numbers bears to Archimedean ordered fields.

In Part I of the present paper, we suggest that whereas R should merely be regarded as constituting an arithmetic continuum (modulo the Archimedean axiom), No may be regarded as a sort of absolute arithmetic continuum (modulo NBG), and in Part II we draw attention to the unifying framework No provides not only for the reals and the ordinals but also for an array of non-Archimedean ordered number systems that have arisen in connection with the theories of non-Archimedean ordered algebraic and geometric systems, the theory of the rate of growth of real functions and nonstandard analysis.

In addition to its inclusive structure as an ordered field, the system No of surreal numbers has a rich algebraico-tree-theoretic structure-a simplicityhierarchical structure-that emerges from the recursive clauses in terms of which it is defined. In the development of No outlined in the present paper, in which the surreals emerge vis-a-vis a generalization of the von Neumann ordinal construction, the simplicity hierarchical features of No are brought to the fore and play central roles in the aforementioned unification of systems of numbers great and small and in some of the more revealing characterizations of No as an absolute continuum."

>> 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?

>
> 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? Please see the further below.

But before you do, please again note that that which makes the set of all real numbers a continuum and have an uncountable cardinality is the set of all those non-computable irrationals, on which any two of them it is impossible to perform repeated addition. And when we teach any mathematics based on the reals or any continuous subset of the reals including using any function in which any domain set or range set is a continuum or otherwise uncountable, we are inescapably using at least an uncountable subset of these non-computable irrationals on which it is impossible to perform repeated addition.

>
>> 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?
>>

>
> 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. And especially make that two non-computable irrationals - see my just above for more on this.

>> 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.
>>

>
> I have other sources for my "repeated addition" meme.
>
> You have refused to connect it with growing and shrinking quantities,

No I have not. Above is just a generalization that applies even when the field in question in not ordered. But when ordered, scaling models the growing and shrinking.

> 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 again and again actually say. Prove me wrong when I say one can't model repeated addition when both factors are irrational. This is especially so when both are non-computable irrationals (see my above points a couple of paragraphs back).

(Note: I say "finite" since a positive real number can be represented as an infinite sum as well as an infinite product.)

>
>> 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.

>>
>
> Thanks to the hyperreals, it sounds like your "continuum" is still
> full of holes.
> All those infinitesimals fell through the cracks hah
> hah.

Again, see the surreals.

Just as having different levels of an infinity does not falsify the idea that the lowest level of infinity is truly an infinity, having different levels of a continuum does not falsify the idea that the lowest level of continuum is truly a continuum.

The open-minded of the mathematics community over the past a little more than a century has simply found that they needed to expand their horizons and thus their definitions of terms like "infinity" and more recently "continuum".

You make fun of the mathematics of the infinite and the continuous, but I think it is good to have an open mind and join open-minded of the mathematics community in doing what I just reported they have done.

Message was edited by: Paul A. Tanner III

Date Subject Author
9/1/12 Jonathan Crabtree
9/1/12 Paul A. Tanner III
9/2/12 kirby urner
9/3/12 Paul A. Tanner III
9/3/12 kirby urner
9/3/12 Paul A. Tanner III
9/4/12 kirby urner
9/4/12 Paul A. Tanner III
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9/5/12 Paul A. Tanner III
9/5/12 Robert Hansen
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9/1/12 Joe Niederberger
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9/3/12 Paul A. Tanner III
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9/5/12 Paul A. Tanner III
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9/5/12 Paul A. Tanner III
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9/10/12 kirby urner
9/10/12 Paul A. Tanner III
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9/8/12 Robert Hansen
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9/8/12 Clyde Greeno @ MALEI
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9/9/12 Paul A. Tanner III
9/9/12 Robert Hansen
9/9/12 Paul A. Tanner III
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9/10/12 Paul A. Tanner III
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9/10/12 Paul A. Tanner III
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9/9/12 Joe Niederberger
9/9/12 Paul A. Tanner III
9/9/12 Wayne Bishop
9/9/12 Paul A. Tanner III
9/10/12 Wayne Bishop
9/10/12 Paul A. Tanner III
9/9/12 Paul A. Tanner III
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9/10/12 Joe Niederberger
9/10/12 Paul A. Tanner III
9/10/12 Joe Niederberger
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