I am not sure whether I reply to Brian, or to Bruno? there are remarks on *my
texts to Brian* without marking the replier and at the end it reads: *
"Bruno"* with no further ado.
Never mind, I want to be short.

"...Rectangles are not found in nature and not are numbers; both are
abstractions of things we see in nature..."
Pray: what things? and how are they 'abstracted into numbers? (Rectangles
etc. - IMO - are artifacts made (upon/within) a system of human
application).
"Yet numbers and rectangles (and many other abstractions) have a
suspiciously good use for modeling in nature"
           ---  " - u s e - ". (?) -----

Equivalence of III + IV as VII? Or in other numbering systems (letters,
etc.) used in various languages? In Bruno's example some time ago the II + I
= III definitely referred to the quantity of the "I" lines. He even went up
to some IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII or similar. Now in my
feeble mind to construct 'symbols' for expressing *how many "I"s there
are*is not the other way around. "3" stands for III, the COUNTED
amount of the
lines and not vice versa.

So: what are those *"naturally occurring"* things that serve for being
abstracted into numbers?
*

"Axioms are statements" - not controversial to what I stated. And please, do
not divert into quite different topics, where you may have a point in some
other aspect. We are talking about numbers, not the masculinity of the US
president.

"Exist" is something to be identified. IMO "physical existence" is a figment
pertinent to the figment of a "physical world" - quite outside of my
position. I "don't permit" physical existence.

To your(?) question after my signature (whoever asked it) I gave already my
apologetic deference conceding to Quentin's retort on that badly applied
sentence of mine. So I repeat it now: "sorry, it does not make sense.
Satisfied?

I have no comment on those paragraphs after the ------------- line.

If I may repeat: so WHAT ARE NUMBERS? (symbols for what? how do they apply
them to quantitative considerations? what if another 'logic' uses them in a
different math (e.g. where 17 is not identifiable as a prime number? Is it
likely that more will be found - as was the zero, or are we in a
"mathematical omniscience" already? Is our restriction to the 'naturals' -
natural, or just a consequence of our insufficient knowledge (caabilities)?

May I quote a smart person: there are no stupid questions, only stupid
answers. I ask them.

John Mikes



On 8/4/10, Brian Tenneson <tenn...@gmail.com> wrote:
>
> John Mikes wrote:
>
> Brian,
> nothing could be more remote for me than to argue 'math' (number's
> application and theories) with you. I thinkyou mix up* 'counting'* for the
> stuff that serves it. As I usually do, I looked up Google for the Peano
> axioms and found nothing in them that pertains to the origination of
> numbers. They USE them and EXPLAIN sich usage. Use what????
>
> Indeed, counting and what I'm referring to as numbers are different.
> Counting is a mental process while numbers have nothing to do with mind
> though the mind may apprehend and understand numbers to some extent.
>
> Counting is not the origin of numbers.  Counting inspired the discovery of
> numbers as elucidated by people like Peano.  Numbers are idealized models
> for the process of counting much like how a rectangle is an idealized model
> for the blueprint of an architectural  structure's foundation.  Rectangles
> are not found in nature and neither are numbers; both are abstractions of
> things we see in nature.
>
> Yet numbers and rectangles (and many other abstractions) have a
> suspiciously good use for modeling things in nature.
>
>
>  I wonder if you have an example where application of numbers is
> extractable from ANY quantity the numbers refer to?
> <Three plus four> is not different from <blue plus loud>, <sound plus
> speed>, *whatever*, meaningless words bound together. UNless - of course -
> you as a human, with human logic and complexity, UNDERSTAND the amount *
> three* added to a *comparable* amount of *four *and RESULT in 
> *sevenpertaining to the same kind of amount.
> *
>
> I only mean to reference the difference between numbers and the quantity
> they point to.  In an important way, <3+4> is different from your other
> examples in that <3+4> can be translated into a language devoid of human
> baggage and symbolically manipulated so as to show an equivalence between
> the symbols <3+4> and <7>.
>
>
>  **
> **
> *Axioms* however sounds to my vocabulary like inventions helping to
> justify our theories. Sometimes quite weird.
> And *Brent* was so right:  *"...I don't think the existence of some number
> of distinct things is the same as the "existence" of numbers...."*  -
> Tegmark's quoted "accounted for..." is not "consists of".
> *To 'explain'   *something by a conceptualization does not substitute for
> the existence and justification of such conceptualization.
>
> Axioms are statements.  Do humans need to exist in order for the statement
> "the galaxy is approximately a spiral shape" to exist?  How about "3+4=7",
> does that require humans to exist in order for the statement to exist?  What
> about the existence of the statement "the president of the US is male"; if
> all the humans were to die out, that statement would still exist.
> Statements are uttered by humans but do not depend on humans for their
> existence.  This is how axioms exist independent of humans, because they are
> statements.  The notation differs and are invented but what is being
> referred to by the symbols is independent of humans.  Moreover, I'm not
> talking about the truth of statements; I'm talking about the statements
> themselves not requiring anyone to utter them in order to exist.
>
> Numbers do not physically exist; so if physical existence is the only form
> of existence you permit, then numbers do not exist... in the same sense that
> math might as well be about Luke Skywalker, who does not exist physically.
> However, math has a suspiciously good use in nature like I said, unlike a
> novel about Luke Skywalker.
>
>
>
> Does it make sense that 'numbers existed' when nobody was around to *K N O
> W  or  U S E??*
> Especially when they did not*  C O U N T*  anything? BTW: what are those
> abstract symbols you refer to as numbers?
> (and this question is understood for times way before humans and human
> thinking).
> Sorry I asked
>
> John M
>
>
>
>
> Does it make sense?  Let me ask you a question.  Way back when, in the
> earliest stages of counting, let's assume there was a point at which a
> hundred thousand was the furthest anyone had counted to.  Now.. Did the
> number 1,000,000 exist at this stage of counting?  I think it did.  A
> million and all of its successors.
>
>
> Bruno,
> -------------
>
> Hmm... Lawvere has tried to build an all encompassing universal
> mathematical structure, but he failed. It was an interesting failure as he
> discovered the notion of topos, (discovered also independently by
> Groethendieck) which is more a mathematical mathematician than a
> mathematical universe.
> Also Tegmark is not aware that Digital Mechanism entails the non locality,
> the indeterminacy and the non cloning of matter, and that DM makes the
> physical into a person-modality due to the presence of the mathematician in
> the arithmetical reality.
> Quanta are special case of first person plural sharable qualia.
>
> -------------
>
> I'm not looking for a truly all-encompassing mathematical structure.  What
> I'm looking for is a mathematical structure in which all mathematical
> structures can be embedded.  By mathematical structure, I mean there is a
> symbol set S consisting of constant symbols, relation symbols, and function
> symbols, and the pairing of a set with a list of rules that interpret the
> symbols.  In Tegmark's papers on "ultimate ensemble TOE" and "the
> mathematical universe," he refers to what I call a mathematical structure as
> a "formal system" (and also mathematical structure).
>
> The structure I'm looking for wouldn't encompass anything that isn't a
> mathematical structure, like a category with no objects/elements.
>
> Tegmark argues that reality is a mathematical structure.  What's cute about
> his argument is that while invoking the concept of a TOE, his argument is
> independent of what that TOE might be.  He defines a TOE to be a complete
> description of reality.  Whether or not this can be expressed in a finite
> string is an open problem as far as I know.  (I doubt it can.)  He argues
> that a complete description of reality must be expressible in a form that
> has no human baggage and I would add to that is something that exists
> independent of humans in the sense that while the symbols used to provide
> that complete description will depend on humans, what is pointed to by the
> symbols is not.
>
> Tegmark argues that reality is a mathematical structure and states that an
> open problem is finding a mathematical structure which is isomorphic to
> reality.  This might or might not be clear: the mathematical structure with
> the property that all mathematical structures can be embedded within it is
> precisely the mathematical structure we are looking for.
>
> I am confident that I have found such a structure but only over a fixed
> symbol set; I need such a structure to be inclusive of all symbol sets so as
> to cast away the need to refer to a symbol set.  The technique I used was to
> use NFU, new foundations set theory with urelements--which is known to be a
> consistent set theory, to first find the set of all S-structures.  Then I
> take what I believe is called the reduced product of all S-structures.  Then
> I show that all S-structures can be embedded within the reduced product of
> all S-structures.  Admittedly, there is nothing at all deep about this; none
> of my arguments are deeper than typical homework problems in a math logic
> course.
>
> My next move is to find justification for the existence of a math structure
> with the important property that all structures can be embedded within it
> --independent of the symbol set-- and thus eliminating the need to refer to
> it.
>
> One thing I wonder is how to define all your notions such as
> "mathematician," "n-brains," "n-minds," and "digital mechanism" in terms of
> mathematical structures.  I'm particularly interested in defining something
> that models awareness and using it to find self-aware structures such as
> "mathematicians."
>
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