John Mikes wrote:

Brian,nothing could be more remote for me than to argue 'math' (number'sapplication and theories) with you. I thinkyou mix up* 'counting'* forthe stuff that serves it. As I usually do, I looked up Google for thePeano axioms and found nothing in them that pertains to theorigination 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.`

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`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 isextractable from ANY quantity the numbers refer to?<Three plus four> is not different from <blue plus loud>, <sound plusspeed>, /_whatever_/, meaningless words bound together. UNless - ofcourse - you as a human, with human logic and complexity, UNDERSTANDthe amount *three* added to a _comparable_ amount of *four *and RESULTin /_*seven* pertaining 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 tojustify our theories. Sometimes quite weird.And *Brent* was so right: /"...I don't think the existence of somenumber of distinct things is the same as the "existence" ofnumbers...."/ - Tegmark's quoted "accounted for..." is not "consistsof"./_To 'explain' _/something by a conceptualization does notsubstitute 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 arethose abstract symbols you refer to as numbers?(and this question is understood for times way before humans and humanthinking).Sorry I askedJohn 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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