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 /_*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 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,
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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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