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.

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"...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." > > -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To post to this group, send email to everything-l...@googlegroups.com. > To unsubscribe from this group, send email to > everything-list+unsubscr...@googlegroups.com<everything-list%2bunsubscr...@googlegroups.com> > . > For more options, visit this group at > http://groups.google.com/group/everything-list?hl=en. > -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-l...@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.