Re: numbers?
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 I or similar. Now in my feeble mind to construct 'symbols' for expressing *how many Is 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
Re: numbers?
On 05 Aug 2010, at 01:18, Brian Tenneson wrote: 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. You may encounter a problem with the notion of 1-person, and 'material' bodies. 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. Computationalism entails something very near such view indeed. It entails also that if such structure make sense, then its cardinality is unknowable by the self-aware beings that could be generated inside. The statement that the cardinality of the mathematical universe is countable or not is absolutely undecidable, from 'inside'. 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. The problem is in defining embedded. I am not sure it makes set theoretical sense, unless you believe in Quine's New foundation (NF). I am neutral on the consistency of NF. With a large sense of embedded I may argue that the mathematical structure you are looking for is just the (mathematical) universal machine. In which case Robinson arithmetic (a tiny fragment of arithmetical truth, on which both platonist and non platonist (intuitionist) is enough. Indeed, I argue with comp that Robinson arithmetic, or any first order specification of a (Turing) universal theory is enough to derive the appearance of quanta and qualia. 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. This again follows from Church thesis, for the 'computationalist' TOE. 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. All right, then. 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. That may be already a lot for non mathematical logicians ... 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. This is done. Everything is defined in term of number and number relation. But it is not asked that the relation is arithmeticaly definable. For example, the ONE of Plotinus
RE: 3-Brain - 1-Mind?
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