Russ, The natural numbers can be described by listing a few axioms for the notion of "successor" (or "the next whole number after this one" or "the operation of adding one") so, in some sense it is a very simple system. Yet all of mathematics can, in some sense be coded into statements bout the natural numbers. Propositions can e given Godel numbers and methods of deduction reduced to simple arithmetic operations. So there are, in some sense, as many theorems about the natural numbers as there are in all of math.
Your question reminded me of the article "The unreasonable effectiveness of mathematics" --which I only know by title. I have never read it, but I have referred to it. Maybe it's time for me to look at it --and the Lorenz article too. --John ________________________________________ From: friam-boun...@redfish.com [friam-boun...@redfish.com] On Behalf Of Russ Abbott [russ.abb...@gmail.com] Sent: Sunday, April 25, 2010 5:45 PM To: The Friday Morning Applied Complexity Coffee Group Subject: Re: [FRIAM] Why are there theorems? Too many interesting comments to follow up. But to Lee, Friam probably doesn't forward attachments. I didn't get the article with your earlier message either. There is an entry in the Stanford Encyclopedia of Philosophy on Evolutionary Epistemology<http://plato.stanford.edu/entries/epistemology-evolutionary/>. It seems from first glance that it makes sense. We -- and all animals -- evolve epistemological capabilities that improve survival. At that level it seems almost tautological. Secondly, your answer to my "question about finding a characterization of "the domains to which mathematical thinking is successfully applied", namely, that they are (or at least necessarily include) the domains for which "effective [i.e., mathematical] thinking" promotes species survival in "the external world in which we have evolved". seems to be contradicted by your own good ideas. Does knowledge about the domains to which they apply promote species survival? (They certainly promote individual survival as a successful mathematician, but that's another matter.) Does knowledge generated by any so-called "pure science" promote species survival? Only by chance, it seems. Besides why should improved species survival be related to the possibility of interesting theorems? The importance to us of a domain is certainly a function of its role in species survival. But why does that suggest that the domain is likely to give rise to sophisticated mathematics? I don't see the connection. -- Russ On Sun, Apr 25, 2010 at 2:07 PM, Eric Smith <desm...@santafe.edu<mailto:desm...@santafe.edu>> wrote: (expressions of ignorance to follow:) I wonder in all this whether there is anything interesting to be said by looking at the relation of syntax to semantics in mathematics, perhaps not in the sense of "applying" language, but rather in the sense of recognizing that mathematics shares syntactic elements with other constructs, which are more primitive than either, and have to do with applying formal descriptions to models of onesself. To be less random and cryptic (with luck): 1. We perform repetitive operations all the time, so our actions "embody" the inductive aspect of the natural numbers in some vague sense. But the natural numbers as a formal construct come into existence when we represent addition-by-one as a syntactic operation. (Here showing my ignorance of what Conway, Knuth, and other number theorists do to show how "real" all these formalisms are.) The point was, one is never supposed to ask "what comes after Z" in the alphabet, while the transition to realizing that one must ask "what comes after 26", sometime between three and four years of age, is the human transition to "understanding" arithmetic, which chimps and monkeys never make, even though they share some of the quantity-sense that is part of the semantic dimension of arithmetic. 2. So now we have the natural numbers as syntactic as well as semantic constructs. Why isn't that all, or why isn't every consequence of it immediately available to us? 2a. [Back to behavior] We break collections into groups all the time, and we compare groups for equivalence. Again, operationally, our actions embody ("en-corp-orate") multiplication and division. When the natural numbers have been created, they present an opportunity for us to do that to them, too. I think of that opportunity as a semantically created thing. Once numbers exist, we can do to them the same things we do to other objects, because they exist in a representation that allows us to think of them as objects. 2b. But grouping and comparing groups of numbers may not yet be multiplication and division. Those become parts of arithmetic when they are assigned a syntactic representation, so that operations are well-defined "without reference" to their semantic antecedents, if I understand the goal of Russell and Whitehead, with all of its reversals etc. The theorems derivable from rules of multiplication and division go from semantic possibilities that could be tested by action, to formal constructs within language, when multiplication and division are made parts of the syntactic construct. 3. From there we encounter a topic that has shown up on this list several times in discussions of emergence: the primes. What brought them "into existence", and why are their identities and properties not immediately available? An algorithm generated inductively from a small number of rules, and guaranteed to stop in finite time, makes prime/non-prime a well-defined distinction. Presumably that distinction doesn't exist in the purely semantic world of action, because it refers to the properties of the algorithm that apply to each particular number; action can only test cases. So primeness seems to rely on the ability to refer syntactically to operations, even though the opportunity to distinguish what can or can't be done to (any particular) set with a certain size is semantically created. So, having taken too much space to say something either obvious or ignorant, should we understand the growth of mathematics as partly an appeal to semantics, to guide us to unforeseen syntax? Then theorems use the new syntax to compactly represent connections that may not have been representable with anything short of simulation, in the earlier syntactic layers. Anyway, Eric ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org
