So, the question is not about people, nor the way people do things. But it is something about where people have been successful, with the recognition that "success" in mathematics typically involves theorems.
Would it be fair to represent your question as: What is it about the way mathematical research domains are defined that leads to the domains to often contain provable properties not-obvious from the method of domain demarcation? For a non-mathematical example: What is it about the domain of inquiry called 'taxonomy' that leads one to the thesis that many current species are descendant from now-extinct species? For a mathematical example: What is it about the domain of inquiry that people call 'natural numbers' that leads one to the thesis that there are a countable infinity of prime numbers? ---------- If I am correct, then I suspect the most straightforward answer to the question "Why are their theorem?" is: Because either: 1) People are bad at demarcating domains of inquiry (i.e., such situations arise unintentionally and unexpectedly), or 2) People find virtue in fuzzy definitions that create the situations in which theorem can occur and are interesting. In mathematics, at this point in History, I suspect people are typically in situation 2. Theorems are possible because mathematicians intentionally demarcate domains in which they expect interesting things to be true, but are net yet sure what the interesting things are. For example, one might define and investigate non-euclidean geometries because one suspected such geometries would have several interesting properties. In the past, I suspect people were more often in situation 1. For example, one might posit that a line has only one parallel line going through any given point, not because it would lead to other interesting theorems, but because one suspected it to be "true" and had not thought through the consequences one way or another. Eric ------------ On Sun, Apr 25, 2010 01:51 PM, Russ Abbott <russ.abb...@gmail.com> wrote: >>In answer to Eric and lrudolph, the answer I'm looking for is not related to epistemology. It is related to the domains to which mathematical thinking is successfully applied, where successfully means something like produces "interesting' theorems. (Please don't quibble with me about what interesting mean -- at least not in this thread. I expect that interesting can be defined so that we will be comfortable with the definition.) What is it about those domains that enables that. > > >-- Russ Abbott >______________________________________ > > Professor, Computer Science > California State University, Los Angeles > > cell: 310-621-3805 > > blog: <http://russabbott.blogspot.com/> > vita: <http://sites.google.com/site/russabbott/> > > >______________________________________ > > > >>On Sun, Apr 25, 2010 at 10:39 AM, <<#>> wrote: > > > >"Evolutionary Epistemology" > > > > ============================================================ >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 > Eric Charles Professional Student and Assistant Professor of Psychology Penn State University Altoona, PA 16601
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