Why are there theorems?
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On Apr 25, 2010, at 11:51 AM, Russ Abbott wrote:
If I get your drift, I think it may have to do with domains with very simple definitions yet have many far reaching results. Almost a success criterion of being approachable and powerful both. Consider graph theory: a graph G = {V,E} where V=a set of vertices, and E is a set of edges connecting two vertices. (This can be made more "mathy" but with very little gain.) A few definitions such as the degree of a vertex is the number of edges it has, and you're off to the races. With nearly no training, many simple theorems can be proved, for example: Prove that in any finite graph, the number of vertices with odd degree is even. Possibly part of this simplicity is how simple a proof can be in the given domain. I like Cris Moore's definition of a proof: a short essay that convinces a skeptic! For the above, for example, you can just use a few statements in english, or resort to Induction. In either case the proof actually convinces and conveys an "a-ha" experience. I rather like what is going on in computer science, your field, in this regard. Many regard an algorithm as best if it is structured upon its proof. Indeed, all algorithms *are* theorems (their inputs, outputs and a statement on the relations between the two) and proofs (the algorithm itself). -- Owen ============================================================ 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 |
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In reply to this post by Russ Abbott
On 25 Apr 2010 at 10:51, Russ Abbott 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. Did you read the article by Lorenz? (I wish *someone* would; so far I haven't had any takers closer to home, which is one reason I sent it to Friam. Content aside, it's a fun article!) It does suggest an answer to your question, I think: humans' capacity for "mathematical thinking" evolved to be useful for human survival in the world; so did humans' capacity for attributing different degrees of "being interesting" to different things and structures in the world; so thinking effectively (i.e., mathematically) about "interesting" things Builds Better Bodies^WSpecies Two Ways. Yes, that answer (or anything along its lines) does leave open that other species might evolve so as to have "minds" that engage in "mathematical thinking" that is quite different from human "mathematical thinking". Lorenz suggests as much (with a rather far-fetched imaginary example of how "counting numbers" might not be "interesting" had things been otherwise). I'm not a philosopher (I'm a mathematician, who has proved quite a few interesting theorems in my day [1]) so I probably shouldn't allow myself to use a word like "epistemology", whose definition I am never quite sure of--much less a coinage like "Evolutionary Epistemology". Let's just take that word off the table for now. Like you, I am interested in "the domains to which mathematical thinking is successfully applied", and I would like to know "what is it about those domains that enables that". It was through my pursuit of a satisfying answer (satisfying to me, of course) that I recently (last week or so) found Lorenz's article. I will now describe the path that got me there. First, I had been (probably since college) vaguely aware of the famous title of the 1960 article by the mathematical physicist Eugene Wigner, "The Unreasonable Effectiveness of Mathematics in the Physical Sciences". Then, in 1978, I read (in an endnote to a review in the Bulletin of the AMS of a book on Wittgenstein) Georg Kreisel's off-hand one-liner in response to Wigner: "(Would it be \textit{obviously} more `reasonable' if we were not effective in thinking about the external world in which we have evolved?)" That response, of course, conflates "effective thinking" with "mathematical thinking", but I can live with that; and it strongly suggests an answer to your 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". (If there are also domains full of *interesting* theorems that don't, and never will, lead to "effective thinking" about any aspect of our world-- and there may be--they can be treated as "spandrels".) I didn't think much more about the subject until four or five years ago, when I was commissioned to write an article on non-standard mathematical models of time that might be useful to psychologists. While doing the (non-mathematical) research necessary for that article, including a lot of observations of psychologists in their native habitats, I noted that no one has ever made a claim for the "unreasonable effectiveness of mathematics in the social sciences", and that anyone who did would be rightly laughed at (except, possibly, in an economics department). Furthermore, most attempts, including attempts by some *very* good mathematicians (like Rene Thom), as well as by a fair number of fraudsters, hacks, and mystagogues, to apply (much) mathematics to (much) social science, have come to nothing (except, in some cases, to pseudoNobel prizes in economics). What is it about the domain of "social science" that seemingly *disenables* any serious use of theorem-thinking? A few weeks ago, I found that Kreisl's point had been made by Lorenz already in 1941--37 years before Kreisl made it, and 19 years before Wigner's article! It was really, really hard to get my hands on Lorenz's paper (for some reason, not a lot of US libraries have German philosophical journals from 1941...), so when I did get it, I wanted to spread it around. As to what it might be about social science that makes it resistant to mathematical thought, maybe it's because life on earth has had a lot longer to adapt to the physical world than to the (human) social world (for all I know, ants have a well-developed mathematics of ant social science). Lee Rudolph Professor of Mathematics and Computer Science Clark University, Worcester MA [1] Leaving aside the precise number of theorems I've proved, interesting or otherwise, I can quite accurately say that I've had at least 4 good ideas, all of which continue to generate new (and--to me!--interesting) theorems (proved and published by others, none of whom belong to the empty set of my own graduate students), in the case of the three oldest of the ideas ("links at infinity", "quasipositivity", and "braided surfaces") over 30 years since I had them--which is *many*, *many* half-lives of the typical Modern Theorem. So I can be interesting to mathematicians, anyway. ============================================================ 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 |
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In reply to this post by lrudolph
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 <[hidden email]> wrote: Eric Charles Professional Student and Assistant Professor of Psychology Penn State University Altoona, PA 16601 ============================================================ 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 |
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In reply to this post by Russ Abbott
No. MATHEMATICAL induction is actually serial DEduction.
I was talking about plain old vanilla philosophical induction: The fallacy is that without deduction, induction can't get you anywhere, and that people who think they are getting somewhere through induction alone are so caught up in an ideology that they cannot see their dependency on deduction.
As I have watched the thread develop, I have been less and less sure that my comment was relevant.
Nick
Nicholas S. Thompson
Emeritus Professor of Psychology and Ethology,
Clark University ([hidden email])
http://www.cusf.org [City University of Santa Fe]
============================================================ 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 |
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In reply to this post by Russ Abbott
Dear Lee,
YOU ASKED: Did you read the article by Lorenz? YOU COMPLAINED: (I wish *someone* would; But did you actually SEND the link to the Lorenz article? It wasnt attached to the message I got. N Nicholas S. Thompson Emeritus Professor of Psychology and Ethology, Clark University ([hidden email]) http://home.earthlink.net/~nickthompson/naturaldesigns/ http://www.cusf.org [City University of Santa Fe] > [Original Message] > From: <[hidden email]> > To: The Friday Morning Applied Complexity Coffee Group <[hidden email]> > Date: 4/25/2010 2:26:49 PM > Subject: Re: [FRIAM] Why are there theorems? > > On 25 Apr 2010 at 10:51, Russ Abbott 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. > > Did you read the article by Lorenz? (I wish *someone* would; so > far I haven't had any takers closer to home, which is one reason > I sent it to Friam. Content aside, it's a fun article!) It > does suggest an answer to your question, I think: humans' capacity > for "mathematical thinking" evolved to be useful for human survival > in the world; so did humans' capacity for attributing different > degrees of "being interesting" to different things and structures > in the world; so thinking effectively (i.e., mathematically) about > "interesting" things Builds Better Bodies^WSpecies Two Ways. > > Yes, that answer (or anything along its lines) does leave open that > other species might evolve so as to have "minds" that engage in > "mathematical thinking" that is quite different from human > "mathematical thinking". Lorenz suggests as much (with a > rather far-fetched imaginary example of how "counting numbers" > might not be "interesting" had things been otherwise). > > I'm not a philosopher (I'm a mathematician, who has proved quite a > few interesting theorems in my day [1]) so I probably shouldn't allow > myself to use a word like "epistemology", whose definition I am > never quite sure of--much less a coinage like "Evolutionary > Epistemology". Let's just take that word off the table for now. > > Like you, I am interested in "the domains to which mathematical > thinking is successfully applied", and I would like to know "what is > it about those domains that enables that". It was through my pursuit > of a satisfying answer (satisfying to me, of course) that I recently > (last week or so) found Lorenz's article. I will now describe the > path that got me there. > > First, I had been (probably since college) vaguely aware of > the famous title of the 1960 article by the mathematical > physicist Eugene Wigner, "The Unreasonable Effectiveness of > Mathematics in the Physical Sciences". Then, in 1978, I read > (in an endnote to a review in the Bulletin of the AMS of a book > on Wittgenstein) Georg Kreisel's off-hand one-liner in response > to Wigner: "(Would it be \textit{obviously} more `reasonable' > if we were not effective in thinking about the external world > in which we have evolved?)" That response, of course, > conflates "effective thinking" with "mathematical thinking", > but I can live with that; and it strongly suggests an answer > to your 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". (If there are also domains full of > *interesting* theorems that don't, and never will, lead > to "effective thinking" about any aspect of our world-- > and there may be--they can be treated as "spandrels".) > > I didn't think much more about the subject until four or > five years ago, when I was commissioned to write an article > on non-standard mathematical models of time that might be > useful to psychologists. While doing the (non-mathematical) > research necessary for that article, including a lot of > observations of psychologists in their native habitats, > I noted that no one has ever made a claim for the > "unreasonable effectiveness of mathematics in the social > sciences", and that anyone who did would be rightly > laughed at (except, possibly, in an economics department). > > Furthermore, most attempts, including attempts by some > *very* good mathematicians (like Rene Thom), as well as by > a fair number of fraudsters, hacks, and mystagogues, > to apply (much) mathematics to (much) social science, > have come to nothing (except, in some cases, to pseudoNobel > prizes in economics). What is it about the domain of > "social science" that seemingly *disenables* any serious > use of theorem-thinking? > > A few weeks ago, I found that Kreisl's point had been made > by Lorenz already in 1941--37 years before Kreisl made it, > and 19 years before Wigner's article! It was really, really > hard to get my hands on Lorenz's paper (for some reason, > not a lot of US libraries have German philosophical journals > from 1941...), so when I did get it, I wanted to spread it > around. > > As to what it might be about social science that makes it > resistant to mathematical thought, maybe it's because life > on earth has had a lot longer to adapt to the physical world > than to the (human) social world (for all I know, ants have > a well-developed mathematics of ant social science). > > Lee Rudolph > Professor of Mathematics and Computer Science > Clark University, Worcester MA > > [1] Leaving aside the precise number of theorems I've proved, > interesting or otherwise, I can quite accurately say that I've > had at least 4 good ideas, all of which continue to generate > new (and--to me!--interesting) theorems (proved and published > by others, none of whom belong to the empty set of my own > graduate students), in the case of the three oldest of the > ideas ("links at infinity", "quasipositivity", and "braided > surfaces") over 30 years since I had them--which is *many*, > *many* half-lives of the typical Modern Theorem. So I can > be interesting to mathematicians, anyway. > > ============================================================ > 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 |
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In reply to this post by Nick Thompson
(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 |
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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. 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 <[hidden email]> wrote: (expressions of ignorance to follow:) ============================================================ 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 |
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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: [hidden email] [[hidden email]] On Behalf Of Russ Abbott [[hidden email]] 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 <[hidden email]<mailto:[hidden email]>> 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 |
Re: Why are there theorems?
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In reply to this post by Nick Thompson
Nicholas Thompson wrote circa 04/25/2010 01:50 PM:
> I was talking about plain old vanilla philosophical induction: The > fallacy is that without deduction, induction can't get you anywhere, > and that people who think they are getting somewhere through induction > alone are so caught up in an ideology that they cannot see their > dependency on deduction. > > As I have watched the thread develop, I have been less and less sure > that my comment was relevant. I think it is. (But as the thread develops, I'm less and less confident that it'll come to anything... Aaaaaaa! I can't believe I might agree with Doug on something. ;-) Going back to: 1) Grant's branch on relations vs. components, 2) Lee's further branch on unity (universe, closure), 3) Russ' inclusion of time and Eric's inclusion of consequence, and 4) Nick's inclusion of fallacious generalization. Sarbajit was right to consult the dictionary. Every valid statement is a "theorem". But when it's just a small stepping stone, we call it a lemma or some other diminutive term. Why? Because math constructs (proofs) are rhetoric. That's all they are, presentations meant to persuade and communicate. Math is a language, first and foremost. And it is used to communicate. So, the main question for "why theorems" has its answer in the larger question, "why communicate?". Any structures that seem to obtain and persist through our acts of communication is a psych- and socio-logical effect of the underlying _carrier_ of that communication. As Lee points out, these "persistent" structures will obviously reflect (if not be total slaves of and epiphenomenal to) the humans that participate. It's a direct effect of the _intent(s)_ of the participants. The concept of psychological induction is necessary in order to follow the rhetoric of whatever proof you're examining. What is the argument (or communication) the author is pursuing? Therein lies your structure, its persistence, its interestingness, relevance, etc. So, Russ' question is way too vague to be answered. You can't generalize across all of math/logic to talk about "why theorems?" any more than you can generalize over all of natural language and ask "why sentences?", unless you're willing to accept the equally vague and useless answer "because we use sentences/theorems to communicate." You have to talk about specific rhetoric. E.g. "why War & Peace?" or at least something like "why number theory?" That way, you can go further and ask what the author's(s') intentions are when building upon that rhetoric. Looking back, I see I didn't explicitly tie in relations vs. components or closure. [sigh] To make it as short as my limited skills allow: o Consequence is composition (even temporally); hence, an author can start with relations or components but, in the end, both are always necessary. o Closure (or circumspection) is necessary for any rhetoric to convince (convict, capture, imprison) the audience. But, more ontologically, any rhetoric is bounded by its context. As Goedel and Tarski point out, a (nontrivial) pure syntax cannot be closed; a higher order language is necessary to complete it. And the highest order language we have is the context in which we're embedded: reality. -- glen e. p. ropella, 971-222-9095, http://agent-based-modeling.com ============================================================ 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 |
Why theorems work. (was Why are there theorems?)
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In reply to this post by Russ Abbott
Russ Abbott wrote circa 04/25/2010 02:45 PM:
> 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. You can only talk about why the structure of any rhetoric _works_ if you explicitly ask what the rhetoric is used for. We use languages, including math, to describe our experiences of the world. Vague touchy feely languages like English are best at describing the vague touchy feely stuff that goes on in our bodies. Those languages have to be good at expressing that stuff. Other languages, like math, are best at describing procedures that we intend to be mimicked in other times and other places, by other entities (including machines). They are not so good at the touchy feely description of experiences, but very good at communicating repeatable procedures. Sophisticated math rhetoric arises and persists more easily in domains that consist largely of repeatable procedures. But that doesn't mean that such rhetoric cannot arise and persist in domains that don't easily submit to such procedure. All it takes is enough participants, all speaking the same (or similar) language and the rhetoric for that domain will eventually cohere and stabilize. Again, though, it will only be as stable as the carrier population that participates. Hence, as the carrier population evolves, the efficiency with which the rhetoric moves between individuals dictates the evolution of the rhetoric. Procedures (like counting) that hop easily between individuals persist (and are probably continually re-invented). Procedures (like showing the convergence of an infinite series) that have less efficient transfer from human to human, risk being lost. So, it's not that any particular rhetoric promotes species survival. It's that the species we're talking about is wired for language and some rhetoric is more transferable than other rhetoric. -- glen e. p. ropella, 971-222-9095, http://tempusdictum.com ============================================================ 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 |
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In reply to this post by glen e. p. ropella-2
The OP's "Too many interesting comments to follow up" sorta sounds like "I've lost interest"! ---- Owen ============================================================ 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 |
Re: Why are there theorems?
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Owen Densmore wrote circa 10-04-26 08:59 AM:
> The OP's "Too many interesting comments to follow up" sorta sounds like > "I've lost interest"! Heh, yeah; but words have consequences! ;-) No (good?) deed goes unpunished. -- glen e. p. ropella, 971-222-9095, http://agent-based-modeling.com ============================================================ 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 |
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I don't follow Glen's 'You can't generalize across all of math/logic to talk about "why theorems?" any more than you can generalize over all of natural language and ask "why sentences?" '
The original intent was to ask why there always seems to be hidden structure -- which is revealed by theorems. It's not the theorems I'm concerned about; it's the hidden structure. -- Russ On Mon, Apr 26, 2010 at 9:29 AM, glen e. p. ropella <[hidden email]> wrote: Owen Densmore wrote circa 10-04-26 08:59 AM: ============================================================ 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 |
Re: Why are there theorems?
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Actually I can follow Glen's line of reasoning (I think).
For example, the way Maths works is that a "theorem" is "proved" by trying to prove a "conjecture". When that approach fails you end up proving a "special case" of the conjecture - which in turn gets elevated to its own status as a "theorem". "Proving" Fermat's Last Theorem took 3 centuries and generated an equal number of theorems for mathematicians to solve/prove. The ultimate perpetual machine to keep mathematicians employed till either the existence of "God" (the grand unified theorem of everything) is proved or we have 33 billion gods (theorems) as we do in India. Sarbajit On Mon, Apr 26, 2010 at 10:15 PM, Russ Abbott <[hidden email]> wrote:
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My take is that contemporary abstract mathematicians have no interest (as mathematicians) in discerning "truth". "The truth" about existence is the business of scientists, philosophers and theologians. Ever since Hilbert's program at the beginning of the twentieth century to "axiomatize" all of mathematics in the manner of Peano (help me out here, mathematicians), pure mathematics has been seen as an abstract exercise in taking arbitrary sets of postulates and reasoning therefrom - without regard to their "truth". The only value judgment involved is whether the results (the theorems that ensue) is "mathematically interesting" - not whether the results matches someone's idea of "reality". If physicists (or anyone else) want to come along and apply some of these models to their interests, then so be it. But proximity to "reality" is not the criteria mathematicians necessarily use for deciding what to pursue. The switch from mathematics being seen as "science" to what I have described above is well epitomized by what happened in plane geometry during the middle of the nineteenth century. Compare Euclidean, Lobachevsky and Riemann geometries. The idea began to emerge to play around with different sets of postulates, reason from there (these are the theorems) and see if you get a result that one can "marvel at". Grant sarbajit roy wrote: Actually I can follow Glen's line of reasoning (I think). ============================================================ 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 |
Re: Why are there theorems?
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In reply to this post by Sarbajit Roy (testing)
sarbajit roy wrote circa 10-04-26 10:59 AM:
> Actually I can follow Glen's line of reasoning (I think). > > For example, the way Maths works is that a "theorem" is "proved" by > trying to prove a "conjecture". When that approach fails you end up > proving a "special case" of the conjecture - which in turn gets elevated > to its own status as a "theorem". "Proving" Fermat's Last Theorem took > 3 centuries and generated an equal number of theorems for mathematicians > to solve/prove. The ultimate perpetual machine to keep mathematicians > employed till either the existence of "God" (the grand unified theorem > of everything) is proved or we have 33 billion gods (theorems) as we do > in India. Yes, exactly. Thanks, Sarbajit. The structure exposed (if you're platonic about it) by the theorems is there because the mathematicians were trying to do something. In this case, prove FLT. As we've said multiple times in this thread, some structure is always there, even as a consequence of the simplest systems. What makes the exposure in one domain greater than that of another domain is the amount and purpose of the people working in that domain. Why is there "hidden" structure? All structure is hidden until it's exposed. Once it's exposed, it's not hidden. What exposes the previously hidden? Attention! Focus! So, again, the answer to your question is to follow the purpose and motivation of the author's(s') rhetoric. -- glen e. p. ropella, 971-222-9095, http://agent-based-modeling.com ============================================================ 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 |
Re: Why are there theorems?
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In reply to this post by glen e. p. ropella-2
<SCREEENNNCKK>
(The sound of Hell freezing over.) On Mon, Apr 26, 2010 at 7:48 AM, glen e. p. ropella <[hidden email]> wrote:
Nicholas Thompson wrote circa 04/25/2010 01:50 PM:
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In reply to this post by Russ Abbott
doug,
Is THAT what it sounds like?
A bit louder than the sound of one hand clapping.
N
Nicholas S. Thompson
Emeritus Professor of Psychology and Ethology,
Clark University ([hidden email])
http://www.cusf.org [City University of Santa Fe]
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In reply to this post by Sarbajit Roy (testing)
On 26 Apr 2010 at 23:29, sarbajit roy wrote:
> Actually I can follow Glen's line of reasoning (I think). > > For example, the way Maths works is that a "theorem" is "proved" by trying > to prove a "conjecture". When that approach fails you end up proving a > "special case" of the conjecture - which in turn gets elevated to its own > status as a "theorem". There's at least one other scenario. The "Poincar\'e Conjecture" was originally just about manifolds of dimension 3. Through the late 1950s, there were many attempts to prove it, with lots of good mathematics generated in the process (and a fair amount of pretty bad stuff, too); the attempts were all (as far as I've ever heard) directed just at that case--dimension 3. In the early 1960s, a pack of (then-)young mathematicians (led by Stephen Smale, but using a variety of techniques besides his) generalized the conjecture, in a natural (and retrospectively obvious) way, to manifolds of every dimension--1 and 2 (where the generalizations had been known before Poincar\'e came along), 3, 4, 5, 6, 7 and up-- and then proved the *other* special cases: first, 7 and up, then 6, then 5. Until the 1980s, 3 and 4 remained unproved; then 4 fell (sort of...there's something left to do). It was only in the last few years that the original conjecture, for dimension 3, was proved (using an enormous number of things that had been developed since the 1960s, some for that or similar purposes, others for quite different purposes). That is, what was first proved had not been conjectured before it was proved (namely, the cases 7 and up); it was a theorem that *historically* "comes out of nowhere" in some sense (even though (a) it certainly came from *somewhere*, namely, "Morse Theory"-- which had, however, been invented for entirely different purposes, namely, problems in the calculus of variations, and was originally only applied to INFINITE-DIMENSIONAL manifolds!; and (b) retrospectively, in "history-as-it-might-have-been", it would have made sense for Poincar\'e to have made his conjecture for all dimensions in the first place...but he didn't, nor did anyone else until Smale came along). Lee Rudolph ============================================================ 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 |
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