Confessions of a Mathemechanic.
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Confessions of a Mathemechanic.
 
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OK. I now confess it: I love math, and feel its a great, very concrete
(hence mechanical) way to work out things, to understand and press on. I have not yet found its peer. Many among us, apparently, feel math is somehow lacking and are building up a fortress to defend against it. I am not of that persuasion. Its a tool, and a good one. No one who accepts mathematics as it is, however, considers it a point of philosophy. We do not argue about it, we try to grasp it. Arguing about it is for those of us who cannot understand it. -- 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: Confessions of a Mathemechanic.
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There are different loves and only
one word for them all. Maybe someone's is better than mine, but, please, don't
burn me like Giordano Bruno at the stake :-) --Mikhail
----- Original Message -----
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Re: Confessions of a Mathemechanic.
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In reply to this post by Owen Densmore
With it, not about it.
Using Math, we open up new worlds, unify conflicts, create new horizons. Oh yeah, you can calculate with it too. Nice, but not necessary. The world may be made of it. Nice, but not necessary. We might be good at it. Nice, but not necessary. C. Owen Densmore wrote: > OK. I now confess it: I love math, and feel its a great, very concrete > (hence mechanical) way to work out things, to understand and press > on. I have not yet found its peer. > > Many among us, apparently, feel math is somehow lacking and are > building up a fortress to defend against it. > > I am not of that persuasion. Its a tool, and a good one. > > No one who accepts mathematics as it is, however, considers it a point > of philosophy. We do not argue about it, we try to grasp it. > > Arguing about it is for those of us who cannot understand it. > > -- 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 > > ============================================================ 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: Confessions of a Mathemechanic.
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In reply to this post by Owen Densmore
Owen Densmore wrote:
> Arguing about it is for those of us who cannot understand it. Hmmm. So no mathematician can also be a philosopher and no philosopher can also be a mathematician. That's an odd position to take in a community of inter-disciplinary people. [grin] I tend to think of all subjects as intertwined to some degree. Sure, there is a kind of "trade school" mathematics (or any subject, really) where people just want to do their job, get their pay, and go home. But there is also a kind of integrative mathematics where one can be both (relatively ;-) facile with the mechanics of math _and_ explore the limits of math. So, I totally reject the claim that arguing about math is for people who can't understand math. Rather, I think "arguing" is a method for learning and comparing one's baroque conceptions to others'. Those of us who don't want to participate should, well ..., not participate. It's also useful to think about the following _old_ bit of wisdom: "Some one will say: Yes, Socrates, but cannot you hold your tongue, and then you may go into a foreign city, and no one will interfere with you? Now I have great difficulty in making you understand my answer to this. For if I tell you that to do as you say would be a disobedience to the God, and therefore that I cannot hold my tongue, you will not believe that I am serious; and if I say again that daily to discourse about virtue, and of those other things about which you hear me examining myself and others, is the greatest good of man, and that the unexamined life is not worth living, you are still less likely to believe me. Yet I say what is true, although a thing of which it is hard for me to persuade you." -- glen e. p. ropella, 971-219-3846, 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 |
Re: Confessions of a Mathemechanic.
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In reply to this post by Owen Densmore
Owen Densmore wrote:
> OK. I now confess it: I love math, and feel its a great, very concrete > (hence mechanical) way to work out things, to understand and press > on. I have not yet found its peer. > > Many among us, apparently, feel math is somehow lacking and are > building up a fortress to defend against it. > > I am not of that persuasion. Its a tool, and a good one. > I'm with you all the way... > No one who accepts mathematics as it is, however, considers it a point > of philosophy. We do not argue about it, we try to grasp it. > until this point... I don't tend to "argue" about mathematics unless you count that chorus of loud voices inside my head, but I do think quite a bit about it, about it's limits, it's place, possible extensions and even alternatives to it. (some of) These discussions here aid me in that cogitation. I spend more than my share of time trying to grasp various parts of mathematics for fun and/or profit, but that doesn't stop me from having philosophical thoughts about the parts (including a top-down understanding) I do grasp. > Arguing about it is for those of us who cannot understand it. > Certainly there is a human tendency to blather on, to speculate, to pontificate (otherwise blogs and mail lists would never have emerged?) about that which we do not understand, but just because we understand something doesn't prevent us from considering it's larger implications and context. Quite the opposite as these recent threads seem to indicate to me? Just my $.02 - Steve ============================================================ 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: Confessions of a Mathemechanic.
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On Jul 17, 2008, at 10:02 AM, Steve Smith wrote:
> <snip> > Certainly there is a human tendency to blather on, to speculate, to > pontificate (otherwise blogs and mail lists would never have > emerged?) > about that which we do not understand, but just because we understand > something doesn't prevent us from considering it's larger implications > and context. > <snip> Good points, Steve. My distinction between "argue" and "grasp" was that amongst math folks you'll hear what sounds like arguing but is instead striving to grasp a difficult point. It is *not* arguing about if math "works" or "is right". And, to be absurdly concrete, consider these threads (using Nabble): 21 Mathematics and Life - Gregory Chaitin Lectures 6 Re: Friam Digest, Vol 61, Issue 16 13 Mathematics and Music 15 Mentalism and Calculus 2 Re: Friam Digest, Vol 61, Issue 13 47 Mathematics and Music 1 MentalismAndCalculus 3 Mentalism and Calculus 8 Mentalism and Calculus ---------------------------------------------- 116 Total messages ..I may have missed a few. That's a LOT of chatter. Hence Doug and I becoming confused .. it was pretty hard to follow. That, added to our not responding correctly to keep threads combined, made it basically impossible. I'd prefer a few "grasp" sessions rather than philosophic debates. -- 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: Confessions of a Mathemechanic.
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In reply to this post by Owen Densmore
On Wed, Jul 16, 2008 at 8:57 PM, Owen Densmore <[hidden email]> wrote:
I suspect a category error: was Goedel's theorem mathematics or an argument about mathematics? -- rec -- ============================================================ 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: Confessions of a Mathemechanic.
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On Jul 17, 2008, at 10:27 AM, Roger Critchlow wrote:
> On Wed, Jul 16, 2008 at 8:57 PM, Owen Densmore <[hidden email]> > wrote: >> No one who accepts mathematics as it is, however, considers it a >> point >> of philosophy. We do not argue about it, we try to grasp it. >> >> Arguing about it is for those of us who cannot understand it. >> > I suspect a category error: was Goedel's theorem mathematics or an > argument > about mathematics? The former. I do admit Gödel creates an interesting problem (not argument) for mathematicians: You MUST be careful about your axioms, and you should be aware of the problems they present. My hazy understanding of Gödel's work is that basically an axiom set can be over-specified (thus creating the potential for both T and !T being provable) or under-specified (T is true but not provable). This is old stuff for linear algebraists. All that said, how many mathematicians are halted in their tracks by Gödel, giving up all as foolish and pointless? Rather they use it as a cautionary tale, much like computer scientists dealing with decidability. -- 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 Owen Densmore
> ..I may have missed a few. That's a LOT of chatter. Yes there is a lot of "chatter", if it weren't for the chorus already living in my head, perhaps it would be absurdly irritating to me too! <grin> > Hence Doug and I > becoming confused .. it was pretty hard to follow. Oh... I misunderstood, you actually tried to follow it all! No wonder! <grin> > That, added to our > not responding correctly to keep threads combined, made it basically > impossible. > Responsible threading! > I'd prefer a few "grasp" sessions rather than philosophic debates. > > Me too. I am not unsympathetic to your (and Doug's) irritation. Many, many, many threads on this list have a ... shall we say ... "masturbatory" quality sometimes? When the circle gets too jerky for me (and I know it doesn't necessarily feel to be that for those who are involved) I often just turn away from it. - Steve ============================================================ 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: Confessions of a Mathemechanic.
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In reply to this post by Owen Densmore
In my mathematical work which involves testing model graphs as hypotheses in
evolved, recurrent neural networks, Gödel's first theorem states that there may be true models that cannot be proven as true in a formal axiomatic system. Thus, "truth" is an underdetermined state when it comes to the application of enumerable axiomatic properties in this arithmetic formal system. When we talk about true and false, we are really talking about two coupled systems 1) false - not false, with 2) true - not true. The intersection yields for "senses"[Dominic Widdows - A Mathematical Model of Word-Sense Disambiguation] of true and false - being TRUE, FALSE, UNDETERMINED, and PARADOX. Gödel's second theorem states that a formal axiomatic system is complete if and only if it is inconsistent. The tack I take is I will go for consistency over completeness any day. But that's just me, and which probably disqualifies me as a philosopher in any canonical, categorically imperative sense. I just Kant do it. Ken > -----Original Message----- > From: [hidden email] > [mailto:[hidden email]] On Behalf Of Owen Densmore > Sent: Thursday, July 17, 2008 10:41 AM > To: The Friday Morning Applied Complexity Coffee Group > Subject: Re: [FRIAM] Confessions of a Mathemechanic. > > On Jul 17, 2008, at 10:27 AM, Roger Critchlow wrote: > > On Wed, Jul 16, 2008 at 8:57 PM, Owen Densmore <[hidden email]> > > wrote: > >> No one who accepts mathematics as it is, however, considers it a > >> point of philosophy. We do not argue about it, we try to grasp it. > >> > >> Arguing about it is for those of us who cannot understand it. > >> > > I suspect a category error: was Goedel's theorem mathematics or an > > argument about mathematics? > > The former. > > I do admit Gödel creates an interesting problem (not argument) for > mathematicians: You MUST be careful about your axioms, and > you should be aware of the problems they present. > > My hazy understanding of Gödel's work is that basically an > axiom set can be over-specified (thus creating the potential > for both T and !T being provable) or under-specified (T is > true but not provable). This is old stuff for linear algebraists. > > All that said, how many mathematicians are halted in their > tracks by Gödel, giving up all as foolish and pointless? > Rather they use it as a cautionary tale, much like computer > scientists dealing with decidability. > > -- 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 ============================================================ 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: Confessions of a Mathemechanic.
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In reply to this post by Owen Densmore
Owen,
> No one who accepts mathematics as it is, however, considers it a point > of philosophy. We do not argue about it, we try to grasp it. I know what you mean, but that what you are talking about is people trying to grasp what theorems follow from given axioms; or what theorems mean; which connections one can draw between disparate areas etc... A quite different question is what axioms to adopt in the first place (foundationalism? yes? no? how?) and, as has been discussed here vividly, the relationship of axiom sets and the theorems to reality. Dismissing philosophy from mathematics does seem rather rash; the discussions in this area are quite interesting. http://plato.stanford.edu/entries/philosophy-mathematics/ And people like Benacerraf, Chihara, Field, Resnik, Shapiro etc (just to pick out a few which come to my mind immediately) have very illuminating publications. What they do is philosophy of math (not: relation math-physics), but as Glen has rightly said, everything is intertwined at one level or another, and I suspect that most of the confusion surrounding applicability of math to reality can be dissolved by getting the philosophy right (and that will include philosophy of math and traditional metaphysics). Reality is not confusing. Our mental models are often not in tune with reality, and that is what is confusing. Cheers, Günther -- Günther Greindl Department of Philosophy of Science University of Vienna [hidden email] Blog: http://www.complexitystudies.org/ Thesis: http://www.complexitystudies.org/proposal/ ============================================================ 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: Confessions of a Mathemechanic.
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In reply to this post by Kenneth Lloyd
Ken,
> proven as true in a formal axiomatic system. Thus, "truth" is an > underdetermined state when it comes to the application of enumerable It is always important to say here that "truth" in respect to Gödel is a mathematical notion (relationship structure/model and formal system), it is often wrongly invoked in philosophical discussion ("Gödel said there can be no truth .. therefor crazy idea etc") > Gödel's second theorem states that a formal axiomatic system is complete if > and only if it is inconsistent. There are perfectly complete and and consistent axiomatic systems. (propositional calculus); heck, even the mega-expressive first order logic (see the completeness theorem). http://en.wikipedia.org/wiki/Completeness_theorem Incompleteness arises when you introduce arithmetic (robinson arithmetic suffices, presburger arithmetic not; in short: you need addition and multiplication in your arithmetic -> with this you can construct gödel numbers, define recursion, and get your (first) incompleteness theorem, from which second follows easily. Cheers, Günther -- Günther Greindl Department of Philosophy of Science University of Vienna [hidden email] Blog: http://www.complexitystudies.org/ Thesis: http://www.complexitystudies.org/proposal/ ============================================================ 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: Confessions of a Mathemechanic.
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On Thu, Jul 17, 2008 at 12:04 PM, Günther Greindl <[hidden email]> wrote:
So, if we want computability, we dispense almost all the real numbers, but if we want completeness, we dispense with all the numbers. -- rec -- ============================================================ 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: Confessions of a Mathemechanic.
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In reply to this post by Günther Greindl
Günther,
I have admitted my ignorance of your domain: a philosophy of science. Apparently, this knowledge is required in FRIAM discussions. Perhaps there is a path between the philosophy behind of, and the science I use in my complexity research. A hint of this path is illustrated in Robert Bishop and Harald Altspacher's "Contextual Emergence in the Description of Properties". http://philsci-archive.pitt.edu/archive/00002934/ Among other issues of description, they state: "The description of properties at a particular level of description (including its laws) offers both necessary and sufficient conditions to rigorously derive the description of properties at a higher level. This is the strictest possible form of reduction. As mentioned above, it was most popular under the influence of positivist thinking in the mid-20th century." The positivist paradigm is inadequate in my opinion. Models can be represented by hybrids of thermodynamic network graphs and neural networks. The point I tried to make in Gödel is that models that are incomplete may be proven, what I call true in some sense is actually more not-false, and models that are unproven may be true. Models are superpositions of other models that may be refuted as demonstrably false and removed from the superposition, otherwise allowed to remain. Therefore, models exhibit uncertainty. Axioms are not models, and models cannot represent axioms. If a model is complete, the ensembles of entangled paths through the superposition will collapse to a trajectory. In my domain, a model may be considered in two types of equilibrium, both where no energy is exchanged with the context. 1) A cold dark model, thermodynamically just above 0 K where no thermodynamic coupling exists between elements - thus entropy is not generated), and 2) a dark model where the model is at thermodynamic equilibrium with its environmental context, but exchanges internal and external couplings. A dark model does generate entropy at its minimal level according to various ambient temperatures. The upshot of the general description above is that perturbations of the model can realize non-analytical solutions (of which there may be infinitely many) - which are impossible with, and completely different than, solving a problem analytically. Re: completeness your propositional calculus, I am ignorant of the concept. Ken > -----Original Message----- > From: Günther Greindl [mailto:[hidden email]] > Sent: Thursday, July 17, 2008 12:05 PM > To: [hidden email]; The Friday Morning Applied > Complexity Coffee Group > Subject: Re: [FRIAM] Confessions of a Mathemechanic. > > Ken, > > > proven as true in a formal axiomatic system. Thus, "truth" is an > > underdetermined state when it comes to the application of enumerable > > It is always important to say here that "truth" in respect to > Gödel is a mathematical notion (relationship structure/model > and formal system), it is often wrongly invoked in > philosophical discussion ("Gödel said there can be no truth > .. therefor crazy idea etc") > > > > Gödel's second theorem states that a formal axiomatic system is > > complete if and only if it is inconsistent. > > There are perfectly complete and and consistent axiomatic systems. > (propositional calculus); heck, even the mega-expressive > first order logic (see the completeness theorem). > http://en.wikipedia.org/wiki/Completeness_theorem > > Incompleteness arises when you introduce arithmetic (robinson > arithmetic suffices, presburger arithmetic not; in short: you > need addition and multiplication in your arithmetic -> with > this you can construct gödel numbers, define recursion, and > get your (first) incompleteness theorem, from which second > follows easily. > > Cheers, > Günther > > -- > Günther Greindl > Department of Philosophy of Science > University of Vienna > [hidden email] > > Blog: http://www.complexitystudies.org/ > Thesis: http://www.complexitystudies.org/proposal/ ============================================================ 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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