Re: [FRIAM] Confessions of a Mathemechanic.
On Thu, Jul 17, 2008 at 12:04 PM, Günther Greindl < [EMAIL PROTECTED]> wrote: > > > 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. > 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: [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
[EMAIL PROTECTED]
Blog: http://www.complexitystudies.org/
Thesis: http://www.complexitystudies.org/proposal/
FRIAM Applied Complexity Group listserv
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Re: [FRIAM] progress v. drift
Again (I hit the pad on my new Macbook and it sent out the e before it was finished.) Nick I believe that math, as is the case with any intellectual tool, has evolved and changed. For example: the development of calculus or algorithms or imaginary numbers Paul ** Get the scoop on last night's hottest shows and the live music scene in your area - Check out TourTracker.com! (http://www.tourtracker.com?NCID=aolmus0005000112) 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: [FRIAM] Confessions of a Mathemechanic.
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 [EMAIL PROTECTED] 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: [FRIAM] Confessions of a Mathemechanic.
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: [EMAIL PROTECTED] > [mailto:[EMAIL PROTECTED] 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 <[EMAIL PROTECTED]> > > 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: [FRIAM] progress v. drift
Don't mind me: I'm just trying out for the position of local curmudgeon (Owen's been slacking in this regard lately). ;-} --Doug On Thu, Jul 17, 2008 at 10:58 AM, glen e. p. ropella <[EMAIL PROTECTED]> wrote: > Douglas Roberts wrote: > > I assume this summary covers the "Mentalism and Calculas" thread as well? > > Allright. I said I'd shut up; but you asked a direct question. [grin] > > As Marcus points out, "thread" is not really a good word for what > happens on a healthy mailing list. It's more like a stretched out clump > of cotton than any kind of thread or noodle. When the scale is coarse, > you can see the cotton as a clump. When the scale is fine, you see all > these fibers running hither and to. > > So, kindasortamaybe, yes, the summary covers some of the Mentalism and > Calculus posts. But, mostly, no, the summary doesn't cover those posts > because those were largely about "category error" and cross-polination > between lexicons, whereas the Mathematics and XYZ clump was about how > math is or isn't distinct as a human effort. > > Then again, that's just my long-winded opinion. > > -- > 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 > 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: [FRIAM] progress v. drift
Douglas Roberts wrote: > I assume this summary covers the "Mentalism and Calculas" thread as well? Allright. I said I'd shut up; but you asked a direct question. [grin] As Marcus points out, "thread" is not really a good word for what happens on a healthy mailing list. It's more like a stretched out clump of cotton than any kind of thread or noodle. When the scale is coarse, you can see the cotton as a clump. When the scale is fine, you see all these fibers running hither and to. So, kindasortamaybe, yes, the summary covers some of the Mentalism and Calculus posts. But, mostly, no, the summary doesn't cover those posts because those were largely about "category error" and cross-polination between lexicons, whereas the Mathematics and XYZ clump was about how math is or isn't distinct as a human effort. Then again, that's just my long-winded opinion. -- 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: [FRIAM] Confessions of a Mathemechanic.
> ..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! > Hence Doug and I > becoming confused .. it was pretty hard to follow. Oh... I misunderstood, you actually tried to follow it all! No wonder! > 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: [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 <[EMAIL PROTECTED]> > 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
Re: [FRIAM] progress v. drift
Thanks, Glen. I assume this summary covers the "Mentalism and Calculas" thread as well? ;-} --Doug -- Doug Roberts, RTI International [EMAIL PROTECTED] [EMAIL PROTECTED] 505-455-7333 - Office On Thu, Jul 17, 2008 at 10:23 AM, Owen Densmore <[EMAIL PROTECTED]> wrote: > Holy cow Glen, that's GREAT, thanks. > > Maybe we should start a tradition of summarizing like this when > threads get rather long. Then Nick can put them into the wiki? > >-- Owen > > On Jul 17, 2008, at 10:08 AM, glen e. p. ropella wrote: > > > > > I'll attempt to identify the core of the recent Mathematics and XYZ > > thread, going back to Nick's original kernel: > > > > Nicholas Thompson wrote: > >> All, One of the running arguments I have with one of my favorite > >> colleagues here in Santa Fe is about whether Mathematics is (or > >> isn't) different from all other intellectual enterprises, such as > >> psychology or philosophy. in that, unlike them, mathematics "adds > >> up," in the long run. Contrary to psychologists and philosophers like > >> me, who are besotted with ephemeral traditions and ideologies, and > >> keep changing the rules of the game, mathematicians have built a > >> structure that is not subject to vicissitudes and whims of > >> intellectual history. (I hope I have represented this argument > >> fairly.) Although I have tried to give him as little comfort as > >> possible, I confess that I have been impressed more and more by this > >> argument as I continue to read accessible works on the history of > >> mathematics. > > > > The core of the question came up several times. In essence, it's > > about > > whether or not progress (or accumulation) is illusory or objectively > > real, and whether math exhibits progress more obviously than other > > domains. > > > > We fleshed out the question by claiming and counter-claiming about > > whether math is a purely social construct or whether it is (and how it > > might be) hooked directly to reality, even to the extent that reality > > may be mathematical. > > > > So, there we are. Was anything achieved in this meandering thread? > > Most certainly. Are the achievements quantifiable? Most definitely > > not. > > > > In any case, I feel the pressure to shut up for awhile. [grin] So, I > > will comply. > > > > -- > > 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 > > > > 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 > 505-670-8195 - Cell 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: [FRIAM] Confessions of a Mathemechanic.
On Wed, Jul 16, 2008 at 8:57 PM, Owen Densmore <[EMAIL PROTECTED]> 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? -- 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: [FRIAM] progress v. drift
Holy cow Glen, that's GREAT, thanks. Maybe we should start a tradition of summarizing like this when threads get rather long. Then Nick can put them into the wiki? -- Owen On Jul 17, 2008, at 10:08 AM, glen e. p. ropella wrote: > > I'll attempt to identify the core of the recent Mathematics and XYZ > thread, going back to Nick's original kernel: > > Nicholas Thompson wrote: >> All, One of the running arguments I have with one of my favorite >> colleagues here in Santa Fe is about whether Mathematics is (or >> isn't) different from all other intellectual enterprises, such as >> psychology or philosophy. in that, unlike them, mathematics "adds >> up," in the long run. Contrary to psychologists and philosophers like >> me, who are besotted with ephemeral traditions and ideologies, and >> keep changing the rules of the game, mathematicians have built a >> structure that is not subject to vicissitudes and whims of >> intellectual history. (I hope I have represented this argument >> fairly.) Although I have tried to give him as little comfort as >> possible, I confess that I have been impressed more and more by this >> argument as I continue to read accessible works on the history of >> mathematics. > > The core of the question came up several times. In essence, it's > about > whether or not progress (or accumulation) is illusory or objectively > real, and whether math exhibits progress more obviously than other > domains. > > We fleshed out the question by claiming and counter-claiming about > whether math is a purely social construct or whether it is (and how it > might be) hooked directly to reality, even to the extent that reality > may be mathematical. > > So, there we are. Was anything achieved in this meandering thread? > Most certainly. Are the achievements quantifiable? Most definitely > not. > > In any case, I feel the pressure to shut up for awhile. [grin] So, I > will comply. > > -- > 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 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: [FRIAM] Confessions of a Mathemechanic.
On Jul 17, 2008, at 10:02 AM, Steve Smith wrote: > > 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. > 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
[FRIAM] progress v. drift
I'll attempt to identify the core of the recent Mathematics and XYZ thread, going back to Nick's original kernel: Nicholas Thompson wrote: > All, One of the running arguments I have with one of my favorite > colleagues here in Santa Fe is about whether Mathematics is (or > isn't) different from all other intellectual enterprises, such as > psychology or philosophy. in that, unlike them, mathematics "adds > up," in the long run. Contrary to psychologists and philosophers like > me, who are besotted with ephemeral traditions and ideologies, and > keep changing the rules of the game, mathematicians have built a > structure that is not subject to vicissitudes and whims of > intellectual history. (I hope I have represented this argument > fairly.) Although I have tried to give him as little comfort as > possible, I confess that I have been impressed more and more by this > argument as I continue to read accessible works on the history of > mathematics. The core of the question came up several times. In essence, it's about whether or not progress (or accumulation) is illusory or objectively real, and whether math exhibits progress more obviously than other domains. We fleshed out the question by claiming and counter-claiming about whether math is a purely social construct or whether it is (and how it might be) hooked directly to reality, even to the extent that reality may be mathematical. So, there we are. Was anything achieved in this meandering thread? Most certainly. Are the achievements quantifiable? Most definitely not. In any case, I feel the pressure to shut up for awhile. [grin] So, I will comply. -- 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: [FRIAM] Confessions of a Mathemechanic.
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: [FRIAM] Confessions of a Mathemechanic.
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
