Mentalism and Calculus
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ROBERT HOLMES SAID This is based on nothing more than reading the entry on categories at http://plato.stanford.edu/entries/categories/ so please take with a pinch of salt... It seems that the tools necessary to construct category systems are severely broken. Specifically, there is no generally accepted method for distinguishing between categories. For example, the Ryle/Husserl method boils down to a highly subjective notion of whether a statement is absurd or not. That means it's perfectly possible for Nick to see a category error ("it's crazy to say that a point can have position and velocity") and me not to see one ("nothing wrong with a point having position and velocity") and we can both be right. IMHO, this means that category theory really can't tell us very much about calculus. NICK THOMPSON REPLIES I had never seen Ryle and Husserl put on opposite sides of a ratio before, so this was very much news to me. I shall study on it and post the passage on the WIKI. . We very close to talking about metaphors or models here, and my standard take on these is very like what you lay out: whether something is a good model for something else depends VERY much on where one is standing as one holds the model up against the thing-modeled. But it seems to me that category errors are more objective than that. If one defines a point as having no extension in space and time, one CANNOT in common sense give it speed and direction in the next sentence, any more than one can divide by zero. I realize that that is not quite what you calculus folks are doing, but's awful damn close.... i.e., it approaches it as the limit. This debate is posted at www.sfcomplex.org/wiki/ComplexityNoodlersCorner Nicholas S. Thompson Emeritus Professor of Psychology and Ethology, Clark University (nthompson at clarku.edu) ----- Original Message ----- From: Robert Holmes To: nickthompson at earthlink.net;The Friday Morning Applied Complexity Coffee Group Sent: 7/9/2008 9:49:11 AM Subject: Re: [FRIAM] Mentalism and Calculus This is based on nothing more than reading the entry on categories at http://plato.stanford.edu/entries/categories/ so please take with a pinch of salt... It seems that the tools necessary to construct category systems are severely broken. Specifically, there is no generally accepted method for distinguishing between categories. For example, the Ryle/Husserl method boils down to a highly subjective notion of whether a statement is absurd or not. That means it's perfectly possible for Nick to see a category error ("it's crazy to say that a point can have position and velocity") and me not to see one ("nothing wrong with a point having position and velocity") and we can both be right. IMHO, this means that category theory really can't tell us very much about calculus. Robert On 7/8/08, Nicholas Thompson <nickthompson at earthlink.net> wrote: All who have patience, Once of the classic critiques of mentalism .... the belief that behavior is caused by events in some "inner" space called the mind ... is that it involves a category error. The term "category error" arises from ordinary language philosophy (I think). You made a category error when you start talking about some thing as if it were a different sort of thing altogether. In other words, our language is full of conventions concerning the way we talk about things, and when we violate those conventions, we start to talk silly. To an anti-mentalist a "feeling" is something that arises when one palpates the world and to talk about our "inner feelings", say, is to doom ourselves to silliness. Feelings are inherently "of" other things and to talk of "feeling our own feelings" is, well, in a word, nutty. As many of you know, I have been engaged in a geriatric attempt to recover what slipped by me in my youth, the chance to understand the Calculus. As I read more and more, it became clear to me that the differential calculus was based on a huge "category error." To speak of a point as having velocity and direction one had to speak of it at if it were something that it essentially wasn't. And yet, of course, the Calculus flourishes. Now the reason I am writing is that I am not sure where to go with this "discovery." One way is to renounce my behaviorism on the ground that category errors ... any category errors ... are just fine. Another way is to start to think of the mind/behavior distinction in some way analogous to the derivative/function distinction. That mind is just the derivative of behavior. For instance, a motive, or an intention, is not some inner thing that directs behavior, but rather the limit of its behavioral direction. A third way, is to wonder about how the inventors of calculus thought about these issues. They, presumably, were steeped in mentalism and it cannot have escaped their notice that they were attributing to points qualities that points just cannot have. Many of the texts have been reading have alluded to the idea that some contemporaries ... perhaps Newton himself ... attributed to the Calculus some sort of mystic properties. I really would like to know more about that. Any intellectual historians out there???? So, I am hoping somebody will help me go in any, or all, of these directions. --Nthompson 04:14, 9 July 2008 (GMT) This noodle, and perhaps some subsequent revisions and commentary, may be found at http://www.sfcomplex.org/wiki/MentalismAndCalculus Nicholas S. Thompson Emeritus Professor of Psychology and Ethology, Clark University (nthompson at clarku.edu) ============================================================ 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 -------------- next part -------------- An HTML attachment was scrubbed... URL: http://redfish.com/pipermail/friam_redfish.com/attachments/20080709/25f8f9e7/attachment.html |
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Nick - the snippet below illustrates the key problem with invoking category errors. I think giving the infinitesimal point speed and direction makes sense and you do not. You see a category error and I do not. So how do we adjudicate? We can't: there's no objective methodology for saying if a category error exists. (BTW, appeals to 'common sense' have as much objectivity as Ryle's invocation of absurdity: not much).
So if there's no remotely objective way of even saying whether we have a category error, then it seems pointless to try and analyse calculus in terms of its category errors. Why use a tool when all the evidence suggests that the tool is broken? Robert On Wed, Jul 9, 2008 at 4:07 PM, Nicholas Thompson <[hidden email]> wrote:
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In reply to this post by Nick Thompson
Robert,
Some how this message got caught in my outbox and you went unchastised for a whole 48 hours.
No! You have gone a bridge to far, unless you are willing to rewrite the role of definitions in axiom systems.
In a system in which a definition is, "a point is a position in space lacking dimension"
you cannot have a proposition that contradicts the definition.
You just cant.
You can REWRITE your definitions, add or subtract axioms, etc, but until you do that, you are just stuck with that Euclidean definition of a point.
I assume that some mathematician is going to write me in a milllisecond and say, "Yeah, yeah. In effect, calculus changed the definition of a point, in the same way that Lobachevski and the Rieman (??) changed the definition of "parallel". . That is how progress is made, you rigid boob!" But then I want to continue to wonder (for perhaps a few more days) what implications this all might have for the concept of mind, because, under the influence of my New Realist ancesters, I have always thought of Consciousness as an extensionless point of view.
Nick
Nicholas S. Thompson
Emeritus Professor of Psychology and Ethology,
Clark University ([hidden email])
============================================================ 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
No, Robert. You have gone a bridge to far, unless you are willing to rewrite the role of definitions in axiom systems.
In a system in which a definition is, "a point is a position in space lacking dimension"
you cannot have a proposition that contradicts the definition.
You just cant.
You can REWRITE your definitions, add or subtract axioms, etc, but until you do that, you are just stuck with that Euclidean definition of a point.
I assume that some mathematician is going to write me in a milllisecond and say, "Yeah, yeah. In effect, calculus changed the definition of a point. That is how progress is made, you rigid boob!" But then I want to continue to wonder (for perhaps a few more days) what implications this might have for the concept of mind. My New Realist mentors taught me to think of consciousness as a point of view. It is a place from which the world is viewed, or at b
Nicholas S. Thompson
Emeritus Professor of Psychology and Ethology,
Clark University ([hidden email])
============================================================ 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
Nick, I think I am beginning to get a glimmer of what you are complaining about. The wording of your definition is ambiguous. How about this one from Google: a geometric element that has position but no extension; "a point is defined by its coordinates" I think you are arguing that since a point has a fixed position, it can't move. The rest of us are talking about a particle (again with no extension) that is moving from one point to another. -Roger On Jul 14, 2008, at 9:28 PM, Nicholas Thompson wrote:
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In reply to this post by Nick Thompson
Roger,
Hmmm! Interesting. Well, I feel I am just being picky, now. But pickiness is what formalisms are about, isnt it???? In my language, I would say that a point is just a position. It's a point of reference. Just as a point of view is a place from where something is seen. I dont know what it would mean to say that the point 0,0 moved? So, to some extent, I am liking this particle, thing, because a particle could be something that could occupy a point. But wait a minute!!! A particle has extension, so lets go with "pointicle". Now we can talk about the motion of this pointicle as we look at smaller and smaller samples of that motion... in fact, make those samples as small as anybody would care to imagine without making them FREEZE the motion of the pointicle. At that point, we could talk about the direction and speed of the particle over that miniscule distance. This seems wiser than to start redefining "particle" which is inherently something that has extension.
But I should be hasty to say, in talking about a category error, and insisting that there was one there, I was not mounting some sort of challange to Newton from my lofty position as a psychologist. On the contrary, I was noticing that psychologists were not the first people to encounter a category error. Newton, did, and on the whole he did rather well with it. I mean, he had a reasonably good career, don't ya think? So, perhaps, category errors play a different role than the devilish one that Ryle and others assigned to them. Perhaps there are good and bad category errors or good or bad USES of category errors. I need to THINK about this.
You have to know that I am inclined to worship mathematicians. My brother was (is, actually) one, and as I was growing up, my parents would beam encouragingly at me and tell me every day that they hoped I would be as smart as my brother. And behaviorists psychologists has always been accused of wanting to reduce psychology to mathematical physics. So, if anything, this project is about casting off these youthful illusions and coming to understand what mathematics REALLY is.
In that connection, did you have a chance to look at the passage cited at www.sfcomplex.org/wiki/MathematicsAndMusic? Here Rothstein quotes Reuben Hersh in support of the idea that mathematics has styles (Kuhnian paradigms?) just like music (or history, or art or psychology, or any of the sloppy disreputable ways that non-physicist intellectuals make their living. Because of my long standing argument with Owen Densmore about formalisms, I think Rothstein is probably taking this point of view too far, but up till now, no mathematician has read the passage that I posted and commented on it, so I don't know for sure.
Reading tonight about non-Euclidian geometries tonight, I was struck by the fact that the Peter Wolff wanted me to know that these geometries were not designed for different surfaces as I had always supposed... rather, they were explorations of what happens when one relaxes certain crucial postulates and propositions of the euclidian system. That they might have relevance to other sorts of surfaces is apparently a SECONDARY consequence. Although the switch between geometries might be a big jump for mathematicians, it is NOT the sort of thing I would identify with a Kuhnian paradigm shift. In fact, everybody seems to have been meticulous in their attempts to maintain continuity as much as possible from one geometry to another. Left to make my own judgement, I think Rothstein is just wrong. All of this would mean that Owen has been right all along. Damn!
By the way, I got through all the mathematical parts of Rothstein's book, but when he started to talk about music I had to bail. As a formalism, sheet music defeated me.
Sorry to go on. Wednesday, I have to make a run to eastern canada to be examined by my older sister who is concerned for my health, so I shall go silent soon, I promise.
I hear there have been deluges in Santa Fe. Acequias that have been dry for a decade brim full with water. Flash floods in the streets! And I am in Massachusetts. I always miss the good stuff.
Nick
Nicholas S. Thompson
Emeritus Professor of Psychology and Ethology,
Clark University ([hidden email])
============================================================ 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
Nick -
So, ummm . . . in a carefully done axiomatization of Euclidean geometry, the terms "point", "line", "plane" (among others . . .) are left explicitly *undefined* . . . See, for example, Hilbert's axiomatization as described here: There are very good reasons for leaving terms such as these explicitly undefined -- this allows a multiplicity of models for a given axiomatic system . . . A book I like on a variety of these issues is "Introduction to Model Theory and Metamathematics" by Abraham Robinson (North Holland Press, 1965) (warning: this is a real mathematics book, probably not for the faint of heart . . . :-) tom On Jul 14, 2008, at 8:28 PM, Nicholas Thompson wrote:
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In reply to this post by Nick Thompson
Nick,
Have you read Penrose's "Emperor's New Mind", "Shadows
of the Mind", and "Road to Reality".
These all explore the relationship between physics,
mathematics and how they relate to / represent the mind.
Ken
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Re: Mentalism and Calculus (step two)
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In reply to this post by Tom Carter
Nick -
OK . . . now that we recognize that terms like "point" are (should more properly be?) left intentionally undefined in the axiomatic systems, we can move to the next step . . . A term like "point" (in an axiomatic theory) is a place where we can make a (temporary?) connection between the axiomatic system and a specific model. So, for example, in R^2 (the real plane), considered as a Euclidean space (a model where the axioms of Euclidean geometry hold), we can make the linkage: (Euclidean) point -> pair of real numbers (x, y) (with this linkage, we can say the R^2 is "a Euclidean space" -- i.e., with this linkage, the axioms of Euclidean geometry hold in R^2) Or, considering R (the real line) as the place where calculus happens, we can make the (different!) linkage: (calculus) point -> function from R to R ( x(t) ) With this "calculus" linkage, we can talk about the "position of the point x at the time t1" as x(t1). We can also talk about the "position of the point x at some other time t2" as x(t2). We can then sensibly say that the "position of the point x" changed from x(t1) to x(t2), and thus it also makes sense to talk about the "point moving from x(t1) to x(t2)" and we can talk about the "velocity of the point at time t1" as d(x(t))/dt evaluated at t1 (or, in other notation, x'(t1) . . .) Note that in this case, the (elaborated) technical term "position of the point x at time t" will have the technical "definition" x(t). The first step is to recognize that being too attached to "definitions" is the origin of suffering (to misquote someone or other :-) In order to do mathematics, we need to be ready to make and unmake attachments as needed . . . Euclid "made a mistake" in thinking that he needed to define terms like "point" and "line" and so, it turns out, he didn't really "define" them, he just left us a legacy of muddled language and ideas in that area . . . people like Hilbert made great progress in clearing up the meta-mathematical muddles that Euclid had left us . . . tom ============================================================ 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: Mentalism and Calculus (The View From Nowhere)
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In reply to this post by Nick Thompson
Nick -
Have you read Thomas Nagel's "The View From Nowhere" ? You might find it amusing . . . tom On Jul 14, 2008, at 8:35 PM, Nicholas Thompson wrote:
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In reply to this post by Nick Thompson
Hey Nick,
I'm not talking about points. I don't care about points. All I'm doing is using the existence of a disagreement about points (you think one thing, I think another) and our inability to resolve it to illustrate my claim that one cannot objectively identify category errors. So identifying supposed category errors in calculus (or anything else for that matter) is probably a fruitless endeavour. Here's what you need to do to show I'm wrong:
I confidently predict that you'll not get past item #1. Ryle tried it, but his argument reduces to the one you are making: saying "It's absurd!" in ever louder tones. IMHO, that just doesn't cut it.
So send me a link to the author and his/her methodology for identifying category errors Robert On Mon, Jul 14, 2008 at 9:35 PM, Nicholas Thompson <[hidden email]> wrote: > No, Robert. You have gone a bridge to far, unless you are willing to > rewrite the role of definitions in axiom systems. > > In a system in which a definition is, "a point is a position in space > lacking dimension" > > you cannot have a proposition that contradicts the definition. > > You just cant. > > You can REWRITE your definitions, add or subtract axioms, etc, but until you > do that, you are just stuck with that Euclidean definition of a point. > > I assume that some mathematician is going to write me in a milllisecond and > say, "Yeah, yeah. In effect, calculus changed the definition of a point. > That is how progress is made, you rigid boob!" But then I want to continue > to wonder (for perhaps a few more days) what implications this might have > for the concept of mind. My New Realist mentors taught me to think of > consciousness as a point of view. It is a place from which the world is > viewed, or at b > > Nicholas S. Thompson > Emeritus Professor of Psychology and Ethology, > Clark University ([hidden email]) > > > > > > ----- Original Message ----- > From: Robert Holmes > To: [hidden email];FRIAM > Sent: 7/12/2008 6:47:34 PM > Subject: Re: [FRIAM] Mentalism and Calculus > Nick - the snippet below illustrates the key problem with invoking category > errors. I think giving the infinitesimal point speed and direction makes > sense and you do not. You see a category error and I do not. So how do we > adjudicate? We can't: there's no objective methodology for saying if a > category error exists. (BTW, appeals to 'common sense' have as much > objectivity as Ryle's invocation of absurdity: not much). > > So if there's no remotely objective way of even saying whether we have a > category error, then it seems pointless to try and analyse calculus in terms > of its category errors. Why use a tool when all the evidence suggests that > the tool is broken? > > Robert > > > > On Wed, Jul 9, 2008 at 4:07 PM, Nicholas Thompson > <[hidden email]> wrote: >> >> <snip> >> >> If one defines a point as having no extension in space and time, one >> CANNOT in common sense give it speed and direction in the next sentence >> >> <snip> >> >> Nicholas S. Thompson >> Emeritus Professor of Psychology and Ethology, >> Clark University ([hidden email]) >> ============================================================ 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
My reference is to that eminent logician, nthompson, who wrote.
that in logic,
Once you have said in your definitions,
"A point has neither extension nor direction"
you cannot start talking about a point's direction or movement with out going back to your definitions and starting over.
consider the following syllogism
(1) All swans are white
(2) Well, except the odd black one.
(3) This bird is black.
(4) This bird is a swan.
"Absurd" in this case means what it means in a reductio argument. Violates the must fundament precepts of logic.
QED
Nick
Nicholas S. Thompson
Emeritus Professor of Psychology and Ethology,
Clark University ([hidden email])
============================================================ 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 the first part, you have just demonstrated Samson
Abramsky's point:
"This dynamic aspect, the interweaving of reasoning and
action, is not adequately catered for by the static conception of
logic."
Samson Abramsky - Christopher Strachey Professor of
Computing, Oxford University (UK)
In the second, simply a variation of Quine's
Paradox.
I cannot see how the two are connected except as a
linguistic game. Then again, I am ignorant in the ways of psychology, but
comfortable in the ways of logic and mathematics.
Ken
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In reply to this post by Nick Thompson
Nick - all you are doing is shouting "Absurd!" in an ever louder voice. Link me to a methodology for assessing the existence/non-existence of a category error and I'll happily have a go at applying it for you - Robert
On Tue, Jul 15, 2008 at 9:58 AM, Nicholas Thompson <[hidden email]> wrote:
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In reply to this post by Nick Thompson
Robert,
Sorry if I am being otiose here, but I am genuinely confused.
Everything I read leads me to the belief that IF one can prove an absurdity from a set of assumptions,then something about the set must be wrong. So the procedure is to assume the truth of a suspect proposition and show that proposition, manipulated correctly by the rules of logic, leads to an absurd conclusion... you know, 2 equals 3, for instance.
Now a category error is simply attibuting properties to an entity ... a thingy of some sort ... that are inviolation of the basic definitions of that entity. In a sense, a category error .... is already a contrary to the logical system that defines it. So I am quite serious when I say that one can detect an error by checking the assertions concerning theoretical entities in a system of thought to see if any of them contradict fundamental assumptions of the system. I cannot imagine that we could disagree there.
You might (but you havent yet)challanged my implication that the category errors identified by Ryle and his ilk fit this pattern. To speak of minds as causing material events might be a category error, on my account, if somewhere "above" in the argument somebody had asserted that all material events are caused by material causes. But has anybody EVER done anything that dumb? I kinda doubt it, and therefore my argument, though correct, has no useful referents.
At that point, we could agree that there is a method for detecting category errors, but nobody has used it yet.
Or at least, I have been unable to come up with an example.
Doesnt that make more sense???
Nick
Nicholas S. Thompson
Emeritus Professor of Psychology and Ethology,
Clark University ([hidden email]
============================================================ 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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