[PEIRCE-L] Re: Stjernfelt: Chapter 9
Jon, Yes, that is exactly it, thank you so much! -- Franklin On Mon, Apr 20, 2015 at 8:30 PM, Jon Awbrey jawb...@att.net wrote: Franklin, This looks like the post you had in mind: BU: article.gmane.org/gmane.science.philosophy.peirce/15796/match=breadth+depth BU:http://permalink.gmane.org/gmane.science.philosophy.peirce/15796 Regards, Jon On 4/20/2015 8:21 PM, Franklin Ransom wrote: Jon, Ben, lists, Whoops! Sorry about that! I guess it just struck me as a Jon kind of thing to do, with the slow reads going on about Peirce's earlier logical works. I apologize for the mistake! -- Franklin On Mon, Apr 20, 2015 at 7:42 PM, Jon Awbrey jawb...@att.net wrote: Franklin, List, I think that was Ben Udell. Regards, Jon http://inquiryintoinquiry.com On Apr 20, 2015, at 7:30 PM, Franklin Ransom pragmaticist.lo...@gmail.com wrote: Cathy, Frederik, lists, Yes, Frederik, that makes sense to me. As I mentioned in my previous post, counting qualities or characters doesn't seem to be helpful. Although it should be possible to enumerate them, to a point, for the purpose of some inquiry. As I recall, Jon Awbrey in the last month or two referenced a text from Peirce about the multiplication of breadth and depth using symbols like 1, 0, and the infinity loop, to distinguish cases such as essential depth and breadth, substantial depth and breadth, the idea of nothing, the idea of being, etc. If infinity was indeed used then, Peirce had certainly contemplated infinite depth and infinite breadth, although perhaps not simply in the sense of counting with no end, but in the direct sense of being that which is without limit, so depth without limit or breadth without limit. -- Franklin On Mon, Apr 20, 2015 at 11:45 AM, Frederik Stjernfelt stj...@hum.ku.dk wrote: Dear Franklin, Cathy, Lists - A small clarification: Peirce's *BxD=A* idea, I think, should not be taken a device for the arithmetic calculation of exact information size - it is rather the proposal of a general law relating Breadth and Depth. His idea comes from the simple idea that when intension is zero, there is no information, while when extension is zero, there is also no information - and that is the relation of the two factors in a product. (It is a bit like his first Boole-inspired definition of universal quantification as a product - he defines truth as 1, falsity as 0, then, in order to be true, each single case of a universal proposition should be true - if any single one of them is false, the total product of them all will be zero.) The BXD=A idea allows him to investigate what happens if intension or extension are in- or decreased, etc. - even if not being able to express that in precise numbers. Best F Den 20/04/2015 kl. 01.14 skrev Franklin Ransom pragmaticist.lo...@gmail.com: Cathy, lists, Well, look at this way: It is possible for there to be objects in the senses which are yet not perceived, because we do not yet have any idea of what it is to which we are looking. It takes a hypothesis to introduce a new idea to us to explain what it is, which hypothesis we can then put to the test. In order to do so, we must determine what kinds of characters to look for (deduction helps here) and then look for existent objects (induction) to learn whether the purported relations between characters obtain in fact, and in this way we come to understand the thing which we are experiencing. It is of course induction which gives us more information; abduction simply gives us the idea which needs to become informed, and deduction is merely explicative, based on relating the idea to other ideas and previously gathered information regarding those ideas. Obviously, we cannot conduct induction without end, because that is a practical impossibility. Our 'sum', as you put it, far from being always an infinity, will very likely never be an infinity in practice, in whatever sense you mean to understand the application of infinity to a 'sum' of information. Of course, as an ideal, where science, the community of inquiry as such, continues to investigate, it is possible for the information of an idea to reach a much greater 'sum' than would otherwise be possible for individuals such as you or me. But it is a commonplace of science that ideas that work and continue to work are understood more thoroughly in their relations to other ideas over the course on inquiry. This means of course that not only the breadth, but also the depth of the idea continues to grow. As a result, typically, rather than tending to make comparisons moot, we start to see a hierarchy of ideas and related sciences appear. Consider this passage: The former [Cows] is a natural class, the latter [Red Cows] is not. Now one predicate more may be attached to Red Cows than to Cows; hence Mr. Mill's attempts to analyze the difference between natural and artificial
Re: Fwd: [biosemiotics:8342] Re: [PEIRCE-L] Natural Propositions, Ch. 10: Corollarial and Theorematic Experiments with Diagrams
Ben, lists, It looks like Ben's post was sent to Peirce-L, but not the biosemiotics list-serv. For biosemiotics list members, please see below for the post to which I am responding. I think that Frederik is largely assuming Peirce's terminology. Peirce uses the words 'schema' and 'diagram' pretty much interchangeably. Yes, Ben, I would have guessed as much. In connection with this, I wonder whether Peirce would have said Kant's schematization in the Critic of Pure Reason is in fact a diagrammatization of some sort. As to the rest of what you had to say, I don't find myself really in any disagreement. I see that Frederik mentioned that whether something is corollarial or theorematic is not relative to a person's intelligence. I would suppose you meant that a mathematician would have a much more advanced logic system available to think about then the (average) schoolchild, in which case your remark would make sense. While Frederik is right to point out that only whether something new or foreign is introduced is what makes the reasoning theorematic (I believe I mentioned in a previous post that this is what is signficant for Frederik, and not so much the complexity of the schema), it is also true that what logic system one is using will affect what counts as corollarial reasoning and what as theorematic; that is, it will affect whether the something new or foreign is really new or foreign to the system, and the typical schoolchild probably has a much simpler logic system to work with than the typical mathematician. Actually, it just struck me that I mentioned that the complexity is not so important for Frederik's account, and you have continued discussing complexity to show how it is important. I'm sorry, I didn't mean to imply that non-triviality is unimportant. It does get mentioned in the text at some point, but does not play a prominent role, not nearly as much as the point that something new or foreign must be introduced into the reasoning. Though, I do wonder somewhat whether non-triviality is connected to a theorem not being easily absorbed into a logic system that could turn what was originally a theorematic reasoning into a purely corollarial reasoning. Perhaps the newer or more foreign the idea, the more nontrivial and fecund it may turn out to be? -- Franklin On Mon, Apr 20, 2015 at 4:18 PM, Benjamin Udell bud...@nyc.rr.com wrote: Franklin, lists, I think that Frederik is largely assuming Peirce's terminology. Peirce uses the words 'schema' and 'diagram' pretty much interchangeably. Here are some key quotes on which Frederik is basing his discussion of the theormatic-corollarial distinction. http://www.commens.org/dictionary/term/corollarial-reasoning I once did a summary (footnoted with online links) of key points (at least as they seemed to me at the time); here it is with a few adjustments of the links: Peirce argued that, while finally all deduction depends in one way or another on mental experimentation on schemata or diagrams,*[1]* still in corollarial deduction it is only necessary to imagine any case in which the premisses are true in order to perceive immediately that the conclusion holds in that case, whereas theorematic deduction is deduction in which it is necessary to experiment in the imagination upon the image of the premiss in order from the result of such experiment to make corollarial deductions to the truth of the conclusion.*[2]* He held that corollarial deduction matches Aristotle's conception of direct demonstration, which Aristotle regarded as the only thoroughly satisfactory demonstration, while theorematic deduction (A) is the kind more prized by mathematicians, (B) is peculiar to mathematics,*[1]* and (C) involves in its course the introduction of a lemma or at least a definition uncontemplated in the thesis (the proposition that is to be proved); in remarkable cases that definition is of an abstraction that ought to be supported by a proper postulate.*[3]* [1] Peirce, C. S., from section dated 1902 by editors in the Minute Logic manuscript, Collected Papers v. 4, paragraph 233, quoted only in part http://www.commens.org/dictionary/entry/quote-minute-logic-chapter-iii-simplest-mathematics in Corollarial Reasoning in the Commens Dictionary of Peirce's Terms, 2003–present, Mats Bergman and Sami Paavola, editors, University of Helsinki. FULL QUOTE: https://archive.org/stream/TheWorldOfMathematicsVolume3/Newman-TheWorldOfMathematicsVolume3#page/n366/mode/1up in The World of Mathematics, Vol. 3, p. 1776. [2] Peirce, C. S., the 1902 Carnegie Application, published in The New Elements of Mathematics, Carolyn Eisele, editor, quoted in Corollarial Reasoning http://www.commens.org/dictionary/entry/quote-carnegie-institution-correspondence-4 in the Commens Dictionary of Peirce's Terms, also transcribed by Joseph M. Ransdell, see From Draft A - MS L75.35-39 in Memoir 19
Re: [PEIRCE-L] Stjernfelt: Chapter 9
Cathy, Frederik, lists, Yes, Frederik, that makes sense to me. As I mentioned in my previous post, counting qualities or characters doesn't seem to be helpful. Although it should be possible to enumerate them, to a point, for the purpose of some inquiry. As I recall, Jon Awbrey in the last month or two referenced a text from Peirce about the multiplication of breadth and depth using symbols like 1, 0, and the infinity loop, to distinguish cases such as essential depth and breadth, substantial depth and breadth, the idea of nothing, the idea of being, etc. If infinity was indeed used then, Peirce had certainly contemplated infinite depth and infinite breadth, although perhaps not simply in the sense of counting with no end, but in the direct sense of being that which is without limit, so depth without limit or breadth without limit. -- Franklin On Mon, Apr 20, 2015 at 11:45 AM, Frederik Stjernfelt stj...@hum.ku.dk wrote: Dear Franklin, Cathy, Lists - A small clarification: Peirce's *BxD=A* idea, I think, should not be taken a device for the arithmetic calculation of exact information size - it is rather the proposal of a general law relating Breadth and Depth. His idea comes from the simple idea that when intension is zero, there is no information, while when extension is zero, there is also no information - and that is the relation of the two factors in a product. (It is a bit like his first Boole-inspired definition of universal quantification as a product - he defines truth as 1, falsity as 0, then, in order to be true, each single case of a universal proposition should be true - if any single one of them is false, the total product of them all will be zero.) The BXD=A idea allows him to investigate what happens if intension or extension are in- or decreased, etc. - even if not being able to express that in precise numbers. Best F Den 20/04/2015 kl. 01.14 skrev Franklin Ransom pragmaticist.lo...@gmail.com: Cathy, lists, Well, look at this way: It is possible for there to be objects in the senses which are yet not perceived, because we do not yet have any idea of what it is to which we are looking. It takes a hypothesis to introduce a new idea to us to explain what it is, which hypothesis we can then put to the test. In order to do so, we must determine what kinds of characters to look for (deduction helps here) and then look for existent objects (induction) to learn whether the purported relations between characters obtain in fact, and in this way we come to understand the thing which we are experiencing. It is of course induction which gives us more information; abduction simply gives us the idea which needs to become informed, and deduction is merely explicative, based on relating the idea to other ideas and previously gathered information regarding those ideas. Obviously, we cannot conduct induction without end, because that is a practical impossibility. Our 'sum', as you put it, far from being always an infinity, will very likely never be an infinity in practice, in whatever sense you mean to understand the application of infinity to a 'sum' of information. Of course, as an ideal, where science, the community of inquiry as such, continues to investigate, it is possible for the information of an idea to reach a much greater 'sum' than would otherwise be possible for individuals such as you or me. But it is a commonplace of science that ideas that work and continue to work are understood more thoroughly in their relations to other ideas over the course on inquiry. This means of course that not only the breadth, but also the depth of the idea continues to grow. As a result, typically, rather than tending to make comparisons moot, we start to see a hierarchy of ideas and related sciences appear. Consider this passage: The former [Cows] is a natural class, the latter [Red Cows] is not. Now one predicate more may be attached to Red Cows than to Cows; hence Mr. Mill's attempts to analyze the difference between natural and artificial classes is seen to be a failure. For, according to him, the difference is that a real kind is distinguished by unknown multitudes of properties while an artificial class has only a few determinate ones. Again there is an unusual degree of accordance among naturalists in making Vertebrates a natural class. Yet the number of predicates proper to it is comparatively small (NP, p.238, quoting Peirce). We can see here that further simplifications are introduced, so taking what is learned about various vertebrates, a new idea, that of vertebrates, appears which simplifies the characters involved. Conversely, species under vertebrates will become much more determinate in terms of their characters, but be simplified with respect to their extension. You said above: Under synechism every real object has an infinite number of attributes, and every meaningful predicate or general term effectively has an
[PEIRCE-L] Re: Stjernfelt: Chapter 9
Franklin, List, I think that was Ben Udell. Regards, Jon http://inquiryintoinquiry.com On Apr 20, 2015, at 7:30 PM, Franklin Ransom pragmaticist.lo...@gmail.com wrote: Cathy, Frederik, lists, Yes, Frederik, that makes sense to me. As I mentioned in my previous post, counting qualities or characters doesn't seem to be helpful. Although it should be possible to enumerate them, to a point, for the purpose of some inquiry. As I recall, Jon Awbrey in the last month or two referenced a text from Peirce about the multiplication of breadth and depth using symbols like 1, 0, and the infinity loop, to distinguish cases such as essential depth and breadth, substantial depth and breadth, the idea of nothing, the idea of being, etc. If infinity was indeed used then, Peirce had certainly contemplated infinite depth and infinite breadth, although perhaps not simply in the sense of counting with no end, but in the direct sense of being that which is without limit, so depth without limit or breadth without limit. -- Franklin On Mon, Apr 20, 2015 at 11:45 AM, Frederik Stjernfelt stj...@hum.ku.dk wrote: Dear Franklin, Cathy, Lists - A small clarification: Peirce's BxD=A idea, I think, should not be taken a device for the arithmetic calculation of exact information size - it is rather the proposal of a general law relating Breadth and Depth. His idea comes from the simple idea that when intension is zero, there is no information, while when extension is zero, there is also no information - and that is the relation of the two factors in a product. (It is a bit like his first Boole-inspired definition of universal quantification as a product - he defines truth as 1, falsity as 0, then, in order to be true, each single case of a universal proposition should be true - if any single one of them is false, the total product of them all will be zero.) The BXD=A idea allows him to investigate what happens if intension or extension are in- or decreased, etc. - even if not being able to express that in precise numbers. Best F Den 20/04/2015 kl. 01.14 skrev Franklin Ransom pragmaticist.lo...@gmail.com: Cathy, lists, Well, look at this way: It is possible for there to be objects in the senses which are yet not perceived, because we do not yet have any idea of what it is to which we are looking. It takes a hypothesis to introduce a new idea to us to explain what it is, which hypothesis we can then put to the test. In order to do so, we must determine what kinds of characters to look for (deduction helps here) and then look for existent objects (induction) to learn whether the purported relations between characters obtain in fact, and in this way we come to understand the thing which we are experiencing. It is of course induction which gives us more information; abduction simply gives us the idea which needs to become informed, and deduction is merely explicative, based on relating the idea to other ideas and previously gathered information regarding those ideas. Obviously, we cannot conduct induction without end, because that is a practical impossibility. Our 'sum', as you put it, far from being always an infinity, will very likely never be an infinity in practice, in whatever sense you mean to understand the application of infinity to a 'sum' of information. Of course, as an ideal, where science, the community of inquiry as such, continues to investigate, it is possible for the information of an idea to reach a much greater 'sum' than would otherwise be possible for individuals such as you or me. But it is a commonplace of science that ideas that work and continue to work are understood more thoroughly in their relations to other ideas over the course on inquiry. This means of course that not only the breadth, but also the depth of the idea continues to grow. As a result, typically, rather than tending to make comparisons moot, we start to see a hierarchy of ideas and related sciences appear. Consider this passage: The former [Cows] is a natural class, the latter [Red Cows] is not. Now one predicate more may be attached to Red Cows than to Cows; hence Mr. Mill's attempts to analyze the difference between natural and artificial classes is seen to be a failure. For, according to him, the difference is that a real kind is distinguished by unknown multitudes of properties while an artificial class has only a few determinate ones. Again there is an unusual degree of accordance among naturalists in making Vertebrates a natural class. Yet the number of predicates proper to it is comparatively small (NP, p.238, quoting Peirce). We can see here that further simplifications are introduced, so taking what is learned about various vertebrates, a new idea, that of vertebrates, appears which simplifies the characters involved. Conversely, species under vertebrates will become much
Re: [PEIRCE-L] Natural Propositions, Ch. 10: Corollarial and Theorematic Experiments with Diagrams
Frederik, lists, I'm dissatisfied with my previous post in this thread, I feel like I've missed the forest for the trees. While I'm not convinced that there's a theorematic applied deduction in the Wegener example, still, the idea of continental drift is not merely a simplifying explanation of the fit between continental coastlines, it's also an idea that anybody would call nontrivial. It involves a complex new idea, and, if true (as it turned out to be), would foreseeably be a basis and foundation for much further discovery. Its nontriviality doesn't give it intrinsic abductive merit in the way that its plausibility does, but said nontriviality still makes it something to be prized if it pans out (as it did). But so far, that's the nontriviality of prospective discoveries, what about a nontriviality of how one got to the abduction of continental drift? I'm trying to think of some parallelism between its abduction and theorematic deduction, so as to analogize the idea of abductive nontriviality to deductive theorematicity. Roughly, something involving nontrivial changes of standing beliefs about geology, changes equivalent to the idea of continental drift. Well, when even I think I'm talking too much, it's time I call it a day. Best, Ben On 4/20/2015 12:58 PM, Benjamin Udell wrote: Frederik, lists, You wrote, My argument, which I may not have made sufficiently clear in the chapter, is that the small step from having spatiotemporal cell phone information represented in long lists of coordinates - and to synthesize that same information in one geographical map, is a corollarial step. Yes, I agree that it's corollarial. I see that I didn't make my agreement clear, sorry about that. I saw it as a case where corollarial reasoning makes clear that which, as a practical matter, was quite obscure. I took it as a case of mere complication, as opposed to complexity in the sense of nontriviality. You wrote, I admit it is more difficult, in general, to precisely extend the corollarial/theorematic distinction to applied cases - but as you can see I did the attempt picking map examples. The central problem for my pov seems to be that in applied cases you should not only include what is given in axiomatics (topographical maps largely respecting Euclidean geometry) but also in the more or less implicit ontological assumptions in the area of application - this is why i count Wegener's map experiment as theorematic. Taken as pure geometry, it is a trivial translation to move South America eastwards to compare its coastline with Africa's - but in terms of geology, it requires the addition of a new idea - namely that continents may move. I think that the mathematical shifting of the South America map to compare its coastline with the Africa map's coastline is required, or at any rate helpful, in order to bring a surprising geological phenomenon to light - the good match. One has ignored geological assumptions in order to do this, and then, looking at it and bringing geological assumptions back into account, one is surprised. Then the idea of an actual geological movement of continents is considered and abduced as a simplifying explanation because it sheds some light how the good fit could have physically happened as a matter of course. 1. Here in the abduction, unlike in theorematic deduction, one has _/concluded/_ in the new element, as opposed to introducing it in order to conclude in something else. 2. Concluding abductively in the proposition of such geological movement, amounts to assuming it as a basis for deducing conceivable practical implications. Here in the predictive deduction, the new element (the hypothetical assumption), unlike the new element that makes a deduction theorematic, is asserted in the original conditions of the deductive problem, not introduced in some construction along the way. 3. The inductive tests of the deduced predictions will tend to support or overturn the hypothetical assumption; now the new element, the hypothetical assumption, is that which the reasoning would conclude by supporting or overturning, the reasoning's thesis, unlike in a theorematic deduction. If we look at the above inquiry cycle as a whole, then the hypothetical assumption, although it is a new element, is that which one seeks to confirm or overturn, and that is not the role played by the new element in a theorematic deduction. Yet, - a theorematic deduction's introduction of a new or outside idea (not part of the problem's explicit conditions or contemplated in the thesis that is to be proved) reminds one of an abductive inference's introduction of a new or outside idea to become the conclusion. And I agree that that's a phenomenon worth explaining. All I can think of at the moment is that it's as if the new element in the theorematic deduction were introduced by a higher-level
[PEIRCE-L] Re: [biosemiotics:8369] Re: Natural Propositions, Ch. 10:
Frederik, lists, I'm not sure, but this appears in my email as a separate thread, having copied posts that I sent to the other thread. Since Frederik replied to my posts on this one, I suppose I'll reply here for now. If this doesn't appear as a new thread to anyone else, then please ignore my comment. Just to be clear, I think that this will definitely be a case of we will just have to agree to disagree. Frederik, you are clearly professionally committed to the a priori; I am constitutionally committed to radical empiricism. Now that you are forewarned about that, I'll say a couple of things about my point of view. I'm not so sure that empiricists like myself have an a priori fear of the a priori. When I look at the philosophy of transcendentalism and its results, the fear strikes me as quite experience-based. One can also think about Peirce's remarks in The Fixation of Belief about the method of the a priori. I'm not, as an empiricist, particularly impressed with logical positivism as a form of empiricism. I believe it a commonplace in classical pragmatism that the theory of experience at play in pragmatism is not the atomistic approach of the British empiricists or their inheritors in logical positivism/empiricism. My understanding is that whether we are talking about Peirce, James, or Dewey, experience is not conceived on the model of a series of distinct, discrete sense impressions or sense-data. Instead, experience is much more complex, in which conjunction and continuity are just as much found in the experience as are disjunction and discreteness--we do not require some outside source to make our experiences appear connected for us in the first place. Certainly the mind works to bring connection and continuity to its experiences. But it does not do this ex nihilo; such connections and continuities work to extend in novel ways connections and continuities already experienced--the mind generalizes what it has been given to work with. So far as I see it, this is the empiricism that classical pragmatism is based upon, and is part of what my take on empiricism amounts to. I'm not entirely sure what is meant by dependence structures of objectivity. I also find your ascription of fallibilism to a priori knowledge as bizarre. Rather than discuss what you have had to say further (this post would become inordinately long), I think it would be best to simplify the matter. Suppose I have a surprising experience, and then develop a hypothesis to explain that experience. Once I have the idea in hand from the hypothesis, I deduce consequences from this hypothesis to the point that I now know how to put the hypothesis to inductive experimentation. Now, at this point, I have not yet conducted any inductions. Is this process, from the gaining of a hypothesis to the deduction of consequences, altogether a priori on your account? -- Franklin On Mon, Apr 20, 2015 at 12:22 PM, Frederik Stjernfelt stj...@hum.ku.dk wrote: Dear Franklin, lists - Sorry for having rattled Franklin's empiricist sentiments with references to the a priori! Empiricists seem to have an a priori fear of the a priori … but no philosophy of science has, as yet, been able to completely abolish the a priori - even logical positvism had to admit logic as a remaining a priori field (reinterpreting that as tautologies, that is true). I should probably have given a note here to my own stance on the a priori - for the interested, I wrote a bit about it in ch. 8 of Diagrammatology (2007). My take on it there comes more from the early Husserl than from Peirce: the a priori has nothing to do with Kantian subjectivity, rather, it consists in dependence structures of objectivity - this makes it subject to fallibilism - the a priori charts necessities - these come in two classes, formal ontology and material ontology - the former holds for all possible objects, the latter for special regions of reality (like physics, biology, society) - no discipline can function without more or less explicit conceptual networks defining their basic ideas - being fallibilist, a priori claims develop with the single scientific disciplines … I happen to think this Husserlian picture (for a present-day version, see Barry Smith) is compatible with Peirce's classification of the sciences where, as it is well known, the upper echelon is taken to be a priori in the sense of not at all containing empirical knowledge while the lower, positive levels inherit structures from those higher ones, co-determining the way they organize and prioritize their empirical material. So, it is in this sense of material ontology that I speak of biogeographical ontology and and the ontology of human culture development involved in Diamond's argument. Given these assumptions, Diamond's argument, so I argue, is a priori. His conclusion that Eurasia privileges the spread of domesticated animals does not depend on the empirical investigation of early cultural
Re: [biosemiotics:8342] Re: [PEIRCE-L] Natural Propositions, Ch. 10: Corollarial and Theorematic Experiments with Diagrams
Dear Ben, lists - Thanks for two mails. The first largely resumes parts of my chapter and indeed Peirce's basic ideas of theorematicity - although it is not entirely correct that P saw his distinction as relative to intellect so that which is corollarial to a grownup will be theorematic to a child. Peirce insisted that theorematicity consists in the addition of something new to a problem - an additional object, manipulation, abstraction, perspective, etc. If once you know which such addition to add in the single case, the remaining problem becomes corollarial, but that has nothing to do with the intellect of the reasoner. A stupid but well-informed person may repeat Euclid's theorematic proof of the angular sum of the triangle, while an uninformed genius may be unable to find that proof - and still the proof requires the addition of auxiliary lines to the triangle no matter what. But you address other important things. It seems as if, to some degree, Peirce without saying it assumes something like the discovery/justification distinction. When saying mathematical reasoning is deductive, this seems to be a justification claim merely, because in the actual procedure of searching for the proof of a theorem, Peirce realizes there may be an abductive trial-and-error phase, particularly in the theorematic cases where it is not evident which new element to add to your problem (is there also something akin to an inductive phase in mathematical proofmaking, e.g. when mathematicians compare and evaluate their result with respect to its potential effects in other areas of math?). So even if mathematics is the science that proceeds by deductive reasoning, there are non-deductive phases in it (discovery), even if the results are deductively valid (justification). My idea with Wegener's map was, of course, to find a theorematic example from applied math. In such a case both the mathematical formalism (here, approximately Euclidean geometry) and the basic assumptions of the material field (geology) must be part of the status quo to which a new element, manipulation, principle etc. should be added. The transformation making the two continent coasts meet is trivial in the Euclidean sense, but the change in underlying geological ontology (from the axiom that continents are eternally stable to the axiom that they float on the surface of the earth) indeed requires the addition of a new idea. In some sense, the radicality of this new idea is eased by the triviality of the transformation in purely geometrical terms. What prompted the idea of generalizing the mathematical notions of corollarial/theorematic to the applied sciences, of course, is Peirce's classificaiton of the sciences where math is number one, implying that all other sciences wihtout exception use mathematical structures - but simultaneously that generalization cannot take place without introdcuding basic principles of those lower sciences, thereby modifying the corr/theor. distinction to some degree because it now has to involve ontological assumptions regarding positive knowledge. But still I think it makes good sense. Best F Frederik, lists, I'm dissatisfied with my previous post in this thread, I feel like I've missed the forest for the trees. While I'm not convinced that there's a theorematic applied deduction in the Wegener example, still, the idea of continental drift is not merely a simplifying explanation of the fit between continental coastlines, it's also an idea that anybody would call nontrivial. It involves a complex new idea, and, if true (as it turned out to be), would foreseeably be a basis and foundation for much further discovery. Its nontriviality doesn't give it intrinsic abductive merit in the way that its plausibility does, but said nontriviality still makes it something to be prized if it pans out (as it did). But so far, that's the nontriviality of prospective discoveries, what about a nontriviality of how one got to the abduction of continental drift? I'm trying to think of some parallelism between its abduction and theorematic deduction, so as to analogize the idea of abductive nontriviality to deductive theorematicity. Roughly, something involving nontrivial changes of standing beliefs about geology, changes equivalent to the idea of continental drift. Well, when even I think I'm talking too much, it's time I call it a day. Best, Ben : Franklin, lists, I think that Frederik is largely assuming Peirce's terminology. Peirce uses the words 'schema' and 'diagram' pretty much interchangeably. Here are some key quotes on which Frederik is basing his discussion of the theormatic-corollarial distinction. http://www.commens.org/dictionary/term/corollarial-reasoning I once did a summary (footnoted with online links) of key points (at least as they seemed to me at the time); here it is with a few adjustments of the links: Peirce argued that, while finally all deduction
Re: Fwd: [biosemiotics:8342] Re: [PEIRCE-L] Natural Propositions, Ch. 10: Corollarial and Theorematic Experiments with Diagrams
Franklin, lists, I think that Frederik is largely assuming Peirce's terminology. Peirce uses the words 'schema' and 'diagram' pretty much interchangeably. Here are some key quotes on which Frederik is basing his discussion of the theormatic-corollarial distinction. http://www.commens.org/dictionary/term/corollarial-reasoning I once did a summary (footnoted with online links) of key points (at least as they seemed to me at the time); here it is with a few adjustments of the links: Peirce argued that, while finally all deduction depends in one way or another on mental experimentation on schemata or diagrams,^*[1]* still in corollarial deduction it is only necessary to imagine any case in which the premisses are true in order to perceive immediately that the conclusion holds in that case, whereas theorematic deduction is deduction in which it is necessary to experiment in the imagination upon the image of the premiss in order from the result of such experiment to make corollarial deductions to the truth of the conclusion.^*[2]* He held that corollarial deduction matches Aristotle's conception of direct demonstration, which Aristotle regarded as the only thoroughly satisfactory demonstration, while theorematic deduction (A) is the kind more prized by mathematicians, (B) is peculiar to mathematics,^*[1]* and (C) involves in its course the introduction of a lemma or at least a definition uncontemplated in the thesis (the proposition that is to be proved); in remarkable cases that definition is of an abstraction that ought to be supported by a proper postulate.^*[3]* [1] Peirce, C. S., from section dated 1902 by editors in the Minute Logic manuscript, Collected Papers v. 4, paragraph 233, quoted only in part http://www.commens.org/dictionary/entry/quote-minute-logic-chapter-iii-simplest-mathematics in Corollarial Reasoning in the Commens Dictionary of Peirce's Terms, 2003–present, Mats Bergman and Sami Paavola, editors, University of Helsinki. FULL QUOTE: https://archive.org/stream/TheWorldOfMathematicsVolume3/Newman-TheWorldOfMathematicsVolume3#page/n366/mode/1up in The World of Mathematics, Vol. 3, p. 1776. [2] Peirce, C. S., the 1902 Carnegie Application, published in The New Elements of Mathematics, Carolyn Eisele, editor, quoted in Corollarial Reasoning http://www.commens.org/dictionary/entry/quote-carnegie-institution-correspondence-4 in the Commens Dictionary of Peirce's Terms, also transcribed by Joseph M. Ransdell, see From Draft A - MS L75.35-39 in Memoir 19 http://www.iupui.edu/~arisbe/menu/library/bycsp/l75/ver1/l75v1-06.htm#m19 http://www.iupui.edu/%7Earisbe/menu/library/bycsp/l75/ver1/l75v1-06.htm#m19 (once there, scroll down). [3] Peirce, C. S., 1901 manuscript On the Logic of Drawing History from Ancient Documents, Especially from Testimonies', The Essential Peirce v. 2, see p. 96. See quote http://www.commens.org/dictionary/entry/quote-logic-drawing-history-ancient-documents-especially-testimonies-logic-histor-5 in Corollarial Reasoning in the Commens Dictionary of Peirce's Terms. The introduction of an idea beyond the explicit conditions of a problem and not contemplated in the thesis to be proved is precisely a 'complexifying' step. One might think of it as a leveraging of imagination to deepen understanding, by which vague remark I'm trying to get at the idea that such complexity is very different from the tedious complication of hundreds or thousands of trivial computations, computations that need to be done sometimes even in pure mathematics, where it is known as 'brute force'. Tedious computation used to be done by people called 'computers' up until computing machines came into use; part of Peirce's burden at the Coast Survey was that there came a time when he had to do his own tedious, lengthy computations and, worse, he found that his computing power was no longer what it was when he was younger; errors crept in. In CP 4.233 (again https://archive.org/stream/TheWorldOfMathematicsVolume3/Newman-TheWorldOfMathematicsVolume3#page/n366/mode/1up) in The Essence of Mathematics, Peirce says, [] Just now, I wish to point out that after the schema has been constructed according to the precept virtually contained in the thesis, the assertion of the theorem is not evidently true, even for the individual schema; nor will any amount of hard thinking of the philosophers' corollarial kind ever render it evident. Thinking in general terms is not enough. It is necessary that something should be DONE. In geometry, subsidiary lines are drawn. In algebra permissible transformations are made. Thereupon, the faculty of observation is called into play. Some relation between the parts of the schema is remarked. But would this relation subsist in every possible case? Mere corollarial reasoning will sometimes assure us
[PEIRCE-L] Re: [biosemiotics:8115] Pragmatism About Theoretical Entities
Dear Jon, lists, Sorry again for an answer a bit belated. Den 17/03/2015 kl. 20.22 skrev John Collier colli...@ukzn.ac.zamailto:colli...@ukzn.ac.za: Thanks, Frederik. I think that to properly call a view Platonist it must reject the existence of particulars in favour of universals. Russell fits this description because fairly early in his (long) career he explicitly rejected particulars, and argued that instances were combinations of “compossible” universals (whence his structuralism, and perhaps a “contraction to individuals”). One can be a Platonist about some domains but not others. For example there are Platonists about numbers and other parts of mathematics (Gödel), and there are the opposite about numbers (Mill and Phillip Kitcher, for example), but not necessarily about scientific laws. Hartrey Field famously rejected numbers altogether, at least with respect to the world of science. His motivation was an extreme nominalism. Peirce was not a Platonist in the sense above, with his distinction between existing and being real. I suppose (no reason to think otherwise so far) that this extends to signs. You're right. But Peircean Existence is not the same thing as individuals being completely determinate - here, he instead invokes the Scotist notion of haecceity to account for existence. As to his more extreme, he means stronger rather than weaker. I think Peirce's idea is that when Scotus says that universals are contracted in particulars, Scotus thereby makes place ofr universals to be reducible to the classes of individuals instantiating them - here Peirce's continuity theory of universals make universals exceed any possible number of realizations of them, hence more strong. But I am not quite sure how he slices it to get a position that is more extreme than (weaker than?) Duns Scotus, which is pretty weak, but still allows universals that are not instantiated. Or perhaps I am missing what he means by ‘extreme’ here. I parted company with my coauthors of All Things Must Go over the existence of structures that don’t interact, for of which in principle we could have no knowledge. This seemed to me to violate a Peircean principle that they started the book with, which is basically the pragmatic principle. Interesting! Best Frederik In any case, we agree on openness of universals. Regards, John - PEIRCE-L subscribers: Click on Reply List or Reply All to REPLY ON PEIRCE-L to this message. PEIRCE-L posts should go to peirce-L@list.iupui.edu . To UNSUBSCRIBE, send a message not to PEIRCE-L but to l...@list.iupui.edu with the line UNSubscribe PEIRCE-L in the BODY of the message. More at http://www.cspeirce.com/peirce-l/peirce-l.htm .
[PEIRCE-L] Re: Triadic Philosophy
Whatever it is. it is the only thing that transcends Reality - in the sense of defining its purpose or the goal of continuity or whatever else you wish to call it. On Mon, Apr 20, 2015 at 12:54 PM, Jon Awbrey jawb...@att.net wrote: Ah, but beauty and truth were always one. Our blurred vision only sees them as two. Making ontology purely a matter of focus. Regards, Jon On 4/20/2015 12:42 PM, Stephen C. Rose wrote: Ontology consists of what is necessary for the achievement of the fusion of truth and beauty. Books http://buff.ly/15GfdqU -- academia: http://independent.academia.edu/JonAwbrey my word press blog: http://inquiryintoinquiry.com/ inquiry list: http://stderr.org/pipermail/inquiry/ isw: http://intersci.ss.uci.edu/wiki/index.php/JLA oeiswiki: http://www.oeis.org/wiki/User:Jon_Awbrey facebook page: https://www.facebook.com/JonnyCache - PEIRCE-L subscribers: Click on Reply List or Reply All to REPLY ON PEIRCE-L to this message. PEIRCE-L posts should go to peirce-L@list.iupui.edu . To UNSUBSCRIBE, send a message not to PEIRCE-L but to l...@list.iupui.edu with the line UNSubscribe PEIRCE-L in the BODY of the message. More at http://www.cspeirce.com/peirce-l/peirce-l.htm .
Re: [PEIRCE-L] Natural Propositions, Ch. 10: Corollarial and Theorematic Experiments with Diagrams
Frederik, lists, You wrote, My argument, which I may not have made sufficiently clear in the chapter, is that the small step from having spatiotemporal cell phone information represented in long lists of coordinates - and to synthesize that same information in one geographical map, is a corollarial step. Yes, I agree that it's corollarial. I see that I didn't make my agreement clear, sorry about that. I saw it as a case where corollarial reasoning makes clear that which, as a practical matter, was quite obscure. I took it as a case of mere complication, as opposed to complexity in the sense of nontriviality. You wrote, I admit it is more difficult, in general, to precisely extend the corollarial/theorematic distinction to applied cases - but as you can see I did the attempt picking map examples. The central problem for my pov seems to be that in applied cases you should not only include what is given in axiomatics (topographical maps largely respecting Euclidean geometry) but also in the more or less implicit ontological assumptions in the area of application - this is why i count Wegener's map experiment as theorematic. Taken as pure geometry, it is a trivial translation to move South America eastwards to compare its coastline with Africa's - but in terms of geology, it requires the addition of a new idea - namely that continents may move. I think that the mathematical shifting of the South America map to compare its coastline with the Africa map's coastline is required, or at any rate helpful, in order to bring a surprising geological phenomenon to light - the good match. One has ignored geological assumptions in order to do this, and then, looking at it and bringing geological assumptions back into account, one is surprised. Then the idea of an actual geological movement of continents is considered and abduced as a simplifying explanation because it sheds some light how the good fit could have physically happened as a matter of course. 1. Here in the abduction, unlike in theorematic deduction, one has _/concluded/_ in the new element, as opposed to introducing it in order to conclude in something else. 2. Concluding abductively in the proposition of such geological movement, amounts to assuming it as a basis for deducing conceivable practical implications. Here in the predictive deduction, the new element (the hypothetical assumption), unlike the new element that makes a deduction theorematic, is asserted in the original conditions of the deductive problem, not introduced in some construction along the way. 3. The inductive tests of the deduced predictions will tend to support or overturn the hypothetical assumption; now the new element, the hypothetical assumption, is that which the reasoning would conclude by supporting or overturning, the reasoning's thesis, unlike in a theorematic deduction. If we look at the above inquiry cycle as a whole, then the hypothetical assumption, although it is a new element, is that which one seeks to confirm or overturn, and that is not the role played by the new element in a theorematic deduction. Yet, - a theorematic deduction's introduction of a new or outside idea (not part of the problem's explicit conditions or contemplated in the thesis that is to be proved) reminds one of an abductive inference's introduction of a new or outside idea to become the conclusion. And I agree that that's a phenomenon worth explaining. All I can think of at the moment is that it's as if the new element in the theorematic deduction were introduced by a higher-level or methodological abduction - 'if I introduce this idea, I might be able to deduce the thesis as a matter of course'. Best, Ben On 4/20/2015 11:31 AM, Frederik Stjernfelt wrote: Dear Ben, Franklin, lists, Den 19/04/2015 kl. 20.05 skrev Benjamin Udell bud...@nyc.rr.com mailto:bud...@nyc.rr.com: Franklin, lists, I agree with Jon, thanks for your excellent starting post. You wrote, [] Why can't corollarial reasoning, which involves observation and experimentation, reveal unnoticed and hidden relations? After all, on p.285-6, Frederik mentions the work of police detective Jorn Old Man Holm and his computer program, which Frederik describes as a practical example of corollarial map reasoning (p.285). In this example, Holm uses the corollarial reasoning to reveal information about the whereabouts of suspects. Doesn't the comparison of the map reasoning with suspects' testimony end up revealing unnoticed and hidden relations? There's a distinction that some make between complexity and mere complication. Corollarial reasonings may accumulate mere complications until the result becomes hard to see, although it involves little if any complexity in, more or less, the sense of depth or nontriviality. I don't know whether there's a theorematic approach to Jørn Holm's
[PEIRCE-L] Triadic Philosophy
Ontology consists of what is necessary for the achievement of the fusion of truth and beauty. Books http://buff.ly/15GfdqU - PEIRCE-L subscribers: Click on Reply List or Reply All to REPLY ON PEIRCE-L to this message. PEIRCE-L posts should go to peirce-L@list.iupui.edu . To UNSUBSCRIBE, send a message not to PEIRCE-L but to l...@list.iupui.edu with the line UNSubscribe PEIRCE-L in the BODY of the message. More at http://www.cspeirce.com/peirce-l/peirce-l.htm .
Re: [PEIRCE-L] Natural Propositions, Ch. 10: Corollarial and Theorematic Experiments with Diagrams
Dear Ben, Franklin, lists, Den 19/04/2015 kl. 20.05 skrev Benjamin Udell bud...@nyc.rr.commailto:bud...@nyc.rr.com: Franklin, lists, I agree with Jon, thanks for your excellent starting post. You wrote, [] Why can't corollarial reasoning, which involves observation and experimentation, reveal unnoticed and hidden relations? After all, on p.285-6, Frederik mentions the work of police detective Jorn Old Man Holm and his computer program, which Frederik describes as a practical example of corollarial map reasoning (p.285). In this example, Holm uses the corollarial reasoning to reveal information about the whereabouts of suspects. Doesn't the comparison of the map reasoning with suspects' testimony end up revealing unnoticed and hidden relations? There's a distinction that some make between complexity and mere complication. Corollarial reasonings may accumulate mere complications until the result becomes hard to see, although it involves little if any complexity in, more or less, the sense of depth or nontriviality. I don't know whether there's a theorematic approach to Jørn Holm's diagrammatization that would show its result in a nontrivial aspect, and anyway its diagrammatic, pictorial presentation already leaves one in no doubt that a pattern is revealed. Certainly the comparison between Holm's map and suspects' testimony may give nontrivial results - but that comparison was not my point - My argument, which I may not have made sufficiently clear in the chapter, is that the small step from having spatiotemporal cell phone information represented in long lists of coordinates - and to synthesize that same information in one geographical map, is a corollarial step. It does not contain any new information which was not already there in the list, but it brings the information together in one conclusive sign so as fo facilitate the charting of e.g. the trajectory of single cell phones on the map. I admit it is more difficult, in general, to precisely extend the corollarial/theorematic distinction to applied cases - but as you can see I did the attempt picking map examples. The central problem for my pov seems to be that in applied cases you should not only include what is given in axiomatics (topographical maps largely respecting Euclidean geometry) but also in the more or less implicit ontological assumptions in the area of application - this is why i count Wegener's map experiment as theorematic. Taken as pure geometry, it is a trivial translation to move South America eastwards to compare its coastline with Africa's - but in terms of geology, it requires the addition of a new idea - namely that continents may move. Best, Frederik - PEIRCE-L subscribers: Click on Reply List or Reply All to REPLY ON PEIRCE-L to this message. PEIRCE-L posts should go to peirce-L@list.iupui.edu . To UNSUBSCRIBE, send a message not to PEIRCE-L but to l...@list.iupui.edu with the line UNSubscribe PEIRCE-L in the BODY of the message. More at http://www.cspeirce.com/peirce-l/peirce-l.htm .
Re: [PEIRCE-L] Stjernfelt: Chapter 9
Dear Franklin, Cathy, Lists - A small clarification: Peirce's BxD=A idea, I think, should not be taken a device for the arithmetic calculation of exact information size - it is rather the proposal of a general law relating Breadth and Depth. His idea comes from the simple idea that when intension is zero, there is no information, while when extension is zero, there is also no information - and that is the relation of the two factors in a product. (It is a bit like his first Boole-inspired definition of universal quantification as a product - he defines truth as 1, falsity as 0, then, in order to be true, each single case of a universal proposition should be true - if any single one of them is false, the total product of them all will be zero.) The BXD=A idea allows him to investigate what happens if intension or extension are in- or decreased, etc. - even if not being able to express that in precise numbers. Best F Den 20/04/2015 kl. 01.14 skrev Franklin Ransom pragmaticist.lo...@gmail.commailto:pragmaticist.lo...@gmail.com: Cathy, lists, Well, look at this way: It is possible for there to be objects in the senses which are yet not perceived, because we do not yet have any idea of what it is to which we are looking. It takes a hypothesis to introduce a new idea to us to explain what it is, which hypothesis we can then put to the test. In order to do so, we must determine what kinds of characters to look for (deduction helps here) and then look for existent objects (induction) to learn whether the purported relations between characters obtain in fact, and in this way we come to understand the thing which we are experiencing. It is of course induction which gives us more information; abduction simply gives us the idea which needs to become informed, and deduction is merely explicative, based on relating the idea to other ideas and previously gathered information regarding those ideas. Obviously, we cannot conduct induction without end, because that is a practical impossibility. Our 'sum', as you put it, far from being always an infinity, will very likely never be an infinity in practice, in whatever sense you mean to understand the application of infinity to a 'sum' of information. Of course, as an ideal, where science, the community of inquiry as such, continues to investigate, it is possible for the information of an idea to reach a much greater 'sum' than would otherwise be possible for individuals such as you or me. But it is a commonplace of science that ideas that work and continue to work are understood more thoroughly in their relations to other ideas over the course on inquiry. This means of course that not only the breadth, but also the depth of the idea continues to grow. As a result, typically, rather than tending to make comparisons moot, we start to see a hierarchy of ideas and related sciences appear. Consider this passage: The former [Cows] is a natural class, the latter [Red Cows] is not. Now one predicate more may be attached to Red Cows than to Cows; hence Mr. Mill's attempts to analyze the difference between natural and artificial classes is seen to be a failure. For, according to him, the difference is that a real kind is distinguished by unknown multitudes of properties while an artificial class has only a few determinate ones. Again there is an unusual degree of accordance among naturalists in making Vertebrates a natural class. Yet the number of predicates proper to it is comparatively small (NP, p.238, quoting Peirce). We can see here that further simplifications are introduced, so taking what is learned about various vertebrates, a new idea, that of vertebrates, appears which simplifies the characters involved. Conversely, species under vertebrates will become much more determinate in terms of their characters, but be simplified with respect to their extension. You said above: Under synechism every real object has an infinite number of attributes, and every meaningful predicate or general term effectively has an infinite number of aspects, so a simple multiplication of B x D is pointless. And yet natural kinds appear, in which certain attributes, predicates, or aspects appear significant, and others do not. It is precisely the work of abduction to simplify what is observed so that what is essential is grasped, and not simply a never-ending multitude of characters. Such simplification is always with respect to a purpose. With respect to natural kinds, such purpose, or telos, is objective, and we see nature all around us selecting certain characters over others as more significant. If this were not true, natural science would be impossible. As to real objects, yes they have an infinite number, but not all of them are relevant to the purpose of interaction with the real object. Certain meaningful attributes are selected for in attention in order to aid conduct with respect to some purpose at hand. Information
[PEIRCE-L] Re: [biosemiotics:8363] Natural Propositions, Ch. 10:
Dear Franklin, lists - Sorry for having rattled Franklin's empiricist sentiments with references to the a priori! Empiricists seem to have an a priori fear of the a priori … but no philosophy of science has, as yet, been able to completely abolish the a priori - even logical positvism had to admit logic as a remaining a priori field (reinterpreting that as tautologies, that is true). I should probably have given a note here to my own stance on the a priori - for the interested, I wrote a bit about it in ch. 8 of Diagrammatology (2007). My take on it there comes more from the early Husserl than from Peirce: the a priori has nothing to do with Kantian subjectivity, rather, it consists in dependence structures of objectivity - this makes it subject to fallibilism - the a priori charts necessities - these come in two classes, formal ontology and material ontology - the former holds for all possible objects, the latter for special regions of reality (like physics, biology, society) - no discipline can function without more or less explicit conceptual networks defining their basic ideas - being fallibilist, a priori claims develop with the single scientific disciplines … I happen to think this Husserlian picture (for a present-day version, see Barry Smith) is compatible with Peirce's classification of the sciences where, as it is well known, the upper echelon is taken to be a priori in the sense of not at all containing empirical knowledge while the lower, positive levels inherit structures from those higher ones, co-determining the way they organize and prioritize their empirical material. So, it is in this sense of material ontology that I speak of biogeographical ontology and and the ontology of human culture development involved in Diamond's argument. Given these assumptions, Diamond's argument, so I argue, is a priori. His conclusion that Eurasia privileges the spread of domesticated animals does not depend on the empirical investigation of early cultural contacts, human migrations or trade routes across the continent - but only on the general knowledge that climate is (largely) invariant along latitudes and that the spead of human cultures involves that of domesticated animals (the two ontologies I claim are involved). As you can see my concept of ontology is deflated - which is also in concert with the ontological commitment in some Peircean ideas (cf. the idea that what exists is what must be there for true propositions to be true, 5.312) - so I do not participate in the analytical quest for the most meagre ontology possible … I would rather say that ontology should comprise general concepts necessary for the sciences at all levels (from elementary particles and genes to empires, wars, media and real estate …) Best F Den 20/04/2015 kl. 04.07 skrev Franklin Ransom pragmaticist.lo...@gmail.commailto:pragmaticist.lo...@gmail.com: Ben, lists, The connection you drew between the first and the fourth definitions of theorematic reasoning is quite interesting; I had not thought of conceptual analysis in quite that way. At least, though, the complexity of the diagram or icon is likely more complicated in the case of theorematic reasoning than in corollarial reasoning. I suppose I somehow think that a theorematic reasoning is often a previous corollarial reasoning but with something novel introduced, which would make the theorematic reasoning straightforwardly more complicated than the corollarial reasoning. Part of my concern about the relationship between theorematic reasoning and abductive inference is that Frederik isn't just attempting to discuss mathematics when treating of theorematic diagrammatic reasoning. Rather, the significance is for all knowledge. Because the mathematical-diagrams are ubiquitous, and because Frederik takes the mathematical diagrams to be a priori, this means that all knowledge includes the a priori as a constituent element. This is a very Kantian move, repeated by C.I. Lewis in his Mind and the World-Order. I am quite wary of this move. I think it very important the way you put the following: The conclusions are aprioristically true only given the hypotheses, but the hypotheses themselves are not aprioristically true nor asserted to be true except hypothetically, and this hypotheticality is what allows such assurance of the conclusions, although even the hypothesis is upended if it leads to such contradictions as render the work futile. And then part of your quote from Peirce: Mathematics merely traces out the consequences of hypotheses without caring whether they correspond to anything real or not. It is purely deductive, and all necessary inference is mathematics, pure or applied. Its hypotheses are suggested by any of the other sciences, but its assumption of them is not a scientific act. There are two things to be said about this. The first is that the hypotheses are originally suggested by experience. The
[PEIRCE-L] The syllogism as a mathemtical category
Hi, A couple of days ago (while jogging, of course), a thought occurred to me that the syllogism may fit the category diagram (also called the 'commutative triangle') I have been using so often on these lists. I checked the idea just now by diagramming it on a piece of paper and found that it indeed seems to work as expected: fg Major Premise Minor Premise Conclusion | ^ | | |__| h Figure 1. The syllogism as a category, with the structure-preserving mappings defined as f = natural process, g = mental process, and h = correspondence. These mappings are thought to satisfy the commutative condition, f x g = h. I am now wondering if this commutative triangle can be viewed as the simplest unit of reasoning in human thought. If so, we may be justified to state that: We think in categories. (05202-15-1) which would be synonymous with We think in signs. (05202015-2) since the Peirceasn sign is itself a commutative triangle, i.e., a category. This leads me to ask whether it would be reasonable to divide all the signs that we use on these lists into three classes -- (i) words (W) (used in most of the posts, e.g., Ben, Edwina, Jeff, Jerry, Gary, Steven, etc.), (ii) diagrams (D) (e.g., Jon, Howard, Edwina, me, etc.), and (iii) mathematical formulas (M) (e.g., mostly me, e.g., see Table 1 below). This WDM trichotomy is an observed fact on these lists and not a theoretical construct. For example, we can readily recognize these three elements in Table 1: W = at least 50% of the symbols appearing in the 6 x 4 table D = the 6 x 4 table itself M = the 8 equations appearing in the table Table 1. A possible relation among entropy, quanta, and information 1. Concept *Entropy* (1) *Quanta *(2) *Information *(3) 2. Field of inquiry Thermodynamics Quantum mechanics Informatics 3. Experiment/Measurement S = ΔQ/T Blackbody radiation spectra Selecting m out of n possibilities 4. Statistical mechanical formulation S = - k∑ pi log pi *Boltzmann-Gibbsentropy *(1866) U(λ, T) = (2πhc2/λ5)/(ehc/λkT – 1) *Planck radiation equation* (PRE) (1900) [1] IP = log2(AUC(P)/AUC(G)) where IP = Planckianinformation [2], AUC = area under the curve, P = PDE, and G = *Gaussian-like equation* [3] y = Ae–(x - µ)^2/(2*σ^2) 5. Mathematical formulation H = - K∑ pi log pi *Shannon entropy*(1948) y = (a/(Ax+B)5)/(eb/(Ax + B)– 1) *Planckian distribution equation* (PDE) (2008) [2] I = A log2 (n/m) *a unified theory of the amount of information*(UTAI) (2015) [4] 6. Emerging Concept A measure of*DISORDER* Quantization of action needed for*ORGANIZATION* A measure of the *ORDER*of an organized system It would be a challenge (and should be possible) to map the WDM trichotomy to the three trichotomies of signs in Peircean semiotics.( https://www.marxists.org/reference/subject/philosophy/works/us/peirce2.htm). All the best. Sung -- Sungchul Ji, Ph.D. Associate Professor of Pharmacology and Toxicology Department of Pharmacology and Toxicology Ernest Mario School of Pharmacy Rutgers University Piscataway, N.J. 08855 732-445-4701 www.conformon.net - PEIRCE-L subscribers: Click on Reply List or Reply All to REPLY ON PEIRCE-L to this message. PEIRCE-L posts should go to peirce-L@list.iupui.edu . To UNSUBSCRIBE, send a message not to PEIRCE-L but to l...@list.iupui.edu with the line UNSubscribe PEIRCE-L in the BODY of the message. More at http://www.cspeirce.com/peirce-l/peirce-l.htm .
[PEIRCE-L] Re: Peirce's 1880 “Algebra Of Logic” Chapter 3 • Comment 7.2
Jim, List,
The form x:y is just Peirce's notation for the ordered pair (x, y).
A 2-point universe like {I, J} provides us with another example of formal
degeneracy (loss of generality) since the number of “diagonal” terms (of the
form A:A) is equal to the number of “off-diagonal” terms (of the form A:B) and
so the case exhibits symmetries that will be broken as soon as one adds another
element to the universe. I am planning to take up a 3-point example in good
time but I wanted to essay the graph-theoretic representation of dyadic
relations first.
Regards,
Jon
http://inquiryintoinquiry.com
On Apr 20, 2015, at 12:46 PM, Jim Willgoose jimwillgo...@msn.com wrote:
jon list.
Very nice work! I got stuck on the large number of objects that could be
attached to a relative and drifted off into prime numbers, factors, subgroups
and the 'conversion' formula(s). It is good to see a valuation for
interpreting some of these things. Given the arrays of zeroes and ones, I
can begin to see how to replace the : sign with various operations that are
less general.
Jim W
Date: Sun, 19 Apr 2015 14:40:33 -0400
From: jawb...@att.net
To: peirce-l@list.iupui.edu
Subject: [PEIRCE-L] Peirce's 1880 “Algebra Of Logic” Chapter 3 • Comment 7.2
Post : Peirce's 1880 “Algebra Of Logic” Chapter 3 • Comment 7.2
http://inquiryintoinquiry.com/2015/04/19/peirces-1880-algebra-of-logic-chapter-3-%e2%80%a2-comment-7-2/
Date : April 19, 2015 at 1:00 pm
Peircers,
Note. This post has a lot of math formatting,
so please follow the link above for a more
readable text.
Because it can sometimes be difficult to reconnect abstractions with
their concrete instances, especially after the abstract types have
become autonomous and taken on a life of their own, let us resort
to a simple concrete case and examine the implications of what
Peirce is saying about the relation between general relatives
and individual relatives.
Suppose our initial universe of discourse has
exactly two individuals, I and J. Then there
are exactly four individual dual relatives or
ordered pairs of universe elements:
• I:I, I:J, J:I, J:J.
It is convenient arrange these in a square array:
⎛ I:I I:J ⎞
⎝ J:I J:J ⎠
There are 2^4 = 16 dual relatives in general over this universe of
discourse,
since each one is formed by choosing a subset of the four ordered pairs and
then “aggregating” them, forming their logical sum, or simply regarding them
as a subset. Taking the square array of ordered pairs as a backdrop, any one
of the 16 dual relatives may be represented by a square matrix of binary
values,
a value of 1 occupying the place of each ordered pair that belongs to the
subset
and a value of 0 occupying the place of each ordered pair that does not
belong
to the subset in question. The matrix representations of the 16 dual
relatives
or dyadic relations over the universe {I, J} are displayed below:
⎛ 0 0 ⎞ ⎛ 1 0 ⎞ ⎛ 0 0 ⎞ ⎛ 1 0 ⎞
⎝ 0 0 ⎠ ⎝ 0 0 ⎠ ⎝ 0 1 ⎠ ⎝ 0 1 ⎠
⎛ 0 1 ⎞ ⎛ 1 1 ⎞ ⎛ 0 1 ⎞ ⎛ 1 1 ⎞
⎝ 0 0 ⎠ ⎝ 0 0 ⎠ ⎝ 0 1 ⎠ ⎝ 0 1 ⎠
⎛ 0 0 ⎞ ⎛ 1 0 ⎞ ⎛ 0 0 ⎞ ⎛ 1 0 ⎞
⎝ 1 0 ⎠ ⎝ 1 0 ⎠ ⎝ 1 1 ⎠ ⎝ 1 1 ⎠
⎛ 0 1 ⎞ ⎛ 1 1 ⎞ ⎛ 0 1 ⎞ ⎛ 1 1 ⎞
⎝ 1 0 ⎠ ⎝ 1 0 ⎠ ⎝ 1 1 ⎠ ⎝ 1 1 ⎠
Relative to the universe {I, J}, the individual dual relatives of
the form A:A are I:I and J:J while the individual dual relatives of
the form A:B are I:J and J:I.
Peirce assigns the name ‘concurrents’ to dual relatives all whose
individual aggregants are of the form A:A. There are exactly 4 of
these and their matrices are shown in the top row of the above display.
All the rest are called ‘opponents’ and their matrices are listed in
the bottom three rows.
Peirce gives the name ‘alio-relatives’ to dual relatives all whose
individual aggregants are of the form A:B. There are exactly 4 of
these and their matrices are shown in the first column of the above
display. All the rest are called ‘self-relatives’ and their matrices
are listed in the right hand three columns.
Notice that the relative 0, represented by a matrix with all 0 entries,
falls under the definitions of both a concurrent and an alio-relative.
References
• Peirce, C.S. (1880), “On the Algebra of Logic”,
American Journal of Mathematics 3, 15–57.
Collected Papers (CP 3.154–251),
Chronological Edition (CE 4, 163–209).
• Peirce, C.S., Collected Papers of Charles Sanders Peirce,
vols. 1–6, Charles Hartshorne and Paul Weiss (eds.),
vols. 7–8, Arthur W. Burks (ed.), Harvard University Press,
Cambridge, MA, 1931–1935, 1958. Volume 3 : Exact Logic, 1933.
• Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition,
Peirce Edition Project (eds.), Indiana University Press, Bloomington
and Indianapolis, IN, 1981–. Volume 4 (1879–1884), 1986.
Resources
• Peirce’s 1870 Logic Of Relatives
