JEFFREY WROTE: How can the examples of measuring silk against a yardstick, comparing biological specimens to a “type-specimen”, and comparing the weight of carbon and gold to hydrogen help us think more clearly about the grounds we having for comparing arguments and saying that one class contains a sample of good inferences and that another class contains a sample of bad inferences. In making such comparisons, we need something more than just a nominal assignment of the term ‘good’ to one class and ‘bad’ to another. Having said that, don’t we need more than an ordinal scale that enables us to make relative comparisons of goodness and badness? How might we arrive in our theory of logic at a standard of measuring the validity of inferences that is richer than a nominal or ordinal scale? After all, we are relying on our standards for comparing arguments for the sake of arriving at conclusions about what, really, is true and false. ++++++ Me:
Jeffery, I enjoyed reading your post. It is thorough and poses some interesting issues. As for the snippet above, it seems to me that in-depth teleological analysis, a logic of purpose, is what is Peirce calls for (in his Methodeutic at least) to accomplish what you describe there. For "what, really, is true and false" depends only secondarily upon what something is true or false "about" and primarily upon the purpose is for making such a determination in the first place. This determination engages inferences as well. Meno Hulswit (sp?) describes what he calls a "developmental teleology," which to me resembles the "away from" direction of an abductive inference, following (as abduction is wont to do), a reticulated (web-like) pattern that is difficult, if even possible, to reverse engineer, especially once the purpose has been thoroughly developed and then stated as a purpose. Of course, one can apprehend purposes by other means: direct replication, reexamination, development of an existing thing or idea. However, for the inquirer seeking to determine "what, really, is true and false," each of the ways of determining purpose (which resemble different ways of making inferences) frames WHETHER or HOW true or false a given inference might be. Thus, it seems to me that an inquirer seeking to measure the truth/falsity of an inference, would need to take special care in the formulation and analysis of purpose and include the generic SORT of purpose ( replicative, review, development of existant, reticulated exploration, etc) in whatever argument is framed for or against an issue, perhaps even including situations for which truth and falisity might be reversed, or might not even apply. Regards, Phyllis Chiasson Jeffrey Brian Downard <jeffrey.down...@nau.edu> wrote: >Gary F., Gary R., Cathy Legg, John Kaag, Jerry, List, > >Jerry says: "My personal feeling about your exposition is that such a view of >material and formal categories leads one into an extra-ordinarily deep >philosophical morass from which you may never emerge." > >At the Congress, several people expressed a worry about falling "down the >rabbit hole" when studying Peirce. The concern was that spending too much >time on the difficult parts of the more challenging essays threatened to pose >insurmountable problems in making sense of what Peirce is up to. > >Despite your warnings, I will have to trust my own judgment in determining >when it makes sense for me to press on when it comes to the more challenging >texts and arguments. My conviction is that Peirce often is trying to teach us >how to employ specific methods in doing philosophy, and that we'll struggle in >our attempts to understand him so long as we lack the experience and skills he >possessed. I don’t know about you, but this puts me in a tough position, >because I seem to lack much of his experience and skills. While Peirce tried >to put many things in the simplest possible terms, he often takes it for >granted that the reader will "actively think" and draw on his sentences as "so >many blazes to enable him to follow the track of the reader's thought." (EP, >301) > >Reading Peirce presents a challenge. As many scholars have pointed out, he >was a remarkably talented logician, and he possessed an intimate familiarity >with the mathematics of the 19th century and its larger history. What is >more, he was a practicing scientist who had a rich understanding of how to do >and not merely read chemistry, astronomy, classificatory biology, and geodesy. > In addition to being a special scientist working in multiple fields, he had a >synoptic sense of the history philosophy and the conceptual landscapes >represented by different philosophical systems—along with a rich appreciation >of the different worldviews that philosophers might try to explore. Above >all, he was a student of methodology, and his aim was to develop a systematic >method for improving the methods of inquiry. > >Turning from these remarks about the difficulties one faces in trying to >understand Peirce's views--especially the more difficult arguments expressed >in the more challenging texts--to the task of reconstructing some of Peirce's >arguments in the text of "New Elements (Kaina Stoicheia)", let's take a look >at the text itself. There are three main sections. The first contains >biographical remarks about the textbook he wrote on the logic of >mathematics--taking topology, projective geometry and metrical geometries as >its subject matter. The second contains a statement of the distinction >between definitions, postulates, axioms, etc. The third, which is the longest >section, is divided into 4 sub-sections. You quote from the fourth and >longest of these subsections. > >What is Peirce doing in the passage you've quoted? It is possible that we are >reading the text somewhat differently. Let me provide a few of comments about >what he is doing in the pages leading up to the passage you've quoted so that >we might clarify some of the differences in our approaches. I note that >you've quoted the passage, but you've said precious little about what you >think is going on here. You refer to an earlier post by Clark, so perhaps I >could turn to what he says at some later time in an attempt to understand your >remarks. > >So, in parts I and II, Peirce starts by referring to his own work on the logic >of mathematics. By the fourth part of section III, he has moved from a >discussion of speculative grammar and critical logic to a series of examples >drawn from the theoretical and the practical sciences. You seem to be >particularly interested in his remarks about the various specific uses of the >concepts of cause and effect, including internal and external causes, along >with material, formal, efficient and final causes. He has an exceptionally >long paragraph on the topic starting on page 313 and ending on 316. The >point of this little foray on the different causes is not to argue for big >metaphysical conclusions. He's made those arguments elsewhere. And, he says >as much: “Yet I refuse to enter here upon a metaphysical discussion.” (EP, ) > >As he points out in the opening sentence of this paragraph, everything he says >here is designed to clarify the distinction between a proposition and an >argument. His goal, I think, is to illustrate how we should go about >classifying different acts of cognition (e.g., as an act of interrogating, >affirming or arguing) and then ascertaining the nature of those acts. So, the >question is something like this: > >1) If the act is one of affirming an assertion, then what is involved in >affirming that the proposition true? > >Or this: > >2) If the act is one of arguing for a conclusion from a set of premisses, >then what is involved in affirming that the argument is valid? > >He is also asking the question: How can we put our questions to nature and >get a reasonable answer? That is, how can we find out what is really the >case? These sound like questions of metaphysics, but he is focusing on a set >of questions that surface in the theory of logic. Namely, what hypotheses >concerning the nature of what is real should we adopt for the sake of >understanding the validity of deductive, inductive and abductive inferences? >He has argued that we need, for the sake of making valid deductive arguments, >to adopt a nominal definition of the real. He sees that induction and >abduction requiring richer hypotheses concerning the real. > >Here are some things that he says about the hypotheses that are required for >the sake of making valid abductive inferences: > > “Abduction . . . is the first step of scientific reasoning, as induction is > the concluding step. >In abduction the consideration of the facts suggests the hypothesis. In >induction the study of the hypothesis suggests the experiments which bring to >light the very facts to which the hypothesis had pointed. The mode of >suggestion by which, in abduction, the facts suggest the hypothesis is by >resemblance, -- the resemblance of the facts to the consequences of the >hypothesis. The mode of suggestion by which in induction the hypothesis >suggests the facts is by contiguity, -- familiar knowledge that the conditions >of the hypothesis can be realized in certain experimental ways. > >I now proceed to consider what principles should guide us in abduction, or the >process of choosing a hypothesis. Underlying all such principles there is a >fundamental and primary abduction, a hypothesis which we must embrace at the >outset, however destitute of evidentiary support it may be. That hypothesis is >that the facts in hand admit of rationalization, and of rationalization by us. >That we must hope they do, for the same reason that a general who has to >capture a position or see his country ruined, must go on the hypothesis that >there is some way in which he can and shall capture it. We must be animated by >that hope concerning the problem we have in hand, whether we extend it to a >general postulate covering all facts, or not. > >We are therefore bound to hope that, although the possible explanations of our >facts may be strictly innumerable, yet our mind will be able, in some finite >number of guesses, to guess the sole true explanation of them. That we are >bound to assume, independently of any evidence that it is true. Animated by >that hope, we are to proceed to the construction of a hypothesis.” (CP >7.218-19) > >Given the fact that the primary subject matter of the “New Elements” essay is >the normative science of logic, let us ask: what are the data (i.e., the >observations) for generating hypotheses in logic and then putting them to the >test? As we seek an answer the question, I believe that we need to focus our >attention on the “data” part of the equation. As he says, “the logician has >to be recurring to reexamination of the phenomena all along the course of his >investigations.” (EP, 311) > >In the paragraphs leading up to his remarks about atomic weights, he considers >the following examples: a psychologist studying the experience of déjà vu, a >logician studying of the experience of similarity and resemblance, a >seamstress buying fabric from a shopkeeper, a homeowner buying a piece of >furniture, and a chemist studying the weight of gold. What is the point of >these examples? Much of Peirce’s attention is fastened on the question of how >we should arrive at a more scientific understanding of the conditions for >making measurements. How should we measure a psychological feeling, or a >length of silk, or a the size of a piece of furniture, or the chemical weight >of an element—or the degree to which one feeling (or other idea) is, logically >speaking, similar to an another. > >In some “comments on “The Basis of Pragmatism in the Normative Sciences,” I >forwarded the claim that Peirce’s phenomenology is, at least in part, an >attempt to answer the following question: what are the formal features in >experience that are necessary for us to draw valid synthetic inferences from >our observations? This is not an easy question to answer. We’re looking for >an answer because we want to understand how it is possible to put the >qualities we’ve observed in a transitive ordering and make comparisons based >on the degree to one resembles or does not resemble another. I’d like to add >the following to what I’ve said thus far: discovering the formal conditions >for putting things in such a transitive order and comparing them are essential >aspects of what is needed to measure them. > >The point he is making about using a yard stick to measure length is analogous >to the point he is making about using a standard for measuring the chemical >weight of gold. In order to make measurements of length, we use something >that is like a rigid bar that can be moved up and down the thing we are >measuring (so that the finite length of the bar does not matter for purposes >of making the measurements). The remark that caught my attention is where he >says that our theory of measurement is based on the idea that we need >something that can serve as a more universal standard. In an effort to make >our standard more universal, scientists have designated one particular bar in >Westminster as the object to which our concept of yard refers. In order to >determine whether or not any other yardstick we might use will lead us into >error, we can—as a matter of principle—compare it to the protypical standard >in Westminster. > >Is this the best way to fix the reference for the concept of a yard? Peirce >thinks it is not the best way to remove some of the errors that will crop up >in the process of making measurements of length. Instead of relying on a >single prototype sitting in a case in Westminster, we should rely on an >average taken from a number of different bars made of different materials and >kept under different conditions (e.g., at different ranges of temperature). >We use the concept of yard in such a way that it refers to the mean length of >them all. This is the same kind of thing that a biologist does when she >compares a number of different specimens and draws up a conception of a >“type-specimen” as a kind of typical thing that has a normal size and shape. > >What is the weight of gold? In saying that it is an elementary chemical >substance having a particular atomic weight of about 197 ¼, we are relying >upon some kind of standard in making the comparison. The standard, of course, >is the atomic weight of hydrogen, which is taken to have a weight of 1. What >is it to say that the weight of hydrogen is 1 unit? His answer is that, in >comparison to air, it is about 14 ½ times lighter. > >In this passage, is Peirce making some kind of metaphysical point about the >deeper “logic” of the chemical elements? I don’t think so. Rather, he is >making a point about what is needed to make comparisons between things—and >then he is asking what is needed to set up a standard for measuring those >things. The system of measurement set up by Dalton in 1803 was a relative >scale that used the weight of hydrogen as the base unit. Technically >speaking, scientists could say that the mass of hydrogen was exactly one only >because it was the serving as the base unit of measurement in a relative >scale. It would not serve the goals of the scientists to say that the concept >of the weight of hydrogen refers to protypical sample stored in a glass case >in Westminster. Rather, the weight of hydrogen, like the length of a yard, >should be taken to refer to a mean over many observations of the relative >weights of gold, carbon, hydrogen and other elements. > >What does this have to do with the normative theory of logic? I believe that >it bears on logic in two ways. First, I believe that an analysis of the >things we observe—in chemistry, biology, the selling of fabric, etc.—requires >us to examine the underlying grounds for making measurements of the various >phenomena. We can draw on mathematics, phenomenology and logic in order to >deepen our understanding of what is necessary to apply one or another kind of >measurement to a given kind of phenomena that has been observed in one or >another of the practical or theoretical sciences. Second, this kind of >question surfaces when we ask what the standards are for analyzing the >phenomena we’re drawing on in the theory of logic. Peirce says as much in his >discussion of what is needed to make something as simple as a comparison >between two qualities of feeling. Take, for instance, a comparison between >two experiences of the color of blue. In the hospital room where I’m sitting >with my daughter, there is a stool and a sheet that have just about the same >hue. From this point on, I will probably refer to this shade of color as >“hospital blue.” When I compare the intensity of the color I experience when >looking at the stool with the color I experience when looking at the sheet, it >seems to me that the color of the stool is remarkably more intense than the >color of the sheet. The two objects are across the room from each other, so >all I can do is to compare the intensity of the one with my memory of the >intensity of the other. What are my grounds for making such a comparison? > >One of the points Peirce is making at this point in subsection 4 is that the >comparison of the intensity of two experiences of the quality of blue is >something that is “measured chiefly by aftereffects.” (EP, 320) He is laboring >over this point, I believe, because he is keenly interested in set of related >issues. Consider, for instance, the following questions: > >1) What is the standard that we can use when comparing the feeling that an >argument is a good inference to the feeling that an argument is an invalid >inference? Isn’t this similar in some respects to comparing the intensity of >a one experience of a feeling of blue to another feeling of blue? Isn’t it >different in other respects? > >2) Once we have formed a class of sample arguments that we take to be good >and a class that we take to be bad, what kind of measurements can be made when >comparing these classes? At the very least, we can apply a nominal scale in >saying that they are labeled as different classes. For the sake of the >logical theory, however, we need a stronger standard of measurement, don’t we? > >3) What is the standard for making the comparison of the goodness or >badness of an argument? Should we take it to be a prototypical argument that >appears to be beyond criticism? Perhaps we should take an argument, such as a >cogito argument, or an ontological argument for God’s reality, or an argument >for the indubitability of the axioms of logic as a prototype, and then place >one or another of these arguments in a glass case in Westminster. I suspect >that this would fail to serve the purpose we have in removing possible errors >from our measurements of the goodness or badness of any given argument. > >How can the examples of measuring silk against a yardstick, comparing >biological specimens to a “type-specimen”, and comparing the weight of carbon >and gold to hydrogen help us think more clearly about the grounds we having >for comparing arguments and saying that one class contains a sample of good >inferences and that another class contains a sample of bad inferences. In >making such comparisons, we need something more than just a nominal assignment >of the term ‘good’ to one class and ‘bad’ to another. Having said that, don’t >we need more than an ordinal scale that enables us to make relative >comparisons of goodness and badness? How might we arrive in our theory of >logic at a standard of measuring the validity of inferences that is richer >than a nominal or ordinal scale? After all, we are relying on our standards >for comparing arguments for the sake of arriving at conclusions about what, >really, is true and false. > >These are the kinds of questions that I’m particularly interested in trying to >answer. My hunch is that, rabbit hole or not, Peirce is pointing us to the >resources needed to answer these kinds of questions. As he points us in a >specific direction, however, he is assuming that we will "actively think" and >draw on his sentences as "so many blazes to enable him to follow the track of >the reader's thought." The real danger is not one of following the blazes >and heading down the rabbit hole. Rather, it is one of sticking with our >personal assumptions and convictions in such a fashion that we make ourselves >impervious to the fruitful suggestions that are around us and, in doing so, >fail to see that we are sitting in a hole of our own making with no sense of >which direction is up and which is down. > >That, at least, is my abiding worry. Hopefully, it is one that will spur me >to active inquiry. > >--Jeff > > >Jeff Downard >Associate Professor >Department of Philosophy >NAU >(o) 523-8354 >________________________________________ >From: Jerry LR Chandler [jerry_lr_chand...@icloud.com] >Sent: Friday, August 22, 2014 3:35 PM >To: Peirce List >Cc: Jeffrey Brian Downard >Subject: Re: [PEIRCE-L] Phaneroscopy, iconoscopy, and trichotomic category >theory > >Dear Jeff: > >Thank you for your exposition on your views on the relations between material >and formal categories. >(From your post below) >First off, if things are sounding mystical to your ears, I hope it is a by >product of the richness of the ideas Peirce is examining--and not a by-product >of the comments I'm offering. > >Your hopefulness is partially realized. And partially not. > >Your may recall Clark’s perceptive’s postings from Kainia Stoichia on CSP >views on causality. In subsequent sentences, CSP gives a crisp example of his >deductions about relations between gold (as a relative weight) when compared >to hydrogen and then to air. >"What is gold? It is an elementary substance having an atomic weight of about >197¼. In saying that it is elementary, we mean undecomposable in the present >state of chemistry, which can only be recognized by real reactional >experience. In saying that its atomic weight is 197¼, we mean that it is so >compared with hydrogen. What, then, is hydrogen? It is an elementary gas 14¼ >times as light as air. And what is air? Why, it is this with which we have >reactional experience about us. The reader may try instances of his own until >no doubt remains in regard to symbols of things experienced, that they are >always denotative through indices; such proof will be far surer than any >apodictic demonstration. From KS. > >This crisp example of material and formal categories (and the logical >phenomena inferred by mathematics) about material categories is worthy of >careful study. He presents a logic of relatives. Classification of categories >inevitably brings forth issues of causality, Aristotelian or otherwise, which >he illustrates. You may find it useful to contrast this example with other >direct examples from biology or medicine as you pursue your thinking about >these matters. > >My personal feeling about your exposition is that such a view of material and >formal categories leads one into an extra-ordinarily deep philosophical morass >from which you may never emerge. For me, the choice of rhetorical terms in >your exposition leads not to calculations but to a Luciferic network of >semantic entanglements. > >Thanks again for clarifying your thoughts. > >Cheers > >Jerry > > > > > >On Aug 22, 2014, at 1:39 AM, Jeffrey Brian Downard ><jeffrey.down...@nau.edu<mailto:jeffrey.down...@nau.edu>> wrote: > >On Wed, Aug 20, 2014 at 3:05 PM, Jeffrey Brian Downard ><jeffrey.down...@nau.edu<mailto:jeffrey.down...@nau.edu><mailto:jeffrey.down...@nau.edu>> > wrote: >Hi Jerry, List, > >First off, if things are sounding mystical to your ears, I hope it is a by >product of the richness of the ideas Peirce is examining--and not a by-product >of the comments I'm offering. > >To a large degree, the answers to the questions you are trying to raise are >going to be found in the larger story that is articulated in the theory of >semiotics. At this point, I am trying to offer some comments on some of >Peirce's explanations and definitions as a kind of run up to the >phenomenological categories--and especially the distinction between the formal >and material aspects of those categories. The general suggestion I'm making >is that Peirce is not providing two entirely separate lists of the categories, >one formal and that other material. Rather, there is a close connection >between the two even if they do not, in experience, match perfectly because >our experience of the material categories of quality, brute fact and mediation >is always so richly complex. My general suggestion may seem controversial >because some interpreters seem to be offering a different reading of the >relevant texts. > >Confining myself to the subject of the phenomenological categories and the >role of mathematics in informing our understanding of the essential formal >elements of the monad, dyad and triad, I do take Peirce to be offering an >account of the elements needed for setting up the frameworks necessary for >referring to grounds, objects and interpretants. One might call them three >interrelated "frames of reference." > >What do the signs that we use in mathematics refer to? Much depends upon >whether we are using the signs to seeks answer to questions in pure or applied >mathematics. Let's consider the case of pure mathematics. What do the signs >used in topology refer to? In the account he offers in the New Elements, the >key operations for setting up a system of mathematical diagrams are those of >generation and intersection. These are the operations used to generate a line >by moving a particle from a point, or for determining the location of a point >on a line by intersecting it with another line. > >As we try to understand the conditions that make it possible for the different >representations to refer, we'll need to be clear in identifying the >representations we're talking about. It is one thing to ask: what does that >particle in the diagram that is being moved refer to? It is another thing to >ask, what does the symbol "particle" refer to? I hope it is clear that the >conditions under which the symbol "particle" refers is dependent, in many >respects, on the conditions under which the iconic particle that is draw on >the page is able to refer. As a hypo-icon, the particle we move as we draw >the line is remarkably rich as a sign. At any time in the act of drawing the >line on the paper, there are qualisigns, sinsigns and legisigns working >together so that the particle can function as a rich sign complex in a larger >process of interpretation. What is more, the particle embodies the idea of a >generator. That is, it embodies a more general rule that determines how we >might generate innumerable other possible lines from the point. This is a >more general rule that enables us to interpret the larger mathematical space >in which the line is being constructed. It enables us to understand how one >line my be transformed continuously to give us a line that is homeomorphic >with the first, or how various kinds of discontinuities might be introduced to >give us another different line altogether. > >I hope you can see that I'm trying to bracket some of the questions you've >raised about the role of real things (i.e., chemical compounds, protein or DNA >molecules, and the like) in serving as the grounds or objects to which one or >another kind of representation might refer. I'm bracketing those questions >for a reason. I'd like to keep the phenomenological analysis of the >conditions under which the signs used in pure mathematics refer free from big >metaphysical assumptions about what is really the case as a positive matter of >fact. There is a long line of philosophers who have tried to import such >metaphysical assumptions into their accounts of the reference and meaning of >the signs used in math and formal logic (e.g., Mill, Quine, etc.), but Peirce >is resisting this move--at least until we're ready to address questions in >metaphysics. Once we are ready and we're using the methods appropriate for >answering questions in metaphysics, we'll need to think about the real nature >of an ideal system of mathematical definitions, hypotheses, theorems, etc., >and what it is for that system to be real as a rich and consistent network of >possible formal relations. > >--Jeff > > >Jeff Downard >Associate Professor >Department of Philosophy >NAU >(o) 523-8354 >________________________________________ >From: Jerry LR Chandler >[jerry_lr_chand...@icloud.com<mailto:jerry_lr_chand...@icloud.com><mailto:jerry_lr_chand...@icloud.com>] >Sent: Tuesday, August 19, 2014 9:28 PM >To: Jeffrey Brian Downard >Cc: Peirce List >Subject: Re: [PEIRCE-L] Phaneroscopy, iconoscopy, and trichotomic category >theory > >Jeffrey: > >Your posts become increasingly mystical. > >This is not a judgement, merely an observation from a philosophy of >mathematics perspective. > >At issue is how to you assign meaning to mathematical symbols. >In particular, in light of K.S. and his comments on the meaning of number in >the context of his description of gold? > >More to the point, does the meaning of mathematical symbols reside in >mathematics itself or do the meanings refer to the reference systems for the >symbol system, that is the application to a particular material reality, such >as the atomic numbers? Or the sequence numbers for a genetic sequence? Or >protein sequence? >In yet other terms, does the concept of order infer a universal meaning or a >meaning dependent on the nouns of the copulative proposition? > >Perhaps you can address these vexing issue? > >Cheers > >Jerry > > > > >On Aug 19, 2014, at 8:28 PM, Jeffrey Brian Downard ><jeffrey.down...@nau.edu<mailto:jeffrey.down...@nau.edu><mailto:jeffrey.down...@nau.edu>> > wrote: > >Gary F., Gary R., List, > >In an effort to think a bit more about the form/matter distinction as it >applies to the phenomenological categories, let me add few comments about an >explanation that Peirce provides concerning the mathematical form of a state >of things. I'd like to add some remarks about this explanation because I >think it offers us a nice way of responding to a concern Gary F. raised. Here >is the concern: > >Gary F. says: "Jeff, I’m interested in your question, 'is there any kind of >formal relation between the parts of a figure, image, diagram (i.e., any >hypoicon) that does not have the form of a monad, dyad or triad?' . . . I >confess that I have no idea how we would go about investigating that question." > >My initial response was: "The answer to the question involves the whole of >Peirce's semiotic--and not just his account of the iconic function of signs. >So Peirce is bringing quite a lot to bear on the question. For starters, >however, I think we should consider the examples he thinks are most important >in formulating an answer. What Peirce sees is that, in mathematics, the >examples we need are as 'plenty as blackberries' in the late summer. (CP >5.483) What do you know, it is late August. Let's go picking." > >As a first stop on our way to the briar patch, let's consider the following >definition from "The Basis of Pragmaticism in the Normative Sciences." > >"A mathematical form of a state of things is such a representation of that >state of things as represents only the samenesses and diversities involved in >that state of things, without definitely qualifying the subjects of the >samenesses and diversities. It represents not necessarily all of these; but >if it does represent all, it is the complete mathematical form. Every >mathematical form of a state of things is the complete mathematical form of >some state of things. The complete mathematical form of any state of things, >real or fictitious, represents every ingredient of that state of things except >the qualities of feeling connected with it. It represents whatever importance >or significance those qualities may have; but the qualities themselves it does >not represent." (EP, vol. 2, 378) > >Peirce suggests that this explanation is "almost self-evident." At this point >in his discussion, however, he merely ventures the explanation as a "private >opinion." I cite this passage because it bears directly on the question of >how our understanding of the mathematical form of something such as a figure >or diagram is supposed to inform our understanding of the formal categories of >monad, dyad and triad (or, firstness, secondness, thirdness)--and how we might >use those categories in performing a phenomenological analysis of something >that has been observed. > >Peirce says that he has introduced this explanation in order to account for >the emphatic dualism we find in the normative sciences. The dualism is >especially marked in logic and ethics (e.g., true and false, valid and >invalid, right and wrong, good and bad), but it is also found in aesthetics. >As such, he is noticing a phenomena that has been widely observed to be a part >of our common experience in thinking about how we ought to act and think, and >he is getting ready to venture a hypothesis to explain what is surprising >about the phenomena. The explanation of the dualism that follows might seem a >bit hard to make out, but I think it is clear that this is what he is trying >to do. > >That might have seemed a bit opaque, so let me try to restate the point. I >think Peirce is drawing on an understanding of mathematical form for the sake >of performing an analysis of a particular phenomenon that calls out for >explanation. We need to see what it is in the phenomena (i.e., the dualism in >the normative sciences) that really calls out for explanation. Otherwise, we >will not have a clear sense of whether one or another hypothesis is adequate >or inadequate to explain what needs to be explained. > >He says the following about his account of the mathematical form of a state of >a things: "Should the reader become convinced that the importance of >everything resides entirely in its mathematical form, he too, will come to >regard this dualism as worthy of close attention?" > >Why does Peirce say that the importance of everything resides in its >mathematical form? On my reading of this passage and what follows in the next >several pages of the essay, I think he is developing the claim I asserted >above. That is, every kind of formal relation that might be found between the >parts of a figure, image, diagram and the space in which such things are >constructed must have the form of what we are calling, in our phenomenological >theory, a monad, dyad or triad. > >It might sound ridiculous to suggest that the dualism present in our >experience of what is valid or invalid as a reasoning or what is right or >wrong as an action can be clarified by using a mathematical diagram, such as a >drawing on a piece of paper of two dots that we might count by saying "one' >and "two," but he says that we shouldn't disregard such a suggestion. He has >argued elsewhere that every observation we might make must involve some kind >of figure or diagram--and the form of such a figure or diagram can be >understood in terms of having the structure of a skeleton set (CP, 7.420-32), >or a network figure (CP, 6.211), or some other kind of really basic >mathematical structure. I refer to those particular mathematical structures >because the first can be applied to things in our experience that are more >discrete in character, and the second can placed over things more continuous >in character. > >Do you buy his claim here? Does the "importance of everything reside in its >mathematical form?" The argument he offers in the rest of section B is worth >a look. > >--Jeff
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