Frederik Stjernfelt • Natural Propositions : 1

Re: Frederik Stjernfelt • Natural Propositions

Frederik,

One small point that I find myself making on a periodic basis:  I think it is better to describe Peirce’s take on logic as “non-psychologism” rather than “anti-psychologism”, the main thing being that logic is a normative rather than a descriptive science. The use of “anti-psychologism” tends to drag in all sorts of implications that are alien to Peirce’s perspective on the relation between the two.

On a related point, especially pertinent to the current run of the bi(o)semiotics literature, is whether “biologism” in logic be just as bad a diversion from the course of Peirce’s semiotics and logic as “psychologism” ever was. Or, to put it more positively, what would it take to place the biological and psychological implementations of sign relations and inquiry processes in their proper perspective?

Regards,

Jon

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C.S. Peirce • Syllabus • Selection 1

Selection from C.S. Peirce, “A Syllabus of Certain Topics of Logic” (1903)

An Outline Classification of the Sciences

180.   This classification, which aims to base itself on the principal affinities of the objects classified, is concerned not with all possible sciences, nor with so many branches of knowledge, but with sciences in their present condition, as so many businesses of groups of living men.  It borrows its idea from Comte’s classification;  namely, the idea that one science depends upon another for fundamental principles, but does not furnish such principles to that other.  It turns out that in most cases the divisions are trichotomic;  the First of the three members relating to universal elements or laws, the Second arranging classes of forms and seeking to bring them under universal laws, the Third going into the utmost detail, describing individual phenomena and endeavoring to explain them.  But not all the divisions are of this character.

The classification has been carried into great detail;  but only its broader divisions are here given.

181.   All science is either,

  • A.  Science of Discovery;
  • B.  Science of Review;  or
  • C.  Practical Science.

182.   By “science of review” is meant the business of those who occupy themselves with arranging the results of discovery, beginning with digests, and going on to endeavor to form a philosophy of science.  Such is the nature of Humboldt’s Cosmos, of Comte’s Philosophie positive, and of Spencer’s Synthetic Philosophy.  The classification of the sciences belongs to this department.

183.   Science of Discovery is either,

  • I.  Mathematics;
  • II.  Philosophy;  or
  • III.  Idioscopy.

184.   Mathematics studies what is and what is not logically possible, without making itself responsible for its actual existence.  Philosophy is positive science, in the sense of discovering what really is true;  but it limits itself to so much of truth as can be inferred from common experience.  Idioscopy embraces all of the special sciences, which are principally occupied with the accumulation of new facts.

185.   Mathematics may be divided into

  • a.  the Mathematics of Logic;
  • b.  the Mathematics of Discrete Series;
  • c.  the Mathematics of Continua and Pseudo-continua.

I shall not carry this division further.  Branch b has recourse to branch a, and branch c to branch b.

186.   Philosophy is divided into

  • a.  Phenomenology;
  • b.  Normative Science;
  • c.  Metaphysics.

Phenomenology ascertains and studies the kinds of elements universally present in the phenomenon;  meaning by the phenomenon, whatever is present at any time to the mind in any way.

Normative science distinguishes what ought to be from what ought not to be, and makes many other divisions and arrangements subservient to its primary dualistic distinction.

Metaphysics seeks to give an account of the universe of mind and matter.

Normative science rests largely on phenomenology and on mathematics;  metaphysics on phenomenology and on normative science.

(Peirce, CP 1.180–186, EP 2.258–259, Online)

Notes

Collected Papers 1

  • Pp. 5–9 of A Syllabus of Certain Topics of Logic, 1903, Alfred Mudge & Son, Boston, bearing the following preface:  “This syllabus has for its object to supplement a course of eight lectures to be delivered at the Lowell Institute, by some statements for which there will not be time in the lectures, and by some others not easily carried away from one hearing.  It is to be a help to those who wish seriously to study the subject, and to show others what the style of thought is that is required in such study.  Like the lectures themselves, this syllabus is intended chiefly to convey results that have never appeared in print;  and much is omitted because it can be found elsewhere.”

Essential Peirce 2(a)(b)

References

  • 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 1 : Principles of Philosophy, 1931.
  • Peirce Edition Project (eds., 1998), The Essential Peirce, Selected Philosophical Writings, Volume 2 (1893–1913), Indiana University Press, Bloomington and Indianapolis, IN.
Posted in Foundations of Mathematics, Logic, Mathematics, Metaphysics, Normative Science, Peirce, Phenomenology, Philosophy, References, Sources | Tagged , , , , , , , , , | 5 Comments

¿Shifting Paradigms? • 4

Re: Harvey Friedman • Foundational Crisis?

2014 Aug 22

Shock and surprise are relative to a prior state of belief.  The belief that mathematics reduces to logic, and that of a purely deductive sort from given axioms, seems to be a fairly recent notion and never universally shared.  All of which brings us back to the question of locating mathematical inquiry in relation to inquiry in general.

Jon Awbrey

Posted in Foundations of Mathematics, Inquiry, Logic, Mathematics, Model Theory, Paradigms, Peirce, Programming, Proof Theory | Tagged , , , , , , , , | Leave a comment

¿Shifting Paradigms? • 3

Re: Harvey Friedman • Good Math

2014 Aug 17

Speaking of mathematics in the context of “general intellectual activity” brings to mind Raymond Wilder’s take on “mathematics as a cultural system”.

I would like to keep that in mind, if on a back burner, and focus on a more directed species of intellectual activity that goes under the name of inquiry, the flower of which species we know as scientific method, however much method or madcap various lights may see in it.  Whatever the case, that brings us to the task of placing mathematical inquiry within the sphere of inquiry in general.

Here I can do no better than recommend C.S. Peirce’s theory of inquiry as one of the best tacks we can take for approaching this task.

Jon Awbrey

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Forest Primeval → Riffs & Rotes

Re: Shifting Paradigms? • (1)(2)(3)(4)(5)(6)

Prompted by the discussion of Catalan numbers on the Foundations Of Math List, I dug up a few pieces of early correspondence and later discussions that bear on the graph theory-number theory correspondences that have occupied me these many years and I began putting reformatted transcripts at the following sites:

These are wiki sites, so please feel free to use the associated discussion pages if you have any interest.

Regards,

Jon

Posted in Algebra, Arithmetic, Combinatorics, Forest Primeval, Graph Theory, Group Theory, Integer Sequences, Lambda Calculus, Logic, Mathematics, Model Theory, Number Theory, Paradigms, Peirce, Programming, Proof Theory, Riffs and Rotes | Tagged , , , , , , , , , , , , , , , , | Leave a comment

¿Shifting Paradigms? • 2

Re: Timothy Chow • Shifting Paradigms?

2014 Jul 31

I can’t remember when I first started playing with Gödel codings of graph-theoretic structures, which arose in logical and computational settings, but I remember being egged on in that direction by Martin Gardner’s 1976 column on Catalan numbers, planted plane trees, polygon dissections, etc.  Codings being injections from a combinatorial species S to integers, either non-negatives \mathbb{N} or positives \mathbb{M}, I was especially interested in codings that were also surjective, thereby revealing something about the target domain of arithmetic.

The most interesting bijection I found was one between positive integers \mathbb{M} and finite partial functions from \mathbb{M} to \mathbb{M}.  All of this comes straight out of the primes factorizations.  That type of bijection may remind some people of Dana Scott’s D_\infty.  Corresponding to the positive integers there arose two species of graphical structures, which I dubbed “riffs” and “rotes”.  See these links for more info:

The On-Line Encyclopedia of Integer Sequences (OEIS)

Jon Awbrey

An interesting tangent to the main subject, but one that I had some ready thoughts on.

Posted in Algebra, Arithmetic, Combinatorics, Foundations of Mathematics, Graph Theory, Group Theory, Inquiry, Logic, Mathematics, Model Theory, Number Theory, Paradigms, Peirce, Programming, Proof Theory, Riffs and Rotes | Tagged , , , , , , , , , , , , , , , | Leave a comment

¿Shifting Paradigms? • 1

Re: Dana Scott • Shifting Paradigms?

2014 Jul 28

This is very interesting to me, but not all my posts make it to the list, so I will spend a few days reflecting on it and post a comment on my blog, linked below. Thanks for the stimulating question.

Jon Awbrey

I’ve been trying to get back to this for over a week now, but there are times when all you can do is document the process, the flow of thought, no matter how slow it goes. So read my ellipsis … and watch this space  

Posted in Foundations of Mathematics, Inquiry, Logic, Mathematics, Model Theory, Paradigms, Peirce, Programming, Proof Theory | Tagged , , , , , , , , | Leave a comment