Writings of Charles S. Peirce: A Chronological Edition, Volume 1: 1857-1866

by Charles S. Peirce, Edward C. Moore, Max H. Fisch

Overview

The PEIRCE EDITION contains large sections of previously unpublished material in addition to selected published works. Each volume includes a brief historical and biographical introduction, extensive editorial and textual notes, and a full chronological list of all of Peirce's writings, published and unpublished, during the period covered.

Product Details

ISBN-13:	9780253372017
Publisher:	Indiana University Press (IPS)
Publication date:	07/19/2010
Edition description:	A Chronological ed.
Pages:	740
Product dimensions:	6.50(w) x 9.30(h) x 1.80(d)
Age Range:	18 Years

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Writings of Charles S. Peirce Volume 8 1890-1892

A Chronological Edition

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By Charles S. Peirce, Nathan Houser, André de Tienne, Jonathan R. Eller, Cornelis de Waal, Albert C. Lewis, Diana Reynolds, Joseph Kaposta, Luise H. Morton, Kelly Tully-Needler, Leah Cummins Guinn

Indiana University Press

Copyright © 2010 Peirce Edition Project
All rights reserved.
ISBN: 978-0-253-37201-7

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CHAPTER 1

Familiar Letters about the Art of Reasoning

15 May 1890 Houghton Library

Stagira, May 15, 1890.

My dear Barbara:

The University of Cracow once conferred upon a very good fellow a degree for having taught the philosophical faculty to play cards. I cannot tell you in what year this happened,—perhaps it was 1499. The graduate was Thomas Murner, of whose writings Lessing said that they illustrated all the qualities of the German language; and so they do if those qualities are energy, rudeness, indecency, and a wealth of words suited to unbridled satire and unmannered invective. The diploma of the university is given in his book called Chartiludium, one of the numerous illustrations to which is copied to form the title page of the second book of a renowned encyclopaedia, the Margarita Philosophica. Murner's pack contained 51 cards. There were seven unequal suits; 3 hearts, 4 clubs (or acorns), 8 diamonds (or bells), 8 crowns, 7 scorpions, 8 fish, 6 crabs. The remaining seven cards were jokers, or unattached to suits; for such cards formed a feature of all old packs. The object of Murner's cards was to teach the art of reasoning, and a very successful pedagogical instrument they no doubt proved.

If you will provide yourself, my dear Barbara, with a complete pack of cards with a joker, 53 in all, I will make a little lesson in mathematics go down like castor-oil in milk. Take, if you will be so kind, the 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 of spades, and arrange these ten cards in their proper order. I mean by this that the ace, or 1, is to be at the back of the pack, the 2 next, and so on, the 10 alone showing its face. I call this the "proper order," because I propose always to begin the count of cards in a pack at the back, so that, in the pack of ten cards you have just been so obliging as to arrange, every card is in its proper place, that is the number it bears on its face is equal to the number of its place from the back of the pack. The face-value of the 2nd card is 2, that of the 3rd card, 3, and so on.

Now let us add 3 to the face-value of each card in the pack. How shall we do that without a printing-press? Why, by simply taking 3 cards from the back of the pack of ten and carrying them to the face. The face-value of card number 1 is now 3 + 1, or 4; that of card 2 is 5, and so on up to card 7 which is 10. Card 8 is 1; but 1 and 11 are the same for us. Since we have only ten cards to distinguish, ten different numbers are enough. We, therefore, treat 1, 11, 21, 31, as equal, because we count round and round the 10, thus:

[ILLUSTRATION OMITTED]

We say 13 and 23 are equal, meaning their remainders after division by 10 are equal. This sort of equality of remainders after division is called congruence by mathematicians and they write it with three lines, thus

13 [equivalent to] 23 (mod. 10).

The number 10 is said to be the modulus, that is, the divisor, or the smallest number congruent to zero, or the number of numbers in the cycle.

Instead of ten cards you may take the whole suit of thirteen, and then, imagining a system of numeration in which the base is thirteen and in which we count

1 2 3 4 5 6 7 8 9 10 Jack Queen King

we have a similar result. Fourteen, or king-ace, is congruent with 1; fifteen, or king-two, with 2, etc.

It makes no difference how many cards there are in a pack. To cut it, when arranged in its proper order, and transpose the two parts, is to add a constant amount to the face-value of every card. So much for addition.

Now how shall we multiply? Suppose we have the pack of ten in its proper order, and wish to multiply the face-value of the cards by 3. We deal out the cards one by one from first to last, into 3 piles, laying them face up upon the table. We first take up the pile the 10, or 0, falls upon, then the next pile, last the third. Putting each pile after the first at the back of that last taken.

We now find in place 1 card 3, or 3 times 1;

in place 2 card 6, or 3 times 2;

in place 3 card 9, or 3 times 3;

in place 4 card 2, congruent to 3 times 4;

in place 5 card 5, congruent to 3 times 5;

in place 6 card 8, congruent to 3 times 6;

in place 7 card 1, congruent to 3 times 7;

in place 8 card 4, congruent to 3 times 8;

in place 9 card 7, congruent to 3 times 9;

in place 10 card 0, congruent to 3 times 10.

Take this pack and multiply again by 3. Multiplying by 3 twice is multiplying by 9. But 9 [equivalent to] - 1.

Accordingly we shall now find

in place 1 card – 1 or 9, in place 2 card – 2 or 8, in place 3 card – 3 or 7, etc.

Multiply again by 3, and since 3 × 9 [equivalent to] 7, we shall find

in place 1 card 7 × 1 [equivalent to] 7,

in place 2 card 7 × 2 [equivalent to] 4,

in place 3 card 7 × 3 [equivalent to] 1,

in place 4 card 7 × 4 [equivalent to] 8,

in place 5 card 7 × 5 [equivalent to] 5,

in place 6 card 7 × 6 [equivalent to] 2,

etc.

Take a pack of 11 cards. We shall now have,

11 [equivalent to] 0

12 [equivalent to] 1

23 [equivalent to] 1

and, in short, to find what any card will be, having performed the necessary arithmetical operation, we subtract the number in the tens place from the number in the units place, and repay anything we borrow in the addition. Thus, suppose we deal into 5 piles, and take up the piles from left to right putting each one at the back of the pile that was at the left of it. We shall now have

in place 1, since 5 × 1 = 5, card 5;

in place 2, since 5 × 2 = 10, card 10;

in place 3, since 5 × 3 = 15 and 1 from 5 leaves 4, card 4;

in place 4, since 5 × 4 = 20 and 2 from 10 leaves 8, and repaying 1 borrowed we have 9, card 9;

in place 5, since 5 × 5 = 25 and 2 from 5 leaves 3, card 3;

in place 6, since 5 × 6 = 30 and 3 from 10 leaves 7, and repaying 1 we get 8, card 8;

in place 7, since 5 × 7 = 35 and 3 from 5 leaves 2, card 2;

in place 8, since 5 × 8 = 40 and 4 from 10 leaves 6, and repaying 1 we get 7, card 7;

in place 9, since 5 × 9 = 45 and 4 from 5 leaves 1, card 1;

in place 10, since 5 × 10 = 50 and 5 from 10 leaves 5, and repaying 1 we get 6, card 6;

in place 11, since 5 × 11 = 55 and 5 from 15 leaves 10, and repaying 1 we get 11, card 11 ([equivalent to]0).

Suppose we now deal again into 9 piles. Now, the last card falls on the second pile. How are we to take up the piles? Answer: After the cards are exhausted, go on dealing in rotation upon the piles to the right of the last single card dealt no longer single cards but whole piles, always taking the extreme left hand one. Thus, in the present case, after the piles are all dealt out, put the left hand pile upon the pile to the right of the Jack, the last single card dealt; that is, put the pile headed by the 6 on that headed by the 4. Then, on the pile one further to the right, that headed by the 9, put the extreme left one headed by the Jack. Next, on the one headed by the 3 put the one headed by the 6, and so on until the piles are reduced to one. You will then find the proper order restored. Why? Because you have multiplied by 5 and by 9, that is, by 45, and 4 from 5 leaves 1, so that you have multiplied the cards in their proper order by 1, which leaves them in their proper order.

I now beg you, my dear Barbara, to take the full pack of 53 cards, and arrange them in their proper order, firsst the spades, second the diamonds, third the clubs, and fourth the hearts, each suit in its proper order,

1 2 3 4 5 6 7 8 9 X J Q K

with the Joker at the face. Deal them out into 12 piles and take up the piles according to the rule. Namely, denoting the Joker by O,

place the pile headed by the X on the pile headed by the 3; then " " " " " " J " " " " " " 4; " " " " " " " Q " " " " " " 5; " " " " " " " K " " " " " " 6; " " " " " " " O " " " " " " 7; " " " " " " " X " " " " " " 8; " " " " " " " J " " " " " " 9; " " " " " " " Q " " " " " " X; " " " " " " " K " " " " " " J; " " " " " " " O " " " " " " Q; " " " " " " " O " " " " " " K.

Next deal the cards out again into 31 piles, and take up the piles according to the rule. Namely,

first, place the pile headed by the Kon the pile headed by the J[??]; then,[ILLUSTRATION OMITTED]

then, place the pile headed by the 8[??] on the pile headed by the 6[??]; [ILLUSTRATION OMITTED]

This restores the original order because 12 × 31 = 372, and 53 into 372 goes 7 times and 1 over; so that

12 31 [equivalent to] 1 (mod 53);

that is, the two dealings are equivalent to multiplying by 1; that is, they leave the cards in their original order.

You, Barbara, come from an ancient and a proud family. Conscious of being raised above the necessity of using ideas, you scorn them in your own exalted circle, while excusing them in common heads. Your cousins Baroco and Bocardo were always looked upon askance in the family, because they were suspected of harboring ideas,—a quite baseless suspicion, I am sure. But do you know that the unremitting study of years has tempted me to favor a belief subversive of your kindred's supremacy, and of those principles of logic that are accepted upon all hands, I mean a belief that one secret of the art of reasoning is to think? In this matter of card-multiplication, instead of conceiving the dealing out into piles as one operation and the gathering in as another, I would prefer a general formula which shall describe both processes as one. At the outset, the cards being in no matter what order, we may conceive them as spread out into a row of 53 piles of 1 card each. If the cards are in their proper order, the last card is the Joker. In any case, you will permit me to call any pile that it may head the Ultima. The dealing out of the cards may be conceived to begin by our taking piles (single cards, at first) from the beginning of the row and putting them down in successive places following the ultima, until we reach the pile which we propose to make the final one, and which is destined to receive all the cards. When in this proceeding, we have reached the final pile, let us say that we have completed the first "round." Thereupon we go back to the pile after the ultima as the next one upon which we will deposit a pile. We may complete a number of rounds each ending with placing a pile (a single card) on the final pile. We make as many as the number of cards in the pack will permit, and we will call these the rounds of the "first set." It will be found useful, by the way, to note their number. Having completed them, we go on just as if we were beginning another; but when we have moved the ultima, let us say that we have completed the first round of the second set. Every round of the first set ends by placing a pile on the final pile. Let us call such a round "a round of the odd kind." Every round of the second set ends by moving the ultima. Let us call such a round a round of the even kind. We make as many rounds of this kind as the whole number of places after the ultima enables us to complete. We call these the rounds of the second set. We then return to making rounds of the odd kind and make as many as the number of piles before the ultima enables us to make. So we go alternating sets of rounds of the odd and the even kind, until finally the ultima is placed upon the final pile; and then the multiplication process is finished.

I will now explain to you the object of counting the rounds. But first let me remark that the last round, which consists in placing the ultima upon the final pile, should always be considered as a round of the odd kind. When you dealt into 12 piles and gathered them up, with the first 48 cards you performed 4 rounds of 12 cards each, and had 5 cards left over. These five you dealt out, making the first round of the second set; and then you transferred these five piles over to the tops of the second five, making another round of the second set. Then from these five piles you dealt to the other two piles twice, making two rounds of the third set. Next the ultima was placed upon the next pile, making a round of the fourth set. Finally the ultima was placed on the last pile which, being a round of an odd set, belonged to the fifth set. So the numbers of rounds were

4, 2, 2, 1, 1.

From this row of numbers, which we will call the M's, we make a second row, which we will call the N's. The first two N's are 0, 1, the rest are formed by multiplying the last by the first M not already used and adding to the product the last N but one. Thus the N's are

0, 1, 4, 9, 22, 31, 53.

The last N is 53. It will always be the number of cards in the pack. Reversing the order of the M's

1, 1, 2, 2, 4

will make no difference in the last N. Thus, the N's will be

0, 1, 1, 2, 5, 12, 53.

Leave off the first M, and the last N will be the number of piles. Thus from

2, 2, 1, 1

we get 0, 1, 2, 5, 7, 12.

Leaving off the last, will give the number of piles into which you must deal to restore the order. Thus from

4, 2, 2, 1

we get 0, 1, 4, 9, 22, 31.

If you deal 53 cards into 37 piles, the numbers of rounds will be

1, 2, 3, 4, 1.

If you deal into 34 piles the numbers will be

1, 1, 1, 3, 1, 2, 1.

If you deal into 33 piles, the numbers will be

1, 1, 1, 1, 1, 5, 1.

If you deal into 32 piles, the numbers will be

1, 1, 1, 1, 10, 1.

If you deal into 30 piles, the numbers will be

1, 1, 3, 3, 2.

You perceive that the object of counting the rounds is to find out how many piles you must deal into to restore the proper order, and consequently by multiplication how many piles you must deal into to make any given card the first.

Going back to 10 cards, if we were to deal them into 5 piles or 2 piles, the piles could not be taken up so as to conform to the rule. The reason is that 5 and 2 exactly divide 10; so that the last card falls on the last pile, and there is no pile to the right of the last card upon which to pile the others. To avoid that inconvenience, we had best deal only with packs having a prime number of cards, or one less than a prime number; for, in the last case, we can imagine an additional last card which remains in the zero place, as long as there is only multiplication, no addition; that is, as long as the pack is not cut.

If we deal a pack of 10 cards into 3 piles twice or into 7 piles twice, we multiply by – 1; for 3 × 3 9 and 7 × 7 = 9, and 9 is one less than 0 or 10. Suppose, then, starting with 10 cards in their proper order we deal them into 3 piles (or 7 piles) and, taking them up according to the rule, next lay them down backs up in a circle, thus:—

[ILLUSTRATION OMITTED]

Then, my dear Barbara, you can say to your little friend Celarent, who is so fond of denying everything, "Celarent, what number do you want to find?" Suppose she says 6. Then, you count 6 places from the 0, say in the right-handed direction. You turn up the 6th card, which is the 8; and you say: "If the 8 is in the 6th place clockwise, then the 6 is in the 8th place counter-clockwise." Thereupon, you count 8 places from the 0 to the left and turn up the 8th card, and lo, it is the 6. Or you might have counted, at first, 6 places to the left and turning up the 6th card, have found the 2. Then you would say "If the 2 is in the 6th place counter-clockwise, then the 6 is in the 2nd place clockwise." And counting 2 places from the 0 to the right, you would again find the 6. The same would hold good if Celarent were to call for any other number.

(Continues...)

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Excerpted from Writings of Charles S. Peirce Volume 8 1890-1892 by Charles S. Peirce, Nathan Houser, André de Tienne, Jonathan R. Eller, Cornelis de Waal, Albert C. Lewis, Diana Reynolds, Joseph Kaposta, Luise H. Morton, Kelly Tully-Needler, Leah Cummins Guinn. Copyright © 2010 Peirce Edition Project. Excerpted by permission of Indiana University Press.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

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Table of Contents

Preface
Acknowledgments
Introduction
1. My Life written for the Class-Book
2. Private Thoughts principally on the conduct of life
3. The Sense of Beauty never furthered the Performance of a single Act of Duty
4. Raphael and Michael Angelo compared as men
5. A Scientific Book of Synonyms
6. Think Again!
7. Analysis of Genius
8. The Axioms of Intuition. After Kant
[Three Essays on Infinity and God]
9. An essay on the Limits of Religious thought written to prove that we can reason upon the nature of God
10. [The Conception of Infinity]
11. Why we can Reason on the Infinite
12. Proof of the Infinite Nature of the Creator
13. I, IT, and THOU: A Book giving Instruction in some of the Elements of Thought
14. The Modus of the IT
15. View of Chemistry: sketched for Young Ladies
16. [A Treatise on Metaphysics]
17. Analysis of Creation
18. SPQR
19. The Chemical Theory of Interpenetration
20. [The Place of Our Age in the History of Civilization]
21. Letter Draft, Peirce to Pliny Earle Chase
22. [Shakespearian Pronunciation]
23. Analysis of the Ego
24. A Treatise of the Major Premisses of Natural Science
25. On the Doctrine of Immediate Perception
26. Letter, Peirce to Francis E. Abbot
On the Logic of Science [Harvard Lectures of 1865]
27. Lecture I
28. Lecture II
29. Lecture III
30. Lecture on the Theories of Whewell, Mill, and Compte
31. Lecture VI: Boole's Calculus of Logic
32. Lecture on Kant
33. Lecture VIII: Forms of Induction and Hypothesis
34. Lecture X: Grounds of Induction
35. Lecture XI
36. Teleological Logic
37. An Unpsychological View of Logic to which are appended some applications of the theory to Psychology and other subjects
38. Logic of the Sciences
39. [The Logic Notebook]
40. Logic Chapter I
The Logic of Science; Or, Induction of Hypothesis [Lowell Lectures of 1866]
41. Lecture I
42. Lecture II
43. Lecture III
44. Lecture IV
45. Lecture V
46. [Lecture VI]
47. Lecture VII
48. Lecture IX
49. Lecture X
50. Lecture XI
51. Memoranda Concerning the Aristotelean Syllogism
52. [On a Method of Searching for the Categories]
Appendix
53. [Diagram of the IT]
Editorial Notes
Bibliography of Peirce's References
Chronological List, 1849-1866
Essay on Editorial Method
Explanation of Symbols
Textual Notes
Emendations
Word-Division
Index