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The Red and the Black Mathematics 15: Lecture 15 Dan Sloughter Furman University October 18, 2006 Dan Sloughter (Furman University) The Red and the Black October 18, 2006 1 / 9

Charles Sanders Peirce Max H. Fisch (in The Play of Musement by Thomas Sebeok): Who is the most original and the most versatile intellect that the Americas have so far produced? The answer Charles S. Peirce is uncontested, because any second would be so far behind as not to be worth nominating. Mathematician, astronomer, chemist, geodesist, surveyor, cartographer, metrologist, spectroscopist, engineer, inventor; psychologist, philologist, lexicographer, historian of science, mathematical economist, lifelong student of medicine; book reviewer, dramatist, actor, short story writer; phenomenologist, semiotician, logician, rhetorician and metaphysician. He was, for a few examples, the first modern experimental psychologist in the Americas, the first metrologist to use a wave-length of light as a unit of measure, the inventor of the quincuncial projection of the sphere, the first known conceiver of the design and theory of an electric switching-circuit computer, and the founder of the economy of research. He is the only system-building philosopher in the Americas who has been both competent and productive in logic, in mathematics, and in a wide range of sciences. If he has had any equals in that respect in the entire history of philosophy, they do not number more than two. Dan Sloughter (Furman University) The Red and the Black October 18, 2006 2 / 9

Pragmatism Identifies our conception of an object with all the possible effects we conceive the object to have. Dan Sloughter (Furman University) The Red and the Black October 18, 2006 3 / 9

Pragmatism Identifies our conception of an object with all the possible effects we conceive the object to have. Example: Hardness - To say an object, such as a diamond, is hard, is to say that the object will act in certain ways under certain conditions, such as scratching objects which are not hard. Dan Sloughter (Furman University) The Red and the Black October 18, 2006 3 / 9

Inferences Deductive arguments: conclusion necessarily follows from the premises. Dan Sloughter (Furman University) The Red and the Black October 18, 2006 4 / 9

Inferences Deductive arguments: conclusion necessarily follows from the premises. Example Dan Sloughter (Furman University) The Red and the Black October 18, 2006 4 / 9

Inferences Deductive arguments: conclusion necessarily follows from the premises. Example Socrates is a man. Dan Sloughter (Furman University) The Red and the Black October 18, 2006 4 / 9

Inferences Deductive arguments: conclusion necessarily follows from the premises. Example Socrates is a man. All men are mortal. Dan Sloughter (Furman University) The Red and the Black October 18, 2006 4 / 9

Inferences Deductive arguments: conclusion necessarily follows from the premises. Example Socrates is a man. All men are mortal. Therefore, Socrates is mortal. Dan Sloughter (Furman University) The Red and the Black October 18, 2006 4 / 9

Inferences Deductive arguments: conclusion necessarily follows from the premises. Example Socrates is a man. All men are mortal. Therefore, Socrates is mortal. Variation: the conclusion might hold a certain proportion of the time. Example An urn contains 6 white balls and 4 red balls. A ball is chosen at random from the urn. Therefore, the chosen ball is white. Dan Sloughter (Furman University) The Red and the Black October 18, 2006 4 / 9

Inferences Deductive arguments: conclusion necessarily follows from the premises. Example Socrates is a man. All men are mortal. Therefore, Socrates is mortal. Variation: the conclusion might hold a certain proportion of the time. Example An urn contains 6 white balls and 4 red balls. A ball is chosen at random from the urn. Therefore, the chosen ball is white. In this case, the conclusion holds 60% of the time. Dan Sloughter (Furman University) The Red and the Black October 18, 2006 4 / 9

Inferences Deductive arguments: conclusion necessarily follows from the premises. Example Socrates is a man. All men are mortal. Therefore, Socrates is mortal. Variation: the conclusion might hold a certain proportion of the time. Example An urn contains 6 white balls and 4 red balls. A ball is chosen at random from the urn. Therefore, the chosen ball is white. In this case, the conclusion holds 60% of the time. That is: probability refers to the proportion of the time the conclusion of A B is true. Dan Sloughter (Furman University) The Red and the Black October 18, 2006 4 / 9

Frequency Interpretation: for the previous example, if we repeat the experiment an infinite number of times, we will draw a white ball 60% of the time. Dan Sloughter (Furman University) The Red and the Black October 18, 2006 5 / 9

Frequency Interpretation: for the previous example, if we repeat the experiment an infinite number of times, we will draw a white ball 60% of the time. But that s not possible: we can repeat an experiment only a finite number of times. What good does it do us to say that a conclusion would be correct 60% of the time if we could repeat the experiment an infinite number of times? Dan Sloughter (Furman University) The Red and the Black October 18, 2006 5 / 9

The red and the black Two decks of cards: one has 25 red cards and one black card, the other has 25 black cards and one red card. Dan Sloughter (Furman University) The Red and the Black October 18, 2006 6 / 9

The red and the black Two decks of cards: one has 25 red cards and one black card, the other has 25 black cards and one red card. You are to draw one card: a red card transports you to eternal felicity, a black card to everlasting woe. Dan Sloughter (Furman University) The Red and the Black October 18, 2006 6 / 9

The red and the black Two decks of cards: one has 25 red cards and one black card, the other has 25 black cards and one red card. You are to draw one card: a red card transports you to eternal felicity, a black card to everlasting woe. Which deck should you choose from? Dan Sloughter (Furman University) The Red and the Black October 18, 2006 6 / 9

The red and the black Two decks of cards: one has 25 red cards and one black card, the other has 25 black cards and one red card. You are to draw one card: a red card transports you to eternal felicity, a black card to everlasting woe. Which deck should you choose from? Obviously, you want to choose from the deck with 25 red cards. But why? Dan Sloughter (Furman University) The Red and the Black October 18, 2006 6 / 9

Without death Peirce, on page 1338: But what, without death, would happen to every man, with death must happen to some man. Dan Sloughter (Furman University) The Red and the Black October 18, 2006 7 / 9

Without death Peirce, on page 1338: But what, without death, would happen to every man, with death must happen to some man. That is: if an event has a positive probability, it must necessarily happen with enough repetitions. Dan Sloughter (Furman University) The Red and the Black October 18, 2006 7 / 9

The community Peirce s resolution (page 1338): It seems to me that we are driven to this, that logicality inexorably requires that our interests shall not be limited. They must not stop at our own fate, but must embrace the whole community. Indispensable requirements of logic (page 1340): interest in an indefinite community, recognition of the possibility of this interest being made supreme, and hope in the unlimited continuance of intellectual activity. Dan Sloughter (Furman University) The Red and the Black October 18, 2006 8 / 9

The community Peirce s resolution (page 1338): It seems to me that we are driven to this, that logicality inexorably requires that our interests shall not be limited. They must not stop at our own fate, but must embrace the whole community. Indispensable requirements of logic (page 1340): interest in an indefinite community, recognition of the possibility of this interest being made supreme, and hope in the unlimited continuance of intellectual activity. Charity, faith, and hope? Dan Sloughter (Furman University) The Red and the Black October 18, 2006 8 / 9

Peirce and Laplace How does Peirce s view of probability contrast with that of Laplace? Dan Sloughter (Furman University) The Red and the Black October 18, 2006 9 / 9