the red and the black - mathematics 15: lecture...

The Red and the BlackMathematics 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.

Pragmatism

I Identifies our conception of an object with all the possible effects weconceive the object to have.

I Example: Hardness - To say an object, such as a diamond, is hard, isto say that the object will act in certain ways under certainconditions, such as scratching objects which are not hard.

Pragmatism

I Identifies our conception of an object with all the possible effects weconceive the object to have.

I Example: Hardness - To say an object, such as a diamond, is hard, isto say that the object will act in certain ways under certainconditions, such as scratching objects which are not hard.

Inferences

I Deductive arguments: conclusion necessarily follows from thepremises.

I Example

I Socrates is a man.I All men are mortal.I Therefore, Socrates is mortal.

I Variation: the conclusion might hold a certain proportion of the time.I Example

I An urn contains 6 white balls and 4 red balls.I A ball is chosen at random from the urn.I Therefore, the chosen ball is white.

I In this case, the conclusion holds 60% of the time.

I That is: probability refers to the proportion of the time the conclusionof A⇒ B is true.

Inferences

I Example

I Socrates is a man.I All men are mortal.I Therefore, Socrates is mortal.

Inferences

I ExampleI Socrates is a man.

I All men are mortal.I Therefore, Socrates is mortal.

Inferences

I ExampleI Socrates is a man.I All men are mortal.

I Therefore, Socrates is mortal.

Inferences

I ExampleI Socrates is a man.I All men are mortal.I Therefore, Socrates is mortal.

Inferences

I Variation: the conclusion might hold a certain proportion of the time.

I Example

Inferences

I An urn contains 6 white balls and 4 red balls.

I A ball is chosen at random from the urn.I Therefore, the chosen ball is white.

Inferences

I An urn contains 6 white balls and 4 red balls.I A ball is chosen at random from the urn.

I Therefore, the chosen ball is white.

Inferences

Frequency

I Interpretation: for the previous example, if we repeat the experimentan infinite number of times, we will draw a white ball 60% of the time.

I But that’s not possible: we can repeat an experiment only a finitenumber of times.

I What good does it do us to say that a conclusion would be correct60% of the time if we could repeat the experiment an infinite numberof times?

Frequency

The red and the black

I Two decks of cards: one has 25 red cards and one black card, theother has 25 black cards and one red card.

I You are to draw one card: a red card transports you to “eternalfelicity,” a black card to ”everlasting woe.”

I Which deck should you choose from?

I Obviously, you want to choose from the deck with 25 red cards. Butwhy?

Without death

I Peirce, on page 1338: “But what, without death, would happen toevery man, with death must happen to some man.”

I That is: if an event has a positive probability, it must necessarilyhappen with enough repetitions.

Without death

I Peirce, on page 1338: “But what, without death, would happen toevery man, with death must happen to some man.”

I That is: if an event has a positive probability, it must necessarilyhappen with enough repetitions.

The community

I Peirce’s resolution (page 1338): “It seems to me that we are driven tothis, that logicality inexorably requires that our interests shall not belimited. They must not stop at our own fate, but must embrace thewhole community.”

I Indispensable requirements of logic (page 1340): “interest in anindefinite community, recognition of the possibility of this interestbeing made supreme, and hope in the unlimited continuance ofintellectual activity.”

I Charity, faith, and hope?

The community

Peirce and Laplace

I How does Peirce’s view of probability contrast with that of Laplace?

the red and the black - mathematics 15: lecture...

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