First Year Seminar, RC. "The Intellectual History of Information: 1680-2001"

This is:http://www-personal.umich.edu/~twod/rc-fys/assignments/study_questions01.html

; &n bsp; Instructor: Tom O'Donnell

Concepts and Questions for Study

Leibniz-- include both [Davis] and [Berlinski] readings

- What was an important difference and advantage of Leibniz' version of the calculus vis-a-vis Newton's? How so? How is this manifested today? According to Davis what was a probable ramification in the continental European Vs English development of mathematics (list some of his examples of advances in 17th Century math)? How does Davis see Leibniz' development of the calculus as a paradigm for another life-long intellectual goal of Leibniz'.

--- What were the central ideas and concepts involved in the discovery/invention of the calculus with which Leibniz grappled? Include discussion and conceptual explanation of limit processes, infinite series, completed/actual infinities (including the opinions of contemporary theologians), the connection between infinite series and finding areas, etc.

--- Discuss the political, social and cultural reality of Leibniz' Germany and his own personal status (social and class status) within European society during his lifetime. And either:

a. In light of this reality, discuss how these factors impacted his education and his philosophical, mathematical, engineering, juridical, diplomatic work and his employment. Include the significance of his trip to Paris -- why he went, who influenced his intellectual development and how, and what was his greatest achievement there. Or,

b. Discuss Leibniz' attitude and interactions with women, especially as intellectuals. Who were these women and what did they interact with Leibniz' on according to Davis, and to Berlinski?

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--- Discuss in detail the three elements of Leibniz' "wonderful idea". Include the distinction between ordinary characteristics and universal system, the calculus ratiocinator, and the components of his program. Be precise. What are models of somewhat similar, though limited systems, in use during his lifetime? What is the fate of Leibniz' program in light of of Frege's system of symbolic logic used today?

--- Discuss and illustrate some of the concepts which Leibniz' contributed to logic. Use the notation used by Davis in describing this.

--- Discuss Leibniz' personal philosophical beliefs, his "Panglossian" attitude towards life. How were these views apparently both a positive and a negative force in his live -- i.e., how did these convictions serve and/or harm him during his lifetime. Who wrote about a Dr. Pangloss; what was this person's attitude?

--Boole-- the [Davis], [Berlinski] andNew Scientistreadings.

- Discuss the social, political and economic times Boole lived and his own social and class origins, and how his personality and habits were or were not a product of the early Industrial Revolution in England. How did his habits and beliefs further or hinder his intellectual work?

--- Explain how Boole applied his new algebra to logic, and explain (according to Davis) how it was that Boole discovered the importance/validity of using the sets
1and0by proceeding in accord with the Aristotelian "principle of contradiction"

--- Explain Aristotelian syllogisms, and how Boole's secondary propositions" went beyond this. What is the algebra of classes, the algebra of inferences. what did Boole say was an advantage of his algebraic methods in logic?

--- Explain binary numbers: how many different numbers can be expressed as a function of the numbers of number of digits (places) in the number? Give analogous examples from base 10 (and/or hexadecimal system).

--- Give an example from Boole's binary logic where n = 3 persons. Using only the operators L ("and"), V ("or") and ~ ("not"), make three clauses with two variables connected by an "or" in each one, and with all the clauses connected by 'and' (this is the 'conjunctive normal form we did in class). How many possible solutions are there? If there is a solution to your problem, what is it?

--- What are the circuit elements which represent "and", "or" and "not"? Rewrite your example above using the corresponding circuit elements. What is the difference between an "or" operator, and an "exclusive or"? What is a "nand" and a "nor"? Make a
truth tablefor all the operators mentioned in this item.

--Peano --the [Berlinski] readings

- How were the long-standing disputes and imprecision about infinite series resolved, what is the present approach using limits to define the sum of an infinite series?

--- Discuss the impetus for Peano to make arithmetic an axiomatic system. What are his postulates?

--Frege --See [Berlinski, chpt. 4] and [Davis, chpt. 3] .

- Discuss, as you did with Boole, how Frege was or was not a "product of his time", including an explanation of the political, economic and social situation Germany was in in his day. How might his later anti-Semitic, anti-Catholic, anti-French and notions of social democratic conspiracies have flowed out of this reality?

--- As opposed to Boole's goal/idea of making logic algebraic, what was Frege's goal? How did he get out of the conundrum of circularity, of not using logic to establish his logic system (called the predicate calculus). Explain the predicate calculus as opposed to the prepositional calculus, including how Frege "got into the propositions" with his "inferential" operator. In relation to Leibniz' program, what is now the "Universal Characteristic of mathematicians exerywhere" [Berlinski, p.69], explain.

--- What are the elements of Frege's system: variables, predicates symbols (for single things and for relationships for 2, 3, 4,...n things), what are the symbols for quantification. What are the symbols used to talk
aboutstatements? Give a truth table for (P Ö Q), and discuss the notion of its "meaning". Explain what the quantifiers " and $ add to the system as compared to Boole's.

--- what is meant by this being a system of "shapes" and "symbols"? What does [Berlinski] mean by his distinction between axioms and axiom schemata" used to talk "about" formulas and predicate symbols?

--- What is an "artificial language" and what is its usefulness?

--- What is the difference between an axiomatic system and a formal system? What do [Davis] and [Berlinski] mean by their references to "machine like" and "the staircase of inference" and constantly saying "check, check", etc.?

--- What, briefly, does Frege have to do with the origins of the philosophy of language, and why?
- How did young Bertrum Russell destroy the foundation of Frege's life's work? Explain the notions of an "extraordinary set" and an "ordinary set".

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