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# 80-110 Nature of Mathematical Reasoning

Spring 2002

Dirk Schlimm

Homework 3

Wednesday, January 30

Due Monday, February 5

1. Deductive validity (2 points)

Assume you are presented with a proof of claim A, and you are convinced that it is a valid proof. What can you say about the truth of the claim A?

2. Forms of arguments (3 points)

Read the handout #2: Anthony Weston, A Rulebook for Arguments, 1992. Chapter IV: Deductive Arguments, pages 46-59.

Give an example of an argument (different than those mentioned in the handout) for each the following forms:
a) modus ponens,
b) modus tollens,
c) hypothetical syllogism,
d) disjunctive syllogism,
e) dilemma.

3. Hume on inductive and deductive reasoning (5 points)

The following two paragraphs are from: David Hume (1711-76). An Enquiry Concerning Human Understanding (1748). Chapter IV: Sceptical Doubts concerning the Operations of the Understanding, Part I.

ALL the objects of human reason or enquiry may naturally be divided into two kinds, to wit, Relations of Ideas, and Matters of Fact. Of the first kind are the sciences of Geometry, Algebra, and Arithmetic; and in short, every affirmation which is either intuitively or demonstratively certain. That the square of the hypothenuse is equal to the square of the two sides, is a proposition which expresses a relation between these figures. That three times five is equal to the half of thirty, expresses a relation between these numbers. Propositions of this kind are discoverable by the mere operation of thought, without dependence on what is anywhere existent in the universe. Though there never were a circle or triangle in nature, the truths demonstrated by Euclid would for ever retain their certainty and evidence.

Matters of fact, which are the second objects of human reason, are not ascertained in the same manner; nor is our evidence of their truth, however great, of a like nature with the foregoing. The contrary of every matter of fact is still possible; because it can never imply a contradiction, and is conceived by the mind with the same facility and distinctness, as if ever so conformable to reality. That the sun will not rise to-morrow is no less intelligible a proposition, and implies no more contradiction than the affirmation, that it will rise. We should in vain, therefore, attempt to demonstrate its falsehood. Were it demonstratively false, it would imply a contradiction, and could never be distinctly conceived by the mind.

Answer the following questions in at least a paragraph each.

a) Try to say in your own words what `relations of ideas' and `matters of fact' are according to Hume?

b) For Hume, Mathematics belongs to the `relations of ideas'. Why? Do you agree?

c) What would Hume reply to the claim `I know for sure that the sun will rise tomorrow'?

d) Explain how the notions of inductive and deductive reasoning introduced in class relate to Hume.