Benutzer:O.tacke/2013/intro math phil

Der MOOC ist leider so langweilig gestaltet und birgt keine didaktische Konzeption, so dass ich ihn abgebrochen habe.

Infinity

Introduction

• Infinity, quite popular subject
• The world seems to be finite (live span, number of children, ...). Therefore many people thought: "Infinity is left to God." Understanding the infinite then is quite an enterprise... (cmp. Nicolaus Cusanus)
• Infinitely possible worlds, God would make one of it the world we're actually living in. => Lecture 3
• Infinitely good could mean having infinitely many good traits.

Arguments and Paradoxes

• (cmp. Zeno of Elea; cmp. Parmenides)
• It appears as if there is something, but really, this might be an illusion (cmp. bending in optics)
• What is an argument? Sequence of statements, first ones are premises, final one is the conclusion (introduced with therefore).
• Logically valid argument => transitivity (Modus Ponens). A logically valid argument doesn't need the premises to be true in order to be logically valid, but if they are, then the conclusion is also true. => Lecture 2, 4
• (cmp. Aristotle) Constructing logically valid argmuents
• Paradox: a logically valid argument in which each premise is plausible if taken by itself, but where the conclusion of the argument is absurd. => method to argue against premises which initially seem to be true.
• Example: Achilles vs. Turtoise

Zeno's Paradox

Example in detail...

Calculus to the Rescue

• modelling the problem in numbers
• boundary issues

Sets

• lists of elements with irrelevant order or repetition of elements
• empty set
• principle of extensionality

Comparing Sets in Terms of Size

• subsets and proper subsets
• bijective functions (or correlation between sets in general, pairing off)
• Example: Galileo Galilei

Diagnosis of Galileo's Paradox

• Galileo's solution: comparing infinite sets is absurd, "equally many" is a critical term.
• There are different ways to define dealing with infinity.

Cantor's Theorem

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Conclusions

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