Lec 1 | MIT 6.042J Mathematics for Computer Science, Fall 2010

TL;DR

This video discusses the concept of proofs and explores different ways of ascertaining truth across various fields, including mathematics, science, society, and religion.

Transcript

The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Who can tell me what a proof is? Any ideas of, ... Read More

Key Insights

• 🏑 Proofs are not limited to mathematics and can be found in various fields such as science, society, and religion.
• 🍽️ Different methods are used to ascertain truth, including observation, experiments, logical deductions, finding counterexamples, examining evidence, relying on inner conviction, and the word of authoritative figures.
• 💁 Axioms are propositions that are assumed to be true and form a basis for proofs.
• 😫 Consistency and completeness are desirable qualities of a set of axioms, but Kurt Godel showed that it is impossible to have both in any set of axioms.

Questions & Answers

Q: What is a proof?

A proof is a chain of statements that are logically supported by previous ones, leading to a set of conclusions.

Q: Are proofs limited to mathematics?

No, proofs exist in various fields. For example, in science, observations and experiments are used to ascertain truth. In society, truth can be established through the word of authoritative figures like judges, juries, religious figures, and bosses.

Q: Can you give an example of how truth is ascertained in society?

In society, truth can be established through methods such as observation, finding counterexamples, examining evidence, and relying on inner conviction or the word of authoritative figures like judges, juries, religious figures, and bosses.

Q: Is there a higher meta-level notion of what a proof is?

Yes, beyond the logical deductions and assumptions of a proof, a higher meta-level notion of a proof is to establish or verify truth across various fields and disciplines.

Summary & Key Takeaways

• The video begins with a discussion of what a proof is and how it is a chain of statements logically supported by previous ones to reach a set of conclusions.

• The concept of proofs is not limited to mathematics but also exists in other fields, such as science, society, and religion, where different methods are used to ascertain truth.

• Methods of ascertaining truth include observation, experiments, logical deductions, sampling, and the word of authoritative figures like judges, juries, religious figures, and bosses.