Exploring the Evolution of Mathematical Foundations: Past, Present & Future 🔍

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Speaker: Harvey Friedman, Department of Mathematics, Ohio State University

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1:16 Agenda
2:13 The foundational life
4:40 Foundational life
6:35 Philosophical life
9:10 Mathematical life, scientific life
10:20 An ambition
11:50 Foundations of mathematics foundations of physical science
12:45 Foundations of applied mathematics
14:33 Profound uneasiness - 1
16:13 Profound uneasiness - 2
17:01 First main step epsilon delta
17:33 Number systems
18:38 Real number system multiple definitions
20:17 Infinite upper shift Kernel theorem
21:05 Rationals and integers
22:30 Grand unification
23:59 Pure set theory immutable objects
25:41 Exploding universe
27:21 Assertions and proofs
28:07 Logical structure
29:27 Proof assistants
30:40 Understanding trivialities
32:03 Strict reverse mathematics
33:45 Fundamental foundation moves: consistency, completeness, incompleteness
34:32 Incosistent systems useful?
36:10 Gödel's second incompleteness Theorem
37:04 Four completeness theorems
38:09 Gödel's first incompleteness theorem
39:16 First mathematically natural incompleteness
40:03 Consistency, and the incorporation of new notions
42:25 Concept calculus
44:33 Completeness, and delicate choice of fragments
45:21 0 - Minimality - 1
45:21 0 - Minimality - 2
47:47 Incompleteness, and concreteness, simplicity, naturalness
48:59 Maximal clique embedding