Logic

Logic is an advanced math elective which combines into a single semester what would typically be taught in a college introduction to logic class with a survey of the history and philosophy of mathematics. Students learn about the syntax and semantics of propositional and predicate logic and prove laws of logic in a natural deduction proof system. We chart the history of math from its origins in the geometry of the ancient world through the invention of calculus and the radical changes away from geometry in the 19th century. We explore some of the crises in the foundations of mathematics and the attempted resolutions at the beginning of the 20th century

Do you feel like you've learned a lot of mathematics in your life so far but don't see how it's all connected together or how it came to be the way it is? What's the connection between geometry and algebra? Where did calculus come from? Which mathematical objects exist? What mathematical statements are true and how do we know? What's the source of our mathematical knowledge? What mathematics has happened in the last 150 years that's not in my textbook?

For answers to these questions, take Logic, a one-semester tour of math history that begins with the mathematics and philosophy of Ancient Greece, studying Euclid's Elements and the axiomatic method, through the discovery of non-Euclidean geometries, the birth of calculus, the transformation of calculus into something totally new, and the various paradoxes of set theory that awaited mathematicians of the 19th and 20th century who tried their best to establish absolute mathematical truth once and for all. It's a humanities class whose subject is mathematics.

Along the way, we will also learn modern propositional and predicate Logic and a formal proof system to secure the reliability of our mathematical reasoning.

Logic Units Outline

Unit I – Propositional Logic

1. Propositional connectives, syntax, truth tables, tautologies, logically equivalent propositions

2. An analysis of the conditional, including converse and contrapositive

3. An analysis of logically valid arguments

4. A formal system of natural deduction to generate logically valid propositional arguments

Unit II – The Mathematics of Ancient Greece

1. The origins of Greek Mathematics in Egypt, Babylonia

2. Ancient Greek civilization, philosophy, society

3. The Pre-Socratics, Socrates, Plato, and mathematical Platonism

4. Proof by Contradiction

5. The earliest records of geometric proof and the discovery of irrational numbers

6. The Axiomatic Method as seen in Euclid’s Elements

7. The necessity of undefined terms in an axiomatic system

8. The philosophy and epistemology of the postulates

9. A close examination of the entire Book I of Euclid’s Elements

10. The proof of the Pythagorean Theorem as the culmination of Book I

11. Attempts to prove the Parallel Postulate

Unit III – Non-Euclidean Geometries

1. Modern attempts to prove the Parallel Postulate

2. Neutral Geometry

3. Many Parallel Postulate equivalents

4. The origins of Hyperbolic Geometry

5. Geodesics and the Angle of Parallelism in the various hyperbolic models

6. The models of Hyperbolic Geometry

7. Positive, negative, and zero Gaussian curvature

8. Spherical Geometry

9. The emergence of multiple incompatible geometric systems and the end of objective mathematical truth

10. Logical Flaws in Euclid’s Presentation of Euclidean Geometry

11. Reasoning from the diagram, Pasch’s Axiom, Betweenness

12. Modern Axiomatizations of Euclidean Geometry

Unit IV – Predicate Logic

1. The syntax of Predicate Logic – predicates, names, variables, functions

2. Quantifiers

3. Translating sentences from English into predicate logic and vice versa

4. Logically equivalent propositions in predicate logic

5. A formal system of natural deduction to generate logically valid propositional arguments

Unit V – Calculus

1. Archimedes and the Method of Exhaustion

2. The historical origins of Algebra in China, India, Arabia, & Persia

3. The introduction of algebra into European geometric mathematics

4. The discovery of calculus by Fermat, Descartes, Cavalieri, Kepler, Wallis, etc.

5. The introduction of algebraic techniques: Newton & Leibniz

6. Fluents, fluxions, infinitesimals and the geometric objects of calculus

7. The abandonment of geometry as a basis for Calculus

8. Cauchy, Bolzano, Weierstrass and the Age of Rigor

9. The ε-δ definition of the limit and the Arithmetization of Analysis

10. Infinite series, the Reimann Integral

11. The Real Numbers as a mathematical foundation for Analysis

12. The Least Upper Bound Axiom of the real numbers

13. The Bounded Monotonic Sequence Convergence Theorem, the Bolzano-Weierstrass Theorem, the Intermediate Value Theorem

14. The rebuilding of Analysis as a deductive axiomatic system

Unit VI – Numbers and Sets

1. The construction of the real numbers as Dedekind Cuts

2. The reduction of number systems to natural numbers

3. The axiomatization of the natural numbers with Peano Arithmetic

4. Proof by induction in Peano Arithmetic

5. Sets and Equinumerosity via Bijections

6. The naturals, integers, and rationals are countable; the reals are uncountable

7. Cantor’s Theorem

8. Transcendental numbers

9. Godel-numbering

10. Transfinite numbers, ordinals, the Continuum Hypothesis

Unit VII – First Order Models

1. The domain of discourse

2. Interpretations of names, predicates, functions

3. Truth in a model

4. Evaluating nonstandard models of arithmetic

Unit VIII – Foundations of Mathematics

1. Hilbert’s Foundations of Geometry and Metamathematics

2. Axioms must be Consistent and Independent

3. Reduction of geometry to arithmetic

4. Frege, modern logic, and Logicism

5. Russell’s Paradox and impredicativity

6. The Axiom of Choice

7. Intuitionism

8. ZFC & modern axiomatic set theory

9. Hilbert’s Program to formalize all mathematics

10. Turing Machines and the formalization of decidability

11. The Halting Problem

12. Godel’s Incompleteness Theorem