Logic, Reason, and the Incompleteness of Mathematics

A fun fact is that the name of this website comes from the book: Logicomix: an Epic Search for Truth.

This article is more or less a review of the book.

Logicomix is a graphic novel(or a comic book) depicting the life of the renowned British mathematician, philosopher, logician, writer, and public intellectual Bertrand Russell, particularly Russell’s quest for fundamental and absolute truth by attempting to establish a new rigid and provable foundation for mathematics.

The book discusses a bunch of intriguing ideas.

The Quest of Solid Foundations

Russell considers the whole subject of mathematics, which dates back to Ancient greek, to be a house built on sand, a home without a solid foundation. Russell thinks that all human mathematical knowledge is unreliable because they are derived from axioms and postulates, self-evident and true but haven’t been proved mathematically.

For instance, Euclid, the father of geometry, builds this branch of mathematics upon axioms and postulates.

Euclid’s Axioms

1. Things which are equal to the same thing are also equal to one another
2. If equals be added to equals, the wholes are equals.
3. If equals be subtracted from equals, the reminders are equals.
4. Things which coincide with one another are equal to one another
5. The whole is greater than the part

Euclid’s Postulates

1. To draw a straight line from any point to point.
2. To produce a finite straight line continuously in a straight line.
3. To describe a circle with any center and distance.
4. That all right angles are equal to one another.
5. That if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the straight lines, if produced indefinitely, will meet on that side on which the angles are less that two right angles.

Therefore, Russell describes the foundation of mathematics as a Shaky World Turtle, as illustrated in the image:

In his decade-long search for a solid foundation in mathematics, Russell completed The Principles of Mathematics and Principia Mathematica, during which He started to realize intuitively that

It’s an endless journey to pursue a solid foundation in math because new axioms need to be introduced to prove an existing axiom. The process goes on and on, never stopping!

As depicted vividly by the following image:

The Incompleteness of Mathematics

Although Russell had an intuitive realization that it’s impossible to get rid of axioms and have a rock-solid foundation for math, it was another mathematical genius Kurt Gödel proved mathematically that

There will always be unanswerable questions and unprovable statements; therefore, arithmetic, and thus any system based on it, is, of necessity, incomplete.

The finding is called Gödel’s First Incompleteness Theorem. Combined with his Second Incompleteness Theorem, Gödel proved mathematically

the limits of provability in formal axiomatic theories.

Russell’s intuition and Godel’s Incompleteness theory confirm one vital point: there will always be unknown and unanswerable questions in mathematical systems and in science and technology built upon mathematics, resonating with Billy Graham’s idea that technology can’t solve human shortcomings as well.

Pacifism

Russell is a pacifist and an opponent of violence of any kind, primarily due to his close encounter with the consequence of wars as a kid.

The public heavily criticized him and even imprisoned him for this.

Russell’s situation is quite relatable to us in 2022, with the raging war between Russia and Ukraine and pacifist people, including Elon Musk, being attacked and criticized.

I am not in any position to judge, but I just find it very interesting that:

People are generally against war when they have encountered the outcomes of battles, directly or indirectly.
Newer generations tend to be subject to nationalistic or other ideological influences and long for wars.
History witnesses the weird cycle of generations of pro and against wars.

Logic and First Principle Thinking

What is logic?

Logic is new and necessary reasoning!

How should we view and approach logic?

Logic is an essential and powerful tool of human minds.
Meanwhile, we should be aware of the limitation of such tools and the finity of science and technology.
Then, use logic, like other tools in life, to make life better.

First principle thinking

With logic, we can reason through first principles for vital questions.