What makes mathematics tick?

In this talk we will enter the backstage of mathematics and answer the question: what is the core mechanism that makes mathematics work? Knowledge of this mechanism is not only to satisfy one’s curiosity; it can also be a powerful tool for making the process of understanding mathematics easier. The next question is: how do we access this tool at home, in the classroom and at the university?

Zurab Janelidze

ALGEBRA

GEOMETRY

Where is Mathematics?

Is Mathematics in the Objective Realm (exists around us) or in the Subjective Realm (exists within us)?

The pure mathematician, like the musician, is a free creator of his world of ordered beauty.

Bertrand Russell

(logic, philosophy)

Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection.

Richard Courant

(Analysis, mathematical physics)

Some things will drop out of the public eye and go away, but there will always be science, engineering, and technology. And there will always, always be mathematics.

Katherine Johnson

(Computations for space missions)

Without mathematics, there’s nothing you can do. Everything around you is mathematics.

Shakuntala Devi

(Mental calculator)

Mathematics exists both in the Subjective Realm and the Objective Realm. It gets activated by exchange of the two realms.

6x + 3 = 33

6x + 3 = 33

6x = 30

x = 5

Step 6. No matter which number x is, if 6x + 3 = 33 is true then x = 5 must be true.

Step 7. The number x = 5 is the only number such that 6x + 3 = 33.

Step 1. Assume that x is a specific number. It can be any number, though. Whatever we will say about x in what follows will not make use of any fact that depends on what number x is.

Step 2. Assume furthermore that 6x + 3 = 33. So x is no longer any number, now x is any number for which 6x + 3 = 33.

Step 3. Then 6x = 30, by subtracting 3 from equal numbers 6x + 3 = 33.

Step 4. Then furthermore x = 5, by dividing equal numbers 6x = 30 by 6.

Step 5. So, if 6x + 3 = 33 is true then x = 5 must be true.

Movement of a mathematical argument

Step 7. The number x = 5 is the only number such that 6x + 3 = 33.

Step 6. No matter which number x is, if 6x + 3 = 33 is true then x = 5 must be true.

Step 5. So, if 6x + 3 = 33 is true then x = 5 must be true.

Step 4. Then furthermore x = 5, by dividing equal numbers 6x = 30 by 6.

Step 3. Then 6x = 30, by subtracting 3 from equal numbers 6x + 3 = 33.

Step 2. Assume furthermore that 6x + 3 = 33. So x is no longer any number, now x is any number for which 6x + 3 = 33.

Step 1. Assume that x is a specific number. It can be any number, though. Whatever we will say about x in what follows will not make use of any fact that depends on what number x is.

What makes mathematics tick is the clausal structure of thought

1, 2, 3, 4, 5, ?

What is the next number?

The clausal structure of mathematical thought is governed by deductive reasoning

Assume furthermore

a = 1, b = 2, c = 3, d = 4, e = 5

Assume a, b, c, d, e, f, … is an arbitrary sequence of numbers

Then f = 6

This clause cannot be obtained by deductive reasoning!

Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country.

David Hilbert

(Algebra, number theory, Foundations, Functional analysis, physics)

If others would but reflect on mathematical truths as deeply and continuously as I have, they would make my discoveries.

Carl Friedrich Gauss

(algebra, number theory, physics)

It is clear that the chief end of mathematical study must be to make the students think.

John Wesley Young

(projective geometry)

Why do children dread mathematics? Because of the wrong approach. Because it is looked at as a subject.

Shakuntala Devi

(mental calculator)

Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding.

William Paul Thurston

(Topology)

In mathematics the art of proposing a question must be held of higher value than solving it.

Georg Cantor

(Set theory)

The only way to learn mathematics is to do mathematics.

Paul Halmos

(logic, statistics, functional analysis)