Introduction Can the brain in principle fully understand itself?

Or is it the case that in order to understand a system S, one always needs a system more complex than S itself?

In this course a precise mechanism will be described, called reflection,

showing that there are systems powerful enough to describe themselves.

Reflection occurs in the living cell, in natural language, in mathematics, in computers and in consciousness.

In all cases reflection has outright dramatic effects: life, language, science, ICT, and wisdom, respectively.

Where there is something powereful, there often is a backside.

In the mentioned examples these are: viruses, paradoxes, incompleteness, undecidability, and mental instability.

Both the powerful positive and negative aspects will be described in the course.

We do not claim that the first question on this page can be answered affirmatively.

But at least the fact that reflection can be fully described in a limited space gives hope.

As a pretaste of things to come we cite from Musil, Der Mann ohne Eigenschaften:

It is understandable that an engineer is preoccupied by his specialty

instead of coming into the freedom and space of the world of ideas,

even if his machines are delivered to the furthest corners of the world;

because he does not need to be more able to apply the daring and novel aspects of the soul of his technique

to his personal soul as that a machine is capable to apply the infinitesimal equations on which it depends to itself.

About mathematics one cannot say this; it is the new method itself, the spirit itself,

in it lay the sources of time and the origin of an immense transformation.

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Lecture notes---Coordinates---Blackboard (Do not click if you already are in Blackboard)