Tao Types

The formal treatment of hierarchical logical levels was carried out by Bertand Russell and Alfred N. Whitehead in the first decade of this century, and appeared in final form in 1910 in the monumental work of mathematical logic with the title of "Principia Mathematica".
A major aim of this work was to preserve classical logic from paradoxes and antinomies. The simplest example is the kind of paradox:

"This statement is false"

If the statement is true then it is false, if it is false then it is true.
By examining the logical structure of this statement is known as having two characteristics simultaneously valid:

• The statement is self-reference, that is it refers to itself. For example the statement 'this apple is red"is not a paradox, since a statement is not an apple, while it may be true or false depending on whether the apple is red or not.
• The statement has the logical structure of the type "assertion of a negation" or -equally- "negation of an assertion", and being self-referential denies himself. For example the statement "this statement is true" is not paradoxical, if it is true is true and if it is false is false.

In the classical Aristotelian logic such a situation is devastating, and is determined by the principle of contradiction, which asserts the falsity of every proposition implying that a certain proposition A and its negation, the proposition that non-A, are both true at the same time and in the same way. In the words of Aristotle (Metaphysics):

« it is not possible to say truly at the same time that the same thing is and is not»

In order to preserve classical logic from paradoxical and self-referential problems Russell and Whitehead was assigned to the hierarchical logical levels, which they called "logical types" special rules, establishing a hierarchy of logical types that can not be broken, especially the rule that objects (elements) of a class (together) are at logical level lower than the class and, to prevent the formation of paradoxes, a class may not have as an element itself or anything that presupposes all the elements of a collection should not be an element of the collection itself.
With this split-level logical types on two different planes, one for the class and the other for its members , along with the rule to prevent logical links between the two planes, Russell and Whitehead intended to prevent the formation of the paradoxes in logic. With this subdivision and rule logic "self-swallowing" classes, in which an element of the class is the class itself,  become meaningless and without any logical validity.
A class of classes, ie, a metaclass, is not really a class, say for example that the set of all concepts is itself a concept is meaningless since it is a 'concept' of a higher logical type. The components of a Russellian hierarchy are among them as an element to a class, a class to a metaclass or one thing to their name.

The work of formal systematization and logical foundation of the Principia Mathematica posed the possibility of creating a unique logical-formal system and organize all of mathematics and then physics. The vision was that romantic due to the success of classical physics, namely that the universe was extremely complicated but fully describable if the logic and the math behind the physics were a complete formal system of description of the physical layer.
Thus arose the question of completeness and consistency of such system; it is complete if all true statements of mathematics are derivable "demonstrable" at its inside, it is coherent-consistent  if no internal contradictory statements can be derived, namely a proposition and its negation. A question of this sort comes in metamathematics, because it is a math survey on mathematics.For this purpose D.Hilbert launched the so-called "Hilbert's program" in the 20s: to demonstrate the completeness and consistency of the PM, or the attempt to axiomatization of mathematics. The main points were: 

• Formalization of all mathematics: all mathematical propositions should be written in a precisely formal language , and handled according to well-defined rules.
• Completeness: proof that all the statements that are true can be proved mathematically in the formalism.
• Consistency: a proof that no contradiction can be obtained in the formalism of mathematics. This test of consistency should preferably use only methods "finitistic" about finite mathematical objects.
• Conservation: proof that all the results of "real objects" obtained using the argument about "ideal objects" (as uncountable sets) can be tested without the use of ideal objects.
• Decidability: there should be an algorithm to decide the truth or falsity of any mathematical proposition.

The Russell and Whitehead's monumental effort to save the classical logic and the Hilbertian program that followed were literally swept away forever by the work of Kurt Gödel in1931, in particular the two incompleteness theorems, not by chance entitled "On formally undecidable propositions of Principia Mathematica and related systems", that demonstrate how a axiomatic theoretical building can not simultaneously satisfy the properties of consistency and completeness, and that no consistent system can be used to demonstrate its own consistency.
During those same years, the development of quantum physics and relativity poses a definitive end to the idea that complete formalization of the physical Universe, together with its logical foundations, was possible.

The work of Russell and Whitehead has today an historical significance, however paradoxes and paradoxical situations exist, as well as in logic, also in life. Its use was, unexpectedly, in the description of the interaction and communication with animals and humans by Gregory Bateson. In his words:

"What Russell and Whitehead had faced was a very abstract problem: logic, which they believed, had to be rescued from the tangles that arise when the "logical types", as Russell called them, are mistreated in their mathematical representation.

I do not know if, while working within the "Principia," Whitehead and Russell had any idea that the object of their interest is essential for the life of humans and other organisms.

Whitehead certainly knew that playing with the types you can have fun and you can make the humor emerge. But I doubt that he ever passed the stage of fun and has come to realize that the game was not insignificant and that would throw light on the entire biology.
While not having to contemplate the nature of the human dilemmas that would have been revealed they avoided - perhaps unconsciously - to arrive at a more general understanding."