Research Catalogue

Descriptive Functions and Intuitive Thought

Mathematics is a science that appeals to the fundamental laws of logic, to what constitutes the structure of the human understanding. None diverges from mathematics without ceasing to think. Remaining blind when presented with a light that shines with brilliance would be very prodigious.

Incapability of describing in mathematics, for this matter, is not an acceptable question. For a philosopher or a scientist, a good description applies to all the objects and satisfies logical laws. Understanding the proof of a theorem in math is to understand each syllogism of which it is composed, and noting that it is correct in accordance with logical rules.

Few would say to have understood, and most would be much more demanding to know if all the syllogisms of a demonstration are correct, and why they are correlated in a particular order, rather than in another. For them, these syllogisms would seem not to be generated by caprice, and not by an intelligence constantly conscious of the goal to be obtained. Doubtlessly, they themselves would not really realize what they want and cannot formulate their desire.

The contrast is explained with the allegory of the bookish trajectories of the sciences. A math book written in Ancient Greek or Egypt or early Enlightenment reasoning might seem lacking an exactitude. What was accepted at that time would be, for instance, that a continuous function cannot change its sign without canceling itself. The idea of a continuous function was at first sensitive, a line drawn with chalk on the blackboard or with a stick on the sand, the simplified image of a complex system.

Intuitive Thought

A lot of things in math used to rely on intuition, and were sometimes false. Intuition couldn't suffice for exactitude. For instance, one of the intuitive rules was that every curve has a tangent, that is to say, that every continuous function has a derivative. This intuitive thought proved to be false.

Intuition has to be left for exactitude and facts. Exactitude could not be established in logic without introducing it into the description. Logic sometimes breeds monsters.

Objects that mathematicians dealt with were described poorly with intuitive senses or imagination. A rough image and not an exact idea on which logic could work.

Logicians had to focus their efforts there. For the uncountable.

Philosophers of mathematics had the ambivalent idea of continuity with intuition, resolved into a complicated system of inequalities relating to integers.

All that remains in analysis is whole numbers, or finite or infinite systems of whole numbers, linked by a network of equalities and inequalities. Mathematics has become arithmetic.

In the last millennium, there have been bizarre functions resembling purposeful functions. No continuity, definitive tangents or derivatives, etc. These strange functions are mostly general, appeared as particular cases.

New functions used to be invented with some practical goal. If the logic was the only light of the inventor, it'd include the most general and bizarre functions. Without these, exactitude was presumed only in phases of the function.

Other than logic, there is also a more subtle reality that constitutes lives of mathematical beings. What would it mean to admire the mason's work in the construction if the architect is not comprehensive? It is intuition that must be demanded for an answer. Logic alone cannot conceive the panorama.