Mathematics consists in the study of numbers, forms or similar objects. About those we can investigate, study, or simply show some kind of interest in one of two possible directions. One, the first, the traditional or commonly used, is the one that has a linear and constructive in its complexity and structure. For example, when we move from natural numbers to integers, fractions, reals and finally complexes. In the same way, when we study addition, from there to multiplication, differentiation, integration, etc. This is done in the mathematics of current use. The other, the second, the least known, is the path of mathematical philosophy. In it one advances not in a linear and pre-established way, as in the mathematics of current use, in the mathematical philosophy, in a totally differentiated way, the path is that of the recursive and challenging analysis, loaded with an unlimited abstraction, looking for new answers and portentos of problems before what traditional and current mathematics takes for granted.

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The same sense

When we say that it is the same sense, we start from the fact that both one and the other, that is, constructive mathematics and mathematical philosophy, refer to the same aspects, that is, mathematical objects. However, the position in one or another perspective will make the difference with respect to the direction, that is, it will be the inclination of the researcher, or student of the mathematical field, which will demarcate the side of the mathematical border where the mathematician is located.
Thus, for example, when the first Egyptian mathematicians who developed a survey of amazing calculations full of pragmatism and applicability, without doubt, made use of the mathematics we have called constructive or current.

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On the contrary, the same geometrical field treated by the ancient Greek geometers, such as, for example, Thales of Miletus and Pythagoras, who transformed those early Egyptian mathematical works into more general postulates and principles, also supported by justifications and demonstrations that For the first time they were part of mathematics and its study, in these cases, undoubtedly, would be dealing with the most authentic mathematical philosophy.

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In correspondence with that visionary, challenging and risky attitude that must accompany any philosopher, regardless of the subfield of that, Thales was called the first great philosopher of history, while Pythagoras is recognized as the initiator of the demonstration in mathematics from the proof of the theorem applied to determine the hypotenuse of a right triangle, better known as The Pythagorean Theorem.