Translated from french by Grok

Mathematics, by virtue of its coherence, creates a world of knowledge. Each lemma, each theorem represents an advance, an extension of this world. And the contours of this world in the making, which have nothing regular about them, depend directly on the mathematical domains to which they are associated.

Naturally, discoveries take place at the frontier of this abstract world, because it is the most perilous place—in the sense that the researcher is not certain of obtaining a result. On the other hand, if one is obtained, it will be, if not revolutionary, at least original.

One of the most interesting tools used in this exploration-creation process is synthesis. Indeed, the organizational power of this procedure can generate, from a heuristic point of view, an idea of generalization that will serve as the basis for conceiving a new conjecture.

In this context, two categories of conjectures are particularly effective. We will give them the following names: conjecture of sublimation and conjecture of transcendence. Since both require significant cognitive activity, they are not accessible to natural intuition. For example, the famous Syracuse conjecture (Collatz conjecture) belongs to neither of these categories.

The conjecture of the first category (sublimation) occurs when the mathematician glimpses certain key points that allow him to construct a demonstration strategy. We find this same type of approach in the game of chess, where combinations are opposed to positions. This way of tackling a new problem enables the researcher not to drown in technical details (like a pebble skipping across water…). Thus, he is not obliged to go through every phase of the future proof’s development. Hence the name conjecture of sublimation. It is to this category of conjecture that Euler’s approach belongs for finding the value of the zeta function at 2 (cf. our article: Pancalism in Mathematics. Thoth, 16, 8/1999). Finally, when an individual regularly uses and exploits this category of conjecture to exist in cognitive space, it is likely that he has reached the level of the first fundamental phase (cf. M-classification. Gift of Fire, 108, 8/1999).

The conjecture of the second category (transcendence) is even more fundamental than that of the first. This time, the process of synthesis is much deeper and requires far broader knowledge, not exclusively from a single domain. Here, the researcher does not use only a local thesis but rather a global theory. From this reflection arises a movement that literally overflows standard techniques. Thus, this movement makes it possible to conceive an idea beyond the reach of the existing world of knowledge. Hence the name conjecture of transcendence. It is to this category of conjecture that the Riemann hypothesis belongs. Finally, when an individual regularly uses and exploits this category of conjecture to exist in cognitive space, it is likely that he has reached the level of the second fundamental phase (cf. M-classification. Gift of Fire, 108, 8/1999).

Post to X