Concentration as a Power of Perceiving

The river Pregel divides the town of Konigsberg into four separate land masses, A, B, C, and D. Seven bridges connect the various parts of town, and some of the town's curious citizens wondered if it were possible to take a journey across all seven bridges without having to cross any bridge more than once. All who tried ended up in failure, including the Swiss mathematician, Leonhard Euler (1707-1783)(pronounced "oiler"), a notable genius of the eighteenth-century.

The lessons of those who are engaged in the mathematics of,  must nurture the powers of intuition to advance the road of uncharted waters so as to be inspired to see nature and what underlies it as if guided by some unseen hand.

How would one tell another of "such a feeling" as they progress on their own journey while having all the tools of their trade in mathematics with them?

Euler was prolific, both in offspring and in intellectual output. He fathered thirteen children, albeit with two wives, and wrote more then eight hundred books and papers in all areas of mathematics. This is all the more astonishing-the part about the papers, that is, not the children-since for a large part of his life he was blind. His power of concentration must have been nothing less then astounding, keeping in mind that he did much of his work without eyesight while screaming kids were scampering around. Late in life he claimed that he had done some of the best work with a baby in his arms and other children playing at his feet.Para 1, Page 54, Poincare's Prize by George G. Szpiro

Outside of themself, one might look to find conducive places sounds, inspirations that would help them on their journey. That journey is usually alone, but if you meet another that has an equal understanding and can help progress you beyond the points on which you are stuck, why would you not collaborate to move forward? To help others move forward?

 The Whole World is a Stage

Euler product formula


Now you must know what sets my mind to think in such abstract spaces. "Probability of seeing a stage in a concert."



A diagram of the Königsberg bridges

Topological ideas are present in almost all areas of today's mathematics. The subject of topology itself consists of several different branches, such as point set topology, algebraic topology and differential topology, which have relatively little in common. We shall trace the rise of topological concepts in a number of different situations.