Nicholas of Cusa and the Idea of True Infinity

I recently attended a presentation on Nicolas of Cusa that addressed some of the examples he used to illustrate his  groundbreaking conception of true or absolute infinity. Upon reflection, I had a few thoughts that connect these examples with major strides forward in science and art. I think it is especially important to show that Cusa is the source of the birth of the scientific method in the Florentine Renaissance. 

The practice of generating "thought experiments" was adopted by Cusa in his Platonic dialogues. It is not arbitrary and no small moment that Einstein's seminal gedankenexperiment that developed the equally groundbreaking conception of relativity partakes of this same essential practice. 

In one example, he shows that a straight line and a circle that expands progressively to infinity are in fact indistinguishable. This great invention of Nicolas is the basis for projective geometry that was adopted and pioneered by the great artists of the Renaissance as the perspective and camera oscura with parallel lines meeting at the infinite "point." 

Albrecht Durer Woodcut 1530

The question of whether there was a substantial difference between a mere mathematical infinity and a true infinity had been the subject of treatises on philosophy especially spurred on since the time that the Islamic Renaissance of the Abbasid Caliphate reached the European Christians. Ibn Sina or Avicenna in the Eleventh Century developed a Socratic proof of God that he called the Necessary Existent. This was the existence of a true infinity.

The development that made possible modern constructive Non-Euclidean geometry likewise arises from Nicolas. Leibniz, who time and again used that principle in his writings and letters, once remarked that the Newtonian notion that an object set into motion without any further forces acting upon it would continue in a straight line was completely an arbitrary assumption. Instead it could be as easily argued that in such an imaginary case the object could continue on a curved trajectory. This is precisely the poetic truthful imagination or thought experiment of a true infinite. 

Likewise, the great work of Georg Cantor on the principle of the Transfinite is admittedly indebted to Cusa. Indeed the different gauges of mathematical transfinite numbers, for example the rational versus the transcendentals, as opposed to Absolute Infinity are acknowledged by Cantor.

Finally, consider that Riemann's universal treatment of the foundations of geometry in his habilitation thesis makes no assumptions of distinguishing the curvature of surfaces that may be of positive, negative or in the Euclidean case zero curvature. Riemann warned in that presentation that the actual measurement of this curvature in the physical sciences must be left to experimentation.  It is this germ of Nicolas of Cusa that allowed for Einstein's theory of general relativity of the curvature spacetime upon which are founded so many of the marvels of 20th century scientific progress.