To me, the world of perfect forms is primary (as was Plato’s own belief) — its existence being almost a logical necessity — and both the other two worlds are its shadows.
Sir Roger Penrose, born on August 8, 1931, in Colchester, Essex, England, is a luminary in the realm of mathematical physics. His journey began with a Ph.D. in algebraic geometry from the University of Cambridge in 1957, and his career has spanned numerous prestigious posts at universities in both England and the United States. His work in the 1960s on the fundamental features of black holes, celestial bodies of such immense gravity that nothing, not even light, can escape, earned him the 2020 Nobel Prize for Physics.
Penrose’s work on black holes, in collaboration with Stephen Hawking, led to the ground-breaking discovery that all matter within a black hole collapses to a singularity, a point in space where mass is compressed to infinite density and zero volume. This revelation illuminated our understanding of these enigmatic cosmic entities.
His work did not stop at the theoretical; he also developed a method of mapping the regions of space-time surrounding a black hole, known as a Penrose diagram. This tool allows us to visualize the effects of gravitation upon an entity approaching a black hole, providing a window into the heart of these celestial mysteries.
Within Penrose’s chapter, “The Godelian Case” (from “The Road to Reality”) the profound implications of Kurt Gödel’s incompleteness theorems are examined in relation to the connection between mathematics and geometry. Specifically, Penrose’s attention centers on the model depicted in Figure 2.1, which portrays a cubic array of spheres. Through this visual representation, Penrose explores the intricate relationship between geometry and mathematical understanding.
By introducing the model of a cubic array of spheres, Penrose highlights the fundamental role of spatial arrangements in mathematical cognition. This geometrical structure serves as a metaphorical embodiment of mathematical concepts, illustrating how spatial configurations can stimulate cognitive processes and facilitate intuitive comprehension of mathematical truths. The intricate interplay between the arrangement of spheres within the model and the underlying principles of mathematics encourages contemplation on the deep-rooted connections between geometry, spatial reasoning, and abstract mathematical thought.
Penrose’s utilization of the cubic array of spheres underscores his broader philosophical framework, which challenges reductionist accounts of human cognition that rely solely on formal systems or computational models. Through this geometrical representation, he advocates for a more holistic understanding of mathematical insight, one that recognizes the essential role of geometric intuition in shaping human understanding.
By looking at the intricate connection between mathematics and geometry, Penrose prompts a re-evaluation of the mechanistic view of cognition, emphasizing the need to incorporate spatial reasoning and intuitive geometrical understanding into comprehensive models of human thought.
(E) Find a sum of successive hexagonal numbers, starting from 1 , that is not a cube.
I am going to try to convince you that this computation will indeed continue for ever without stopping. First of all, a cube is called a cube because it is a number that can be represented as a cubic array of points as depicted in Fig. 2. 1 . I want you to try to think of such an array as built up successively, starting at one corner and then adding a succession of three-faced arrangements each consisting of a back wall, side wall, and ceiling, as depicted in Fig. 2.2.
Now view this three-faced arrangement from a long way out, along the direction of the corner common to all three faces. What do we see? A hexagon as in Fig. 2.3. The marks that constitute these hexagons, successively increasing in size, when taken together, correspond to the marks that constitute the entire cube.
This, then, establishes the fact that adding together successive hexagonal numbers, starting with 1 , will always give a cube. Accordingly, we have indeed ascertained that (E) will never stop.
Penrose’s work is characterized by a profound appreciation for geometry. His father, a biologist with a passion for mathematics, introduced him to the beauty of geometric shapes and patterns at a young age. This early exposure to geometry shaped Penrose’s unique approach to scientific problems, leading him to develop new mathematical notations and diagrams that have become indispensable tools in the field. His creation of the Penrose tiling, a method of covering a plane with a set of shapes without using a repeating pattern, is a testament to his innovative thinking and his deep understanding of geometric principles.
His fascination with geometry extended beyond the realm of mathematics and into the world of art. He was deeply influenced by the work of Dutch artist M.C. Escher, whose intricate drawings of impossible structures and infinite patterns captivated Penrose’s imagination. This encounter with Escher’s art led Penrose to explore the interplay between geometry and art, culminating in his own contributions to the field of mathematical art. His work in this area, like his scientific research, is characterized by a deep appreciation for the beauty and complexity of geometric forms.
In geometric cognition, Penrose’s work has the potential to make significant contributions. His unique perspective on the role of geometry in understanding the physical world, the mind, and even art, offers a fresh approach to this emerging field. His belief in the power of geometric thinking, as evidenced by his own ground-breaking work, suggests that a geometric approach to cognition could yield valuable insights into the nature of thought and consciousness.
Objective mathematical notions must be thought of as timeless entities and are not to be regarded as being conjured into existence at the moment that they are first humanly perceived.
I argue that the phenomenon of consciousness cannot be accommodated within the framework of present-day physical theory.
His Orch OR theory posits that consciousness arises from quantum computations within the brain’s neurons. This bold hypothesis, bridging the gap between the physical and the mental, has sparked intense debate and research in the scientific community.
Penrose’s work on twistor theory, a geometric framework that seeks to unify quantum mechanics and general relativity, is a testament to his belief in the primacy of geometric structures. This theory, which represents particles and fields in a way that emphasizes their geometric and topological properties, can be seen as a metaphor for his views on cognition. Just as twistor theory seeks to represent complex physical phenomena in terms of simpler geometric structures, Penrose suggests that human cognition may also be understood in terms of fundamental geometric and topological structures.
This perspective has significant implications for the field of cognitive geometry, which studies how humans and other animals understand and navigate the geometric properties of their environment. If Penrose’s ideas are correct, our ability to understand and manipulate geometric structures may be a fundamental aspect of consciousness, rooted in the quantum geometry of the brain itself.
The final conclusion of all this is rather alarming. For it suggests that we must seek a non-computable physical theory that reaches beyond every computable level of oracle machines (and perhaps beyond).
— Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness
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