February 15, 2025
Consider the question: What if this philosophical construct (presented within an image, attached below) were plotted on a three-dimensional mathematical graph? Only then would the scientific facts behind it be fully justified.
Take, for example, the Penrose Triangle—an "impossible" figure first conceptualized 62by Swedish artist Oscar Reutersvärd in 1934 and later rediscovered in the 1950s by physicist Roger Penrose. Described as "impossibility in its purest form," this figure was popularized by Penrose and further explored in the works of M.C. Escher. It appears to be a solid structure composed of three straight sections of square beams, seamlessly joined at right angles.
However, its paradoxical nature cannot be conclusively analyzed through mere physical observation and philosophical interpretation. Instead, by mathematically plotting the enclosed three-dimensional image onto a precise three-dimensional graph, its inherent absurdity becomes evident. The deceptive illusion, which seems structurally feasible in a two-dimensional representation, is mathematically exposed as an impossible configuration in three-dimensional space.
Addressing the statement: "Abstract mathematics is based on logical principles rather than empirical validity. It does not rely solely on physical evidence."
When we add one apple to another, we perceive two apples. However, the concept of "two" itself is an abstract mathematical construct rather than a directly observable physical entity. The sum exists as a logical principle within mathematics, not as a tangible proof in itself.
All real numbers—1, 2, 3, and beyond—are fundamentally conceptual, created within the framework of mathematical reasoning rather than derived from physical evidence. While mathematical concepts often align with physical reality, their foundation is purely abstract, shaped by human perception and logical consistency rather than empirical observation.
Mathematics is not confined to the physical universe; its abstract principles hold universally, independent of space, time, or physical existence. It is the fundamental language of the cosmos—objective, unique, and uninfluenced by human divisions such as culture, religion, or race.
Physicists seek mathematical formulations to explain the origins, structure, and dynamics of the universe. In doing so, they attempt to decipher the underlying mathematical order that governs both the observable universe and any potential realities beyond it. Understanding the universe is, in essence, understanding its mathematical nature.
For philosophy to be scientifically meaningful, it must be grounded in abstract mathematical logic. Without mathematical rigor, philosophical reasoning remains speculative and cannot be accepted as a scientific discipline.
Conclusion
This distinction underscores a fundamental contrast between philosophy and science: while philosophy interprets illusions conceptually, science—through mathematical rigor—reveals their underlying reality.
Attachment: the image as stated in the above mentioned text.
February 21, 2025
Revisiting the Penrose Triangle :
Karl Pfeifer, Your statement, "Not quite so impossible if you know what you're doing. It's a matter of point of view," appears to distort the discussion.
The three-dimensional construction of the object clearly represents its original structure, as shown in the image Penrose triangle.jpg. Meanwhile, the other image, Penrose Two-dimensional.jpg, illustrates an illusionary perception from a specific viewpoint where the object appears to have all its ends joined. In reality, only two ends are connected, while the third remains separate.
This illusion results from a two-dimensional projection of a three-dimensional structure, making the impossible geometry seem plausible from a particular perspective. Additionally, the image enhances the illusion more prominently than what one would perceive when viewing the physical object up close.
Your comment suggests an unnecessary willingness to accept or entertain an absurdity as a real possibility. This tendency is often observed in individuals who prioritize appearing overly insightful rather than applying nuanced thinking to understand objective reality
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