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.
Image
Abstraction vs. Reality
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.
https://www.theguardian.com/artanddesign/2015/jun/20/the-impossible-world-of-mc-escher
Not quite so impossible if you know what you're doing. It's a matter of point of view: https://www.instructables.com/Penrose-triangle-modified/
Yes but this is a projection on the plane Not the triangle itself Btw nobody says it's impossible There are numerous reports where authors demonstrate how the triangular shape curved on the plane looks like the PT The author himself was an artist
Penrose Triangle | Shadows and reflection destroy the magic | fdecomite | Flickr
https://en.wikipedia.org/wiki/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.
Soumendra Nath Thakur
You wrote: "Your comment suggests an unnecessary willingness to accept or entertain an absurdity as a real possibility."
For me, visual absurdities are real only as a misconstrual of the content of a perception. In other words, the illusion or misconstrual is real qua psychological phenomenon, but what it psychologically purports is an absurdity and not a real state of affairs. However, what underlies the illusion is something real and what that is might be shown by changing one's point of view, as demonstrated in the video. The video demonstrates that what prima facie looks to be impossible is, seen in a different light, just another humdrum possibility. No fancy mathematical construction required for that demonstration.
Karl Pfeifer
You stated, "For me, visual absurdities are real only as a misconstrual of the content of a perception."
However, your response suggests an implicit willingness to entertain absurdity rather than objectively analyse the phenomenon through careful observation and scientific reasoning.
The distinction here is clear: the three-dimensional structure of an object accurately represents its original form, while an illusory image presents a deceptive perception from a particular viewpoint. This illusion arises from a two-dimensional projection of a three-dimensional structure, creating the appearance of an impossible geometry. However, upon closer examination and a change in perspective, the illusion is easily deconstructed, revealing the objective truth.
A scientifically minded individual should instinctively question an impossible geometry rather than passively accept the illusion as a genuine possibility. Treating an illusion as if it holds some deeper reality, rather than acknowledging it as a perceptual misinterpretation, leans more toward belief than scientific inquiry. The pursuit of truth requires a willingness to reassess perspectives critically rather than entertain impossibilities as though they hold fundamental legitimacy.
Best regards
Soumendra
I don't want to interrupt this discussion but the object under question can be analyzed by both analytic and synthetic geometry It's an interesting topic to discuss from the perspective of projective geometry so would be Picasso's works for example Just a few observation points The object under question is a three dimensional shape projected onto the plane under the following transformation:\left|\begin{matrix}
1& & \\
& 1& \\
& & 0
\end{matrix} \right|
This is an orthogonal projection (there are others) but with reference to the object in question the answer is given here:
https://en.wikipedia.org/wiki/3D_projection:
Limitations of parallel projection See also: Impossible object
Soumendra Nath Thakur
Confronted with a Penrose-triangle image, there are two possibilities for me. It is either (1) merely a drawing or else (2) a photo or motion picture image. If (1) there's no mystery; it's essentially 2-D, even if it's good enough to fool some folks into misconstruing it as representing a 3-D possibility. The drawing is what it is. To regard it as an illusion of anything is a departure into psychology. If (2), that is if the Penrose-triangle image is a photo or motion picture frame, one must also acknowledge that it resembles a state of affairs in the real world that is the cause of the image. This is not passive acceptance of the illusion (i.e. of what is illuded). Someone of sound mind who watches the video1 from the beginning up to and including the frame in which the full Penrose-triangle image finally appears will not be swayed into accepting it as a depiction of a genuine impossibility. The video is a demonstration of a possibility just because it depicts an actuality.
You wrote, accusing me of: "Treating an illusion as if it holds some deeper reality, rather than [?] acknowledging it as a perceptual misinterpretation...." Yet you also accurately quoted me as saying "visual absurdities are real only as a misconstrual of the content of a perception." I don't see a relevant difference between misinterpretations and misconstruals in the present context. Both are real psychological phenomena that don't represent reality.
_________________
1 The video: https://www.instructables.com/Penrose-triangle-modified/
Perspective on the Penrose Triangle:
Dear Dr. Karl Pfeifer
The Penrose Triangle is essentially a two-dimensional image designed to create the illusion of a three-dimensional structure, often engaging the philosophical mind.
From a mathematical perspective, understanding its underlying concept takes no more than a moment. When I first encountered the image, I instinctively visualized it within a three-dimensional coordinate system and immediately recognized its inherent inconsistency—being fundamentally a 2D construct.
Given this, extensive speculation seems unnecessary.
Best regards,
Soumendra Nath Thakur
Soumendra Nath Thakur
I share your initial instinct, which of course is more or less endemic to us 3-D critters in our phenomenological 3-D lifeworld. Nevertheless, I like to envisage uncommon possibilities. It occurs to me that there might be other fun mathematical perspectives besides the kneejerk obvious one. What would you think of a Penrose Triangle as a sort of Möbius strip with three kinks and a few extra twists as a consistent possibility? (Is topological equivalence the right term for something like that?) And I wonder whether consistent interpretations might be found in the weird n-dimensional geometries of string theory.
Any mathematicians care to weigh in?
Dear Dr. Karl Pfeifer,
I appreciate your enthusiasm for exploring mathematical perspectives. However, my earlier assessment already involved a mathematical approach, as I used a three-dimensional coordinate system to analyze the Penrose Triangle and determine its fundamental inconsistency. This was not a mere dismissal of the illusion but a rational conclusion based on mathematical reasoning.
Mathematics should serve to clarify objective truths rather than legitimize illusions through unnecessary abstraction. Once the nature of the Penrose Triangle has been decisively understood, further speculation—such as invoking topological equivalence or higher-dimensional geometries—becomes redundant. While such abstract mathematical frameworks have their place, they are not required for resolving a straightforward question about visual illusion.
Philosophy benefits greatly from mathematics, as it provides clarity and precision. However, introducing complex mathematical detours when they are not needed does not necessarily strengthen philosophical inquiry. Rather, it risks obfuscating a settled issue.
Thanks and regards,
Soumendra Nath Thakur
For your entertainment 🤩
https://archimedes-lab.org/2021/07/14/possible-impossible-cube/
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