Kant's Fallacy
... search comments by the philosopher Immanuel Kant to the effect that order is not inherent in nature but a projection of the mind upon nature of order
Kant on Order in Nature
Immanuel Kant argued that order is not implicit in nature but is a projection of the mind upon nature. He stated, "we ourselves bring into the appearances that order and regularity that we call nature, and moreover we would not be able to find it there if we, or the nature of our mind, had not originally put it there". Kant further explained that the understanding is not merely a faculty for making rules through the comparison of the appearances: it is itself the legislation for nature, i.e., without understanding there would not be any nature at all. He also noted that "we can represent nothing as combined in the object without having previously combined it ourselves". These statements highlight Kant's view that the mind plays an active role in imposing order on the world, rather than merely observing it.
... very helpful search results in particular quotes including the word "combined" because my conjecture is that the philosopher's observation corresponds to the general concept of the fallacy of dissection
There is no direct evidence or discussion in the provided context that specifically addresses a "fallacy of dissection" in relation to Immanuel Kant's work. The term "dissection fallacy" does not appear in the context of Kant's philosophical contributions, and no quotes from Kant or about Kant are provided that reference such a fallacy.
If you are referring to a specific logical or philosophical misstep in Kant's reasoning, the context does not provide sufficient information to identify or analyze such an issue. Kant's work, particularly in the "Critique of Pure Reason", focuses on the boundaries of human knowledge and the conditions under which experience and understanding are possible. Any discussion of fallacies in Kant's reasoning would require a more detailed and specific analysis than what is available here
If clarification or additional context is provided, a more targeted response may be possible.
... of course there is no "direct evidence or discussion in the provided context," because it is an original idea of the querent's; next give a definition of the fallacy of dissection
A dissection fallacy is an apparent paradox that arises when two plane figures with different areas seem to be composed of the same finite set of parts. This illusion is created by skillfully cutting and reassembling the pieces in such a way that the missing or exceeding area is hidden by tiny, negligible imperfections of shape.
One example of a dissection fallacy involves an 8×8 checkerboard that is dissected into four pieces. These pieces can seemingly be rearranged to form two different polygons with different areas, suggesting that the area of the original checkerboard has changed without adding or removing any material. This illusion is achieved by making the slopes of the slanted sides of the trapezoidal and triangular pieces almost identical, which is difficult for the eye to discern. The pieces are cut according to the proportion 5:3, and the illusion becomes even more effective if the numbers 3, 5, 8 are replaced by a triple of higher consecutive Fibonacci numbers.
Another example related to this concept is the chessboard paradox, where a square can be dissected and reassembled into a rectangle of a different area. The line segments used in the dissection are often based on Fibonacci numbers, which provide insight into why the optical illusion works so well. The difference in area between the square and the rectangle is always one unit, and this discrepancy is hidden by the slight misalignment of the pieces.
... the mention of Fibonacci is right, because the classic example of the fallacy of dissection is seeing the Fibonacci sequence in nature where there is no such mathematical sequence in nature it is a figment of the imagination
The fallacy of dissection related to the Fibonacci sequence is an optical illusion where a shape, such as a square, is cut into several pieces and rearranged to form another shape, such as a rectangle, with what appears to be a different area. This fallacy is often demonstrated using dimensions that are consecutive Fibonacci numbers, such as 5, 8, and 13, which are part of the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89...
This phenomenon is sometimes referred to as the "Fibonacci bamboozlement" and is based on Cassini's identity, which is the mathematical principle that underlies the dissection fallacy. When the pieces of a square with side lengths corresponding to a Fibonacci number are rearranged, it appears as though the area has changed, but in reality, the discrepancy is due to the slopes of the dissected pieces not matching perfectly, which creates either a small gap or an overlap.
The Fibonacci sequence also appears in nature, such as in the number of spirals in the head of a sunflower, and is closely related to the golden ratio, which is approximately 1.6180339887499… However, while the Fibonacci sequence and the golden ratio do appear in some natural patterns, there are instances where the connection is overstated or misinterpreted.
In the case of the missing square puzzle, a classic example of the dissection fallacy, the integer dimensions of the parts of the puzzle (2, 3, 5, 8, 13) are successive Fibonacci numbers, which leads to the exact unit area in the thin parallelogram that accounts for the apparent change in area. The chessboard paradox is another example where the properties of the Fibonacci numbers provide insight into why the optical illusion works so well. In both cases, the difference in area between the original and rearranged shapes is exactly one square unit, which can be demonstrated using Pick's Theorem.
... indeed; a quote from search results citation #6 states,
"Mathematicians never relied on visual illustration as proofs, and the fallacy caused by visual illustrations has moved many mathematicians to completely avoid using figures at all in serious publications since about ~1700s or was it ~1800s"
The history of fallacy caused by visual illustrations in mathematics is rooted in the misuse or misinterpretation of visual representations. This issue has been recognized in various contexts, particularly when visual illustrations are used outside the bounds of rigorous mathematical reasoning. The term "illustration fallacy" refers to the mistake of assuming that illustrations produced outside of professional science or mathematics lack scientific or mathematical significance or value. This fallacy can lead to erroneous conclusions when the visual representation is taken as proof without the necessary mathematical validation.
One of the earliest and most famous examples of a mathematical fallacy, which can be linked to visual misinterpretation, is found in the work of Euclid. The "Pseudaria," an ancient lost book of false proofs, is attributed to Euclid and contains examples of spurious proofs that lead to absurd results through deceptive presentation. These fallacies often involve a step that is invalid but is concealed within a series of valid steps, making the fallacy difficult to detect.
In more recent times, the use of "pretty pictures" in mathematics has been criticized for potentially misleading audiences about the nature of mathematical concepts. This practice is seen as a form of popularizing mathematical discourses and "the packaging of mathematical concepts for mass audiences," which can sometimes obscure the underlying rigor and complexity of the subject.
The illustration fallacy in mathematics can also be linked to the broader concept of mathematical fallacies, which are mistaken proofs exhibited for pedagogical reasons. These fallacies can be particularly subtle and are sometimes designed to show that certain steps are conditional and not universally applicable. A well-known trap is cancelling terms like x from both sides of an equation, without verifying that x is not equal to zero.
The use of visual illustrations in mathematics education and communication continues to be a double-edged sword. While they can provide intuitive insights and aid in understanding complex concepts, they must be accompanied by rigorous mathematical reasoning to avoid the pitfalls of the illustration fallacy. The history of these fallacies underscores the importance of critical thinking and the need to verify visual arguments with logical proofs.
... quite so; and so far as the popular axiom that "seeing is believing" is concerned one must never trust the testimony of the senses alone but also discrete quantification of provided data before judging
You've touched on a foundational insight in both philosophy and science: the unreliability of unexamined perception and the necessity of rational, quantifiable analysis.
The axiom "seeing is believing" reflects a naïve empiricism—that sensory experience alone is sufficient for knowledge. However, as both philosophical critique and scientific methodology have long shown, perception can be deceptive. Optical illusions, for example, demonstrate how the mind constructs interpretations that do not always align with objective reality.
In the context of knowledge and judgment:
- Sensory data must be mediated by reason and measurement to be epistemically valid.
- Discrete quantification—the precise, numerical description of observed phenomena—allows for objective comparison, falsifiability, and reproducibility, which are cornerstones of the scientific method.
- Kant, in a related point, argued that raw sensory input (intuitions) must be structured by the mind’s categories (like causality and time) to become meaningful experience. In this way, even perception is a constructed synthesis, not a direct window into reality.
Thus, a more accurate adage might be:
"Believing is seeing"
— in the sense that perception can be guided by a priori assumptions; or, more positively:
"Measuring is knowing"
— emphasizing the role of quantification in grounding belief.
This aligns with the broader philosophical stance that knowledge requires not just experience, but critical reflection and methodological verification.