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Continuing the discussion Categorizing with metaphor, analogy, generalization, and abstraction my next question is how two concepts metaphor/analogy equivalent to symmetry(change without change) .If yes how? Give me some examples

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The concept of symmetry as "change without change" is already a "poetic" way to describe the fundamental concept of symmetry, which encompasses various forms like Reflectional Symmetry, Rotational Symmetry, Translational Symmetry, Scale Symmetry, and more. In other words, symmetry involves the constancy of fundamental qualities within an object or pattern despite undergoing different transformations. Therefore, if two analogous concepts fulfill the following conditions: 1) the definition of a transformation can be established, and 2) the results of these transformations are identical to each other, then symmetry can be defined. Merely applying "change without change" for discussing these matters lacks the necessary rigor.

Your insight that analogy aligns with generalization might be useful here. When there are two analogous concepts, A and B, their relationship represents a generalization—that is, two abstracted outcomes established over similar specific instances. Consequently, we can potentially "construct" a ratio function for the extracted predicate here and attempt to uncover another symmetry. However, I am doubtful whether this can be applied to any pair of arbitrary analogous concepts with a fixed transformation. Therefore, I perceive the responder's answer as negative.

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The terms metaphor and simile come from literary studies. A simile is a figure of speech in which the poet says "X is like Y" (e.g. "My love is like a red red rose/... (https://www.poetryfoundation.org/poems/43812/a-red-red-rose)) A metaphor is a figure of speech in which the poet says "X is Y" (e.g., April is the cruellest month,... (https://www.poetryfoundation.org/poems/47311/the-waste-land)). If you ignore the difference between "is like " and "is", both metaphor and simile are analogies,

In a metaphor/simile, the poet asserts a similarity between two dissimilar things. Finding a deep similarity in apparently unrelated/dissimilar things is what Coleridge called 'esemplastic imagination', what Arther Koestler calls 'bisociation' as what is common to humour, science, and poetry (https://www.themarginalian.org/2013/05/20/arthur-koestler-creativity-bisociation/) (https://www.themarginalian.org/2013/05/20/arthur-koestler-creativity-bisociation/)

When we find what is shared between two things, we are finding something deeper, something more abstract than what is at the surface. When we say that right angled triangles and equilateral triangles are both triangles, we are identifying the more abstract property shared between these two, in spite of the differences. When we say that triangles, quadrilaterals, triangles, pentagons and hexagons are all triangles, we are finding a still deeper/abstract similarity.

When we see two photographs, we compare them by talking about the similarities and differences. But if we are told that they are photographs of the same person, we describe it in terms of what remains unchanged and what has changed. What happens here is the statement of similarity and difference along the dimension of time (change).

As concepts in mathematics, symmetry is what is unchanged in a transformation, and a transformation is what is changed against the backdrop of what remains constant. In biology, they use the term homology to refer to symmetry (What biologists call bilateral symmetry is just one kind of symmetry.

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