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To elaborate, I would like to take the definition of square as example, the square is shape with four equal sides and either two sides form a 90 degrees angle, while we can not directly see it. What we see is square with four 10 inches sides or other sizes. That means it is impossible to imagine what a general square look like. I think this conclusion is against common sense, however, I cannot figure out where I am wrong

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  • How is a square defined?
    – Hudjefa
    Mar 9, 2023 at 7:53
  • it is not 90 degrees angel, it is pi/2 Mar 9, 2023 at 8:21
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    This issue was encountered by Locke in his idea of a general triangle, "neither oblique nor rectangle, neither equilateral, equicrural, nor scalenon, but all and none of these at once", for which he was criticized by Berkeley. A solution later offered by Kant was that the idea of a general triangle is not an image at all, hence need not be imagined. It is rather a "schema", a sort of mental algorithm for generating and/or recognizing triangular images, which can produce "all of these", but "none at once". Abstraction consists in developing such an algorithm from observed instances.
    – Conifold
    Mar 9, 2023 at 8:35
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    In evolution we learned to evade all lions, not just a particular one. We 'generalize' by noting a "good enough" match. It is probably the first task we built computer "neural networks" to perform. Not that complex. I don't see the obstacle to imagining a general square, it simply scales in size, something our vision system is very used to. Give your billion years of life experience more credit :-)
    – Scott Rowe
    Mar 9, 2023 at 11:18
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    Greeks gave us the idea of geometric quantities not tied to specific numbers, "abstracting" from numbers tied to physical measurements to line segments and lengths expressed in ratios for example. Areas, volumes, angles and lengths as ratios can all be defined without specific numbers. Philosophers loved this idea and kept made sure it spread. Gouvêa, William P. Berlinghoff, Fernando Q. Math through the Ages. MBS Direct, American Mathematical Society, 2020. It means we can think about these things without sense data or empirical measurements. This is just a brief overview of the genesis
    – J Kusin
    Mar 9, 2023 at 20:48

2 Answers 2

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What we see is square with four 10 inches sides or other sizes.

Not quite. You see the square, and the four sides of the square, but you don't actually see that they are 10 inches long.

One 1km-sided square seen from a distance of 10km looks the same as a 10cm-sided square seen from a distance of one meter.

Similarly, we can imagine a particular shape, but we cannot imagine a particular distance. However, we can imagine two similar shapes one larger the other smaller.

So, it seems that our perception and imagination are already relativistic. We do not perceive distances in the scientific sense, we perceive something like relative position and size.

And this seems to make it easy to imagine a square.

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The mind's job is to solve cognitive problems. "Where shall I go for lunch? Is this thermos large enough to hold all my tea? Can I form a square with the same area as this circle? Should I be concerned about coyotes in this forest?"

To solve these problems, the mind has a lot of different tricks. Forming a mental picture is a powerful trick, which can help decide where to go for lunch or whether the thermos can hold the tea, but it's not the only trick.

We might use verbal, logical, or numerical reasoning: "the cafe is more expensive than the pizza place. I can't square the circle because pi is a transcendental number."

We might rely on mere habit.

We might try to think about a general scenario by forming mental pictures of lots of different examples, observing common trends through the different examples, and then using verbal/logical reasoning to try to prove the trends always hold. This trick is useful especially in mathematics, when thinking about abstract objects such as the squares you mention.

We can say that we understand something, if we are able to easily solve "enough" cognitive problems involving that thing. For example, a person understands algebra if they can easily work out algebra problems on paper. A person understands (some aspects of) dogs if they are skilled at interacting with and caring for dogs. A veterinarian understands more aspects of dogs if they are also skilled at diagnosing and treating health problems in dogs.

So, to understand what the general concept of a square is, means that we are able to easily solve all sorts of cognitive problems involving squares, such as mathematically proving things about squares. Picturing examples of squares is often useful to this understanding, but the picture alone isn't enough; we must unify such pictures with logical, verbal thinking, including knowledge of some mathematical theorems relevant to squares, before we can say we understand the general square.

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  • +1 For suggesting semantic grounding of abstractions revolves around a collection of operations regarding various elements of the extension and intension of a term.
    – J D
    Mar 9, 2023 at 21:49
  • One of my favorite quotes: "Thought presents alternatives. It was not meant to solve problems or decide things." - Zulaikha Mahmud. Relying only on the mind is perilous.
    – Scott Rowe
    Mar 10, 2023 at 16:16

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