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My view is that mathematical entities are not physical or visual objects, so they do not look like anything. Is this view correct? I would love to know whether there are philosophers who claim mathematical entities, like say the number 1 or the empty set, actually look like something.

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    What about geometry...?
    – Joseph Weissman
    Dec 19 '20 at 16:53
  • What d'you mean 'look like' something, this seems like very imprecise language? E.g. Does representing something in the physical world count? Does a triangle 'look like' something physical? Etc. Dec 19 '20 at 17:27
  • Some mathematical entities have visual representations, geometric shapes, graphs, knots, commutative diagrams, even sets and numbers. Aristotle believed that any thought requires images, including mathemtical thought. See SEP, Visual Thinking in Mathematics for modern views.
    – Conifold
    Dec 19 '20 at 21:57
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The parts of mathematics that deal with geometry contain mathematical objects that can be visually represented. In the realm of algebra, equations can be mapped into visual form as graphs which are handy in understanding their characteristics; but I do not think those representations are quite the same thing as what you are asking about.

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Great question! And the short answer is yes, they do! Or they should, at any rate.

There are two major philosophical problems that need to be addressed before we arrive at the conclusion above:

  1. The so-called "mathematical realism" -- do mathematical entities exist independently of the human mind?

    Which, itself is a special case of the "hard problem" of consciousness because a math entity is a form of qualia (that's why 2 + 2 = 5 is wrong (false) by definition, just like red = blue is wrong by definition).

    Math entities being qualia means they are subjective -- you have "9" and I have mine, just like you and me seeing red as our very own, subjective experiences. And only because we are looking at the same thing (e.g. the same red apple1) we end up with the same ideas of "red".

    Or, even in more general terms, all that -- math, qualia, our thoughts, our consciousness -- all that is information. And information exits as physical records of it -- like, in your brain, for example (and no, "physical" ≠ "paper", tho some people here, apparently, believe otherwise).

  2. How do we think? What is the nature of the thought process? Well, thinking is a form of daydreaming, unless you are p-Zombie.

    (But then you aren't thinking in the first place -- rather, you are "having thoughts" courtesy of the neural net AI in your head, acting as a chat-bot. We all have it, the AI -- think of it as an autopilot. The problem with p-Zombies is that their autopilots are on their own since forever. Since their used-to-be-conscious Self, still just a child, was too tired trying to make any sense of this crazy world, so they decided to take a nap. They've been napping ever since.)

    But when you do think, you daydream your "9" (it's a light shade of green in my case), and the empty set, and the rest of it. It means they do look like something, but that appearance can vary a lot between individuals.

1 BTW, why it's always an apple?! Eve, Newton, Jobs... Now that's the question!

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