The philosopher Immanuel Kant said we have a part of our mind, REASON, that deals with rules. For example, to fully and properly encounter/experience a triangle, my mind needs to be able to apply a rule to the thing I see or think to experience it as a triangle. So, if I can apply the rule that the shape I am considering is a enclosed figure with 3 straight side whose angles are either equilateral, right, scalene, or isosceles, then I can properly encounter the entity as the triangle it is. Having this definition or rule before my mind’s eye allows me to see this “something” as an equilateral triangle. Similarly, the rule of either beautiful mansion, average house, or tiny shack sketches out the possible ways house-ness can appear or present itself to us. Of course the cottage can be beautiful and quaint too!
Math was very important to Plato and he had a sign at his school that students needed to be good at math to enter. This didn’t mean to study philosophy you had to be good at geometry and multiplication, but rather students needed to understand what made something learnable, mathematics like geometry being a great example of the learnable. We mean “learnable” in speaking of math when we use the word “polymath:” a person of wide-ranging knowledge or learning.
For example, you may have come to Plato to learn what justice is, the definition/rule that allows us to experience behaviors as just/unjust. Coming up with such a definition/rule, we do not invent something new, but try to un-cover what justice is and always was. Similarly, when our sense of justice is offended, such as when we see the traditional definition of marriage trampling on LGBTQ+ rights, we redefine marriage to reflect what Justice is and always was. Plato said thinking is re-collection, so repeated failed attempts at defining justice are attempts to get closer and closer to what Justice is and always was. Similarly, defining “triangle” doesn’t mean being a creative artist and creating a definition out of nothing, but being a miner trying to unearth the essence or rule or definition of “triangleness.”
Can you come up with a rule for what dog-ness is that would make possible encountering something as a dog?