Triangles or other geometric concepts do not exist in the real world, their approximations do. Needless to say, I am not a follower of Plato.
I think it is clear that I did
not say geometrical concepts exist in the world.
However, the unintentional misstatment is problematic for reasons other than the fact that it is a misstatement. I will flesh out this matter because it bear an interesting relevancy to the God is Square Circle argument, which should be evident as my discussion proceeds.
First, geometrical concepts do in fact exist in the real world, if (a) they exist in the human mind and (b) the human mind is said to exist in the real world.
I don’t think any one will want to deny “a”.
If one chooses to avoid agreeing with the statement that says “geometrical concepts exist in the real world” by denying “b” then he must demonstrate that the human mind exists elsewhere than in the real world. This could be very difficult to do since the real world that exists externally to man is the same world in which man exists.
Second, in case anyone is experienceing
angst because the conclusion above that says geometrical concepts exists in the real world, seems inescapable, then I can ease your mind by assisting you with logically avoiding this conclusion.
When we reflect on geometrical concepts we grasp their perfect nature. We can conceive of the perfect circle, the perfect triangle, etc. On the other hand, no geometrical object existing in the external world can be said to be perfect. So, what is the origin of our ideas of perfect geometric figures?
One answer is the Platonic position that ideas are innate. Another solution is the Aristotelian position that says all knowledge originates with the senses: there is nothing in the mind that was not first in the senses.
I’ll go with Aristotle on this one. If all knowledge originates in the senses, and perfect circles do not exist in the external world for our senses to perceive, then how do we account for the intellect’s idea of a perfect circle? The mind abstracts the universal attributes from the particular, imperfect circles in the world that are perceived by the senses. Particular circles perceived by the sense vary in circumference, color, and so on. But there is something common to all circles that makes them circles and accounts for the reason why they are referred as circles despite their accidental variations.
The mind’s concept of circle is not of this or that particular circle but of circle itself, or we can say the concept is of circleness. It is a universal and applies to all particular circles as pertains to what is required of a circle to be a circle: its essential attributes.
In comparison, the senses perceive directly the particular circles in their particularity or individuating qualities. The sense do not know the universal. Nothing that exists in the external world is a universal. There is no universal circle existing in the physical world.
Particular and Universal
Everything that exists in the physical world is a particular thing, this atom, this tree, this dog, this triangle. It would be an absurdity to think of a physical object as a universal. Universal is radically contrary to the physical, which everyone agees can only be particular in nature.
However, as we have seen, the concepts in the mind must be universals. If they were not, it would be impossible to have a discussion about perfect geometric figures, or any anthing else whatsoever.
From the facts that everything physical must be a particular thing, and that geometrical concepts are universals, it necessarily follows that geometrical concepts are non-physical.
Accordingly, we can say that geometrical concepts do not exist in the real world in the same way that a tree exists in the world.
So, there it is. I fulfilled my offer of showing the way out of having to maintain that perfect geometric figures in the real world.
The word “in” is a spatial concept based on physical reality. When we say the the mind or its concepts are “in” the world or “in” this or that man, we are not using the word in the sense as when we say that a tree exists in the real world, because the mind, of which concepts are modifications, are not physical or spatial, they are ‘above’ the physical. They are metaphysical reality. What is metaphysical can be said to be in the real world, in the sense that metaphysical nature really exists. In fact it has more reality than does physical reality.