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A short essay on God and Four Sided Triangles

One of the problems that exercised theologians during the middle ages was the question of whether God could envision or, indeed, make a four sided triangle or a three sided square or some such monstrosity. At first sight the problem seems trivial or foolish, but it really is an instance of a more fundamental problem: Is God bound and limited by the laws of reason and logic or can She overcome them or alter them as She chooses. On one hand God is omnipotent; She may do anything She pleases, logic and reason notwithstanding. On the other hand it seems inconceivable that anyone, even God Herself, can maintain that 1=2.

At first sight it would seem that there is no theological problem here. It is a well founded principle of theology that revelation is superior to reason; that God may work miracles if She so chooses; that the universe is an artifact of Hers that She may alter without regard as to its internal logic or laws. The trouble is that logic and reason, truth and beauty, are not in the universe per se. That is, there is not thing that is truth, no object that is reason. They are absolutes, ideals, outside the universe, affecting and shaping the character of the universe but unaffected by it. In fact, their existential character is much like that of God Herself, and it widely held that these ideals are aspects of God. That is the crux of the problem; for if God is Truth and Truth is God, then how could God conceive of that which is not truth. On the other hand, it is held that God is omnipotent and therefore may do as She chooses, including thinking untruths. So you see, the four sided triangle is an aspect of a fundamental theological problem.

Now it seems to me that the answer must be in the affirmative; that God can make four sided triangles if She so chooses. My answer is based on the observation that it is in fact possible to conceive of four sided triangles and defend their existence. I suspect that She has chosen not to do so, but I argue that it is within Her powers, Let us consider for a moment a conversation between those redoubtable and hoary combatents, Sage and Simp:

Sage: Greetings Simplicitus. See thou the line figure I have inscribed in the sand. I have been musing on it for an hour or more.
Simp: I see it, oh Sage. But to tell the truth, I know not what to make of it, for it seemeth to me to a shaky and irregular circle. What is it supposed to be?
Sage: Alas, my infirmities grow, and my hand is palsied with age, so the drawing it somewhat irregular. It is a representation of a geometrical figure. As you know, matter and man are corruptible whereas the content of geometry is incorruptible. So it is that this figure fails of good form whereas that which it represents is clear and true.
Simp: So I understand, oh Sage. But I remain puzzled; what form is it that your line represents?
Sage: Why, it is a four sided triangle.
Simp: I am not sure I heard you rightly, oh Sage. You speak of a four sided triangle whereas it is well known that triangles possess three and only three sides. I have hear it argued that God may conceive of such a thing even though it is not possible; but such a thing is surely beyond man.
Sage: So have I understood. You may appreciate my amazement at realizing that such a construction was possible and that its conception was with the powers of a man's mind.
Simp: But sire, I find this incomprehensible. I cannot conceive of such a thing. Nor, if it were possible, do I see it implemented in this line, which I see only as a rather shapeless form.
Sage: Well then, let me make things clear. Let me poke three holes in the sand to represent the three vertices. You see.
Simp: Why yes, I see. There are now three points in the sand that are the vertices of three angles.
Sage: Well then. Joining each pair of vertices is a line that we shall call the side of a triangle. Now we count these sides and we see that there are four of them.
Simp: But sire, surely you jest. They are readily counted and there are but three of them. Let me make it clear. You agree that I am holding up three fingers.
Sage: With those reservations well known to philosophers everywhere concerning sensory perception, yes, I agree.
Simp: Well then, sire, I place one of each of these fingers one to a line, thusly. Now would you not say that for each finger there is a line and vice versa?
Sage: Why, yes, I would say that.
Simp: Why then, as there are three fingers, and there is one line for each finger, there must be three lines.
Sage: Why no, Simplicitus, I sorrow to see your error. Thou has three fingers pointing at four lines.
Simp: But sire, did thou not say that there were as many fingers as there were lines?
Sage: Why yes, I agreed to that.
Simp: Well then, it would seem that thou art maintaining that three is the same as four.
Sage: Oh no, I would never maintain that. The difference between three and four is clear and self evident - no one would dispute that.
I think we can leave them here. It is clear that their dispute can go on indefinitely. Sage is going to say, in one way or another, that three and four are the same. Simp is going to catch him at his trick and point out that Sage is, in effect, saying that three and four are the same. Sage is going to deny that he is saying any such thing. Simp is going to go back and ask what he did say if it wasn't sayiing that three and four are the same. Sage is going to assert the same thing in a different form. Round the mulberry bush they go again. Simp can never establish that Sage is being inconsistent because Sage always changes his position to avoid the immediate inconsistency while retaining the basic inconsistency by returning to it whenever he gets the chance.

Sage's rather heavy handed little game with Simplicitus is based on a principle well known to mathematicians; it is perfectly possible to be consistent within a formally inconsistent system provided you never carry inconsistent lines of reasoning to the point of inconsistency. This is done in the foundations of Mathematics wherein traditional Mathematics, which is consistent and not totally rigorous, is based on Set Theory, which is rigorous and inconsistent.

Now it is presumed, at least by Mathematicians, that Mathematicians know what they are doing when they are skirting the edge of inconsistency. However we have many instances of people who engage in inconsistent conceptual frames of thinking and don't know that they are doing so. Such people are called psychotics, or, sometimes, politicians.

Now surely god, if she chose to, could engage in such modes of thought if She wished. However She would have great advantages for She could shape the universe to Her will. Not only could She create a formally inconsistent universe, but She could also rig things so that the thinking beings in Her universe could not detect it.

For all we know that's exactly what She did. Maybe triangles "really" do have four side and we have been tricked into incorrectly perceiving them as having three sides. Certainly we would have no way of ever knowing. I kind of like to think that She did. To me it seems like the sort of trick that a God that could create a universe having theologians and politicians would pull.


This page was last updated February 1, 2006.
It was reprinted from Personal Notes #1
Copyright © 1969, 2006 by Richard Harter

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