Paradox of the Stone - Part Deux

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Brian Bosse

"The Brain"
Hello Eveyone,

Sometime last summer there was a discussion regarding the question: Can God create a stone too heavy to lift? Someone was good enough to provide two articles dealing with the subject. One article was written by a guy named Wade, and was critical of the other article written by Mavrodes. At the time I concluded that Wade's criticism was good, and shelved the two papers. Recently, I went back to the two papers, and reviewed my thoughts. I have now come to the contrary conclusion that Wade did not understand Mavrodes' argument and that Mavrodes was spot on. I have produced my own defense against the skeptic's question, and would like your thoughts and/or criticisms.

The Problem

Question: Can God create a stone too heavy for Him to lift?

At the heart of this question is a skeptic’s argument against the existence of an omnipotent God. The essence of the argument is as follows:

Case (1) – If God exists, then He is omnipotent. If He is omnipotent, then He can create a stone too heavy for Him to lift. If He can create a stone too heavy for Him to lift, then it is possible for there to be a stone God cannot lift. If it is possible for there to be a stone God cannot lift, then God cannot be omnipotent. If God is not omnipotent, then God does not exist.

Case (2) – If God exists, then He is omnipotent. If He is omnipotent, then He can create a stone too heavy for Him to lift. If He cannot create a stone too heavy for Him to lift, then God is not omnipotent. If God is not omnipotent, then God does not exist.

Since cases (1) and (2) exhaust all possibilities, and since both cases conclude that God does not exist, then the inevitable conclusion is that God does not exist.

The Solution

It is clear from the Scriptures that God is omnipotent and in some sense limited. For instance, the Scriptures teach that God cannot lie and that He cannot sin. In fact, God cannot do anything contrary to His character. As such, when the Bible speaks of God being omnipotent it does not understand ‘omnipotence’ to mean “able to do anything.” St. Thomas Aquinas pointed out that God’s omnipotence should only be construed to range over objects, actions, or states of affairs whose descriptions are not self-contradictory.[1] George I. Mavrodes[2] clarifies this point when he says, “My failure to draw a circle on the exam may indicate my lack of geometric skill, but my failure to draw a square circle does not indicate any such lack. Therefore, the fact that it is false (or perhaps meaningless) to say that God could draw one does no damage to the doctrine of His omnipotence.”[3] St. Thomas Aquinas and Professor Mavrodes have outlined a possible refutation of the skeptic’s argument. If it can be shown that the object – a stone too heavy for God to lift – is a self-contradictory object, then God’s not being able to create such an object “does no damage to the doctrine of His omnipotence.” In order to make this determination we need to explain what we mean when we say an object is self-contradictory.

Self-Contradictory Objects

It seems to be fairly uncontroversial that a square circle would be a self-contradictory object. As such, let’s analyze why such an object is considered self-contradictory. Assuming Euclidean Geometry, the following are definitions for a circle and a square…

Square: A regular polygon with four sides.

Circle: A set of points in a plane all equal distant from a given point.

When we say that object ‘x’ is a square circle we are in effect asserting that the object is a regular polygon with four sides and at the same time a set of points in a plane all equal distant from a given point. The can be symbolically represented as…

(1) (S(x) ∧ C(x))

From (1) we can derive the following propositions:

(a) Given object M, if (r, s) is a point on M, then there is at most one more point on M whose x coordinate is r. (This follows from the property of being a circle.)

(b) Given object M, if (r, s) is a point on M, then there is at least one more point on M whose x coordinate is r. (This follows from the property of being a square.)

(c) Given object M, if (r, s) is a point on M, then if there is another point whose x coordinate is r, there are an infinite number of such points.​

We have derived a contradiction. (c) contradicts (a). However, (c) is based only on the assumption that there is another point whose x coordinate is r. What happens if we assume the opposite, namely that there is not another point whose x coordinate is r? That means there is only one such point, and this contradicts (b). What we have shown is that given (1) we can derive contradictions. In short, we were able to show that…

(2) (S(x) → ¬C(x))

This gives us the key insight into what we mean when we say that an object is self-contradictory. An object is self-contradictory if and only if the object has two properties that are contradictory.

The Incoherent Stone

Consider the object that has the following two properties: (1) ‘stoniness’, and (2) “too heavy for God to lift.” In order for this object to be self-contradictory, then (1) must be inconsistent with (2) in some sense. From a Christian perspective, all things have been created by God and are sustained by Him (Col. 1:16-17). As such, any object that is a stone is an object created by God. All objects created by God are objects that can be lifted by God. Therefore, any object that has the property of ‘stoniness’ is an object that can be lifted by God. From this, it can be shown that “a stone too heavy for God to lift” is self-contradictory. The Christian argues (via Mavrodes and Aquinas) that since such self-contradictory objects cannot exist, then God cannot create such objects, and God’s not being able to create such objects in no way militates against His omnipotence.

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[1] St. Thomas Aquinas, Summa Theologica, Part I, Q. 25, Art. 3.

[2] Dr. Mavrodes, a former Professor of Philosophy at the University of Michigan, has served as President of the Society for Philosophy of Religion and the Society of Christian Philosophers, and as a member of the Executive Committee of the American Theological Society. He has held editorial positions with American Philosophical Quarterly, Faith and Philosophy, and The Reformed Journal.

[3] George I. Mavrodes, “Some Puzzles Concerning Omnipotence,” The Philosophical Review, Vol. 72, No. 2. (Apr., 1963), pp. 221
 

Presbyterian Deacon

Puritan Board Graduate
I'm reminded of the old story I heard while in Bible College.

An atheist asked a systematic theology professor, "Can God create a rock so dense he can't move it?" The professor replied, "He already has! Your head."

:rofl:
 

Civbert

Puritan Board Junior
The Incoherent Stone

Consider the object that has the following two properties: (1) ‘stoniness’, and (2) “too heavy for God to lift.” In order for this object to be self-contradictory, then (1) must be inconsistent with (2) in some sense. From a Christian perspective, all things have been created by God and are sustained by Him (Col. 1:16-17). As such, any object that is a stone is an object created by God. All objects created by God are objects that can be lifted by God. Therefore, any object that has the property of ‘stoniness’ is an object that can be lifted by God. From this, it can be shown that “a stone too heavy for God to lift” is self-contradictory. The Christian argues (via Mavrodes and Aquinas) that since such self-contradictory objects cannot exist, then God cannot create such objects, and God’s not being able to create such objects in no way militates against His omnipotence.

I think you are on the right path, but I think there is another solution. And it does not need a strictly "Christian perspective". The contradiction is in the concept of a stone with a mass so great that it can not be lifted by any force. We can simply look at the idea of force and mass and see that a mass too great to be moved (or two masses two great to be separated) by any definable force is self contradictory and therefore incoherent.

Think of it this way. The idea of a stone too heavy to be lifted, is like finding a number that is greater than the largest natural number. But by definition, for every natural number n, there is a number m that equals n+1. So, there is no number greater than the highest natural number, because there is no highest natural number.

The question is a logical trap. All you need to demonstrate is that the question implies a self-contradiction is and is therefore incoherent. There is no direct answer to the question because the question itself is nonsensical.

Now a coherent question would be, can God cause a contradiction to be true. And the only rational answer would be that it is unknowable because a true contradiction is rationally inconceivable. As a Christian we can reply that our concept of God, revealed to us in Scripture, can not lie and is not irrational; so we have no reason to say God can cause a necessarily false conclusion to be true.
 

Brian Bosse

"The Brain"
Hello Civbert,

I think you open up interesting lines of possibilities. Let's assume Newtonian Physics as a framework. Let M(x) be the mass of some object 'x'. Let M(y) be the mass of some object 'y'. Let 'G' be the universal gravitational constant, and D(x,y) be the distance between the center of mass of x and y. The weight of 'x' relative to 'y' would be defined as: W(x)=(G*M(x)*M(y))/(D(x,y)^2). When we say x cannot be lifted, we are saying that there is no force greater than W(x).

What is interesting about this is that W(x) is realtive to M(y). If W(x) is maximal, then M(y) cannot exceed M(x). (They could be the same.) Not only this, W(x) is relative to the distance between the centers of mass for objects 'x' and 'y'. This distance must be the minimal distance possible. What does that mean? Density now comes into play. If an object is very dense, then the space an object takes up is smaller. This means its center of gravity can be closer to another object than say another less dense object could be. So, we are now talking about maximal density. In other words, in order to have a maximal W(x), you need another object that has the same properties of mass, both objects need to be maximally dense, and they need to be maximally close.

From this, can we derive a contradiction?

Brian
P.S. I suspect even thinking along this line will still end up back at God.
 
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Civbert

Puritan Board Junior
Hello Civbert,

I think you open up interesting lines of possibilities. Let's assume Newtonian Physics as a framework. Let M(x) be the mass of some object 'x'. Let M(y) be the mass of some object 'y'. Let 'G' be the universal gravitational constant, and D(x,y) be the distance between the center of mass of x and y. The weight of 'x' relative to 'y' would be defined as: W(x)=(G*M(x)^2*M(y))/(D(x,y)^2). When we say x cannot be lifted, we are saying that there is no force greater than W(x).

What is interesting about this is that W(x) is relative to M(y). If W(x) is maximal, then M(y) cannot exceed M(x). (They could be the same.) Not only this, W(x) is relative to the distance between the centers of mass for objects 'x' and 'y'. This distance must be the minimal distance possible. What does that mean? Density now comes into play. If an object is very dense, then the space an object takes up is smaller. This means its center of gravity can be closer to another object than say another less dense object could be. So, we are now talking about maximal density. In other words, in order to have a maximal W(x), you need another object that has the same properties of mass, both objects need to be maximally dense, and they need to be maximally close.

From this, can we derive a contradiction?

Brian
P.S. I suspect even thinking along this line will still end up back at God.

:eureka: You know what this is! It just came to me. It's a kind of Zeno's paradox! As the density goes up (approaches infinite), and the distance gets smaller (i.e. approaches zero), the force to "lift" increases infinitely. It's a limitless equation which doesn't converge on a maximum. The question can only be answered if there were a definite maximum.

The equations we have now are empirical - and scientific measures break down when things get too dense or small. I think we are talking about black-holes here. So our physics break down. But I don't think the logic does. I think it's an infinite loop with no definable end. It's still n+1. :think:
 

Brian Bosse

"The Brain"
Hello Civbert,

Hold onto your hat, Anthony. Assuming Newtonian Physics, the following is the formula for the weight of an object ‘x’ relative to an object ‘y’ from a given distance between their centers of mass:

W(x) = M(x)*G*M(y)/D(x,y)^2

Let’s assume that the physical universe is such that there is a limit to density, i.e. there is a maximal density. For the sake of discussion, let's say the maximal density is such that for every kilogram of mass for an object ‘x’ the object occupies a spherical space with the diameter of 1.35*10^-32 meters (this is an approximation of the Planck Density – the density of the universe one unit of Plank time after the Big Band explosion).

Assume we have two objects of maximal density that are touching each other. (Think of two spheres – one on top of the other.) Both objects have the same mass (M), and both objects have the same maximal density. The distance between the two centers of mass is precisely the diameter of one of the spheres. Therefore, the distance between the two sphere is M*1.35*10^-32 meters. The weight of each sphere would then equal…

W=M*G*M/(M*1.35*10^-32)^2
W=M*G*M/M*M*1.82*10^-64
W=G/1.82*10^-64
W=3.66*10^53N

This is an extremely interesting result! Given a maximal density, we have determined the maximal weight that any object could possibly have, and the mass of the object makes no difference!!!!!!!! The only things that play a part are the two universal constants of maximal density and gravitation. The reason for this is that the more massive (mass) the objects become the farther away their center of gravities become, and this completely offsets any gains under the constraints listed above.

As such, given the universe as described above, there is an upper bound as to how much an object can weigh. The question then becomes can God create a force that exceeds 3.66*10^53N? If so, then there is no object, given the constraints of our universe, that cannot be lifted.

I suppose the next question is if God can create a universe so that an object cannot be lifted? Again, this ultimately goes back to God. What do you think?

Brian
 

Grymir

Puritan Board Graduate
Thank's y'all. This is great. You've updated and made present an old argument in today's terms that a simpleton like me can understand.

I give people a hard time when they ask me how many angels can dance on the head of a pin as an example of useless theological debates, and I show them the important theological consecquences that flow from said question and how it reveals the theology of a potentional pastoral candidate.
 
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