Today, Thomas P. Sheahen contemplates that koan-like question "can God make a square circle?"

And, as he observes, God cannot, at least not...

"within the constraints of the geometry that we usually mean when using the words 'square' and 'circle'. But look carefully at the phrase '...we usuall= y mean...' -- it refers to the body of knowledge classified as common sense. It's important to notice that this phrase states a limitation, and conveys =
a restriction upon God's action, based entirely upon a human way of thinking. To say that God has to think the same way we do is a very severe limitation - one that most of us would never ascribe to God, once we take note of how limited our own minds are."

Moreover, Sheahen notes that these human limitations "creep into language, become customary in thought processes, and gradually become 'accepted wisdom', then 'ultimate truth.' It is a commonplace limitation of humans to grasp one manner of thinking, and then assume that is the only possible way of thinking. This error usually arises because of the characteristics of communication via language--hidden assumptions are built in, and these introduce constraints that may be unwarranted. A mathematical illustration is used to convey this point."

And thus the problem of the circle, the square, and God that we are contemplating today.

Thomas P. Sheahen hold B.S. and Ph.D. degrees in physics from the Massachusetts Institute of Technology. Throughout his career, he has specialized in energy-related research in collaboration with private-sector companies and universities, putting scientific principles to work in difficult situations. He has worked for Bell Telephone Laboratories, the National Bureau of Standards, and Science Applications International Corporation, among others. As a Congressional Science Fellow and a Senior Policy Analyst with the Office of Technology Assessment, he worked on energy-related legislation. The consistent thread in his work has been to find ways to resolve seeming conflicts in science by advancing to a higher level of thinking. Sheahen has been a Visiting Professor at John Carroll University, and taught the Templeton-supported course Issues in Religion an= d Science. Since reading The Phenomenon of Man in 1963, he has remained confident that science and religion are fully compatible, but both need to advance to a higher level of thinking.

One final comment, if the mathematical notation is unclear, please contact me for a copy of the original paper.

--Stacey E. Ake <ake@metanexus.net>

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Subject: Can God Make a Square Circle? From: Thomas P. Sheahen Email: <tsheahen@alum.mit.edu>

Recently, an article in America magazine[1] argued that God's options i= n creation were limited: "God could no more make a dynamic, living material world in which bad things do not happen than God could make a square circle= , or a rock too big to lift. It would be just as much a logical and physical contradiction." Without joining the discussion about constrained evolution, here I wish to draw attention to the way our own limited ability to think gets translated into a supposed limitation upon God.

Human limitations creep into language, become customary in thought processes, and gradually become "accepted wisdom", then "ultimate truth." I= t is a commonplace limitation of humans to grasp one manner of thinking, and then assume that is the only possible way of thinking. This error usually arises because of the characteristics of communication via language--hidden assumptions are built in, and these introduce constraints that may be unwarranted. A mathematical illustration is used to convey this point.

Common Sense vs. Mathematics

Can God make a square circle? Not within the constraints of the geometr= y that we usually mean when using the words "square" and "circle". But look carefully at the phrase "...we usually mean..." -- it refers to the body of knowledge classified as common sense. It's important to notice that this phrase state a limitation, and conveys a restriction upon God's action, based entirely upon a human way of thinking. To say that God has to think the same way we do is a very severe limitation - one that most of us would never ascribe to God, once we take note of how limited our own minds are.

Setting aside the differences between various languages, all human beings convey ideas to one another via language, which is rooted in culture and thought, and which has some inherent assumptions. When everyone has the very same set of assumptions lying beneath their processes of thought and language, what is called common sense is subject to the condition known as general bias. As Lonergan[2] has discussed, "Common sense... is incapable o= f analyzing itself, incapable of making the discovery that it too is a specialized development of human knowledge ..." The way out of general bias involves "...confronting human intelligence with the alternative of adoptin= g a higher viewpoint or perishing."[3]

The language of mathematics is particularly well-suited to taking an upward step and adopting a higher viewpoint. In mathematics, assumptions ar= e not hidden but are plainly stated up front. The very fact that it is abstract releases mathematics from the presuppositions of everyday thinking= . The purpose of the example given here is to illustrate how adopting a highe= r viewpoint overcomes the bias that leads us to think God is limited in some way.=20

The question at hand is "How do you make a square circle?" As we usuall= y mean these terms, you can't. But mathematics allows additional meanings, an= d the combination of all those meanings comprises the higher viewpoint.

Changing Coordinates

In the ordinary realm of everyday thought, we customarily live with a perception of nature given by Euclidean Geometry; that is the natural state of our culture, language and thought patterns. If we stay in that realm, then it is impossible to make a circle square. But we can add a new structure of thought, a higher viewpoint: once we step up to the level of Analytic Geometry, then in Cartesian coordinates (x, y) the circle is defined by

R^2 = x^2 + y^2 (1)

where R is a fixed number. Alternately, still within Analytic Geometry, we can convert to cylindrical coordinates (r, =A2), using the transformation x =
r cos=A2 and y = r sin=A2, and write =20 r = R for all values[4] of =A2, -=BC =BE =A2 =BE +=BC (2)

In these drawings, the one on the left is most definitely what we customarily call a circle, because we automatically think in terms of Cartesian coordinates. Implicit in looking at that figure is an x-y coordinate system superimposed upon it. That's simply the way humans think, how we ordinarily understand things. In Analytic Geometry, equation (1) above describes it correctly.

But now consider the drawing on the right [a square], and superimpose upon it a coordinate system in which the horizontal coordinate is the angle =A2, ranging from a limit of -=BC on the left to +=BC on the right; and the vertical coordinate is the radius r. Let r = zero denote the line across th= e bottom and r = R denote the line across the top. Equation (2) states that a circle has r = R for all possible values of =A2 and that's exactly what is shown in the right-hand drawing. With the coordinates labeled in this way (cylindrical coordinates), the drawing on the right most definitely defines[5] a circle.

Of course you will object, "Hey, that's not what I had in mind!" and that is precisely the point. You didn't think of it because of a priori limitations that are part of your customary outlook, your standard condition. Human beings carry around with them all sorts of limitations of thought, culture and language.

Dealing with Limitations

Sometimes, human beings are able to recognize the limitations and constraints built into everyday life by language, culture, and modes of thinking, and then defer to God's superior wisdom. One way that some people have gotten beyond their own limitations is by turning to some authority to learn about God, such as Scripture. Of course, where there is no recognitio= n of a limitation, then there is no motivation to look further, whether via authority, a higher viewpoint, or a combination of the two. The fight between various proponents of either science or religion is often based on disdain for the other person's source of authority.

In mathematics and physics, we frequently defer to authority. For many arcane items, we read a proof or a derivation once, nod in agreement, and then just look it up whenever needed. The interested reader can find an example of this turning-to-authority in the appendix. The point made there is that either adopting a higher viewpoint or resorting to authority gives you the right answer. Of course, you have to have confidence in your authority for that path to work. Similarly, when you step up to a higher viewpoint, your confidence in that transition must be built upon some knowledge base that includes the recognition that your previous viewpoint was insufficient.

Conclusion

Too often human beings assume that God is subject to the same limitations we have. Because our ability to think is limited and constrained, it is easy to fall into the trap of thinking that God is likewise constrained.

Not only can God make a square circle, humans can too. The trick is to adopt a higher viewpoint, to allow your mind to rise above conventional way= s of thinking.=20

The message to be taken away from our mathematical example is this: for very many forms of human endeavor, we are all stuck in a limited and constrained world without "coordinate transforms." We don't have the skills and understanding necessary to see things as God does. Therefore it's best to be deferential, and avoid any notion of God that comes with human limitations.

Appendix: Area

The following mathematics is provided to illustrate the important point about resorting to authority, or adopting a higher viewpoint.

Suppose we wanted to know the area enclosed by a circle. To find the area using Cartesian coordinates, we must integrate over all elements of area: I= I dx dy. That can get tough in a hurry:

1. One way is to draw lots of lines on the paper and add up lots and lot= s of tiny squares, eventually finding out that the answer is =893.14 R^2 =89 R^2.

2. Another way is to a) observe by symmetry that all four quadrants are the same; b) note that the integral over dy ranges over 0 < y < ymax such that=20

x^2 + y(max)^2 = R^2;

then, c) carry out a modest amount of calculus, so that when the integratio= n over dy is finished, the remaining integral over dx has the form

I (R^2 - x^2 )^1/2 dx

with x running from 0 to + R. At this point, d) most people just look it up in a table of integrals. That is resorting to authority, the mathematical equivalent of Scripture.

3. However, if you are able to step up from the use of Cartesian coordinates and adopt cylindrical coordinates, then the area is the integra= l over all r and _, given by =20 IIdr rd=A2 = Irdr Id=A2 = (R^2/2 )(2=BC) = =BCR^2 ,

and we all recognize that result as the area of a circle. Note that this result was achieved easily by working in cylindrical coordinates, *not* in Cartesian coordinates.

Of the three methods sketched above, obviously the simplest way is to transform to cylindrical coordinates and be done in a minute. But what is a human being to do who has no knowledge of such a coordinate transform? He's stuck with either slowly and painfully counting out the answer, never being sure of getting it right, or else resorting to authority. That happens a lo= t outside the field of mathematics, too.

The purpose of this mathematical exercise is to underline how the limitations of human language and thought lead directly to certain strategies in the pursuit of knowledge.

Notes:

1. D.P. Domning, "Evolution, Evil and Original Sin", America (12 Nov 2001) 2. B.J.F. Lonergan, "Insight: A Study of Human Understanding" (Longmans-Green:1958), chapter 7, section 8, p. 226. 3. Lonergan, Ibid., ch. 7, section 8.4, p. 235. 4. When dealing with angles, =B1=BC = =B1180=9A are the extreme values. 5. For math sticklers: it's a circle with an infinitesimally small dot in the middle.

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Published   2002.12.03