Van Huyssteen J.W. (editor) Encyclopedia of Science and Religion

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GEOMETRY,MODERN:THEOLOGICAL ASPECTS

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a number of embryos, implanting only those that

are unaffected by the deleterious gene.

As genetic testing becomes more sophisticated,

it offers great promise for advances in diagnostic

and preventive techniques. As the understanding

of the complex relationship between genes and

environmental factors increases, it is hoped that

drugs can be developed that will prevent a wide

range of late onset diseases. Some envision a day,

for example, when individuals with a genetic pre-

disposition for Alzheimer’s disease will be able to

take prescribed drugs preventing the onset of the

disease, or at least mitigating its effects.

Although genetic testing undoubtedly benefits

many people, it also raises a number of important

ethical, pastoral, and religious issues. There are

concerns over privacy. Some worry that genetic

testing will be used to discriminate against individ-

uals in employment or insurance coverage. There

are concerns over the moral status of the fetus and

embryo. Although prenatal testing may prevent the

birth of children suffering from severely debilitat-

ing illnesses, the techniques also entail destruction

of selected fetuses and embryos. More broadly, ge-

netic testing raises intriguing implications for theo-

logical anthropology. How will a burgeoning

knowledge of human genetics, as well as the abil-

ity to manipulate genes, inform religious accounts

of what it means to be human?

See also BIOTECHNOLOGY; DNA; GENETIC

DETERMINISM; GENETICS; HUMAN GENOME

PROJECT; NATURE VERSUS NURTURE;

OCIOBIOLOGY

Bibliography

Chapman, Audrey R. Unprecedented Choices: Religious

Ethics at the Frontiers of Genetic Science. Minneapo-

lis, Minn.: Fortress Press, 1999.

Cole-Turner, Ronald. The New Genesis: Theology and the

Genetic Revolution. Louisville, Ky.: Westminster John

Knox Press, 1993.

Cole-Turner, Ronald, and Waters, Brent. Pastoral Genetics:

Theology and Care at the Beginning of Life. Cleve-

land, Ohio: Pilgrim Press, 1996.

Peterson, James C. Genetic Turning Points: The Ethics of

Human Genetic Intervention. Grand Rapids, Mich.:

Eerdmans, 2001.

BRENT WATERS

GEOCENTRISM

In geocentric worldviews, the earth is the center of

the universe. The ancient Greek philosopher Aris-

totle (384–322

B.C.E.) thought of celestial bodies as

beautiful and pure, travelling on the surface of per-

fect spheres, and of the earth as an imperfect place

that had fallen to the center of the universe. In the

second century

B.C.E., Ptolemy adjusted the geo-

centric theory with epicycles (orbits imposed on

the orbits of the planets) and eccentrics (orbits that

were centered to the side of the universe) so that

the theory was better able to predict the orbits of

the sun, moon, and stars. The geocentric view of

the universe was replaced by the heliocentric (sun-

centered) view that was pioneered by Nicolaus

Copernicus (1473–1543), adopted and defended by

Galileo Galilei (1564–1642), and much refined by

Johannes Kepler (1571–1630), who discovered the

elliptical nature of planetary orbits.

See also ANTHROPOCENTISM; COSMOLOGY, PHYSICAL

ASPECTS; GALILEO GALILEI

DENIS EDWARDS

GEOMETRY,MODERN:

HEOLOGICAL ASPECTS

The discovery of mathematics in deep Antiquity,

together with its essential pair, geometry, was an

important factor shaping rationalistic tendencies of

the European spirit. From Plato’s belief that “God

geometrizes” through Einstein’s conviction that the

goal of science is nothing else but “to discover the

mind of God,” interaction between geometry and

theology continued with change a changing rate

and intensity.

Middle Ages through the nineteenth century

During the Middle Ages theology formed the natu-

ral environment for the sciences. For instance, the

shift in theology from the understanding of God’s

presence in the world in terms of “his power” to

the understanding of his omnipresence in terms of

“all places” fostered the gradual emergence of the

modern idea of space extending to infinity. This

process culminated with the French philosopher

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and mathematician René Descartes (1596–1650),

who identified matter with only one of its attrib-

utes, extension: body is nothing but an extended

thing. Descartes was doubtless inspired by his

monumental discovery of analytic geometry—the

first really important discovery in geometry after

Euclid and Apollonius. In Descartes’s view, sci-

ence, which should be done “in a geometric man-

ner” (more geometrico), is concerned with ex-

tended bodies, thus leaving to philosophy the

realm of consciousness.

In the seventeenth century a kind of fusion oc-

curred between science and theology (called

physico-theology) to an extent unheard before. This

is clearly seen, for instance, in the writings of Isaac

Newton (1642–1727). In creating his concept of ab-

solute space Newton was a direct successor of for-

mer disputes on God’s omnipresence. Newtonian

absolute space, which “in its own nature, without

relation to anything external, remains always simi-

lar and immovable” (Principia; 1687), has three at-

tributes: homogeneity, immobility, and infinity,

which qualify it as both the universal arena for

physical processes and the “sense organ” of God

(sensorium Dei). The enormous successes of New-

ton’s physics overshadowed his theology and only

the former function of the Newtonian space con-

tinued to exercise its influence on subsequent gen-

erations of thinkers.

Newton’s absolute space as an arena for phys-

ical processes constituted an inherent element of

the mechanistic worldview, and it came as a shock

when it turned out that Euclidean space is not the

only possibility. The dispute concerning Euclid’s

“fifth postulate” lasted from antiquity. The ques-

tion was whether the fifth postulate has to be ac-

cepted as an independent assumption or could be

deduced from other postulates. Many proofs of the

fifth postulate produced during the centuries in-

variably turned out to fail. Around 1830, three

mathematicians—Nikolai Ivanovich Lobachevsky

(1793–1856), Janos Bólyay (1802–1860), and Carl

Friedrich Gauss (1777–1855)—demonstrated inde-

pendently but almost simultaneously that one can

obtain a new geometry, a geometry that is ab-

solutely consistent from a logical point of view,

based on the negation of Euclid’s fifth postulate.

This shows that Euclid was right: The fifth postu-

late is an independent assumption and cannot be

derived from other postulates. This long expected

conclusion was overshadowed, however, by the

fact that a new non-Euclidean geometry was pos-

sible. Soon it became manifest that by playing with

axioms an infinite number of geometries could be

created. In fact, in the second half of the nine-

teenth century many new geometric systems were

created and extensively explored. The philosophi-

cal significance of this mathematical revolution was

comparable to that of Copernicus (1473–1543): Hu-

mans are not only creatures from the outskirts of

the universe, but even the universe, at least con-

ceptually, is not unique; it is a member of an infi-

nite family of geometric universes.

German mathematician Georg Friedrich Bern-

hard Riemann (1826–1866) in his 1854 inaugural

lecture created a broad conceptual setting for mod-

ern geometry, which admitted more than three

spatial dimensions. He also foresaw its physical ap-

plications: The world, with all its physical fields,

could be but a system of fluctuating geometries.

Relativity

At the end of the nineteenth century, peoples’

imaginations were fed with multidimensional geo-

metric pictures. Some philosophers started specu-

lating on “other dimensions” as living places for

spirits, and the popular writer Edwin A. Abbot pub-

lished a book in 1884 entitled Flatland, the princi-

pal aim of which was criticism of Victorian Eng-

land, but which in fact inspired both philosophers

and scientists to deal with new geometric spaces.

With the advent of the special and general the-

ories of relativity the concept of space-time entered

the imaginary requisites of popular and philosoph-

ical literature and became a powerful tool of sci-

entific investigation. From then on, geometry

would not only deal with the problem of space but

also with at least some aspects of the time prob-

lem. Consider only two such problems that have

repercussions in theological matters. The first

problem concerns the nature of time flow and its

relationship to eternity. The theory of relativity fa-

vors, but does not require, a picture of space-time

as existing in one totality with the idea of the flow-

ing time being only a “projection” of human psy-

chological experience onto the world. Such a pic-

ture is consonant with the traditional idea of God’s

eternity (going back to Augustine of Hippo

[354–430

C.E.] and Boethius [c. 480–c. 526 C.E.]) as

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existence outside time rather than existence in time

flowing from minus infinity to plus infinity. The

second problem concerns the interpretation of the

initial singularity appearing in some solutions of

Einstein’s equations describing the evolution of

cosmological models. The question whether such a

singularity (for instance the one corresponding to

the Big Bang in the standard cosmological model)

could be identified with God’s act of creation was

once heatedly discussed. The prevailing view at

the start of the twenty-first century is that such in-

terpretations should be postponed (if they are

methodologically legitimate) until a trustworthy

quantum cosmology becomes available

Rapid progress in relativity theory, especially

during the second half of the twentieth century,

greatly contributed to the development of geome-

try. New physical problems required the sharpen-

ing of known geometric methods and the inven-

tion of new ones. In fact, the necessity to consider

more and more abstract spaces gradually led to the

broadening of the notion of geometry itself. The

process of the geometrization of physics has

changed both physics and geometry.

Noncommutative geometry

It seems that a long dialogue between science and

religion has made people more cautious about

drawing theological conclusions from scientific

premises, but there is still one lesson the theolo-

gian can learn from this process. The degree of

generalization of spatial and temporal concepts

one meets in geometry and its applications to

physics is a good warning against anthropomor-

phisms in theological language.

One notable achievement in geometry at the

end of the twentieth century is the creation and

rapid progress in the so-called noncommutative

geometry, which has some roots in the mathemat-

ical formalism of quantum mechanics. One of its

aims is to deal with spaces that are intractable with

the help of the usual geometric methods. Non-

commutative spaces are, in general, purely global

entities; no local concepts have, in general, any

meaning. For example, the concept of point, as a

typically local concept, has no meaning in many

noncommutative spaces. The number of attempts

to apply noncommutative geometry to physics, for

instance to create a fundamental physical theory, is

constantly increasing. Some such attempts can

have a profound philosophical meaning. For ex-

ample, it is possible to create a model of the fun-

damental physical level in which there is no space

and no time in their usual senses (space consisting

of points and time consisting of instants, which are

local concepts) and yet, in spite of this, an authen-

tic dynamics (i.e., equations modeling behavior of

physical systems under the action of forces) can be

defined in them. Even if such models will turn out

to be false, they demonstrate, by being logically

consistent, that time (in the usual sense as transient

succession of events) is not the necessary condi-

tion for an authentic activity. This seems to falsify

the claim of some theologians that the idea of an

active agent existing outside the flow of time is

contradictory in itself.

Conclusion

To conclude, it could be said that although in the

past there were many direct influences coming

from geometry to theology, it seems unlikely that

this will happen in the future. However, one could

expect an indirect influence. Modern geometric

methods and their application to physics and

other natural sciences doubtless shape people’s

sense of rationality, and this feeling for the rational

will continue to be a powerful source of theolog-

ical inspirations.

See also BIG BANG THEORY; EINSTEIN, ALBERT;

ATHEMATICS; NEWTON, ISAAC; PHYSICS,

UANTUM; RELATIVITY, GENERAL THEORY OF;

INGULARITY; SPACE AND TIME

Bibliography

Boyer, Carl B. A History of Mathematics. New York:

Wiley, 1968.

Connes, Alain. Noncommutative Geometry. New York:

Academic Press, 1994.

Davies, Paul C. W. The Mind of God: The Scientific Basis

for a Rational World. New York: Simon and Schuster,

1992.

Funkenstein, Amos. Theology and the Scientific Imagina-

tion from the Middle Ages to the Seventeenth Century.

Princeton, N.J.: Princeton University Press, 1986.

Kline, Morris. Mathematics in Western Culture. London:

Penguin Books. 1977.

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Torrance, Thomas F. Space, Time, and Incarnation. Ox-

ford: Oxford University Press, 1969.

MICHAEL HELLER

GEOMETRY:PHILOSOPHICAL

ASPECTS

The theological and religious importance of geom-

etry needs to be addressed in conjunction with the

much wider question of the relationship between

those religious aspirations that strive to lay hold

upon abstract eternal truths embedded and em-

bodied in God and others that emphasize the im-

portance of contingent, temporal, and ephemeral

features of existence. To some extent this antithe-

sis reflects the differences between the ancient

Greek philosophers Plato (427–347

B.C.E.) and

Aristotle (384–322

B.C.E.) in their attitudes toward

the status of mathematical objects.

For Plato, neither geometrical objects such as

points, lines, and circles, nor arithmetical objects

such as numbers could be conceived as existing in

the physical world. Since two shapes could not be

the same, nor two objects equal, concepts such as

shape and number had to belong to a realm be-

yond sense and experience, the realm of forms or

ideas. Aristotle rejected this notion, preferring to

think of geometrical and arithmetic objects as re-

ductive abstractions from experience that give rise

to mental generalities. During the Middle Ages, the

Christian philosopher Thomas Aquinas was to

make much of this in terms of intellective and ab-

stractive knowledge, and in their turn John Duns

Scotus (c. 1265–1308) and William of Ockham (c.

1285–1347) were to contribute to the debate by re-

lating the issues to questions of universal and par-

ticular knowledge. Even so, the issue of the

grounding of geometrical truth did not challenge

the self-evident truth of Euclidean geometry; that

had to await the advent of non-Euclidean geome-

tries and the philosophical criticisms of John Stuart

Mill (1806–1873) during the nineteenth century.

Euclidean geometry

Greek mathematics culminated around 300

B.C.E.

in Euclid Alexandria’s Elements, whose achieve-

ment was to treat geometry axiomatically through

a rigorous system of deduction. This abstraction

reflected the value placed upon eternal ideas by

the platonic school, and rid geometry of reliance

upon particular instances of such things as circles

and lines. Euclid’s achievement was to classify

rather than discover the theorems he systematized.

He was able to see that the entire edifice of geom-

etry could be captured in a deductive system based

upon five foundational assumptions or postulates.

Granted those assumptions, no reference to the

physical world was required, and the truths of the

theorems he was able to deduce became tautolog-

ical. Albert Einstein (1879–1955), writing long after

the monopoly of Euclidean geometry had been

broken, reiterated essentially the same point about

the relationship between geometry and experience

in his essay of that name where he observed that

only one assumption is required in addition to a

geometrical system: the further postulate that it is a

model for the real world.

Whether the configuration and behavior of the

physical world conforms to a deductive geometri-

cal system is nonetheless an open question. If it

does, there is a remarkable harmony between an

abstract construction of the human mind and the

workings of the world—part of what contemporary

physicist Stephen Weinberg has called “the unrea-

sonable effectiveness of mathematics,” but the ef-

fectiveness of geometry may say more about the

limitation and consistency of human thought and

action than it does about the behavior of the

world. As the Italian philosopher Giambattista Vico

(1668–1744) put it in the Nuovo Scienza (New Sci-

ence, 1725), the dilemma arises from the funda-

mental question of the relationship between the

“found” and the “made” (verum et factum).

Euclidean geometry dominated mathematics

for the subsequent two thousand years. Problems

posed in antiquity that provided the stimuli for the

development of enormous areas of mathematics—

construction of a square with area exactly that of a

given circle, the doubling of the volume of the

altar at Delos, or the trisection of an angle (all to

be solved using only straightedge and com-

passes)—remained unresolved until modern times,

when all three were proved to be impossible. The

influence of Euclidean geometry permeated edu-

cation, architecture, science, and literature. The

mediaeval trivium and quadrivium made geome-

try an essential ingredient of education. The Italian

poet Dante Alighieri (1265–1321) designed the

structure of Hell, Purgatory, and Heaven around it

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in his epic poem The Divine Comedy. First Ptolemy

(c. 85–165

C.E.) and then Nicolaus Copernicus

(1483–1543), Johannes Kepler (1571–1630), and

Isaac Newton (1642–1727) devised their world sys-

tems upon it. Prohibition of images in Islam en-

couraged the intricate geometrical patterns used to

decorate mosques.

Perhaps the most significant change in the

study of geometry occurred with René Descartes’s

(1596–1650) invention of algebraic or analytic

geometry as described in the Discourse on Method

(1637) by envisaging geometrical figures superim-

posed on a grid, thereby making their properties

susceptible to algebraic analysis. It is impossible to

exaggerate the importance of this change, for it al-

lowed geometry to be integrated into the calculus

as discovered by Newton and Gottfried Wilhelm

Leibniz (1646–1716), and so into the emerging the-

ories of the physical world. Immanuel Kant

(1724–1804) took it as an obvious a priori truth in

the Critique of Pure Reason in 1781 that the sum of

the angles of a triangle is 180 degrees, and Euclid

continues to be taught throughout the world as a

quintessential example of a deductive system to

assess the potential of young mathematicians. Carl

Friedrich Gauss (1777–1855), prompted by his

study of curvature, was experimenting with alter-

native geometries, and the nineteenth century saw

them proliferate through the work of Nikolai

Ivanovitch Lobachevsky (1793–1856), Gauss’s

pupil Georg Friedrich Bernhard Riemann (1826–

1866), and others.

Non-Euclidean geometries

From Euclid’s own time there had been persistent

attempts to deduce the “fifth postulate”—often

cited as “through any point not in a given line one

and only one line can be drawn parallel to the

given line”—from the other four. These attempts

continued until beyond Gauss’s time, but he grad-

ually became convinced that they were futile, that

the fifth postulate was independent of the others,

and therefore that it could be modified to produce

non-Euclidean geometries. Lobachevsky was simi-

larly obsessed with proving the fifth postulate, but,

unlike Gauss, between 1826 and 1829 he worked

on and eventually published his discovery of non-

Euclidean geometry, thus dealing a blow to the

Kantian system similar, as Carl Boyer puts it, to the

impact on Pythagoreanism of the discovery of in-

commensurables. The “Copernicus of geometry,”

Lobachevsky was the first to generate an entirely

consistent and coherent geometry that rejected Eu-

clid’s fifth postulate (although Janos Bólyay

[1802–1860] also developed one almost simultane-

ously), but even he was so bemused by its

counter-intuitive properties that he called it “imag-

inary” geometry. Non-Euclidean geometry none-

theless remained an obscure mathematical curios-

ity until taken up and generalized by Riemann. He

realized that geometry need not be based upon

quasi-Euclidean postulates at all, but could be re-

garded as a set of n-tuples (co-ordinates) com-

bined according to certain rules. These rules define

a metric and give rise to different kinds of Rie-

mannian “space” governed by tensors. These

spaces were to prove fundamental to the revolu-

tion in physics brought about by Einstein.

The realization that there were non-Euclidean

geometries shook the foundations of mathematics

and contributed to the demise of absolute founda-

tionalism in philosophy, even though the discovery

of different and mutually exclusive axiomatic

geometries gave new momentum to the study of

deductive systems based upon “foundations.”

Whereas for Kant and his predecessors there had

seemed to be no element of choice in the determi-

nation of the geometry of the world because there

was only one geometry, and that Euclid’s, after

Gauss and Lobachevsky it became necessary to

add the postulate that Einstein was to remark on,

that a particular geometry should also be chosen as

the geometry of “real world.” As he was to show in

his theory of General Relativity, the “natural”

geometry of the universe is not Euclidean at all,

but Riemannian.

Implications for theology

In theological terms, the ubiquity and power of

geometry have often been regarded as evidence

for the work of God, whose mind has come on

such a basis to be thought of as a perfect deductive

system. But there are serious difficulties with such

a view. One concerns the parallelism between de-

duction and determinism; another concerns the

problem of the found and the made.

Deductive systems self-consciously avoid the

introduction of any new material whatever; proof

involves rendering explicit what is already implicit

by applying the rules of deductive logic. A uni-

verse governed by physical laws equivalent to

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such deductive systems would be deterministic;

there would quite literally be “no new thing under

the sun” (Eccles. 1:9). At best, as occurs when im-

plicit truths are rendered explicit by the articulation

and proof of a new theorem in geometry, people

would find themselves surprised by the unfore-

seen; but any freedom, either for God or for

human beings, would be illusory, the outworking

of an implicit and inevitable necessity.

Since Kant, people have been forced to take

seriously the notion that what they regard as the

intelligibility of the world is in reality the inner co-

herence of their modes of thought: in Vico’s lan-

guage, the intelligibility of the made, not the

found. Objections to the employment of geometry

as a model for the world reiterate this view in the

context of doubts about the fine structure of the

world and the limits of observation. Such doubts

have been reinforced by the development of quan-

tum mechanics and relativity, which suggest that

human intuitions about the world are mistaken (al-

though even these suggestions are still to be con-

strued within a conceptual system, and not as

grounded in a noumenal world).

Nonetheless, the expansion of elementary

geometry into analytic geometry, topology, and

linear algebra, preserves the sense of the “unrea-

sonable effectiveness of mathematics,” and sug-

gests that, all doubts to the contrary notwithstand-

ing, there may remain some sense in Newton’s

designation of the universe as the divine senso-

rium, albeit construed as a creation embodying the

structure of a divine geometry, and intelligible only

to a divine mind.

John Stuart Mill (1806–1873) was one of the

first to challenge Kant’s view that the a priori truths

of geometry are necessary consequences of the

possibility conditions of rational thought, in other

words, that to be rational people have to think the

world, amongst other things, in Euclidean terms.

Mill did not know of non-Euclidean geometry, but

attributed the apparent inescapability of Euclidean

geometry to paucity of imagination, and its domi-

nation to the kinds of experiences to which human

beings are susceptible. Mill seems to have been

vindicated by the predilection of physical theory—

in both quantum mechanics and relativity—for

non-Euclidean geometries that defy everyday

human intuitions.

Towards the end of the nineteenth century

fundamental changes in philosophy of mathemat-

ics occurred, most notably the articulation of logi-

cism by German mathematician and philosopher

Gottlob Frege (1848–1925). Frege attempted to re-

duce arithmetic to logical categories by employing

the theory of sets and the non-Euclidean geome-

tries of Riemann that discard the intuitive notions

of line and plane familiar from Euclid in favor of

abstract n-tuples governed by arbitrary rules.

Attempts to reduce mathematics to logic were

associated with Frege’s attack on psychologism, it-

self a descendant of Kant’s view that the nature of

intelligible reality is governed not by properties of

an objective world but by the rules of thought.

Often called a modern platonism, Frege’s work

struggled to ground mathematics in an inviolable

world independent of experience. Unfortunately,

by adopting the theory of sets, Frege fell foul of

Bertrand Russell’s (1872–1970) celebrated paradox

that the “set of all sets that are not members of

themselves” both is and is not a member of itself,

thus demonstrating that there is an apparent antin-

omy in the theory of sets.

Quite apart from its relevance to ontology and

epistemology, and thus to theology, geometry has

played a major role in the more everyday devel-

opment of religion. Closely associated with the ed-

ucated and priestly classes, with astronomy and

astrology, with numerology and mysticism, geom-

etry has repeatedly had an impact on the way peo-

ple have viewed the order and mystery of the

world. The Pythagoreans regarded number as the

basis of all knowledge and truth, many religions

and cults have seen mystical significance in the

properties of geometrical shapes, especially the

golden rectangle/ratio and the pentangle, widely

employed in magic.

The fundamental religious importance of

geometry nevertheless emerges from questions of

the relationship between divine creative purpose,

the structure and operation of the natural world,

and the conceptual capacities of the human mind.

If, as Einstein suggested, “the only unintelligible

thing about the world is that it is intelligible,” there

seems to be an intrinsic harmony between all three,

and geometry seems able, at least in the limit, to

embody the structure of the world as found. If, as

others suggest, following Kant, “the only intelligible

thing about the world is that it is unintelligible,”

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GOD

— 377—

then geometry is a fabrication designed to render

the world intelligible at the expense of misrepre-

senting its intrinsic structure, and geometry can do

no more than embody a structure of the world as

made in the image of the human mind.

See also KANT,IMMANUEL;MATHEMATICS;NEWTON,

SAAC;PHYSICS

Bibliography

Boyer, Carl B. A History of Mathematics. Princeton, N.J.:

Princeton University Press, 1985.

Conway, John H., and Guy, Richard K. The Book of Num-

bers. New York: Springer-Verlag, 1996.

Coxeter, H. S. M. Introduction to Geometry. New York:

Wiley, 1961.

Gulberg, Jan. Mathematics: From the Birth of Numbers.

New York: Norton, 1996.

Heath, Thomas L. The Thirteen Books of Euclid’s Elements,

Books 1 and 2. New York: Dover, 1956.

Hofstadter, Douglas R. Gödel, Escher, Bach: An Eternal

Golden Braid. New York: Basic Books, 1999.

Stewart, Ian. Flatterland: Like Flatland, Only More So.

London: Macmillan, 2001.

JOHN C. PUDDEFOOT

GERMLINE INTERVENTION

See GENE THERAPY

GLOBAL WARMING

Three important indicators suggest that the Earth’s

climate is going through a period of global warm-

ing: (1) an increase in atmospheric temperatures

near the Earth’s surface; (2) an increase in the sur-

face temperature of the Earth’s oceans; and (3) an

increase in sea levels. Since global weather patterns

are extraordinarily complex, with different systems

influencing one another, the effects of global warm-

ing will vary from region to region. For instance, as

global warming continues, some regions should

have dramatic increases in annual precipitation lev-

els, whereas other regions should have dramatic

decreases—even desertification. Within the science

and religion literature, discussions of global warm-

ing occur most frequently within ecological ethics.

Ethicists draw from the climatology sciences to in-

form their reflection and analysis.

See also ECOLOGY; ECOLOGY, ETHICS OF; ECOLOGY,

ELIGIOUS AND PHILOSOPHICAL ASPECTS;

COLOGY, SCIENCE OF

RICHARD O. RANDOLPH

GOD

Taken in its subjective sense, the word God refers

to whatever is the object of one’s ultimate concern.

Thus one might judge about a person, “Money or

power is his god.” But one can also ask whether

his “god” really is God, whether what he treats as

god possesses the properties one would expect in

an object of ultimate concern. In this second or

more objective sense, then, God refers to whatever

is truly ultimate: the greatest being, the highest ob-

ject of belief, the ground of all being. Most often,

to believe in God means to believe that the ulti-

mate reality is personal. That is, the divine pos-

sesses all the positive features that one associates

with “mind” (intellect, will, self-consciousness, and

perhaps emotions), but possesses them in an infi-

nitely higher and more perfect form than humans

do. For virtually all theists, God is understood as

the creator of all things. For most theists, God is

also understood as providentially involved in guid-

ing the world subsequent to its creation.

Two major sources have added more specific

content to the notion of God. The various religious

traditions have developed extensive beliefs about

the nature of God, the actions and self-revelation

of God in the world, and the sorts of ethical and

moral principles that most correspond to the divine

nature. In a similar fashion, but not always in lock-

step, the philosophical traditions have reached

conclusions on what most appropriately count as

attributes of God, how (if at all) the divine could

be known, and why an infinite God could never

be fully comprehended by finite knowers. Theolo-

gians have combined features from both of these

approaches. They draw on beliefs from one or

more of the religions, while analyzing and refor-

mulating these beliefs using conclusions and con-

ceptual tools developed by philosophers over the

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GOD

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centuries. The result is a spectrum of positions on

whether there are many gods or only one, on what

it means to say that God is personal, and on how

God is related to the world.

A brief history of God

Before there was belief in one God (monotheism),

there was belief in many gods (polytheism). The

earliest cultural remnants show humans relating to

parts of the natural world (mountains, bodies of

water, thunder and lightening, changes in climate)

as if they were the product of personal forces.

Finding reasons for natural events was perhaps the

first step toward science, which gives explanations

based on impersonal forces rather than on super-

natural agents.

As cultures became more sophisticated, the

gods took on personalities distinct from natural ob-

jects. Some of this evolution is visible in the He-

brew Bible, an authoritative text for Jewish, Chris-

tian, and Muslim views of God. Yahweh, the God

of Abraham and his clan, was “a jealous God”

(Exod. 20:5) who would allow “no other gods” be-

fore him (Deut. 5:7). Gradually the Israelites real-

ized that Yahweh was “a great King above all

gods” (Ps. 95:3), indeed so all-encompassing that

there could be no other gods: “For I am God, and

there is no other” (Isa. 45:22). Hence, the three

Western monotheisms came to hold that God’s

power must be unlimited (omnipotence), as must

be God’s perception (omnipresence), God’s

knowledge (omniscience), and God’s goodness

(omnibenevolence). Yahweh must be the sole cre-

ator of all that is. All must stem from God, and God

must have created all out of nothing (creatio ex ni-

hilo). God became the ultimate ground and expla-

nation of all things, the One who alone is worthy

of worship.

In addition to this shared basis, the Western

monotheisms also evidence important differences,

regarding, for example, whether the divine nature

is trinitarian (three-in-one) or not. Even if the full

variety of specific beliefs about God cannot be

treated in this entry, the differences remain vital for

many believers. Indeed, many would resist the no-

tion of “generic theism.” That is, many would say

that they are not believers in God in general but

believers in “the God of Abraham, Isaac, and

Jacob,” or disciples of Allah as he revealed himself

to the prophet Mohammed, or believers in the

Holy Trinity of “God the Father, His Son Jesus

Christ, and the Holy Spirit.”

God and contemporary science

Some leading philosophers and scientists (for ex-

ample, in the twentieth century, Bertrand Russell,

Antony Flew, Edward O. Wilson, and Richard

Dawkins) hold that belief in God as an explanatory

principle is incompatible with science. Clearly, if

science entails some form of metaphysical natural-

ism (physicalism, materialism, or nontheistic emer-

gence), then all forms of theism are excluded; be-

lief in a single act of divine creation would be no

better off than the belief that one must sacrifice to

the rain god. By contrast, other leading scientists

are theists and find no conflict between their reli-

gious belief and the practice of science.

Among the latter group one finds stronger and

weaker claims. For example, many hold that sci-

ence and personalist theism are at least compatible

and can coexist without contradiction or tension.

Perhaps science explains the “how” of the uni-

verse, theism its ultimate “why.” Perhaps divine ac-

tions concern only the “before” and “after,” the

moment of creation that led to the existence of

physical laws and the final act that establishes “a

new heaven and a new Earth” (Rev. 21:1). Or per-

haps God-language refers to the ground of all ex-

istence and all value but can never be used to ex-

plain any particular thing or event.

Others make stronger claims: The order in the

universe is best understood as an expression of

the nature of God. Without God one cannot finally

make sense of the lawfulness and mathematical

simplicity of the physical world ( John Polking-

horne), or of the evolution of intelligent life (theis-

tic evolutionists and Intelligent Design theorists

such as William Demski), or of human rationality

and morality (Alvin Plantinga). It is argued that the

fundamental physical constants are “fine-tuned” so

as conjointly to make it possible, or even likely,

that intelligent life would emerge, and that a su-

pernatural agent offers the best explanation of this

fact. To use Robert John Russell’s distinction, they

argue either that the universe is consistent with

what the believer in God would expect (the theol-

ogy of nature) or that the fine-tuning of physical

constants actually provides evidence that God ex-

ists (natural theology).

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GOD

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Those who find science and theism in conflict

suggest two different answers. One group re-

sponds that belief in God has to be eliminated, or

at least radically modified so that it fits into the

gaps left by science and makes no claims incom-

patible with it (the “god of the gaps”). For exam-

ple, theistic language could be viewed as an ex-

pression of a cultural, emotional, or psychological

particularity, similar to one’s manner of dressing or

speaking. If God-talk makes no truth claims, it can-

not conflict with scientific results. Another group

responds that the results and methods of science

should instead be set aside whenever they conflict

with theological truths. Religious fundamentalism

may employ scientific-sounding language, as in

“young Earth creationism”; it may refute science

by appeal to scriptural texts; or it may associate

God with “truths beyond the reach of reason” seen

only through “the eyes of faith.”

In summary, the differences in the logic of sci-

entific theories and God-language are generally ac-

knowledged. Proponents differ on whether the dif-

ferences are tensions and, if so, how serious they

are. Should the tensions be minimized, bringing

science and religion into the greatest consonance

possible, or should they be maximized, making the

contrasts as stark as possible?

Issues on God and science

The God-science relationship has continually fasci-

nated reflective persons for its alternating reso-

nances and dissonances.

The problem of divine action. For Jews, Chris-

tians, and Muslims, God creates the world, sustains

it in existence, and acts providentially to bring

about divine purposes. Far from being deists, these

traditions espoused miracles (supernatural inter-

ventions into history that set aside natural law). In-

deed, the miracle of the resurrection lies at the

center of Christian faith. But such miracles are by

definition inaccessible to scientific study; indeed,

they seem to imply the negation of scientific results

and methods. Contemporary efforts to minimize

the conflict include developing noninterventionist

accounts of divine action in the world, reducing

God’s role to a single all-encompassing act, and of-

fering fully naturalized reworkings of the tradi-

tional religions that eschew all miracle claims.

Evidences for and against God. Do human be-

ings inhabit a cosmos that displays the signs of

creation by a benevolent, omnipotent deity? Some

say no. Vast regions are cold and uninhabitable;

does all this exist just for the sake of intelligent an-

imals on one planet? Entropy means the universe

will wind down; what sign is there of “a new

heaven and a new Earth”? Finally, why would a

benevolent God allow such incredible evil, suffer-

ing, and wastefulness of life—both in the natural

world and at the hands of man?

Others argue that the cosmos does display

signs of creation by God. Could a random origin

and evolution have produced beings capable of

rational thought and moral action? The improba-

bility suggests design. Moreover, they argue, the

result is different in kind from physical evolution;

consciousness, rationality, and morality are better

explained by a “first cause” that itself possesses

these features. The universe possesses a mathe-

matical simplicity that evokes a religious (or quasi-

religious) response from many scientists, and a

beauty that for some is both awe-inspiring and

sublime. The argument for God as the best expla-

nation becomes more compelling when supple-

mented with personal religious experience of the

divine or, in Immanuel Kant’s phrase, of “the moral

law within.”

God and specific scientific results. In cosmol-

ogy, the “singularity” of the Big Bang seemed to

offer support for a doctrine of creation. In Jim Har-

tle and Stephen Hawking’s quantum cosmology,

however, there would be no t = 0 (time equals

zero), hence no time at which God could create.

Perhaps creation could be understood as the con-

tingency of the world on God, even if there were

never a “moment of creation,” as Robert John Rus-

sell posits.

Neo-Darwinian evolution involves random ge-

netic variation and selective retention by the envi-

ronment. Denying evolution seems impossible, but

theists have argued that the process may be

“guided” by God in ways not yet fully visible or

understood. Sociobiology and evolutionary psy-

chology also challenge the ontological uniqueness

of the human animal and hence challenge claims

that humans are created “in the image of God.”

The neurosciences can increasingly reconstruct

the neural correlates of cognitive functions. Will

they someday be able to detect the neurological

footprints of God’s interactions with individuals?

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GÖDEL’S INCOMPLETENESS THEOREM

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Might they discriminate between genuine and

counterfeit experiences of God? Or will God’s in-

teractions with the world always escape human de-

tection and rational analysis?

“God beyond God,” experience, and mystery

The history of the interrelations between God and

science mirror something of the history of God and

philosophy. Like philosophy, science uses its ana-

lytic tools to falsify an ever larger number of spe-

cific claims about God. Yet neither can verify the di-

vine, and neither can rule out God’s existence. The

experiences of something transcendent, someone

divine, remain; hence room remains for conceiving

God in a way that conflicts with neither science nor

philosophy (the Transcendent Other, the “God be-

yond God”). New philosophical theologies, such as

panentheism, can reformulate traditional claims

about God’s relationship with the world in new and

more adequate ways. In the end, the question of

God remains part of the ultimate mystery that faces

humans in their walk between birth and death.

See also CREATIO EX NIHILO; DIVINE ACTION;

MERGENCE; GOD OF THE GAPS; MONOTHEISM;

ATURAL THEOLOGY; OMNIPOTENCE;

MNIPRESENCE; OMNISCIENCE; PANENTHEISM;

HEISM; THEOLOGY

Bibliography

Armstrong, Karen. A History of God: The 4000-Year Quest

of Judaism, Christianity, and Islam. London: Heine-

mann, 1993.

Clayton, Philip. God and Contemporary Science. Grand

Rapids. Mich.: Eerdmans, 1998.

Clayton, Philip. The Problem of God in Modern Thought.

Grand Rapids, Mich.: Eerdmans, 2000.

Davies, Paul. God and the New Physics. New York: Simon

and Schuster, 1983

Moltmann, Jürgen. God in Creation: A New Theology of

Creation and the Spirit of God. Minneapolis, Minn.:

Fortress Press, 1993.

Peacocke, Arthur. Theology for a Scientific Age: Being and

Becoming—Natural, Divine, and Human. Minneapo-

lis, Minn.: Fortress, 1993.

Polkinghorne, John. Belief in God in an Age of

Science. New Haven, Conn.: Yale University

Press, 1998.

PHILIP CLAYTON

GÖDEL’S INCOMPLETENESS

THEOREM

By the early part of the twentieth century, the work

of mathematical logicians such as Gottlob Frege,

Bertrand Russell, and Alfred North Whitehead had

honed the axiomatic method into an almost ma-

chine-like technique of producing mathematical

theorems from carefully stated first principles (ax-

ioms) by means of clear logical rules of inference.

In 1931, however, Kurt Gödel (1906–1978), an Aus-

trian logician, uncovered a surprising limitation

inherent in any axiomatic system intended to pro-

duce theorems expressing the familiar mathemati-

cal properties of integer arithmetic.

Gödel developed a method, whose reach was

slightly extended by J. Barkley Rosser in 1936, that

shows how, given any such (consistent) system of

axioms, one can produce a true proposition about

integers that the axiomatic system itself cannot pro-

duce as a theorem. Gödel’s incompleteness result

follows: Unless the axioms of arithmetic are incon-

sistent (self-contradictory), not all arithmetical

truths can be deduced in such machine-like fash-

ion from any fixed set of axioms. This result, that

here consistency implies “incompleteness,” has

striking implications not only for mathematical

logic, but also for machine-learning (artificial intel-

ligence) and epistemology, although its precise

significance is still debated.

Gödel’s work was stimulated by a program of

the mathematician David Hilbert (1862–1943),

whose goals included showing that the consis-

tency of higher mathematics need not be based

solely upon faith in the reasonableness of its ax-

ioms and methods, but could be established using

means no more questionable than those of ele-

mentary arithmetic. The link Gödel discovered be-

tween consistency and completeness of elemen-

tary arithmetic, however, led him to a further result

strongly suggesting that this goal, as originally en-

visioned by Hilbert, is unattainable. Although rela-

tively few nonspecialists have mastered Gödel’s

proof, many general readers have attained an ap-

preciation of his argument through popular ac-

counts of its connection with familiar paradoxes

involving self-reference.

See also ARTIFICIAL INTELLIGENCE; MATHEMATICS;

ARADOX, SELF-REFERENCE

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