Inklings Forever: Published Colloquium Proceedings 1997-2016 Inklings Forever: Published Colloquium Proceedings 1997-2016

Volume 7

A Collection of Essays Presented at

the Seventh Frances White Ewbank Colloquium

on C.S. Lewis & Friends

Article 11

6-3-2010

Mere Mathematics: The Role of Mathematics in the Apologetic Mere Mathematics: The Role of Mathematics in the Apologetic

Works of C.S. Lewis Works of C.S. Lewis

Matt D. Lunsford

Union University

Available at: https://pillars.taylor.edu/inklings_forever/vol7/iss1/11

This Essay is brought to you for free and open access by the British Author Collections at Pillars at Taylor

University. It has been accepted for inclusion in Inklings Forever: Published Colloquium Proceedings 1997-2016 by

an authorized editor of Pillars at Taylor University. For more information, please contact pillars@taylor.edu.

Clive Staples Lewis (1898-1963) was one of the intellectual giants of the 20th century and argu-

ably the most influential Christian author of that period. In spite of his own personal lack of suc-

cess in the area of mathematics, C. S. Lewis exhibited a lofty appreciation of the discipline as

demonstrated by his numerous references to mathematics and to mathematical objects, and by his

recurrent use of mathematical terminology, in his apologetic writings. This paper will explore

Lunsford, Matt D. ―Mere Mathematics: The Role of Mathematics in the Apologetic Works of C.S.

Matt D. Lunsford, Union University

Clive Staples Lewis (1898-1963) was one of the intellectual giants of the 20th century

and arguably the most influential Christian author of that period. Lewis was born in Belfast,

educated at Oxford, and taught medieval and Renaissance literature at both Oxford and Cam-

bridge. As a scholar, he made significant contributions to the areas of literary criticism, child-

ren’s literature, and fantasy literature. His conversion to Christianity is well documented in his

autobiography Surprised by Joy, and gave rise to a body of apologetics works. In spite of his

own personal lack of success in the area of mathematics, C. S. Lewis exhibited a lofty apprecia-

―could never have gone very far in any science because on the path of every science the lion ma-

thematics lies in wait for you.‖ (Lewis, Surprised by Joy 137) In his early training at Oldie’s

School, Lewis credits only some geometry and grammar as accomplishments. His tutelage later

under Kirk (Mr. Kirkpatrick) proved indispensable for Lewis’ ratiocination skills. It was with

Kirk that he prepared for his first attempt at Responsions, a required examination at Oxford that

included elementary mathematics. Lewis was not successful on his first attempt and continued

to prepare for the exam with Mr. Campbell. His preparation included algebra, a subject for

which Lewis had a personal dislike – ―devil take it!‖ (Lewis, Surprised by Joy 187) He never

passed Responsions; however, due to his service in World War I, he was granted a waiver. Lew-

is claims that, without this exemption, his career at Oxford would have concluded prematurely.

fy a point of difficulty for the reader.

In Miracles, Lewis states that rational thought and the conscience of man are not products

of the system of Nature. He refused to accept a ―behavioristic theory of logic, ethics, and aes-

thetics.‖ (Lewis, Surprised by Joy 208) This led him to consider the relationship between ma-

thematics and three laws: the laws of thought, the laws of morality (Natural Law), and the laws

of nature.

In perhaps his greatest compliment to the discipline, Lewis states, ―Pure mathematics is

the type of successful thought.‖ (Lewis, God in the Dock 65) To him, the laws of thought were

seen to be self-evident and could not be changed, for to modify the laws of thought would, in es-

sence, nullify the ability to reason and thus leave one in the situation of not being able to know

simple rules of arithmetic follow deductively from self-evident axioms, just as rational thinking

something human beings have made up for themselves and might have made different if they had

liked?‖ (Lewis, Mere Christianity 24)

Suppose one wants to put this Reason to work to discover truths about the universe. How

can one be sure that a belief is actual truth and not just wishful thinking? To address this ques-

tion, Lewis uses an analogy from arithmetic. ―Suppose, I think, after doing my accounts, that I

have a large balance at the bank. And suppose you want to find out whether this belief of mine

is 'wishful thinking'. You can never come to any conclusion by examining my psychological

condition. Your only chance of finding out is to sit down and work through the sum yourself.

When you have checked my figures, then, and then only, will you know whether I have that bal-

ance or not. If you find my arithmetic correct, then no amount of vaporing about my psycholog-

ical condition can be anything but a waste of time. If you find my arithmetic wrong, then it may

discover the psychological causes of the error.‖ (Lewis, God in the Dock 272-273) So, accord-

ing to Lewis, the logical procedure needed to correct a mistake in arithmetic displays a prototype

of successful rational argumentation. Lewis was so bothered by the modern method of debate

which assumes that one is wrong and then argues why he is wrong rather than demonstrating that

he is wrong, that he gave it a name – ―Bulverism‖. (Lewis, God in the Dock 273)

What does Reason have to say about the truth claims of Christianity? Lewis draws upon

his arithmetical analogy: ―But, of course, being a Christian does mean thinking that where Chris-

tianity differs from other religions, Christianity is right and they are wrong. As in arithmetic –

there is only one right answer to a sum, and all other answers are wrong: but some of the wrong

answers are much nearer being right than others.‖ (Lewis, Mere Christianity 43) In a different

rence, to know a lot about the universe they live in. If, on the other hand, I swallow the scientific

cosmology as a whole, then not only can I not fit in Christianity, but I cannot even fit in science.‖

(Lewis, The Weight of Glory 105-106)

Can one really conceive of an alternate set of moral laws? In Mere Christianity, Lewis

answers, ―Think of a country where people were admired for running away in battle, or where a

man felt proud of double-crossing all the people who had been kindest to him. You might just as

well try to imagine a country where two and two made five.‖ (Lewis, Mere Christianity 19) He

then adds, ―It seems, then, we are forced to believe in a real Right and Wrong. People may be

sometimes mistaken about them, just as people sometimes get their sums wrong; but they are not

another, the differences are not really very great – not nearly so great as most people imagine –

and you can recognize the same law running through them all‖ (Lewis, Mere Christianity 24-25)

and 2) there is a standard in both mathematics and Natural Law which is independent of personal

or public opinion. As Lewis writes, ― The moment you say that one set of moral ideas can be

better than another, you are, in fact, measuring them both by a standard, saying that one of them

conforms to that standard more nearly than the other. But the standard that measures two things

is something different from either. You are, in fact, comparing them both with some Real Mo-

rality, admitting that there is such a thing as a real Right, independent of what people think, and

that some people's ideas get nearer to that real Right than others.‖ (Lewis, Mere Christianity 25)

Assuming that there is a Real Morality, how can an individual use this fact to make prop-

er moral decisions? Just as constructing a rational argument requires knowledge of the laws of

of facts, 2) the recognition of self-evident truths (which Lewis calls intuition), and 3) the logical

arrangement of ―facts so as to yield a series of such intuitions which linked together produce a

proof of the truth or falsehood of the proposition we are considering.‖ (Lewis, The Weight of

Glory 54) Lewis uses another mathematics analogy, this time from geometry, to illustrate this

process. Now the geometric proof is the prototype. If a correct geometric proof is well crafted,

then ―each step is seen by intuition, and to fail to see it is to be not a bad geometrician but an

idiot.‖ (Lewis, The Weight of Glory 54) Lewis does add that, ― You can invent a simpler proof,

that is, a simpler concatenation of intuitable truths. But when you come to an absolute inability

to see any one of the self-evident steps out of which the proof is built, then you can do nothing.‖

(Lewis, The Weight of Glory 55) While admitting that moral decision-making does not admit

the very possibility of progress demands that there should be an unchanging element....I take it

we should all agree to find this sort of unchanging element in the simple rules of mathematics. I

would add to these the primary principles of morality. And I would also add the fundamental

doctrines of Christianity.‖ (Lewis, God in the Dock 45) Hence, for Lewis, the three realms of

mathematics, morality, and Christianity exhibit instances of static knowledge that will never be

replaced. As for progress, Lewis issues this warning: ―If you are on the wrong road, progress

means doing an about-turn and walking back to the right road; and in that case the man who

turns back soonest is the most progressive man. We have all seen this when doing arithmetic.

When I have started a sum the wrong way, the sooner I admit this and go back and start over

plain man's picture of space, turn out to be the shadow: numbers are the substance of our know-

ledge, the sole liaison between mind and things.‖ (Lewis, God in the Dock 46) Mathematics

provides the language for expressing the laws of nature, which are the result of observed consis-

tency and assumed uniformity in the universe. Lewis argues that by using only the method of

historical probability, ―we cannot say that uniformity is either probable or improbable.‖ (Lewis,

Miracles 165) Moreover, Lewis maintains that, ―Three conceptions of the 'Laws' of Nature have

been held. (1) That they are mere brute facts, known only by observation, with no discoverable

rhyme or reason about them. We know that Nature behaves thus and thus; we do not know why

she does and can see no reason why she should not do the opposite. (2) That they are applica-

tions of the law of averages. The foundations of Nature are in the random and lawless. But the

number of units we are dealing with are so enormous that the behavior of these crowds (like the

As the laws of nature follow inductively from the observation of regularity, it remains a

possibility that the laws could be violated from the outside. In fact, Lewis claims that none of

the three theories prevents the Supernatural from invading Nature. The first two theories are eas-

ily addressed as the first gives no rhyme or reason why things are as we observe and thus no rea-

son why they should continue in the same pattern, and the second, which depends on the law of

averages, will work only for undoctored Nature and the question of whether or not miracles oc-

cur is precisely the question of whether Nature is ever doctored. As for those who hold to the

third theory, Lewis claims that even this theory does not prevent the Supernatural from invading

Nature: ―If the laws of Nature are necessary truths, no miracle can break them: but no miracle

needs to break them. It is with them as with the laws of arithmetic. If I put six pennies into a

of nature have been violated but simply that the antecedent is no longer A but is really A’. In

other words, as long as nothing from outside of nature interferes, one expects the universe to

obey these laws. If, however, something were to interfere, that would not be breaking the laws

of nature, as those laws were never meant to account for such things.

What is the relationship between Reason (which follows from the laws of thought) and

Nature (which demonstrates its own laws)? Lewis describes the connection by appealing to the

mathematical idea of a relation that is ―unsymmetrical‖. (Lewis, Miracles 39) A relation is

simply a set of ordered pairs that is a model for almost any type of association between objects

(people, animals, things, etc.). For example, suppose that Joe and Sue are siblings with common

tion; however, (Bill, Joe) and (Bill, Sue) would be. Notice that neither (Joe, Bill) nor (Sue, Bill)

would be in this second relation as neither Joe nor Sue is the father of Bill; thus the relation ―be-

ing a parent of‖ fails to have symmetry. Lewis claims that an analogous asymmetrical relation-

ship exists between Reason and Nature. Reason can act upon Nature to change it, but the reverse

is not possible. For example, Reason can alter physical nature through the use of mathematics

(e.g. bridges, air conditioning, engineering) and can alter psychological nature through argu-

ments applied to our emotions. However, Nature has no such claim on Reason. When nature

attempts to interfere with human consciousness, this simply is to produce Nature and to suspend

Reason as ―Nature is quite powerless to produce a rational thought: not that she never modifies

our thinking but that the moment she does so, it ceases (for that very reason) to be rational.‖

(Lewis, Miracles 38)

you still get straight lines, but many lines make one figure. In a three-dimensional world, you

still get figures but many figures make one solid body. In other words, as you advance to more

real and more complicated levels, you do not leave behind you the things you found on the simp-

ler levels: you still have them, but combined in new ways – in ways you could not imagine if

you knew only the simpler levels.‖ (Lewis, Mere Christianity 142) Lewis suggests that Chris-

tians meet difficulties in their Faith that render them in ways like an inhabitant of Flatland trying

to understand a solid object. In particular, Lewis uses the correlation of dimensions as an analo-

gy for the concepts of the Trinity, time and eternity, and temporal versus eternal existence.

The doctrine of the Trinity espouses the triune personality of one Being. Lewis compares

this incomprehensible concept of one Being consisting of three Persons to the geometric fact that

would either imagine the six squares coinciding, and thus destroy their distinctness, or else im-

agine them set out side by side, and thus destroy the unity. Our difficulties about the Trinity are

of much the same kind.‖ (Lewis, Christian Reflections 79-80) In contrast, Lewis comments that

the Pantheist, even though he may claim a super-personal God, in actuality conceives of a sub-

personal God ―as though the Flatlanders thought a cube existed in fewer dimensions than a

square.‖ (Lewis, Miracles 136) Instead of a Being with a real character of its own, his God ―be-

comes simply 'the whole show' looked at in a particular way or the theoretical point at which all

the lines of human aspiration would meet if produced to infinity.‖ (Lewis, Miracles 131)

Lewis proposes that God is not at all in the human timeline. God sits above, beyond in

nity, Lewis remarks, ―If we think of time as a line–which is a good image, because the parts of

time are successive and no two of them can co-exist; i.e., there is no width in time, only length–

we probably ought to think of eternity as a plane or even a solid. Thus the whole reality of a

human being would be represented by a solid figure.‖ (Lewis, The Problem of Pain 125) Eterni-

ty is depicted as at least two-dimensional when compared to one-dimensional time and the totali-

ty of human existence is seen as three-dimensional.

In exploring the relationship between temporal and eternal life, Lewis writes, ―Suppose

that the earthly lives she and I shared for a few years are in reality only the basis for, or prelude

to, or earthly appearance of, two unimaginable, supercosmic, eternal somethings. Those some-

things could be pictured as spheres or globes. Where the plane of Nature cuts through them–that

is, in earthly life–they appear as two circles (circles are slices of spheres). Two circles that

Moreover, in the essay ―Transposition‖, Lewis puts forward the juxtaposition of a richer

system to a poorer system to further explain the relationship between the spiritual life and the

natural life. Lewis gives an example of the richer and poorer that is readily experienced, namely

emotions and sensations. The emotional life is ―richer‖ than the life of sensations because hu-

man nerves produce the same sensation to express more than one emotion. For instance, both

joy and sorrow often yield tears. It is impossible to find a one-to-one correspondence between

such systems and ―the transposition of the richer system into poorer must, so to speak, be alge-

braical, not arithmetical.‖ (Lewis, The Weight of Glory 77) The most famous example, claims

Lewis, is from the art of drawing. ―The problem here is to represent a three-dimensional world

on a flat sheet of paper. The solution is perspective, and perspective means that we must give

notion of Transposition from above ―as we all do in the case of emotion and sensation or of the

three-dimensional world and pictures, and as the spiritual man does‖ (Lewis, The Weight of

Glory 81-82) otherwise one will reach incorrect conclusions. For without Transposition, the nat-

ural life will appear to be all there is. ―The brutal man never can by analysis find anything but

lust in love; the Flatlander never can find anything but flat shapes in a picture; physiology never

can find anything in thought except twitching of the grey matter. It is no good browbeating the

critic who approaches Transposition from below.‖ (Lewis, The Weight of Glory 81)

Lewis claims the principle of Transposition might also enlighten the doctrine of the In-

plane figures no truths of solid geometry.‖ (Lewis, Miracles 178) Once again Lewis uses the

concept of dimensionality to elucidate his ideas. In this analogy, the Divine Incarnation is as a

proposition in solid geometry that generalizes this truth in plane geometry – humans exist as

composite moral rational creatures, purely natural in many ways but nonetheless more than just

natural beings. Conversely, just as no truths of solid geometry are revealed by plane figures,

there remain facts beyond human comprehension: ―I do not think anything we do will enable us

to imagine the mode of consciousness of the incarnate God. That is where the doctrine is not

fully comprehensible.‖ (Lewis, Miracles 177)

Furthermore, Lewis offers that the principle of Transposition might illuminate the doc-

trine of the resurrection of the body. Lewis contends that the New Nature that is being created

through the Son, is interlocked in ways with the Old Nature, in a manner similar to the way that

obey us as our cortex now does.‖ (Lewis, Miracles 250) With the resurrection of Christ, ―a

wholly new mode of being has arisen in the universe,‖ (Lewis, Miracles 241) says Lewis, a body

that belongs to the category of New Nature and that ―is differently related to space and probably

time, but by no means cut off from all relation to them.‖ (Lewis, Miracles 241) As for the com-

plete expression of redeemed humanity, Lewis proposes, ―It is like when you throw a stone into a

pool, and the concentric waves spread out further and further. Who knows where it will end?‖

(Lewis, The Great Divorce 106)

The two categories, namely the relationship between mathematics and specific laws and

secondly the employment of geometry and dimension, have been thoroughly examined. Through

comparison and contrast, analogy and illustration, simile and metaphor, concepts and terminolo-

Lewis could not tame the lion mathematics, he was able to appreciate and articulate the beauty

and power of the discipline he never mastered, and that is true genius.