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Mathematics and the beauty of God

David Watts david.watts@manchester.ac.uk The University of Manchester

What is our aim ?

Emphatically not (for example) to presume to “prove” God mathematically ...

But to illustrate the relevance and plausibility of Christian Trinitarian Theism for the philosophy of mathematics ...

The joys of mathematics…

…lots and lots and lots of maths.

Why address this subject ? ■  A counterblast to the utilitarian view of the University. ■  Mathematics has been viewed wrongly [even by Christians]

as a classic case where Christian belief is irrelevant. ■  ‘Cultural Mathematics’ books are now among the best-sellers. ■  Chaos theory & Fractals have widened the applicability of maths. ■  Mathematical software [Maple, Mathematica, etc] removes

drudgery from study + research, & enhances geometrical insights. ■  People educated to imagine multi-dimensional space

& parallel worlds. ■  New opportunities for communication of Christian perspectives.

“Man’s chief end is to glorify God and to enjoy him for ever”.

1 0 1 1 0 1

1 0 1 1 0 1

1 0 1 1 0 1

1 0 1 1 0 1

1 0 1 1 0 1

1 0 1

1 0 1

1 0 1 1 0 1 1 0 1 1 0 1

1 0 1 1 0 1 1 0 1 1 0 1

1 0 1 1 0 1 1 0 1 1 0 1

1 0 1 1 0 1 1 0 1 1 0 1

1 0 1 1 0 1 1 0 1 1 0 1

1 0 1 1 0 1 1 0 1 1 0 1

1 0 1 1 0 1 1 0 1 1 0 1

1 0 1 1 0 1 1 0 1 1 0 1

1 0 1 1 0 1 1 0 1 1 0 1

1 0 1 1 0 1 1 0 1 1 0 1

1 0 1 1 0 1 1 0 1 1 0 1

1 0 1 1 0 1 1 0 1 1 0 1

1 0 1 1 0 1 1 0 1 1 0 1

1 0 1 1 0 1 1 0 1 1 0 1

1 0 1 1 0 1 1 0 1 1 0 1

1 0 1 1 0 1 1 0 1 1 0 1

1 0 1 1 0 1 1 0 1 1 0 1

1 0 1 1 0 1 1 0 1 1 0 1

1 0 1 1 0 1 1 0 1 1 0 1

1 0 1 1 0 1 1 0 1 1 0 1

1 0 1 1 0 1 1 0 1 1 0 1

1 0 1 1 0 1 1 0 1 1 0 1

1 0 1 1 0 1 1 0 1 1 0 1

1 0 1 1 0 1 1 0 1 1 0 1

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What is mathematics?

... discovery, ‘invention’, exploration & application of abstract patterns ...

NB MacTutor History of Mathematics Archive, at: http://www-history.mcs.st-andrews.ac.uk/history/index.html

Exploration & imaginative reconstruction of the numerical, spatial, & cognate domains of created reality –

seen in its order, symmetry, variety of pattern & change.

The logical-mathematical proof of theorems remains vital.

or expressing it

more theologically

Mathematics: discovery versus original invention

Where do the structures of maths come from?

Are they merely the creation of the human mind, imposing its categories upon a formless world? or do mathematical structures have a prior existence?

The former view is highly influential and is particularly associated with the Enlightenment philosophy of Immanuel Kant, whereas the latter view is often traced back to Plato, and his theory of forms.

This is a serious crime if one believes that our mathematical theories are merely elaborate mental constructs, precariously hoisted aloft. Then rigour becomes the nerve-racking balancing act that prevents the entire structure from crashing down around us.

Tristan Needham, Visual Complex Analysis. (1997) OUP. p. xi

But suppose one believes, as I do, that our mathematical theories are attempting to capture aspects of a robust Platonic world that is not of our making.

“My book will no doubt be flawed in many ways of which I am not yet aware, but there is one "sin" that I have intentionally committed, and for which I shall not repent: many of the arguments are not rigorous, at least as they stand.

I would then contend that an initial lack of rigour is a small price to pay if it allows the reader to see into this world more directly and pleasurably than would otherwise be possible”. The story of a mathematician - John Forbes Nash Jr.

What do biblical writers find beautiful? ■ Crowns – Is. 28:5; 62:3; Ezek. 16:12; 23:42. ■ Garments – Josh 7:21; Is. 52:1 ■ Ornaments – Ezek 7:20 ■ Voices – Ezek 33:32 ■ Houses – Isaiah 5: 9 ■ A city – Lam 2: 15 ■ The elevation of a mountain – Ps. 48:2

■ ... God “has made everything beautiful in its time” (Eccles 3: 11 RSV)

just a few examples...

What is beauty?

■  An aesthetic quality; names what we find attractive, satisfying and excellent in an object or person.

■  With visual art – perceived through senses. ■  With literature – perceived by mind + imagination. ■  Ingredients of (artistic) beauty :

◆ Unity, balance, symmetry, harmony of parts ◆ or not subject to such analysis - so for the biblical writers - ◆ But - used by extension to indicate a generalised

positive response to someone or something. ◆ Includes inner beauty of character or personality.

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‘Beauty’ as applied to God:

■ A more metaphorical usage ... ■  Positive qualities of artistic beauty provide language ◆ for identifying the perfection of God ◆ & the pleasure the believer finds in that perfection,

as a spiritual response. ■ But beauty does not define God; God defines beauty.

Moses in Psalm 90: 17 utters a prayer that the divine beauty be communicated to God’s covenant people:

“May the beauty of the Lord rest upon us”.

God’s beauty is closely associated with his glory ...

“In that day the Lord of hosts will be a crown of glory and a diadem of beauty to the remnant of his people”.

(Isaiah 28:5)

In the NT [Hebrews 1: 2f]

It is a quality of Jesus Christ, the radiance of God’s glory.

Beauty was apparent for OT Israel in the Tabernacle & Temple – next to nature, their most vivid experience of aesthetic beauty.

“One thing I ask of the Lord, this is what I seek: that I may dwell in the house of the Lord

all the days of my life, to gaze upon the beauty of the Lord

and to seek him in his temple”. [Psalm 27:4]

In this view of the Sanctuary in the Second Temple, we can see the three major vessels, the Menorah, the Table of the Showbread and the Golden Incense Altar.

A priest can be seen tending to the lamps of the Menorah, and the High Priest kneels to prostrate himself towards the Holy of Holies.

The high priest – beautiful in his ‘uniform’ – was a microcosm of the Temple

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The jewelled breastplate of the high priest represented the 12 tribes of Israel ...

This imagery is developed further in the NT vision of the City of God ...

... and suggests one link between divine beauty and mathematics ...

10And he carried me away in the Spirit to a mountain great and high, and showed me the holy city Jerusalem, coming down out of heaven from God, 11 having the glory of God: her light was like unto a stone most precious, as it were a jasper stone, clear as crystal:

Revelation 21:10ff

12 having a wall great and high; having twelve gates, and at the gates twelve angels; and names written thereon, which are [the names] of the twelve tribes of the children of Israel:

A crystal is a suitable object to illustrate divine beauty ...

Inference : In what does the beauty of a crystal consist?

... unity, balance, symmetry, harmony of parts ...

... but mathematical Group Theory underlies the specific beauty [symmetry] of crystals ...

M

k  Z

Γ

Γ

X

K

K WLU

Xk  y

M

Γ

k  X

It permits us to determine theoretically the total set of possible unit cells & lattice types for crystals

Continuous Symmetry of a perfect sphere: Kh

NB symmetry and broken symmetry

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Rotational subgroup: I 2 3 4 2

5 5 5 5 3 3 2{ ,6 ,6 ,6 ,6 ,10 ,10 ,15 }E C C C C C C C

A virus crystal

Dodecahedra: Ih

2 3 4 2 3 7 9 55 5 5 5 3 3 2 10 10 10 10 6 6{ ,6 ,6 ,6 ,6 ,10 ,10 ,15 , ,6 ,6 ,6 ,6 ,10 ,10 ,15 }E C C C C C C C i S S S S S S σ

Symmetry in physics & mathematics ■  In physics, symmetry generates considerable interest.

New impetus through “Supersymmetry” & “Superstring” revolutions [1&2].

■  Symmetry discernable even amidst change. [Barrow, Theories of Everything, p. 20f]

■  2 complementary approaches: ◆  a priori: Foundational Symmetry Principles

[eg that nature is symmetrical] implies new laws of nature. ◆  a posteriori: derives symmetry principles empirically from phenomena.

■  Their interaction seen (eg) ◆  in interpreting Lorentz transformations. ◆  supersymmetry theories of Fermions.

■  Some symmetries (initially?) appear accidental. eg. Fock symmetries of an electron in a Coulomb potential; (Atkins & Friedman MQM p.58, 95, 122-163).

■  Pierre Curie’s Principle (1894): “An effect cannot posses a greater symmetry than its cause”. But symmetry ‘breaking’ shows it is not always transmitted from cause to effect.

Groups with very high symmetry: multiple high fold rotation axes The platonic solids: polyhedra constructed from regular polygons with all vertices and edges equivalent: 5 possibilities only.

icosahedron

Pythagoras and mathematical harmony:

1. ‘harmony of the spheres’

“only 5 regular convex solids can be circumscribed by a sphere”.

A

P

Q

B

To the Pythagoreans, the pentagon was a symbol of the universe

The point of intersection P or Q of two diagonals divides each in its ‘golden ratio’ or ‘section’ or mean or ‘divine proportion’

[f]

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2. The ‘golden section’ l

a b

l aa b

φ = =

[f]

2.l b a=

Hence, if a = 1 and l = x 11

x xx+ =

2 1 0x x− − =

the negative root 1 5 0.61803'2

φ −= = −

the positive root 1 52

1.61803x φ += =≡

. ' 1φ φ = −

Consider a line of length, l

l l aa l

+=

generalising

f is “the most irrational number” ie. the least well approximated by rationals …

11 11 11 111 ....

φ = ++

++

+

Recursive (fractal) definition:

11

where11

X

XX

φ = +

= +

Unexpected encounters with the golden mean

1. Consider a sequence of integers formed according to the following rule

1,3,4,7,11,18,....eg.

eg. 3, 4, 1, 5, 6, 11, 17, 28, 45− +

1Lim( ) 1.618...n nu u φ+ = =n→∞

2. The simplest such sequence is the Fibonacci Series 0, 1, 1, 2, 3, 5, 8, 13, 21, ...

Then take the ratio of consecutive terms

... a totally unexpected appearance of f in a non-geometrical context!

1 2n n nu u u− −= + 1 1 1

1 2 1 1 1 3 3

1 1 4 4 6

1 1 5 5 10 10

1 1 6 6 15 15 20

1 1 7 7 21 21 35 35

Pascal’s Triangle – contains the Fibonacci series

p.131f

Ubiquitous in statistics and probability theory ...

Logarithmic spiral – produced by a Fibonacci series

. . Cotr a eϑ α= ( 2) Cote π αφ =

p.172

galaxy M51 30 million light-year distant

Hurricane Isabel

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Ingredients of mathematical beauty & pleasure

■ The alternation of tension & relief [perplexity & illumination].

■ The realisation of expectation. ■  Surprise at the unexpected. ■ The perception of unexpected relationships. ■ Enjoyment of patterns. ■  “brevity is the soul of wit”.

[eg. Fermat’s last theorem; Goldbach’s conjecture].

p. 81f

... also applicable to music ...

The golden mean and music Simple musical intervals can stimulate some emotional response in the hearer with negligible training.

This the ‘tip of the iceberg’ in the numerous inter-relationships between mathematical patterns and music.

p.173f

Creation & beauty in nature

■ The beauty of God is an important theme in the Psalms, & in many Christian writings, eg of Jonathan Edwards (1703-58) & H.U. von Balthasar (1905-88).

■ Beauty in pure mathematics has long been appreciated. ◆  fractals have led to a renewed popular interest.

■ This interest is significant also for natural sciences. ■ Beauty of theories often derives from their symmetry.

Mathematical gifts have been widely distributed among the peoples of earth. And some Islamic, Jewish & Christian mathematicians have made highly significant contributions ...

Some noted Christian mathematicians of 16th -19th C

■ Galileo Galilei (1564 – 1642) ■  Johannes Kepler (1571 – 1630) ■ Blaise Pascal (1623 -1662) ■ Leonhard Euler (1707 – 1783) ■ Augustin Louis Cauchy (1789 – 1857) ■ George Boole (1815 – 1864) ■ George Gabriel Stokes (1819 – 1903) ■ Georg Friedrich Bernhard Riemann (1826 – 1866)

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“I render infinite thanks to God for being so kind as to make me alone the first observer of marvels kept hidden in obscurity for all previous centuries”.

Galileo Galilei, 1609 AD William Blake (1757-1827)

To see a World in a Grain of Sand And a Heaven in A Wild Flower Hold Infinity in the palm of your hand And Eternity in an hour ...

. 1 0ieπ + =

Euler’s formula – one of the most elegant equations in mathematics.

•  It connects the five fundamental constants of mathematics (e, p, i, 0, 1). [Imaginary number i is the square root of -1].

•  It includes the three most important mathematical operations (addition, multiplication and exponentiation).

•  It includes the constants symbolizing the four major branches of mathematics – arithmetic (0,1), algebra (i), geometry (p) and analysis (e).

The number e is both irrational and transcendental. The irrationality of e was proved by Euler in 1757. Charles Hermite proved it is transcendental. If a number is said to be transcendental, it means that it cannot be a solution of a polynomial equation with integer solutions.

What type of number is e?

Albert Einstein said "It is the greatest mathematical discovery of all time".

"The natural exponential function is identical with its derivative. This is the source of all the properties of the exponential function and the basic reason for it importance in applications”.

e can also be defined as the function that equals its own derivative.

Cauchy’s residue theorem for contour integration in the complex plane ...

i

-i

1 -1

Augustin Cauchy (1789-1857)

f (z)dz = 2π .i. Res

j=1

n

∑C!∫ ( f ,a)

“I remember clearly the day in the autumn of 1965, during my Complex Variables class as a senior at M.I.T., when I learned that the result of contour integration was two pi i times the sum of the residues.

For me, it was about as close to a revelation as I had received up to that time in my life”.

12 . . Res( , )

n

jji f aπ

=

= ∑

Henry Fritz Shaefer III Director, Center for Computational Quantum Chemistry, The University of Georgia

f (z)dzC!∫

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The beauty of theories often associated with their symmetry:

■ Maxwell’s equations: electromagnetism. ■  Schrödinger’s equation: quantum mechanics. ■ Balmer’s equation: Interpretation of atomic spectra. ■ Einstein’s general theory of relativity. ■ Dirac’s approach to relativistic quantum mechanics. ■ Yang-Mills equation: SU(2) gauge symmetry of isospin. ■ Higgs field equation: symmetry breaking ■ The logistic map: chaotic dynamics.

Bernhard Riemann (1826 – 1866)

2ds g dx dxµν µ ν=

Spatial distances are defined by a metric (a generalisation of Pythagoras):

From the Riemann curvature tensor of 4-D space-time

Lawrie p.36

... his mathematics was indispensable to Einstein

Emmy Noether’s Theorem (1918)

“For every continuous symmetry there is a corresponding conserved quantity; [such as electric charge]

and vice versa”.

So a broken symmetry should produce a broken conservation law.

1882 - 1935

This led to the appreciation of gauge (phase) symmetry

,, ,

4 0r rrd Xdx

dxν σρ σ σ ρν

ρ ν ρ ν

η δη η

ε ⎧ ⎫⎛ ⎞∂ ∂⎪ ⎪− − Ψ =⎜ ⎟⎨ ⎬⎜ ⎟∂ ∂⎪ ⎪⎝ ⎠⎩ ⎭∫ L LL

“I had the feeling that, through the surface of atomic phenomena, I was looking at a strangely beautiful interior, and felt almost giddy at the thought that I now had to probe this wealth of mathematical structures nature had so generously spread out before me. …

Werner Heisenberg, 1971. Physics and beyond.

[ ], .x x xx p xp p x i= − = h

... despite a rather materialist outlook, reported ...

... If nature leads to mathematical forms of great simplicity and beauty – coherent systems of hypotheses, axioms, etc - …we cannot help thinking they are “true”, that they reveal genuine features of beauty”.

( ) ( 0)A xi e mx

xµ µµγ ψ⎡ ∂ ⎤⎛ ⎞− + =⎜ ⎟⎢ ⎥∂⎝ ⎠⎣ ⎦

The Dirac Equation of Relativistic Quantum Field Theory

is the wave function( ) xψwith 4 components ( ), ( ), ( ), ( )e e p px x x xψ ψ ψ ψ↑ ↓ ↑ ↓

A(x) fields, with various subscripts, are the electromagnetic potentials

0

1 0 0 00 1 0 00 0 1 00 0 0 1

γ

⎛ ⎞⎜ ⎟⎜ ⎟=⎜ ⎟−⎜ ⎟⎜ ⎟−⎝ ⎠

1

0 0 0 10 0 1 00 1 0 01 0 0 0

γ

−⎛ ⎞⎜ ⎟−⎜ ⎟=⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

2

0 0 00 0 00 0 0

0 0 0

ii

ii

γ

⎛ ⎞⎜ ⎟−⎜ ⎟=⎜ ⎟−⎜ ⎟⎜ ⎟⎝ ⎠

3

0 0 1 00 0 0 11 0 0 00 1 0 0

γ

−⎛ ⎞⎜ ⎟⎜ ⎟=⎜ ⎟⎜ ⎟⎜ ⎟−⎝ ⎠

gµ are rotation-matrices for the four space-time coordinates

Inscribed within Westminster Abbey ...

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0 0 1 3 1 3 2 2

0 0 2 2 1 3 1 3

1 3 1 3 3 2 0 0

2 2 1 3 1 3 0 0

( )i 0 ( ) ( )( )0 i ( ) ( )

0( )( ) ( ) -i 0( ) ( ) 0 -i ( )

e

e

p

p

xeA m i e A A ieAxeA m ieA i e A Axi e A A ieA eA m

ieA i e A A eA m x

ψψψ

ψ

↑

↓

↑

↓

⎛ ⎞∂ − + − ∂ + ∂ + + ∂ +⎛ ⎞⎜ ⎟⎜ ⎟∂ − + −∂ − − ∂ −∂ + − ⎜ ⎟⎜ ⎟ =⎜ ⎟⎜ ⎟∂ − ∂ − − −∂ − ∂ − +⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟∂ + ∂ − ∂ − − ∂ + +⎝ ⎠⎝ ⎠

Paul Dirac (1902 – 1984) ... despite his militant atheism, Dirac used ‘beauty’ as a key criterion in developing his remarkable equation for the electron and anti-electron - (thus predicting the existence of the positron; which subsequently led to the experimental discovery of various forms of anti-matter).

equivalent form ...

"Even if there is only one possible unified theory, it is just a set of rules and equations… What is it that breathes fire into the equations and makes a universe for them to describe?"

"Why does the universe go to the bother of existing?"

Stephen Hawking,

A Brief History of Time

“… We have already considered with disfavour the possibility of the universe having been planned by a biologist or an engineer; …

Sir James Jeans (1877 – 1946) The Mysterious Universe (1930)

…from the intrinsic evidence of the creation, the Great Architect of the Universe now begins to appear as a pure mathematician”.

A thousand years of theological disputes nurtured the habit of analytical thinking that could be applied to the analysis of natural phenomena.

Freeman Dyson - theoretical physicist - writes ... “Western science grew out of Christian theology. It is probably not an accident that modern science grew explosively in Christian Europe and left the rest of the world behind.

Sir Michael Atiyah

Atiyah’s index theorem “... opened up tunnels between areas of mathematics that did not appear to be related ....”

“It was like archaeologists discovering the same characteristic patterns in tombs from two seemingly unrelated cultures”

Progressive Unity appears in

mathematical research

eg. in development of geometric algebra

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Theological significance of mathematical symmetry ■  There is an immediate affinity between symmetry & order & beauty. ■  Curie’s Principle [CP]:

“(some) symmetries are transmitted from a cause to its effect” This is congruent with Thomistic Natural Theology (13th C).

■  For Aquinas, there exists a “likeness (similitudo) to God” within the created order, due to being caused by God.

■  There are certain physical or metaphysical fingerprints in nature providing the basis for an inductive argument to God’s existence & some aspects of his nature.

■  One fundamental quality of God is “perfection”. According to Aquinas, this is partially reflected in nature. ‘Perfection’ has both moral & aesthetic connotations, including those of symmetry [embodied in both structural features & theoretical constructs].

■  CP suggests a perfect original cause from which all effects derive their perfection.

Although Platonism as a whole is quite distinct from Christian Theism, it embodies insights which - when detached and purged from their original context - are congruent with Christian belief.

■  the logical and /or temporal pre-existence of mathematical forms is acknowledged before their embodiment in the rhythm of the physical universe;

■  they pre-exist in the eternal mind of God.

Returning to the topic of Plato’s mathematical realism ...

■ Thus, human mathematical research is primarily the discovery of mathematical patterns embodied within the universe by God. In the words of Johannes Kepler, it is “thinking God’s thoughts after him”.

■ Even in the most abstract branches of mathematics, as in the elucidation of (say)10 or multi-dimensional space, free creation of such structures by the human mind is not absolute, but only creaturely, and is thus relative to God’s contemplation of all possible worlds, from which he has freely chosen the contingent world(s) that do have existence.

Concluding theological observations

■  as Ecclesiastes 3:11 expresses, there is within the human heart a restless desire for a beauty, which the world itself can never fully satisfy.

■  but Christian faith & life is defined by a tension between the “already” and the “not yet”, so there is a present and partial enjoyment of divine beauty accessible now, which shall be wondrously consummated hereafter in the life everlasting.

■ The provisional and partial nature of our present understanding requires, among Christian scholars, humility and openness-to-new-ideas ...

“For now we see in a mirror dimly, but then face to face.

Now I know in part, then I shall know fully,

even as I have been fully known”.

1 Corinthians 13: 12

■ Thus it is possible to enjoy mathematics now as a good gift of God himself – that clearly manifests the divine beauty.

■ Yet such enjoyment - if not coupled with genuine love and worship of the Giver – will be tainted by idolatry and ultimate frustration.

St. Paul’s critique of pagan society in Romans 1: 20ff remains pertinent to many ‘centres of learning’ today:

“ ...although they knew God, they did not honour him as God or give thanks to him, but they became futile in their thinking ...”

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The incorporation of all God-centred human culture – inclusive of mathematics – into the City of God

is expressed in the Revelation of St John: 21: 24

“... by its light will the nations walk, and the kings of the earth

will bring their glory into it ...”

By contrast ...