Einstein wondered why is it that we can make sense of the universe. This is a question that today’s guest author, Jennifer Siggers, has also asked. Jennifer is a mathematician based at Imperial College London who applies her skills to biological problems. She is also a Christian, and her faith leads her to ask what mathematics can reveal about God. Whether you appreciate the power of numbers or not, it is fascinating to see where this line of thinking can lead. To find out more about Jennifer’s faith and work, see God in the Lab: How Science Enhances Faith.
Jennifer Siggers
Numbers have fascinated me since I can remember.
We all discover beauty in different things in life, according to our personalities. Many find beauty in music, art and scenery – and some find it in abstract mathematical phenomena and how they link to real life. I think the emotional response to these different types of beauty is similar. From an early age, I began to discover beauty in mathematics.
At primary school I was fascinated by exact powers of 2: 1, 2, 4, 8, 16, 32 and so on. These numbers have several interesting properties. For example, you can prove that 2^(2n) – 1 is divisible by 3, whereas 2^(2n – 1) – 1 is not.
In sixth form, I developed an interest in complex numbers, which has remained with me. De Moivre’s theorem states that e^(iθ) = cos θ+ i sin θ relates the trigonometric cosine and sine functions to the exponential, which I found difficult to get my head around but curiously appealing.
Later at university, I learned that if you throw a spinning cuboid box and it is rotating around its longest or its shortest axis, it will rotate stably. If it is rotating around the intermediate axis the rotation is unstable. What I liked was that this fact could be both proved mathematically, and also demonstrated simply and convincingly in front of our very eyes using the nearest convenient cuboid (which happened to be a textbook)!
Nowadays, I love the way that even simplified mathematical models done on the back of an envelope can give us insights into the mechanisms underpinning heart disease, sight loss and the like.
My love of mathematics has led me to wonder why these things work out in such a beautiful way. Why are relationships between numbers full of patterns like this? Why can we have a hope of finding them?
The Bible tells us that before God created, ‘the earth was formless and empty’ (Genesis 1:2). So Christians understand that everything (including the laws of nature and mathematics) was created by God in its entirety. Later in Chapter 1 of Genesis we read that everything in God’s original creation was good.
Since we are all created with different personalities, we have our own unique tastes. We are also created in the image of God, and God’s creation is good, so it’s not surprising that we find aspects of his creation beautiful. Indeed we are told in Ecclesiastes 3:11 that ‘[God] has made everything beautiful in its time.’
But what does this tell us about God, and what should we be doing about it? Let me highlight three things.
First of all, seeing this sort of beauty reminds Christians that God is creator. When we see something beautiful, we know that it is only like that because that is how God created it. It is a wonderful method by which God helps us to remember him in our daily lives – not by criticising us as we so often forget him, but by gently showing us something of himself in a way that gives us great enjoyment, as well as pointing to him.
Secondly, it gives us insight into God’s character. Just as listening to a great piece of music reminds us that God is the ultimate composer and musician, so understanding a piece of abstract mathematics shows us that God values order. He makes things work by setting up natural laws that only he can break (since he created them!), and we can learn more about him and get to know him better as we study these amazing phenomena.
Thirdly, the fact that God’s creation displays many examples of beauty points others to the existence of God. As Psalm 19:1 says, ‘the heavens declare the glory of God’, and in fact all of God’s creation declares God’s glory. ‘Declare’ is a strong word to use, implying as it does that God’s existence and purpose is being shouted from the rooftops for all to see. Romans 1:20 states this even more strongly: ‘For since the creation of the world God’s invisible qualities – his eternal power and divine nature – have been clearly seen, being understood from what has been made, so that people are without excuse.’
For Christians, therefore, the beauty of the universe can be used to show something about God’s character. If a friend tells me that he or she loves something about the universe, I might say something like, ‘Yes, that is awesome, and for me it is a reminder of the way God works in the world, that he created an orderly universe and loves beautiful things’.
We have an awesome, fantastic God who is more wonderful than we can imagine and who creates amazing things that we enjoy. Let’s make the most of them and give him glory!
It has always impressed me that the laws of Nature (that which God has designed into its fabric) are so elegant and parsimonious. Why are symmetry and conserved dynamical quantities so tightly coupled? Because Nature 'understands' (permit me the anthropomorphism) Hamilton's principle of Least Action (leading to the Euler-Lagrange Equations) and Noether's Theorem (which is essentially Lie group algebra applied to functionals). All of classical mechanics follows from these two principles, with a classical Lagrangian chosen to reproduce Newton's 2nd Law. In a sense, Hamilton's principle is more fundamental than Newton's.
Now add in Lorentz Covariance ( Special Relativity). Starting from the Lagrangian for a relativistic free particle, consider how to add in a potential term in a fully Lorentz covariant way. It turns out that there are only two ways to do it.
Add the potential term outside the square root, and electromagnetism emerges. Extend that with first and second quantization, and QED emerges, and more generally, Quantum Field Theory. Add the potential term inside the square root - this takes you down the path to General Relativity, as the only way to do this is to modify the metric tensor. (For a full treatment, pick up a copy of Anton Zee's "Einstein Gravity in a Nutshell").
Now, historically, classical mechanics and then electromagnetism came first, then Special Relativity, then General Relativity, and traditionally that is the way we teach it to students. But turn it around, and one sees how the laws of Nature are 'derivable' from such simple and elegant principles. Where did those basic principles come from? How did Nature come to know them? Perhaps there is yet a more fundamental set of principles that we don't know about yet, but then the same question arises.
Next time a skeptical scientific colleague asks for something that supports Psalm 19 or Romans 1:20, point them to this, and ask them to explain where the principles came from. Ask them if they now see why Christians maintain that our faith is based on reasonable, rational inferences from what we know.
Thank you for this very thoughtful comment and a good recommendation for teachers to "turn it around, and one sees how the laws of Nature are 'derivable' from such simple and elegant principles." That is elegant! :)
Certainly as physics teachers, teaching an advanced undergraduate course for physics students, the opportunity to do this is one that should not be passed up. When I came across this in Zee's textbook, following along with the math, seeing the tensorial definition of the EM field tensor just pop out from the variational principle was breathtaking. It really is a profound result, and I immediately thought of Psalm 19.
It's too bad that explaining this to our mathematically challenged (and I don't say that in a pejorative way) brothers and sisters in Christ is so difficult.
Jennifer -- You have such interesting and profound comments here. Would you consider becoming a member of ASA and our CWIS affiliate? We would love to have you formally join us! Your $85/yr dues is only $7/month, and goes a long way to help sustain ASA financially.
Victoria -- I see that you are already an ASA member; thanks so much!
Jennifer AND Victoria -- Would either or both of you be interested in presenting on this topic at the Azusa, CA conference in July 2016? That would be fascinating. Please consider offering an abstract by Feb 15; further info is on the ASA home page or at http://c.ymcdn.com/sites/network.asa3.org/resource/resmgr/AnnMtg2016/2016CallForAbstracts.pdf . Thanks for considering!
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Posted Saturday, December 19, 2015