Disclaimer: This is not intended to be an attempt to ‘prove God mathematically’, but merely how I see the role and presence of God in something that I spend a lot of my life doing. This is an issue that must be reconciled if I desire to meaningfully say that Jesus IS my life. This is how I, personally, do and see mathematics as a Christian. Again, I do not think or claim that the God revealed through Jesus is the mathematical conclusion of anything here, but rather the consistent fulfilment and personal reality of mathematics in my life.

Abstract: Last Sunday, I had the opportunity and privilege to co-run (with Andi) a seminar for undergraduate mathematics students at church aimed at addressing the issue of “thinking Christianly” and “taking every thought captive for Christ” (2 Cor 10:5) with respect to our studies and in particular, our field of mathematics. A helpful summary of our whole time together can be found on Andi’s blog (https://aqw20.wordpress.com/), but I felt that it might be useful to independently put up the thoughts I had written down in preparation for my section of the seminar. This is my largely unedited (due to time constraints) draft, were I sought to discuss where God specifically ties into and relates to mathematics and why I believe mathematics, contrary to popular belief, is not independent from Him. I will go about this by discussing the origin, limitation, dependence, appreciation, and objective of mathematics. Obviously, this was targeted at university mathematics students, but aside from some of the jargon and examples, it should be easily accessible to anybody interested.

He is the image of the invisible God, the firstborn of all creation. For by him all things (including maths) were created, in heaven and on earth, visible and invisible, whether thrones or dominions or rulers or authorities – all things were created through him and for him. And he is before all things, and in him all things hold together (including maths)And he is the head of the body, the church. He is the beginning, the firstborn from the dead, that in everything (including maths) he might be preeminent. For in him all the fullness of God was pleased to dwell, and through him to reconcile to himself all things (including maths), whether on earth or in heaven, making peace by the blood of his cross.” – Colossians 1:15-20 (ESV) (with sacrilegious insertions by yours truly) .

Origin: In some fields (e.g. biology, cosmology, philosophy) the question of origin plays a huge role. Mathematics tends to be, in general, rather unconcerned about this. Mathematicians are more focussed about the mathematical world, and I would say that mathematics comes, in some sense, from the mind. I am not talking about the question ‘Is mathematics discovered or invented?’, but rather that definition, deduction, abstraction, proof, and even description of observations happen in our minds. This points us to the true origin of the mind and rationality, and in particular, the ‘origin’ of maths. There is no reason we should be able to reason, let alone do maths or even hold this discussion right now. Why should our minds be trustworthy? Why should our minds seek to serve truth and not only survival or nothing at all? Rationality and logic cannot have meaningful existence unless we have an absolute rationality. It can only make sense with respect to truth with a capital T.

Limitation: On a more pure side, mathematics is not ‘complete’. We cannot reduce everything into a self-contained system or description. In fact, mathematics may be the only field that proves its own limitedness. You may have heard of Gödel and his incompleteness theorems (a course in Part C at Oxford if you’re interested) upsetting Hilbert’s program (an attempt to completely describe mathematics).

“Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete, i.e., there are statements in the language of F which can neither be proved or disproved in F.” – Raatikainen (on Gödel’s First Theorem).

Essentially, mathematics cannot be complete – there will always be things you cannot know in your system. If you add these things to your set of assumptions, something else will arise that cannot be known.

Speaking of assumptions, we are forced to do maths with and under assumptions (or axioms), e.g., commonly Zermelo-Fraenkel + Choice. Assumptions, by definition, are not proven, though we try to justify them (by abstraction or generalisation). We will discuss this a bit more in the next part.

To the mathematician, the lack of completion is deeply unsettling. Imagine if we could not complete \mathbb{Q} to \mathbb{R}! Herein lies a need for a completeness, or rather, THE Completion of everything.

Remark: I would like to remark that as mathematicians, we are ‘used to’ the idea that most things (i.e. anything based on empirical evidence) cannot be proven in the same ‘absolute’ way as in mathematics. Hence, if we are to have any life at all that is not under a crushing depression of uncertainty with respect to love, relationships, history, arts, and faith e.t.c. we learn that things may and can be found true through many different ways. No true Christian claims that the God revealed through Jesus can be mathematically proven, but we can learn to accept this through other evidence and experience, and thus find contentment in this completion.

Note that this does not mean we are to give up the pursuit of knowledge or the seeking of answers. A rather crude example might be (with the roles of proof/evidence reversed) that you may be certain and content that a solution exists for a certain PDE via functional analytic or variational methods, and yet still meaningfully pursue an explicit solution or description via separation of variables or numerical analysis.

Dependence: On a more applied side, mathematics attempts to describe the world (e.g. physics, economics). Often concepts in pure maths arise out of generalisation and abstraction of the natural, observable world. For example:

  • Numbers (1,2,3 apples) -> Number Theory -> Algebraic Geometry
  • Addition (1 apple and 1 apple gives 2 apples) -> Ring and Group Theory -> Homology and Category Theory
  • Shapes (shape of an apple) -> Geometry -> Topology
  • Continuity in time (ignoring physical uncertainty) -> Calculus -> Analysis

The basis in the real world is often what provides ‘justified assumptions’ or ‘reasonable definitions’. But all this (pure indirectly and applied directly) rests on the large assumption that the universe is consistent and not subject (in general) to randomness and chaos, at least, not to a level that renders attempts to define and describe it completely useless. There is an intrinsic consistency of nature and mathematical description that depends on an Unknown. Though of course, as Christians, we believe that the would-be Unknown has made himself Known and that the consistency of the universe is inherently tied up in and founded upon this unchanging Foundation.

Appreciation: There is, without a doubt, beauty in mathematics. This drives many of the world’s past and present mathematicians (myself included). There is beauty in efficiency (e.g. E=MC^2, e^i\pi +1 = 0), elegance (e.g. geometric proof of Pythagoras’ Theorem), ‘coincidence’ (Galois correspondence and the insolvability of the quintics), usefulness (number theory and cryptography), and ingenuity (personal favourites as an undergrad – the proofs of Urysohn’s Lemma and Cauchy’s Theorem on triangles in Rudin).

The recognition of beauty is effectively an appreciation of goodness abstracted, which points to that which is Utterly Good and Infinitely Glorious – Supreme Beauty himself.

Objective: Perhaps we can summarise these things by looking at the objective of mathematics. Loosely defined, the goal of mathematics is to provide a ‘true’ (rational, consistent, finite) description of both abstract and tangible realities. The process includes motivations/generalisations, definitions/rules, guesses/proofs, and to a mathematician, the presentation, satisfaction, and admiration of veracious beauty.

Note: I am aware that the rest of this post below is by no means an immediate logical conclusion. As I have said before, a ‘proof’ of the Christian God is not what I am trying to present, but merely my own personal reflection, and one that I think other Christians mathematicians would and should echo. (Don’t forget, this was for a Christian seminar.)

This all comes together, or is “subversively fulfilled” in the God revealed by Jesus in the Gospel – the beautiful Source and Truth on which all of mathematics is founded and completed.

(I forgot to include the following part in the actual seminar even though I wrote it down previously, woops!)

Conclusion: We have seen that mathematics is impossible without God. This means that doing mathematics without acknowledging him amounts to “cosmic theft” from God: knowing Him but not honouring Him, taking His gifts and rejecting the Giver, worshipping and serving the created things rather than the Creator (Romans 1:21-22,25).

May this, then, be our prayer: “Heavenly Father, we thank you that we can come before your Genius despite our limitations, and through Jesus, we have access to the Creator of the Universe, and in particular, mathematics. We ask that you reveal yourself in our studies and research, as we “think Your thoughts after You” (Kepler). We pray this in the name of Jesus and for His sake and glory. Amen!”