Bonhoeffer and Mathematics

This coming Spring, Wheaton College will sponsor a Theology Conference based on the witness and ethics of Dietrich Bonhoeffer.  We all know Dietrich Bonhoeffer for his work with the Confessing Church and the Nazi Resistance movement, his involvement in which led to his execution at the Flössenburg Concentration Camp only 3 weeks before the end of the war.  Unfortunately, his death left him theologically enigmatic.  That conservative evangelicals and the progressive liberals can both claim with veracity Bonhoeffer’s identification with their end of the theological spectrum is, in my opinion, evidence of that.  His actions were also rather contradictory at times.  His dedication to pacifism came from his friendship with Jean Lasserre, who was a French reformed pastor in New York studying at Union Theological Seminary.  Yet he was tied to the plot to assassinate Hitler.  During his postdoctorate year at Union, Bonhoeffer was an ardent opponent of the social gospel… but social ethics were very much important to Bonhoeffer, made even more so during his years in prison, and even hints of liberation theology were beginning to emerge from his later letters from while in prison.  Gustavo Gutiérrez drank a few cups from the well of Bonhoeffer.  Suffice it to say, Bonhoeffer is difficult to theologically pin down, and I do not think it is helpful to try to claim him as an evangelical or a progressive liberal.  One thing for sure is that Bonhoeffer was a martyr, and I do not think any evangelicals and liberals have any right to claim him for themselves unless they are willing to follow in his example.  In this day and age, nobody from the West can really do that.

Bonhoeffer, it should be noted, began his theological ministry in the academy at the University of Berlin, then one of the top schools in the world.  Back then, everyone wanted to go to Germany to study.  It was only due to the Nazis that a lot of the brilliant scholars moved to the United States and taught at the Ivy League institutions that the center of cutting-edge academic thought shifted in favor of the United States.  Needless to say, Bonhoeffer was rubbing elbows with the best and brightest professors in theology, and walked the same halls as famous academic figures in many fields.  Perhaps for that reason, in one of his lectures on Genesis, Bonhoeffer made a curious remark about mathematics.  First of all, he talks about how the stars and the galaxies seem to just operate on their own, irrespective of the human condition.  They belong to the world of the fixed – the stars didn’t change when sin entered humanity.

But then…

Human beings participate in the world of what is fixed… for they know numbers.  Humankind in the middle remains a knowledge of numbers, of the unchanging, the fixed, of that which is apparently unaffected by humankind’s fall.

The exposition is not long, but too long for a mere blog post.   The important thing to note is that Bonhoeffer did not view mathematics as inherent truth.  The temptation is to think that because math is undebatable (1+1 = 2, no matter how you see it), it can serve as a foundation for human thought or existence.  Indeed, math has somewhat of an air of eternality and truth to it that seems quite transcendent.  Bonhoeffer argues two points, and the brilliance of it lies in the fact that he was able to bridge somewhat the gap between mathematics and theology such that we can’t see each as mutually exclusive from the other.  

First, the creation narratives suggest that all creation is given not just life, but their existence.  It is easy to think in a deistic fashion that God created and then left the world to take matters into their own hands.  God, rather, upholds creation continually.  We exist because God upholds continually our existence.  Mathematical relationships hold because their relationships are upheld by God.  In other words, F = MA or E = MC2 are true not because we have proven them to be so; we merely discovered that F = MA.  Why must F = MA instead of F = MA-2?  Because those relationships were issued by God.  The power of God is such that if God chooses, F can = MA-2.  Bonhoeffer puts it “…like everything else, [numbers] are God’s creatures and so receive their truth wholly from the Creator.”  In putting it in this fashion, we see the influence of Karl Barth in his work already.

That was the argument from theology.  But secondly, Bonhoeffer notes that even mathematical truths themselves are not inherently consistent.  Bonhoeffer was not a mathematician, nor did he ever claim to be one.  But one of his friends studied mathematics before he switched to theology after hearing Karl Barth lecture. Furthermore, Bonhoeffer was at the University of Berlin where much of the discussions took place regarding the nature of mathematics.  For most of the 19th century, the center of mathematical thought was German, notably at the universities of Berlin and Göttingen, although very respectable faculties exist at other schools.  Göttingen was the home institution of the eminent mathematician Karl Friedrich Gauß.  Berlin was the home institution of Karl Weierstraß, as well as many other scientists who made their marks in their respective fields (Albert Einstein, Wassily Leontief, Ernst Schrödinger, Gustav Hertz, among many others).  Note, as I said before, many of these scientists moved to the United States as a result of WWII where they established their careers, some of them at Harvard and Princeton, which is partly why those schools remain at the forefront of mathematical thought today.  In any case, Bonhoeffer could not have missed the debate at any rate.


The debate in question concerned Georg Cantor's set theory which, itself, was part of a larger debate concerning the epistemology of mathematics.  Where does the number 1, for example, get its "one-ness"?  From whence does 2 derive its "two-ness"?  And how do we know that 1 + 1 = 2? German mathematics has largely built up to this point.  Mathematics since Gottfried Leibniz has progressively become more and more abstract as mathematicians seek to uncover the foundational principles undergirding the mathematics that are used.  Bonhoeffer seemed to suggest in Creation and Fall that such an abstraction not only detracts from the usefulness of mathematics, but also pulls mathematics away from its creaturely status, making it somehow a tempting alternative for God.  Cantor's solution was his set theory and his theory of the transfinite numbers.  It was controversial at the time, drawing criticism from other notable mathematicians such as Leopold Kronecker and Henri Poincaré. Cantor was himself a rather enigmatic figure. But however strong the criticism, Kronecker and Poincaré were unable to disprove his set theory.

Unfortunately, not all is well with Cantor.  Cantor himself found several exceptions to his set theory. It was a British mathematician/philosopher, Bertrand Russell, who disproved Cantor's thesis with a counterexample.  While Russell himself proposed a fix to accommodate his paradox, Kurt Gödel, a logician who had emigrated to Princeton, NJ to work at the Institute of Advanced Studies, came up with his Incompleteness Theorems, essentially showing that any proofs that are arithmetic in nature (such as those concerning Cantor's work) there will be a statement within that is true, but nonetheless unprovable.  In essence, Cantor's thesis cannot be resolved to mathematical satisfaction.  But more importantly, Kurt Gödel's theorems suggest that mathematics itself cannot be completely consistent.  Bonhoeffer would agree, but go further and assert that its consistency and truth-bearing can only be upheld by God alone.


So what?  Well, if mathematics is indeed a creaturely science, then I suggest that it falls under human dominion.  Now, I am not conceding to Dominionism, the idea that the world should ascribe to a fundamentalist theocracy.  Rather, the fact that mathematics is a system of thought which attributes its existence and logic to God, that God commanded humanity to rule over the earth suggests that we are to do the same to mathematics.  What, then, does it meant to "rule over the earth"?  We live in a world that is heavily premised on the use of natural resources in achieving technological improvements (note: this applies to socialist governments as well).  And so, "ruling over the earth" implies efficient use of resources.  There is, in other words, a one-way relationship between humankind and the earth; we decide how to manage the earth for our benefit.  Earth, by the way, doesn't mean natural resources.  It includes all created orders, from trees to technology, from ibexes to iPods.  Thus, to rule over the earth doesn't mean we abstain from eating meat, or cutting down trees at all, or that we stop using our iPads, but that we eat rightly, cut down trees rightly, use our iPads and Facebook accounts rightly.  The key environmentally-charged word is "sustainability".   


Bonhoeffer, on the other hand, sees human dominion over the earth in terms of our freedom from the earth, which allows us to rule over it.  But by ruling over it, we are freed from it - the earth is like a servant to its master - we are free to administrate over it.  This does not mean we chop down all the trees in the world, but that we chop down only what is necessary.  The beauty of paper recycling is that we now can use this technology to be more effective in our administration over the trees (in other words, there will be less trees cut down as a result).  Our freedom from earth doesn't mean we are divorced from the earth, because it is the earth that upholds human flourishing.  But sin messes things up to the point where we are not ruling over the earth, but it's the other way - the earth dominates us.  We no longer know how to use things aright.  We spend our days on Facebook lamenting on how boring our days are.  We let the earth define who we are instead of the other way around.


Now what does all this have to do with mathematics?  What we need to realize is that mathematics, a creaturely system of thought, is a creature of the earth.  As such we are to rule over it, to make use of it rightly.  Now, mathematics is not a resource like trees, and so we have to muse over what mathematics is for.  In a nutshell - mathematics is an objective description of certain aspects or relationships in the world.  The number 1 derives its value from God, but we use it to describe one-ness.  We see a lone individual standing in the middle of a field, and so we say that there is "one" person standing there.  1 describes the individual at that place and time, and the individual, by virtue of being alone in the field, assumes a meaning of "one".  But suppose someone joins him/her.  That individual, assuming he/she allows the new person to join, sheds the identity of "oneness" and assumes one of "twoness".  Thus, we now can say that there are 2 people in the field.  Note, then, that to describe the "joining", we come up with the concept of addition.  So, 1 + 1 = 2 is a symbolic way of telling a story of two identities of "oneness" that join to assume a new identity of "twoness".  This symbolism allows us to apply 1+1=2 to many other situations.  One car + one car = two cars, for example.


The goal of pure (or, abstract) mathematics is to ascertain a mathematical foundation for this symbolism.  Of course, it is important and useful, but at some point, if we go to the heart of trying to mathematically prove the oneness of 1 using mathematical logic, we can only fail because the oneness of 1 is imputed by God.  Mathematically, we cannot prove that one is an identity that is imbued by a divine God.  This is why Leopold Kronecker famously declared that "God created the integers - all else is the work of man."  Cantor has tried to prove the mathematical consistency of his transfinite set theory, and even that has failed.  Interestingly, Cantor had discussions of his theory with Roman Catholic theologians, even at one point writing a letter to Pope Leo himself.  A cardinal responded with some reservations, suspecting that Cantor's theory was tantamount to pantheism.  In the end, and possibly to shore up the legitimacy of his theory, he reconfigured his theory to be a divinely-inspired concept.  God was the mathematician, and he - Cantor - was merely the messenger.

All this sounds abstract, but there is an important application.  In 1947, Paul Samuelson's important  Foundations of Economic Analysis was published.  Borrowing from the mathematical relationships ascertained in thermodynamics, he assumed that all rational human beings act in a self-interested manner, and, thus, act in a utilitarian fashion.  From that, we can ascertain how humans make their choices based on their preferences and any limitations (e.g. how much money they have, etc.)  Now, I do not wish to put down Samuelson's discovery.  It was, and remains, a powerful way of analyzing human behavior because we all do make such utilitarian decisions from time to time.  Whenever we go shopping for our things (e.g. our food), we make utilitarian decisions as to which food to buy, for example, because we buy food for ourselves - no point buying food for decoration, for example.  Analysis of such behavior allows companies to make necessary decisions so wastage is minimized.   But when this utilitarianism is taken to the level of Gary Becker, whose 1976 book on the economics of human behavior earned him the Nobel Prize, it becomes worrying.  Marriage becomes a fundamentally utilitarian task - I choose to marry someone because he/she maximizes my utility.

Cantor's theory falls within the real of pure mathematics, having very little application to non-mathematical fields and yet it is unprovable to mathematical satisfaction.  What makes human behavior even more trickier to mathematize is the nature of human behavior - we are created beings, not manufactured computers.  That is why there are recessions and periods of growth - if everything were completely predictable, the markets would have accounted for it.  There is always an aspect of human behavior that not even the mathematized world can put a finger on.  Thus, it is important that mathematics be used to describe human behavior only when its employment accurately describes behavior.  In other words, economists need to be aware of the limitations of the utility-maximizing methodology before employing them.  The dangers of applying that methodology in places where it was never meant to be applied only creates a sense of false certainty, as if we have mastered the understanding of human behavior.

But in reality what happens is that the mathematics masters us, and we begin to conform to what mathematics dictates human behavior to be.  And so we begin to become utilitarian in everything we do. The tell is not difficult - churches begin to measure growth empirically, and cater their ministry towards increasing empirical growth instead of spiritual growth.  This has been documented - consult the work of Laurence Iannaccone, a student of Gary Becker.  Churches grow because they provide services which are in demand by the people.  People choose to attend one church over another based on how much it satisfies his or her own utility.  Now, this may certainly explain churchgoing behavior, but in no way does this explanation become the modus operandi of the church.  In other words, the church does not exist to cater to peoples' spirituality, but it exists to worship God.  Worshipping God is not something that can be reduced comfortably into a mathematical relationship.

What I am not criticizing is the usefulness of mathematics.  What I am wary about is the misconstrual of mathematics as a Truth.  Christians believe in a God who is Truth.  Thus, the cardinal who responded to Georg Cantor was rightly suspicious of endorsing his transfinite set theory outright.  The problem is not that Cantor's set theory was useless or wrong, but that by endorsing a mathematical theory that supposedly cements the internal consistency of mathematics, he [the cardinal] is endorsing the view that there is another truth that is derivable outside of God.  If mathematical truth is derivable outside of God, then humanity risks wrongly relying on mathematical truth as an epistemological foundation of human life.   In economics, we see that such an idea leads to more questions, some of it befuddling and devoid of mathematical explanation.  Thus, "a knowledge of numbers that is godless ends in paradox and contradiction."