Mathematics and Christianity

1. You've got to be kidding...

Is there such a subject as "Mathematics and Christianity?" The question came up for me once again as I discussed the issue of faith and mathematics, and how (and if) the christian faith enters into one's teaching and professional work as a mathematician with a recent graduate our PhD program, who has taken a job at CMU (Canadian Mennonite University) here in Winnipeg.  I previously had taught for four years at Fresno Pacific University in California, a similar university with a Mennonite, and therefore evangelical christian, orientation, and hopefully some of the wisdom I had thereby acquired was of some value to him.  One of the questions every professor at FPU was asked to address in their first year is how their faith and their academic discipline intersect, and how this affects their day-to-day teaching.  Part of the job application process is a lecture open to the entire institution's faculty in which one addresses this question.  So I have had some time to reflect on this question.

Today I am teaching in a completely secular environment, under quite different assumptions.  Yet I remain an evangelical christian.  I can deny neither the mathematical nor the christian component of my life.  Perhaps surprisingly to one who has not thought about the question, there is considerable interplay between the two.  While my teaching mandate here does not include (indeed, in some ways precludes) an overt expression of my faith in the classroom, there are many ways in which I believe my professional work, including teaching, research and service, is affected by the fact that I am a christian; this is true for me in the same way as it ought to be true for a christian who also happens to be a plumber, businessman, laborer, single mom living on welfare, or child dying of an awful disease.

However, this is not the kind of connection I wish to address in this document, important as it is.

Instead, I wish to discuss something that vaguely falls into the category of "Philosophy of Mathematics" and vagely under the heading "christian world view".  I have almost nothing to say that falls into the category of "Science and Religion" -- that is, religion in general.  My perspective is specifically Christian, and I will not pretend to speak for other faiths; there are others much better qualified to do so than myself.  So far this description of my intentions is a bit scary to me -- I get awful visions of "crusaders madly battling to raise the cross of christendom over the pagan land of mathematics" -- an image, and caricature I hope I can dispel for my reader as well as myself with what I have to say.

I am also a lay christian, so I speak not as an "official" representative of christianity (except insofar as all christians are supposed to be representatives for their faith), but as a member of the "rank and file", reporting from the particular trench in which he finds himself.

My principal thesis is basically quite simple:  In the Gospel of John, Jesus responds to Pilate's questioning, "So, you are a king?" by saying that he is indeed a king, but that his kingdom is not of this world -- it consists of truth, and his subjects are those who follow the truth.  In another place, he says "I am the Way, the Truth and the Life".  In another, "You shall know the truth and the truth will set you free"; elswhere, "the true worshipers will worship the Father in spirit and in truth".  Math and christianity meet in focussing on the primality of truth.

Now, the truth of which Jesus speaks in the passages to which I refer above (and many others) is not a synonym for mathematics, by any means, but it means first of all one thing to me -- that a christian is one whose first allegiance is to the "kingdom" that consists of truth.  While many christians may seem very dogmatic, one who is true to his roots as a christian is committed to what is true, whether or not it happens to coincide with what he has believed thus far.  We should refuse to follow lies, and we must insist on carving our way along paths of established truths, insofar as we are given light, and to the extent that we are able; when we act with only partial light, we do not act in direct violation of known truth.  We cannot dogmatically hold to beliefs in the face of clear, verifiable facts to the contrary.  I take this as to apply to the very tenets of the faith itself (including, paradoxically, the truth doctrine I am espousing here).  Indeed, if beliefs fail to stand up to rigorous analysis, they fail to provide an adequate foundation for life, and are not worth adhering to.

Mathematics, at least the variety of math (more on this later) with which I deal, concerns itself with propositions known, or accepted, as true.  It has been said that math is the science of "if..then", that is, there are no conclusions in math other than those arising from some assumptions, and so there is no universal truth in mathematics.  There is some truth in this statement, but it contains a fatal flaw -- "if..then" is itself a form of truth; either the relationship between the antecedent and the conclusion is "true" or it is not.  If this relationship founders, so does alltruth, so we had better be careful how eager we are to knock out this foundation stone.  As you will see, with my definition of mathematics, this relationship itself falls within the realm of mathematics, and so there is
no difficulty imposed by accepting that mathematics (at least as we practise it) begins with initial assertions (axioms, etc.) and procedes by logical deductions from there.

Rightly viewed, mathematics is a gold standard for truth.  It is one way of establishing unassailable truths.  I usually get one of two main responses to this claim:  i) either an outright denial of it (I believe this response is badly misguided -- it is either made carelessly, in sheer ignorance of the facts, or with great articulation and quasi-clarity of thought in one of the great mind-bending exersises of "postmodern" deconstructionism, an issue I will deal with, albeit indirectly, in due course); or ii) an acceptance that mathematics indeed can establish unassailable truths, but that such truths as it deals in are largely irrelevant to the issues that form the meat-and-potatoes of daily human existence and certainly not to the larger issues such as Love, Eternity, Divinity, Morality and Purpose in life (this response seems much more reasonable than the first, and I do not blame those who respond in this way at all, but I shall argue that it is based on a fundamental misunderstanding of the nature of Mathematics, and I will hope to do justice to my case for this, because I realise it is an uphill battle, since it requires undoing a great deal of "education" about the subject that is part of the subtext of our culture).

In any case, even if you don't accept my above statements about mathematics, clearly mathematics is about truth, of a sort, and it is about establishing the validity of that truth, whether effective in doing so or not.  That, and my preceding statements about the nature of christianity, are all one needs in order to understand my claim that there is some interplay between the two, at least in the life and mind of an evangelical christian who is also a mathematician.  I will attempt to show here that this interplay is not trivial, that it has some significance, not just to me but to anyone who cares -- and perhaps even to those who do not care.

2. What is Mathematics?

Mathematicians are sticklers for proper definitions.  We generally won't even discuss something in math class unless an ironclad, indisputable and unambiguous definition for it has first been established.  For without proper definitions, everything that follows is made of jello.

But -- have you ever heard a mathematician define "mathematics"?  Most don't try, but I have heard several; the result is often laughable.

The boldest definition of mathematics by a mathematician that I have heard is that of Reuben Hersch:  "mathematics is what mathematicians do".  Ingenious, no?  Ingenious, yes, but absolutely useless!  Worse than meaningless, this definition carries the unacceptable implication that mathematics is no more, and no less, than an arbitrary collection of human activities.  The reader of Hersh's extended articulation of this idea -- his book "What is Mathematics, Really?" -- will find, after hundreds of exhausting pages of philosophical mumbo-jumbo, that he eventually reaches the natural conclusion to which this particular definition leads:  that math is a cultural artifact, and mathematical truth is arbitrary, a mere consensus, subject to change and revision over time and at the whims of those forming the consensus.

The definition I shall give, in contrast, does not refer to mathematicians, or to any other group of humans at all -- humans are quite irrelevant to the question of what mathematics is.  Indeed, so is the earth, and the very cosmos.  In this sense, mathematics is transcendent.  (PLEASE, do not read rabid mysticism into such statements in this document -- by "transcendent", I simply mean "beyond"; mathematics is not dependent on any particular material or temporal manifestation, it's truths go beyond these things.  I know no better way to say this than to say that it is transcendent.  My exact meaning for this phrase should become clear as we proceed.)

What is a chemist, but one who studies the science (or what can be known) of chemicals?  And what is the Biologist, but one who studies the science of biological systems?  And what is the physicist, but one who studies the science of physical systems?  Each branch of science delves into what can be known about a particular aspect of the world that is open to our exploration and yields information to our means of exploration, and whose truths can be resolved by appropriate tests of validity?  Yet it is defined not by the activities of the practitioners of the discipline but by the objects of study toward which those activities are directed.

I contend that the word "science" is a bit unfortunate because it tends to denote only those things which canbe known.  But a physicist, for example, does not generally mean this when he speaks of physics -- it would lead back into Hersh's error, of limiting the meaning of the field to the boundaries of human endeavor.  What the physicist really means by "physics" is the set of facts about physical systems that completely describe how they operate and what they are.  Perhaps these facts can all be known; perhaps not.  Perhaps it cannot even be known which parts can be known...but "physics" refers to the entire set, whatever they may be, knowable or not.

Similarly, mathematics is a science -- not in the limited sense of "what can be known", but in the broader sense of "the facts, whether they be knowable (to humans) or not".  By this I mean "...also whether they be representable by arrangements of mathematical symbols, or not".  This understanding of the word "science" is important to my definition -- more important than it probably is for chemists, biologists or physicists.  Also, please don't misconstrue my definition as saying that mathematics is onlya science -- it is also an art, and it is also a humanity, as I argue elsewhere; in these cases, however, one focusses not on mathematics proper, but on mathematical activity, or on human appreciation of, or ability in, mathematics.  (The main shortfall of most definitions of mathematics is that they limit the scope of "mathematics" to some subset of human endeavor, or potential human endeavor, ultimately falling into the trap of Hersch.)  I will treat mathematics primarily as a science, with "science" taken in the above sense.

With all of the above understood, here is my definition of mathematics.

Mathematics is the science of structure, form and relationship.

Mathematics, unlike the three physical sciences mentioned above, does not study any one particular aspect of the physical world -- it studies that which lays behind and undergirds all physical manifestations of all phenomena -- structure.  I also mention form and relationship, not because I don't realise that these three overlap, but because I feel it is necessary to properly delineate the scope of the subject and to avoid potential misunderstanding as to the intended uses of the definition.

This definition explains why all scientists use mathematics in their discipline -- it is indispensible because one cannot do science without establishing patterns and facts that deal with structure, form and relationship.  It is, in a way, what science is about.  At the very least, this is why mathematics provides the "language of science", as is commonly said.

There is nothing without structure, form, or relationship.  At least, if there is, we can never know it.  For, if we know it, then it has a relationship, first to our conscious thoughts, and second to the symbols in terms of which we are able to think of it.  If it "exists", then it has some relationship to some realm in which it "exists".  If it is something, then it has some form, be it abstract, or material, or spacial, temporal, conceptual, and so on.

Suppose something is "formless, and void", as the world is said to be in Genesis I, or as some artists attempt to represent in their paintings or sculptures.  Even so, this is a form, as much as 0 is a number and the empty set is a set, and thinking about nothing is thinking about something.  To think that "formlessness", if it is a state attained by any existential thing, lends that thing the property of not having any form is a mistake -- at least it is a misunderstanding of the way in which I intend to use the word "form" in the definition above.  For "formlessness" means to us "lack of identifiable form".  If there is such a thing as "identifiable form", then this can be defined and delineated.  Avoidance of such forms is a constraint upon the form of such an object, and therefore qualifies as a category of form -- forms which are not (human-)identifiable.   Deliberately try to conceive something with no form; suppose you are successful.  You have still failed, for the very consideration you needed in order to make this attempt, and evaluated it, required that you ask and answer questions about form, or the lack thereof -- both of which qualify as part of the "science" of form.  Form cannot be escaped.

Keith Devlin defines mathematics as "the science of patterns".  Then, one objects, what about things that have no patterns?  "Patternlessness" is itself a pattern, of a higher sort, in the same sense as formlessness is a form -- for, the only meaning we can give to such a term is that it is merely the avoidance of a certain class of patterns...this avoidance is, in itself, conformity to a "pattern", of sorts.  Suppose something has successfully avoided this class of patterns.  To successfully describe this property of that thing, we must, of necessity, refer to those patterns.  So this property and our knowledge (if it is knowable) of it, falls squarely within considerations native to the science of patterns.

(I far prefer Devlin's definition to Hersh's; it is, in a way, a specialization of mine, and is certainly in the same spirit.  But reading Devlin's work on the subject it becomes clear that he would answer the above objection in a manner opposite to mine, for it appears that, by "pattern", he means "humanly perceived pattern", which is a step in the direction of Hersch. I wonder if he has clearly thought out the implications of this definition and how fairly it delineates the boundaries of mathematics, and whether he would change his definition, or his application of it, after such an exercise?)

Structure cannot be escaped.  What about a changing structure, like an exploding bomb?  Why, if you mean a material structure in our 3D world that changes in time, then altogether it can be viewed as a 4D structure.  What about a random structure?  As long as you give "random" a precise meaning, and that meaning says something like "something is random if it does not have such-and-such a structure", then it indeed has a structure, which can be defined in terms of this avoidance, as with "formlessness" and "patternlessness" above.

"Well, now", you say, "you have changed the rules of the game on the fly; you have made your definition so general that it no longer has any meaning -- it is as bad as Hersch's definition, but for the opposite reason -- it is not connected to any particulars at all!"

The "disconnection from particulars" in my definition is its strength; it reveals precisely what it is that makes mathematics unique among the sciences, for mathematics is not about any particular instance of any particular object, material or otherwise -- it is about those characteristics shared by all things.  (NOTE:  this is not intended to be a New Age-type declaration of all-in-oneness, just a baldfaced claim about the generality of the subject of mathematics.)

Incidentally, it is this very generality that makes mathematics so useful.  Solve one equation, say the quadratic equation x2+3x+2=0, and you have solved the essential partof a myriad of problems in economics, biology, chemistry, physics, epidemiology, quantum mechanics, general relativity, problems not yet thought of by humans, and problems concerning things not in any way affecting this universe.  As everyone with experience doing word problems knows, the hard part is usually not the mathematics, it is finding the mathematics -- putting the words about some specific situation into its most general context, keeping only that information relevant to the actual problem.  When the problem is boiled down into these terms, it magically becomes mathematics.  That is because mathematics is the bare bones of reality.  It is what is left when all the particulars are erased.  The process of stripping the words away to reveal the essential core has been called abstraction.

I wish to make yet another subtle adjustment to our language at this point.  In spite of my discussion above, by abstractionI shall not mean the act of stripping off particulars, or the thing which is arrived at by doing so; I think this is an error, or at least it is an error to treat mathematics as "abstraction" in this sense.  Another mistake is to use "abstraction" as a synonym for "idea" or "concept" in this context.

The first error would have mathematics obtained only by squeezing reality until no more juice comes out.  But mathematics does not proceed, in this way, from the material world.  Quite the opposite -- the universe, while not literally proceeding outof mathematics, exists only with due respect to all the truths within mathematics that lay down rules about how physical systems and their internal forms, structures and relationships, must behave.

The second error is similar; like Hersch's definition, it leaves mathematics as a byproduct of the fickle human mind.

In spite of the dictionary meaning of the word, I shall use the term abstractionto refer to such things as have no physical or material particulars, though they may (or may not) be taken as categories into which particular phenomena and objects fall, such as: number, measure, sphere, line, tesseract, continuity, connectedness, consistency, matrix, variable, set, ordered lattice, interval, loyalty, goodness, computability, and so on.  At the exact definition of this term I yield to some arm-waving, for I have no reference points, other than some classical examples, from which to describe the concept, and having flatly refused, above, to accept two of the standard connotations of the term.  Here's another, inadequate, attempt:

Abstraction is "possibility before (or in the absence of) realization".

Take it or leave it.

Mathematics is abstraction (not the verb, the noun) -- as its meaning ends up after the way I have hammered it above.  Not "abstraction from" anything, merely "abstraction" as a pre-existent state, a mold into which all specific things must fit.  This is what we study when we study mathematics, the skeleton on which all other things hang.

So is mathematics a branch of philosophy?  NO.  Mathematicians don't need to concern themselves with the nature of the field of mathematics, but can go ahead and study the subject, even in blissful ignorance of its boundaries, even as a field naturalist with a deep interest in butterflies does not compromise his integrity as a biologist by focussing on the object of his interest instead of the philosophical category into which it fits.  But you may view my ramblings above as a bit of marginally competent philosophical chatter.  I'm afraid it is the only way for me to make clear what I mean when I speak of mathematics, and I do apologize for the parts of it that sound like mumbo-jumbo; I hope I have been articulate enough that a slow and careful reader can grasp my intentions.

3.  So what does this have to do with Christianity?

If you're still with me, and want to know how this all comes out, write to me ... writing takes time, and I have other things to do with my time, obviously, and this article is a long-term work in progress.  If anyone cares where this is all going, I may be motivated enough to push it to completion.

Feb. 2003:

It's been several months since I wrote the note above, and several people have asked for the "rest" of this article.  Unfortunately, part of the reason I stopped writing was that I realised how much more there was to do and this third section became enormous in my mind as I realised all the things that needed saying. I think it is only fair to any readers, if I am to leave this on a public page, to at least outline some of the things I hope to write about, and sketch things in as I have time.  Necessarily, the document will then evolve as I take shots at it, but my general thoughts will be accessible to those wishing to consider them.

So...please...consider anything that follows merely a sketch of ideas, in various degrees of completion, and anything that appears here may change or disappear as the document evolves.  It will not, anytime in the future, have any semblance of order; my first priority is simply to get some of these thoughts on paper.

In answer to the title question for section 3, I will refer back to my identification of the central tenet of christianity (in my introduction (which is, of course, not an orthodox way to define christianity, and would rightly be considered heretical if taken the wrong way.)  I will not say for the moment what this "truth" I refer to is, except that it is, in some direct way, associated with who Christ is (in the mind of a christian).  I will take truth as an atomic concept, and instead give a couple of personal anecdotes that relate to truth as a concept that intersects the two worlds of an evangelical christian who studies mathematics.  I will eventually work this into a full-fledged answer to the question...Both incidents are factual, but I see no point in identifying the individuals of whom I speak; I am sure they represent many others.

Scene I -- An isolated christian mathematician

As many have found, the internet is an arena well-suited to striking up acquaintances with others of a like mind.  Several years ago, I had a chance meeting with the web page of a fellow christian mathematician who lived overseas and who was struggling with the relationship between these two areas of his life.  I should not say that he saw a conflict himself, but that he had some difficulty explaining his christianity to his mathematical friends, and his enthusiasm for mathematics to his christian friends.  Some extracts from his correspondence with me:

...when I entered graduate school, I ...lost almost all support from the christians around me.  Although I have my church life, I found that they can't support me...when I have some conflict between ways ot thinking in mathematics and our belief---even the pastor...didn't think that it is really a problem...when I decided to enter graduate school...I was challenged by some of my [christian] friends:  "Why do you plan to hide in the ivory tower?  It is no use for you to be a mathematician!  You have to go into society and serve others!"  I am a little bit angry for their comment, partly because they seem to misunderstand mathematics and the work or mathematicians.  But sometimes I also make the same comments to myself!  Is it really not my mission to be a mathematician...?  I prayed to God this morning:  "Help me to be a good and honest mathematician, so that I can love you more.  Perhaps, this is the mission you gave me!" mentioned "humanistic mathematics".  Frankly, this is what I thought when I was an undergraduate.  I tried to explain why I still believed in something although I am a math student.  I treated the doctrine as axioms.  This seems to solve my conflict...but I found that this argument has serious problems [now that I am a] graduate student.  If I follow this kind of thought I will become a "relativist", and I cannot say anything about truth and reality...

There are at least four difficulties evident here; i) one is that there is a fear of mathematics, and more generally, of science, among many evangelicals; ii) there is a sense that mathematics is a frivolous endeavor, wasteful of one's resources for doing good in the world; and on the other side iii) there is a sense (by the "mathematical side") that a mathematician ought to be too rational to believe in such things as God--particularly in the christian formulation of God.  Finally, iv) there is a feeling that the job of the mathematician is to reduce all human activity to the level of axioms and thus rationalize one's life.  As my friend discovered, to do this from a purely formalist perspective leaves one with no definite answers---any one axiom system is as good as any other (consistent set of axioms), and one is led, at best, to relativism along this road.

I take up each of these briefly.  As for i), I can only concur that my friend's observations appear correct.  Many evangelicals simply do not understand mathematics and/or science, and there is a great fear of the unknown.  There is a fear that, if math or science was allowed free reign, they might do serious damage to one of the foundations of our faith and it will come crumbling down.  This is my observation, but I will not say that there is a valid reason for christians to feel this way.  Like most fears, it is based not on the known, but on the unknown.

I have a simple twofold answer.  The first part is that all truth is God's truth, or at least consistent with the doctrine of God.  I don't put this forward like the medievals as a test of truth---ie dismiss anything inconsistent with doctrine as a falsehood---but by way of stating what must be obviously the case if Christ's statements about his relationship to truth are taken as foundational.  The second part is Christ's declaration that, not only has he come that we may know the truth, but that the truth will set us free.  So, a christian should not be afraid to face truth, no matter what form it comes in.  The antidote for false truths, or partial truths, or half-truths, is not retreat into dogma and "blind faith", but into more, clearer truth and the faith of open eyes and clear thinking.

The second problem is more profound.  Is mathematics a frivolous activity?  Unlike a plumber or even a musician or football player, it is hard to see the immediate benefit of our work.  If I play all day with equations and symbols in my mathematician's paradise, at the end of the day, whom have I helped?  As a christian I am called to serve others.  The obvious answer is that those of us in the profession at universities are also teachers, and so help others daily who are learning mathematics.  But this answer is not enough, for we are only helping them to learn mathematics---if math is not an inherently useful thing, then what ultimate good is served by doing so?

To this I answer that the pursuit of truth is inherently a good end in and of itself.  That I thoroughly enjoy the pursuit does not detract from its value.  From the outside it is easy to dismiss the truths of mathematics as frivolous pursuits and of less value than, say, the pursuit of truth in a criminal court or relating to some justice issue.  But it is precisely there that the aspiring critic reveals his or her misunderstanding of mathematics.  A review of my definition of mathematics will reveal that mathematics is the underlying structure for all phenomena found in the universe, and beyond.  This includes all such human activity, and more.

The problem is not finding significance in the mathematics, but in bringing that significance, in the broadness of its generality, down to a level we can understand and appreciate.  This is the problem of Applied Mathematics---to show how mathematical truths and methods can be used to solve problems in the real world.  It is a great study to follow the path of mathematical discoveries from a recreational tinkering in someone's head to some important application.

Minkowski space comes to mind (there are myriads of similar examples).  Here is an exotic space conceived only as a solution to a rather esoteric problem:  is it necessary to admit the parallel postulate among Eulid's axioms of geometry?  Minkowski (and others) discovered that, if the parallel postulate is replaced with other statements, one can obtain interesting and surprisingly beautiful alternate geometries---like, and yet unlike that of Euclid---that become fascinating playthings for the mathematician.  But are these geometries of any use?  In the case of Minkowski space, the answer came when Einstein formulated his Theory of General Relativity.  He discovered some laws that dictated how the universe would evolve and had very strong reasons to believe that these laws were correct, but there was one problem---the laws say that the universe violates the flat geometry of Euclid.

Einstein learned, perhaps by chance, of Minkowski space, and thereby found a setting in which his laws work.  And today, broadly speaking, we treat the shape of the universe, not as a manifestation of Euclidean space, but (approximately, and locally) as Minkowski space.  A frivolity becomes the framework of the universe, and it's all in a day's work in mathematics.  When one grasps the enormous consequences of each tiny advance in mathematics, it is hard not to be awed by the great significance of mathematical work, whether one is a pure or applied mathematician.  To work in this field is a tremendous privilege, and we deal in a currency of unfathomable worth.

What about problem iii)?  Should a mathematician be too rational to believe in God?  This question presupposes, of course, that there is something irrational about belief in God.  On what basis must we say this?  You might say that it does not follow from the axioms, and I would ask, "what axioms do you have in mind"?  For relativism cuts both ways.

When Laplace finished his great Celestial Mechanics, a treatise on the working of the heavens, he was challenged by none other than Napoleon, how he could write a great work about the mechanisms of creation without ever once mentioning the creator?  Laplace responded "Monsieur, Je n'ai pas besoin de cette hypothese", or "Sir, I have no need of that hypothesis".  One may argue along Laplace's line that, for sure, there is no obvious harm to science in having a God-axiom, but it is completely unnecessary in order to have a complete conception of the universe---it is therefore an irrelevant hypothesis.

Whether or not one grants the second half of this line of reasoning, one must ask whether the point of having a belief in God is to fill in gaps in one's scientific knowledge.   If so, then one has fallen into the old line of the "God of the Gaps"---the God who exists mainly as an explanation for otherwise unexplainable phenomena.

A believer in the God of the Gaps is necessarily in an uncomfortable position in a civilation in which knowledge of the world increases.  For this god will find himself continually retreating from the light of human discovery, always seeking the shadows and hiding in dim corner beyond the latest advance of science.  Followers of such a god find themselves continually in need of revisions of their beliefs.  Or they are thrust by necessity into debates to defend the "unknowabilty" of fields even as they fall increasingly under the scrutiny of science.  Much of the so-called creation/evolution debate strikes me of having this kind of character---that is, the better half of the debate does (the other half comes when proposals are made for "scientific" theories designed to support a god-hypothesis, and with no support that goes beyond attacking competing explanations; this is the more degenerate side of the debate, but it is a problem entirely separate from the one I am currently addressing; perhaps I will come around to discussing it later.)

I hold this perspective partly to blame for the negative response of some evangelicals to scientific advances (my point i) above).

The main difficulty with the God of the Gaps, from a christian perspective, is simply that it is foreign to God as he is presented in the bible.  The god of the bible does not hide in shadows delineated by man's ignorance---quite the opposite, he declares that all that is hidden will be made known (e.g., Lk 8:17), and that in him there is no darkness (meaning, in context, ignorance).  He broadly invites mankind to explore his truths, to come and reason, to pursue knowledge and wisdom.

This is not a timid god or one that evaporates, spectre-like, in the light of day, but one who openly declares himself and walks among us.  As the apostle John put it, "The light shines in the darkness, but the darkness has not understood it...the true light that gives light to every man was coming into the world.  He was in the world, and though the world was made through him, the world did not recognize him."  (Jn 1:4-10, NIV) The God portrayed in the bible has very much the opposite character one expects of the God of the Gaps---he offers truth, light, knowledge and insight, and meets us in person to present it.  Ignorance, on the other hand, is the principal human response---it is not that God is inconsistent with truth, merely that humans avoid acceptance of that truth, or light, preferring darkness.  From this perspective a christian who pursues the sciences, such as mathematics, is working in concert with God, part of whose business is revealing truth to mankind.

Here is an exciting realisation---the christian scientist or mathematician is not in the business of avoiding intersections between his field and "ultimate question", but that he is participating with God in bringing light to bear in realms once dominated by the darkness of ignorance (and, as a Corollary, so are scientists who don't happen to be christians, whether they see it this way or not).

Finally, we have question iv):  should we not limit our lives to that which can be derived "mathematically" from some set of fixed laws, or axioms?  In a sense, the answer to this question is "yes"---presupposing we have full knowledge of such axioms and full power to carry out the requisite mathematical derivations.  But there are many reasons why, operationally, the answer must be "no"; I list four:

1.  We do not, in fact, posess knowledge of all the "axioms" of life.

2.  We certainly do not have the mathematical power to work these axioms out into imperatives concerning daily choices in our lives.

3.  Ask whether this requirement is true for the routine choices of daily life, and you will see the absurdity of it.  Must the (christian or non-christian) mathematician have mathematical certainty whether to wear the red socks or the blue each day, or whether to take the stairs or the elevator to his office?  Clearly not.  We do not live our lives with such (quasi-)mathematical nonsense.  It is not necessary to attempt to resolve every question of life as if it were a quadratic equation---even if such a thing were possible, in principle.

4. There is a very good reason to doubt that axiomatics are sufficient to resolve all truths in the pure realm of mathematics (namely Godel's incompleteness theorem, which I shall perhaps address later).  How much less the truths of life.  This is not to deny the ability of mathematics (in the generality I claim for the field in the first section of this document) to address such questions---it is merely a denial of the centrality of the axiomatic method to the global domain of mathematics.  It is now understood how limited axiomatics are in this regard and even tightly minded formalists (and dare I suggest even those of the constructivist school) have now to admit that math pushes beyond the chains imposed by any measly set of axioms.

So problem iv) is not really a problem at all, it is an illusion.

Scene II -- A (non-christian) deist/theist mathematician

I will call this person James -- not his real name.  I knew him during my years in graduate school; he was a loner, in more ways than one.  Coming from a remote area, James had not grown up among academics and was clearly not particularly comfortable among them.  But he was a talented mathematician starting out his career.  One day James gave me a paper he had written; it had the mark of a young mathematician in its bluntness and naive construction, but the ideas were fresh and interesting (though not close to my own field, so I could not judge the value of the work).  I was particularly struck by the Acknowledgement he gave at the end of the paper.

Often one acknowledges people who have contributed to a paper, perhaps by giving advice or technical assistance, but not of the sort that would justify a coauthorship.  Graduate students will thus often acknowledge their supervisor; a person relying on the assistance of a programmer for minor details of work described in a paper, or, receiving helpful advice from a colleague who reads an early draft, will mention this in an acknowledgement.  James, however, acknowledged God in his paper.  I don't recall the exact words, but he wrote something like,

"The author wishes to acknowledge God, who knew all these things first."

As a christian I was impressed with the boldness of such a declaration; these days one generally does not so brashly acknowledge God's role in a research publication. James' acknowledgement was exactly the right thing to say, from a christian perspective.  But one does not offer such boldfaced declarations as part of serious published work because the inevitable ridicule will detract from the work itself and ultimately not serve either the cause of God or promote the value of the academic work to which it is attached; scholarship ought to stand on its own merit, not because of this or that association with some higher (or baser) cause.  This is the principle of not "casting pearls before swine".  There is simply no point in offering such acknowledgement in an arena in which it would only invite ridicule.

I see a related problem with the phenomenon of "christian athletics".  It is not wrong, either from a christian or from an athletic perspective, to publicly acknowledge God, but there is something wrong with using one's athletic prowess to gain a platform for this proclamation, or to attempt to promote God by association with the laurels we win.   Similar comments necessarily apply to "christian beauty queeens", or "christian performing artists".  I recall a christian musician who recognized this tension and cleverly parodied this mentality in a song: "I'm a star, Lord, I'm a star; with me you'll go far".  God does not need our recommendation to "get his foot in the door".  His mission to this world does not need all-star sponsorship.  As the Apostle Paul said, "God has chosen the foolish things of the world to shame the wise; God chose the weak things of the world to shame the strong.  He chose the lowly things of this world and the despised things---and the things that are not---to nullify the things that are, so that no one may boast before him."  (I Cor. 1:27-29)

I was impressed with James' acknowledgement, however, because it was so simply and innocently proferred.  I mentioned to him that it was refreshing to see a reflection of this sort in an academic work.  He responded by telling me of his background; his family was associated with some religions (I think he mentioned two different religious backgrounds).  He lived, however, in a country that promoted atheism rigorously, and he was faced, throughout his upbringing, with agressive challenges to any kind of religious talk.  Like many in his country, then, he learned to internalize his thoughts about ultimate questions and worked out his own views about such things.  He came to the conclusion, after many sorts of deliberations, that there must be a god, that he is creator of all that there is, and that this god is transcendent, omniscient, and either benevolent, or in the very least, not evil.

I probed further into James' beliefs and it appears that he felt it was unlikely that God had any particular kind of personality; he was more like a force that pervades the universe.  But, perhaps as a cosmic consciousness.  God, being creator, was not equated with the universe, and was not contained within it.  Those familiar with the distinction between Theism (belief in a personal god) and Deism (belief in a supreme being devoid of personality) will recognize that James fell somewhere in the middle of the spectrum, but slightly on the Deist side of the fence.

Christianity, of course, is about as Theist as one can get.  Nevertheless, it is possible for a Theist and a Deist to have a meaningful conversation about the nature of God, and that is what James and I had, over a period of weeks.

One day I asked James if he thought God had any particular characteristics that defined him as a God -- perhaps he would say that God was "this" kind of god rather than "that" kind.  What I had in mind was some of the catch-words and phrases that Christians and others might use, such as "justice", "compassion", "locality/nonlocality", "omnipotence", "imminence", and so on.  Did God play favorites, or does he treat all equally?  Is he concerned with the affairs of men, or is he only concerned with the larger motions of the universe?  If concerned about men, what about sparrows?  Worms?  Amoeba?  Does he like sports?  Does he like music?  Does he have a sense of humor?  In other words, what is God really like, in his view?

James considered the question, but not for very long.  He said that he didn't think that God had anything that would qualify as a personality.  The things that delineated God from elements of the creation were some abstract and very general properties.  It certainly would be improper, in his view, to attribute to God any passions or sentiments, such as compassion or love.

So I asked James how he came to the conclusion that God must be the creator of the universe.  Because it is self-evident, he says, that the universe must have a cause---something cannot come from nothing, and there must be a reason why there is something rather than nothing.  So I asked, "Why?  If God has no passions, no sentiments, no motivations, then why did God create the universe?"  If one appeals to God as a first cause then one must attribute something to God that acts as motive, that explains, in terms of the characterisitics of God, why he would do such an extravagent thing---or one is again left with the problem of an absent first cause, but a worse one, in which a God frivolously, or for no good reason, creates the universe, with no interest in it and for no purpose.

James clearly had not ever considered that question, so it was the end of our discussion for the day.  We didn't speak for some time; the next time he came to see me, he asked what my answer to the question was.  I responded that, unlike his view, in the christian view, God is a person, and has definite personality traits.  For example, he is good; he identifies with the lowly and the oppressed.  He actively promotes justice, promises to ultimately avenge all wrongdoing and reward all good.  He calls a people to be his own and identifies himself with them -- and more.  Although God is far beyond our comprehension there are some basic traits that allow us to understand some of what motivates him. Although our thinking about these things is imperfect, we can examine the better parts of our own souls and use them, by way of analogy, as a window into his eternal being.

So why did God create the universe?  I think it is because he was lonely.  Sorry, it doesn't sound profound enough to be God-like.  It also sounds like God has a weakness; maybe that is a bad thing to say about him.  Nevertheless, the God who makes himself weak and vulnerable for the sake of others is very much part of the Christian story, so I don't think is completely wrong to describe him as "lonely" even though it is surely also not exactly right.  It may be the only way we can humanly conceive of his motivation.

God is lonely, and he desires relationships.  He desires a relationship with you.

The more I think of the lonely God, the more I think it fits the christian worldview---I'm afraid I can't find chapter and verse to support it, it is just something that "seems right".  But I haven't found any biblical passages that address the question; the bible is somewhat silent about this particular "why?".  It is consistent with the God who seeks out relationships with men and women, and who sacrifices to guarantee these relationship.  It explains why God is our friend, and why he desires to adopt us into a family relationship.  The thought that God created the entire universe, in its immenseness, just to commune with mankind, seems too wonderful to accept, and perhaps it is wrong; perhaps there are countless other races and civilizations whom he loves in the cosmos.  But the christian conviction is that God loves the "world", not only in the grand scale, not only by the countless millions, but right down to the individual scale -- he knows my name, and he knows yours; he knows us deeply, intimately, better than we know ourselves.  The psalmist wrote, "O Lord, you have searched me and you know me.  You see me when I sit and when I rise; you perceive my thoughts from afar.  You discern my going out and my lying down; you are familiar with all my ways.  Before a word is on my tongue you know it completely, O created my inmost being; you knit me together in my mother's womb.  I praise you because I am fearfully and wonderfully made...My frame was not hidden from you when I was made in the secret place.  When I was woven together in the depths of the earth, your eyes saw my unformed body.  All the days ordained for me were written in your book before one of them came to be..."  (Ps. 139, selections).

I sometimes wonder---what was there about our conversation that was peculiar to an interchange between mathematicians?  I'm not sure I can answer that confidently, but I do know that it was in the context of a presentation of mathematical ideas that our conversation came about.  And, how we both thought through these questions reflect our minds and the way we think of things.  My question about God concerned structure, form and relationship (suitably interpreted).  Yet our thoughts are, I think, basic to those of people of almost any background who engage in such a discussion.  There was little new in our thoughts by way of content, but perhaps they were articulated in a helpful way because of our mathematical way of thinking...I don't know.

But I like to think so.

4.  Some tangents that need to be discussed.

I have lots on the backburner, so I'm going to revert to list mode for now.

- Perhaps most interesting, my fledgeling theory of "The Three Causes".  Is there such a thing as Free Will?  Is it a necessary part of Christian doctrine?  Is there a distinctly "mathematical" approach to this question?  My ideas on this topic are unfolding slowly, but I have ended up discussing it with those who want to go deeper into the material herein, so I figured it is probably a priority for inclusion here.
- The fears and misgivings, and misconceptions some christians have about mathematics, science and academics in general.
- Some grave misconceptions many academics have about christianity.
- The damage often done to the image of christianity by well-meaning persons representing themselves as scientists (or mathematicians) and hoping to lend an air of scientific (or mathematical) authority to various christian dogmas.
- The occasional abuse of (quasi-)mathematical methods to "prove" christianity
- How a mathematician can use a mathematical perspective to probe questions about eternity, transcendence, and so on.
- Some thoughts on the "logic" of some classic anti-theistic arguments, such as the Problem of Evil and whether God can create something so unmovable that he cannot himself move it.
- Some thoughts about popular philosophical nonsense such as whether a tree falling in a forest, with nobody nearby, makes a sound.
- On the triumph of science (I like Asimov's article "The Relativity of Wrong", which neatly lays to rest the tired argument that, since science is constantly updating how it views the universe, we cannot trust its conclusions)
- On the gaps in our knowledge (Similarly, I like mathematician Roger Penrose's keen insight into some of the gaping holes left even by the wonderful cosmological theories we have today with their breathtaking accuracy.  Research scientists will be in business for quite some time to come; if Penrose is right, science will find itself focussing more and more on fundamental questions as it tries to penetrate age-old questions such as, "what is the true nature of consciousness?".)
- Does knowledge invalidate faith or extinguish wonder?    I cannot recall the author, but I don't think I can improve (though I will elaborate) on the following quote:  "The larger the island of knowlege, the longer the shoreline of wonder."
- On whether or not mathematics is dehumanizing, or antithetical to the human experience.  Another favourite quote:  "If you think that rationality is cold, you must ask yourself exactly what is so warm about irrationality".
- How do science and mathematics address/inform a christian's pursuit of his faith?
- What great tasks cry out for labor on the part of christian mathematicians?
- On the fallacy of "Christian Scholarship" as superior to the simple life simply lived. (ironically, it probably takes a christian scholar just to consider the question...  Ultimately I conclude, like Paul, that "...whatever was to my profit I now consider loss for the sake of Christ.  What is more, I consider everything a loss compared to the greatness of knowing Christ Jesus my Lord, for whose sake I have lost all things.  I consider them rubbish, that I may gain christ and be found in him, not having a righteousness of my own...but that which is through faith in Christ---the righteousness that comes from God and is by faith..." (Php. 3:7-9).  One day our work will all burn, and all that will be left is the relationships we have established with Christ and with others.  All else fades in importance as our perspective recedes from our limited circumstances in this world.)
- On what aspects of christianity make it particularly interesting to a mathematician
- the cosmology of Christ and God (both what is in the bible, and what isn't -- on the latter side there is a famous quote by Galileo, "The bible tells us not how the heavens go, but how to go to heaven".  Actually Galileo was quoting a prominent christian theologian of the day, but that does not diminish the aptness of the quote.  On the former side, I observe that, although many "cosological connections" in the bible can be summarily dismissed, there remain some incredibly suggestive subtexts that I find fascinating to speculate upon, and I shall do so -- probably to my own peril...)
- On "garden-variety platonism"; both a christian and a mathematician's perspective.

R. Craigen