Several years ago I made the assertion on this blog that evangelicals should "think Christianly" about their work and fields of study. I also claimed that we are merely fooling ourselves if we believe that we can approach our vocations with a sense of religious neutrality. Naturally, some people were skeptical. Even those who agreed with my general point did not see, for example, how there could be a particularly Christian view to hard subjects like mathematics.
While I certainly understand their hesitation, I do in fact believe there is a Christian view of mathematics. Indeed, I believe that there is a distinctly Christian view of everything.
The reason this idea seems so foreign (if not downright absurd) is that most of our theories about the world have only a minimal pragmatic affect on how we actually live our lives. Both my neighbor and I, for example, may get sunburned even if we different beliefs about the sun. The fact that I think it is a ball of nuclear plasma while he believes that it is pulled across the sky in a chariot driven by the Greek god Helios doesn't change the fact that we both have to use sunscreen. It is only when we move beneath the surface concepts ("The sun is hot.") to deeper levels of explanation ("What is the sun?") that our religious beliefs come into play.
Even the concept that 1 + 1 = 2, which almost all people agree with on a surface level, has different meanings based on what theories are proposed as answers. These theories, claims philosopher Roy Clouser, show that going more deeply into the concept of 1 + 1 = 2 reveals important differences in the ways it is understood, and that these differences are due to the divinity beliefs they presuppose.
But before we can see why this is true, let's review what constitutes a religious belief.
A belief is a religious belief, says Clouser, provided that (1) It is a belief in something(s) or other as divine, or (2) It is a belief concerning how humans come to stand in relation to the divine. The divine, according to Clouser, is whatever is "just there." He contends that self-existence is the defining characteristic of divinity, so that the control of theories by a belief about what is self-existent is the same as control by a divinity belief and thus amounts to religious control of all theories.
Whether we refer to it as being self-existent, uncaused, radically independent, etc., it is the point beyond which nothing else can be reduced. Unless we posit an infinite regress of dependent existences, we must ultimately arrive at an entity that fits the criteria for the divine.
Different traditions, religions, and belief systems may disagree about what or who has divine status, or whether such an ontological concept should be considered a "religious belief." But what they all agree upon is that something has such a status. A theist, for instance, will say that the divine is God while a materialist will claim that matter is what fills the category of divine. Therefore, if we examine our concepts in enough detail, we discover that at a deeper level we're not agreeing on what the object is that we're talking about. Our explanations and theories about things will vary depending on what is presupposed as the ultimate explainer. And the ultimate explainer can only be the reality that has divine status.
Returning to our example, we find that the meaning of 1 + 1 = 2 is dependent on how we answer certain questions, such as: What do "1" or "2" or "+" or "=" stand for? What are those things? Are they abstract or must they have a physical existence? And how do we know that 1 + 1 = 2 is true? How do we attain that knowledge?
Let's look at the answers proposed by four philosophers throughout history:
Leibnitz's view -- When Gottfried Wilhelm Leibniz, an inventor of the calculus, was asked by one of his students, "Why is one and one always two, and how do we know this?" Leibnitz replied, "One and one equals two is an eternal, immutable truth that would be so whether or not there were things to count or people to count them." Numbers, numerical relationships, and mathematical laws (such as the law of addition) exist in this abstract realm and are independent of any physical existence. In Leibnitz's view, numbers are real things that exist in a dimension outside of the physical realm and would exist even if no human existed to recognize them.
Russell's view -- Bertrand Russell took a position diametrically opposed to Leibnitz. Russell believed it was absurd to think that there is another dimension with all the numbers in it and claimed that math was essentially nothing more than a short cut way of writing logic. In Russell's view, logical classes and logical laws -- rather than numbers and numerical relationships -- are the real things that exist in a dimension outside of the physical realm.
Mill's view -- John Stuart Mill took a third position that denied the extra-dimensional existence of numbers and logic. Mill believed that all that we can know to exist are our own sensations -- what we can see, taste, hear, and smell. And while we may take for granted that the objects we see, taste, hear, and smell exist independently of us, we cannot know even this. Mill claims that 1 and 2 and + stand for sensations, not abstract numbers or logical classes. Because they are merely sensations, 1 + 1 has the potential to equal 5, 345, or even 1,596. Such outcomes may be unlikely but, according to Mill, they are not impossible.
Dewey's view -- The American philosopher John Dewey took another radical position, implying that the signs 1 + 1 = 2 do not really stand for anything but are merely useful tools that we invent to do certain types of work. Asking whether 1 + 1 = 2 is true would be as nonsensical as asking if a hammer is true. Tools are neither true nor false; they simply do some jobs and not others. What exists is the physical world and humans (biological entities) that are capable of inventing and using such mathematical tools.
For each of these four philosophers what was considered to be divine ("just there") had a significant impact on how they answered the questions about the nature of the simple equation. For Leibnitz it was mathematical abstractions; for Russell it was logic; for Mill is was sensations; and for Dewey it was the physical/biological world. On the surface we might be able to claim that all four men understood the equation in the same way. But as we moved deeper we found their religious beliefs radically altered the conceptual understanding of 1 + 1 = 2.
What all of the explanations have in common, what all non-theistic views share, is a tendency to produce theories that are reductionist -- the theory claims to have found the part of the world that everything else is either identical with or depends on. This is why the Christian view on math, science, and everything else must ultimately differ from theories predicated on other religious beliefs.
How insightful is your analysis of the belief 1+1=2 when it's also just as valid as an understanding of 2+2=4 and 4+4=8? It seems like you've given an excellent argument for why religious beliefs have an impact on a *philosophy* of mathematics as opposed to mathematics.
No mathematics professor debates 1+1=2. But what about the debate concerning Fermat's Last Theorem? If you can demonstrate how religious beliefs enter that debate, then you'd have an interesting post.
Or, more relevant to the present day, the postmoderns:
Jacques Derrida.
If language is only a construct to meet a need, like Dewey, as cited, then the words are only metaphors for something else, something more elemental. Deconstructing to the basis for "1+1=2" will help us arrive at the intent and need for each component. The "1" may or may not be necessarily numeric, depending upon the assigned metaphor to the expression "1". Derrida finds no meaning in any component other than what is assigned by the readers and authors. It's all just metaphor.
Even outside of Christianity, a stable epistemology can lead to quality mathematics. It existed in the Islamic world as well as the Hindu world. I don't know that Christianity has any ownership on this matter. But what is important to note is that, with the entrance of relativism, especially in the extreme forms of Dewey and Derrida, knowledge and understanding can and often do suffer.
A companion study might be a review of the changing relativistic views and any degradation of math education that has resulted. Or, if it hasn't suffered, why education is so relativistic on the one hand, but so foundationalist with regard to mathematics -- why it is "schizophrenic" with regard to anything that might be considered "absolute".
Collin
http://evangelicalperspective.blogspot.com
OK so I'm not sure what the Christian take on this is supposed to be? Your definition of devine as 'just there' seems like a good start but how does it apply to mathematics really? For any particular field, you're going to get to a 'just there' point which may or may not really be the devine 'just there'. It may very well be the point at which you have reached the boundaries of the field and now other fields take over.
For example, if you subscribe to the theory that 1+1=2 is sensation then math is 'just there' as a sensation just like aroma is 'just there' as a basic element in gourmet coffee brewing. But it isn't the devine 'just there'. There is a whole field of study devoted to sensations and how we perceive them. No doubt that too has a 'just there' point where you keep going.
Of course Christianity would assume there is a final 'just there' point as would other theisms and philosophies but it would seem that 'just there' point may be as far away from mathematics as it is from brewing coffee or simply speaking English. (Speaking English is another example, at some point English is 'just there' as a language but that point has little to do with Christianity. The language a non-Christian like Christopher Hitchens speaks is the same as CS Lewis. One isn't speaking 'non-Christian English' while the other is speaking 'Christian English').
Collin,
Why don't you read an entry for contextualism? That might have some answers for you about how to go about doing mathematics while at the same time entertaining doubts about philosophical beliefs once taken for granted.
http://www.iep.utm.edu/c/contextu.htm
I discuss the rhetorical strategy employed by Clouser and Carter here. In short, the choice of "divine" to describe what's "just there" is a savvy little move.
What all of the explanations have in common, what all non-theistic views share, is a tendency to produce theories that are reductionist -- the theory claims to have found the part of the world that everything else is either identical with or depends on. This is why the Christian view on math, science, and everything else must ultimately differ from theories predicated on other religious beliefs.
I think there is some equivocation going on here. You explain how theistic and non-theistic views differ, but then you conclude that the Christian view of math must differ from theories predicated on other religious beliefs. You're seem to be equivocating Christianity with theism and other religions with non-theism.
You still haven't described a "distinctly Christian" approach to mathematics. How do the features that distinguish Christianity from other forms of theism impact our view of mathematics?
Indeed what's non-Christian, for example, about holding that mathematics is a sensation that is 'just there' and then saying our sensations derive from some more fundamental 'just there'? It seems as if a Christian, when limiting himself to talking about mathematics, could end up in any of the schools you described as well as others.
If one Christian can honestly approach mathematics as a
Leibnitz and another can honestly approach it as a Millite or Deweite then it would seem that there is no particularly Christian approach to mathematics anymore than there's a Christian appraoch to brewing coffee.
I think this is the fundamental difficult with the piece. Even if you grant that underlying a philosophy of mathematics is something we might consider a religious belief or set of beliefs (considered broadly) that fact doesn't *entail* that there is a *distinctly* Christian philosophy of mathematics.
Nancy Pearsey makes the same mistake in her book. She starts off really well by saying that we need to fashion a Christian worldview. She finishes, though, by dictating what that looks like. There is very little liberty in telling me that to be a Christian means I have to sign off on every jot and tittle of the Discovery Institute.
How dare they?
http://www.zeitgeistmovie.com
A Christian world view is important, but I think it's also important not to get bogged down in the details of some things.
For instance, 1+1 is still 2, no matter what any of those men though about it, and no matter how we decide to think about it.
What does change is how we decide to use and apply this simple truth. How is it incorporated into our Christian world view, and how it reacts with other things we have incorporated.
A Christian world view is one that takes things and creatively makes them better and more than they were before we got our hand on them. But the foundation is always God's work, as he is the origin of all.
OK so what's a Christian world view and how is it different from just a simple Christian view?
Joe wrote:
While I certainly understand their hesitation, I do in fact believe there is a Christian view of mathematics. Indeed, I believe that there is a distinctly Christian view of everything.
I don't see what the 'distinctly Christian' view of 1+1=2 is unless it's a generic "because God wants it that way" implicitly tacked onto every statement of fact and conclusion.
In that cse the only thing distint thing about a "distinctly Christian" math textbook is that it will have a lot more pages than its "distinctly atheist" counterpart.
I don't see what theism has to do with anything. Leibniz was a Platonist on this issue, and he was also a theist, and so he grounded the necessary truths of mathematics in God's nature, as Thomas Aquinas and others had before him. But Platonism itself doesn't have anything to do with theism. It just means that numbers exist transcendently, apart from the concrete world and even if no concrete objects existed. Russell's view involves abstract, transcendent numbers also, even if he defined them in terms of sets. Contemporary naturalists like W.V. Quine and Crispin Wright have done the same thing. But you could just as easily be a non-reductionist Platonist as a reductionist one, all without accepting the existence of any divinity. These are just necessary existents in the same way that sets and logical truths are. Where does theism come in?
Marco,
I didn't ask the questions because I don't know.
For further study and clarification (and answers to objections) on this topic...
we recommend not only Clouser's book "The Myth Of Religious Neutrality," but also "Knowing With The Heart".
You can find a number of Roy Clouser's other writings online here.
I recommend starting with:
1) Reason and Belief in God
2) The Uniqueness of D's Program...
3) On the General Relation of...
4) A Sketch of D's Philosophy of Science...
5) Is God Eternal?
6) Prospects for Theistic Science [&responses]
Collin,
Awesome about having everything figured out. So what are the answers? I'm curious... but the answers can't be 1.) Everyone in the world is stupid except for me or 2.) Sin.
Marco,
Why so hostile?
The answer, the problem we face, is our various forms of Platonism. While it seemed to server the west well in the past (Aquinas) it has brought us much grief of late (Kant).
Collin
I'm with Jeremy here. I basically think there *are* differences, to be sure, between various ways of intending "1+1=2" as an utterance, and various ways of using it, etc. But it's not clear to me that Leibniz was either especially good or bad, for example, or that Mill's POV as expressed above was any more or less concordant with theism than that of Leibniz.
Nor are the comments very helpful. Why should the recondite disputations of career math guys be more absorbing than the really legitimate question of whether an archetypal 'it just is' fact is really an instance of interpretation?
And how is it possible to use "Dewey and Derrida" in the same sentence as exemplars of some common form of "relativism"? Look up "essentialism," friend.
Joe's opening statement was excellent: even basic 'facts' are instances of interpretation, appropriation, and exploitation when they take place historically, that is, when they are involved in our (intersubjectivity) lives. The interesting part is that some of them interact *very heavily* with our other beliefs; and some, like '1+1=2', do not. How I intend '1+1=2' seems to have fewer practical effects than whether I hold it to be true.
Cheers,
PGE
Collin,
My response came across as snarkier than I intended. I'll be honest, though, I have no idea what your response means.
PGE,
The reference to Fermat's last theorem obviously didn't imply that math is more interesting than philosophy. It's rather that they are different.
If Joe is talking about how a Christian goes about doing math (or any other "secular" field), which is really the point of talking about forming a Christian Worldview (because, really, how many of us are professional philosophers?), then the point is difficult to make.
If Joe is merely claiming that religious beliefs have some effect on the philosophy of mathematics, well, then that strikes me as a truism.
If Joe is merely claiming that religious beliefs have some effect on the philosophy of mathematics, well, then that strikes me as a truism.
Perhaps but it seems like the "Christian view of mathematics", whatever that is, is no different than any other view of mathematics. Mathematicians of whatever stripe will have the same answers, the same theorems etc. They will differ on what we call philosophy. Did God make the numbers, did the numbers make God or are both illusions? These aren't really mathematics questions.
The link Joe provided had some interesting stuff but I'm pressed for time to read it in detail. Some things jumped out:
The presumption seems to lie with an accurate understanding of "anything revealed by God". I would suggest the reverse rule would be more useful for religion. Any reading of the Bible is wrong if it contradicts reality. Since reality contains any revelation by God to man, the revelation itself must be consistent with reality. If, for example, your 'reading of the revelation' seems to say the earth is 10,000 or less years old then that is wrong because reality says otherwise (true you could try the argument that the revelation is a lie...but then it isn't technically a revelation).
The second proposal is all important truths can be found in scripture. He rightfully discounts this belief as getting things backwards. It still persits in some quarters such as the "Bible Code" which assumes the Bible will answer anything you want if you just figure out how to unscramble the letters (sort of like a Magic 8 Ball).
Joe,
This is an excellent piece.
You tackle a tricky subject and do it justice.
I happened to be thinking about the meaning of "1 + 1 = 2" just this past weekend, and I'd like to share with you what I came up with.
"1" is the unit of counting. That is to say, when we go, "1, 2, 3, 4" we are creating a sequence of thing-ies called numbers which is increasing by the amount "1".
"2" is the number which comes after one in the counting sequence.
"1 + 1 = 2" is a statement about the numbers "1" and "2", and about the acts of "counting" and "adding". When we say "1 + 1 = 2", we are saying that "Adding the counting unit ("1") to itself ("1" again) is the same as counting from the number "1" to the number "2".
This is something that we learn at so early an age that it very quickly becomes bloody obvious and it is hard for us to see what is being said at all. In fact, we all have forgotten that we learned this, we have forgotten that there was a time when we didn't know that "1 + 1 = 2". It's kind of like the geometrical fact that two straight lines can only intersect in one point -- our reaction is, "Well of course, how could it be otherwise?"
However, if our universe is curved, like the surface of a sphere, then two straight lines can in fact meet in more than one point. Likewise, if addition is defined as what happens when you add two velocities (such as walking on a moving train), then one plus one does not equal two -- due to Einstein's relativity, "1 + 1" is only equal to about "1.9999999999990" (or some other number, depending on how fast the number "1" is defined to be).
So the statement "1 + 1 = 2" is not necessarily a reflection of immutable reality. It is, rather, a consequence of our definitions of "1", "2", and "+", and our definitions of "equal" and "counting" as well. It is a statement that says, "If you define '1', '2', '+', and '=', in a certain special way (which happens to correspond to the ways in which nearly 100% of all people define them), then '1 + 1 = 2' is a logical consequence of those definitions."
So where do all these numbers and statements exist? Is there a Platonic realm of ideal mathematical concepts?
Probably not. Numbers and other math concepts are just ideas that are useful for dealing with reality, due to the tendency of many things in nature to exist in discrete units that can be counted. Countless natural phenomena are able to be counted: stars, grains of sand, people, days and nights, and so on. They all have a countable or discrete aspect to them, and that is why whole numbers are an everyday concept. And that is how math ideas are technical concepts that have been derived from ordinary everyday concepts, which have been around forever because of the nature of the world we live in.
So here is where the math came from:
Reality, for the last 10 billion years or so, has manifested itself in all kinds of discrete, whole-number forms and ways.
People have observed and interacted with the whole-number aspect of reality since before there were people -- that is to say, our primate ancestors knew what a discrete whole was before we did. When people developed language, they invented words for numbers and for math actions such as counting and adding. And when people invented math, they used the numbers and the ideas they had already been using to write logic-statements such as "1 + 1 = 2", which are now known as "equations".
So math is a product of the way our brains automatically comprehend reality, with a big dose of logic - assisted by the manipulation of special symbols - thrown in. It's a specialized kind of writing and thinking.
So it's certainly a tool, and it's certainly based on logic, but as you point out, it involves a whole lot more than that as well.
Thanks for the interesting post, sir.
Why do we speak of "Christian" mathematics, and
of the need for Christians to develop a distinct
approach? For reasons that have nothing to do with
theory, and everything to do with the human heart.
We seek an Origin somewhere--if not in a divinity
transcendent from creation and its order, then in
something within creation and its order. This is
the predisposition of fallen man: to inordinance;
to elevate -something- in creation to a place of
usurping dominance. This is what it is to idolize,
whether the idol be a sculpture, a dictator, or a
theoretical concept. Insofar as regeneration of the
heart is needed for becoming free of one's idols,
it is for -existentially- reasons that Christians
would lead the work.
Some may consider this claim to be the heights of
hubris! While I take it to be true, my intent is
not to argue this point, but to explain how the
adjective "Christian" ought to be understood in
connection with mathematics. It is probably a
point of confusion.
I expound this in the following (written before
the first two paragraphs. Please forgive the
redundancies.)
--
Clouser does not assert that one needs, in principle,
to be a Christian, in order to have a "Christian" view
of mathematics (or any theory). His argument, if I
understand it aright, is that we are prone to view
mathematics (or any theory) with preferential regard
to some aspect of the order creation. This preferential
regard (or slant, or bias) does not give mathematics
its proper due, as mathematics qua mathematics is
thereby reduced to mathematics qua something else.
The "Christian" view of mathematics is simply a
non-reductionist view. It is, (says me now), not
a static view, but one that naturally grows both
with the subject, and with a growing understanding
of the created order. It is naturally informed by
an (historically developing) "Christian" ontology,
committed to a non-reductionist understanding of
of the entire order of creation (in the broadest
terms, with the special sciences working out the
manifestation of this order in special ways -- eg
mathematical, biological, economic order.)
This ontology does not demand commitment to the
Christian faith. The "religious" requirement (the
requirement of the heart) is simply to not elevate
any aspect of the creational order to be "boss"
over the rest in one's theoretical reflections,
usurping the unique relative autonomy of the other
aspects. (One recognizes a psychological dimension
to mathematics without reducing mathematics to a
psychic phenomenon). The ontology is, with respect
to how the term is unusually applied, "religiously"
neutral, and does not involve special pleading.
It has "an empirical character, apart from its view
on the Origin of the creation and its lawfulness."
(MD Stafleu, Time and Again (preface), referring to
physics in particular.)
- Largo
Joe, some years ago one of the concocters of the "Outcomes Based Education" approach made the statement, "In our curriculum, one plus one doesn't necessarily equal two if the student doesn't feel good about that answer."
Even "facts" aren't facts anymore once the First Fact has been abandoned: "In the beginning, God…"
Really? Who was the person who said that? When did they say it? What was the context and what authority did they have? Did they write it up as a textbook? A lesson plan? The curriculum's of a kindergarten, grade school or high school?
A fast review of outcomes based education on wikipedia doesn't seem to suggest such a statement would easily square with it (http://en.wikipedia.org/wiki/Outcomes_Based_Education).
The article on standards based mathematics also doesn't seem to lend much support for such a view. (http://en.wikipedia.org/wiki/Standards-based_mathematics).
I know, it isn't always easy to remember when or where you heard or read something so if you can't cite who said it maybe you could tell us about the context you heard or read it in? Was it at a school meeting? Reading a journal article? Perhaps a textbook on education theory or something?
Since you place get importance on facts I think you should be able to help us get a few more facts about your quote.
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