Archive for June, 2010

Mathematics in the New Creation

June 20, 2010 2 comments

One of the most amazing passages in the bible is near the end of the book of Revelation, where John describes his vision of the new heavens and the new earth:

1Then I saw a new heaven and a new earth, for the first heaven and the first earth had passed away, and the sea was no more. 2And I saw the holy city, new Jerusalem, coming down out of heaven from God, prepared as a bride adorned for her husband. 3And I heard a loud voice from the throne saying, “Behold, the dwelling place of God is with man. He will dwell with them, and they will be his people, and God himself will be with them as their God. 4 He will wipe away every tear from their eyes, and death shall be no more, neither shall there be mourning, nor crying, nor pain anymore, for the former things have passed away.” 5And he who was seated on the throne said, “Behold, I am making all things new.” (Revelation 21.1-5)

So God is planning on remaking the world, and he himself intends to dwell in it somehow. He who was seated on the throne, the king of the new creation, is Jesus (of course) and he is making all things new. All things. So it occurs to me, from time to time, that this activity that I occupy much of my waking life with–that is, mathematics–may be among the things that God will remake in the new creation. At least, I hope it isn’t destined for the scrapheap of history. I don’t think it is, though it may be drastically altered. So I have been thinking about what this will be like.

In general, the idea of God’s redeemed creation is that it will be similar to the world as it is now, and much of it will be the fulfillment or completion of what is already here. However, in spite of this, this world is really only a dim shadow of what is to come. We have to always remember that the new creation will be unimaginably beautiful and alive. All the effects of sin will be removed. It will be a place of peace and boundless love, and God, the source of all such things, will be gloriously present. With this basic set up in mind, let’s consider mathematics. For those of you who don’t actually do mathematics, I hope this will make some sense!

One thing I sometimes wonder about is how difficult mathematics will be in the new creation, and how much of the process of learning mathematics will be retained. For those of you who are unaware, much of my mathematical life is spent in a grueling process of attempting to assimilate and understand new mathematical constructs in order to, among other things, apply them to the problems I am thinking about. Now, this process certainly has its rewards, but it can also be quite frustrating. So I wonder if anything like this will still take place in the new creation.

It is conceivable that God would simply reveal beautiful mathematical truths to us, “fully formed,” so to speak. Yet I think it is somewhat doubtful that he would take away the learning process entirely. I think it likely that our ability to assimilate new ideas will be greatly increased–perhaps increase without bound throughout eternity, in parallel to the increasing beauty of God that is to be revealed–but I think there will always have to be a process involved. In general, I don’t think learning will ever be something that we no longer have to do. There will always be things we don’t know, since we will always be finite beings. I think the frustration will be gone–along with the feelings of failure, self-doubt, or anger–but the journey, the adventure, will be retained. Indeed, it will be greatly enhanced.

For learning, we will need teachers as well as students–though perhaps the roles will not be quite the same as on this earth. Most of the joy in mathematics comes from the moment one comprehends a new truth, or an idea crystallizes clearly for the first time. That, and the ability to share such moments with others. I think that God will give increase to both of these things. I’m not sure what structure, from earth, will remain. I’m talking about the basic format here. Some of it seems worth keeping: at the barest level, chalkboards, I hope, will still be there (Along with “good chalk” as my adviser is fond of saying. If you don’t know what this is, don’t worry about it.).

One relief, I think, is that there will certainly be no opportunity to worry about the passage of time, as there is in an earthly mathematical career. One of the sadder realizations that every mathematician has early on is that there is no way to even come close to learning all the details of the mathematics that he or she is interested in. We have to pick and choose, and we always feel like we could have done more, or need to do more. But, in heaven, there will be no more concern about what areas of mathematics are worth pursuing, worth fitting in to one’s lifetime. So if I, and a few mathematical friends, would like to explore completely the proof of deep and beautiful theorem, a project that might take several years, we could simply go ahead and do it.

When we are free from sin and thus free from pride, all of the silly competitiveness of academia will be gone. This will be another great relief: not to have to worry about one’s thesis, publications, academic reputation, or tenure! And not to possess a personal “body of work” at all except insofar as God has perhaps ordained in order to show to others the beauty of the individual minds he has created. I do suspect he will indeed do this. Euler’s theorems belong to Euler in that there is something of his mind, the first human mind that conceived of them, in the theorems themselves–or rather, they tell us about him. But also about Him, the God who thought the theorem first, and made it part of Euler.

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Rescuing science from scientism

June 16, 2010 9 comments

I recently had a very interesting discussion on a message board. The topic of the thread had to do with an article about “respecting the differences between science and religion.” I’ve found that authors who write things with such titles, in general, have a distorted idea of both science and religion, particularly in relation to their respective epistemologies–that is, what each enable us to know, and what sorts of truths each is and ought to be concerned with. But these things are not well understood these days.

Somehow, in the course of the thread, I found myself taking what might seem like a difficult position. Some of you may have read my previous posts on creation and evolution, and know that I am not a young earth creationist (YEC–this is the idea that God created the universe less than 10,000 years ago, with life pretty much as it is in its present form). In spite of this, I found myself defending the idea that young earth creationism is science, and in particular that creationists follow the scientific method.

People found this claim surprising; indeed, the charge that is always laid against YECs is precisely that they do not follow the scientific method. This is seen to be their chief difficulty–that they ignore the facts and the evidence, and instead base their beliefs about the world on blind faith. I think this picture of them is appealing to people because it reinforces the dichotomy of reason vs. faith, and correspondingly of science vs. religion. Yet, if you look closely at the creationists, ignoring the evidence is not really what they are doing. Indeed, they actually seem quite obsessed with the evidence, the very same body of data that evolutionists study.

The creationists are doing something, and it is clearly quite similar to what ordinary scientists do. The difference, I think, is this obvious point: creationists are committed to the idea, mentioned above, that the earth is less than 10,000 years old. This, and a few other things that they take to be axiomatic about the natural world, shape all of their further reasoning, all of the deductions they make from the data they are presented with.

Well, you might say, this is scientific blasphemy! The whole point of science is to take nothing for granted, to test every hypothesis, to hold everything up to the scrutiny of experimentation. Well, I say, not really. This is an imaginary ideal. In every age, and every community that practices something like science, there will be things taken for granted, propositions about the natural world that are unquestioned, perhaps that have never even occurred to anyone to question, which nevertheless shape the science that is practiced. These ideas are not based on any experiment or known data. Rather they flow out of the worldview and imagination of the community that practices the science.

So I believe young earth creationists are best understood as a particularly insular community that is doing legitimate science within a particular cosmology that the rest of the world has (probably rightly) abandoned. This, I realize, is a somewhat idealistic take–creationists are guilty of many intellectual sins. But, then again, so are we all. The irony, I would argue, is that evolutionists are equally dogmatic about another proposition that is also not established by any experiment: the uniformity of the natural world. This is the belief that unknown causes have not interfered with the natural world in an essential way, throughout the course of its history. This is a basic axiom necessary for the evolutionary paradigm to begin its exploration. A lot more could be said here, and a lot needs to be qualified, but this is a topic for another post.

What I want to focus on now is how to repair the negative effect, in general, of a certain philosophical mindset towards science that is practically taken for granted today by most people. It is the idea that is expressed well by my favorite philosopher, Owen Barfield, writing about the Copernican revolution:

When the ordinary man hears that the Church told Galileo that he might teach Copernicanism as a hypothesis which saved all the celestial phenomena satisfactorily, but ‘not as being the truth’, he laughs. But this was really how Ptolemaic astronomy had been taught! In its actual place in history it was not a casuistical quibble; it was the refusal (unjustified it may be) to allow the introduction of a new and momentous doctrine. It was not simply a new theory of the nature of the celestial movements that was feared, but a new theory of the nature of theory; namely, that, if a hypothesis saves all the appearances, it is identical with the truth. (p. 50, Saving the Appearances, by Owen Barfield)

I love this quote, and I wish with all my heart that more people today understood its significance. Truly modern science effectively began with the notion that, if a scientific theory “saves all of the appearances,” which means that it can predict all phenomena it is concerned with accurately, then it can be identified with the truth. This was the real shift in thinking that was at the heart of the scientific revolution. So from this standpoint, the scientific revolution is not responsible for rapid progress in science directly–rather it is responsible indirectly through the zeal of those who pursue it, through the belief that they were attaining ultimate truth by doing so. This is the idea that has impressed itself on Western consciousness, and it is of great significance to how all of us look at the world today.

And I love the fact that this new philosophy is what the Church, cautious as it always is, was actually opposed to–this is not how the mythology of the scientific revolution is told today. We are taught that the Church opposed the Galileo because he had undermined a cosmology that was taught in the bible. In fact, what was actually being opposed, what was seen as a real danger, was a philosophical shift. And today we see the consequences of that shift–Barfield considers the worldview that developed to be a modern form of idolatry, a sort of worship of the natural world through confusing our representations about it with the truth. In its ugliest form, it is what I would call “scientism.” And this is a philosophy that is casually assumed, without any justification, in news articles, academic journals, and much of our popular entertainment.

Scientism is distinct from science: science is rational and creative, it explores, questions, and delights in knowledge. Scientism is dogmatic and conquering. Science is playful; scientism is utilitarian. Science adores creation for its own sake; scientism worships its own understanding of creation. Now, scientism hails the “scientific method” as the source of the success of science. And I believe that it does so in order to contrast itself to superstition, which is its method of caricaturing religion. I think that this little move has done a great disservice to our understanding of what science is, and what scientists actually accomplish. It causes us to miss the imaginative and human elements to science.

We must come to see that it is not really the scientific method that makes science great. Nor is it totally what makes it progress. This is only one ingredient among many others. Just as mathematics is more than just formal proofs (yet everything must be subjected to them), science is more than a formal method. The scientific method, to the extent that it is even well defined, can only ever really be a filter, a way of making sure our ideas make sense and match up with reality. But much more goes into doing science: scientists engage in a characteristic way of thinking, a frame of mind that considers nature abstractly, trying to explain things in a causal and mathematical framework.

I want to restore the human element to science, and to help us to stop taking ourselves so seriously in regards to it. We need to stop thinking of it as some sort of “ultimate truth project.” Rather, we should rejoice that we are small explorers of a glorious and complex creation. We discover patterns and beauty, we come up with interesting ways to explain them. When these models fail us, we update them, still recognizing the beauty of the old models, and still taking hints from them as to how to go about exploring new things. Creation is our friend, but not our god–it is endlessly fascinating, but exploring it is only one of the reasons why we are here.

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Advice on math graduate school (my mistakes!)

June 5, 2010 Leave a comment

After n years of pursuing a Phd in mathematics, where n is greater than or equal to 5, it appears that I may actually be finishing soon. I’m not done yet, but I’m starting to see the light at the end of the tunnel.

For a while, I’ve been wanting to write down some of my thoughts on how a person can go about learning mathematics, navigating mathematics graduate school in general, and eventually doing mathematical research. This last thing is something I am quite new to, so I don’t have too many thoughts in that regard. But I think I’ve learned quite a bit about the first two in these n years. At least, I’ve made quite a few mistakes, and I’d like to think I’ve learned from them.

I will present my advice in list form (I like lists).

1. Don’t be overly rigorous. But don’t be un-rigorous either.

I list this one first, because it was the lesson that took me longest to learn. There is an important balance to strike in learning  mathematics, and not grasping this balance can lead to disaster. Basically, there are two ways to go wrong in terms of how much we try to understand things: we can look at things too closely, or not closely enough.

I’ve always had absolute confidence in my own understanding of what constitutes a mathematical proof. Consequently, when I first started reading mathematical texts, and attending lectures, I insisted on abiding this standard. I would scrutinize every line of my notes; I would rewrite proofs in textbooks until the entire logic seemed clear to me.

As you might imagine, I got stuck a lot. For one thing, the fact is that many textbooks leave a lot out, and no math lecture ever contains all the details. What took me a long time to realize is that this is OK. You don’t always have to have all the details. Sometimes it is best not to fill in the gaps in proofs yourself–this can be horribly time-consuming, and is often less enlightening than you think it will be. What matters most is being able to understand what is going on, to have the right picture in your head so that you can really work with the objects in consideration.

It certainly matters a lot that you do understand what a proof is, and how to really make one yourself. Thus the opposite error, of being un-rigorous, can get you into a lot of trouble too. I get the feeling that some people read mathematical proofs and don’t actually realize that there are steps missing. Then they try to emulate this style of writing themselves–they copy the style without understanding the actual substance that underlies it. I think this is probably the origin of a lot of the nonsense mathematics out there.

So definitely you should know what a proof is, and especially know how to tell when you have proved something yourself. But, if you focus too much on the details of a particular textbook, you will never get far enough to understand why the authors are doing what they are doing anyway. Consequently:

2. Don’t read linearly.

Here is a great way to waste a few months of your life: decide that you are going to sit down with a giant tome (or even a thin little book) on a difficult subject, and read straight through it. I have embarked on this project several times, and have never been successful. Often the first few pages go well. Then it all falls apart.

What I have come to understand is that when people write a mathematics text, they aren’t usually thinking about the time constraints of the reader. They are just outlining the subject in the way that they find interesting. They put things together as they see them–all the examples and theorems that they think are important to developing the subject, or help them understand what is going on. This is a beautiful thing, and a lot can be gleaned from it if you know how to approach the book. But it can also lead you into trouble if you don’t.

I think it is helpful to look at things like this. In a given day, you only really have a finite amount of mental effort you can exert in trying to understand new things. What you are trying to do is maximize the insight gained with respect to this mental effort. So if I see a part of a proof that looks like its going to be a lot like something I’ve seen before (though one should be cautious in making this judgment), I skip it. There is no sense in banging my head on it; its far better to work out the parts that are new to you, that look like they have the potential to give you a new insight into what is going on. Sometimes I give up on trying to understand a proof altogether. I move on to something else. It is OK to do this! It is far better that to waste hours doing practically nothing, all for the sake of some false sense of completeness.

Rather than reading a text straight through, consider just looking at the definitions and major theorems first. Try reading a proof if you think it looks interesting–don’t worry too much about the previous results that it depends on. Take those for granted (but do understand what they say), and if you find yourself interested in them, you can always try to read the proofs later.

Then maybe try some exercises. This brings me to:

3. Exercises are more important for your understanding of a subject than are learning the proofs of major theorems.

Perhaps an analogy will help illustrate this one. Suppose you want to learn how to play golf. What you could do is get a video of Tiger Woods golfing in one of his classic performances. You could then find the exact golf course on which that performance took place, and try to replicate what Woods did: at each hole, you could try to swing exactly as he did in the video. When the ball doesn’t land in the same spot as in the video, you could pick it up, bring it back, and try again. You agree with yourself to move on as soon as you’ve basically replicated the shot that he took.

Of course, this is a ridiculous plan–assuming you aren’t already a great golfer, such a project would probably be endless. And here is the point: it wouldn’t be an effective way to learn how to golf. You’d be much better off by picking up a golf club, and asking someone for advice on how to swing. You could spend a few weeks just taking practice swings. Then you could incorporate putts, or whatever (I actually don’t know how to play golf). It would take a long time, and you’ll probably never get as good as Tiger Woods, but eventually you could get to the point where you are capable of something at least resembling a respectable round of golf. And you’d get to the point where, by watching Tiger Woods, you might actually learn something.

Math is similar. The problem is that, even if we learn what the great mathematicians have done, by going through it step by step, we don’t really learn how to think like them. We don’t learn a lot by examining the details of a difficult theorem if we aren’t yet accustomed to the ideas that go into it. It is good to know what has been done–this is why we should read theorems, and also get a general idea of where they came from. But it’s by doing exercises that we learn how to think like the people who came up with these theorems–it is by doing the exercises that we gain a feel and an intuition for the objects involved, and where we learn what it means to actually try to prove something.

I’d go so far as to say that it is far better to do all of the exercises in a book, and read none of the proofs, than to do the opposite–read each proof carefully and do none of the exercises. Another fun thing you can do is this: when you come to theorems that have a proof that looks short, try to work it out for yourself. Even if you get stuck you gain insight into what the challenge is.

4. Feel free to learn lots of theorems, ideas, and definitions–even if you don’t learn the details.

This is basically just a more general expansion on the advice above about reading textbooks. I think there are some important points to clarify here, and some nuances. There is always a danger of learning nothing but “what has been done”–of becoming an encyclopedia of famous results that, if it really came down to it, you couldn’t reproduce any of the details or even really say what the theorems you can easily quote are actually about. This should be avoided. I’m talking more about have a sense of what is going on in mathematics as a whole.

In my next post, I hope to talk about classes, seminars, and talking to other mathematicians. For now though, I think I should go work on my thesis…

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