Judging from the questions people ask scientists who are also Christians, I take it that popular opinion holds that modern science and belief in miracles are fundamentally incompatible. I don't know why this is exactly, but I'm sure it's all David Hume's fault.
The argument seems to go something like this. Science is based on the understanding that all of nature exhibits certain regularities, called laws, and that by examining its parts through repeatable experiments, we can learn more about it. If miracles exist, then this regularity doesn't hold. Yet we know that science keeps telling us more and more about the way the universe really is. So miracles don't exist.
But that doesn't follow. If science is based on studying things which are repeatable, then science will tell you about those aspects of the universe which tend to repeat themselves. One will find it extremely difficult to prove that the whole universe follows (and always has followed) certain laws without any irregularity. If you say that we have not found any such irregularity yet, you are begging the question. First of all, a lot of people disagree with you, based on the amount of testimony that exists claiming miracles have occurred. Secondly, your sample size is vanishingly small--especially if modern science is in fact correct about the size and age of the universe. The only way you could possibly extrapolate from such a small sample size is to assume what you want to prove.
On the other hand, I confess that despite my religious affections (or perhaps because of them, as I'll explain shortly), I am attracted to the idea of a globally regular universe--that is, a universe without miracles. This would appear to put me in quite a predicament, and I'm not always sure how to overcome it.
My reason for being attracted to this idea is entirely aesthetic. The most incomprehensible thing about the universe, Einstein said, is that it is comprehensible. That is, the most incomprehensibly beautiful thing. You start with a very short list of axioms, you derive a mathematical theory, and then you watch in awe as physical objects actually seem to obey this theory, as if the Creator of the universe were some sort of divine mathematician. And you realize that whatever inherent limitations this puts on you and your life--for instance, I suppose it means death for us humans is probably inevitable--it is simply pure joy to see that at the heart of all reality is supreme rationality, such that only through the slow and painstaking efforts of the greatest minds can human beings start to glimpse the underlying principles. If we have a purpose in this world, it is to be the products of such a divinely rational order.
Aside from sheer awe, however, there is a big payoff to living in a world guided by universal, comprehensible laws. Through science, we can master the world, creating technologies which push back against death and disease, increase our comfort, and allow us to live more fulfilling lives. If this is starting to sound like some sort of modern secular religion, is that at all surprising? Haven't we, thanks to the scientific revolution, in fact stumbled onto something quite extraordinary? All religions have features both attractive and repulsive, and this new scientific religion is no different. It may not promise eternity or redemption, but it promises both awe of the transcendent and practical means by which we can live meaningful lives. That is hardly something to scoff at.
Now if miracles are real, then this glorious vision is tainted, if not shattered. Not only does it ruin the idea of perfectly uniform mathematical laws governing the universe, but it even makes us wonder why we put so much effort into understanding the universe when there is a much easier way. If God can simply cure diseases and raise the dead by the uttering a word, why doesn't he? Why do we slave away trying to understand laws which are not really laws at all, when all the while God could step in and just fix everything whenever he likes? It seems both sacrilegeous and immoral. Only a divine bully with no respect for transcendent beauty could possibly intervene at such irregular intervals, while hiding in the dark the rest of the time.
Thus the problem of miracles reduces to the problem of evil: how is God's existence compatible with the presence of suffering, death, and disasters in this world? These latter are most certainly consistent with the laws of the universe--laws cannot be broken, even if it would suit our purposes to do so. But a personal God, capable of intervening--how can he allow it?
The other side is not without objection. There is, for instance, the problem of good: how can we make the concept of goodness intelligible in a world governed entirely by impersonal, unyielding laws? We can push back against suffering and death, yes, but to what end? Short life and disease do not threaten modern people nearly as much as boredom, depression, and even suicide. Try as we might to create our own meaning, anything we invent without any reference to a transcendent source eventually appears, well, meaningless.
I am inclined to think that neither of these problems can really be "solved." So I find my reflections on this matter humbling, both as a mathematician and as a Christian. On the one hand, as much as many of us would love to claim science as the banner of objective truth, the reality is that the vision driving us is every bit as religious as Christianity or any other major world religion, with just as many weaknesses. On the other hand, as a Christian I need to consider this modern, secular religion to be a real contender for my heart and soul. When Einstein spoke of a "cosmic religious feeling," he wasn't kidding.
Whether one of these religions will ultimately succeed in shaping our civilization in the future surely depends in part on how we answer serious philosophical questions like that of miracles. But I believe the more important factor is the human heart, which, once it receives and adopts a certain of vision of the world, will follow it far and wide, for reasons far beyond the intellect's comprehension. For the Christian's heart, of course there are miracles. And for the modernist's, of course there are not. And for those of us somewhere in between, I suppose there is the hope that one day we'll know for sure.