David Hume is the greatest philosopher Britain has produced and an intellectual hero to many atheists. His arguments against religion are clear, incisive and devastating. The only fly in the ointment is the very strong evidence that he wasn't an atheist at all, but an agnostic.

There is a thread of uncertainty than runs through all Hume's works on religion, yet this shadow of doubt does not lead Hume to pull his punches. The Hume who I hope emerges over the next eight weeks is an exemplary model of how the most pressing and telling critiques of religion not only cannot, but should not, attempt to deliver any fatal blows.

A second recurrent theme of this series will be how Hume had a knack of getting to the heart of issues, even when he lacked a great deal of information that we have now. For example, his arguments about the origins of the universe, which we'll come to in week three, are just as telling post-Darwin as they were in the 18th century.

Hume's writings on religion are scattered across several works. Apart from his Dialogues Concerning Natural Religion and Natural History of Religion, we'll also be looking at his essay On the Immortality of the Soul and two chapters from An Enquiry concerning Human Understanding, section XI and, first, perhaps the best-known, his essay Of Miracles, which comprises section X.

"It is strange that such prodigious events never happen in our day," Hume imagines a "judicious reader" saying of miracles. We might think the same today, but many would evidently disagree. From Hindus flocking to see statues of Ganesha drink milk in temples all over the world in 1995, to the miracle of the plane crash in the Hudson river, belief in the miraculous is not confined to the "ignorant and barbarous nations", as Hume put it.

But what makes an event a miracle? Hume was very precise about this. It is "a transgression of a law of nature by a particular volition of the deity, or the interposition of some invisible agent." Hume didn't seem to think this was very controversial, since he originally relegated his definition to a mere footnote.

Two hundred and fifty years later, however, this gloss is more contentious. It should not be. Hume was right, and any attempt to make a miracle anything less destroys the phenomenon it strives to name.

Many resist the definition because they do not see why a miracle should not operate within the constraints of physical laws. Take the plane crash in the Hudson, for example. For it to be miraculous, there is no need for the plane to have defied gravity, or Newton's three laws of motion. What made it a miracle was that, for everyone to have survived, an unfeasibly unlikely series of events had to have occurred, but all individually within the bounds of physical possibility.

This explanation will not do. If the series of causes and effects which comprised the crash proceeded without any interference from a divine agency, then there was only a miracle in the figurative sense of an extremely unlikely and fortuitous sequence of events. But if God did so much as even hold up a sub-atomic collision here or delay a transfer of energy there, then the laws of nature have indeed been violated, for physical events would have been prevented from having their usual causes, as physical laws say they must.

"Laws of nature" are not simply limits on how bodies can behave which leave room for them to do any number of specific things, like cosmic speed restrictors on lorries, or thermostatic switches. They go deeper than this. Every single physical event in the universe obeys them. God needn't make something travel faster than the speed of light to break them; all he need do is make it travel faster than its mass and acceleration determine it should do.

Because someone is sure to mention it, let me briefly dismiss the last refuge of the pseudo-rational mystery-monger: quantum physics. Not all causes have determinate effects, you might think. Some physical reactions are merely probabilistic. Given that this is the case, could God not create miracles by exploiting this uncertainty? A cause could have one of two effects, and by making sure it has one rather than the other, God steers nature without changing its laws.

This answer is just as hopeless as the standard one. If quantum effects are indeterminate, then the only way for them to occur within the laws of nature is indeterminately. The moment God steps in, he changes the effect from indeterminate to determinate – he tears up nature's rule book.

Hume is therefore right. Miracles are violations of the laws of nature. The merely extraordinary is not miraculous.

That so many find this so hard to accept reflects the fact that humans have problems with conceiving the unlikely. That very improbable things will often happen is entirely predictable from the sheer number of opportunities each day provides for exceptions to the norm to occur. We also tend to overestimate how improbable things are. The miracle of the Hudson, for instance, is far less of a miracle when you think about the time, expense and expertise put into pilot and cabin crew safety training, and aircraft design.

So now we've defined what a miracle truly is, can we say any have ever occurred, or even could occur? I'll discuss Hume's answer next week.

Read more blogs from the 'How to believe' series