How can a chemical that causes cancer also cure it? That's the paradox scientists at the University of Dundee think they have solved. Arsenic has been thought to be effective in treating leukaemia since the eighteenth century, but little was understood about how this poison could have therapeutic benefits. The Dundee scientists think they now have the answer.

However, for many people, what's most striking about this story is not the solution but the paradox itself. Human beings have been intrigued by paradoxes since at least the time of the ancient Greek philosophers, who formulated several that people still puzzle over today. Apparently, water-tight reasoning tells us one thing, while another argument, or experience, tells us another. Both sides of the contradiction can't be true, but both appear to be equally rock-solid. It's the intellectual equivalent of an optical illusion, when what we know and what we see just don't match up.

People tend to respond to paradoxes in two ways. One is to be intrigued by the puzzle and become engrossed in the search for solutions. But in my experience, the more common reaction is to see them as evidence that rationality is not all it's been cracked up to be and we would be better off if we freed our minds from the tyranny of binary logic. If logic says kill can't also be cure, but experience shows it is, then so much for logic.

There are plenty of historical examples wheeled out to support this view. For instance, it is said that according to the laws of aerodynamics, bumble-bees can't fly. Fortunately, bees don't read physics and buzz around anyway. It has also been thought that scientific laws meant humans could never be airborne, that you would suffocate if you travelled in a train, or that vacuums were impossible. If we had allowed ourselves to be constrained by what reason told us was possible, we'd still be living in the stone age.

It's a forceful argument but totally wrong. What limits us is not rationality but over-confidence about our ability to use it, and an excessive faith in the state of current science. Train travel and vacuums only appeared to contradict the dictate of reason. Now we understand how both work, we find that there are perfectly rational explanations for both. As for the bumble-bee, its defiance of science is just an urban myth. Crude application of now superseded laws of aerodynamics might make its flight baffling, but do the calculations correctly using up-to-date science and nothing is paradoxical about the insect at all.

Likewise, the arsenic "paradox" was solved in Dundee by scientists who, true to the best values of rationality and experimental method, broke down the components of arsenic and identified what each specific molecule did. The paradox of "arsenic cures and causes cancer" was explained as "some molecules in arsenic cause some cancers, but other molecules destroy others". Far from defying logic, the paradox is only properly understood if we reason logically to solve it.

The same rigorous way of thinking needs to be applied to other seeming paradoxes, such as that of homeopathy. Conventional science says that it cannot work because the supposedly active substances are too diluted to have any effect. To which the homeopaths reply that "the memory of water" may not be understood, but it's real, and it is simple narrow-mindedness to say otherwise.

The homeopaths are right to say that just because something seems impossible by current science, it doesn't mean it is. That's why arguing that homeopathy cannot work by definition is misguided. But that doesn't mean reason just takes a holiday and we all celebrate the triumph of natural mystery over human knowledge. We still have to use our heads to decide if homeopathy actually does work, and on that front, it just hasn't proved itself. Defying current science is not enough to show something is false, but we still need some good reasons to be persuaded it is true.

The clash between what science says should work and what actually does creates paradoxes that we struggle to resolve. But these are challenges for reason, not to it. Such contradictions provide no comfort for those who like to believe without evidence or deny the achievements of human inquiry. The true nature and value of reason is distorted both by those who belittle rationality, and those who think it has already fully mapped out the territories of the possible and impossible.