Reason destroys itself by Roger Penrose in New Scientist

Reason destroys itself

• 23 July 2008
• From New Scientist Print Edition. Subscribe and get 4 free issues.
• Mathematician Roger Penrose

6: Reason destroys itself

Even in formal mathematics, reason breaks its own rules, says mathematician Roger Penrose. Watch a related video.

DO WE know for certain that 2 plus 2 equals 4? Of course we don't.

Maybe every time everybody in the whole world has ever done that calculation and reasoned it through, they've made a mistake. Maybe it isn't 4, it's really 5.

There is a very, very small chance that this has happened. But now I am talking about probabilities:

I am using mathematics again, on the basis that it makes sense in the first place. So my reasoning is circular and, as often happens, you have to stand back and ask, well, what is reasonable?

We have to distinguish this question from pure Reason - with a capital "R". Philosopher David Hume notoriously argued, applying the rules of that formal Reason, that induction is impossible: however often we add 2 to 2 and get 4, that does not logically tell us anything about the result next time. And doubt does permeate all our reasoning, or should. But at some point we have to stand back and ask what is reasonable - with a small "r".

When we ask what is reasonable from our direct experiences, we are finding and applying general rules about behaviour which are consistent. In mathematics we demand absolute formal consistency. But often, even when you're doing mathematics, you explore something by trying to prove it, get stuck, and say, oh, maybe it's not true, and look for a reasonable path for your next round of applying pure Reason.

The mathematician Kurt Gödel went through a process like this as he destroyed any hope that formal Reason could be universal. In 1900 David Hilbert proposed that we should build the whole of mathematics on formal, logical rules. But in 1931 Gödel showed that if you have any set of trustable rules that are computationally checkable (a machine could go through them and see whether you've applied them correctly), then statements exist that you have to accept as well, according to the rules, but that you cannot arrive at by means of those rules.

Gödel used the word "intuition" to describe how you get to those statements. For me, that word has connotations which are not quite appropriate: I'd say we need understanding and insight. Gödel's result is one of the things some people correctly use to pick holes in Reason. Another is the very unreasonable behaviour of quantum mechanics where, for example, particles seem to be able to be in two places at once. This doesn't match our normal experience, so we say it's unreasonable.

“Quantum mechanics is one thing people use to pick holes in Reason”

Personally, I think there is something not quite right about quantum mechanics. And quantum mechanics seems to be to blame for some people who'd like to be exempt from Reason being able to say: "Science is full of these contradictions and unreasonableness - and so I can more or less say anything."

You had people saying we shouldn't turn on the Large Hadron Collider experiment because a small probability exists that it might create black holes that would annihilate Earth. Sensible scientists say that this is ridiculous, there's no chance. On the other hand, there's a small chance that accepted theory is wrong, so there is a chance!

That's one example of people putting unreasonable standards on being sure. Another is climate change. Most scientists say that the climate is changing and it's changing for quite clear reasons - because we are pumping in all this carbon dioxide, for one. There's no puzzle, we can see it happening, and we can see why it's happening. But a few don't agree. True, sometimes the small minority turns out to be right. But it doesn't mean you should do what the small minority say. The majority is a majority for good reasons.

To me, Reason is essential for human discourse and all forms of enquiry, whether legal or scientific or mathematical. It is absolutely central. But we have to be reasonable about it.

Read about all the problems with reason in our special issue

Roger Penrose is a mathematical physicist at the University of Oxford, UK

From issue 2666 of New Scientist magazine, 23 July 2008, page 49