I EXPECTED the talk to be way above my head, spiralling into the swirling abstractions of the glass-bead game, touching on higher mathematics, fugal resonances, polyphonic patterns, and then moving into the enthralling and baffling technicalities of the piano opening, the gambit and the gambit declined.

But no. It was all about blood and killing. It was all about emotion. The conversation was not cerebral at all, it was entirely visceral. The glorious algebra of it all was something that may, perhaps, be discussed another day. Right now, the talk was all of killing and of being killed.

He’s gone for the jugular. He’s bleeding to death before our eyes. He’s landed the knockout blow. He’s on the ropes. He’s a dead man walking. That’s the move, that was lethal, that’s killed him. He can’t go on, he’s taken too much punishment. It was all blood and guts, as if we were in the middle of the shoot-out in Taxi Driver.

And all the time, the two contestants sat there in their neat, clean, and noticeably unbloody suits. Looking at each other. Looking away. Looking back.

And then, all at once, a whir of action. One contestant stretched out his hand — and then snapped it back again. A few minutes later, the other contestant stood up — and walked about a bit. And I tell you quite truthfully that it really was enthralling. Because what they were doing was not above my head at all.

They were trying to smash each other to bits. And I have seen that happen with Mike Tyson, Malcolm Marshall and Michael Schumacher. The way they were trying to smash each other, yes, that was indeed above my head. But I had no problem whatsoever in relating to the struggle for supremacy between Garry Kasparov and Anatoly Karpov in New York.

It was the World Chess Championship. I subsequently spent a lot of time covering another World Chess Championship, between Kasparov and Nigel Short in London, a bout of ritual bloodletting that was sponsored by The Times. I remember meeting the rather likeable Short several times. Although he managed to keep a few of the ironies about himself, the impression he increasingly gave was of a man being clubbed silly.

And this was tough for a man who had proved his own hardness of mind time and again in years of metaphorical combat. But he had met someone harder and, at times, was made to look like a 12-year-old playing with a grown-up. Short was in a wrestling-with-your-dad situation. He was a man of suffering, that much was inescapable.

The British Chess Federation is trying to persuade Sport England that chess is a sport. This is an interesting academic argument but for one thing: if you are reckoned to be a sport, you can get funded and you can get tax breaks. That makes it a serious business rather than a saloon-bar discussion. Definition is crucial. Chess has a huge international following, but in this country, we don’t think of it as a sport. My reports on the Kasparov-Short encounter appeared on the news pages.

So what about darts? That is everybody’s test case. But Sport England has, indeed, recognised darts as a sport. Clearly, great athleticism is not the definition of sport. Sport England demands that applicants show “physical agility”. An activity that you mostly do lying down can therefore qualify as a sport. Rifle shooting, for example. Most sports, as we understand the term, are about movement. Shooting is about absence of movement. If you can reach a state in which, at the moment of truth, nothing moves but your finger, you are shooting well. It is a high and rare skill, a test of physical and mental ability and it is an Olympic sport.

It is also one of the few sports in which a large Scotch gives you a distinct advantage, drink in well-judged quantities being a notorious nerve-settler. Accordingly, shooters are drugs-tested for booze. Which brings us back to darts. Darts is also a test of stillness and accuracy, and a well-judged intake of booze is a genuine performance enhancer.

Watch Phil Taylor throwing. It is like a frieze on a Greek drinking vessel, body still, shoulder rock-steady, a classical study of physical perfection. The arm moves only from the elbow, a perfect technique. But what gives him the edge is his mind. Taylor’s mental strength gives him both his physical steadiness and his ability to hit doubles and close out legs and sets and matches. In short, darts is a physical skill that is backed by mental strength. That describes darts, it may very well work towards a definition of sport.

I have seen this combination of physical skill and mental strength in David Beckham, Martin Johnson, Nasser Hussain. I have also seen it in Katarina Witt, Dame Ellen MacArthur and Lucinda Green, skater, sailor and three-day eventer. It is a combination that makes sport watchable, beauty and intensity in the action, the revelation of character, of mental strength in the narrative of the event.

There is resistance to the idea of darts being a sport, because it doesn’t seem to be physically demanding. There is some cultural throwback involved here, a notion that sport has to be good for you, that sport has to make a person morally and physically better. This goes back to the Victorian idea of sport as something that stops boys masturbating or, still worse, leaping into bed with each other. “The rest was only sending you all to bed dead tired,” the redoubtable headmaster says in Rudyard Kipling’s Stalky & Co. If a game doesn’t make you tired, then there is resistance to the idea that it is a sport. Try telling that to Kasparov and Short.

Sporting people are still keen to see sport as something that is virtuous. Virtuous because it is tiring, at the very least. We must recognise that this pursuit of virtue is a confusion when we come to definitions. All the same, if we stick with the notion of sport as a marriage of physical skill and mental strength — and I must confess that I am happy with it — chess seems to be a loser.

And yet chess can claim to teach certain kinds of virtues: how to think, how to plan, how to recognise patterns, how to win, how to lose. But virtue is not enough to make chess a sport. Football, darts, shooting and ice skating — they are sports. But chess is not a sport, not in the terms that I suggest.

That doesn’t stop chess from being a good thing, whether we are talking about virtue or pleasure. It doesn’t stop chess being about the pursuit of excellence, it doesn’t stop chess masters from being people of strength and skill. They are just not sportsmen. It is not my definition that is wrong. It is the categorisation. Chess is a mindsport. It shouldn’t be compared to darts and football, but with bridge, Scrabble, poker, spoof, paper-scissors-stone. And as such, respected, and given the financial support and the tax breaks it craves.

So we’d better close with that definition. Sport: a competitive activity that rewards a combination of physical ability and mental strength. Or can anyone do better?