WE lost something when, as a culture, we jettisoned the old folk, saying: "Sticks and stones may break your bones, but words will never hurt you." Over the last half-century, with the rise of the notion of self-esteem and increasing numbers of studies into bullying and emotional pain, we appear to have decided that there is no value in saying: "They're only words." Nowadays, researchers investigating bullying at Iowa State University tell us that "sticks and stones may break your bones, but words can kill".

Today, words proliferate. We are surrounded by a digital babble, of twitter and other social media and, for our own good health, we cannot afford to give these words any more power than they already have by taking them too seriously. We can't afford, for instance, to give Twitter trolls any more sense of empowerment than they already have, by considering their words worthy of police and legal attention. We drop the old "sticks and stones" adage at our peril.

Last week saw the climax of what some have called the "social media Olympics", when stories surfaced that a 17-year-old Twitter troll had been arrested after allegedly sending a tweet telling diver Tom Daley, whose father died of cancer last year, that "you let your dad down" by failing to clinch a medal. The spur for Reece Messer's arrest had been over a later, more menacing, tweet and he was handed a harassment warning. His father declared that he had ADHD and "needed help, not punishment". It is possible that, like many internet trolls, his behaviour was the equivalent of someone standing on a street corner and wildly ranting – the kind of tirade that most of us would ignore as mutterings drifting out on the wind, but which are taken more seriously when frozen in the digital air.

Social media is like a vast, virtual playground with all the fun, but also all the ugliness, that we associate with the school playground. But it differs from that world in that it is escapable. No-one has to go there. If you can't take the vile insults and slanderings, you can switch it off. In fact, this is what many celebrities have been doing for decades. They don't look at newspapers or peruse reviews; they just get on with life and ignore what the world appears to be saying. And, indeed, some Twitter users have started doing the same. Last week, Blue Peter presenter Helen Skelton closed her account because of abusive messages, saying: "Turns out I don't have very thick skin after all." Swimmer Rebecca Adlington has given up Twitter for the Olympics, as has gymnast Louis Smith who said: "You can get 100 nice ones but there is that one message that you do not want to see."

However, others decline to turn off – as if, somehow, they expect that the trolls will be silenced. Channel 4 presenter Kirstie Allsopp said recently that she had called in the police over "utterly vile and unacceptable" messages from two anonymous Twitter users. On one tweet she stated: "I will not accept being told to shoot my own womb or bleed to death with a spade in my vagina by anyone. Bullying is unacceptable."

Horrible, misogynistic and vile as this is, is it really bullying? Isn't it more like the random drivel of the societal unconscious? Nasty to hear, but not to be taken personally. These are, surely, the words that once only whirled inside the heads of others, but now stream out to the world via keyboards.

Meanwhile, Twitter has also made celebrity status available to those who otherwise would never have found it, or indeed courted it. You can be afflicted with stalkers, evil fan mail or general abuse without having ever made so much as a low-budget TV talent show performance. Trolling thrives in a world which has generated polarities of fame and anonymity, and where nameless, unrecognised nobodies get to spit on those who have some kind of glory. Having these people arrested surely only increases their sense of power.

Given this, people should perhaps think carefully before signing up to Twitter with a view to cultivating a following. Are you really thick-skinned enough to handle what other people might say about you? In some ways tweeting is an extreme sport of our digital times, the equivalent of virtual white-water rafting, not to be entered as if it were a casual row down a stream.

Those who take it up should observe its perils, and look, for example, to those who survive it well. In this context, diver Tom Daley has shown himself a champion. His own response to the tweets was a virtuoso gentle slap-down, a retweeting of the original post, saying: "After giving it my all ... you get idiots sending me this." Daley didn't shut down his Twitter account and nor, according to his coach, has he been affected by the abuse.

Police have admitted that they "can't control Twitter". Meanwhile, Twitter, a pro-free speech company, doesn't want to monitor or intervene in messages posted by its users. How, then, do we keep calm and carry on?

We live in an age when debates over freedom of speech, political correctness and bullying have made words seem more potent than ever. All the more reason to consider that it may be time to take the "sticks and stones" message seriously again. The point of the old saying, after all, was not that it represented a truth, but that it was an encouraging exhortation. Even in my own childhood, I can remember finding it useful. Words can hurt – there's no denying it – but you can also choose to shrug off those words. You can choose to block a troll. You can choose to say, as Eliza Doolittle did in Pack Up Your Troubles, the ultimate tweet song: "I don't care what the people may say; what the people may say about me."