My father was well beyond the danger point of a dive when a weight slipped from his jacket and spiralled downwards into the gloom.

We had arrived at the quayside to discover our Red Sea pleasure boat laden with US soldiers on leave from peacekeeping duties in the Gulf. The platoon's NCO proved to be a seasoned diver. His charges were novices. A 19-year-old private from Arkansas, paired with my father, then 70, for the safety check and first dive of the day, failed to secure the weights that are essential in controlling a diver's surfacing speed.

The mishap illustrated one of many rules that the sea subverts: on dry land a healthy teenager will invariably outperform a septuagenarian in any physical exercise. Underneath the waves, age is as fluid as water. Confronted with a problem, the soldier lost control of his buoyancy, flailing like a fish in a net, while the older man, an emeritus professor of theatre history, calmly recalibrated their dive plan, communicating the change in the limited vocabulary of scuba sign language. Only once we were back on board was the natural order restored, as the senior grouched about the teen, who stretched out in the heat, blithely unaware of the animus he had attracted.

But the natural order is itself in flux. My father still dives in his 80s, still lectures, researches, writes books. People are living longer, sometimes much longer. Across the developed world the average lifetime has lengthened by 30 years since the beginning of the 20th century. The fastest-growing segment of the world population is the very old, with the number of centenarians up from a few thousand in 1950 to 340,000 in 2010 and projected to reach nearly 6 million by 2050. You might have thought we'd use all that extra time to squeeze in a few additional stages of life – from the seven ages of man observed by Shakespeare, when life expectancy at birth was below 40, to maybe 10, 12, 15 stages now that a man born in Stratford-upon-Avon looks forward to an average span of 76.9 years.

Here's the crazy, counterintuitive thing: the ages of man are actually eliding. Youth used to be our last hurrah before the onset of maturity and dotage, each milestone benchmarked against culturally determined expectations. Those expectations are now swirling and re-forming like glassfish in a current. What that means is that the premises on which our governments legislate are outdated. Our economies are based on data that no longer applies. There is a profound disconnect between how we imagine life and how it actually unfolds.

The meanings of age have become elusive; visual clues untrustworthy. Children dress like louche adults. Their parents slouch around in hoodies and trainers. Rising phalanxes of Dorian Grays rely on exercise, diet and cosmetic procedures to remain transcendentally youthful. Glowing teens and twentysomethings are propelled by some of those same procedures into a semblance of premature ageing, their sculpted, frozen faces timeless rather than fresh. Female celebrities don't grow old; they vanish.

And there's another reason our perceptions of age have come adrift: the disappearance of death in the developed world. If we're lucky we may be in the middle of our lives before we see death up close and then it's usually medicalised. Polite societies don't dwell on death; we're expected to dab our eyes and get on with the business of living. In the absence of legitimised outlets, our orphan emotions attach to public bereavements, depositing flowers at the gates of Kensington Palace and composing Facebook tributes for total strangers. But there's also a growing trend to believe we can control death. Assisted suicide is becoming just another lifestyle choice, at least in the abstract. Moreover, science has already added decades to our lives and must surely be on the verge of adding many more. Barely a day goes by without a media report of a breakthrough that seems to promise another lifestyle choice: to defer death altogether.

Things change. Amortals don't, not at the core. These are the swelling ranks of people – and I am one – who live agelessly, doing and consuming many of the same things from teens into old age. For us, the concept of age-appropriate behaviour has little meaning. We don't structure our lives around the inevitability of decline and death because we prefer to ignore it. Perpetual motion is a hallmark of the condition; we are prone to overwork, to adventuring. Nothing banishes those pesky intimations of mortality more effectively than illicit sex or emotional drama or some high-octane combination of the two. Unwitting revolutionaries, we assume all options remain open, from youth into old age, and may be startled if we get round to having children at all to find ourselves reliant on fertility treatments or adoption agencies or surrogates to help us to do so (as a result having fewer kids than non-amortals, but sometimes in batches of two or more). We never consider ourselves too young to pair up, break up, launch businesses, take on the world or too old for fresh commitments, old habits, the latest technologies or new diversions.

David Battiscombe is a 59-year-old English bass-player-turned-property lawyer: "If somebody said tomorrow I had to stop practising law and do something else it wouldn't faze me in the least. And I think whenever I stop doing this I would expect to do something else, probably something completely different. It's the old Woody Allen dead shark thing, you can't stay still. That's just the kiss of death. A shark, if it stops moving, dies."

Battiscombe took up running in 2009 and last month completed the London marathon, a few months shy of his 60th birthday. Other amortals might mark that milestone with a different kind of marathon, a celebratory bender that leaves them mewling and puking like infants. The world is used to the spectacle of baby boomers, who never imagined they would one day wake up to find themselves in danger of being marginalised by the youth-oriented culture they helped to create, challenging shibboleths about age.

Yet amortal impulses are by no means confined to boomers. Amortality has reached a tipping point and is spreading through all the generations. For a dystopian vision of the ageless world we now inhabit you have only to watch hypersexualised teens competing with hypersexualised sexagenarians on any of the talent shows that have become Saturday night TV staples. The svengali behind many of the shows is Simon Cowell, who is on the evidence of his own utterances one kind of amortal archetype. "All the things I used to like as a kid I still like," he told Piers Morgan in an intimate chat in front of 6 million viewers. "Genuinely, my tastes haven't changed at all."

The evergreen Cowell – Botox, he once said, is a routine, "like cleaning my teeth" – has more than a touch of the Peter Pan about him. He has fathered no children. His plans to marry, after his first-time engagement aged 50, show no signs of reaching a speedy fruition. Death is unconscionable to him. "I can't go to funerals and stuff like that. I find it very difficult to deal with that kind of reality. I shut myself off totally because it affects me so badly," he confided in the same interview.

"The big surprise for me is that age is just a number. It's a number without meaning," mused Hugh Hefner. Aged 85, he's preparing to marry a woman 60 years his junior.

Amortals such as Cowell and Hefner may appear marooned in Neverland but they are only part of the story. The actor Richard Wilson became the personification of grumpy old age as Victor Meldrew in the sitcom One Foot in the Grave. In real life he's as ageless as Mick Jagger or Meryl Streep. "I find myself looking at people and thinking, 'Oh, look at that poor old man,' and I realise that they're probably younger than me," says Wilson, 74. "Because my image of myself is not of an old man, I get quite shocked when people call me an old man." When he was first offered the part of Meldrew, he turned it down, he tells me. "I didn't see myself playing older people yet."

"[When] I'm 70, and I pick up the phone and I'm talking to some young spark of 30 who's at the top of his game, will he pay any attention to me?" asks Bob Geldof, rhetorically. "No, but if I was still, say, having hit TV shows or hit records or still politically active in the proper sense, still arguing and being listened to, then they probably would. So age disappears in direct proportion to the vitality of your ideas."

Retirement isn't a proposition that appeals to amortals unless life after work promises to be busier and better than the life before. And the impulse to keep working isn't such a bad thing, given the changing profile of the world's population. In Europe, the 60-pluses are projected to make up 37% of the population by the middle of this century. In some countries, two-fifths of citizens will be in their seventh decade or beyond.

One of those countries will be Greece. Among the austerity measures its government proposed as it struggled to manage the country's debt crisis was to raise the retirement age – from 53 to 67. Angry Greeks immediately flooded the streets to protest an outrage that threatened their right to be paid to do nothing for an average of 27 years. (Despite having the highest smoking rate in the world, the average Greek life span hovers around 80 years.)

Amortals are more inclined to celebrate the lifting of compulsory retirement ages and to deplore the ageism that seriously disadvantages older job seekers. Here's someone who reluctantly left her job at the Royal Court Theatre only when her employers discovered she was beyond pensionable age, and went on to establish a freelance public relations consultancy: "The other day I was washing my hair in the shower and they were blabbing on the Today programme about the problems with being able to afford to take care of the old. And I was half-listening, and I suddenly thought, I'm an elderly person – I'm 77 – and this is about me, but I don't feel any connection to it at all. And it isn't a money thing because financially I'm the least secure I've ever been in my adult life. It's really that I don't feel like an elderly person. I used to worry quite a lot about what people would think when I turned up to do jobs, because you know, the expectation, but I've completely got over that. I just think well, I am who I am and it doesn't seem to bother anyone. Nobody has ever said: 'Gosh, do you think you're up to this?' Never, never, never."

The speaker is my mother. She is amortal, like my father (her ex-husband; both parents are long remarried to younger partners) and many in our wider family. It's no coincidence that quite a few relatives work in entertainment. It's an industry that commodifies the desire for distraction, and it's rammed with people whose greatest talent is to distract themselves. But genes don't make us amortal; our socialisation does that, and the elements of that socialisation have changed dramatically as the culture of deference died, and traditional forms of authority ceded dominance to a makeshift cast of celebrities and scientists and a globalised, digitised range of cultural influences. Family, too, has lost traction as a vehicle for cultural transmission. As generational differences erode, relationships between adults and children change, often becoming closer. But if there is a weakness to amortal parenting, it is that amortal parents may be better friends to their kids than they are parents.

In the vanguard of the amortally driven transformation of family life, you'll find celebrities endorsing these changes through their own adventures in family-building. "I feel just as hungry today as the day I left home," remarked Madonna. Her appetite for adding to her family extended beyond her 50th birthday. Elton John unburdened to OK! magazine about his decision to start a family with his partner David Furnish. "David said: 'Well what about surrogacy?' I said: 'You know what? Why not?' I'm 62 at this point, but I feel 40."

Celebrities are the most effective sales force of our age, popularising ideas and behaviours. They're committed to the institution of marriage, for example. It's the institution of being married that's palled. And they increasingly use their celebrity power to shift products, too. Avatars of extended youth are often hired as the public faces of the multibillion-pound, multi-faceted global industry that is devoted to combating ageing and death or exploiting our passionate desire to do so. When we see famous actresses apparently defying age thanks to patented nostrums and programmes and potions, we react like envious customers watching Meg Ryan's phoney orgasm: "I'll have what she's having."

The problem is that not even scientists can agree on the causes of ageing or the possibilities of an antidote. A majority of mainstream scientists are pessimistic about the possibilities for unabated life extension. The world's verifiably longest-lived person, Jeanne Calment, died at 122 in 1997, and that may be close to the edge of the possible human span. But there have been flurries of excitement around discoveries that seem to hold out the promise of slowing ageing. Last November, Nature magazine published a study showing that mice with suppressed production of an enzyme called telomerase aged swiftly but could be rejuvenated if the telomerase supply was restored. And there is a chorus of dissenting voices promising that if we work out ways to live long enough, we'll be able to live for ever. Ray Kurzweil, for example, puts his faith in nanotechnology, the development of machines tinier than atoms that could be deployed in the human body to repair the ravages of time. Kurzweil's impressive record as an inventor (he developed the first flatbed scanners, optical character-recognition software, print-to-speech and speech-recognition technologies, as well as making fine keyboards found in many music studios), together with his unnerving habit of issuing outlandish predictions that later prove true, mean only the foolhardy would dismiss his forecasts out of hand.

He has signed up to have his head cryonically frozen after death, envisaging resuscitation in a more technologically advanced future, but he's not "super-enthusiastic" about refrigeration; it is, he says, a back-up plan. He is perched on a sofa in his office in Wellesley, near Boston, surrounded by awards, posters for two films centred on his transhumanist ideas, photos with people even more successful than he. "I have enough trouble pursuing my interests while I'm alive and kicking," he says. "It's hard to imagine doing that when you're frozen, but proponents of it say it's better than the alternative. Really, my plan is to avoid dying, I think that's the best approach."

For Kurzweil – 62 at the time of the interview last year, "biologically more like 41" – that effort involves a Spartan diet, exercise and handfuls of vitamins and around 150 supplements daily. Many amortals can't be bothered to put the work into staying vibrant, trusting instead to boffins like Kurzweil to deliver us from the clutches of our own biology.

Unfortunately there's no firm evidence that they will do so, even in our longer lifetimes. Amortals may be assailed by depression or left unprepared when the gap between our ageless sense of self and the reality of ageing yawns. I would urge my family and friends to drink elixirs or open their veins to restorative swarms of nanobots if I thought that would grant them even a few additional years. Instead I cling to the hope that by eating well and taking exercise, engaging and being engaged, they will at the very least challenge Jeanne Calment's record for longevity.

Stretching the health span to match the life span is a goal to which any healthy human can aspire, assuming he or she has access to the fundamentals of life and the modern health and sanitation systems that have already extended our lives. The blind watchmaker has saddled some of us with defects likely to thwart that ambition; accidents will happen; we inhabit a fractious planet in which violence is distributed as blindly as genes. Those eventualities are largely unpredictable and unavoidable, but the rest is down to us.

And amortality promises benefits as well as perils. Amortals will not step aside for younger talent, but our compulsion to keep working may provide at least partial relief from the economic strains of a greying population.

Moreover, research has shown that attitude does play a significant role in determining how we age. How you feel often becomes who you are. Boredom and its ugly sisters – detachment, isolation, rigidity – can drain life of joy, as surely as sickness and poverty.

When I asked my father about the regimen that keeps him fit enough for diving, he quoted an American folk song: "Oh, it's beefsteak when I'm hungry/Rye whiskey when I'm dry/If a tree don't fall on me/I'll live till I die." In fact, like many amortals, he doesn't easily or entirely accept his mortality. His impulse to outrun death is one of the reasons he's ageing pretty well.

Catherine Mayer's Amortality: The Pleasures and Perils of Living Agelessly is published on 12 May by Vermilion, priced £12.99

Are you amortal? Answer the following 10 questions and find out

Tick the answer that most closely matches your immediate response.

1. The difference between the age you are and the age you feel is:

a) More than 10 years

b) Up to 10 years

c) I feel the age I am

d) No idea. Age is an irrelevance

2. Describe your typical Sunday:

a) I often have to catch up on work but I also find time for friends

b) I usually have a lazy day and watch TV

c) Church, household chores and family pursuits

d) Hard to say, but as busy as the rest of the week

3. The late thirties and forties are the best ages to have babies:

a) I agree, and I/my friends have done just that/are planning to do that

b) I'm not sure that's true. It's probably better to have kids when you have more energy

c) I disagree. It's unfair on the kids who have older parents

d) I haven't got round to thinking about babies

4. Your friend's new lover is 25 years younger than your friend. You:

a) Don't think the age difference matters. They're soul mates

b) Understand the attraction but the age gap means the relationship is unlikely to last

c) Think "there's no fool like an old fool"

d) Hadn't noticed

5. You are permanently stressed and worried. You turn to:

a) Therapy. I start by fixing myself

b) Retail therapy. It always makes me feel better

c) Prayer. It always makes me feel better

d) Stress? The only thing that makes me stressed is having nothing to do

6. Your ideal holiday would be:

a) Action-packed. An adventure

b) A mixture of relaxation and sight-seeing

c) I like to spend quiet time at home

d) I prefer work trips. They're a better way to get under the skin of a country

7. Your fitness routine involves:

a) Guilt. I belong to a gym but I rarely go

b) I exercise/play a sport but I'm not as fit as I used to be

c) At school/college I played a vigorous sport

d) I don't really need to exercise because I am always running to the next appointment

8. When you think about retirement you:

a) Shudder. I never want to stop working

b) Wonder if I'll ever be able to afford to retire

c) I am looking forward to my golden years

d) I haven't really thought that far ahead yet

9. The following statement best describes your attitude towards getting old:

a) I don't intend to get old the way people used to.

b) It comes to us all, I suppose...

c) I look forward to the dignity and wisdom of age

d) I haven't really thought that far ahead yet

10. Every life ends in death. Discuss:

a) Maybe, but science is on the point of finding ways to make us live longer

b) That's true but I'd really rather not think about it

c) And death is the beginning of eternal life

d) I haven't really thought that far ahead yet

ANSWERS

Mostly As

Bona fide amortal: You have strong amortal inclinations and are already living agelessly, at least in some respects.

Mostly Bs

On the way to amortality: You have latent amortal tendencies that may well develop in future.

Mostly Cs

Mortal soul: You have significant immunity to amortality and are unimpressed by many of its manifestations.

Mostly Ds

Amortal to the max: You're so detached from external concepts of age that you probably don't even realise how agelessly you live.