Image copyright Getty Images Image caption IBM computer Deep Blue beat Gary Kasparov at chess in 1997 by 3.5 games to 2.5 games

There are moments that live on in business history.

One of them is the cry: "Mr Watson come here, I want to see you," spoken by Alexander Graham Bell back in 1876, in the world's first telephone conversation.

Another significant moment was the day in 1997 when the IBM computer called Deep Blue beat the then world champion Gary Kasparov at chess.

And then another IBM moment in 2011 when an even more intelligent computer called Watson -after the IBM founder Thomas Watson and his IBM chief executive son Thomas - won the TV game Jeopardy against human competition.

These last two IBM contests demonstrate - we're told - big advances in machine intelligence.

Foreigners have to take the most recent one on trust - Jeopardy is not a familiar game outside the USA, and how clever you have to be to win it is not understood globally.

Anyway, the Jeopardy win got the technology community excited that a threshold moment had been passed on the computing roadmap set out by the late British genius, Alan Turing.

His so-called Turing Test predicted that one day machines would be able to interact with human beings in a way that it would not be possible to tell whether the other party to the interaction was man or machine. At least on the screen.

Image copyright Science Photo Library Image caption Alan Turing predicted that computers would one day be able to fully interact with humans

In one way there is nothing difficult to understand about this progress in computing. It was implicit in Moore's Law, laid down 40 years ago by Gordon Moore, co-founder of silicon chip giant Intel.

Moore's Law points out that computer power on a chip doubles roughly every two years. It's not so much a law as a roadmap for the whole computer industry. This machine progress - still continuing - naturally means that the rate at which computers can crunch data is expanding at a similar speed.

When an IBM computer played a human at draughts, or checkers, in 1959 under the supervision of Arthur Samuels, it did not need - and did not have - as much computing power as the chess-playing Deep Blue.

Draughts is a simpler game. Its whole repertory of potential moves and counter moves is far smaller than chess. But computer power was very limited then.

Jeopardy, I'm told, is a different matter, rather more than just one step up in complexity.

The questions are allusive and unstructured, they come from all over. Big Blue's Jeopardy victory was therefore a breakthrough moment for members of what is now known as the artificial intelligence community.

But is the superior number crunching that computers can now routinely carry out real intelligence or simulated artificial intelligence (AI)?

Computers taking over?

Many companies big and small are now pursuing the holy grail of artificial intelligence - at its starkest, thinking machines. Most are shrouding their efforts in secrecy, IBM isn't.

Watson is now being marketed as a tool for people to explore and use. In New York, there's an impressive building near the city's so-called Silicon Alley devoted to demonstrating Watson, and finding uses for its apparent intelligence.

Image copyright Getty Images Image caption IBM's Watson computer won US quiz show Jeopardy

A new cluster of AI specialists is emerging in New York. Some of them are financial market algorithmic whizz kids redeployed after the crisis.

Some are refugees from AT&T's famous Bell Labs over the river in New Jersey. It was there that the transistor was developed in 1947.

Bell Labs also did a lot of work on speech recognition for telephone networks... something that is obviously allied to machine intelligence.

At Memorial Sloan Kettering Hospital in New York, I went to hear from a famous cancer specialist who is using Watson's data-gathering skills to expand hugely his own knowledge base, and bring him instant news of developments in his field that may be relevant to the symptoms he feeds in to it.

Some people I've heard from recently think that we will soon enter an era when computer diagnosis using machines will so improve on human diagnosis that medicine will move swiftly into machine intelligence world. That's what the Indian-born venture capitalist Vinod Khosla told me in Silicon Valley, California, last year.

But the New York specialist I met was convinced that a human doctor would remain at the centre of things. He would use Watson greatly to expand his understanding, but not to do its own independent diagnosis.

When machines might outstrip humans as thinkers - is making a lot of headlines. But the people closest to it are wary of the claims made by experts such as Ray Kurzweil, chief engineer at Google, that the human race will sometime soon be eclipsed by intelligent machines.

Mr Kurzweil has long been convinced that one year (maybe 2050) computers will have evolved to be as clever as we are. Two years later - following the drum beaten by Moore's Law - they will be twice as clever.

Image copyright Getty Images Image caption Will robots, such as Aldebaran Robotics' Pepper, eventually be more intelligent than humans?

At which stage it would be logical to hand over to them, since they know more than we do, and will continue to improve.

In spite of Mr Kurzweil's concept of this takeover point, which he calls the singularity, most of the other people I've been listening to think that AI is not a fixed threshold point in the evolution of computer power.

It is a reflection of the ever-increasing ability of computers to search and do pattern recognition in an ever-increasing store of data. The concept of AI reflects this burgeoning power of the computer to cope with stuff.

Each step on the way, each computerised victory over humans in checkers, or chess, or Jeopardy, looks like a material step towards the ultimate - machines that are as intelligent in every way as are we mortals.

Image copyright Getty Images Image caption The co-founder of Intel said that computing power would double every two years

But crunching data, and learning from that, is only one of the things that human beings have mastered - to a certain extent.

And for many of us, we see that the mastery of big data by computers is clever, but not as clever as human intelligence.

It may be that this process goes on for a long time; ever more impressive thresholds will be crossed by computers such as Watson. Progress towards AI, but never the achievement of real artificial intelligence itself.

My tablet computer's spelling corrector is demonstrating that, every sentence I write.

The accompanying radio documentary to this feature will be broadcast on Radio Four's In Business programme on Thursday, 7 May at 20:30 BST, and again at 21:30 BST on Sunday 10 May. It will able be available via the In Business podcast page.