Mr Ellison told a Paris computer conference that the PC was about to decline in importance, just as the mainframe has. Instead, we would have simple terminals that combined a TV and a phone. If we wanted anything complicated, we would get it downloaded over the phone. "The personal computer," he opined, "is a ridiculous device."
Mr Gates disagreed, reckoning that people would not want dumb terminals, and he noted the security problems when information was downloaded over the phone - problems evident with the Internet.
His view is unsurprising as Microsoft makes most of its money by selling software for PCs, but there is an element of incongruity about the way software is distributed. Just a couple of weeks ago, Microsoft launched Windows 95. How did people get this new product? They went to a shop, bought a cardboard box with some pieces of plastic inside, paid for it by handing over money or maybe signing a credit-card chit, took it home and then spent half an hour unpacking it and loading it into their machines.
In other words, although the product is a set of electronic signals, something unimaginable to any previous generation, it was bought in exactly the same way as the Romans bought a new scroll.
Mr Ellison objects to the inappropriate nature of the PC: a machine that itself is unnecessarily powerful for the job in hand. Far more sensible, he would say, to get the software you need - and only the software you need - down the line. But for ordinary punters, the principal objection, surely, is less the excessive power (which comes very cheaply), but the fact that PCs do all sorts of wonderful things you don't want them to do, but don't do very well the things you do.
When computer experts cannot agree, call in the economists! Or rather, since theoretical economists are renowned for their ability to disagree, the economic historians. Seen in a long historical perspective, what we have here is a product, the PC, which is still at a very early formative stage. And there are scores of examples from similar periods in the past of how difficult it is to see how a particular invention will ultimately be used. When wireless was invented, people thought its main use would be for contact between ships: no-one realised it would become an entertainment medium, too.
We also find it hard to judge whether a product will be principally a personal one, or whether it will be mainly used commercially. At the beginning of the century, many people assumed cars would remain so complicated to drive that most people would have to use chauffeurs. Then, in the 1920s and 1930s, they made the opposite mistake with planes: that the private plane would follow the private car as a mass consumer item.
Further, there are plenty of examples of technology that exist for years with limited commercial applications but are suddenly transformed into mass consumer items by a fall in the price: the fax is the best recent example.
How does the PC fit in to this historical pattern?
There are three different technologies all seeking to carry the signals from one place to another and competing to be the main method of processing and communicating information - the phone, the television, and the personal computer. The phone can have a screen (although the technology is not yet available to give a decent quality picture), and the phone companies are gearing up to offer all sorts of services down the line. The television is becoming interactive (although in practice the services the cable companies offer, such as home shopping or booking theatre seats could be delivered in many other ways, including the humble phone). And you can now make phone calls, send faxes and watch films or TV on your computer.
So this clash between Messrs Gates and Ellison is over a tiny corner of the battlefield: is the intelligence in the PC or in the network? The economist looking at the whole battlefield would say that there are there are two very simple questions, the answers to which will determine the rate of take-up of the competing technologies. The first: what do people want - or more precisely, what are they prepared to pay for any particular product or service? The second: at what price can that product or service be manufactured and delivered?
The key point here is that the technically possiblity of producing something does not mean it is worth doing so if it costs more than people are prepared to pay. Peter Schwartz, author of The Art of the Long View, pointed out to a London seminar last week that Japan had spent $9bn (pounds 5.8bn) developing a system of analogue high-definition television. But take-up has been tiny, and the system is now being overtaken by digital processes. So all that money, not to mention the opportunity cost - what might have been done with the resources had they been deployed elsewhere - has been entirely wasted.
What do we know from history about people's preferences? Surely the most obvious lesson is that we will pay an enormous amount for mobility and entertainment. People will pay very large proportions of their income to have cars - rationally, another "ridiculous device", for a family car will exceed the speed limit by up to 50 per cent and is typically only used for one hour out of the 24. People will also pay large sums to watch movies and sport or listen to popular music. People also want to communicate while on the move - witness the boom in mobile phones.
And what can we judge about the cost of producing goods and services? Advances in technology, particularly in electronics, mean that goods will carry on becoming relatively cheaper, but rising labour costs mean that many services will become relatively more expensive.
So putting power into PCs is very cheap, and as a result PCs will go on getting more powerful until we reach the stage, in perhaps another 10 years, when there simply is no point in making them more powerful because they will do more than anyone would want them to do. Because writing software takes human time, it will go on becoming more expensive. So instead of buying software that is only used occasionally, the tendency will be for people to rent it when they need it - just as people rent a video - except the software would be delivered down the line.
Conclusion one: both Bill Gates and Larry Ellison are right, for PCs will go on getting more powerful, but they will also become more of a terminal that receives software from a central point.
Conclusion two: the big game is not between two software firms. Rather it is between the phone companies, the media groups, and the computer industry as a whole. And that game is wide open.