The article claims that if we were to suddenly and instantly switch ALL ENERGY PRODUCTION in the world over to nuclear power, given the current efficiency of nuclear power plants and the current amount of KNOWN uranium reserves, it would work for about six years. This is far from the implication that if we were to start phasing out petroleum and replacing it with nuclear power we would get about six years into it before we ran out of uranium.
On this unreasonable basis he rejects nuclear power as a viable option. Furthermore, his article includes a pathetically weak, three-pointed discussion of criticisms of his suscribed theory. I intend to expand on his discussion.
1) "First, it is argued that there are plenty of good-quality uranium deposits available, that reserves are abundant, and that they will become more so when demand strengthens. But there is little to support this. From the 1960s to the 1980s, exploration for uranium deposits was intensive; most that was there to be found was found. Some small deposits doubtless remain to be discovered, but the geology of uranium is now well known: there are almost certainly no major new discoveries ahead." [1]
He claims that there is "little to support" the existance of more good-quality uranium reserves, but then offers even less to support their nonexistance. He offers the heavy exploration from the 60s to the 80s as evidence that everything that is there was already found. I believe his implication is that we couldn't find anything more, and so we stopped looking.
But as mlorrey commented, uranium is about as common as the element tin. Because of this abundance, uranium is cheap right now, and has been for some time. There has been no impetus to find new uranium deposits. According to the 2001 "Analysis of Uranium Supply to 2050" released by the International Atomic Energy Agency in Vienna, the quantity of uranium in highest-confidence reserves is over 3.2 million tons [2]. According to a 2005 report by the World Nuclear Association, there are 3.5 million tons [3] of known recoverable uranium. If David Fleming's assertion that no new major discoveries have taken place between the 80's and today is correct, then this number has remained nearly unchanged for the past twenty years. Twenty years ago, 3 million tons of uranium would have looked like thousands of years worth of energy. Why spend money finding more when you already have more than you'd ever plan on using?
Something doesn't make sense in Fleming's logic, and I'd hardly trust his assertion that no more uranium will be found, especially when the experts predict the total availible amount of conventionally recoverable uranium is on the order of 9.7 million tons [3].
2) "Second, critics point out that uranium is an abundant element; there is plenty of it in the earth's crust and in seawater. But in both cases the energy needed to extract it would be more than could ever be recovered." [1]
This is categorically wrong. Unless I am mistaken, uranium in the earth's crust is extracted by mining, and the Japanese have shown that it can be extracted from seawater for a price of about $300 per kilogram [2]. Current uranium prices are on the order of $30 to $40 per kilogram, so this not economically practical as long as we have cheap uranium at our disposal.
Since this $300/kg price includes the energy cost, and 1 kilogram of raw uranium after roughly $1000 worth of processing yields about 40,000kWh of power [4] at today's efficiencies, it's obviously possible to recover all of the energy and more from seawater uranium.
3) "Third, there is the argument that we could use uranium more efficiently by developing breeder reactors, which would be 100 times as efficient as today's thermal reactors. But after 50 years of extremely expensive research, they are still not technically feasible." [1]
Fifty years is gross and dishonest mischaracterization. Fifty years ago was 1956, two years after the first nuclear power plant was finished, and a little over ten years after the first bombs were made. I'd hardly say that modern fast breeder reactors have had 50 years of devoted research.
New nuclear technologies, like all techologies, take time, money, and sociatal impetus to come to fruition. But even if fast breeder reactors don't become a reality within the next ten years, new reactors have shown that it is possible to improve the efficiency of the process by substantial amounts.
Lastly, if you stand by David Fleming's above assertion, I would point out the subtle hypocracy of assuming there will be no major developments in nuclear power while championing wind power, a technology that will need many major developments before it will be capable of replacing other energy sources.
References:
[1] David Fleming, "Nuclear Confusion", Prospect Magazine (
http://afr.com/articles/2005/06/23/1119321845502.html)
[2] "Analysis of Uranium Supply to 2050", International Atomic Energy Agency
[3] "Supply of Uranium", World Nuclear Association, (
http://www.world-nuclear.org/info/inf75.htm)
[4] "The Economics of Nuclear Power", (
http://www.uic.com.au/nip08.htm)
After all looking after the short term and sod the future is what got us in the miss to start with 
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