Today on Meet the Press, Paul directly related the marked reduction in government spending from 1945 and the years after to a growing post-war economy.
While it is true private investment started increasing to higher levels in the latter part of the 40's, it is not at all clear that that is due to the spending cuts of the government from post-WWII. The US economy really didn't hit it's stride until the 1950s. And certainly, for the first 3 years immediately after the War, GDP growth was negative or stagnant. And even the 1950s there were three significant recessions suggesting that the growth that occurred in the 50s was is spurts. Indeed, the dramatic growth in living standards really didn't occur until the 60s.
So, take Dr. Paul's comments with a grain of salt as there are lots of economic changes that occurred between the end of the War and the early 50s that can explain the relative good times of the 1950s-1960s including a dramatically growing and changing manufacturing sector (and an increased move out of agriculture) and a booming suburban housing market. And particularly with regard to the housing boom, perhaps the biggest explanation is a post-Keynesian one: the rapid increase in the late 50s and onward of non-revolving credit used to buy durable goods (the stuff put in those new cookie cutter homes and the cars in the driveways). The crux of Paul's argument is that the reduced spending of the government from 1945 onward freed up resources for the private sector. But with the onset of heavy uses of credit, citizens didn't need present-day resources, they could borrow from future-day resources instead. And thus began our credit-heavy economy.
This point is key so I will restate: in our modern economy, we use uncertain future resources to finance growth, not present resources. To quote Minsky, this is why "stability is unstable."