In Hamming's new book, there's a section on doing 'back of the envelope math," a snippet of the page is below. The problem I'm having is that nowhere are several of the variables defined, making the math hard to follow. I'm guessing I'm missing something obvious. What's a? What's T? I assume e is this guy.
He has assumed a functional form for the number of scientists at a given time. You pick some year at $t=0$. Then $a$ is the number of scientists alive that year. $b$ is the inverse of the time it takes for the number to increase by a factor of $e\approx 2.71828$. We need two data points to establish $a$ and $b$. Say we know there were $100,000$ scientists in $1900$ and $200,000$ in $1920$ (completely made up numbers). We could decide to measure $t$ in years since $1900$, and since $e^0=1$ we have $a=100,000$. Then $$200,000=100,000e^{20b}\\2=e^{20b}\\ \log 2=20b\\ b=\frac {\log 2}{20}\approx 0.3466$$ $T$ is the time you measure, which could be today. The integral in the numerator computes how much knowledge there was $17$ years ago and the one in the denominator computes how much knowledge there is today. Since knowledge doubles every $17$ years, this ratio is $\frac 12$.
He then assumes that each scientist produces knowledge at the same rate (or at least on average, but over all of history) and asks whether the two statements are compatible. If knowledge doubles every $17$ years, half of all the scientist-years in history must be the last $17$ because half of all the knowledge was produced in the last $17$ years. I suspect the answer is they are not because half of all the scientist-years were long enough ago that the scientist is dead by now. You can rescue this by saying that scientists today are less productive of knowledge than scientists in the past because being a scientist in the past was much harder so only the very productive scientists could make a career of it.