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April 11, 2020
When Maxwell translated his abstract symbols back into reality, they predicted that electricity and magnetism could propagate together as a wave of invisible energy moving at the speed of light. In a matter of decades, this revelation would change the world.
Calculus is an imaginary realm of symbols and logic; nature is an actual realm of forces and phenomena. Yet somehow, if the translation from reality into symbols is done artfully enough, the logic of calculus can use one real-world truth to generate another.
Truth in, truth out.
geometers
He theorized that under certain circumstances, light passing through matter could stimulate the production of more light at the same wavelength and moving in the same direction, creating a cascade of light through a kind of chain reaction that would result in an intense, coherent beam.
It’s incredible but true: Even in the subatomic realm where Newtonian physics breaks down, Newtonian calculus still works. In fact, it works spectacularly well. As we’ll see in the pages ahead, it has teamed up with quantum mechanics to predict the remarkable effects that underlie medical imaging, from MRI and CT scans to the more exotic positron emission tomography.
This result for the area of a circle, A = rC/2, was first proved (using a similar but much more careful argument) by the ancient Greek mathematician Archimedes (287–212 BCE) in his essay “Measurement of a Circle.”
In fact, many of the greatest pioneers of the subject did precisely that and made great discoveries by doing so. Logical, no. Imaginative, yes. Successful, very.
During the Inquisition, the renegade monk Giordano Bruno was burned alive at the stake for suggesting that God, in His infinite power, created innumerable worlds.
“When you have eliminated the impossible, whatever remains, however improbable, must be the truth.”
jargon
fervor
astutely
famous textbook by Muhammad Ibn Musa al-Khwarizmi (c. 780–850 CE), whose last name lives on in the step-by-step procedures called algorithms.
mathematicians square the errors at each point to make the negative ones become positive. That way, they can’t possibly produce any spurious cancellations.
method of least squares.
All of which raises an extremely important general point: Patterns are what make compression possible in the first place. Only patterned data can be compressed. Random data cannot.
smolder
In this example, the exponential function 2x comes into play. Specifically, if we measure time in units of 20 minutes, the number of bacteria after x units of time would be 1000 × 2x cells.
Similar exponential growth is relevant to all sorts of snowballing processes, from the multiplication of real viruses to the viral spread of information in a social network.
There’s no mathematical reason for preferring 10 over any other base. It’s a traditional favorite because of an accident of biological evolution: we happen to have ten fingers. Accordingly, we have based our system of arithmetic, the decimal system, on powers of ten.
Logarithms were the supercomputers of their era.
It’s freaky. Our minds fool us into believing that 1 is as far from 2 as 2 is from 4, and as 4 is from 8, and so on. We somehow sense frequency logarithmically.
