Vector: A Surprising Story of Space, Time, and Mathematical Transformation

by Robyn Arianrhod

It took experimental physicists a quarter of a century to verify in the lab Maxwell’s prediction of radio waves, and it took a hundred years to detect Einstein’s gravitational waves. That’s an indication of how far ahead of the game these vector- and tensor-based theories were.

It took experimental physicists a quarter of a century to verify in the lab Maxwell’s prediction of radio waves, and it took a hundred years to detect Einstein’s gravitational waves. That’s an indication of how far ahead of the game these vector- and tensor-based theories were.

It’s as if the act of describing physical reality mathematically creates a magnifying glass, revealing, through mathematical patterns, underlying physical attributes that had long lain hidden.

It’s as if the act of describing physical reality mathematically creates a magnifying glass, revealing, through mathematical patterns, underlying physical attributes that had long lain hidden.

One of my goals is simply to show just how long it takes—and how much intercultural cooperation is needed—for sophisticated mathematical ideas to develop.

One of my goals is simply to show just how long it takes—and how much intercultural cooperation is needed—for sophisticated mathematical ideas to develop.

Mesopotamia

Mesopotamia

blithely,

blithely,

Mary Somerville

Mary Somerville

pattern and generality.

pattern and generality.

This may sound obvious today, but it took three and a half thousand years for mathematicians to move from solving quadratic equations—“quadratic” derives from the Latin for “square,” so quadratic equations are those whose highest power is x2 (the unknown multiplied by itself, as the ancients put it)—to solving “cubic” and higher equations. These higher-degree equations are much more difficult, of course; but part of the reason solutions didn’t come easily was that algebra was tied to words and concrete images for such a very long time.

This may sound obvious today, but it took three and a half thousand years for mathematicians to move from solving quadratic equations—“quadratic” derives from the Latin for “square,” so quadratic equations are those whose highest power is x2 (the unknown multiplied by itself, as the ancients put it)—to solving “cubic” and higher equations. These higher-degree equations are much more difficult, of course; but part of the reason solutions didn’t come easily was that algebra was tied to words and concrete images for such a very long time.

What he was getting at is that generalization is far easier in symbols than in words. And when you can generalize—when you can see common patterns that apply to an unexpectedly wide range of problems—you can make extraordinary progress in science and technology as well as mathematics.

What he was getting at is that generalization is far easier in symbols than in words. And when you can generalize—when you can see common patterns that apply to an unexpectedly wide range of problems—you can make extraordinary progress in science and technology as well as mathematics.

With the rise of algebraic symbolism, a new kind of abstract thinking arose, which opened the way for the development of calculus—and

With the rise of algebraic symbolism, a new kind of abstract thinking arose, which opened the way for the development of calculus—and

for it was perhaps the first time that mathematics had been used to predict the very existence of a physical phenomenon, rather than explaining what was already known.

for it was perhaps the first time that mathematics had been used to predict the very existence of a physical phenomenon, rather than explaining what was already known.

(cosθ)2 + (sinθ)2 = 1. Euler realized that you could factorize this equation as: (cosθ + isinθ)(cosθ − isinθ) = 1.

but they should remember that the moving power of mathematical invention is not reasoning, but imagination.”

Incidentally, it was at a British Association meeting in 1833 that the term “scientist” was first introduced, in order to put scientific and mathematical researchers on the same professional footing as artists.

historical journey

thanks to an entry in the newly bound Encyclopaedia Britannica.

So, part of what I’ve been trying to show throughout this story is just how long it takes for ideas to develop and find their best form.

Maxwell said that a mathematician often gets so tired with all his calculating that he has no energy for thinking—whereas

We need more than high-tech comfort, though, if we want a meaningful life. We need a sense of wonder, too.