Lately, I have been listening to this podcast series that has completely changed how I see the history of science.

In the series, Viktor Blåsjö, a professor and an emerging historian of mathematics, told a mind-blowing story about Galileo which is nothing like what we are usually told about. We have been told by the teachers, by the media, and by the mainstream science philosophers, that Galileo was the “father of modern science.” But to the contrary, Galileo was not only poor in mathematics, but also a dilettante physicist. Let me name a few things from the podcast series and you will know how interesting it is:

• Galileo couldn’t even solve some easier math problems, let alone understanding the new developments in math and physics of his contemporary mathematicians. That was why he tried to gain fame by attacking Aristotle from 2000 years ago, like beating a dead horse.
• Galileo picked on Aristotle also because he could not understand Archimedes, the true master of math and physics from Aristotle’s time. Framing Aristotle as the great authority of physics makes it easy for Galileo to defeat. Had he picked on Archimedes, his whole arguments would just fall apart.
• Contrary to the common belief that Galileo took experiements and obervations seriously, he often manipulated the data just to prove his points, made baseless and wrong guesses, and even plagiarized results from other people.
• Why haven’t the mainstream philosophers and historians debunked Galileo already? Because they also lack the skill to appreciate the great work of the mathematicians since Archimedes. Nowadays, 20 times more papers about Aristotle than about Archimedes are being published under the category of history of science.

Of course, all of these sound like wild claims if you are hearing them for the first time. So I would encourage any curious minds to go listen to the podcast or read the corresponding monograph then judge for themselves.

This new interpretation of Galileo has solved one of my long-standing puzzles for me. In all the serious science textbooks, the most important discoveries are often associated with great mathematicians (such as the principles and laws associated with Archimedes, Kepler, Newton, Bernoulli, Euler, Gauss). Why then has Galileo, “the father of modern science,” never been linked to any of those physical laws, other than the folklore of the “Pisa experiment”? Because he lacked the mathematical ability to do serious physics.

Studying the history of mathematics allows us to understand the more true history of science. But there are more reasons to study the history.

Why study history of mathematics?

We know that Leibniz was one of the inventors of calculus. But do you know that Leibniz did not actually prove the fundamental theorem of calculus? In fact, he did not worry much about the foundation of calculus, certain not the “rigorous definition” of derivatives.

What Leibniz cared most about was the transcendental curves, that is, the graphs of transcendental functions. According to the inspiring book Transcendental Curves in the Leibnizian Calculus, again by Blåsjö, “the problem of transcendental curves was to him (Leibniz) the guiding star for the better part of his mathematical works throughout his life.”

To Leibniz, functions like $\log(x)$ and $\cosh(x)$ are nothing but notations; the claim that $y=\cosh(x)$ solves the equation $y’'=y$ makes no sense. The important thing to Leibniz was that someone can actually graph a function like $\log(x)$, or calculate its value given any input $x$. After all, what’s the point of writing the symbol “$\log(x)$,” or naming a function “hyperbolic,” if you can’t even find their values? Therefore, to Leibniz, an expression like $\int_1^x \frac{1}{t}\,dt$ makes much more sense than $\log(x)$; being able to draw a catenary by hanging a chain is more important than simply writing $\cosh(x)=(e^x+e^{-x})/2$.

The same can be said about the other common transcendental functions such as $\sin(x)$ and $\cos(x)$. Indeed, students are being told that sinusoidal functions are “elementary functions,” but how “elementary” is it if you can’t even tell what $\sin(1)$ is right away? People still love to see animations, such as this one and this one, that can help them understand what trigonometric functions are.

The discussions above give us another important reason to study the history of mathematics. History tells us what are the “natural” things to pay attention to when we first teach or learn a mathematical concept. Knowing why we need transendental functions in the first place is understanding; telling people that these functions are elementary is indoctrination.

True education is about understanding. Unfortunately, a lot of the schooling nowadays are about indoctrination (and probably day-care as well).

Right way to study history

In Blåsjö’s History of Mathematics course, he contrasted two ways of studying history of thought:

• The bad way: to enforce our own beliefs and ways-of-thinking on the people in the past. For example, a bad historian might ask:
  • Who first discover this thing that I believe in?
  • When did people start thinking like us?
• The good way: to embrace the diversity of thoughts, be curious about different perspectives. For example, a good historian might ask:
  • Why do people used to think differently?
  • Why did their way of thinking make more sense at their time?

Historians who enforce their own beliefs on people from the past cannot find out the true history of Galileo, let alone the true history of science. Teachers who enforce their own thoughts on students cannot inspire anyone to understand.

Next time you are told that somethings are conventional or have been the way they are, just remember this powerful quote from an amused chimp:

If the news are fake, imagine history.