One hundred years ago, Einstein gifted his general theory of relativity to humanity, packaged in a box of very abstract, incredibly mathematical, and generally hard-to-solve nonlinear differential equation. One hundred years later, we are celebrating the detection of gravitational waves (GWs), one of its many predictions that include the big bang, supernovae, black holes, gravitational lensing, gravitational time dilation, … and for those who value a discovery only by its practical application, GPS.

The detection of GWs has been expensive (over $600 million so far), time consuming (started in 1992), and labor intensive (the paper, published in Physical Review Letters announcing the discovery of GWs, has over 1000 authors ). Has it been worth it?

One answer to the question takes us back another 50 years from the time that Einstein wrote his equation. In 1865, James C. Maxwell gifted another abstract, mathematical, and hard-to-solve set of differential equations to humanity. These equations, among other things, predicted the hitherto undetected electromagnetic waves (EMWs). These waves were detected in 1887, only eight years after Maxwell’s early death at the age of 48. The detection of EMWs was inexpensive (the cost of a microwave oven in a university laboratory), relatively quick (started in 1885, only two years before their discovery), and labor cheap (Heinrich Hertz was the single author). In the intervening thirteen decades, we have invented radio, television, computers, smart phone and a host of other gadgets, all of which would not exist without EMWs.

A second answer to the question takes us back another two hundred years. In 1666, Isaac Newton gave us the law of gravity and applied it to the solar system in yet another abstract mathematical equation whose solution required no less a Herculean intellectual task than the *invention* of calculus and differential equations.

A third answer puts us in the Golden Age of Greek science and mathematics. Euclid, Archimedes, Aristarchus, Eudoxes, Pythagoras, and all of the great geniuses of that period engaged in highly abstract mathematical ideas and inventions such as trigonometry to answer the seemingly practically useless questions posed by the motion of planets and stars. In their quest for wisdom, they even faced great dangers on their journey of hundreds of miles — sometimes on foot — to learn the knowledge acquired by Egyptian and Babylonian priesthood.

So, I ask again: Was all of this worth it?

To answer the question definitively, we have to go even further … much much further. Was it worth the effort of the first homo erectus to bash together two stones (and it *did* require effort because their brain was not developed enough to make the bashing as simple as it is for us today)? This question is relevant because, if the practical application of later discoveries is any indication, our distant ancestors at first bashed stones together, not because it had any practical application, but simply because their opposable thumbs enabled them to do so. It is very likely that the first attempts produced nothing. It then took a while before chips with sharp edges were produced. And the discovery of the utility of the sharp-edged byproducts probably came later, perhaps tens even hundreds of years later.

The idea of stone bashing was an extremely abstract idea, on a par with the mathematics that went into the general theory of relativity, and required the intelligence of the Einsteins of two million years ago!

No one questions the importance of the skinning of animal kills in the evolution of mankind, although at the time of its discovery, no one could anticipate the utility of the idea of stone bashing. No one questions the importance of Greek mathematics and its role in the development of post-Renaissance science. No one questions the enormous impact EMWs have had on our civilization, and by extrapolation, the significance of the theoretical framework developed by Maxwell and mathematicians and physicists that followed him.

GWs are the sharp-edged chips of bashing the stones of general theory of relativity! No one knows where they will lead us, just as no one knew where EMWs would lead us 150 years ago, and no homo erectus knew where the (abstract, theoretical) idea of bashing two stones together would lead them. But one thing is clear:

The GWs of today are as crucial to our evolution as the sharp-edged byproducts of stones were two million years ago.

The difference between the two is that one or a few members of our race discovered the connection between the bashing of stones and its utility in skinning animal kills, while the collaboration of over a thousand physicists from over forty countries was involved in the discovery of the GWs. Our evolution is no longer driven by the accidental discovery of a few isolated individuals, but by the global effort of thousands of scientists.

Another difference — or should I call it *similarity* — is merely in the level of sophistication. While sharp-edged stones were the outcome of the initial idea of bashing two stones together, GWs are the outcome of the initial mathematical idea encapsulated in the general theory of relativity.

Mathematics is to the evolution of modern humans as the idea of bashing two stones was to the evolution of homo erectus.

Keeping this in mind is important in the teaching of mathematics to our children and the crucial role it plays in our evolution. It also tells us that this teaching ought to impart the *skill* of mathematical reasoning and know-how to our children just as homo erectus imparted the *skill* of bashing stones to their children.

If our race does not annihilate itself, two million years from now, our progeny will look at our highly abstract and sophisticated mathematics we are using to invent theories like relativity and the standard model of fundamental particles, and place it in as primitive a category of ideas as we place the stone bashing of our two-million-year old ancestors today.