John Horgan is an award winning science writer / journalist,
well known for his *Cross-check* column in *Scientific American*.
He is also the author of several books, including
*The End of Science*,
published by Addison Wesley in 1996,
and republished by Basic Books in paperback.
This book asks whether the age of great discovery is behind us.
He speaks of revolutions, scientific revolutions,
and perhaps there are no more.
They do seem to have come fast and furious, most of them in the past 200 years.
Here are the revolutions, as I see them.
This is my list, not Horgan's.

- Spherical Earth
- Heliocentricism
- Universal Gravity and Orbital Mechanics
- Atomic Theory of Matter
- Periodic Table of Elements
- Linnaean Taxonomy
- Thermodynamics
- Genetics and Heritability
- Evolution
- Capitalism
- Vaccinations
- Electromagnetism
- Germs and Disease
- Plate Tectonics
- The Subconscious Mind
- Special and General Relativity
- Gödel's Incompleteness Theorem
- Quantum Mechanics
- Expanding Universe
- Antibiotics
- Turing Machines and Undecidability
- Standard Model in Particle Physics
- Big Bang

Horgan suggests that there are no more revolutions to be had, and science itself may soon be reduced to mere puzzle solving and adding details to existing theories. This opinion / philosophy is sometimes dubbed Horganism, which is also his Twitter handle. Some find his projections disconcerting, even pessimistic, but our emotions have no bearing on whether things are, in fact, true. We must approach this question objectively - well - as objectively as we can. To this end, he asked one of his detractors the following.

"Even if you don't accept that science is close to being finished now, do you think that science is an infinite process, like literature or even mathematics, or do you see any limits, and possible end point to the process?"

I have to admit, if the laws of nature are finite, then some day we will know most or all of them, and on that day, Horgan will be right.
And yet I don't think it is quite that simple.
I suspect people will be learning, and enjoying, and advancing the fields of science for as long as the human race survives,
but the law of diminishing returns is, I believe, inescapable.
Perhaps he should have titled his book *The Slowdown of Science*.
Let's look at each field in turn.

If any field is finite, it is probably physics. We have successfully broken matter down into atoms, and atoms into subatomic particles, and these particles into quarks, but I don't believe this continues forever. There is a bottom to matter, and if it isn't quarks, it lies one or two levels below. At the same time, there are only so many forces, these also mediated by particles. Physics will some day be known, either in one grand unified theory or as a jumble of overlapping theories with arbitrary constants forming a patchwork that isn't pretty. The universe is what it is. If you allow for a multiverse then physics becomes infinite once again, with different rules in each universe, but we can only observe, test, and verify the laws of our own universe. Speculating on the form and structure of other universes is somewhere between math, religion, and philosophy. Therefore I am willing to concede that physics is finite, but that doesn't mean we will ever know it all. Remember that you can approach a limit without reaching it, as each scrap of knowledge is more and more difficult to develop, comprehend, and verify using the technology we have at hand. The Large Hadron Collider is one of the most ambitious engineering projects ever undertaken - existing solely to elucidate the laws of physics. Supersymmetry proponents claim we have not found the predicted particles, only because are machines lack the necessary power, and they may be right - we simply don't know. So on we go, perhaps forever.

If carbon was missing from the periodic table, then chemistry might also be finite - so many orbitals, so many bonds, so many molecules. Then again, if carbon was missing from the universe, we wouldn't be here to talk about it. Well carbon is present, and the range of chemistry and biochemistry stretches beyond our imagination. From polymers to nanotubes to proteins, the array of molecules seems endless, and perhaps it is. If the combinations are in fact finite, they exceed the size and age of the universe, so we may as well call it infinite. With chemistry as a base, biology is also infinite, especially if you include exobiology. This does not mean knowledge marches on at the same pace forever; it does not. The law of diminishing returns dogs physics, chemistry, biology, and every branch of science, and even the arts. Still, there is always something over the horizon. A cure for cancer, or the majority of cancers, may be within our grasp in just a couple centuries. Horgan would not call this a revolution - it does not replace an erroneous paradigm with a correct one - yet such a discovery is just as important to mankind as some of the revolutions that have come before. Science is worth pursuing, even if all the revolutions are behind us, even if the next steps are slow and steady.

Unlike all the other topics in this article, I have actually studied mathematics, so perhaps my speculations are worth a little more in this arena. As Horgan points out, math is infinite. There is always a new theorem to discover and prove. Some of these theorems are interesting and some are not. Mathematicians don't have an objective definition for "interesting", nor do we have criteria to determine whether a proof is "elegant". These concepts are a bit like pornography - we know it when we see it. However, I'm pretty sure there are an infinite number of interesting theorems, although fewer and fewer of them will have elegant proofs. Some proofs go on for thousands of pages, such as The Classification of finite Simple Groups, and some proofs require computer verification, such as the Four Color Theorem. In the world of math, we have already picked the low hanging fruit, but the trees are tall, and there is always something else to prove. As the proofs get harder, results come slower, demonstrating once again the law of diminishing returns. Hundreds of mathematical geniuses across two centuries collaborated to prove Fermat's Last Theorem, and the resulting proof is understood, in its entirety, by just a couple hundred people in the world. (Sadly, I am not one of them.) See the Nova special for more details. What about the Riemann Hypothesis? This may require thousands of mathematicians spanning several centuries, and perhaps that proof, if and when it comes, can only be understood by a dozen people at a time. Ten thousand years from now, a gifted mathematician might spend 250 years (who knows what our life span will be at that time) learning all there is to know about algebraic topology, so that she can come up with one or two original results. These may be new theorems, or steps along the way to a large and unsolved conjecture. I hope this does not discourage our descendants from studying math, for the joy is in the learning, and the exploring, and there is always something new to discover, even if it takes most of your life to get there.

Turning to the arts,
there are an infinite number of songs to sing, and books to write, and movies to produce,
but most of these aren't very interesting.
People will start to repeat, and unfortunately, they already do.
How many songs are covers, and how many movies are remakes?
In my opinion, damn few of these are better than the original.
*War of the Worlds*,
*The Ten Commandments*,
*The Day the Earth Stood Still*,
*The Music Man*,
*Flight of the Phoenix*,
*The Sound of Music* -
these are all classics that should have been left alone!
(Occasionally a remake is worth its salt, e.g. *Freaky Friday* and *Oceans Eleven*.)
Even sequels rarely compare with their originals.
The arts are subject to exhaustion, just like the sciences.
It becomes harder and harder to come up with a new plot, a new idea, a new melody, a new theme.

"What has been will be again, what has been done will be done again;
there is nothing new under the sun."

Ecclesiastes 1:9

An S curve is not a precise mathematical function.
Rather, it is a function that starts low, rises quickly, then flattens out at a higher level.
It is shaped like the letter S, if you stretch your imagination and the letter S at the same time.
Pull the top half of the S to the right and the bottom half to the left, so that it becomes a function,
y as a function of x.
The middle of the S becomes the inflection point,
where the graph stops curving up and starts to curve down again.
It is still rising, always rising, but the slope decreases as you move to the right, and the graph flattens out once again.
An example is sin(x) from -π/2 to π/2.
The y coordinate starts at -1, then the graph slowly rises,
attains its steepest slope at the origin, the inflection point,
then flattens out again as it rises to 1.
This S curve has a finite domain, but we are more interested in curves whose domain is the entire x axis.
An example is ex for x ≤ 0,
and 2 - e-x for x ≥ 0.
The curve approaches 0 as x goes negative, and 2 as x goes positive,
as though 2 is as much as you can know.
(0,1) is the inflection point.
The knowledge acquired per year is maximized at this point.
Horgan's title, *The End of Science*,
suggests a curve like this, with a roof.
Yet I don't believe this is what he means.
I think he is proposing an S+ curve, and if he is, I agree.
This curve might look like ex for x ≤ 0,
and log(x+1) + 1 for x ≥ 0.
Assume this curve represents the state of knowledge, and rewind 200,000 years.

Our ancestors had some knowledge, knowledge that is properly called science. Perhaps they knew that "Red next to yellow will kill a fellow, red next to black is a friend of jack.", (though not using words), to distinguish between the venomous Coral snake and several harmless mimics such as the Scarlet snake. Certainly they knew how to make stone tools to obtain prey and defend themselves from predators and rival tribes. Their knowledge remained virtually flat from generation to generation. What would it be like to live in that time? Your world is your extended family, and nature, and hardship, and tools, and survival, and perhaps the feeling of love as you select a mate. There is no understanding of the past, and no anxiety about the future. If you are one in a thousand, you might craft a better tool, or build a better trap for small game. Perhaps this knowledge is lost when you die, or perhaps you are able to pass it on to your children, whereupon the knowledge curve rises just a bit, just a tiny fraction higher above the x axis. And so it goes, generation after generation, millenium after millenium.

In the recent past, humans developed language and agriculture, and the curve started to bend upward. We were really on an upward swing through the Babylonians, Greeks, Egyptians, and Romans, but the progress of civilization is not as monotonic as the S+ curve suggests. Most of Europe fell into the Dark Ages for 1,000 years. Dedicated individuals preserved knowledge as best they could, while advances were made in China and the Middle East. Finally the curve shot upward again in the period known as the Renaissance. This ushered in most of the scientific revolutions listed above. As we enter the 21st century, we have passed the inflection point, and knowledge is starting to flatten out. Technology continues at a brisk pace, but science, according to Horgan, is slowing down, as it inevitably must. My mother, who has very little education, but is very observant, expressed this thought to me when I was still a teen-ager, around 1978.

"My parents, your grandparents, have seen more change than any other generation ever has or ever will. When they were born in 1904, people were riding in horse and buggy. Now they see men walking on the moon."

In other words, the 20th century is the inflection point, when acquired knowledge per year was at its maximum. I think my mother is right, and if you don't take Horganism to an extreme, or take his ideas out of context, I think John Horgan is right as well.

I fear the arts have passed their inflection points as well. Music entered a "Cambrian Explosion" in the 60's and 70's, and by the 80's it was beginning to flatten out. Yes, there are some excellent bands making new music today, but not 50 innovative groups on each side of the pond, and I don't think today's music will be on the radio 50 years from now, whereas the Beatles will still be heard. Call me a 70's snob if you wish, that's fair. In a few decades we'll know if I'm right or wrong. In the same way, I believe classical music passed its inflection point around 1910. Sure I like George Gershwin, and Aaron Copland, but I still maintain that classical music is already flattening out. We will never see anything comparable to the Bach Beethoven Mozart explosion again.

Horgan may be off by a few decades, at most a century, regarding the precise inflection point of science,
but it's hard to see how he could be wrong in concept.
Science and math and the arts must reach a slowdown phase eventually,
as they ride the gently rising branch of the S+ curve.
I hope our descendents will not be discouraged a million years from now, when knowledge seems to be flat,
when no one has discovered anything new or interesting in several generations.
Keep plugging away, because there are still new things to learn, and new songs to write.
If you are not the one to discover a new truth, or write a new book,
you can still revel in the beauty of everything that has gone before.
Explore Gauss' proof of quadratic reciprocity,
listen to Beethoven's seventh symphony
and Boston *More Than a Feeling*,
and look at the rings of Saturn through a telescope.
You might not be on the inflection point, but it's sure better to be on the upper branch than the lower branch - sooooo much better!

John Horgan also published *The End of War* in 2012.
In this case I hope he is talking about a complete and abrupt cessation, rather than a slowdown.
War needs to end, today, and I believe it can.
Find more information at
WorldBeyondWar.org.

The incredible complexity of the abc conjecture