As typical with contrarians, Ramón y Cajal said some things that held up well and others that didn't. In the same book "Advice for a Young Investigator" that this excerpt is from he also gave his view of theorists: "Basically a theorist is a lazy person masquerading as a diligent one because it is easier to fashion a theory than to discover a phenomenon"!
Well, it's true that we can't really tell how serious he was being. And it is worth remembering he was a neuroscientist who studied neurons. He was probably thinking of people who made complex theories about "how the brain works" without ever designing experiments to test them, not physics theorists.
Didn't we recently confirm gravitational waves by checking out a couple interacting black holes, originally theorized by Einstein 100 years ago? I think even Einstein would agree that it was much harder to discover it than to theorize it.
The experiment would never have been done without the theory, and a long line of experiments suggesting that Nature and the theory had a lot to do with one another, first. We knew that gravitational waves, or something very much like them, happened due to the binary pulsar work by Hulse and Taylor in 1974.
Einstein's advance was quite spectacular, and early.
Without the theory, how well do you think the funding submission for the experiment would have gone? Two laser interferometers a couple of kilometres long in specially built tunnels, with super sensitive custom equipment costing millions of dollars running for years just on the off chance? Sure, here’s the cheque!
So their method for finding the signal was to calculate the signal, then subtract it from the data and see if the residual noise looks like noise...how is that good science? It seems like it would be too easy to make your data fit the theory.
There is a cool documentary about how Einstein was trying to prove relativity theory, and it involved a lot of work, special telescope, traveling, war... I think it was done at third attempt.
Imagine if he would have died before proving it ...
My point is, he didn't stop in the theory, but also designed an experiment and worked a lot to get it proved.
Cajal's statement clearly doesn't apply to him.
I'll try to find the video, or if someone remembers please link it.
I would say that successful theorists are exactly those who discover phenomena. Saunders Mac Lane is perhaps the epitome in mathematics of someone who was guided by phenomena.
This is why category theory was not discovered, it was reverse engineered! The reverse engineering steps were:
3. Natural transformations
2. Functors
1. Categories
Edit: Of course, when he said theorist I think he meant people who don't experiment physically.
Yes, but that would be a more generic description. All of mathematics abstracts previously partially known knowledge.
When we went from 1 coconut -> the set {1}, then we were being really abstract for the times.
But I think your point is that category theory synthesises group theory, linear algrebra, topology, etc. into one concept, which was very much the spirit of the origins of category theory. However, Mac Lane and Eilenberg thought that their diagrams were just an aid to mathematics (much like a Venn diagram, Cayley diagram or a Feynman diagram). But when they realised that natural transformations are so ubiquitous and fundamental, then they realised that their graphs were not just a useful shorthand, but in fact would lead to a whole new type of mathematics. When people thought (not Mac Lane though) category theory was "abstract nonsense" they were making this mistake of thinking that the diagrams are illustrations rather than concrete mathematics.
In the same way, you might thing that {1,2,3} is just an illustration, but in fact it is a rigorous shorthand for a very specific set.
The real meat behind category theory are things like natural transformations and adjunctions. But to get to category theory from there, you do a kind of reverse engineering.
I would say theories and discoveries go hand in hand, or maybe leapfrog each other. They both travel in the direction of the fog at the edge of our understanding.
