Crazies say they validate and test their idea, but they actually suffer from severe confirmation bias where they summarily dismiss any counterexample or problem.
> Would you be able to spot a Ramanujan from the dross?
Yes. We would do this by analyzing their arguments for logical errors, then testing their new theory, then hopefully proving their new theory. The difference between a crazy and genius is the results their new ideas produce.
I don't think you appreciate how crazy Ramanujan was in his time. While he was somewhat appreciated in India (not at any high level though), when he reached out to professors in the UK, the first several thought he had no ability to become a mathematician. They were used to seeing mathematical proofs, and Ramanujan not providing any made him appear fraudulent. The only one who understood his brilliance was G.H. Hardy, also considered an eccentric by his peers.
In Ramanujan's words, "an equation for me has no meaning unless it expresses a thought of God". In mathematician circles of the early 20th century, this was an obscene statement. Even today, can you imagine a researcher in any field saying something like that, and being taken seriously?
Yet Ramanujan is probably one of the greatest mathematicians of all time. In Hardy's word, he had "never met his equal, and can compare [Ramanujan] only with Euler or Jacobi".
To think that "we" (whoever you mean by that) can recognize sheer talent of Ramanujan's type is, frankly, arrogant. If anything, the ivory towers are even more enclosed - research is more and more about the quantity of publications, not about the quality of ideas. A recent Nobel Laureate stated that if he had not received the Nobel Prize, his university would have probably fired him, for lack of a consistent publication record.
So no, I don't agree with your statement. While the scientific method you describe for approaching nature has worked tremendously well for understanding the world, it still completely fails to capture the core of what it is that makes a person a true genius.
Which only proves the point further - Hardy could have easily selected one of the other 999 cranks, yet by some intuition he knew that Ramanujan was the real deal.
After seeing Ramanujan's theorems on continued fractions on the last page of the manuscripts, Hardy said the theorems "defeated me completely; I had never seen anything in the least like them before", and that they "must be true, because, if they were not true, no one would have the imagination to invent them".
Ideas are cheap and proof is expensive. It’s impossible to give all ideas equal consideration. You have to apply a filter, it’s inevitable. The question is just what filter you apply.
Ramanujan was missed by at least one maths professor. Also, there are mathematical proofs such as that of Fermat's last theorem, or the claimed proof of the abc conjecture by Shinichi Mochizuki for which verification is very difficult.
I'll agree that the costs don't come close to replicating work by LIGO or CERN.
There's a difference between a flawed proof (e.g. Wiles's original proof of FLT which was found to contain a gap) and outright crankery, though. There are more mathematical cranks than most people would think, and their arguments usually fall apart rather easily.
yes, but it depends on the field. ramanujan had the great advantage of working in a field where ideas can be tested with pencil and paper. if you show a purported ramanujan a lacuna in his proof, or if you can't find one, that's pretty good evidence
it's much harder to skim the dross off a leonardo, to correct your mixed metaphor, because siege engines and flying machines require empirical testing, and that's expensive. within the hypothetico-deductive method you can tell whether someone's logic is incorrect (just as with ramanujan) but if they're proposing a different hypothetical basis, you often have to test it empirically
often you can proceed from their proposed hypotheses to obviously empirically false conclusions. but historically that hasn't been an especially good guide; in the 19th century kelvin proved that the sun was younger than the earth, demonstrating that something was wrong with his hypothetico-deductive framework, but it took quite a while to figure out what. and today relativity and quantum mechanics are widely considered irreconcilable
Would you, though? Ramanujan famously left out his proofs quite often, relying on his intuition which was especially phenomenal. Many accomplished mathematicians did not see his genius for what it is and it took the support of Hardy to really further Ramanujan's career and accomplishments.
Maybe not me personally, but more accomplished, smarter mathematicians would. His results were checked as being correct and he did publish papers in which he showed some of the steps and this was before Hardy. T
Ramanujan was so ahead of his peers that they didn’t even know how to assess him or his work.
After seeing Ramanujan's theorems on continued fractions on the last page of the manuscripts, Hardy said the theorems "defeated me completely; I had never seen anything in the least like them before",[76] and that they "must be true, because, if they were not true, no one would have the imagination to invent them".[76] Hardy asked a colleague, J. E. Littlewood, to take a look at the papers. Littlewood was amazed by Ramanujan's genius. After discussing the papers with Littlewood, Hardy concluded that the letters were "certainly the most remarkable I have received" and that Ramanujan was "a mathematician of the highest quality, a man of altogether exceptional originality and power".[74]: 494–495 One colleague, E. H. Neville, later remarked that "not one [theorem] could have been set in the most advanced mathematical examination in the world".
The current norms of pedagogy permit incremental improvements, but they fail to handle talent that is far and away better than the current top of the field. In fact they find ways to prohibit people from shaming them in such a way, which is understandable. But unfortunately, a Ramanujan today would probably be relegated to “remedial education” or institutionalized.
As for his results being checked, they were - after nearly a century in some cases:
As late as 2012, researchers continued to discover that mere comments in his writings about "simple properties" and "similar outputs" for certain findings were themselves profound and subtle number theory results that remained unsuspected until nearly a century after his death.
What I learned from reading about Ramanujan is that true geniuses are not just incrementally ahead of their field - they are so far ahead, it is literally unfathomable to laypeople and experts alike. The best approach is to step aside and try to nurture the talent without interfering in their methods and ways.
Would you be able to spot a Ramanujan from the dross?
You say that "they" ... something ... but ... they ... something ... disagree with the world. Care to flesh this out with an anonymised example?