... Niels Bohr
If we gauge the "success" of the famous Mathematicians and Physicists in terms of batting averages, the first three chaps we'll discuss would have been 0.300 hitters, which is respectable, but the last would have hit 0.995 and made Babe Ruth look like a chump in comparison.
Ah, but what the "wild speculators" did RIGHT made up for all their wrong, wildly speculative ideas, and the one thing our .995 hitter did wrong would set back an important branch of Physics for 30 years.
The first two were professional Physicists, the last two professional Mathematicians. Let's begin:
1) George Gamow (1904 - 1968)
I love Gamow. IMO he deserved one, maybe two Nobel Prizes in Physics.
Before this age of Kaku, Hawking, Greene, more Kaku, DeGrasse Tyson, and still more Kaku (will the man EVER stop writing?!), and Sean Carroll, there was Carl Sagan. Before Sagan, Issac Asimov. But before Asimov, George Gamow was the top popularizer of Physics. As it says in his Wikipedia entry:
Gamow was a highly successful science writer, with several of his books still in print. He conveyed the excitement of the revolution in physics and other scientific topics of interest to the common reader. Gamow himself prepared the illustrations for his books, which added a new dimension to and complemented what Gamow intended to convey in the text. Wherever it was essential, he used mathematics.
In 1956, he was awarded the Kalinga Prize by UNESCO for his work in popularizing science with his Mr. Tompkins... series of books (1939–1967), One Two Three ... Infinity, and other works.
Gamow was working on a textbook entitled Basic Theories in Modern Physics, with Richard Blade, but it was not completed before he died. He wrote a book entitled My World Line: An Informal Autobiography, which was published posthumously in 1970.
What he got wrong: Ylem
Ylem is a term which was used by George Gamow, Ralph Alpher, and their associates in the late 1940s for a hypothetical original substance or condensed state of matter, which became subatomic particles and elements as we understand them today. The term ylem was actually coined by Ralph Alpher. 
It reportedly comes from an obsolete Middle English philosophical word that Gamow's assistant Ralph Alpher came across while thumbing through a dictionary, which means something along the lines of "primordial substance from which all matter is formed" (which in ancient mythology of many different cultures was called the cosmic egg), and derives from the Greek hylem, "matter". Restated, the ylem is what Gamow, et al., presumed to exist immediately after the Big Bang. Within the ylem, there were assumed to be a large number of high-energy photonspresent. Ralph Alpher and Robert Herman made a scientific prediction in 1948 that we should still be able to observe these red-shifted photons today as an ambient cosmic microwave background radiation (CMBR) pervading all space with a temperature of about five Kelvin (when the CMBR was actually first detected in 1965, its temperature was found to be three Kelvin). It is now recognized that the CMBR originated at the transition from predominantly ionized hydrogen to non-ionized hydrogen at around 400,000 years after the big bang.
In a taped interview available from the American Institute of Physics, Gamow talks about YLEM:
...this is YLEM. All this photography was done when I was still in Washington. When this series of origin of elements, alpha, beta, and gamma worked, I remember I bought the bottle of Cointreau to celebrate and wrote YLEM on it and then photographed the bottle and put my head as a genie coming out of it.
Weiner: What is YLEM in this case?
Gamow: A mixture of protons, neutrons and electrons.
Weiner: But the letters stand for what?
Gamow: You can look in the Webster dictionary. This is a word—I think it's an old Hebrew word, but Aristotle was using it—in Webster dictionary it says "material from which elements were formed".
Weiner: The primordial substance?
Gamow: The primordial substance, yes—ylem.
Weiner: The thick soup or whatever you want to call it?
Gamow: I mean this is the old Hebrew word meaning something like "space between heaven and earth".
2) John Archibald Wheeler (1911 - 2008)
Known for: Being Richard Feynman's Grad school mentor
I love Wheeler, too. Heck I love all these guys, why else do you think I'm writing about them? See how that works? But genius alone isn't enough, you must still be right, and who is right 100% of the time? Nobody.
What he got wrong: Promoting his grad student Hugh Everett III's CRAZY Many-Worlds Interpretation of Quantum Mechanics
I'm not going to give MWI any more space than it deserves. And it's not just me, believe that. I'm sure the maths are fine, but does it describe "reality"? I highly doubt it. To be fair, David Deutsch has a "different" take on MWI. Is that wrong too? I don't know. Generally speaking, I hate ALL interpretations of QM, but Copenhagen, weak though it may be, is still king IMO until someone proves otherwise.
3) Roger Penrose (1931-2032)
The single most adorable man on Planet Earth today, and most days. Maybe ever.
I don't like word "Quanglement". What it means, Quantum Entaglement, is fine. But I will never say "quangled." It just sounds stupid. Read the following an see if you can do so without laughing whenever you come across "quangle" or "quangled."
He was almost ... Never .... Wrong. Once he was though. It involved Quantum Entanglement.
What he got wrong: He wrote a fine textbook on QM that was hostile to Entanglement. Grete Hermann, mathematician and one of the few female members of Emmy Noether's students known as "Noether's boys", noted it first. Actually, von Neumann's proof was rigorous and true, but his assumption was not. Hermann was right, even Heisenberg agreed, but the year was 1934-35. Because she was a woman, any man backing her up risked his reputation. Also, von Neumann's greatness was well-established, and to take on that great man was a reputation risk in its own right.
So nothing was done about von Neumann's false assumption until John Stewart Bell discovered the same error, in the 60's.
And the rest as they say, is Quanglehistorical.
For further reading:
- Chapter 16, "The Age of Entanglement" by Louisa Gilder
- The Physical Review - The First Hundred Years - A Selection of Seminal Papers and Commentaries
- Roads To Reality - Comparing Penrose and Wolfram by J. Andrew Ross