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To make it more formal, I am looking for striking historical examples of objects or concepts that were well known in a field and perceived as different, but later discovered to be the same. I am interested in how they happen and what impact they have, because they are perhaps the clearest symbols of scientific success, and the simplest cases of what Arthur Koestler called bisociation, "perceiving of a situation or idea in two self-consistent but habitually incompatible frames of reference", which underlies discovery process in general.

This is potentially a "big list" question, but I expect that striking "bidentifications" that had a major impact are very rare. Below are three examples of what I have in mind, but examples in any natural or social sciences would be interesting.

Hesperus is Phosphorus (Venus)

Perhaps the prototypical such example, the discovery is anecdotally attributed to Pythagoras (6th century BC). Careful observation of the night sky reveals that the bright star appearing in the evening after the sunset is the same as the one seen before the dawn. Had to be done before the motion of Venus could be modeled.

Brahistohrone is tautochrone (cycloid)

"But the reader will be greatly amazed, when I say that exactly this cycloid, or tautochrone of Huygens, is our required brachistochrone." With these words Johan Bernoulli in 1697 presented his solution to the problem that attracted the attention of Newton and Leibniz, among others, and launched the calculus of variations. Perhaps, in some part because Bernoulli's example encouraged the motivating idea that solutions might be otherwise known entities. His method of solution, the optico-mechanical analogy, was itself an example of bisociation, which played a role in discovering the variational principles of mechanics. The brahistochrone is the curve constrained to move along which a heavy particle will reach the end the fastest. And the tautochrone is the curve that the particle starting at any intermediate point of will reach the end of in equal times. Today we should be even more amazed that the fastest curve is the curve of equal times because even a slightest variation in the conditions of the problem destroys the correspondence.

Coal is diamond (carbon)

In 1796 after burning diamonds (!) in oxygen Tennant discovered that only carbon dioxide was produced as a result. Which meant that diamond is chemically identical to coal, a conclusion that evaded Lavoisier. A century later in 1893 Moissan stirred a major controversy claiming that he transmuted coal into diamond in his furnace. Today his claim appears more credible, and of course diamond synthesis is an industry.

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Such "bidentifications" of concepts in math with major impact are what some important mathematical ideas are all about. Here are some examples, and the list could very easily be extended.

1) Logarithms show multiplication on $\mathbf R_{+}$ is a disguised form of addition on $\mathbf R$, and this is why logarithms were such an important computational aide for centuries.

2) The introduction of coordinates in the plane by Descartes showed many types of curves are basically the same thing as algebraic equations in two variables, which allowed geometric problems to be solved algebraically, algebraic problems to be solved geometrically, and allowed spaces of dimension greater than three (and the associated geometric language) to be defined by using ordered lists $(x_1,\dots,x_n)$ with $n > 3$.

3) Computing areas and tangents are the same problem in reverse, as codified in the Fundamental Theorem of Calculus. The importance of this discovery is clear to anyone who knows calculus.

4) Exponential and trigonometric functions are essentially the same thing when allowed to be functions of a complex variable. This streamlines the theory of Fourier series when you allow complex coefficients, since it lets infinite series in sines and cosines be written as series in powers of $e^{ix}$.

5) The discrete groups preserving certain automorphic functions are the same as groups of motions in non-Euclidean geometry. Poincare's discovery of this while stepping onto a bus is famous, and it led to his model of the hyperbolic plane using complex variables.

6) Probability lacked an axiomatic mathematical foundation until the 1930s, when Kolmogorov identified the basic concepts of probability theory such as events, probabilities, random variables, and independence, with notions of measure theory that came out of the work of Lebesgue from thirty years before. Briefly put, probability = measure theory when the whole space has measure 1.

7) Modular forms and representations of Galois groups were studied separately, even by Artin and Hecke in the same department at the same time, but the fact that they are two different ways of getting at the same thing (constructing the same $L$-functions in different ways) was not appreciated until several decades later in the context of the Langlands program.

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  • $\begingroup$ I was kind of trying to limit the scope of "bidentifications" versus more diffused forms of bisociation as being about objects and easily presentable in a form A is B, but many of your examples still qualify. $\endgroup$ – Conifold Jan 22 '15 at 22:08
  • $\begingroup$ Since you allowed concepts previously thought of as different and later recognized to be the same, I felt that all these math examples (and so many more) really do fit. For instance, tangent lines and areas are not literally the same thing, but the development of calculus is built off the fact that problems of both types can be turned into each other, so that conceptually they are the same even though they don't look at all alike. You wrote that "many" of my examples still qualify, so are there particular ones you don't think qualify, or are you just not sure what some of them are about? $\endgroup$ – KCd Jan 22 '15 at 23:38
  • $\begingroup$ True, I didn't make it precise, and it's hard to separate conceptual "objects" like brachistochrone from concept concepts anyway :) We can say derivative of antiderivative is the original, but it gives away the punchline, and in terms of tangents and areas there'd be lots of subordinate clauses. I am happy with all of them, it's just harder to turn some into A is B where A and B are "mathematical objects". $\endgroup$ – Conifold Jan 23 '15 at 1:03
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I would say that the most famous discovery of this sort was the explanation, why "gravitational mass" and "inertial mass" are the same. (The fact was so well known since Galileo that nobody really noticed that it is some strange coincidence. This remarkable coincidence is explained in General relativity).

The only competing example is that light and electromagnetic waves are the same, but it does not quite fit: electomagnetic waves (other than light) were not detected until after Maxwell predicted them. But anyway, he discovered that optics and electromagnetism are about the same thing.

Much earlier discovery of the same kind is that Sun is one of the stars, and Earth is one of the planets.

EDIT. It did not occur to me until I read KCd's answer that examples from mathematics also qualify:-) But in mathematics there are so many, that this will be indeed a VERY big list. So I limited myself to physics/astronomy.

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  • $\begingroup$ Nice examples! We can still say inertia is gravity charge, and light is electricity :) But isn't the equivalence principle a postulate, how does GR explain it? I thought Mach's principle was meant as an explanation, but Einstein ultimately rejected it. $\endgroup$ – Conifold Jan 22 '15 at 22:22
  • $\begingroup$ The space allowed me to comment does not permit to explain how Einstein explained this. But shortly speaking, by interpretation of gravity as a property of the space itself, rather than some "charge". And yes: Mach probably was the first to see that there is a problem here. And made an attempt of explanation. $\endgroup$ – Alexandre Eremenko Jan 22 '15 at 22:27
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My pick is Apatosaurus/Brontosaurus.

A short run-down:

Around 1877, Othniel Charles Marsh discovered a dinosaur skeleton that he named Apatosaurus. In 1879, he found a different skeleton that seemed completely different from the first. Marsh named it Brontosaurus. Throughout the next few decades or so, skeletons were incorrectly assembled and mixed up due to Marsh's mistake (he had also incorrectly added the Apatosaurus head, as well); eventually, it was realized that a mistake had been made. Exactly 100 years later, the correct head was inserted, thereby definitively dissipating the myth that Brontosaurus was a distinct genus.


Update

A recent study has concluded that Brontosaurus is a different genus than Apatosaurus. Peer review is needed to determine the validity of the result, but it seems that the two were different all along.

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I hope my memory of this is right...

James Clerk Maxwell came up with some differential equations to describe electricity and magnetism. As usual when differential equations are solved, there was a certain constant that appeared. Measurements showed the constant had value $3 \times 10^{10}$ cm/sec ... the speed of light. Nowadays we call light an "electromagnetic radiation", acknowledging its connection to electricity and magnetism.

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    $\begingroup$ This is true. He found a differential equation like a wave equation, and he computed the velocity of that wave, finding a value close to the then-accepted estimate for the speed of light. See physics.stackexchange.com/questions/1574/… $\endgroup$ – KCd Jan 27 '15 at 2:37
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Matrix mechanics and the Schrödinger wave formulation of quantum mechanics. A brief history of the mathematical equivalence between the two quantum mechanics, Carlos M. Madrid Casado, Lat. Am. J. Phys. Educ. Vol. 2, No. 2, May 2008.

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This might not be quite what you mean, but I would suggest NP-complete problems.

There have been many, many problems, including Sudoku, scheduling, graph colouring, SAT, and thousands of other, which have all been proved to be NP-complete, so turn out to be "equivalent".

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