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Wikipedia’s “List of conjectures” page lists 128 mathematical conjectures. All conjectures are called “conjecture” except two: Riemann hypothesis and Schinzel’s hypothesis.

I think there is a consensus that the Riemann hypothesis is a conjecture not a hypothesis. For instance, Mathworld says “the Riemann hypothesis is a deep mathematical conjecture...” If so, why is the Riemann conjecture called the Riemann hypothesis?

Do you know any sources that trace the story of how the conjecture of Riemann came to be called a hypothesis? I know that in his 1859 paper Riemann did not state the “Riemann conjecture” as a formal conjecture in the form we use it today.

Am I wrong to think that “hypothesis” is a concept used in life sciences? "Hypothesis" is not a mathematical concept. When a conjecture is proved it becomes a theorem. We cannot say the same for hypothesis.

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    $\begingroup$ No clear difference in use... See Conjecture as well as Definition of conjecture. $\endgroup$ May 14 at 6:38
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    $\begingroup$ But also Hypothesis has a mathematical use: "to denote the antecedent of a proposition, like in the proposition "If P, then Q"." Thus, we can say "the hypotheses of a theorem". $\endgroup$ May 14 at 6:39
  • $\begingroup$ @MauroALLEGRANZA I see your point but I still think there is too much ambiguity in common usage. For instance, are there any formal conditions that a statement must satisfy before being classified as a conjecture? This question hsm.stackexchange.com/questions/6957/… has historical examples of conjecture that was helpful to me. $\endgroup$
    – zeynel
    May 14 at 7:15
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    $\begingroup$ On the use of "conjecture" in mathematics and elsewhere see Mazur, Conjecture. He tracks its first use to (Winson's 1902 English translation) of Hilbert's 1900 address that stated his 23 problems. He did not apply it to Riemannsche Vermutung, but von Koch did in 1902. Heath in his translation of Elements uses "Riemann hypothesis" for what Saccheri called "hypothesis of the obtuse angle". Possibly, this was transferred to continuum and Riemann "hypotheses". $\endgroup$
    – Conifold
    May 14 at 8:07
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    $\begingroup$ Your comment about it not being a mathematical concept seems a bit far-fetched. The word hypothesis is widely used in statistics as the null hypothesis and the alternative hypothesis. Whether statistics is part of mathematics could be debated of course. $\endgroup$
    – mdewey
    May 14 at 13:25

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Naming is up to convention and broad agreement, and subject to variable forces and coincidences, such as choices by translators.

Felix Klein describes the Riemann Hypothesis (RH) as follows in his "Entwicklung der Mathematik in dem 19ten Jahrhundert"):

Die Grundlage dieser Arbeit ist die sog. $\zeta$-Funktion, über deren Nullstellen Riemann bestimmte Vermutungen aussprach, die trotz eifrigsten Bemühens von Seiten der verschiedensten Mathematiker noch immer nicht alle bewiesen sind.

Notice, the use of "Vermutungen" here. This can be very sensibly translated as "conjecture" today. In German, RH is to this day known as "Die Riemannsche Vermutung", though it is also known as "Riemann Hypothese". But is it perhaps that indeed these are substitutable terms, at least in German understanding, and as indicated by Conifold the term conjecture is actually modern predating some of these shifts and changes (and is anglo-centric).

I believe that there is another effect at play here. The notion of having a clear singular categorical term like "conjecture" is new. See for example Hilbert's publication on his famous problems. The term "hypothesis" does not appear in the German 1, though it does appear in the English translation 2 where the German term "Voraussetzung" is translated as "hypotheses". The introduction of Hilbert's essay is very valuable for this kind of discussion as it encodes Hilbert's grappling for abstract mathematics in the awareness that some axioms, or foundational assumptions may have been made, while trying to fend off too enthusiastic use of them from physical intuition. The term "Vermutung" appears only once as "Kroneckerschen Vermutung" (which BAMS does translate as "Kronecker's conjecture". Apparently the first known use of conjecture per Conifold's comment. It's noteworthy that the term conjecture also only appears this once in the article.). Regarding the Riemann Hypothesis, Hilbert writes: "äußerst wichtigen Behauptung von Riemann" ("extremely important claim by Riemann" (my translation) compare "exceedingly important statement of Riemann" (BAMS translation)). To me the variability in coinage simply means that there is not one category, but claims, assumptions, guesses are described with variable terms all indicating that they are not certain or proven.

Incidentally, there is a recently solved "Hauptvermutung" that also does not go by the English conjecture in its name, but by broad agreement has kept its German coinage (for a possible German root of the term see Alexandroff and Hopf "Topologie I" Springer (1935)). "Haupt" here means main or fundamental, and Vermutung is simply a German term for conjecture, assumption, or guess. The full term "Hauptvermutung der kombinatorischen Topologie" translates literally to "fundamental conjecture of combinatorial topology". There are numerous terms where English-speaking mathematics has fun using German nomenclature. See for example "Nullstellensatz".

What the words "hypothesis" means (and English and German understanding does not necessarily have to fully agree here!) and how they fit into mathematical thinking has shifted when foundational axiomatization became a program and goal with Russell and Hilbert. It is a mistake to translate concepts naively across these lines.

Riemann's own famous paper on the foundations of geometry is titled: "Über die Hypothesen, welche der Geometrie zu Grunde liegen" contains the word hypothesis. The physical and mathematical disciplines were not as categorically separated as they are these days, so there is no surprise that a term like hypothesis appears here. Incidentally in the text the term appears only once. Similar usage of "hypothesis" can also be found in Poincaré's book "La Science et l'Hypothèse" (Science and Hypothesis), where Euclidean, Riemann's and Lobachevsky's ideas are contrasted as basing on different "hypothesis". This makes sense when there is a grappling for the right set of axioms, and in fact a big part of the shift was the departure from Euclid's parallel axiom. Part of the recognition here, both in Riemann and Poincaré is that geometric axioms as related to the physical world are based on hypothesis of space in physical reality. I.e. they are, at least in their minds, not pure abstraction and axioms.

Incidentally none of this actually explains why it is normative today to say Riemann Hypothesis. The earliest known use of Riemann Hypothesis to me in the English speaking literature is:

In it the exact phrase "the Riemann Hypothesis" appears numerous times. It's my best guess that this coinage simply stuck.

Correction: Conifold and KCd brought to my attention that an abstract by Hardy of 1914 mentions Riemann Hypothesis in English (predating the above!) and that there are earlier mentions of Riemann Hypothesis in French by Littlewood in 1912. See comments below.

P.S. As von Koch (1902) was mentioned. He first calls it "Satz von Riemann" with a footnote that states that it the "Satz" (theorem) has not been rigorously proven. Later he refers to the statement as "Die fundamentale Riemannsche Behauptung", mirroring Hilbert's earlier German nomenclature. He does use the word "Hypothese" but not with attribution to Riemann on page 463. In other words von Koch uses three different terms to describe RH.

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    $\begingroup$ Although this isn't related to the way the problem got labeled as a "hypothesis" in English, the word "conjecture" in Russian is гипотеза (gipoteza), which looks just like the word "hypothesis" but really means "conjecture", e.g., the Hodge conjecture, Poincare conjecture, and Birch and Swinnerton-Dyer conjecture are all named in Russian as гипотеза of Hodge, Poincare, and Birch and Swinnerton-Dyer. $\endgroup$
    – KCd
    May 14 at 19:53
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    $\begingroup$ You say a 1916 Hardy and Littlewood paper is the earliest English reference you know mentioning the term "Riemann hypothesis". An earlier 1912 paper by Littlewood uses it in French: Quelques conséquences de l’hypothèse que la fonction ζ(s) de Riemann n’a pas de zéros dans le demi-plan $R(s) > 1/2$ , C.R. Acad. Sci. Paris Sér. I Math. 154 (1912), 263–266. See biodiversitylibrary.org/item/31526#page/269/mode/1up. The term l'hypothèse de Riemann appears several times in the paper. $\endgroup$
    – KCd
    May 14 at 20:44
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    $\begingroup$ Hardy used "Riemann hypothesis" in scare quotes in a note in Proceedings of the London Mathematical Society from 1914 On the Zeros of Riemann's Zeta Function, probably meaning that the term was not yet coined in English. I agree that it likely got entrenched by Hardy-Littlewood before "conjecture" became standard. $\endgroup$
    – Conifold
    May 14 at 21:10
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    $\begingroup$ Landau has Riemannsche Vermutung in his review of von Koch 1902 (which Zentralblatt mislabels as 1901). He then keeps using it in Handbuch der Lehre von der Verteilung der Primazahlen (1909) and subsequent papers. So he is probably responsible for German coinage. $\endgroup$
    – Conifold
    May 14 at 21:46
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    $\begingroup$ The usage by von Koch in his 1901 paper is also hypothèse only in the sense of "assumption" (namely the assumption that $R(\rho) = 1/2$) and he didn't write l'hypothèse de Riemann. $\endgroup$
    – KCd
    May 14 at 21:46

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