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could anyone please clarify me the meaning of the term 'hypothesis'?

with relation to terms 'reasoning' and 'assumption' ?

Many thanks

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A hypothesis is just a guess that something is true. In science you often make a hypothesis and then test it by doing an experiment. –  littleO Nov 9 '12 at 6:28
    
Thanks for your reply. you meant hypothesis is a positive assumption then? –  nish1013 Nov 9 '12 at 6:54
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You can also hypothesize that something is false. When we make a hypothesis, we recognize that it might be wrong, and quite often we eventually discover a hypothesis we had made was wrong. It really is just a guess, but usually with some amount of evidence (not proof) that the guess is correct. –  littleO Nov 9 '12 at 7:11
    
Probably I should say "educated guess" rather than "guess". –  littleO Nov 9 '12 at 8:01
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2 Answers 2

"Hypothesis" is one of those words that have a whole slew of interrelating meanings in different contexts, but no overarching simple definition that describe all of them exactly.

The most primitive and original meaning of "hypothesis" appears to be simply whatever we're reasoning FROM in a logical argument. This belongs to a view of logic that says the purpose of logic is not so much to establish absolute truths, but to find out what follows from what else. Then whatever something follows from is the hypothesis of the argument that the "something" indeed follows.

Note that at its root the word doesn't imply anything about why we chose to use that particular hypothesis as a starting point. It might be something we know is true, or something we think is true (and hope to prove true at some later time). But it could equally well be something we think is false and are trying to prove false by deriving something we already know is false from it. Very often a hypothesis will be something we know to be sometimes true and sometimes false, and the point of our reasoning is to find out more truths that will necessarily hold in those situations where the hypothesis happens to be true.

As a technical term within logic, this meaning of "hypothesis" is exactly synonymous with "assumption". And there are many other contexts where "hypothesis" and "assumption" are equally good words, since both derive their auxiliary meanings from this core logical meaning. In contexts where this is not the case -- that is, for purposes where you can only say "hypothesis" but not "assumption", or vice versa -- it is mostly a matter of historical accident which of the words have won out in each case. Attempting to formulate a general rule about which kinds of meaning "hypothesis" is better at expressing than "assumption" appears to be a fool's errand. They need to be learned one by one.

A very closely related logical meaning is that for any statement or claim of the form "if A, then B", A is said to the be hypothesis of the claim. This usage transfers the idea of a "hypothesis" from the argument that establishes $A\Rightarrow B$ to the naked assertion that $A\Rightarrow B$. "Assumption" is possible here, but appears to be less common than "hypothesis", especially if the claim is written symbolically rather than a theorem statement in prose.

The first "hypothesis" many students of mathematics (or CS) encounter under that name is the induction hypothesis. This is just the meaning from the previous paragraph: In order to prove $P(n)$ for all $n$ by induction, we need to prove $P(0)$ (or $P(1)$) as well as $P(n-1)\Rightarrow P(n)$. The "induction hypothesis" $P(n-1)$ here is called a hypothesis simply by virtue of being to the left of a $\Rightarrow$. During the actual induction step of the argument it becomes a "hypothesis" in the above original form.

Less mathematically, but occurring in the sciences in general, a hypothesis is a proposed truth. The connection with the logical meaning is that the argument for a scientific hypothesis often goes like this: "We have observed A, B, C. If such-and-such were true, then A, B, C is what we would observe. Therefore such-and-such is a possible explanation for our observations." The italicized part is where such-and-such becomes a "hypothesis" in the logical sense. From this usage the word broadens into meaning a proposed explanation that we're currently investigating how well works, and from there to mean any kind of speculation which doesn't yet have enough evidence in favor of it to be a "theory".

In the Riemann Hypothesis, it is more or less this latter sense of "hypothesis" that is at play. The naming of the RH is somewhat idiosyncratic; today it is much more common to call such things "conjectures" in mathematics. But names tend to stick once established.

A propos of Riemann, his famous inaugural lecture that launched differential geometry was entitled "About the hypotheses that lie at the foundation of geometry". Here "hypotheses" means what we'd call "axioms" in modern usage; again the connection to the logical core is immediate.

And while we're about giants, Newton declared "I'm not creating hypotheses". He was again referring to the logical meaning, but sideways, since his point was that he wasn't proposing any logical argument at all that would have his law of gravity as a consequence, and therefore he did not, in particular, need any hypothesis to reason from in such an argument.

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A hypothesis, in mathematics is just another word for conjecture, and conjectures are based on heuristic arguments, calculations, similar results etc. They are almost never simply guesses. In spite of induction and computer science tags, an induction hypothesis is different (a little). You try a few cases (typically refereed to as base cases) and than assume that a statement holds for some arbitrary integer, say $k$ -- this is the induction hypothesis, which you then use to prove that $k + 1$ case is also true. However, if you strongly believe in something (or you are given a homework problem) you usually consider one or two base cases.

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Many thanks for your reply. So hypothesis is something already reasoned/proved ? a 'proposition'? –  nish1013 Nov 9 '12 at 6:55
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No, a hypothesis is a conjecture, there might be good reasons to assume its validity, but it does not mean it is true. A proposition is a theorem, which might appear before an important theorem. A proposition is a theorem which seems unrelated to the topic or has no obvious motivation but is used somewhere in the proof of an important theorem. –  glebovg Nov 9 '12 at 8:06
    
However, for some reason mathematicians do not like change and prefer old names. For example, Bertrand's postulate or Bertrand's conjecture is no longer a conjecture -- it is a theorem, which Chebyshev proved, so some people refer to it as Chebyshev's theorem. I think it just takes a while to get used to new names. Another example is Poincare conjecture, which is actually a theorem. –  glebovg Nov 9 '12 at 8:16
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