Conjectures, Hypotheses, Axioms

[Leo Authersh]

Intuitive representation of the differences amongst Conjectures, Hypotheses and Axioms.

@fresh_42 @FactChecker @WWGD

 

[FactChecker]

My two cents:
Axiom: A statement that will be assumed true without question in the following discussion. Can be used at will. The goal is to determine what would follow from the axioms, rather than to question the axioms.
Conjecture: A statement that is believed to be true. The case for and against its truth will be examined.
Hypothesis: A statement whose truth needs to be questioned and tested. There is less implied belief in it than in a conjecture. In fact, it may be something that is believed to be false and is expected to be disproved.

The distinction between conjecture and hypothesis is not formally defined.

 

[fresh_42]

Intuitive representation of the differences amongst Conjectures, Hypotheses and Axioms.

1. conjecture - unproven statement which is expected to be true by evidence like exemplary calulations or failed searches for counterexamples, e.g. $NP \neq P$. The literally translation would be "a casting (throw) together (of facts, etc.)".$\\$ $\\$
2. hypothesis - unproven assumption without any expectations whether true or false (however, often false), e.g. an assumption in an indirect proof that is supposed to be false. It is more of a tool in a reasoning than a statement on its own, e.g. https://en.wikipedia.org/wiki/Thesis,_antithesis,_synthesis. The literally translation would be "imputation" $\\$ $\\$
3. axiom - unproven statement which is set to be true by definition. It is somehow one of the fundamental bricks a building of conclusions is constructed upon. This leads to a true chain of conclusions and not necessarily to a true statement in an absolute sense. The attempts to find such absolute truths, which can be used as basis for all other calculations failed and had to fail. An axiom is the closest we can get to something absolutely true. The literally translation would be "tenet = it holds".

So all of them are unproven, and they have different roles within logical deductions.
One could shortly write them by
$$
\text{ axioms } \Longrightarrow \stackrel{\text{use of hypothesis }}{\ldots \ldots} \Longrightarrow \stackrel{\text{ unknown way }}{\ldots \ldots} \Longrightarrow \text{ proof of conjecture or counterexample }
$$