[paul cotter (OP)]

It's a long time since I have had any formal scientific tuition. What I am interested in refreshing is the hierarchy of scientific thought. I tend to use terms like axioms, theorems, theories, hypotheses, conjectures etc in a rather flippant manner and I am asking for a rigorous evaluation of how scientific thought develops. PS I realise these terms can apply to most disciplines, not just science,

[alancalverd]

We start with observations and develop hypotheses that model them and predict the next observations. When we have gone through the loop a few times we can identify theories (a guess at the mechanism behind the obs) and laws (robust mathematical projections of what will happen in the wider universe, regardless of the mechanism).

Thus we have a law of gravitation F = GMm/r²  that is good enough to put a satellite into orbit, but no idea of how bodies actually attract each other apart from the theory that mass bends spacetime.

Underlying all this is the common language of mathematics which is based on the mutual acceptance of certain axioms such as those of Euclidean geometry (similarity, identity, parallelism, etc) with useful theorems (statements that can be rigorously proved as long as the axioms are true) such as Pythagoras.

As far as I can see, a conjecture is a mathematician's hypothesis: a  theorem that looks reasonable but hasn't been proved.

The difference is that mathematical proof is singular and positive (if A then B, QED) whereas scientific proof  is continuous and negative (we haven't found a system that doesn't obey Newton's laws).

[Eternal Student]

Hi.

I tend to use terms like axioms, theorems, theories, hypotheses, conjectures etc in a rather flippant manner

    Well many of these terms are flexible in their application.   Which one you chose to apply may depend on things like which country or culture you were brought up in and/or which sphere of human endeavour you are working in.

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There's a dictionary definition which we shouldn't really ignore.   Let's try and put your terms in an order from the strongest to the weakest:

Axiom:
a statement or proposition which is regarded as being established, accepted, or self-evidently true.

Theorem:
a general proposition not self-evident but proved by a chain of reasoning; a truth established by means of accepted truths.

Theory:
a supposition or a system of ideas intended to explain something, especially one based on general principles independent of the thing to be explained.

Hypothesis:
a supposition or proposed explanation made on the basis of limited evidence as a starting point for further investigation.

Conjecture:
an opinion or conclusion formed on the basis of incomplete information.
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   All definitions taken from the "Oxford Languages" dictionary which Google has built in to its short English Dictionary as of 2023.    (More information:    https://languages.oup.com/google-dictionary-en/    )
    For more scientific definitions use a specialist scientific dictionary.  As mentioned, most of these terms are flexible and very subject specific.  Mathematics through to Social Science will all have different accepted uses and understandings of the terms.
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     @alancalverd  has already made some effort to talk about the use of the terms in Science,  although I'd say that was very biased toward physics.
    @alancalverd  added the term "law".   That would usually be put above a theory and below a theorem, so it's a theory with so much evidence for it that it is almost certainly true.   Alternatively, you might consider a "law" as something that can be considered as an axiom and allows a system to be developed based on those axioms.   For example, Newton's laws of motion are the axioms of Newtonian Mechanics and it doesn't matter if Newton's laws were found to be wrong  (and according to General relativity they are).   What you can derive from Newton's laws is still valid in the system we call Newtonian Mechanics.   (Whether that is a good enough model for some real world situation is a different issue).
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Underlying all this is the common language of mathematics which is based on the mutual acceptance of certain axioms such as those of Euclidean geometry (similarity, identity, parallelism, etc) with useful theorems (statements that can be rigorously proved as long as the axioms are true) such as Pythagoras.

    This post is already too long,  so I won't discuss Mathematics and axiom systems.
    There's no obligation that Science is based on Mathematics or must be formulated as some mathematics.  Physics often is but Biology doesn't easily fit that description or stereotype.   Darwin's theory of evolution isn't formulated as a relationship between some numerical quantities.

Best Wishes.

[alancalverd]

For example, Newton's laws of motion are the axioms of Newtonian Mechanics and it doesn't matter if Newton's laws were found to be wrong

Spoken like a true mathematician.

It is indeed the case that the whole of newtonian mechanics can be rigorously derived from Newton's laws, but the world would be a very different place if they were wrong!

"Just Six Numbers" (Martin Rees, 1999) explains the profound significance of the six critical ratios that determine the observed size shape and history of the entire universe provided that Newton was right.

[Eternal Student]

Hi.

Sadly I don't have a copy of "Just Six Numbers", so I can't comment on that.   I may acquire a copy.

but the world would be a very different place if they were wrong!

   - but we do already know that they are wrong.   At best, Newtonian mechanics is a good approximation in the low speed and weak field limit of GR.
     Rather than theorising about regions near black holes that we may not have measured,  we do have some good data about the precession of Mercury.  GR explains this adequately while Newtonian mechanics (including Newtonian gravity) does not.   You don't have to use GR and maybe something like MOND would be a lighter adjustment but it is still some modification.
 
Best Wishes.

[alancalverd]

No disagreement.But how would the world look if F = GMm/r^2.1?

The "laws" of physics are all circumscribed by boundary conditions such as incompressibility and the like, which actually makes them very like good statute law, where "normally" and "reasonable" make life tolerable. But whilst hard cases make bad law, they make interesting physics!

[Bored chemist]

But how would the world look if F = GMm/r2.1?

Absent.
Only an inverse square law provides a stable orbit.