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Are axioms nothing but assumptions and, if yes and in accordance to Occam's Razor, should they be minimized?

When postulating scientific theorems which, unlike axioms, are subject to the scrutiny of proof but do factor in axioms, are there formal systems which at least attempt to quantify the amount of axioms, or assumptions, that are antecedents to the theorem? Put simply, can we sort theorems based on how much of their input are axiomatically accepted assumptions?

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    I think the short and obvious answer would be sure, but sometimes it's simply impossible to minimize. But yes, definitely axioms are assumptions, that are needed logically to push a theory further, assumptions which can't be proven. – Yechiam Weiss Mar 14 '18 at 17:11
  • How about a formal system that quantifies theorems or even ideas based on the amount of axiomatic input? – amphibient Mar 14 '18 at 17:15
  • what is your goal here? Minimizing the axioms? Choosing theories based on the amount of axioms in them? – Yechiam Weiss Mar 14 '18 at 17:19
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    We do not postulate theorems: we prove them from axioms. – Mauro ALLEGRANZA Mar 14 '18 at 18:08
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    For math, there is a specific program of research: Reverse mathematics. – Mauro ALLEGRANZA Mar 14 '18 at 18:09
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Regarding "in accordinate with Occam's Razor", you should not minimize assumptions — in fact, you should maximize them.

Occam's razor is not about assumptions, but 'entities'; e.g. the quote

Entities are not to be multiplied without necessity. Source: Wikipedia

The point of the Razor is about having too many degrees of freedom — if your theory has a lot of flexibility to accommodate observation, then we haven't really learned much if you make observations consistent with the theory.

If we have the alternative, a theory that is inflexible and can only accommodate a narrow range of observations, then we learn a great deal when the theory consistently agrees with observation!

Given a theory, adding an additional axiom can only decrease the number of results consistent with the theory! The only effect an axiom can have is to let you prove new statements — and that means the negations of those statements are no longer consistent with the theory!


There is an obfuscating factor here, though; for some reason things are often discussed in the opposite direction.

For example, consider passing from the description of 3-space via Euclidean geometry to a description via a general Riemannian manifold.

In this comparison, Euclidean geometry is obtained by adding the additional assumption "the metric tensor is given by this specific formula". However, for some reason, this comparison would more often be phrased in the opposite direction: that Riemannian manifolds are obtained by adding the assumption "suppose the metric tensor is not given by the standard formula".

Passing to the Riemannian manifold description of 3-space has definitely multiplied entities, but it has reduced the amount of assumption you have to make about the nature of 3-space.

  • what's an entity in this context ? if you don't put it in a context, it's a useless term. – amphibient Mar 15 '18 at 15:08

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