Axiomatic system vs Formal System

Question

What's the differences beetween Axiomatic systems and Formal Systems ? What i think is that an Axiomatic system is simply a less strict version of a Formal System, as the answer to this question says.

And, for example, linear algebra stands on an axiomatic system, instead First Order Logic stands on a formal system.Is it true ?

What do you think ?

According to a reasonable defintion, an Axiomatic system "is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems." If so, Spinoza's Ethica, ordine geometrico demonstrata is an axiomatic system. It is not a Formal system; compare it with e.g Post canonical system. — Mauro ALLEGRANZA, Mar 11 '18 at 17:38
The biggest difference may just come down to whoever uses the words. There's not any clean mathematical distinction in which the two terms have separate, well-established, broadly recognizable meanings. Instead, authors use various terms ("axiomatic system", "formal system", "theory", "logic", "formal logic", "logical system", ...) to mean a variety of related things. Of course, in philosophy or other areas it might make perfect sense to distinguish the terms somehow, but the way they are distinguished is likely to depend on the author. — Carl Mummert, Mar 11 '18 at 18:57
Does this answer your question? Clarifying the definition of an axiomatic system — tryst with freedom, Mar 09 '22 at 14:00

Peter Smith · Answer 1 · 2018-03-13T13:34:11.123

Terminology is fluid (as Carl Mummert remarks). But in many/most people's hands, axiomatic vs non-axiomatic and informal vs formal mark orthogonal distinctions.

You can have informal and formal axiomatic systems (Euclid vs first order Peano Arithmetic). You can axiomatic vs non-axiomatic systems, whether fully formal or otherwise (e.g. axiomatic and natural deduction systems of first order logic).

Axiomatization is a matter of how some theoretical apparatus is organised: do we lay down some "starter" propositions, and then some rules for deriving more propositions? Or do we, e.g. regiment just using derivation rules?

Formalization is a matter of how stringent we are in specifying that apparatus -- usual informal mathematical standards of rigour or the sort of specifications we could (in principle) feed to a computer for formal checking?

I said Euclid, not Euclidean. Of course we can formalize Euclidean geometry as e.g. Hilbert did. But the Euclid we learnt at school could be said to be an axiomatic but not fully formalized theory. — Peter Smith, Oct 23 '21 at 15:57

Axiomatic system vs Formal System

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