What is the difference between theory and theorem?

Answer 1

A theorem is a step in a mathematical theory that allows, starting from hypotheses, to demonstrate rigorously some conclusion.

A theory is composed by five elements in general

- axioms;

- primary entities;

- logical rules;

- derived entities;

- theorems.

axioms and primary entities are the base of the theory: they are objects that are not defined from within the theory and relationships among these objects that are simply assumed three.

In standard geometry the point, the straight line, the plane are primary objects while the assumption that the straight line that passes y two given points always exists and it is unique is an axiom.

Every theory starts from axioms and primary objects. Logical rules are the rules that are assumed to guarantee that, if premises are true, the consequences derived using such rules are also true. Standard logical rules are generally assumed like

A=B => B=A

and the so. In mathematics standard mathematical demonstration rules like the modus ponens are also adopted. In selected theories, probability rules are also used as logical rules.

Derived objects are objects defined inside the theory, like the triangle or the polygon in standard geometry.

We have just see what a theorem is: it is the instrument allowing to derive new properties from objects axioms and already derived properties.

If we deal with a scientific theory, like a physical theory, a mathematical structure is not enough to use the theory to interpret nature. We must have the so called interpretation scheme allowing us to translate experimental results into relationships between the objects of the theory and back.

For example we must connect the theory object we call electron, with its properties like the motion equation, the charge and so on, with experimental observations like interference figures in electrons beam experiments that we interpret at the theory light as the sign of electrons presence.

Answer 2

A theorem is a step in a mathematical theory that allows, starting from hypotheses, to demonstrate rigorously some conclusion.

A theory is composed by five elements in general

- axioms;

- primary entities;

- logical rules;

- derived entities;

- theorems.

axioms and primary entities are the base of the theory: they are objects that are not defined from within the theory and relationships among these objects that are simply assumed three.

In standard geometry the point, the straight line, the plane are primary objects while the assumption that the straight line that passes y two given points always exists and it is unique is an axiom.

Every theory starts from axioms and primary objects. Logical rules are the rules that are assumed to guarantee that, if premises are true, the consequences derived using such rules are also true. Standard logical rules are generally assumed like

A=B => B=A

and the so. In mathematics standard mathematical demonstration rules like the modus ponens are also adopted. In selected theories, probability rules are also used as logical rules.

Derived objects are objects defined inside the theory, like the triangle or the polygon in standard geometry.

We have just see what a theorem is: it is the instrument allowing to derive new properties from objects axioms and already derived properties.

If we deal with a scientific theory, like a physical theory, a mathematical structure is not enough to use the theory to interpret nature. We must have the so called interpretation scheme allowing us to translate experimental results into relationships between the objects of the theory and back.

For example we must connect the theory object we call electron, with its properties like the motion equation, the charge and so on, with experimental observations like interference figures in electrons beam experiments that we interpret at the theory light as the sign of electrons presence.