Usually an axiomatic system does not stand alone, but other systems are also assumed to hold. For example, we will assume:
1. the real number system,
2. some set theory,
3. Aristotelian logic system, and
4. the English language.
We will not develop any of these but use what we need from them.
One of the pitfalls of working with a deductive system is too great a familiarity with the subject matter of the system. We need to be careful with what we are assuming to be true and with saying something is obvious while writing a proof. We need to take extreme care that we do not make an additional assumption outside the system being studied. A common error in the writing of proofs in geometry is to base the proof on a picture. A picture may be misleading, either by not covering all possibilities, or by reflecting our unconscious bias as to what is correct. It is crucially important in a proof to use only the axioms and the theorems which have been derived from them and not depend on any preconceived idea or picture. Pictures should only be used as an intuitive aid in developing the proof, but each step in the proof should depend only on the axioms and the theorems with no dependence upon any picture. Diagrams should be used as an aid, since they are useful in developing conceptual understanding, but care must be taken that the diagrams do not lead to misunderstanding. Two exercises in Chapter Two illustrate this point: (1) A false proof that all triangles are isosceles. (2) A faulty proof of a valid theorem.
Usually not all the axioms are given at the beginning of the development of an axiomatic system; this allows us to prove very general theorems which hold for many axiomatic systems. An example from abstract algebra is: group theory → ring theory → field theory. A second example is a parallel postulate is often not introduced early in studies of Euclidean geometry, so the theorems developed will hold for both Euclidean and hyperbolic geometry (called a neutral geometry).
Certain terms are left undefined to prevent circular definitions, and the axioms are stated to give properties to the undefined terms. Undefined terms are of two types: terms that imply objects, called elements, and terms that imply relationships between objects, called relations. Examples of undefined terms (primitive terms) in geometry are point, line, plane, on, and between. For these undefined terms, on and between would indicate some undefined relationship between undefined objects such as point and line. An example would be: A point is on a line. Early geometers tried to define these terms:
point Pythagoreans, “a monad having position"
Plato, “the beginning of a line"
Euclid, “that which has no part"
line Proclus, “magnitude in one dimension", “flux of a point"
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Answer:
Usually an axiomatic system does not stand alone, but other systems are also assumed to hold. For example, we will assume:
1. the real number system,
2. some set theory,
3. Aristotelian logic system, and
4. the English language.
We will not develop any of these but use what we need from them.
One of the pitfalls of working with a deductive system is too great a familiarity with the subject matter of the system. We need to be careful with what we are assuming to be true and with saying something is obvious while writing a proof. We need to take extreme care that we do not make an additional assumption outside the system being studied. A common error in the writing of proofs in geometry is to base the proof on a picture. A picture may be misleading, either by not covering all possibilities, or by reflecting our unconscious bias as to what is correct. It is crucially important in a proof to use only the axioms and the theorems which have been derived from them and not depend on any preconceived idea or picture. Pictures should only be used as an intuitive aid in developing the proof, but each step in the proof should depend only on the axioms and the theorems with no dependence upon any picture. Diagrams should be used as an aid, since they are useful in developing conceptual understanding, but care must be taken that the diagrams do not lead to misunderstanding. Two exercises in Chapter Two illustrate this point: (1) A false proof that all triangles are isosceles. (2) A faulty proof of a valid theorem.
Usually not all the axioms are given at the beginning of the development of an axiomatic system; this allows us to prove very general theorems which hold for many axiomatic systems. An example from abstract algebra is: group theory → ring theory → field theory. A second example is a parallel postulate is often not introduced early in studies of Euclidean geometry, so the theorems developed will hold for both Euclidean and hyperbolic geometry (called a neutral geometry).
Certain terms are left undefined to prevent circular definitions, and the axioms are stated to give properties to the undefined terms. Undefined terms are of two types: terms that imply objects, called elements, and terms that imply relationships between objects, called relations. Examples of undefined terms (primitive terms) in geometry are point, line, plane, on, and between. For these undefined terms, on and between would indicate some undefined relationship between undefined objects such as point and line. An example would be: A point is on a line. Early geometers tried to define these terms:
point Pythagoreans, “a monad having position"
Plato, “the beginning of a line"
Euclid, “that which has no part"
line Proclus, “magnitude in one dimension", “flux of a point"
Euclid, “breadthless length"
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