About Foundations of Geometry
Truth Tables. In this topic we take a careful look at making mathematical statements and determining their validity. Our main tool will be the use of truth tables. But first we will consider what a mathematical statement is and how to write clear and precise mathematical statements. Then we learn how to do computations with statements so that we can determine tautologies, contradictions, and contingencies.
Truth Table Examples. In this topic we illustrate the process of building truth tables. In terms of a truth table, statements p and q are called variables By definition, a proposition whose value is true for all cases of all variables is called a tautology; a proposition whose value is false for all cases of all variables is called a contradiction; and a proposition whose value depends on a value of a variable is called a contingency. Note that all four operations, And, Or, Not, and Implication are all contingencies. It is interesting to also note that we can take a combination of contingency statements and build a contradiction or a tautology. Recall that the contrapositive of is the statement and the converse of is the statement. In this topic we will construct the truth tables for three tautologies, three contradictions, and the contingencies.
Axiomatic Method. The problem is to erect the entire structure of Euclidean geometry upon the simplest foundation possible; i.e. to choose a minimum number of undefined elements and relations and a set of axioms concerning them, with the property that all of Euclidean geometry can be logically deduced form these without further appeal to intuition.
Euclid's Common Notions, Postulates, And Definitions. This topic of geometry states the postulates and definitions that Euclid used in his Elements. The early Greek approach to the axiomatic method did not call for listing of undefined terms (or primitives) and as such, Euclid failed to realize the need to leave certain terms undefined. Euclid's perception of a point was physical, "a point is that which has no part." This and other definitions are more accurately called descriptions rather than definitions. Euclid used the term postulate to refer to assumptions that were specific to geometry. Common notions (often called axioms), on the other hand, were assumptions used throughout mathematics and not specifically linked to geometry.
Birkhoff's Postulates. Birkhoff was the first to build the real number system into the foundations of Euclidean geometry. In doing so; he only needed four axioms instead of Hilbert's sixteen axioms. Birkhoff's approach has gained acceptance even though it does not produce new theorems in Euclidean geometry. His axioms do produce a logical equivalent version of Euclidean geometry with fewer needed axioms and so his approach is appealing. One postulate is the usual assumption that two points determine a unique line. His postulates of Line Measure and Angle Measure is where the real numbers enter; he assumes that the points on a line (and angles through a point) can be put into a correspondence with real numbers and real numbers mod 360 in a way that is compatible with distance measurement and angle measurement, respectively. In particular, no mention of betweenness is needed in Birkhoff's axioms because a point being between two others can be defined in terms of distance and real numbers. This topic of geometry states Birkhoff's axioms as published in the Annals of Mathematics in 1932.
Hilbert's Undefined Terms, Definitions, and Axioms . It wasn't until after the discovery of non-Euclidean geometry that mathematicians began examining the foundations of Euclidean geometry and formulating precise sets of axioms for it. The problem was to erect the entire structure of Euclidean geometry upon the simplest foundation possible; i.e. to choose a minimum number of undefined elements and relations and a set of axioms concerning them, with the property that all of the Euclidean geometry can be logically deduced form these without further appeal to intuition. Hilbert's approach does address Euclid's lack of attention to the notion of undefined terms and the concepts of incidence, betweeness and congruence. An example of Hilbert's precision and detail was to distinguish between a line and a line segment, as Euclid did not. This topic details Hilbert's undefined terms and preliminary definitions which can be used to provide the basis for traditional Euclidean geometry. A famous quote from Hilbert: "One must be able to say at all times-instead of points, lines, and planes---tables, chairs, and beer mugs."
MacLane's Postulates. In the 1960's Saunders MacLane proposed a new set of axioms that are more like Birkhoff's than Euclid's or Hilbert's axioms. Like Birkhoff's axioms, MacLane's axioms also use the real numbers; and so has fewer axioms than Hilbert's axioms did. MacLane's approach also does not generate any new theorems in Euclidean geometry but rather simplifies previous attempts at erecting the foundations of Euclidean geometry. The main differences are that MacLane uses a function to measure distance between points (called a metric function) and the so called Continuity Axiom, which incorporates the Crossbar Proposition into Euclidean geometry as an axiom. This topic of geometry states MacLane's axioms, from which all of Euclidean geometry can be proven.
The SMSG Postulates. In an attempt to generate a list of axioms that are more accessible to younger students the School Mathematics Study Group (SMSG) developed a system of axioms that was specifically designed for the use in high school geometry courses. These axioms, which do give rise to all theorems in Euclidean geometry, are not minimal in nature and are meant to move the student almost immediately to more interesting and less intuitively obvious results. The principle is this: if fewer axioms are proposed (such as the Hilbert, Birkhoff, MacLane axioms), then the more elementary (and obvious) results are needed to "start up" the theory of Euclidean geometry. The idea is to get younger students involved in more interesting results in a timely manner. So even though some of the SMSG axioms are redundant, they do achieve the desired effect of almost immediately being able to state significant results. This topic of geometry states the SMSG axioms, from which all of Euclidean geometry can be proven.
The UCSMP Postulates. The University of Chicago School Mathematics Project (UCSMP) developed a system of axioms for Euclidean geometry that are still widely used today in most high school geometry textbooks. These axioms are not minimal; that is, some of the axioms can be proven given some of the other axioms. The idea is to get younger students involved in more interesting results in a timely manner. For example, the UCSMP axioms incorporate a transformational approach via the "Reflection Postulate", which asserts that certain transformations exist and have specified properties. Since understanding geometric relationships in terms of functions of points and angles involves a deeper level of understanding for a rigorous level to be achieved, students benefit by being exposed to a not-so rigorous treatment of transformational result in Euclidean geometry. This topic of geometry states the UCSMP axioms, from which all of Euclidean geometry can be proven.
Euclid's Book One . In the Classical Greek view, a proof should culminate in a full statement of what had been proved. Thus Euclid's proof would have ended with a complete restatement of the conclusion of the proposition. Heath omits this reiteration of the conclusion and simply replaces it with "etc." Also Heath uses the Latin abbreviations Q.E.F. and Q.E.D. for the phrases, "Being what was required to do" and "Being what was required to prove", respectively. There are many philosophical differences between Euclid's work in the Elements and what we consider today; for example, Euclid thought of a line what we today would think of as a curve.
Incidence with Hilbert Axioms. Hilbert partitioned his axioms for Euclidean geometry into five groups (connection, order, congruence, parallels, continuity) and his axioms of connection (or incidence) will shed some light on his undefined terms. Hilbert's axioms went a long way towards helping people understand that the axiomatic approach to mathematics was a viable and much needed mathematical enterprise in its own right. Many axiomatic systems have been created and even recreated for subjects that are widely understood in the mathematical community. Hilbert's axioms are close in spirit to Euclid's postulates because they do not associate real numbers with segments; i.e. there is no way to measure a line segment. This section details Hilbert's incidence axioms and illustrates some basic propositions that can be proven directly from these axioms.
Betweenness with Hilbert Axioms. A purpose of the Hilbert Betweenness Axioms (Hilbert's Order Axioms) is to give meaning to the undefined term between; seeing as between is undefined, we have only the axioms to guarantee that the points in this geometry behave in a way that is consistent with our interpretation of "betweenness". This topic of geometry points out that there are exactly two sides to a line in the Euclidean plane (Half-Planes Proposition), and that lines are not circular (Betweenness Property), and how to decompose a line into its parts (Linear Decomposition and Line Separation Propositions). The interior of an angle and the Crossbar Proposition are also detailed. Some of these results were taken for granted by Euclid.
Congruence with Hilbert Axioms. A purpose of the Hilbert Congruence Axioms is to give meaning to the undefined term congruence; seeing as congruence is undefined, we have only the axioms to guarantee that the points in this geometry behave in a way that is consistent with our interpretation of "congruence". This topic of geometry points out that the Hilbert Congruence Axioms do give segment and angle congruence as congruence relations. Addition and subtraction of segments and angles are detailed. More relations are defined for angles and segments and trichotomy properties are detailed. The ASA Congruence Criterion and Isosceles Criterion are proven.
Interior and Exterior Angles With Hilbert Axioms. This topic proves the well-known Alternate Interior Angle and Exterior Angle Propositions much as Euclid did for the Euclidean plane. However, Euclid had gaps in these proofs; in fact, his gaps may not be noticed until trying to follow his proofs on a sphere. This topic of geometry uses the Crossbar Proposition to close the gap in the Exterior Angle Proposition. Also the SAA Congruence Criterion and the Hypotenuse-Leg Congruence Criterion are detailed. Propositions concerning midpoints, bisectors, and relations between sides and angles of a triangle are proven.
Saccheri-Legendre Theorem With Hilbert Axioms. This topic defines a Dedekind cut and proves the Dedekind Axiom implies the Archimedian Axiom. After introducig the measure of a segment and an angle, the triangular inequality and the Saccheri-Legendre Theorem are proven.
Cite this as:About Foundations Of Geometry
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/about-foundations-of-geometry.html
