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האקסיומות של הגיאומטריה )@ לפנ"הס033( האלמנטים של אאוקליד Definitions: 1. A point is that which has no part. 2. A line is breadthless length. 3. The extremities of a line are points. 4. A straight line is a line which lies evenly with the points on itself. 5. A surface is that which has length and breadth only. 7. A plane surface is a surface which lies evenly with the straight lines on itself. 10. When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands. 15. A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another; 23. Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction. Postulates: 1. To draw a straight line from any point to any point. 2. To produce a finite straight line continuously in a straight line. 3. To describe a circle with any centre and distance. 4. That all right angles are equal to one another. 5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. Common Notions: 1 Things which are equal to the same thing are also equal to one another. 2. If equals be added to equals, the wholes are equal. 3. If equals be subtracted from equals, the remainders are equal. 4. Things which coincide with one another are equal to one another. 5. The whole is greater than the part. אקסיומות פילוסופיה של המתמטיקה Proposition 1: On a given finite straight line to construct an equilateral triangle. Proof: Let AB be the given finite straight line. Thus it is required to construct an equilateral triangle on the straight line AB. C D A B E With centre A and distance AB let the circle BCD be described; [Post. 3] again, with centre B and distance BA let the circle ACE be described; [Post. 3] and from the point C, in which the circles cut one another, to the points A, B let the straight lines CA, CB be joined. [Post. 1] Now, since the point A is the centre of the circle CDB, AC is equal to AB. [Def. 15] Again, since the point B is the centre of the circle CAE, BC is equal to BA. [Def. 15] But CA was also proved equal to AB; therefore each of the straight lines CA, CB is equal to AB. And things which are equal to the same thing are also equal to one another; [C.N. 1] therefore CA is also equal to CB. Therefore the three straight lines CA, AB, BC are equal to one another. Therefore the triangle ABC is equilateral; and it has been constructed on the given finite straight line AB. (Being) what it was required to do. :חלקי ההוכחה - הצגה,ניסוח - הכנה specification מיקוד,הגדרה בניה או הכנת הכלים הוכחה מסקנה .1 .2 .0 .4 .5 .6 :ספרי האלמנטים ) יסודות (לא כולל המעגל- גיאומטריה של המישור.1 על יחסים בין שטחים- גיאומטריה של המישור.2 יסודות: המעגל- גיאומטריה של המישור.0 חסום וחוסם: המעגל- גיאומטריה של המישור.4 ) תורת הפרופורציות (אאודוקסוס.5 תורת הפרופורציות בהקשר של גיאומטריה במישור.6 אריתמטיקה.9-7 גדלים אינקונמנזורביליים.10 יסודות- גיאומטריה של המרחב.11 על גופים מסוג מסויים- גיאומטריה של המרחב.12 ) על הגופים הרגולריים (אפלטוניים- גיאומטריה של המרחב.13 Leo Corry - 2011 -2- סמסטר א' תשס"ה אקסיומות פילוסופיה של המתמטיקה Grundlagen der Geometrie - 1999 - האקסיומות של הילברט I. Axioms of Incidence ()חלות: 1. For every two points A, B there exits a line a that contains each of the points A, B. 2. For every two points A, B there exists no more than one line that contains each of the points A, B. 3. There exist at least two points on a line. There exist at least three points that do not lie on a line. 4. For any three points A, B, C that do not lie on the same line there exists a plane [alpha] that contains each of the points A, B, C. For every plane there exists a point which it contains. 5. For any three points A, B, C that do not lie on one and the same line there exists no more than one plane that contains each of the three points A, B, C. 6. If two points A, B of a line a lie in a plane [alpha], then every point of a lies in the plane [alpha]. 7. If two planes [alpha], [beta] have a point A in common, then they have at least one more point B in common. 8. There exist at least four point which do not lie in a plane. II. Axioms of Order: 1. If a point B lies between a point A and a point C, then the points A, B, C are three distinct points of a line, and B then also lies between C and A. 2. For two points A and C, there always exists at lest one point B on the line AC such that C lies between A and B. 3. Of any three points on a line there exists no more than one that lies between the other two. 4. Let A, B, C be three points that do not lie on a line and let a be a line in the plane ABC which does not meet any of the points A, B, C. If the line a passes through a point of the segment AB, it also passes through a point of the segment AC, or through a point of the segment BC. III. Axioms of Congruence: 1. If A, B are two points on a line a, and A' is a point on the same or on another line a' then it is always possible to find a point B' on a given side of the line a' through A' such that the segment AB is congruent or equal to the segment A'B'. In symbols AB = A'B'. 2. If a segment A'B' and a segment A"B", are congruent to the same segment AB, then the segment A'B' is also congruent to the segment A"B", or briefly, if two segments are congruent to a third one they are congruent to each other. 3. On the line a let AB and BC be two segments which except for B have no point in common. Furthermore, on the same or on another line a' let A'B' and B'C' be two segments which except for B' also have no point in common. In the case, if AB = A'B' and BC = B'C' then AC = A'C'. Leo Corry - 2011 -3- סמסטר א' תשס"ה אקסיומות פילוסופיה של המתמטיקה 4. Let angle(h,k) be an angle in a plane [alpha] and a' a line in a plane [alpha]' and let a definite side of a' in [alpha]' be given. Let h' be a ray on the line a' that emanates from the point O'. Then there exists in the plane [alpha]' one and only one ray k' such that the angle(h,k) is congruent or equal to the angle(h',k') and at the same time all interior point of the angle(h',k') lie on the given side of a'. Symbolically angle(h,k) = angle(h',k'). Every angle is congruent to itself, i.e., angle(h,k) = angle(h,k) is always true. 5. If for two triangles ABC and A'B'C' the congruences AB = A'B', AC = A'C', angleBAC = angleB'A'C' hold, then the congruence angleABC = angleA'B'C' is also satisfied. IV. Axiom of Parallels: (Euclid's Axiom) Let a be any line and A a point not on it. Then there is at most one line in the plane, determined by a and A, that passes through A and doesn't intersect a. V. Axioms of Continuity: (Archimedes' Axiom or Axiom of Measure) If AB and CD are any segments, then there exists a number n such that n segments CD constructed contiguously from A, along the ray from A through B, will pass beyond the point B. (Axiom of Line Completeness) An extension of a set of points on a line with its order and congruence relations that would preserve the relations existing among the original elements as well as the fundamental properties of line order and congruence that follow from Axioms I-III, and from V,1 is impossible. Hilbert's Comments: If Archimedes' Axiom is dropped [non-Archimedean geometry], then from the assumption of infinitely many parallels through a point it does not follow that the sum of the angles in a triangle is less then two right angles. Moreover, there exists a geometry (the non-Legendrian geometry) in which it is possible to draw through a point infinitely many parallels to a line and in which nevertheless the theorems of Riemannian (elliptic) geometry hold. On the other hand there exists a geometry (the semi-Euclidean geometry) in which there exists infinitely many parallels to a line through a point and in which the theorems of Euclidean geometry still hold. From the assumption that there exist no parallels it always follows that the sum of the angles in a triangle is greater than two right angles. Leo Corry - 2011 -4- סמסטר א' תשס"ה אקסיומות פילוסופיה של המתמטיקה :ציטוטים של הילברט על השיטה האקסיומטית )1995( מתוך קורס על יסודות הגיאומטריה.1 Also geometry [like mechanics] emerges from the observation of nature, from experience. To this extent, it is an experimental science.... But its experimental foundations are so irrefutably and so generally acknowledged, they have been confirmed to such a degree, that no further proof of them is deemed necessary. Moreover, all what is needed [in geometry] is to derive these foundations from a minimal set of independent axioms and thus to construct the whole edifice of geometry by purely logical means. In this way [i.e., by means of the axiomatic treatment] geometry is turned into a pure mathematical science. In mechanics it is also the case that all physicists recognize its most basic facts. But the arrangement of the basic concepts is still subject to a change in perception ... and therefore mechanics cannot yet be described today as a pure mathematical discipline, at least to the same extent that geometry is. We must strive that it becomes one. We must ever stretch the limits of pure mathematics further afar, on behalf not only of our mathematical interest, but rather of the interest of science in general. )1933( מתוך מכתב לפריגה.2 After a concept has been fixed completely and unequivocally, it is on my view completely illicit and illogical to add an axiom—a mistake made very frequently, especially by physicists. By setting up one new axiom after another in the course of their investigations, without confronting them with the assumptions they made earlier, and without showing that they do not contradict a fact that follows from the axioms they set up earlier, physicists often allow sheer nonsense to appear in their investigations. One of the main sources of mistakes and misunderstandings in modern physical investigations is precisely the procedure of setting up an axiom, appealing to its truth (?), and inferring from this that it is compatible with the defined concepts. One of the main purposes of my Festschrift was to avoid this mistake. )1935( מתוך קורס על יסודות המתמטיקה.3 The edifice of science is not raised like a dwelling, in which the foundations are first firmly laid and only then one proceeds to construct and to enlarge the rooms. Science prefers to secure as soon as possible comfortable spaces to wander around and only subsequently, when signs appear here and there that the loose foundations are not able to sustain the expansion of the rooms, it sets to support and fortify them. This is not a weakness, but rather the right and healthy path of development. Leo Corry - 2011 -5- סמסטר א' תשס"ה