Indeed, the Greek, Eratosthenes, accurately calculated the diameter of the Earth around 200 BCE. Contrary to popular opinion, Columbus was NOT unusual in that he believed the Earth was round. Any reasonably educated person of the time (and for geography, that would certainly include sea going captains) knew the Earth was round. Columbus was unusual in that he was one of a minority who believed the Earth was much smaller than Eratosthenes calculation. That was based on a purely "stylistic" argument- it just didn't make sense that all the land (Europe, Asia, and Africa) would be on one side of the Earth leaving only open ocean on the other side. It simply didn't occur to him that there would be two whole continents, unknown to him, on that side!Deveno said:Of course the Euclidean plane isn't the surface of a sphere, but remember also: the ancient Greeks did not know the world was round, either (some suspected it, of course).
If lines and points are just abstract objects, then theoretically there could be a line with only one point on it or no points at all. This does not contradict axiom 2).solakis said:1) There exist at least two different points on each straight line
2) There is exactly one line on two different points
But the 2nd doesn't imply the 1st axiom??
Deveno said:Not necessarily. To see why, remember that "point" and "line" are "undefined" terms. Of course, what we hope to "achieve" with our axiom system, is a description of "the Euclidean plane" but other interpretations are possible, and we want to rule these other interpretations out, for the time being.
To give an idea of how this might fail to be true, suppose we have two points on a sphere (like, you know, our, um, PLANET). If we draw a "line segment" (on a great circle, that is to say an intersection of a plane passing through "the center of the sphere" and the surface of the sphere) connecting these two points and extend it indefinitely, we actually get a "circle".
Now imagine two points on opposite sides of the equator. The equator itself is then a "line" that passes between them...but, so is a perpendicular circle that connects them and passes through the north and south poles. So we have to be careful about how we "interpret" what "line" and "point" mean; if "line" actually means: "great circle on a sphere", axiom 2 isn't TRUE, even if "point" has its usual (intuitive) meaning.
Of course the Euclidean plane isn't the surface of a sphere, but remember also: the ancient Greeks did not know the world was round, either (some suspected it, of course).
Evgeny.Makarov said:If lines and points are just abstract objects, then theoretically there could be a line with only one point on it or no points at all. This does not contradict axiom 2).
HallsofIvy said:Indeed, the Greek, Eratosthenes, accurately calculated the diameter of the Earth around 200 BCE. Contrary to popular opinion, Columbus was NOT unusual in that he believed the Earth was round. Any reasonably educated person of the time (and for geography, that would certainly include sea going captains) knew the Earth was round. Columbus was unusual in that he was one of a minority who believed the Earth was much smaller than Eratosthenes calculation. That was based on a purely "stylistic" argument- it just didn't make sense that all the land (Europe, Asia, and Africa) would be on one side of the Earth leaving only open ocean on the other side. It simply didn't occur to him that there would be two whole continents, unknown to him, on that side!
(It has occurred to me that perhaps Columbus knew perfectly well that the Earth was that large and expected to find new land. He just told Ferdinand and Isabella what they wanted to hear!)
Who says that all lines have to be obtained using axiom 2? That is, it may seem that as we take all possible pairs of point and apply axiom 2 to get a line through them, we obtain all lines. But why should this be true? I may say, as in a supermarket, "Oh, and through in this chocolate and call it a 'line'". Unless forbidden elsewhere, I can call any object a "line", and it does not have to do anything with existing points.solakis said:If there is exactly one line on two different points .then there are at least two different points on a line.(these are the points that the line passes thru)
solakis said:Eratosthenes not only measered the equator ,but he did so with an error of less than one percent.
A truly remarkable approximation
.johng said:Here's my two cents, for what it's worth.
Loosely speaking, a model for an axiom system is a concrete interpretation of the axioms so that each axiom is true in the model. There are much more formal descriptions of a model, but the idea has intuitive appeal for me.
If a system has a model, then the system is consistent. That is, it is impossible to prove both a statement S and its negation. Again, I can believe this on an intuitive level. In a concrete interpretation, certainly at most one of S and its negation can be true.
An axiom A is independent in a system \(\displaystyle \Sigma\), if the remaining axioms do not imply A. One way to prove A is independent is to give models both for \(\displaystyle \Sigma\) and the system \(\displaystyle \Sigma^{\prime}\) where this second system is the same as \(\displaystyle \Sigma\), except A is replaced by the negation of A. Now could A be proved with the remaining axioms? If it could, then both A and its negation would be true in the second model, clearly absurd.
For the problem at hand, consider model 1: {1,2,3}= the set of points P. L={{1,2},{1,3},{2,3}} the set of lines. Clearly both axioms are satisfied by this model.
model 2: Same points as 1, but L={{1,2},{1,3},{2,3},{1}}. This easily is a model for the system (not 1) and (2). So axiom (1) is independent; i.e. it is impossible to prove (1) from (2).
I do not care whether your intention was to beat up the greeks,particularly the present day greeks, or not.Deveno said:'Twas not my intention to beat up the Greeks, per se.
Deveno said:I am aware of Erarosthenes' calculation, and indeed, it was possible with tools available at the time to measure the curvature of the earth.
Aristotle was not a "rough" contemporary of Euclid but of Plato who was his teacherDeveno said:What is also just as remarkable, was that the abstraction of "earth-measuring" was something completely planar (there were Greeks such as Menelaus of Alexandria who saw that geodesics were the natural analogue of lines on curved surfaces). Perhaps the Greeks were actually more sophisticated than we know, and anticipated the development of manifolds, and therefore worked in "flat geometry" because locally, surfaces are pretty much their tangent planes.
Much of spherical trigonometry was apparently developed by Muslim mathematicians in order to calculate more accurately the proper direction and distance from Mecca, which was seen as very important.
Perhaps "ancient Greeks" is too broad a term: it is my understanding that commonly held views of a spherical Earth occurred somewhere around 350 BC or so (cf. the position held by Aristotle, a "rough" contemporary of Euclid) , but Greek geometry was well-established at least 2 centuries before this.
Euclid's work will remain one of the brightest spots in human thought not only for the study of its subject matter ( surface and solid geometry) ,but for the application ,for the 1st time in human history, of the axiomatic method in constructing the 1st finite logical deductive system.Deveno said:Most of Euclid's Elements, is not original work by him (as is the case with many mathematical treatises today) but rather an exposition and codification of already well-known works (although many of the proofs are attributed to Euclid, and at least some, if not all of the later chapters).
Deveno said:I remember reading in Herodotus references to the "ends of the earth", which implies he believed it HAD edges. He lived in the 5th century BC.
solakis said:I do not care whether your intention was to beat up the greeks,particularly the present day greeks, or not.
No tools my friend ,just a vertical pole, knowledge and logical conclusions
Aristotle was not a "rough" contemporary of Euclid but of Plato who was his teacher
Plato was the student of Socrates ,the greatest philosopher of all times .
Aristotle died in 322 Bc at the age of 62.
Euclid wrote the "elements " at 300 BC
One needs a life span to even start to study the monumental works of those two giants ,Plato and Aristotle.
Aristotle thru his book "The Organon " founded Logic
Try to study .not just read . "The laws" of Plato
This man said it all .
Euclid's work will remain one of the brightest spots in human thought not only for the study of its subject matter ( surface and solid geometry) ,but for the application ,for the 1st time in human history, of the axiomatic method in constructing the 1st finite logical deductive system.
Aristotle did recognised the need of the axiomatic method in every scientific field,here are his own words:
"Every demostrative science must start from indemostrable principles;otherwise,the steps of demonstration would be endless.of these indemonstrable principles some are (A) COMMON TO ALL SCIENCES,OTHERS ARE(b) PARTICULAR .OR PECULIAR TO THE PARTICULAR SCIENCE;(a)the common principals are the axioms,mostcommonly illustratedby the axiom that,if equalsbe subtracted from equals, the remainders are equals.In (b)we have first the genus or subject matter,the EXISTENCE OF WHICH MUST BE ASSUMED "
cAPITAL LETTERS ARE MINE.
The 'Organon" of Aristotle and the "Elements" of Euclid are the two cornerstons of our Western Civilation.
But even before the exposition of the rules of logic and the axiomatic method ,"Ancient Greeks" made the greatest discovery of all times that led to the discovery of the rules of logic and the axiomatic method.
Do you happen to know what that is??
Many historians of today theydo not even know what is the difference between mass and volume.
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Deveno said:Both of these latter cultures developed forms of logical thinking completely independent from Greek culture.
Axioms are statements that are assumed to be true without proof. They serve as the foundation of a geometric system and are used to derive other theorems and definitions.
Axioms provide the basic rules and assumptions in geometry that help us understand and describe the properties of shapes, angles, and space. They also allow us to make logical deductions and prove theorems.
Yes, there are different sets of axioms in geometry. The most commonly used set is Euclid's five postulates, which form the basis of Euclidean geometry. However, there are also non-Euclidean geometries that use different sets of axioms.
No, axioms cannot be proven because they are assumed to be true without proof. However, their implications and theorems derived from them can be proven using logical reasoning and other axioms as building blocks.
Axioms provide the fundamental rules and assumptions that allow us to apply geometric principles to real-world situations. They help us understand and analyze the properties and relationships of objects and spaces in the physical world.