MHB Geometry Book: 2 Axioms & Their Implications

[solakis1]

In a Goemetry book i read the following two axioms.

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]

Re: Geometrikal axioms

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).

 

[HallsofIvy]

Re: Geometrikal axioms

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).

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!)

 

[Evgeny.Makarov]

Re: Geometrikal axioms

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??

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).

 

[solakis1]

Re: Geometrikal axioms

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).

I did not ask what a straight line is ,but i simply asked whether the 2nd axiom implies the 1st irrespectively whether the 2nd axiom is true or not.

By the way do you have any Galop of the time showing that the Greeks did not know that the Earth was round ,but some suspected it??

Or, is this a conclusion by reading ancient greek books.

 

[solakis1]

Re: Geometrikal axioms

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).

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)

Is not that correct??

 

[solakis1]

Re: Geometrikal axioms

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!)

Eratosthenes not only measered the equator ,but he did so with an error of less than one percent.

A truly remarkable approximation

 

[Evgeny.Makarov]

Re: Geometrikal axioms

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)

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.

 

[Deveno]

Re: Geometrikal axioms

Eratosthenes not only measered the equator ,but he did so with an error of less than one percent.

A truly remarkable approximation

'Twas not my intention to beat up the Greeks, per se. I am aware of Erarosthenes' calculation, and indeed, it was possible with tools available at the time to measure the curvature of the earth.

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. 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).

**********

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.

**********

Also, you guys, seriously, no one here has any sense of humor?

**********

As far as I know, most mariners knew by the time of Columbus, that the Earth was round. The Portuguese deserve special mention in this, being accomplished sea navigators.

(Sigh). Of course, logical as ever, Evgeny's answer is far better than mine. The point being (a point I perhaps made somewhat poorly) that what we take to be "lines" isn't dictated by the axioms. In fact, if *anything* obeys the axioms, we may as well call that thing "a line".

For example, if we call a line any pair of elements from a given set $S$ and $S = \{x\}$ is a singleton subset, then $S$ clearly obeys axiom 2, as there is only one possible line: $(x,x)$. However, $S$ does not obey axiom 1, so axiom 1 is not a consequence of the second (but could be given OTHER assumptions).

 

[johng1]

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 $$\Sigma$$, if the remaining axioms do not imply A. One way to prove A is independent is to give models both for $$\Sigma$$ and the system $$\Sigma^{\prime}$$ where this second system is the same as $$\Sigma$$, 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).

 

[solakis1]

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 $$\Sigma$$, if the remaining axioms do not imply A. One way to prove A is independent is to give models both for $$\Sigma$$ and the system $$\Sigma^{\prime}$$ where this second system is the same as $$\Sigma$$, 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).

.

Using the machinery for independence you just mentioned can we prove if the axiom:

There are at least two
points on any line and at least one point not on the line

proves axiom 1??

 

[solakis1]

Re: Geometrikal axioms

'Twas not my intention to beat up the Greeks, per se.

I do not care whether your intention was to beat up the greeks,particularly the present day greeks, or not.

I am aware of Erarosthenes' calculation, and indeed, it was possible with tools available at the time to measure the curvature of the earth.

No tools my friend ,just a vertical pole, knowledge and logical conclusions

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.

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 .

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).

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??**********

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.

Many historians of today theydo not even know what is the difference between mass and volume.

**********

 

[Deveno]

Re: Geometrikal axioms

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

Personally I would consider a vertical pole a tool, cf. the present-day usage of stadia in surveying.

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 .

My understanding is that Euclid's actual life-span is not known, and that the best estimate puts him living at c. 300 BC. This makes him a "rough" contemporary of Aristotle (384-322 BC). The dates we have of the Elements' writing are conjectural, at best. It is possible both were alive concurrently, and the opposite may also be true. Take your pick.

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.

I note dryly that not all civilization is Western, and that some might find the emphasis derogatory. If the Chinese had not lost their enthusiasm for seafaring so early, it may well have been a quite different world we live in. The same could be said of various other empires that flourished world-wide throughout the ages.

I'd like to note in passing that one feature of Aristotelian logic is application of the "Law of the Excluded Middle" (Metaphysics, book 3), an assumption that has come under increasing scrutiny and attack in the last 2 centuries (an example of a non-Aristotelian logic is so-called "fuzzy logic" employed with so much success in manufacturing and AI).

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??

No idea, unless you mean some observation of Pythagoras (or one of his followers, or maybe some unknown Babylonian, historical records from 600 BC get pretty sketchy). Some ascribe the result to Indian mathematicians, still other ascribe to Chinese mathematicians. Both of these latter cultures developed forms of logical thinking completely independent from Greek culture.

Many historians of today theydo not even know what is the difference between mass and volume.

**********

I would argue that every time-period encompasses a wide variety of beliefs. It is hard to judge from our vantage point which ones were "commonly accepted" and which ones were held with some skepticism.

I seem to have touched a nerve with you, for some reason that makes you want to dispute things I say. You're certainly welcome to do so, I am as prone to error and fallacy as anyone.

Mind you, I dabble in mathematics more than I do in say, history or philosophy. I read Plato's Republic many, many years ago, but I am certainly no expert on the subtleties of various Hellenist philosophers.

Honestly, I just wanted to point out, that what your two original axioms implied, depend on what you take to be "lines". Nothing more. If I had johng's pithiness, I would have wrote what HE did. In fact, he touches on a subject that troubles me greatly:

We don't have a model for ZFC (often taken as the foundational axiom system for most, if not all, of mathematics). And the reason why this disturbs me, is that unless a model can be found (and there are growing reasons to believe this cannot be done), I have zero reason to believe that sets even exist AT ALL (this doesn't stop me from using them, I just have to keep reminding myself I don't actually know what I'm doing).

 

[solakis1]

Re: Geometrikal axioms

Both of these latter cultures developed forms of logical thinking completely independent from Greek culture.

I tried for so long to find an 0ld Chinese book of logic ,discribing a form of logic completely different of the logic developed by the Ancient Greeks ,but my trial was negative.

Perhaps, you will be so kind to let me know,the existence of such book(s)