Thread: definition of some concepts in geometry.

Hello dear friends:
I study electronics at the university but I am very interested in understandig the exact definition of some basic concepts in math. for example I really like to know & understand the exact & mathematical definition of following elements in geometry:
1.point
2.line
3.distance

actually I think that there is a close correlation between geometry & calculus so understanding these concepts will help me studying the calculus better.

Any help that you can give me would be appreciated.
with respect
YASHAR

Originally Posted by yashar

Hello dear friends:
I study electronics at the university but I am very interested in understandig the exact definition of some basic concepts in math. for example I really like to know & understand the exact & mathematical definition of following elements in geometry:
1.point
2.line
3.distance

Excellent question.

First you said you understand the exact mathematical definition of these terms, that is not true! There are no defintions for these terms. A point, line and plane and distance are referred to as undefinable. The problems with Euclidean geometry is that it cannot be really completely considered pure math because these important concepts are not definied, ever. This is why the first 4 postulates cannot be proven since they rely on mathematical defintions of a point and line which we never defined. That is way geometry is not complete. As opposed to other fields of math where everything is on a solid foundation. David Hilbert tried to improve geometry based on this flaw but he never fully succedded, though he was able to get rid of pictures and diagrams which are used in geometry. You should understand that geometry is more intended for engineers and scientists so it really does not need such a strong foundation. Among mathematicians geometrical proves are not considered true proofs because they lack these definitons and important proofs.

Mathematicians think of a point as an ordered pair (x,y) in R^2 (in fact this can be generalized to higher dimensions). Where x and y are real numbers. That is it. Because it captures the applied meaning of a point. In a similar fasion mathematicans define distance as a metric,
d(x,y)=sqrt(x^2+y^2). All because it captures the idea of an applied meaning.

Originally Posted by ThePerfectHacker

Excellent question.

First you said you understand the exact mathematical definition of these terms, that is not true! There are no defintions for these terms. A point, line and plane and distance are referred to as undefinable. The problems with Euclidean geometry is that it cannot be really completely considered pure math because these important concepts are not definied, ever. This is why the first 4 postulates cannot be proven since they rely on mathematical defintions of a point and line which we never defined. That is way geometry is not complete. As opposed to other fields of math where everything is on a solid foundation. David Hilbert tried to improve geometry based on this flaw but he never fully succedded, though he was able to get rid of pictures and diagrams which are used in geometry. You should understand that geometry is more intended for engineers and scientists so it really does not need such a strong foundation. Among mathematicians geometrical proves are not considered true proofs because they lack these definitons and important proofs.

Mathematicians think of a point as an ordered pair (x,y) in R^2 (in fact this can be generalized to higher dimensions). Where x and y are real numbers. That is it. Because it captures the applied meaning of a point. In a similar fasion mathematicans define distance as a metric,
d(x,y)=sqrt(x^2+y^2). All because it captures the idea of an applied meaning.

Euclid in "Elements" gave only brief descriptions, not definitons, of those elements just enough to give reader intuitive sense what it is.

He described point and line:
1) "A point is that which has no part."
2) "A line is breadthless length.".

It is obvious that those sentences can't be considered as definitions.

Hilbert also doesn't define them, but starts his "Foundations of Geometry" with:
Let us consider three distinct systems of things. The things composing the first system,
we will call points and designate them by the letters A, B, C,. . . ; those of the second, we
will call straight lines and designate them by the letters a, b, c,. . . ; and those of the third
system, we will call planes and designate them by the Greek letters alpha, beta, gamma,. . . The points
are called the elements of linear geometry; the points and straight lines, the elements of
plane geometry; and the points, lines, and planes, the elements of the geometry of space
or the elements of space.

which is even smaller explanation (if it is explanation at all) of point, line and plane then Euclid's explanation.

Hi again:
First I want to thank you for the answers. what ThePerfectHacker says is very interesting. you say that the geometry has not a strong mathematical foundation (owing to the lack of definitions of point,line, ...) but there are many concepts in math (especially in calculus) that are defined using geometry or are related to it, for example: angle, trig functions, the integration which calculates the area, the derivative which calculates the slope of the tangent line of a curve & ... you also said that the mathematicans dont accept geometrical proves. now I have another question:
Is it possible to introduce & study concepts like angle, trig functions, limit & continuity, integration (the area) , derivative (the slope of tangent line) & in general the calculus (specially the 3 dimentional calculus) just without any geometry?
this is the question that I have been thinking to it for a long time & I will be really grateful if your help me.

thanks
Yashar

Originally Posted by yashar

Is it possible to introduce & study concepts like angle, trig functions, limit & continuity, integration (the area) , derivative (the slope of tangent line) & in general the calculus (specially the 3 dimentional calculus) just without any geometry?[[/COLOR]r

The things in red are all geometric concepts.

I don't understand the controversary behind point, line, and plane.
They are defined in the same way that addition, subtraction, multiplication, and division are defined. Tell me, Can you define addition without using subtraction, multiplication, or division?

Originally Posted by yashar

Is it possible to introduce & study concepts like angle, trig functions, limit & continuity, integration (the area) , derivative (the slope of tangent line) & in general the calculus (specially the 3 dimentional calculus) just without any geometry?

Calculus is what enigneers/physicists call it. It used geometric approaches for easiness.

Mathematicians refer to it as real analysis. And yes all those geometric concepts are no longer existant. For example, a function is simply treated as a set. A sine and cosine are definied through their series expansion (or through the unique solution to the differencial equation y''-y=0). Concepts of tangents are not treated as lines touches curves rather as limits. And area is not treated geometrically but rather than a Riemann Integral (or Lebesque).

See, pure mathematicians do not use geometry in proofs

Originally Posted by Quick

The things in red are all geometric concepts.

I don't understand the controversary behind point, line, and plane.

There are a number of postulates dealing with them that require proof. For example, the parrallel postulate (revisited) cannot be proved thus it gives difficulty to us (mathematicians).

They are defined in the same way that addition, subtraction, multiplication, and division are defined.

Manure! That is how they teach you at school.

Tell me, Can you define addition without using subtraction, multiplication, or division?

Yes, I can, but you cannot.

Anyway, first we define the natural numbers, then rational, then real, then complex (mathematically of course).

A way to think of natural numbers is by "cardinality". After you define what a finite set is (a set is just a collection of certain objects called elements. For example, the set of all states. The set of all colors. The set of all primes....). You can define a natural number as the size (cardinality) of a finite set. The empty set (for example the set all man who are woman is non-exisitant) has cardinality of zero.
(See Peano Axioms).
Then you can define addition as the cardinality of the union of two set not having the same element.

Then you can define rationals are a/b b not zero. Note, / does not denote division the expression a/b is simply a way of expressing a rational number.
Then you define
a/b+c/d=(ad+bc)/bd
(See Field of Quotients)

Real numbers are tricky. But in analyisis it is done. There are two ways.
(See Cauchy sequences or Dedikind Cuts).

These pages might be difficult for you but at least you will see the way mathematicians think and define above the lesser mortals.

Originally Posted by ThePerfectHacker

There are a number of postulates dealing with them that require proof. For example, the parrallel postulate (revisited) cannot be proved thus it gives difficulty to us (mathematicians).

Manure! That is how they teach you at school.

Yes, I can, but you cannot.

Anyway, first we define the natural numbers, then rational, then real, then complex (mathematically of course).

A way to think of natural numbers is by "cardinality". After you define what a finite set is (a set is just a collection of certain objects called elements. For example, the set of all states. The set of all colors. The set of all primes....). You can define a natural number as the size (cardinality) of a finite set. The empty set (for example the set all man who are woman is non-exisitant) has cardinality of zero.
(See Peano Axioms).
Then you can define addition as the cardinality of the union of two set not having the same element.

Then you can define rationals are a/b b not zero. Note, / does not denote division the expression a/b is simply a way of expressing a rational number.
Then you define
a/b+c/d=(ad+bc)/bd
(See Field of Quotients)

Real numbers are tricky. But in analyisis it is done. There are two ways.
(See Cauchy sequences or Dedikind Cuts).

These pages might be difficult for you but at least you will see the way mathematicians think and define above the lesser mortals.

TPH, the problem with what you said is that the term set cannot be defined. Saying a set is a collection of objects is not a rigorous definition. What is a collection? It's just another word for set.

The book Topology by James Dugundji starts off on page 1 with set theory, which is the foundation of all that follows. The third paragraph begins "Rigorously, the word set is an undefined term in mathematics." So all of Topology or Real Analysis is based right at the beginning on an undefined term. This is no different than having point an undefined term.

Many terms in mathematics must remain undefined. They are called primitive terms. A field in mathematics starts with identifying some primitive terms, then defines axioms and other terms based on the primitives, and goes from there using logic to develop implications and theorems.

PS: Dugundji's book was written partly to be a reference for the advanced mathematician and is excellent. It is out of print, but your library may have a copy. Spend a hour with it and you may decide it will be worth the $100 for a used copy for your personal library.

PH, when you break everything down as far as you can go, you end up with postulates (I realize you know what postulates are, but since I just learned them I'm going to italicize it anyway ) And postulates are merely laws defined by logic. You can define addition with sets (whatever that means, brief overview of the peano axioms and all I got was that they were laws of arithmetic formed by postulates) but then you have to define sets, and then you have to define the definitions of sets, until you can't use anything to prove stuff, and when that happens, what then? I'll tell you, Mathematicians get picky...

Originally Posted by Quick

PH, when you break everything down as far as you can go, you end up with postulates (I realize you know what postulates are, but since I just learned them I'm going to italicize it anyway ) And postulates are merely laws defined by logic. You can define addition with sets (whatever that means, brief overview of the peano axioms and all I got was that they were laws of arithmetic formed by postulates) but then you have to define sets, and then you have to define the definitions of sets, until you can't use anything to prove stuff, and when that happens, what then? I'll tell you, Mathematicians get picky...

Yes, they do get picky. But they need to.

Originally Posted by ThePerfectHacker

The empty set (for example the set all man who are woman is non-exisitant) has cardinality of zero.

Obviously TPH has not considered that the set transgenders belongs to the set of all men who are women...

-Dan

Originally Posted by JakeD

TPH, the problem with what you said is that the term set cannot be defined. Saying a set is a collection of objects is not a rigorous definition. What is a collection? It's just another word for set.

I know. I did not mentioned it might have been useless to Quick.
(this is the reason with the problems of naive set theory, "set of all sets" since we never defined the term it leads to problems. Zermelo-Frankael axioms fix the problem. So having a not definied term is not so bad).