A Point Cannot Be Measured

mathxyz · 2024-05-18 14:25:15

In geometry we learn that a point cannot measured. Why is that the case?

Bob · 2024-05-18 19:37:24

The idea comes from Euclid. He wrote out a set of geometry axioms for an idealised version of geometry. He then showed how to derive many theorems from these and wrote a book "The Elements" that has been used by mathematicians ever since.

Not only does it give us many famous results (eg the angles of a triangle add up to 180) but his axiomatic approach has given mathematicians the way to develop any mathematical theory.

He defined a point as having a position but no size. Lines have length but no width. It's an idealised view. When you mark a point on paper and look at it under magnification you can see your point does have size.

Bob

KerimF · 2024-05-19 01:20:44

For instance, this is indeed one of the proofs that being non-existent for not being well-defined doesn't imply being non-useful in one's life.

mathxyz · 2024-05-19 07:51:26

Bob wrote:

The idea comes from Euclid. He wrote out a set of geometry axioms for an idealised version of geometry. He then showed how to derive many theorems from these and wrote a book "The Elements" that has been used by mathematicians ever since.
Not only does it give us many famous results (eg the angles of a triangle add up to 180) but his axiomatic approach has given mathematicians the way to develop any mathematical theory.
He defined a point as having a position but no size. Lines have length but no width. It's an idealised view. When you mark a point on paper and look at it under magnification you can see your point does have size.
Bob

Thank you, Bob. Always very informative and educational replies.

mathxyz · 2024-05-19 07:52:52

KerimF wrote:

For instance, this is indeed one of the proofs that being non-existent for not being well-defined doesn't imply being non-useful in one's life.

A point cannot be measured is found in most if not all geometry textbooks.

KerimF · 2024-05-19 22:38:25

mathxyz wrote:

KerimF wrote:
For instance, this is indeed one of the proofs that being non-existent for not being well-defined doesn't imply being non-useful in one's life.
A point cannot be measured is found in most if not all geometry textbooks.

Sorry for not being clearer.
Can we see THE point as defined in Geometry? Of course, we can't point out to something and say this is THE point as it is defined in Geometry.
And isn't this unseen point very useful if it is included in some scientific studies? Of course, it is (to me in the least).

I mean, this is the logical answer to those who insist that one 'should ignore' an idea (and its related studies) if it refers to something or entity that cannot be seen (dur to lack of complete definition, for example).
After all, we can't blame them. They surely didn't have the chance or the time to learn and use Geometry for example.

mathxyz · 2024-05-20 00:20:11

KerimF wrote:

mathxyz wrote:
KerimF wrote:
For instance, this is indeed one of the proofs that being non-existent for not being well-defined doesn't imply being non-useful in one's life.
A point cannot be measured is found in most if not all geometry textbooks.
Sorry for not being clearer.
Can we see THE point as defined in Geometry? Of course, we can't point out to something and say this is THE point as it is defined in Geometry.
And isn't this unseen point very useful if it is included in some scientific studies? Of course, it is (to me in the least).
I mean, this is the logical answer to those who insist that one 'should ignore' an idea (and its related studies) if it refers to something or entity that cannot be seen (dur to lack of complete definition, for example).
After all, we can't blame them. They surely didn't have the chance or the time to learn and use Geometry for example.

What else in the world cannot be measured?

Bob · 2024-05-20 04:00:14

The quality of mercy is not strained.

Bob

mathxyz · 2024-05-20 05:53:29

Bob wrote:

The quality of mercy is not strained.
Bob

God's mercy?

Bob · 2024-05-20 07:29:59

The Merchant of Venice - Portia's speech, script by Shakespeare

Bob

AnthonyRBrown · 2024-05-20 22:05:54

mathxyz wrote:

KerimF wrote:
mathxyz wrote:
A point cannot be measured is found in most if not all geometry textbooks.
Sorry for not being clearer.
Can we see THE point as defined in Geometry? Of course, we can't point out to something and say this is THE point as it is defined in Geometry.
And isn't this unseen point very useful if it is included in some scientific studies? Of course, it is (to me in the least).
I mean, this is the logical answer to those who insist that one 'should ignore' an idea (and its related studies) if it refers to something or entity that cannot be seen (dur to lack of complete definition, for example).
After all, we can't blame them. They surely didn't have the chance or the time to learn and use Geometry for example.
What else in the world cannot be measured?

Love,Hope,Faith

If a point cannot be measured? then what's the point!

Bob · 2024-05-20 23:40:57

Ha ha! Nice answer.

Whilst most things in maths can be measured, it's an error to assume that {the set of things that cannot be measured} is empty. Here's a few:

the focus of a parabola
the place where a curve meets its asymptote
the null set
the number of tesselating tiles needed to cover an infinite plane

In Euclidean geometry (which is where we came in) it is necessary that a point has no dimensions. It exists because it is where two lines cross, but, if it had a size, that would mess up all the axioms.

Many years ago I did the "draw a triangle; measure the size of its angles; add them up; and consider what do you notice" with a class. One student, who was clearly destined for great things, said "The angles add up to about 180, sometimes 179, sometimes 181." Based on the class experiment this was a correct conclusion. But Euclidean geometry requires that the angle sum of a triangle is exactly 180. It doesn't work if you start worrying about the accuracy of the measurement.

Bob

AnthonyRBrown · 2024-05-21 04:57:28

Ok so I have an idea on how a point can be measured...

So we attached the amount of ink that was going to be used to write a point,to the end of a needle then we drop the ink onto an Atomic weighing scale,this should give us an accurate measurment of a point!

Last edited by AnthonyRBrown (2024-05-21 04:58:19)

Bob · 2024-05-21 05:55:41

In Euclidean geometry lines and points are drawn with magic ink*. No matter the degree of magnification a point always looks the same, a dimensionless position. Worse than that, the magic ink is weightless so no weight device can determine a non zero weight for it.

Bob

* This ink is available from Bryan Thwaites' sale of useful equipment for schools teaching mechanics. Other items include weightless pulleys, frictionless surfaces and non stretchable strings.

AnthonyRBrown · 2024-05-21 07:00:42

Cool!

mathxyz · 2024-05-21 08:59:32

AnthonyRBrown wrote:

Ok so I have an idea on how a point can be measured...
So we attached the amount of ink that was going to be used to write a point,to the end of a needle then we drop the ink onto an Atomic weighing scale,this should give us an accurate measurment of a point!

A point cannot be measured is a definition in Geometry that must be accepted as truth.

AnthonyRBrown · 2024-05-21 17:38:30

mathxyz wrote:

AnthonyRBrown wrote:
Ok so I have an idea on how a point can be measured...
So we attached the amount of ink that was going to be used to write a point,to the end of a needle then we drop the ink onto an Atomic weighing scale,this should give us an accurate measurment of a point!
A point cannot be measured is a definition in Geometry that must be accepted as truth.

There is no such thing as "must be accepted"! where I come from,for me that is like a Red flag to a Bull
My whole life has been driven by proving that wrong! and many others who naturally think outside the box,can you imagine how many great inventions,acheivments,scientific breakthroughs,etc. that would have been missed if all.those people took notice of the "must be accepted"

mathxyz · 2024-05-21 18:23:32

AnthonyRBrown wrote:

mathxyz wrote:
AnthonyRBrown wrote:
Ok so I have an idea on how a point can be measured...
So we attached the amount of ink that was going to be used to write a point,to the end of a needle then we drop the ink onto an Atomic weighing scale,this should give us an accurate measurment of a point!
A point cannot be measured is a definition in Geometry that must be accepted as truth.

There is no such thing as "must be accepted"! where I come from,for me that is like a Red flag to a Bull
My whole life has been driven by proving that wrong! and many others who naturally think outside the box,can you imagine how many great inventions,acheivments,scientific breakthroughs,etc. that would have been missed if all.those people took notice of the "must be accepted"

Perfect. Now you can prove mathematically why a point has no measure. We are all waiting for this great mathematical feat.

AnthonyRBrown · 2024-05-21 20:44:43

mathxyz wrote:

AnthonyRBrown wrote:
mathxyz wrote:
A point cannot be measured is a definition in Geometry that must be accepted as truth.

There is no such thing as "must be accepted"! where I come from,for me that is like a Red flag to a Bull
My whole life has been driven by proving that wrong! and many others who naturally think outside the box,can you imagine how many great inventions,acheivments,scientific breakthroughs,etc. that would have been missed if all.those people took notice of the "must be accepted"
Perfect. Now you can prove mathematically why a point has no measure. We are all waiting for this great mathematical feat.

mathxyz I think you have misunderstood the type of person I am,I am not driven by "trying to prove mathematically why a point has no measure."

I am driven by proving mathematically why a point has a measure!

So the first questions for you is...Can the point you are talking about be seen? or is the point you are talking about an imaginary point?

mathxyz · 2024-05-21 22:21:05

AnthonyRBrown wrote:

mathxyz wrote:
AnthonyRBrown wrote:

There is no such thing as "must be accepted"! where I come from,for me that is like a Red flag to a Bull
My whole life has been driven by proving that wrong! and many others who naturally think outside the box,can you imagine how many great inventions,acheivments,scientific breakthroughs,etc. that would have been missed if all.those people took notice of the "must be accepted"
Perfect. Now you can prove mathematically why a point has no measure. We are all waiting for this great mathematical feat.

mathxyz I think you have misunderstood the type of person I am,I am not driven by "trying to prove mathematically why a point has no measure."
I am driven by proving mathematically why a point has a measure!
So the first questions for you is...Can the point you are talking about be seen? or is the point you are talking about an imaginary point?

Ok. Cool. Take your pencil or pen and tap a piece of paper once, quickly and hard. You will see a point on the paper. Now, measure the point. If you do that successfully, you will be the first human on the planet to find the measurement of that tiny, little dot. There you go: by point I mean dot.

AnthonyRBrown · 2024-05-22 01:14:08

Ok do you want me to measure the size of the point? or measure where the point is on the paper?

mathxyz · 2024-05-22 08:48:24

AnthonyRBrown wrote:

Ok do you want me to measure the size of the point? or measure where the point is on the paper?

I can measure where the point is on the paper with a simple ruler. Measure the circumference, radius and diameter of the DOT, you know, the DOT on your paper. Find the area of the DOT, the DOT itself.

KerimF · 2024-05-23 14:16:24

Back to math, 'infinity' and 'an imaginary number' cannot be seen (or measured) outside one's mind. But their use, in solving certain problems, is very important, for one reason or another.

(Sorry, one may think that an atheist who insists to see something to believe its importance in his life cannot also believe in the importance and existence of many math's elements )

AnthonyRBrown · 2024-05-23 23:35:10

mathxyz wrote:

AnthonyRBrown wrote:
Ok do you want me to measure the size of the point? or measure where the point is on the paper?

I can measure where the point is on the paper with a simple ruler. Measure the circumference, radius and diameter of the DOT, you know, the DOT on your paper. Find the area of the DOT, the DOT itself.

OK! There are two solutions to this

Solution (1) If you use a felt tip pen as an example to make a point on a piece of paper,then it can be measured because it exists!

Solution (2) If you use a reference point as a point,then it cannot be measured because it does not exist!

A.R.B

Math Is Fun Forum

#1 2024-05-18 14:25:15

A Point Cannot Be Measured

#2 2024-05-18 19:37:24

Re: A Point Cannot Be Measured

#3 2024-05-19 01:20:44

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#4 2024-05-19 07:51:26

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#5 2024-05-19 07:52:52

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#6 2024-05-19 22:38:25

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#7 2024-05-20 00:20:11

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#8 2024-05-20 04:00:14

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#13 2024-05-21 04:57:28

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