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No problem

What is the exact difficulty in defining a point in Euclidean geometry?

There is no difficulty and no problem, it's just the way it is defined.

Meaning

"A point is a mathematical object with no shape and size."

The definition literally means that a point has no "shape and size" (i.e., it occupies no space, it has no body, no interiour, no color, no temperature, no feature, whatever other word you would like). It literally has nothing except the few numbers that define/describe it.

"A point is a mathematical object" means that it exists purely mathematically; there is no physical equivalent to a point. It is a mathematical object, which means it is (ina purely abstract thing which we argue about in the frame of some mathematical system.

For example, in the simple 2-dimensional plane we are used to when discussing these things in a school setting, for example) a combination of two numbers (which also are mathematical objects, not physical). Or in the setting of Euklid, by just reasoning about the relationships between points, lines etc. in an abstract plane without even giving coordinates to them.

Not a point

As a comparison: what is not a point? If you take your pencil and make a little impression on your piece of paper, this is not a point. That is a pencil mark, or a circle if you so wish. It has non-mathematical properties, for example the minuscule thickness of the layer of paint you addded to the paper, or the radius of the circle, its color, the type of molecules making it up, etc. etc. Even if you go to the lowest possible scales, i.e., placing a (classical, not quantum-mechanical) atom somewhere with a raster tunnel microscope, that atom will still take up a spherical amount of space and will not be a "point".

No problem

What is the exact difficulty in defining a point in Euclidean geometry?

There is no difficulty and no problem, it's just the way it is defined.

Meaning

"A point is a mathematical object with no shape and size."

The definition literally means that a point has no "shape and size" (i.e., it occupies no space, it has no body, no interiour, no color, no temperature, no feature, whatever other word you would like). It literally has nothing except the few numbers that define/describe it.

"A point is a mathematical object" means that it exists purely mathematically; there is no physical equivalent to a point. It is a mathematical object, which means it is (in the simple 2-dimensional plane we are used to when discussing these things in a school setting, for example) a combination of two numbers (which also are mathematical objects, not physical).

Not a point

As a comparison: what is not a point? If you take your pencil and make a little impression on your piece of paper, this is not a point. That is a pencil mark, or a circle if you so wish. It has non-mathematical properties, for example the minuscule thickness of the layer of paint you addded to the paper, or the radius of the circle, its color, the type of molecules making it up, etc. etc. Even if you go to the lowest possible scales, i.e., placing a (classical, not quantum-mechanical) atom somewhere with a raster tunnel microscope, that atom will still take up a spherical amount of space and will not be a "point".

No problem

What is the exact difficulty in defining a point in Euclidean geometry?

There is no difficulty and no problem, it's just the way it is defined.

Meaning

"A point is a mathematical object with no shape and size."

The definition literally means that a point has no "shape and size" (i.e., it occupies no space, it has no body, no interiour, no color, no temperature, no feature, whatever other word you would like). It literally has nothing except the few numbers that define/describe it.

"A point is a mathematical object" means that it exists purely mathematically; there is no physical equivalent to a point. It is a mathematical object, which means it is a purely abstract thing which we argue about in the frame of some mathematical system.

For example, in the simple 2-dimensional plane we are used to when discussing these things in a school setting, a combination of two numbers (which also are mathematical objects, not physical). Or in the setting of Euklid, by just reasoning about the relationships between points, lines etc. in an abstract plane without even giving coordinates to them.

Not a point

As a comparison: what is not a point? If you take your pencil and make a little impression on your piece of paper, this is not a point. That is a pencil mark, or a circle if you so wish. It has non-mathematical properties, for example the minuscule thickness of the layer of paint you addded to the paper, or the radius of the circle, its color, the type of molecules making it up, etc. etc. Even if you go to the lowest possible scales, i.e., placing a (classical, not quantum-mechanical) atom somewhere with a raster tunnel microscope, that atom will still take up a spherical amount of space and will not be a "point".

Source Link
AnoE
AnoE
  • 839
  • 4
  • 10

No problem

What is the exact difficulty in defining a point in Euclidean geometry?

There is no difficulty and no problem, it's just the way it is defined.

Meaning

"A point is a mathematical object with no shape and size."

The definition literally means that a point has no "shape and size" (i.e., it occupies no space, it has no body, no interiour, no color, no temperature, no feature, whatever other word you would like). It literally has nothing except the few numbers that define/describe it.

"A point is a mathematical object" means that it exists purely mathematically; there is no physical equivalent to a point. It is a mathematical object, which means it is (in the simple 2-dimensional plane we are used to when discussing these things in a school setting, for example) a combination of two numbers (which also are mathematical objects, not physical).

Not a point

As a comparison: what is not a point? If you take your pencil and make a little impression on your piece of paper, this is not a point. That is a pencil mark, or a circle if you so wish. It has non-mathematical properties, for example the minuscule thickness of the layer of paint you addded to the paper, or the radius of the circle, its color, the type of molecules making it up, etc. etc. Even if you go to the lowest possible scales, i.e., placing a (classical, not quantum-mechanical) atom somewhere with a raster tunnel microscope, that atom will still take up a spherical amount of space and will not be a "point".