On Monday, January 6, 2020 at 8:12:33 AM UTC-5, Eram semper recta wrote:
He also wrote:
In “The Elements”, Euclid defines a point as “that which has no breadth or width”, and a line as “that which lies evenly with itself”.
The very next thing he does is completely ignore those (terrible) definitions, and he never once refers to them for the rest of this monumental book. He never uses them, never mentions them, never says “so AC is a line because it lies evenly with itself”.
Instead, he posits a few axioms that are satisfied by points, lines, circles and the relationships between them (such as incidence), and everything from this point onward is drawing conclusions from those axioms.
This is one of the most brilliant, brilliant moves in the history of human thought.
In the realm of mathematics, an object is what it does (I keep quoting Tim Gowers with this phrase, and I will likely do so many more times). The only thing that matters about points, lines, real numbers, sets, functions, groups and tempered distributions is the properties and features and rules they obey. What they “are” is of no concern.
Not only does it not matter, it gives math immense power when we realize that the same conclusions that hold for our “usual” points and lines hold for anything that satisfies the same axioms, whether or not it looks or smells or quacks like a point or a line....
> In fact, Euclid writes NO such thing! A "point is defined as that which has no part". What Euclid meant is clear from the Platonic philosophy that spawned mathematics: the "point" is merely the concept of place or location. It asks the question "Where?"
>
> Typical misrepresentation and lack of attention to detail by this mainstream moron Amit. In fact Euclid defines both a line and straight line as follows
>
> 2. And a line is a length without breadth.
> 3. And the extremities of a line are points.
> 4. A straight-line is (any) one which lies evenly with points on itself.
>
> Firstly, Euclid was attempting to write these things down for the first time in human history - he was a pioneer and pioneers don't always do a perfect job, but Euclid was pretty close to perfect! All it takes is a genius like me to understand.
(HA, HA!!) A "genius" who, even at his advanced age, is STILL struggling with basic elementary-school arithmetic. (See below)
>
> From my revised Elements:
So NOT Euclid's original definition. Thought so.
>
> 1. A point is the concept of location or place or position. It has no dimension or size or extent.
> 2. A line is a path between two points whose chief attribute is *length* or *distance*.
> 3. A straight line is the shortest path between any two points in a plane.
>
So, this is NOT from Euclid, but from the math "genius," John Gabriel. (HA, HA, HA!!!)
>
> --The very next thing he does is completely ignore those (terrible) definitions, and he never once refers to them for the rest of this monumental book. He never uses them, never mentions them, never says “so AC is a line because it lies evenly with itself”.-- Amit
>
> LIE. What follows is based ENTIRELY on these foundation objects. Geometry is about DISTANCES and their relationships.
>
> . IS NOT a point
> ___ IS NOT a line
> o IS NOT a circle
>
>
Ummm... Is this supposed to be a reference to Euclid definition of a line? (Hint: It isn't.)
>
> --Instead, he posits a few axioms that are satisfied by points, lines, circles and the relationships between them (such as incidence), and everything from this point onwards is drawing conclusions from those axioms.-- Amit
>
> FALSE. There are NO axioms or postulates in the Elements
[snip]
It seems you have missed the entire point of Euclid's "Elements," Troll Boy. Maybe it will make more sense when you have completed your high school education.
Even at his advanced age (60+?), John Gabriel is STILL struggling with basic, elementary-school arithmetic. As he has repeatedly posted here:
"1/2 not equal to 2/4"
--October 22, 2017
“1/3 does NOT mean 1 divided by 3 and never has meant that”
-- February 8, 2015
"3 =< 4 is nonsense.”
--October 28, 2017
"Zero is not a number."
-- Dec. 2, 2019
"0 is not required at all in mathematics, just like negative numbers."
-- Jan. 4, 2017
“There is no such thing as an empty set.”
--Oct. 4, 2019
“3 <=> 2 + 1 or 3 <=> 8 - 5, etc, are all propositions” (actually all are meaningless)
--Oct. 22, 2019
No math genius, our JG!
Interested readers should see: “About the spamming troll John Gabriel in his own words...” (lasted updated December 2019) at
https://groups.google.com/forum/#!msg/sci.math/PcpAzX5pDeY/1PDiSlK_BwAJ
Dan
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