point in math

Math is invented so that it represents the concepts we are trying to represent, as simply as possible. Assuming that each number corresponds to one location on a number line agrees with what we expect from generalizations of the real world, and makes calculations easy, so we go with that (and have for a very long time).

Now, perhaps you could invent a new kind of geometry in which there was such a thing as a "semi-point" and there were two numbers in the same place on a number line; but you probably couldn't get anyone to try using it, because it would be too cumbersome with no benefits. That's not to say it would necessarily be "wrong".

Mathematicians do invent new mathematical objects or systems, just to see what will happen! And sometimes the results turn out to be useful, even if they weren't trying to make something that corresponds to the real world. But if what they invent is not either useful or interesting, their paper will just gather dust, even if it's perfectly valid, because there will be no motivation for anyone to pursue it further.
 
Hi guys, I totally know that I already opened this thread before, and I will not discuss more .. but just for my heart and my soul to be satisfied..... at the end, in briefly, a point isn't anything .. yeah? I mean if I want to calculate a quantity between A and B (lets assume quantity represented a distance ) so that quantity is irrelevant to A and B, I mean irrelevant to point A and B .. right? thanks alot
 
Ryan$ said:
Hi guys, I totally know that I already opened this thread before, and I will not discuss more .. but just for my heart and my soul to be satisfied..... at the end, in briefly, a point isn't anything .. yeah? I mean if I want to calculate a quantity between A and B (lets assume quantity represented a distance ) so that quantity is irrelevant to A and B, I mean irrelevant to point A and B .. right?
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Here is a favorite quote of mine. “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.” Albert Einstein in Geometry & Experience, 1929. Einstein was warning against being to literal in referring to thing mathematical. Almost all axiom systems begin with a listing of undefined terms. Point is on many, many of those lists.
You asked about distance. What you may not realize is that distance is a measure usually call a metric. \(\displaystyle d(P,Q)\) is the distance between points
\(\displaystyle P\;\&\;Q\) BUT that has very strict rules: 1) \(\displaystyle d(P,Q)\ge 0\) 2) \(\displaystyle d(P,Q)=d(Q,P)\) & 3) \(\displaystyle d(P,Q)=0\) if and only if \(\displaystyle P=Q\).
Absolute value is a metric. \(\displaystyle |x-y|\) is the distance from \(\displaystyle x\text{ to }y\).
If \(\displaystyle |x-7|<4\) then \(\displaystyle x\) is within four units of seven or \(\displaystyle 3<x<11\) note that 7 is midpoint and \(\displaystyle 2(4)=8\) is the diameter of that interval.
But more to your point, \(\displaystyle |X|=|X-X|=|0|=0\) the distance of \(\displaystyle X\text{ to itself is }0\), the measure of a point is \(\displaystyle 0\).

So lighten up, take Einstein's advice to heart.
 
Ryan$ said:
… will not discuss more …
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Okay. But I will … :p

… a point isn't anything .. yeah? …
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A point is not anything physical, in the real world.

However, a point is something. As people have said, a point is a concept, an idea, a useful model for things like numbers, locations, intervals, graphs, etc.

?
 
Hi guys, I know that I already opened a thread about that subject but still struggling and by you I really boost myself.

when I imagine point, then I imagine a black box which if I split one of its points then its place will be white so it has dimension .. and that's wrong analogy .. can anyone help me how should I imagine point?! thanks alot
 
Ryan$ said:
when I imagine point, then I imagine a black box which if I split one of its points then its place will be white so it has dimension .. and that's wrong analogy .. can anyone help me how should I imagine point?!
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In mathematics it is impossible to imagine a a point. Point is an undefined term. A point is something that just is.
In a famous example of a finite geometry bee hives are points.
 
Hi guys, I really want to verify about something it might be silly but I face it every time, and I want to verify if I just alone facing it or actually it's likely to others.
for example once I face something like subtraction such as : 4-5 then in my mind I imagine it like I have something continuous like this ----------------------------------------------- which its length is 5 and if I want to subtract 4 then I just remove 4 units from that line, what's confusing me that the mutual area or "point" is found between the removed area (the empty) and the reminder area .. exactly what I mean ----------------------------------- ------ , the right line is the subtracted area, and once I removed the left line (which it's like 4 because we are doing 5-4) then the point between the line removed and the left line is mutual between two lines(left and right) so doesn't it matter and change the quantity of 5-4 ?! I mean I claim that it would be matter because there's one point left from the ( "removed line" = 4 ) on the reminder amount of 5-4 ... and it's found because it's mutual between two lines so if I removed it from the left line , it would be still on the right line, doesn't that matter?!
 
A point has no dimension. From your previous posts, it seems you are stuck on the thought that they do.

May I suggest you think of 5 - 4 as:

Put 5 oranges out on your table and take away 4. There's 1 left.
 
HI I'm totally with you, but I'm facing that thinking that point has dimension ..in other words my mind is telling me like this "so this mutual point has dimension" and then I say no no it doesn't have dimension .. and then I solve ... I'm asking if I the only one facing that or it's standard if so, I should go to the doctor for that abnormal thinking ..
 
what I'm trying to say, sometimes I learn something new like point doesn't have dimension etc ... but while solving my mind taking me to the opposite "struggling .. " is that normal thinking process?! or it's something abnormal to me ?!
 
I do think you are over-thinking things - although that's not necessarily a bad thing. Definitions are definitions are definitions.
 
Harry_the_cat said:
I do think you are over-thinking things - although that's not necessarily a bad thing. Definitions are definitions are definitions.
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what do you mean by "over thinking things" ? so how can I relate to things?! or think about things? you mean there's no need to think more than as it's?! take thinks not deeply just as it's ?!
 
Hi guys, first of all, I know that I already opened this subject before, but again I want to close that gap, so I believe by you guys I will overcom on that gap.

"point" in math is nothing, in other words doesn't have dimension, so I totally convinced with that ! but my problem is how can I imagine it? I mean if I'm solving a problem and I imagine a points , how can I imagine them? If I would imagine a point then I make it as it has a dimension, so how actually I imagine it? for instance lets take I have distance like this:
1------------------------------20 , and I want to divide it by half , in other words I will put a "point" on the half of that distance in other words 1--------------------*---------------------20 , so I imagine that point which divide the distance by half like this .. but if so then the point "*" has a dimension because as you see in the graph it has dimension ..so how can I imagine it? what's make me harder to understand the point is, how can imagine it and manipulate it on math?! for example I try to imagine a point in a black circle, but if I take a point on he black circle , then the black circle would have a point of "empty-white" so it's also wrong analogues because it makes a point with dimension .. !! any help how can I imagine point over math?
 
Ryan$ said:
If I would imagine a point then I make it as it has a dimension, so how actually I imagine it? for instance lets take I have distance like this:
1------------------------------20 , and I want to divide it by half , in other words I will put a "point" on the half of that distance in other words 1--------------------*---------------------20 , so I imagine that point which divide the distance by half like this .. but if so then the point "*" has a dimension because as you see in the graph it has dimension ..so how can I imagine it?
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If you divided the line into 2 segments 10 each, how long is the "point"? 20 - 10 - 10 = 0.
You can draw the point however you like, it's still 0 size.
 
Ryan$ said:
... I imagine that point which divide the distance by half like this ...

1--------------------*---------------------20

... but if so then the point "*" has a dimension because as you see in the graph it has dimension ...
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The * is not the actual point; it is only your representation of it. (Nobody can draw an actual mathematical point because it's not a real object.)

You never replied to my question, from about two months ago: What is the smallest positive Real number?

?
 
Hi guys ! something facing me since I started learning math .. which is:

lets assume I have bottle, its size 2L so the maximum height signed as "-20" (be careful that's " - " isn't menus it's just a line)
I want to calculate the amount between the maximum height of the bottle to the half of the bottle, which it's 0.5*2L=L .

So what should I do is, 2L-L=L ! all is fine, but once again what about the line L itself , I mean the line at height L ,am I subtract it also from 2L?!
the line at height L I mean that over this line there's height L, so since I do 2L-L am I subtract the line itself of height L also?!

thanks
 
Math deals with idealizations. The line has no width so there is nothing to subtract.

You keep trying to turn math into physics. They are different subjects completely.
 
"whenever I have sec/sec then sec/sec=1"
If it helps, think of sec/sec as 1 sec/1 sec. It does not change the problem.
 
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