The Definition of Parallelism
The geometry inside a plane includes straight lies of infinite lengths falling inside a plane. It is regarded as a fact that two points inside a plane can define a unique straight line. By intuition we assume that one point and one line also can define a unique straight line if we demand the defined line to be parallel to the given line.
However, parallelism has to be defined. According to an about 2000 years old definition parallel lines have no common point. This definition is based on a common point that does not exist. A correct definition should be based only on concepts that really exist. Therefore, when the number of common points takes the value zero and the concept no longer exists we are not allowed to use the concept in a definition. We should base a definition on existing concepts and describe what these concepts really do and not on something that they do not do, because there are a lot of other things that they do not do.
The disappearance of common point is more like a consequence of parallelism and instead the concept constant separation (or equidistance) is a better representation for the essence of parallelism. We can therefore find a better definition based on constant separation. This means that any arbitrarily chosen point on one of the lines always is on the same distance to the other line. Based on this definition we can prove that one point and one line can define a unique line parallel to the given line. From a correct definition we find that our intuition was correct.
Why was parallelism explained in that way 2000 years ago and why has the definition survived for such a long time? Perhaps both questions have the same answer. The reason can be that we beleave too much in simplicity (Occam’s Razor) and therefore have a very high ambition to describe a complex relation in very few words. The older definition really is very short. The human desire to be very short may have fooled us to use a concept that does not exist in a definition for a long time.
Conclusions
We cannot define a concept by concepts that do not exist and not by actions that never are done.
John-Erik Persson
john.erik.persson@gmail.com
naturalphilosophy.org/site/johnerikpersson
Written by John-Erik Persson
Hi John,
Interesting point (pun intended). Your understanding of constant separation is extremely similar to my own of constant infinitesimal slices of area. Doesn’t look like you are participating in the conference this year?
Jeff
I am not coming to the conference.
Do you think that definitions based on not existing concepts absurd? Not doing something is just an effect of parallelism. We should instead state what it is that causes parallelism.
John-Erik
Not that I think parallelism is a great concept for physical theories but I would define it as a (poor) geometric attempt to express a lack of change or difference, i.e. something that doesn’t spatially or temporally change.
Jeff
Yes, parallelism should not be a part of physics, but misundrstandings have made it to be so. You should state CONSTANT separation and not separation different from zero. You should not define based on INFINITE lines either.
John-Erik
Hello sir I am a learning schollar I would like to know the deffinition of the term constant parallelism
Dean
I do not know what you mean. I have not said anything like ‘constant parallelism’. I said that parallelism can be defined by constant separation.
regards
John-Erik
“definition should be based only on concepts that really exist.”
so what is the name of that philosophy??
Roger
The name is not important. However, you must use concepts from the real world. That is very important. If the concepts does not exist it cannot imply any limitations on behavior.
John-Erik
“The name is not important.”
what philosophy is it where the name is not important?
also when you “when the number of common points takes the value zero and the concept no longer exists”
what do you mean by that; zero exists, are you saying it does not exist?
Roger
No common points means that there exist not any points at all. Besides you cannot in a definition demand lines to be drawn to infinity. You cannot state what they are NOT doing in a definition.
You can define crossing lines by exactly ONE common point. You cannot define parallel lines by negating this statement. It is not a definition to list a lot of things lines are NOT doing.
John-Erik
John-Erik
“Common point” is a concept, “zero” is a concept; they are both concepts that can be said to exist as relating to things used in description of physical reality. You say “you cannot in a definition demand lines to be drawn to infinity” – how do you work that out; at least tell me what philosophy you are working from that makes you believe all these sort of things.
Roger
http://youstupidrelativist.com/01Math/05Dim/03Z2Parallel.html
“you stupid relativists” – sounds like if you have a plan to win friends and influence people, then that plan needs a lot more work.
Roger
If the crossing point no longer exists you cannot use it in a definition.
Have you realized that there are 2 kinds of parallelism. Separated with no common points and coincident with all points in common. The conventional definition defines only separated parallelism.
John-Erik
Roger
You cannot define by listing what NOT is done. You cannot use infinite concepts either.
Do you know that there are 2 kinds of parallelism:
Separated with no common points
Coincident with all points in common.
Parallelism is both but the common definition defines only separated.
John-Erik
John-Erik
“You cannot define by listing what NOT is done. You cannot use infinite concepts either.” – Why not? It just sounds like to me you have deviant philosophy. I wish you would tell me the name of it.
Roger
Roger A
Thank you for demanding more details. I think they are needed.
I think that a definition as a part of a theory should be possible to falsify as Popper said. Therefore searching for a common point in an infinite line takes an infinitely amount of time. We must demand test-ability in finite time and therefore only segments of finite length are allowed in a definition. We should not be allowed to demand something to be not existing since not existing cannot be proved. Many failures in detection of an ether does not prove the ether to be not existing. After many failures we can be forced to stop testing but that does not mean that we have gained any new information.
Another indication (not a proof) of error is that Euclid’s definition is valid for separated parallelism but not for coincident. Therefore Euclid has to say that the defining point must be outside the defining line. If we define by constant separation it does not matter if the defining point is on or outside the defining line.
I have not studied philosophy very much so I cannot classify myself. However, I have read a little from Karl Popper.
I think that the problem with parallelism id that the solution is so simple that geometers could not beleave it.
I have a question:
In my opinion we can state that PARALLEL LINES HAVE CONSTANT SEPARATION. This definition captures the essence of parallelism. Euclid’s definition describes (in my opinion) rather an effect of parallelism. Separation in a point on one of the lines is simply the distance to the other line. This can be called equidistance also. My question to you is:
Why do you not like CONSTANT SEPARATION?
Regards from John-Erik
Bill
Thank you for this interesting link.
My opinion is:
No negation in a definition
No infinite elements in a definition
What is your opinion about this?
John-Erik
John Erik
You ask me “Why do you not like CONSTANT SEPARATION?” – that was nothing about what I was asking; so I don’t see why you want to ask than other than as diversion. I was asking what philosophy you were going by. Now you seem to tell me you go by Popper. And you totally misrepresent that philosophy. You say “I think that a definition as a part of a theory should be possible to falsify as Popper said.”- which is false, he wanted theories to be falsifiable, not definitions. You say “We must demand test-ability in finite time” – far as I know, Popper never said anything like that. You just have misunderstood Popper’s philosophy.
Roger
Roger
Thank you very much for your answer.
No. I said that I could not classify myself. Not that I was a Popperian.
No, I said that the theory became not falsifiable due to the definition.
What do you think about the definition I suggested? I think it is important to discuss suitable definitions. Do you not think so?
I also think that we should not demand an infinite process.
With the best regards from
John-Erik
John-Erik
Whatever nameless philosophy you go by, you didn’t seem to use Popper’s philosophy properly. Definitions are not falsifiable, so should not be muddled with idea of what a theory is. You say “What do you think about the definition I suggested?”- lots of different definitions can be generated from lots of different philosophies. I can’t make sense of what thinking process you are going through. You say “I also think that we should not demand an infinite process.”- that would discard a lot of maths: real analysis, calculus,…. I don’t see any benefit in doing that. Anyway, good luck with your enterprise.
Roger
Roger
I said that I did not use Popper at all so I cannot be doing it in a wrong way.
Thank you for wishing me good luck.
However, I will repeat again what the central and most important point is:
We should define concepts by means of points that really EXIST or by crossings that are really DONE. We should not define by NEGATION. Do you have more to say about this important thing?
Best regards from
John-Erik
We should not define by not existing concepts.
John-Erik
John-Erik
You did try to use Popper earlier when you said quote “I think that a definition as a part of a theory should be possible to falsify as Popper said. Now you say “I said that I did not use Popper ” – just means it is pointless continuing
Roger
Roger
It is not important if Popper and I have the same opinion on one point.
The important question that I asked is:
Can we use a not existing concept in a definition? This is the central question you should try to answer. It is also important that we discuss things where we do not agree. If we only talk about where we agree we never will come forwards. So your different opinions are very well come. Thank you.
With best regards from
John-Erik
Roger
Can we use a not existing common point in a definition?
John-Erik Persson
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