Prove a point exists

[hatsoff]

That was going to be my original response, though I would not have stated as clearly as you did.

However, my second thought was that line assumes the existence point (point being logically prior to line), and therefore any argument to prove the existence of point from the existence of line implicitly presupposes what it attempts to show. Hence that argument does not hold.

What say you?

If a theorem can be proved from a set of axioms, then those axioms collectively logically imply the theorem, yes. However, the axiom declaring the existence of a line does not individually imply the existence of a point. For example, if we had axiom I1 but not I2, then we could declare that there is no line l with l nonempty. Or if we only had axiom I2 but not I1, then we could say that there are no lines and thus no points.

 

[Friar_David_O.Carm]

My point was that if your abstract thought refers to something that you can point at, even though you cannot see it, then you have still proven that the concept (in your mind) really refers to something in the real world == exists.

In the long run, all existence is determined by your mind’s concern of it even though that seems like I’m saying that if you don’t concern yourself with it, it doesn’t exist, I am not saying that.

Your mind is actually NOT yours to dictate existence to. All your conscious mind can do is provide ordered categories for relevant concerns about existence. Thus your conscious mind, by merely declaring that a category is open and defining that category as something that you can point to in the outside “real” world, establishes that the category is valid and hence refers to an “existence” (ei. something that has effect). Even a “blank space” has affect. The blank space exists. Just as the point exists.

I don’t know. Seems you are making a kantian argument here.

I do not buy into the philosophy of Kant.

 

[James_S_Saint]

Well, I don’t argue other people’s philosophies. The real question is whether you believe in logic and how it is associated to both mind and reality.

Do you see the truth of this part of the explanation;

Even a “blank space” has affect. The blank space exists. Just as the point exists.

 

[Friar_David_O.Carm]

Well, I don’t argue other people’s philosophies. The real question is whether you believe in logic and how it is associated to both mind and reality.

If we are working from different philosophical schools of thought then we can not agree, that is a fact.

Do you see the truth of this part of the explanation;

Even a “blank space” has affect. The blank space exists. Just as the point exists.

Please explain what you mean by “affect”.

A point that resides as an abstract thought in your mind does not exist if it has no physical representation.

I believe in logic but the way it is used by some philosophical schools of thought I do not agree with.

 

[James_S_Saint]

Please explain what you mean by “affect”.

You’re kidding me…?

“To affect” is to restrict or force change between two entities. The property of “affect” is the ability to affect. It is the ability to bring “effect” - the resultant change.

A point that resides as an abstract thought in your mind does not exist if it has no physical representation.

No. The concept of a point exists in your mind. We are talking about the actual point in space.

A volume can be said to exist. You can point at it even though there is nothing within it. A point is similar. You can point at it by pointing literally anywhere even though there is nothing within it.

 

[Friar_David_O.Carm]

You’re kidding me…?

“To affect” is to restrict or force change between two entities. The property of “affect” is the ability to affect. It is the ability to bring “effect” - the resultant change.

No I am not kidding you. We must use precise language and not knowing how someone defines a term being used can cause issues.

With your definition of “affect” I do not see how empty space has an “affect”. Empty space is nothing and nothing does not cause a change.

No. The concept of a point exists in your mind. We are talking about the actual point in space.

A volume can be said to exist. You can point at it even though there is nothing within it. A point is similar. You can point at it by pointing literally anywhere even though there is nothing within it.

That is what I was trying to find out. From what the OP posted it was not clear whether he was talking about a mathematical point, which does not exist in space, or an actual point in physical space.

From your replies, though, I think we are operating from different schools of philosophical thought.

I think I will move on as I really do not have the time, nor the inclination, to continue on with this.

 

[itinerant1]

If a theorem can be proved from a set of axioms, then those axioms collectively logically imply the theorem, yes. However, the axiom declaring the existence of a line does not individually imply the existence of a point. For example, if we had axiom I1 but not I2, then we could declare that there is no line l with l nonempty. Or if we only had axiom I2 but not I1, then we could say that there are no lines and thus no points.

Define line.

 

[hatsoff]

Define line.

Lines are usually defined as sets, the members of which (if they exist) are points. They are further characterized by the usual geometric axioms of incidence, order, etc.

 

[itinerant1]

Lines are usually defined as sets, the members of which (if they exist) are points. They are further characterized by the usual geometric axioms of incidence, order, etc.

Depending on the definition used, though, line is axiomatic, and includes within its notion that of point. If point is included as essential to the meaning of line, then I’m not sure how line can be used to prove the existence of point. Line presupposes the existence of point. One does not prove that which is presupposed in the argument.

 

[itinerant1]

Depending on the definition used, though, line is axiomatic, and includes within its notion that of point. If point is included as essential to the meaning of line, then I’m not sure how line can be used to prove the existence of point. Line presupposes the existence of point. One does not prove that which is presupposed in the argument.

Addendum: I am not sure I grasp the idea of line not composed of points. At least in the geometrical real, line seems to presuppose point.

 

[hatsoff]

Depending on the definition used, though, line is axiomatic, and includes within its notion that of point. If point is included as essential to the meaning of line, then I’m not sure how line can be used to prove the existence of point. Line presupposes the existence of point. One does not prove that which is presupposed in the argument.

The “notion” of a point is not the same as the existence of a point. You can define an abstract metric space in which points do not exist.* So, if we wish to formally show that a point exists in a space, we require a proof, albeit a trivial proof.

*- Let M={}. That’s it. It follows that MxM is an empty set and f:MxM to R an empty function, and unless a geometric axiom conflicts with this characterization, it will hold consistent with no points extant.

 

[itinerant1]

The “notion” of a point is not the same as the existence of a point. You can define an abstract metric space in which points do not exist.* So, if we wish to formally show that a point exists in a space, we require a proof, albeit a trivial proof.

*- Let M={}. That’s it. It follows that MxM is an empty set and f:MxM to R an empty function, and unless a geometric axiom conflicts with this characterization, it will hold consistent with no points extant.

Granted that the notion or concept of a point and the existence of a point are not identical. Yet, when we attempt a proof of the existence of point in space, we must have some idea of in what that existence consists. What type of existence does it have. Merely ideal existence?

Also, there must be some correspondence to our concept of point and the point whose existence we try to prove. We attempt to prove that which corresponds to what exists in the mind conceptually. Point does not exists in matter, but is abracted from physical matter and space. Hence, in reliance on the abstracted notion, we are back to the concept, or the abstract concept of point, again.

The same holds true for line. Line, as defined in geometry is not found in the external world. There are no perfect lines, or even circles, or triangles in the real world, only their approximations. The perfect line exists only conceptually, and that concept is derived from the notion of point, a line consisting in a limited or unlimited series of points.

Hence, the notion of line presumes the notion of point as integral to its definition. So, I end up with my same conclusion.

 

[eucharisteo]

In math, I don’t believe the existence of “point” can be proven. Point is an axiom or postulate. An axiom is a starting assumption from which propositions are logically derived. Axioms are not derived by principles of deduction or demonstrable by formal proofs. There is nothing that a point logically follows from, and therefore its existence cannot be proven.

May the question was show that a point exists, or something like that. Unless you teach this subject I’m not sure you realize where I’m coming from here. I’m trying to remember and maybe soon look for my old notebook.

 

[eucharisteo]

Thanks for all the comments. I plan to dig further into this. October 1, was the day of my annual review and my supervisor(s) recommended me to pursue a degree in engineering for licensing. They use me like one so why not get paid like it. I’ve also been pulling my children out of Catholic schools for several disturbing reasons that we discovered.

 

[itinerant1]

May the question was show that a point exists, or something like that. Unless you teach this subject I’m not sure you realize where I’m coming from here. I’m trying to remember and maybe soon look for my old notebook.

I often answer questions from a different perspective. Confuses the heck out of people. In any case, it has been fun, and I hope all the best for you. Sorry about the Catholic school thing, whatever it is about. Many Catholic schools these days are Catholic in name only.

 

[James_S_Saint]

In school texts, there is usually a specific “proof” that is considered “the proof” even though in reality everything has many proofs. You might find what you need better on a logic or mathematics forum where many students ask such questions and have student aids commonly available to give them the precise answer. There are many on the Net.

 

[Shredderbeam]

I don’t think “points” are made of concrete.

“See that? That’s a tree.”
“How do you know that’ thing is a tree?”
“Because that thing is what we call a tree.”

“Oh, well, see this point between my fingers?”
“I don’t ‘see’ anything. How do you that’s a point?”
“Because that’s what I call a ‘point’”

Ok, I see what you mean, but what I meant was that a point is essentially an abstraction, just as “tree” is an abstraction.

 

[eucharisteo]

Ok, I see what you mean, but what I meant was that a point is essentially an abstraction, just as “tree” is an abstraction.

I believe that we’re supposed to define a point and then show using some method that it you can show a point probably exists in or like using “Limits”. My guess. But Thanks again everyone. The last math forum I found I emailed the guy that solved a math problem and showed him why he did it incorrectly. But that was algebra.

 

[Ghoti]

You know, I’ve read a few forums where I defy you prove that *any *point exists…

 

[Friar_David_O.Carm]

Ok, I see what you mean, but what I meant was that a point is essentially an abstraction, just as “tree” is an abstraction.

A tree is not an abstraction. While the word that we assign to the object is tree, the object is there not not an abstraction. If someone had a different name for what we call tree (which does happen as other languages call it something other than tree) that does not change the object in any way.