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Why is “points exist” not an axiom in geometry?

A model of geometry with the negation of Pasch’s axiom?Why is the Generalization Axiom considered a Pure Axiom?Tarski-like axiomatization of spherical or elliptic geometryHilbert's Foundations of Geometry Axiom II, 1 : Why is this relevant?Why is “lies between” a primitive notion in Hilbert's Foundations of Geometry?Alternatives to Fano's Axiom in Projective SpaceAxiom of Choice — Why is it an axiom and not a theorem?Replacing axiom SAS by AAS in neutral geometry.Redunduncy of Pasch's Axiom of Hilbert's Foundations of GeometryModel of ordered plane with the negation of Pasch's axiom

16

1

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I am not sure why "points exist" is not an axiom in geometry, given that the other axioms are likewise primitive and seemingly as obvious.

geometry axioms

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edited 17 hours ago

Peter Mortensen

559310

asked 19 hours ago

user10869858user10869858

1047

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  Why not "lines exist" then points would be derived objects? Not an answer, just a thought.
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  – Paul
  19 hours ago
  

  
  
  
  


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  If the question is about Euclid's Elements specifically, there's alot missing in those axioms. For example, in the very first proof, Euclid assumes that two circles draw at a certain distance from one another must intersect.
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  – Jack M
  18 hours ago
  

  
  
  
  


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  2
  

  
  
  
  
  
  

  
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  See David Hilbert, The Foundations of Geometry (1899), page 2 : "Let us consider three distinct systems of things. The things composing the first system, we will call points and designate them by the letters $A, B, C$ ; those of the second, we will call straight lines and..."
  $endgroup$
  – Mauro ALLEGRANZA
  16 hours ago
  

  
  
  
  


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  Defining too much detail can quickly devolve into a problem wonderful put by Lewis Carrol. en.wikipedia.org/wiki/What_the_Tortoise_Said_to_Achilles platonicrealms.com/encyclopedia/Carrolls-Paradox
  $endgroup$
  – Duck of Death
  6 hours ago
  

  
  
  
  

add a comment | 

16

1

$begingroup$

I am not sure why "points exist" is not an axiom in geometry, given that the other axioms are likewise primitive and seemingly as obvious.

geometry axioms

share|cite|improve this question

edited 17 hours ago

Peter Mortensen

559310

asked 19 hours ago

user10869858user10869858

1047

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Why not "lines exist" then points would be derived objects? Not an answer, just a thought.
  $endgroup$
  – Paul
  19 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  If the question is about Euclid's Elements specifically, there's alot missing in those axioms. For example, in the very first proof, Euclid assumes that two circles draw at a certain distance from one another must intersect.
  $endgroup$
  – Jack M
  18 hours ago
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  See David Hilbert, The Foundations of Geometry (1899), page 2 : "Let us consider three distinct systems of things. The things composing the first system, we will call points and designate them by the letters $A, B, C$ ; those of the second, we will call straight lines and..."
  $endgroup$
  – Mauro ALLEGRANZA
  16 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Defining too much detail can quickly devolve into a problem wonderful put by Lewis Carrol. en.wikipedia.org/wiki/What_the_Tortoise_Said_to_Achilles platonicrealms.com/encyclopedia/Carrolls-Paradox
  $endgroup$
  – Duck of Death
  6 hours ago
  

  
  
  
  

add a comment | 

$begingroup$

I am not sure why "points exist" is not an axiom in geometry, given that the other axioms are likewise primitive and seemingly as obvious.

geometry axioms

share|cite|improve this question

edited 17 hours ago

Peter Mortensen

559310

asked 19 hours ago

user10869858user10869858

1047

$endgroup$

share|cite|improve this question

edited 17 hours ago

Peter Mortensen

559310

asked 19 hours ago

user10869858user10869858

1047

share|cite|improve this question

edited 17 hours ago

Peter Mortensen

559310

asked 19 hours ago

user10869858user10869858

1047

edited 17 hours ago

Peter Mortensen

559310

edited 17 hours ago

Peter Mortensen

559310

asked 19 hours ago

user10869858user10869858

1047

asked 19 hours ago

user10869858user10869858

1047

2 Answers
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31

$begingroup$

In the presentation I have most convenient to me (Hilbert's axioms), the axioms for plane geometry start with a trio of "axioms of incidence". One of those axioms is "There exist three non-collinear points". That axiom certainly includes the existence of points.

share|cite|improve this answer

answered 19 hours ago

jmerryjmerry

11.3k1426

$endgroup$

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  5
  

  
  
  
  
  
  

  
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  +1. For a specific reference, see Wikipedia's "Hilbert's Axioms" entry. In that listing, the third Incidence axiom reads "There exist at least two points on a line. There exist at least three points that do not lie on the same line." Further, the eighth Incidence axiom reads "There exist at least four points not lying in a plane."
  $endgroup$
  – Blue
  19 hours ago
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The version I'm working from (Hartshorne's textbook) organizes it a little differently. The other two incidence axioms in that version are "For any two points $A,B$, there exists a unique line $l$ containing them", and "Every line contains at least two points". (The axioms from #4 on in the Wikipedia version are axioms for three-dimensional geometry)
  $endgroup$
  – jmerry
  19 hours ago
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
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  I still would find it interesting to know why it's left out in so many places then.
  $endgroup$
  – user10869858
  18 hours ago
  
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
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  @jmerry: Probably every text organizes things a little differently. :) I just thought OP would benefit from access to some version of a "complete" reference, in case there are additional questions about what a comprehensive axiomatic system may-or-may-not cover.
  $endgroup$
  – Blue
  18 hours ago
  

  
  
  
  


• 
  
  
  

  
  5
  

  
  
  
  
  
  

  
  $begingroup$
  @user10869858: What do you mean by "so many places"? Note that Euclid's Elements itself is notoriously lacking in covering all the necessary logical bases; on the other hand, Hilbert's axioms may overwhelm a new learner. It's unsurprising (though perhaps a little unfortunate) that introductory treatments (say, high school textbooks) might make certain compromises, confident that advanced courses and/or readily-available resources (like Math.SE! ;) will fill in any gaps.
  $endgroup$
  – Blue
  18 hours ago
  
  
  

  
  
  
  

 | 
show 1 more comment

3

$begingroup$

Getting a bit meta here, but the point of axioms is not to play pedant. They meaningfully differentiate between scenarios that might be considered. If the geometry of a system that didn't have points was a thing, then maybe there would be a field with an axiom of this nature.

Side note: often the mathematical fields of interest to the outside world are those that mimic something in real life. There, it's obvious that "the triangle equality holds". However that's not to say in maths it is clear that would always be the case, and nothing for which it doesn't hold is interesting.

share|cite|improve this answer

edited 11 hours ago

answered 13 hours ago

ANoneANone

1553

New contributor

ANone is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  You can't just say an axiom is unnecessary because the case where it doesn't hold is boring.
  $endgroup$
  – user2357112
  10 hours ago
  

  
  
  
  


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  Strongly disagree with your first sentence. Axioms have various purposes (and the term "axiom" is used with different meanings), but one of their most important purposes is to "play pedant" and provide a thorough logical foundation for a subject. This is certainly the case for the traditional use of axioms in geometry.
  $endgroup$
  – Eric Wofsey
  8 hours ago
  

  
  
  
  

add a comment | 

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31

$begingroup$

In the presentation I have most convenient to me (Hilbert's axioms), the axioms for plane geometry start with a trio of "axioms of incidence". One of those axioms is "There exist three non-collinear points". That axiom certainly includes the existence of points.

share|cite|improve this answer

answered 19 hours ago

jmerryjmerry

11.3k1426

$endgroup$

• 
  
  
  

  
  5
  

  
  
  
  
  
  

  
  $begingroup$
  +1. For a specific reference, see Wikipedia's "Hilbert's Axioms" entry. In that listing, the third Incidence axiom reads "There exist at least two points on a line. There exist at least three points that do not lie on the same line." Further, the eighth Incidence axiom reads "There exist at least four points not lying in a plane."
  $endgroup$
  – Blue
  19 hours ago
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The version I'm working from (Hartshorne's textbook) organizes it a little differently. The other two incidence axioms in that version are "For any two points $A,B$, there exists a unique line $l$ containing them", and "Every line contains at least two points". (The axioms from #4 on in the Wikipedia version are axioms for three-dimensional geometry)
  $endgroup$
  – jmerry
  19 hours ago
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I still would find it interesting to know why it's left out in so many places then.
  $endgroup$
  – user10869858
  18 hours ago
  
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @jmerry: Probably every text organizes things a little differently. :) I just thought OP would benefit from access to some version of a "complete" reference, in case there are additional questions about what a comprehensive axiomatic system may-or-may-not cover.
  $endgroup$
  – Blue
  18 hours ago
  

  
  
  
  


• 
  
  
  

  
  5
  

  
  
  
  
  
  

  
  $begingroup$
  @user10869858: What do you mean by "so many places"? Note that Euclid's Elements itself is notoriously lacking in covering all the necessary logical bases; on the other hand, Hilbert's axioms may overwhelm a new learner. It's unsurprising (though perhaps a little unfortunate) that introductory treatments (say, high school textbooks) might make certain compromises, confident that advanced courses and/or readily-available resources (like Math.SE! ;) will fill in any gaps.
  $endgroup$
  – Blue
  18 hours ago
  
  
  

  
  
  
  

 | 
show 1 more comment

31

$begingroup$

In the presentation I have most convenient to me (Hilbert's axioms), the axioms for plane geometry start with a trio of "axioms of incidence". One of those axioms is "There exist three non-collinear points". That axiom certainly includes the existence of points.

share|cite|improve this answer

answered 19 hours ago

jmerryjmerry

11.3k1426

$endgroup$

• 
  
  
  

  
  5
  

  
  
  
  
  
  

  
  $begingroup$
  +1. For a specific reference, see Wikipedia's "Hilbert's Axioms" entry. In that listing, the third Incidence axiom reads "There exist at least two points on a line. There exist at least three points that do not lie on the same line." Further, the eighth Incidence axiom reads "There exist at least four points not lying in a plane."
  $endgroup$
  – Blue
  19 hours ago
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The version I'm working from (Hartshorne's textbook) organizes it a little differently. The other two incidence axioms in that version are "For any two points $A,B$, there exists a unique line $l$ containing them", and "Every line contains at least two points". (The axioms from #4 on in the Wikipedia version are axioms for three-dimensional geometry)
  $endgroup$
  – jmerry
  19 hours ago
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I still would find it interesting to know why it's left out in so many places then.
  $endgroup$
  – user10869858
  18 hours ago
  
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @jmerry: Probably every text organizes things a little differently. :) I just thought OP would benefit from access to some version of a "complete" reference, in case there are additional questions about what a comprehensive axiomatic system may-or-may-not cover.
  $endgroup$
  – Blue
  18 hours ago
  

  
  
  
  


• 
  
  
  

  
  5
  

  
  
  
  
  
  

  
  $begingroup$
  @user10869858: What do you mean by "so many places"? Note that Euclid's Elements itself is notoriously lacking in covering all the necessary logical bases; on the other hand, Hilbert's axioms may overwhelm a new learner. It's unsurprising (though perhaps a little unfortunate) that introductory treatments (say, high school textbooks) might make certain compromises, confident that advanced courses and/or readily-available resources (like Math.SE! ;) will fill in any gaps.
  $endgroup$
  – Blue
  18 hours ago
  
  
  

  
  
  
  

 | 
show 1 more comment

$begingroup$

In the presentation I have most convenient to me (Hilbert's axioms), the axioms for plane geometry start with a trio of "axioms of incidence". One of those axioms is "There exist three non-collinear points". That axiom certainly includes the existence of points.

share|cite|improve this answer

answered 19 hours ago

jmerryjmerry

11.3k1426

$endgroup$

share|cite|improve this answer

answered 19 hours ago

jmerryjmerry

11.3k1426

answered 19 hours ago

jmerryjmerry

11.3k1426

answered 19 hours ago

jmerryjmerry

11.3k1426

3

$begingroup$

Getting a bit meta here, but the point of axioms is not to play pedant. They meaningfully differentiate between scenarios that might be considered. If the geometry of a system that didn't have points was a thing, then maybe there would be a field with an axiom of this nature.

Side note: often the mathematical fields of interest to the outside world are those that mimic something in real life. There, it's obvious that "the triangle equality holds". However that's not to say in maths it is clear that would always be the case, and nothing for which it doesn't hold is interesting.

share|cite|improve this answer

edited 11 hours ago

answered 13 hours ago

ANoneANone

1553

New contributor

ANone is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  You can't just say an axiom is unnecessary because the case where it doesn't hold is boring.
  $endgroup$
  – user2357112
  10 hours ago
  

  
  
  
  


• 
  
  
  

  
  5
  

  
  
  
  
  
  

  
  $begingroup$
  Strongly disagree with your first sentence. Axioms have various purposes (and the term "axiom" is used with different meanings), but one of their most important purposes is to "play pedant" and provide a thorough logical foundation for a subject. This is certainly the case for the traditional use of axioms in geometry.
  $endgroup$
  – Eric Wofsey
  8 hours ago
  

  
  
  
  

add a comment | 

3

$begingroup$

Getting a bit meta here, but the point of axioms is not to play pedant. They meaningfully differentiate between scenarios that might be considered. If the geometry of a system that didn't have points was a thing, then maybe there would be a field with an axiom of this nature.

Side note: often the mathematical fields of interest to the outside world are those that mimic something in real life. There, it's obvious that "the triangle equality holds". However that's not to say in maths it is clear that would always be the case, and nothing for which it doesn't hold is interesting.

share|cite|improve this answer

edited 11 hours ago

answered 13 hours ago

ANoneANone

1553

New contributor

ANone is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  You can't just say an axiom is unnecessary because the case where it doesn't hold is boring.
  $endgroup$
  – user2357112
  10 hours ago
  

  
  
  
  


• 
  
  
  

  
  5
  

  
  
  
  
  
  

  
  $begingroup$
  Strongly disagree with your first sentence. Axioms have various purposes (and the term "axiom" is used with different meanings), but one of their most important purposes is to "play pedant" and provide a thorough logical foundation for a subject. This is certainly the case for the traditional use of axioms in geometry.
  $endgroup$
  – Eric Wofsey
  8 hours ago
  

  
  
  
  

add a comment | 

$begingroup$

Getting a bit meta here, but the point of axioms is not to play pedant. They meaningfully differentiate between scenarios that might be considered. If the geometry of a system that didn't have points was a thing, then maybe there would be a field with an axiom of this nature.

Side note: often the mathematical fields of interest to the outside world are those that mimic something in real life. There, it's obvious that "the triangle equality holds". However that's not to say in maths it is clear that would always be the case, and nothing for which it doesn't hold is interesting.

share|cite|improve this answer

edited 11 hours ago

answered 13 hours ago

ANoneANone

1553

New contributor

ANone is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

share|cite|improve this answer

edited 11 hours ago

answered 13 hours ago

ANoneANone

1553

New contributor

ANone is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

answered 13 hours ago

ANoneANone

1553

New contributor

ANone is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

answered 13 hours ago

ANoneANone

1553