Alternative to Euclid's Concept of Straight Line

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Discussion Overview

The discussion centers on alternatives to Euclid's concept of a straight line, particularly in the context of geometry as presented in "The Elements." Participants express dissatisfaction with Euclid's definitions and seek other perspectives or formulations that might offer clarity or rigor.

Discussion Character

  • Exploratory
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • One participant expresses dissatisfaction with Euclid's definition of a straight line as the shortest distance between two points, seeking alternatives.
  • Another participant provides a link to an online version of "The Elements" that includes critical commentaries on Euclid's work.
  • A different participant notes that Hilbert's axiomatic geometry, which includes 20 axioms, offers a more rigorous framework compared to Euclid's 5 axioms, suggesting that Euclid relied more on common sense in his proofs.
  • Concerns are raised about the vagueness of Euclid's definition of a straight line, indicating that this is a common sentiment among participants.
  • One participant expresses confusion regarding the concept of dimensions, particularly the third dimension, indicating a broader interest in foundational geometric concepts.

Areas of Agreement / Disagreement

Participants generally express dissatisfaction with Euclid's definitions, indicating a shared sentiment, but they propose different alternatives and frameworks, suggesting that multiple competing views remain. The discussion does not reach a consensus on a preferred alternative.

Contextual Notes

Participants highlight limitations in Euclid's definitions, particularly regarding vagueness and reliance on common sense, but do not resolve these issues or provide definitive alternatives.

Who May Find This Useful

This discussion may be of interest to those studying geometry, particularly historical perspectives on geometric definitions and axiomatic systems, as well as individuals exploring foundational concepts in mathematics.

pjunky
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I'm not getting satisfied with Euclid's concept of straight line and shortest distance b/w to given points is a Straight line,At present I'm studying "The Elements".

Can you people know any alternatives on this topic?
I'm interested to study them.


Thanks:shy:
 
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pjunky said:
I'm not getting satisfied with Euclid's concept of straight line and shortest distance b/w to given points is a Straight line,At present I'm studying "The Elements".

Can you people know any alternatives on this topic?
I'm interested to study them.

By today's standards, Euclid wasn't entirely rigorous. In 1899, Hilbert gave a course on axiomatic geometry from his own formulation. Whereas Euclid describes geometry in 5 axioms, Hilbert does the same with no less than 20.

http://en.wikipedia.org/wiki/Hilbert_axioms

Euclid relied more heavily on common sense in proofs. Granted, all proofs require an amount of common sense, it's clear that Hilbert assumed even less.
 
pjunky said:
I'm not getting satisfied with Euclid's concept of straight line and shortest distance b/w to given points is a Straight line,At present I'm studying "The Elements".

Can you people know any alternatives on this topic?
I'm interested to study them.


Thanks:shy:

As for Euclid's definition I.4 of the straight line, it is NOTORIOUSLY vague, so you are not alone.

(For the a-culturalists here on PF, check out the link)
 
wow..! thank you all people
I think I'm alone but so many are here,its great.
I'm very confusing with dimension concept too,not satisfying with concept of 3rd Dimension.
 

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