[FOM] Understanding Euclid
Jeremy Avigad
avigad at cmu.edu
Sat Dec 6 12:42:08 EST 2008
Of course, there is a lot that Euclid does not state in his postulates
and common notions, but, rather, takes to be "diagrammatically obvious,"
or implicit in his definitions. Ed Dean, John Mumma, and I have recently
finished a detailed analysis of Euclidean inference, where we try to
spell out the details:
"A formal system for Euclid's Elements"
http://arxiv.org/abs/0810.4315
That might help shed some light on where and how non-spherical
assumptions enter into his reasoning.
Best wishes,
Jeremy
> Date: Thu, 04 Dec 2008 23:02:21 -0600
> From: William Tait <williamtait at mac.com>
> Subject: Re: [FOM] Understanding Euclid
> To: pratt at cs.stanford.edu, Foundations of Mathematics <fom at cs.nyu.edu>
> Cc: William Tait <williamtait at mac.com>
> Message-ID: <C17EBFB4-5D56-4747-9CF3-A734471EE3F6 at mac.com>
> Content-Type: text/plain; charset=US-ASCII; format=flowed; delsp=yes
>
> Vaughan,
>
> The converse of Postulate 5 is derivable in Euclid. He does define two
> lines to be parallel when they do not intersect. The converse of
> Postulate 5 then is: If two lines are perpendicular to a third, then
> they are parallel. If they intersected, the three line segments would
> form a triangle. Two of its interior angles are, by assumption, equal
> to two right angles and so the three interior angles are more than two
> right angles, contradicting Proposition 32 of Book 1.
>
> Proposition 32 does not hold in spherical geometry.
>
> Bill
> On Dec 3, 2008, at 5:15 PM, Vaughan Pratt wrote:
>
>> Euclidean geometry is standardly understood as the geometry of the
>> plane, more generally of flat or uncurved space. To make this stick
>> however, Euclid's fifth postulate should be phrased as an equivalence:
>> two lines fail to meet if and only if a third line intersecting them
>> both meets them at the same angle (or any equivalent phrasing thereof,
>> e.g. that the interior angles on the same side of the cutting line sum
>> to two right angles, or that a point P and a line L determine a unique
>> line through P parallel to L).
>>
>> Wording the postulate in this way then allows "parallel lines" to be
>> defined equivalently as lines that don't meet, or as lines that cut
>> any
>> third line if at all in the same angle. Before contemplating
>> weakening
>> the Parallel Postulate in any way, one should first decide which of
>> these one is going to take as the definition of "parallel," because
>> the
>> respective predicates denoted by the two definitions diverge when it
>> is
>> weakened. Here I'll follow the custom of defining parallel lines to
>> be
>> lines that don't meet.
>>
>> When the Parallel Postulate is omitted altogether, other geometries
>> such
>> as hyperbolic and elliptic satisfy the first four postulates. In
>> particular those postulates are true on the sphere, for example the
>> Earth's surface so modeled.
>>
>> Euclid required just the only-if direction of that equivalence, that a
>> line cutting parallel lines meet them at the same angle. His phrasing
>> more precisely was of the contrapositive, couched in terms of two
>> adjacent angles summing to less than two right angles, which one might
>> interpret constructively as the Triangle Postulate, that a base and
>> two
>> such angles always determine a (not necessarily unique) triangle,
>> analogously to the other two odd-numbered postulates, that two points
>> determine both a line and a circle (in those cases uniquely). Yet
>> another phrasing is that *at most* one line can pass through a point P
>> parallel to a line L, also constructive but in this case partially and
>> uniquely instead of totally and nonuniquely, concomitant with point-
>> line
>> and epi-monic duality.
>>
>> In this weaker form the Parallel Postulate is true not only on the
>> plane
>> but on the sphere when lines are understood to be great circles or
>> geodesics. Hence all of Euclid's postulates as originally stated are
>> true on the sphere.
>>
>> The great search for a counterexample to the Parallel Postulate only
>> makes sense for Euclid's statement of it. Its phrasing as an
>> equivalence (or even just as the contrapositive, two lines cut by a
>> third at the same angle don't meet) admits the sphere as an obvious
>> counterexample worth not even brownie points at the level of
>> mathematical sophistication at which the search was conducted. The
>> searchers were clearly well aware of the soundness of Euclid's
>> postulates for the sphere. With regard to the preference for the
>> one-directional version of the Parallel Postulate, Euclid and the
>> searchers were surely on the same page.
>>
>> I would welcome pointers to sources of information and opinion bearing
>> on either of the following questions.
>>
>> 1. All intuition about Euclidean space per se demands that the
>> Parallel
>> Postulate be bidirectional, in order to rule out positive curvature as
>> well as negative when proving theorems true on the plane but false on
>> the sphere. Given this, why were Euclid and the searchers in
>> agreement
>> about stating the Parallel Postulate in its weakened one-directional
>> form? Was Euclid covertly hoping that his theorems would all hold for
>> surveying projects of a scale where the earth's curvature was
>> significant, or did it merely not occur to him that he might need the
>> converse at some point?
>>
>> 2. Proposition 47 of Book 1 of the Elements states and proves
>> Pythagoras's theorem. This is clearly false on the sphere, which
>> contains an equilateral triangle all three of whose angles are right.
>> This is not the earliest counterexample: Proposition 32 shows that the
>> angles of a triangle sum to two right angles. Euclid is evidently not
>> playing by his own rules, and is appealing to unstated postulates.
>> Does
>> he appeal somewhere to the converse of his fifth postulate? If so how
>> did he fail to notice he was doing so? This would make him an early
>> user of the Fallacy of the Converse, from P implies Q infer Q
>> implies P,
>> and an embarrassingly visible one at that. Proclus's Commentary
>> raises
>> only relatively finicky concerns, I don't know who first pointed out
>> this more glaring discrepancy but it's hard to imagine anyone
>> seriously
>> working on the independence of the Parallel Postulate without being
>> aware of it. It was surely common knowledge by the time Hilbert
>> embarked on his program to make Euclid's system more rigorous by
>> formalizing it.
>>
>> Vaughan Pratt
>>
>> --I was raised to be rigorous, not formal.
>> _______________________________________________
>> FOM mailing list
>> FOM at cs.nyu.edu
>> http://www.cs.nyu.edu/mailman/listinfo/fom
>
>
>
> ------------------------------
>
> Message: 2
> Date: Fri, 5 Dec 2008 04:07:28 -0500 (EST)
> From: "Michael J Barany" <mjb245 at cornell.edu>
> Subject: Re: [FOM] Understanding Euclid
> To: "Foundations of Mathematics" <fom at cs.nyu.edu>
> Message-ID:
> <35031.82.69.67.60.1228468048.squirrel at webmail.cornell.edu>
> Content-Type: text/plain;charset=iso-8859-1
>
> I don't have a direct answer to your question, but I can point out that we
> shouldn't be surprised that obvious fallacies or counter-examples may have
> persisted for even thousands of years. Lakatos's Proofs and Refutations is
> full of such examples, including during the time when the parallel postulate
> was most actively being challenged. Counterexamples were proposed to Cauchy's
> theorem that limits preserve integrals (without stating the uniform limit
> condition) some half a century before they were widely accepted as genuine
> counterexamples. It wasn't that mathematicians were being misguided or
> unthorough,... just that a lot of things seem more obvious looking backwards
> than looking forwards.
>
> Smiles,
>
> Michael
>
>
>> Euclidean geometry is standardly understood as the geometry of the
>> plane, more generally of flat or uncurved space. To make this stick
>> however, Euclid's fifth postulate should be phrased as an equivalence:
>> two lines fail to meet if and only if a third line intersecting them
>> both meets them at the same angle (or any equivalent phrasing thereof,
>> e.g. that the interior angles on the same side of the cutting line sum
>> to two right angles, or that a point P and a line L determine a unique
>> line through P parallel to L).
>>
>> Wording the postulate in this way then allows "parallel lines" to be
>> defined equivalently as lines that don't meet, or as lines that cut any
>> third line if at all in the same angle. Before contemplating weakening
>> the Parallel Postulate in any way, one should first decide which of
>> these one is going to take as the definition of "parallel," because the
>> respective predicates denoted by the two definitions diverge when it is
>> weakened. Here I'll follow the custom of defining parallel lines to be
>> lines that don't meet.
>>
>> When the Parallel Postulate is omitted altogether, other geometries such
>> as hyperbolic and elliptic satisfy the first four postulates. In
>> particular those postulates are true on the sphere, for example the
>> Earth's surface so modeled.
>>
>> Euclid required just the only-if direction of that equivalence, that a
>> line cutting parallel lines meet them at the same angle. His phrasing
>> more precisely was of the contrapositive, couched in terms of two
>> adjacent angles summing to less than two right angles, which one might
>> interpret constructively as the Triangle Postulate, that a base and two
>> such angles always determine a (not necessarily unique) triangle,
>> analogously to the other two odd-numbered postulates, that two points
>> determine both a line and a circle (in those cases uniquely). Yet
>> another phrasing is that *at most* one line can pass through a point P
>> parallel to a line L, also constructive but in this case partially and
>> uniquely instead of totally and nonuniquely, concomitant with point-line
>> and epi-monic duality.
>>
>> In this weaker form the Parallel Postulate is true not only on the plane
>> but on the sphere when lines are understood to be great circles or
>> geodesics. Hence all of Euclid's postulates as originally stated are
>> true on the sphere.
>>
>> The great search for a counterexample to the Parallel Postulate only
>> makes sense for Euclid's statement of it. Its phrasing as an
>> equivalence (or even just as the contrapositive, two lines cut by a
>> third at the same angle don't meet) admits the sphere as an obvious
>> counterexample worth not even brownie points at the level of
>> mathematical sophistication at which the search was conducted. The
>> searchers were clearly well aware of the soundness of Euclid's
>> postulates for the sphere. With regard to the preference for the
>> one-directional version of the Parallel Postulate, Euclid and the
>> searchers were surely on the same page.
>>
>> I would welcome pointers to sources of information and opinion bearing
>> on either of the following questions.
>>
>> 1. All intuition about Euclidean space per se demands that the Parallel
>> Postulate be bidirectional, in order to rule out positive curvature as
>> well as negative when proving theorems true on the plane but false on
>> the sphere. Given this, why were Euclid and the searchers in agreement
>> about stating the Parallel Postulate in its weakened one-directional
>> form? Was Euclid covertly hoping that his theorems would all hold for
>> surveying projects of a scale where the earth's curvature was
>> significant, or did it merely not occur to him that he might need the
>> converse at some point?
>>
>> 2. Proposition 47 of Book 1 of the Elements states and proves
>> Pythagoras's theorem. This is clearly false on the sphere, which
>> contains an equilateral triangle all three of whose angles are right.
>> This is not the earliest counterexample: Proposition 32 shows that the
>> angles of a triangle sum to two right angles. Euclid is evidently not
>> playing by his own rules, and is appealing to unstated postulates. Does
>> he appeal somewhere to the converse of his fifth postulate? If so how
>> did he fail to notice he was doing so? This would make him an early
>> user of the Fallacy of the Converse, from P implies Q infer Q implies P,
>> and an embarrassingly visible one at that. Proclus's Commentary raises
>> only relatively finicky concerns, I don't know who first pointed out
>> this more glaring discrepancy but it's hard to imagine anyone seriously
>> working on the independence of the Parallel Postulate without being
>> aware of it. It was surely common knowledge by the time Hilbert
>> embarked on his program to make Euclid's system more rigorous by
>> formalizing it.
>>
>> Vaughan Pratt
>>
>> --I was raised to be rigorous, not formal.
>> _______________________________________________
>> FOM mailing list
>> FOM at cs.nyu.edu
>> http://www.cs.nyu.edu/mailman/listinfo/fom
>>
>
>
>
>
> ------------------------------
>
> Message: 3
> Date: Fri, 05 Dec 2008 01:13:25 -0800
> From: Vaughan Pratt <pratt at cs.stanford.edu>
> Subject: Re: [FOM] Understanding Euclid
> To: Foundations of Mathematics <fom at cs.nyu.edu>
> Message-ID: <4938F0B5.8020209 at cs.stanford.edu>
> Content-Type: text/plain; charset=ISO-8859-1; format=flowed
>
> Bill,
>
> Derivable from what? Proposition 32? In my message I fingered
> Proposition 32 as a counterexample to spherical geometry along with
> Proposition 47. We're in agreement that Proposition 32 does not hold in
> spherical geometry.
>
> The difficulty I'm having is that I don't see how any of Euclid's five
> postulates could fail in spherical geometry. My point was that
> propositions such as 32 and 47, which clearly don't hold in spherical
> geometry, can't follow logically from postulates all of which hold in
> spherical geometry.
>
> Which one of Euclid's five postulates fails in spherical geometry?
>
> Vaughan
>
>
>
> William Tait wrote:
>> Vaughan,
>>
>> The converse of Postulate 5 is derivable in Euclid. He does define two
>> lines to be parallel when they do not intersect. The converse of
>> Postulate 5 then is: If two lines are perpendicular to a third, then
>> they are parallel. If they intersected, the three line segments would
>> form a triangle. Two of its interior angles are, by assumption, equal to
>> two right angles and so the three interior angles are more than two
>> right angles, contradicting Proposition 32 of Book 1.
>>
>> Proposition 32 does not hold in spherical geometry.
>>
>> Bill
>> On Dec 3, 2008, at 5:15 PM, Vaughan Pratt wrote:
>>
>>> Euclidean geometry is standardly understood as the geometry of the
>>> plane, more generally of flat or uncurved space. To make this stick
> [...]
>>> 2. Proposition 47 of Book 1 of the Elements states and proves
>>> Pythagoras's theorem. This is clearly false on the sphere, which
>>> contains an equilateral triangle all three of whose angles are right.
>>> This is not the earliest counterexample: Proposition 32 shows that the
>>> angles of a triangle sum to two right angles. Euclid is evidently not
>>> playing by his own rules, and is appealing to unstated postulates. Does
>>> he appeal somewhere to the converse of his fifth postulate? If so how
>>> did he fail to notice he was doing so? [...]
>
>
> ------------------------------
>
> Message: 4
> Date: Fri, 05 Dec 2008 13:47:19 -0500
> From: Colin McLarty <colin.mclarty at case.edu>
> Subject: Re: [FOM] FOM Understanding Euclid
> To: fom at cs.nyu.edu
> Message-ID: <f09a6892137aa.137aaf09a6892 at cwru.edu>
> Content-Type: text/plain; charset=us-ascii
>
> ----- Original Message -----
>>From Vaughan Pratt <pratt at cs.stanford.edu>
> Date Wed, 03 Dec 2008 15:15:17 -0800
> To Foundations of Mathematics <fom at cs.nyu.edu>
> Subject [FOM] Understanding Euclid
>
> asked about how Euclid excluded the geometry of great circles on a sphere.
>
> Euclid's postulate that a line segment can always be extended was
> understood to mean "extended to new points," i.e. a segment can always
> be extended without returning to itself. That is how he proves
> Proposition 16, the first of his propositions that fails in the sphere
> geometry: "In any triangle, if one of the sides is produced, then the
> exterior angle is greater than either of the interior and opposite angles."
>
> I know that Johann Heinrich Lambert also took the postulate that way,
> much later, and I believe pretty much everyone did before the 19th
> century.
>
> best, Colin
>
>
> ------------------------------
>
> Message: 5
> Date: Fri, 5 Dec 2008 13:53:42 -0500 (EST)
> From: Alasdair Urquhart <urquhart at cs.toronto.edu>
> Subject: Re: [FOM] Understanding Euclid
> To: Foundations of Mathematics <fom at cs.nyu.edu>
> Message-ID: <Pine.LNX.4.64.0812051342100.5134 at apps0.cs.toronto.edu>
> Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed
>
>
> The fact that Euclid states the Parallel Postulate as an implication,
> not an equivalence is very likely related to the fact that
> the converse of the Fifth Postulate is proved as a theorem
> (Euclid I.27), deduced as a consequence of I.16 without the
> use of the Fifth Postulate. It is well known that Euclid
> tries to go as long as possible without using the Parallel
> Postulate; it is in line with this procedure that he would
> state the postulate in as weak a form as possible.
>
> Proclus called attention to this:
>
> This [Postulate 5] ought even to be struck out of the
> Postulates altogether; for it is a theorem involving many
> difficulties ... and the converse of it is actually
> proved by Euclid himself as a theorem. [Heath's edition
> of the Elements, Volume 1, p. 202]
>
> Alasdair Urquhart
>
>
>
>
>
>
> ------------------------------
>
> Message: 6
> Date: Fri, 05 Dec 2008 13:10:55 -0800
> From: John Edward <jeedward at yahoo.com> (by way of Martin Davis
> <martin at eipye.com>)
> Subject: [FOM] TMFCS-09 call for papers
> To: fom at cs.nyu.edu
> Message-ID: <200812052110.mB5LAwZ9016723 at nlpi025.prodigy.net>
> Content-Type: text/plain; charset="iso-8859-1"; format=flowed
>
>
> TMFCS-09 call for papers
>
> <?xml:namespace prefix = o ns = "urn:schemas-microsoft-com:office:office" />
>
> The 2009 International Conference on Theoretical
> and Mathematical Foundations of Computer Science
> (TMFCS-09) (website:
> <http://www.promoteresearch.org/>http://www.PromoteResearch.org
> ) will be held during July 13-16 2009 in Orlando,
> FL, USA. We invite draft paper submissions. The
> conference will take place at the same time and
> venue where several other international
> conferences are taking place. The other conferences include:
>
> ? International Conference on Artificial
> Intelligence and Pattern Recognition (AIPR-09)
>
> ? International Conference on Automation,
> Robotics and Control Systems (ARCS-09)
>
> ? International Conference on
> Bioinformatics, Computational Biology, Genomics
> and Chemoinformatics (BCBGC-09)
>
> ? International Conference on Enterprise
> Information Systems and Web Technologies (EISWT-09)
>
> ? International Conference on High
> Performance Computing, Networking and Communication Systems (HPCNCS-09)
>
> ? International Conference on Information Security and Privacy (ISP-09)
>
> ? International Conference on Recent
> Advances in Information Technology and Applications (RAITA-09)
>
> ? International Conference on Software
> Engineering Theory and Practice (SETP-09)
>
> ? International Conference on Theory and
> Applications of Computational Science (TACS-09)
>
>
>
> The website
> <http://www..promoteresearch.org/>http://www.PromoteResearch.org
> contains more details.
>
>
>
> Sincerely
>
> John Edward
>
> Publicity committee
>
>
>
>
>
>
>
>
>
>
>
> ------------------------------
>
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom
>
>
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