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July 18

Question from an old dunce about Euclid's V° axiom

Hello Learned Ones ! I just enjoyed Agora (film) , and wonder at the V° Euclid axiom (or postulate ?) twice hinted at in the film by Hipatia of Alexandria, when she wants to quiet down her students. It seemed to me it could very bluntly be "digested" as : "if 1 + 1 = 2 , then 1 = 2" ... Where did I goof ? (I know, I'd have tried to study some maths during those 60 last years, but I seemed to be able to fare quite OK without them, & we didn't really like each other, so...) . Thanks beforehand for your answers , t. y. Arapaima (talk) 07:12, 18 July 2013 (UTC)[reply]

Euclid's fifth axiom is the parallel postulate. Why you would think that the parallel postulate implies that 1 = 2 escapes me, I'm afraid. Looie496 (talk) 07:22, 18 July 2013 (UTC)[reply]

The film Agora never refers to Euclid's fifth axiom. Instead, the film was referring not to the axioms of Euclid's Elements, but to his "common notions", and not to the fifth one, but to the first one: "Things which equal the same thing also equal one another", basically, a form of the transitive property. In the film, Hypatia uses it to argue that two people who are friendly/allied with her should also be friendly/allied with each other. In symbols, if a = b and b = c, then a = c. It has nothing to do with 1 + 1 = 2. —SeekingAnswers (reply) 21:55, 18 July 2013 (UTC)[reply]

Didn't I tell you I was an old dunce ? Thanks a lot, & you can use my example to scare the young ones who don't like maths ... Arapaima (talk) 08:09, 19 July 2013 (UTC)[reply]

Terminology

Is there a word (more specific than "symmetry") for the concept that indistinguishable things must have identical properties? For example, in the case of a regular polygon, it is impossible for any side to have a different property from any other side because there is no way to tell the sides apart. I.e., if hypothetically one side had a different property then it would be impossible to specify which side that was, therefore the premise of different property is wrong. Is there a name for this concept/method? 86.179.115.136 (talk) 20:48, 18 July 2013 (UTC)[reply]

Congruency? — Quondum 21:11, 18 July 2013 (UTC)[reply]

It sounds like you're talking about Reductio ad absurdum (for the proof method). There's various other ways of phrasing what you'r asking, none of them that great as an alternative. Is this the case you're interested in, or were you looking to use the term elsewhere? You may want to read: Isomorphism, Invariant (mathematics), Equality (mathematics). You may also want to read the article Dihedral group.Phoenixia1177 (talk) 07:27, 19 July 2013 (UTC)[reply]

If they are truly impossible to distinguish, then they are intensionally equal. So all properties are the same. But this is essentially a tautology. If they had different properties, then it would have been possible to distinguish them by their properties. Sławomir Biały (talk) 08:57, 19 July 2013 (UTC)[reply]

All of mathematics is essentially a tautology, isn't it? 86.128.1.121 (talk) 11:39, 19 July 2013 (UTC)[reply]

Math is the truth about numbers and shapes. — 79.113.210.34 (talk) 02:09, 20 July 2013 (UTC)[reply]

It's more like the truth about some user-defined things that we hope are equivalent to numbers and shapes. — Preceding unsigned comment added by 86.171.174.107 (talk) 20:46, 20 July 2013 (UTC)[reply]

While technically true, this seems like sophistry. The point is that saying that they are indistinguishable means the same thing as that they have the same properties. There is nothing further to say on the matter. Sławomir Biały (talk) 06:25, 20 July 2013 (UTC)[reply]

See identity of indiscernibles. It is important to distinguish between identity and equality. Two sides of a regular polygon are equal in length but they are not equal in orientation or position, so they are not identical.. Gandalf61 (talk) 07:29, 20 July 2013 (UTC)[reply]