Quest for equality
in [Math]
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From: BD on 19 Jun 2010 12:10 Hero <Hero.van.Jindelt(a)gmx.de> wrote: > Hero wrote: > >> So, in 1 + 1, is 1 here added "to itself"? > > Most of you know this proof and as it is so convincing, the > accompanying commentary is digested without chewing: > Proposition 1.1. Let (G; o ) be a group, then > there is exactly one neutral element > and each element of G has exactly one inverse. > > Proof. Let e; ~e element of G be neutral elements of (G; o ). > Thus, for every x element of G we have > x o ~e = e o x = x, and therefore, > e = e o ~e = ~e, > " " > ~e neutral e neutral > > and hence, there is exactly one neutral element. > http://iamwww.unibe.ch/~halbeis/4students/gtln/sec1.pdf > > Give it a thought. It states, if there are two neutral elements, then > they are equal. > Nowhere is forbidden, that there are more equal elements. > The 'conclusion' : "there is exactly one neutral element" > is a confusion of identy with equality. > > And with rational numbers there have to be several ones, several twos, > several twelves, ..., otherwise terms like > 1 + 1 can not be constructed. Your name is mentioned more than once in the above. Does that imply there are more than one of you? Perhaps you should consider changing your name from Hero to Zero to make your numerosity clear.
From: Hero on 19 Jun 2010 13:12 BD wrote: > Hero <Hero.van.Jind...(a)gmx.de> wrote: > > Hero wrote: > > >> So, in 1 + 1, is 1 here added "to itself"? > .... > > Your name is mentioned more than once in the above. > Does that imply there are more than one of you? Hei, Bill. No. It is your newsreader, which is bi. Google is mono. > Perhaps you should consider changing your name > from Hero to Zero to make your numerosity clear. You wouldn't write this, if you had read, what I wrote. After all, I don't consider Your first name as short for a number with nine naughts. With friendly greetings Hero
From: Hero on 21 Jun 2010 10:40 Hero wrote: > The 'conclusion' : "there is exactly one neutral element" > is a confusion of identy with equality. Let's continue the quest. Through the ages there were myriads of Ones dealt with, calculated with. Aristoteles Metaphysik is about the plurality of Ones. Euclid wrote in the elements: "A number is a multitude composed of units." Nowadays there seems to be only exactly one 1 left over. Frege developed this into a problem in "Die Grundlagen der Arithmetik". He starts asking: "Can we place in the equality 1 + 1 = 2 both times for 1 the same thing, let's say the moon? It rather seems, that we have to place for the first one something different as for the second."(my translation) http://www.ac-nancy-metz.fr/enseign/philo/textesph/Frege.pdf And the quest is still on. Recently Berry Mazur asked "When is one thing equal to some other thing?" http://www.math.harvard.edu/~mazur/preprints/when_is_one.pdf He more or less avoids his difficulties with equality by switching over: "The major concept that replaces equality in the context of categories is isomorphism." "...objects need only be given up to unique isomorphism, this being an enlightened view of what it means for one thing to be equal to some other thing." When he is done the question arises: Objects are considered 'up to isomorphism', why not considering objects, for example numbers, 'up to equality'? After all, this is in the title of Barrys text. And we saw, that the proof of the 'uniquness' of the neutral element in a group is not a proof of 'exactly one and no other one' but a proof 'up to equality'. With friendly greetings Hero
From: Hero on 22 Jun 2010 13:21 Hero wrote: > Frege ... > "Can we place in the equality 1 + 1 = 2 both times > for 1 the same thing, let's say the moon? I think, this should convince even the last one of you: By prime factorization we get 1960 = 2^3 * 5^1 * 7^2, so 1960 has three 2s, one 5 and two 7s, composed with multiplication. This is an example of a multiset (remember Gerry wrote about it). When 2^3 has three 2s in multiplication, so 5 * 4 has five 4s added up. So 1+1 has two different 1s! .............. Let me conclude: identical >------< different /|\ /|\ | | | | \|/ \|/ equal >--------< not equal The opposites or contraries identy and difference can also be described with self, individual, unique, exist in singular (the moon ,a person)- divers, another, distinct from. The opposites or contraries equality and unequality can also be described with the same, similar - inequality. What is identical is also equal. What is not equal is also different. But: What is different can be equal or can be not-equal. What is equal can be different or might be identical. With friendly greetings Hero
From: Hero on 16 Jul 2010 15:10
Hero wrote: > > identical >------< different > /|\ /|\ > | | > | | > \|/ \|/ > equal >--------< not equal > > The opposites or contraries identy and difference can also > be described with > self, individual, unique, exist in singular (the moon ,a person)- > divers, another, distinct from. > > The opposites or contrariesequalityand unequality can also > be described with > the same, similar - inequality. > > What is identical is also equal. > What is not equal is also different. > > But: > > What is different can be equal or can be not-equal. > What is equal can be different or might be identical. Talking on a general level about 'equality' one should not forget, that in math this is nearly always a specified equality. Most often numerical and more general quantitative equality and unequality is considered, often without mentioning this. Dividing a square by a diagonal into two equal triangles is understood in its context. Equality of form and size and unequality of location is important here. One might see a contradiction in taking two somehow different 1's and adding them, as there is just one 1 in the set of natural numbers. It's one 1 in this set'up to equality'. Considering these numbers as placeholders for objects, one can add two 1-euros and it is sufficient to know, that they are different; it's not necessary to describe in what they are distinct. So in the set of natural numbers numerical equality and unequality matters. With friendly greetings Hero |