Question

wikipedia says :
"The first proof of the existence of irrational numbers is usually attributed to a Pythagorean (possibly Hippasus of Metapontum), who probably discovered them while identifying sides of the pentagram.[8] The then-current Pythagorean method would have claimed that there must be some sufficiently small, indivisible unit that could fit evenly into one of these lengths as well as the other."
why does it say "indivisible unit" ? why is this necessary for it to be indivisible. we can divide it in half . and there are twice the number of copies in each .

Answer 1

Plato thought a point was a fiction and called a point the beginning of a line. This leads to the idea of an indivisible line. Remember that the Greeks were not into infinity and infinite processes so the idea of infinite divisibility (leading to a point?) would have been something bad for them.

So irrationality would have been a bit of a shock to the system.

Answer 2

what is an indivisible line? oh a line segment that is shrunk to a point?

im reading this online:
"The Pythagoreans believed that magnitudes could always be measured using whole numbers, which would imply that lengths are not infinitely divisible"

why is this the case?

Answer 3

ie Ratios.

Answer 4

oh i think if its infinitely divisible, you can never find a common unit that measures both of them

Answer 5

That's the point, the Greeks didn't believe in infinite divisibility, worst case you would end up with a ratio (aka a rational)

Answer 6

im trying to do a formal proof.

but its false to say that you need an indivisible unit

"The then-current Pythagorean method would have claimed that there must be some sufficiently small, indivisible unit that could fit evenly into one of these lengths as well as the other."

Answer 7

A formal proof of what?

The Greeks believed there was always some smallest indivisible unit that would produce a rational result (even if they couldn't measure it).

Answer 8

any unit you find that measures two quantities can be divided

Answer 9

what you do you mean?

the greeks always believed in an indivisible element. can you give me an example

Answer 10

Today, with your axiom system, yes.

In Greek times, not allowed.

Answer 11

Atom.

Answer 12

Atom, from Greek atomos, meaning indivisible.

Answer 13

so everything is a whole number copy of the atom

Answer 14

Correct (even if they were not able to measure or prove the existence of it, they posited it, like an axiom).

Result, no irrationals.

Answer 15

It's not that strange, you can always get as close as you want to some quantity by constructing a rational with the right numerator and denominator.

Answer 16

, so the same issue is with rationals,

if magnitudes could always be measured using rational numbers, this would imply that lengths are not infinitely divisible.

Answer 17

That's right, except that they didn't have algebra, so they used ratios of lengths to measure things (a strictly geometric approach).

Answer 18

actually i dont see it.

if pythagoreans believed given any length x, there exists a k such that x = nk. , then when n=1, we have a contradiction when x < k , since we have run out of whole numbers

Answer 19

for instance if x = k/2

Answer 20

You are trying to use modern day algebra.

Greeks used a unit length, this doesn't have to be 1 (it can be anything).

Answer 21

isnt there a unit length , a different one for each pair of lengths?

Answer 22

what does the word "unit" mean? that its a common unit of measurement for all measurements?

Answer 23

The ratio of the lengths is the important thing....

Answer 24

what does the word unit imply

Answer 25

Today, 1

Then, a measure.

Answer 26

You have to change your whole way of thinking if you want to think like the Greeks did.

Answer 27

a measure?

Answer 28

yes i want to think like greeks

Answer 29

but i checked wikipedia, and it said a unit implies a standard or a measure, like a unit meter

Answer 30

That's today's definition of a measure.

Answer 31

but our units are divisible, we can have centimeters, millimeters. in greek times i dont think you could divide a unit

Answer 32

whats the greek definition of measure ?

Answer 33

Imagine you draw a triangle in the sand and it is Pythagorean.

The Greeks would say that the sides are in the proportion 3:4:5 and you can count the 5 in 3's or in 4's or in 8's if you want.

Answer 34

Let me see if I can get you a quote from a translation of Euclid that will show you what I mean...

Answer 35

you cant count 5 in terms of 3

Answer 36

are you good at math proofs
i want to show
n! > n^k for n positive integers, and k fixed positive integers. for sufficiently large n

Answer 37

Have a look at this (Euclid Book 5, Definitions).

Answer 38

http://aleph0.clarku.edu/~djoyce/java/elements/bookV/bookV.html#defs

Answer 39

this word measure is ambiguous

Answer 40

It's ambiguous today, then it is just a convenient piece of wood, if you like (a blank ruler, a length).

Answer 41

the euclides definition of measure seems to be division?

Answer 42

For your proof, maybe:

http://en.wikipedia.org/wiki/Factorial (the Rate of Growth section).

Answer 43

A ratio, by definition (today), IS division. For the Greeks, it's a comparison.

Answer 44

In physical terms (as opposed to mathematical) the Greeks may well turn out to have got it right.