(6th March 2013, 23:07)John V Wrote: Thoughts?

The ratio was discovered. We still have not invented a way to represent it.

(6th March 2013, 23:07)John V Wrote: Thoughts?

The fundamental form of this question is, Were numbers discovered or invented?

It is a fairly simple answer once you recognize there are different kinds of numbers even though they appear to be the same.

Counting numbers are different from numbers used for calculation, numbers that operate upon each other. One can look at a number of things and count them as four things. One cannot add four apples and four bananas and four people and get twelve unless one abstracts to twelve things. One you abstract you are no longer counting but operating on numbers. Those are different kinds of numbers.

Counting is a trick we discovered but observing small numbers is something most animals can do. The can see the difference between one and two. Chimps can observe the difference up to five if I remember correctly. But none can abstract to calculate.

The trick of counting is an invention of sorts, very useful, but it is merely tagging what we see. What we do is no different from hanging numbered cards on things until we run out of things to hang cards on and then use the last number. It is a way to get beyond what we can observe without counting which is less than ten for most people. See chimps above.

So really all we did was come up with a gimmick to count more than we can observe. That is one kind of numbers. A rather uninteresting kind as the focus in the trick used.

Manipulating numbers is an incredibly recent event. Pi was "discovered" by the Greeks but they use geometry.