mardi 1 septembre 2009

Two more than two make four, but why?

One can analyse the proposition with its constituents by counting different ways, but ultimately the sentence reposes on some intuitions:

- numbers are exact, and each is different from each of the other
- numbers differ minimally by one
- the more ones that go into one number, the more different ways there are of subdividing it

This can of course be put into syllogisms, but they would miss the point that intuition is at the heart of reason. I will however give them.

Two is one more than one. Self-evident intuition, since definition.
Three is one more than two. Dito.
Four is one more than three. Dito.

One more than one is two. Conversion of first, since definition and definiend are interchangeable. et c for other two definitions. Now we go:

Two more than two is one more than one more than two.
One more than two is three.
Two more than two is one more than three.
One more than three is four.
Two more than two is four.

Simply speaking, four will subdivide into (if only two numbers that may be equal):

one more than three
two more than two
three more than one.

There is one reason which will not do:

"two is 2.00 and not 1.99 or 2.01

2.00 + 2.00 add exactly up to 4.00

whereas 1.99 + 1.99 would add up to only 3.98 deviating into direction of 3.00 and 2.01 + 2.01 add up to 4.02 deviating into direction of 5.00,

but in 4.00 there is no such deviation, and therefore it shall stick to the four which is exactly between 3.00 and 5.00"

A This is true but non-essential.
B This is derivative.
C Fractions are not arithmetic proper anyway, but into applied geometry or the science of proportion and measures, like in commercial mathematics or carpentry (you may have 4.02 € or 3.98 m. but there is no such number as 4.02 or 3.98, these things are monetary exchange values or size measures, but the value of one item is one value 4.02 times as great as the € and not 4.02 values and the size of one length is one size 3.98 times as great as the metre, not 3.98 sizes).
D It is petty.
E It invites some wizeacres to go further and deny the existence of even 3, 4 and 5 because 4.00... up to 10 billion zeros and so on for 2, 3 and 5 is never in fact verified in any measurement, because measurements are never that exact.

Hans-Georg Lundahl
Paris V, Mouffetard
St Symeon Stylites day
as counted in G. Kal.
YooL 2009