a model which assumes numbers are related by straightness and light|
As a pragmatic device for relating numbers to spatial dimensions it is a simple and convenient model.
'Real' single-component numbers... from the largest to the smallest... are thought of as being on a linear continuum.
One simply imagines a 'straight' line on which a 'zero' is indicated and other +ve and -ve integers are marked at
equal unit intervals in both directions away from the zero 'origin'.
Integers ('counting' numbers) and fractions can be indicated on the line in their appropriate scaled position.
The numbers a = -2, b = -0.5, and c = 11/4 could be indicated on a number line thus:
For many purposes this is a very useful modelling device and can be utilized in practical situation with little further reflection.
There are, however, certain metaphysical problems lurking in the model as well as difficulties emerging when the
concept of number is extended.
To begin with, just exactly what is meant by a 'straight' line?
Not 'curved' presumably... but 'curvature' and 'straightness' are inextricably related.
One cannot judge 'curvature' without reference to 'straightness', just as one cannot judge 'heaviness' without reference to 'lightness' or
'happiness' without reference to 'unhappiness'.
They are all relative concepts.
The oft repeated suggestion that a straight line is the shortest distance between two points might initially seem logically plausible,
but measuring the distances involved in order to select the shortest requires us to know whether the ruler was straight or not to begin with.
Tensioning a 'weightless' thread approximates a 'straight' line but that is all.
It might serve as a useful practical estimate of 'straightness', but it is hardly going to serve as a definition.
Eventually we are drawn to some sort of variation of the carpenters technique, of sighting along the edge of whatever is being checked
Whether one uses MarkI eyeballs or the latest laser projection technology, it is the propagation of light...
whatever its nature might be...
that ends up being used as the ultimate standard of 'straightness'.
Claims are often made that the realm of mathematical modelling is not subject to actual physical verification,
but granting this exemption almost always results in metaphysical dilemmas when the models are imported back into the real world.
The suggestion that a simple number line is somehow 'one-dimensional' is thus seen to be an abstraction fantasy.
Light only operates in the 'space' of reality.
The checking of a 'one-dimensional' line can only be done in '3-dimensional' space-time.
Another seemingly obvious assumption, of supposing there to be a unique 'point' on a number line which can be associated
with each and every number, also turns out to be a fiction.
As long as the numbers being considered are restricted to certain integers and fractions, all appears plausible,
but more sophisticated computational procedures result in numerical entities which do not fit comfortably into the scheme.
Multiplying a number by itself is called squaring so that for example:
+3 x +3 = +9 and -3 x -3 = +9 is usually written (± 3)2 = +9
The inverse process of obtaining a number which multiplies by itself to equal the
given number is called 'finding the square root':
√ (+9) = ± 3 or √ (+25) = ± 5 or even √ (+5.76) = ± 2.4
But what about the square root of two? What is the value of √ 2 ?
Whatever it is, it has to multiply by itself to equal 2. i.e. (√ 2)x(√ 2) = 2
We could try and find it approximately by trial and error.
Since 12 = 1 and 22 =4
we would expect it to be between 1 and 2.
Suppose we try 1.5 as an approximation for √ 2.
1.5 x 1.5 = 2.25 so that is obviously too big.
Half way between 1 and 1.5 is 0.75 so lets try that.
0.75 x 0.75 = 0.5625 which is too small.
OK, now lets try half way between 0.75 and 1.5, which is 1.125.
1.125 x 1.125 = 1.265625 which is better but still too small.
Continuing this way, after quite a few more attempts we keep getting closer.
1.4142 x 1.4142 = 1.99996164 which is within about 2 thousandths of a percent.
Very good for a lot of purposes, but certainly not exact.
Eventually we discover that it is not possible to find an exact ratio of two
integers for this elusive intellectual entity.
For this reason it is called 'irrational'.
We still mark it on the number line as if it had a specific position, but
in fact we are only ever able to find rational numbers which are just bigger
and just smaller. Quite strange really.