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On Flags and Numbers

I hope the seasoned mathematician, or anyone to whom the following is known or obvious,
will forgive me for labouring the topic, but I believe many find it vexed,
and there are one or two fundamental issues embedded subtly
within it that even the most seasoned may wish to revisit.

Natural Counting




It is natural to assign numbers to intervals of a line as we make them, or as they are made–we simply count them as we go. We usually assign the counts to the points marking the intervals, rather than to the intervals themselves, with the tacit understanding that the 'n' of the n'th point in a range also refers to  the interval between that point and the "one before"–the (n-1)'th point.

Do note carefully that,
because we are working projectively,
and preserving incidence alone,
we are here counting intervals,
not sizes.

Where to Start?

However, since the range extends indefinitely on the line in either sense from any point, any point in the range can be labeled '0' (zero), with the simple meaning, "We will start counting here!" We can do this because points are without size, so we have as many places available for counting as we wish, going in either sense from any start. We surely won't run out of them.

Which way?

But it is obvious that we will have a problem with knowing, just from the numbers themselves, in which sense we are counting points, because the numbers are the same either way.

In what follows, please bear firmly in mind that
a point is not a line,
a line is not a point,
and that, though either may be incident with the other,
neither consists of the other.

While it is NOT true that, for a given measure in a line,
there is a count for every possible selection of a point of the line,
we can be quite sure of the converse –
that a point is always available as a candidate for counting/numbering


Given a measure in a line
there is no guarantee that a number of its intervals
will fit between
two distinct, randomly-selected points.
on that line.

Continually subdividing the intervals into fractions need not lead to a fit of a whole number of intervals between our points, because there are positions not selected by the measure.

(Fractional division is equivalent to increasing the number of integral intervals, as one may eliminate the fractional character simply by recasting the sub-divisions as whole units. That is, by rescaling.)

In fact, the only sub-division projectively available (that is, found by using a straight-edge alone) is, ‘by two’ – bisection. After this, by serial, repeated division-by-two of such sub-divisions, division by powers of two –

20=1, 21=2, 22=4, 23=8, 24=16, 25=32, .......

– becomes possible, and, clearly,

only points falling exactly at such subdivisions
can be noted and counted by the
numbering system built on these subdivisions
(that is, by this “
calibration” )

All other points in the line must simply go undetected. Such points must be unfittable and uncountable—and hence, unmeasurable.


On the other hand,
Given the points

(and the invariants), it will always be possible to find a measure with intervals that do fit. By conscious construction, we take the interval between the two, given points as an interval of the measure.

We can extend this to one given invariant, and a maximum of three randomly-selected points. The position of the second invariant is determined by the selection of these first three points, and this in turn determines the position of the fourth point, and indeed the positions of all other points in the measure. This is an expression of the Fundamental Theorem.

A randomly-selected fourth point could be, but need not be, in the appointed fourth position.

If we insist that an unfittable point must have a number (which, as we perhaps now see, may not be an altogether wise thing to do), then we must accept that that number is not determined by the given measure.

Flags (Signs)

With all the above clearly in our minds, we can say points-cum-intervals in the measure "before", or, "to the left of", zero should carry a little flag (conventionally, '-') indicating that, "we are counting up, but backwards/leftwards", or, "we are lying to the left of zero".

We could just as well put these flags on the intervals, "counting up, rightwards/forwards", or, "lying to the right of zero".  It doesn't much matter where they go, provided we remember that they are just there to keep us orientated, and provided that we stick to our decisions about how to use them.

The animation shows "+" flags on the numbers running right from zero. These are very often simply omitted, as they are supposed to be "tacit", or "understood", meaning, 'if a plus sign is not there, the number is positive.'

This is sometimes unhelpful, as zero and infinity are always unsigned. Moreover, occasionally we want to refer only to the value of a signed number (the so-called "absolute" value, e.g., |x|). The confusion is compounded by statements like, "Absolute values are always positive", which is simply untrue: they are neither positive nor negative.

We also need flags because, as we have seen, we are assigning the same numbers twice, in two different senses.

The issue of flagging and numbering is a source of enormous and endless confusion, especially in classrooms, in which it may for instance be heard that, 'quantities can be less than nothing', which is nonsense, or that, 'minus times minus is plus', which seems like nonsense.

If we say,
"lying to the left or right of"
we are speaking  positionally.

We are specifying where something is - passively.
Passive Flags

On the other hand,
if we say,
"going rightwards/leftwards,
we are speaking

We are doing something - actively.

Active Flags

So our flags are also used twice, like the numbers: they can be both passive signs and active operators.

So, for example, when we multiply, we operate, and when, operating, we multiply minus by minus, we are simply issuing the command, "Reverse!  Go the other way round!  Stop what you're doing, and do the opposite!"  We were "going left", or whatever the second minus meant, before, so on multiplying by the first minus, we will be "going right", or positively, after.  Minus times a minus is a plus.  So simple, so easy to forget!

By this flagging scheme, neither zero nor infinity carries a flag--both are unsigned.
Both mark a place where transition may be made from the section with one sign to the section with the other. Indeed, their roles may be swapped.  If we are to leave "plus territory" and enter "minus territory", or vice versa, we may do so only by way of one of these two places (this serves as a reminder that one point does not divide a line; two do).

Please do notice
also that the numbers, as quantities,
are always either zero, or more than zero, never less. They simply cannot be less.

Mention has been made here only of lineal measure
- intervals of stepping from point to point along a "fixed" line. -
Nothing is said about angular measure
- intervals of rotation of a line around a "fixed" point. -
These ideas are dual.
We will find that the two kinds of measure are dual, too.

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