On flags and Numbers

On Flags, Numbers and Continuity

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 measure as we make them, or as they are made–we simply count them as we go. We usually assign the counts to the end-points of 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.

Where to Start?

However, since the range extends infinitely 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 we must have the same number of digits (namely, ∞) available going in either sense counting from any start—and 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.

Continuity

In what follows, do please keep firmly in mind that
a point is not a line,
and
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, in a given measure in a line,
there will always be a count- or number-per-point,
we can be sure of the converse
- that a point is always available as a candidate for counting/numbering -
because by definition the line is continuous.

To see this, suppose that it is not continuous—that is, suppose that there are points missing from it,
with the meaning that there are places that the line may visit from which points are absent.

What will be where these points are not? Presumably, gaps. Of what will the gaps consist?
Certainly they must consist of places, just those places where the points are missing.

But what are places? By definition, they are points! To say that a point can be absent from a place contradicts the definition of a point, as point and place are one and the same thing. It follows immediately that points cannot be missing.

Putting it another way, there is nowhere that isn't somewhere,
so there is nowhere that isn't a point.

If they are “missing”, then, it must be because we choose
- or some selection-system, such as an equation, or a function, or an algorithm,
or even a physical law, -
“chooses” to ignore (deselect) them.

Conversely, if they are present, they have been selected by the line,
and the points in a measure are a selection from that selection.

Given a measure
there is no guarantee that an integral number of its intervals will fit between two randomly selected, distinct points. Continually subdividing the intervals into fractions, which is equivalent to increasing the number of integral intervals (as one may eliminate their fractional character simply by recasting the equal sub-divisions as whole units), need not lead to an exact fit, because there are always positions not selected by a 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,
or,
forwards/backwards",
we are speaking operationally.
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 numbers, are always either zero, or more than zero, never less. They simply cannot be less.