Clarifying the Nature of Ordinal Numbers (2)

As we have seen, with the linear (analytic) interpretation of number, the ordinal notion is effectively reduced in cardinal terms.

Therefore we must move to a circular (holistic) interpretation, so that the ordinal aspect - while necessarily existing in relationship to the cardinal - yet preserves its own distinct identity.

When one reflects on it, the ordinal notion can only have meaning with respect to a group of numbers that are defined in a cardinal manner.

So for example one can validly seek to interpret the ordinal notions of 1st and 2nd (in the context of 2), the ordinal notions of 1st, 2nd and 3rd (in the context of 3), the ordinal notions of 1st, 2nd, 3rd and 4th (in the context of 4), the ordinal notions of 1st, 2nd, 3rd, 4th and 5th (in the context of 5) and so on.

So again to illustrate, if we have a class of 20 pupils (in cardinal terms), then we can provide ordinal rankings in an examination for 1st, 2nd, 3rd, .....20th positions.

What is fascinating here is that each ordinal position (e.g. 1st) acquires a uniquely distinct meaning (as the cardinal number of the group changes).

And in everyday terms, we can readily accept the relative meaning of such ordinal positions. For example, one can instinctively appreciate that 1st in a 2-horse race does not perhaps match the achievement of 1st in a 20-horse race!

So ordinal positions have a merely relative meaning (that keeps changing depending on context).

One could perhaps also inquire regarding the meaning of 1st (in the context of 1). In fact it is quite instructive! If we now consider for example a 1-horse race, the horse that comes in 1st equally comes in last! So this highlights the truly circular (paradoxical) nature of ordinal positions (that possess a merely relative validity).

Thus the question then arises as to how we can give an indirect quantitative interpretation for all these relative notions of ordinal numbers (which inherently are qualitative in nature).

And the answer is through simply taking successive roots of 1.

So for example, in the simple case where the cardinal group = 2, we can obtain an indirect quantitative expression for 1st and 2nd in this context.

Now to obtain the two roots of 1, strictly we must consider the two following equations

x² = 1¹ and  x² = 1²  respectively.

So what we are in effect seeking here is to reduce the Type 2 notion of number in a Type 1 manner.

Crucially therefore, we are attempting to "convert" the Type 2 notion indirectly in a Type 1 manner.

Therefore, though our original expression represents the "higher" holistic dimension of 2 (indicated by the power of x) the result from obtaining the two roots, represents the 1-dimensional "conversion" (that defines the Type 1 quantitative approach to numbers).

Indeed we can express the two results as fractions (in the Type 2 system).

So the first root is x = 1^1/2 and the second root is x = 1^2/2 = 1¹

In fact though not strictly required in this simple case, the general procedure for obtaining all the various roots of 1 is provided through the Euler identity (which we have already addressed at length in past blog entries).

e^ix  = cos x + i sin x = 1

Then when x is expressed in radians as 2π = 360 degrees.

e^2iπ  = cos 2π + i sin 2π =  1 (i.e. 1¹).

So 1^1/2  = cos π + i sin π  = – 1.

Thus the two roots of 1 (i.e. of  1¹ and 1² respectively) are  – 1 and + 1.

So these results express in an indirect circular quantitative manner, the notions of 1st and 2nd respectively (in the context of 2).

Therefore, for example, to indirectly express in a quantitative manner the notions of 1st, 2nd and 3rd (in the context of 3) we would now obtain the 3 roots of 1 - strictly the 3 roots of 1¹ and 1² and 1³ - respectively (i.e. 1^1/3, 1^2/3 and 1^3/3) which are – .5 + .866i,   – .5 – .866i and + 1 respectively.

Note that the last root in each case, representing the 2nd in the context of 2 and the 3rd in the context of 3 respectively, = 1 (i.e. + 1). And in general terms, this will always be the case for the nth root of n.

So the reduction of ordinal to cardinal notions in conventional mathematical terms arises from sole consideration of this limiting case.

Thus in the case of 1, only the last unit ( which in this case is also the 1st) is considered. Then in the case of 2 units, only the last (i.e. 2nd) is considered; in the case of 3 only the last (i.e. 3rd) considered, and so on.

So we could say that the true holistic meaning of the ordinal relates to the other roots in each case
(≠ 1). And in general, through obtaining the n roots of 1 (strictly 1¹ , 1², 1³,..... 1ⁿ ), we can obtain fascinating indirect quantitative expressions for all possible ordinal numbers, 1st to (n – 1)th respectively, (within the range defined by the given cardinal number, n.