The noun and the adjective, one relates to the other and enhances our subjective understanding of the other. However, the one is not the other, and to confuse the one as the other is both felicitous and fallacious. Felicitous in that the whole of the mathematical arithmetic is based on such a confusion. Fallacious in that one cannot learn anything about nouns by studying adjectives and vice versa. What is to be had in learning is by the marrying of the two, and the pertinence of the one to the other in describing yet greater combinations.
Hamilton may not be the first to clearly derive arithmetic from the ground up starting with sequential relations, but he is the first modern to do so in the full description of the vector nature of measurement, bringing into the subjective description what thitherto had been discounted as irrelevant in computation that being orientation or direction.
On the Comparison of any one effective Step with any other, in the way of Ratio, and the
Generation of any one such step from any other, in the way of Multiplication; and on the
Addition, Subtraction, Multiplication, and Division of Algebraic Numbers in general,
considered thus as Ratios or as Multipliers of Steps.
21. The foregoing remarks upon fractions lead naturally to the more general conception of algebraic ratio, as a complex relation of any one effective step to any other, determined by their relative largeness and relative direction ; and to a similarly extended conception of algebraic multiplication, as an act (of thought) which enlarges, or preserves, or diminishes the magnitude, while it preserves or reverses the direction, of any effective step proposed. In conformity with these conceptions, and by analogy to our former notations, if we denote by a and b any two effective steps, of which a may be called the antecedent or the multiplicand, and b the consequent or the product, we may employ the symbol b a to denote the ratio of the consequent b to the antecedent a, or the algebraic number or multiplier by which we are to multiply a as a multiplicand in order to generate b as a product: and if we still employ the mark of multiplication ×, we may now write, in general, b = b × a : (163.) a or, more concisely, b = A × a, if b = A, (164.) a
21. The foregoing remarks upon fractions lead naturally to the more general conception
of algebraic ratio, as a complex relation of any one effective step to any other, determined
by their relative largeness and relative direction ; and to a similarly extended conception of
algebraic multiplication, as an act (of thought) which enlarges, or preserves, or diminishes
the magnitude, while it preserves or reverses the direction, of any effective step proposed.
Hamilton's treatment of arithmetic on the way to his theory of couples is "stand out"! It does not retrace historical development it set fractions and number in general in ts full context: the adjective to magnitudes which possess orientation and sequence. There is a use of number in general as an adverb to actions on these magnitudes, including rotation deformation and lineal translation.
So what do we do with magnitudes? we translate, rotate magnify and deform and multiple and divide in a particular instance that is called adding and subtracting, And we compare.
Now the greeks coined the word "Logos" to describe the very first comparison when the Monad divided. When there was only one there was no comparison . When Monad divides the first thought is "logos!": comparison . From then on it is ratio upon ratio comparison upon comparison. This spreading comparison is called "Kairos", ratio upon ratio, ratio related to ratio: proportion.
Thus ratio and proportion, logos and kairos are all that described reality whether it is static or dynamic. The use of fractions adjectively describes this fractional change, either by growth or diminishing. The change is effected by iterative actions.
Ratios of magnitudes as monads or metrons gives rise throigh iteration to adjectives of counting and measuring, which spin off to develop an astonishing arithmetic, which reveals complex relations and actions in relationships. The act of naming an arrangement of magnitudes is the act of converting from sequenced ordering to naming a stage in a sequence. The development of number therefore is the development of names for various sequence stages. Hamilton in naming a relationship a fraction and giving each fraction a name in a naming scheme creates "a number". But the "numbers" he creates are "numbers" based on adjectives not magnitudes.
Hamilton realises that magnitudes can be ratioed but not called a number. However we can subjectively form a sequence of adjectives and these can be called "numbers". What can we do directly with the "objecive" magnitudes in front of us? Apart from compare we can do the affine transformations, and we can observe them change. At each change a new arithmetic is formed.Thus for each "fraction" of change an arithmetic can be devised. These changing arithmetics enable us to model change by comparing changing fractions of a form. This is the notion of the differential calculus.
The changing forms produce changing ratios. Each ratio is the basis of an arithmetic but the changing magnitudes themselves are an "arithmetic of change". Newton had a vision of these changing magnitudes as an arithmetic and he called them fluxions, magnitudes that change. To think of them as numbers was to do damage to the ideas of "number".
Newton called them fluxions but treated them as "numbers", "numbers that vanished!" They are not numbers, they are ratios of magnitudes, and dynamic magnitudes at that. Objectively they represent forms placed in a relationship for comparison.
Calculus is not therefore based on number but on forms in comparison.
Eudoxus in his theory of proportions set out the method of comparisons which Euclid exposits in book 6
His interest was concentrated on magnitudes. He was familiar with spherical rotation as were many philosopers and there measure o rotation was the motion on a sphere with a reference frame established by Babylonian observational data of star positions which he assigned to relative spheres of rotation. The point however is not usually drawn out that the magnitude of measurement for rotation was understood and utilised. This magnitude was subject to his theory of proportion. The magnitude used is and was arc length and chord comparison with the diameter.
So Eudoxus comparisons were not just about the dimensions of a form!
In fact one must consider any form as apriori to the measuring tools used on it, and the tools must be seen to be suggested and derived from the comparing of the forms directly. Thus if i took a sphere and compared it with a sphere in Athens, almost immediately the comparison of spheres is a comparison of relative position to myself: translation; i may hen consider the relative orientation of the spheres to myself, to each other, to the stars etc:rotation; and i may then also consider the relative magnitudes of the spheres:magnification; finally i may compare whether they are both as spherical as each other:deformation. For each of these comparisons i would need to derive and utilise a metron/monas for which i could then immediately derive a euclidean arithmoi gematria.
The one form that brings all these considerations together is the compass-multivector network. Describing and defining forms in terms of combinations of compass vector networks thus enables me to deal in a complete way with space and any object in it and made of it, and my subjective apprehension of it.
This apprehension is based on direct magnitudes comparison, not on the comparing of numbers themselves.
The invariant relations i seek once i have made these comparisons is what enables me to describe space and spatial transformations. In such a scenario "fractions" mean something about a spatial relationship determined by ratios of magnitudes.