¿How Did Newton Write his Mathematical Models, without Equations, because lacking a System of Metrical Units?

 

 

 

 

 

 

 

¿How Did Newton Write his Mathematical Models,

lacking

a System of Metrical Units?

 

 

 

 

 

 

 

 

 

 

 

Dear Reader,

 

Based upon our experience of physics texts today, we might be prone to assume that Newton’s «magnum opus», his 1687 «Principia» – short for «Philosophiae Naturalis Principia Mathematica» in Latin, or Mathematical Principles of Natural Philosophy in English translation – is chock full of equations.

 

But it is not.

 

Instead, it is full mainly of geometrically-expressed proportions.

 

¿Why?

 

Newton’s formulations in the «Principia» drew upon the ancient Greek tradition of Euclid, and the somewhat more dynamical-geometrical representations of Archimedes.

 

Although Newton was highly-skilled at algebra – and with algebraic equations – they were only beginning to become scientifically [“philosophically”] accepted in Newton’s time.

 

There was also the practical, technical problem that the expression of physics relationships by algebraic equations requires a system of “physical units”, a metrological system, which hardly existed during Newton’s lifetime.

 

But there is evidence that there were more philosophical, and pedagogical, reasons for Newton’s choice of proportions over equations.

 

Newton apparently felt that algebraic equations tended to obscure the physical meanings that he was pursuing and presenting in the «Principia»; that such, abstract, equations tended to block the visualizations and intuitions that helped to convey the concrete physical realities to which his proofs pertained, and that made his proof steps follow more intuitively.

 

In her book Newton’s Principia: The Central Argument, published in 2003, Dana Densmore carefully describes Newton’s eschewal of equations, presentations via ratios and their proportions, and the delicate differences that this involves.

 

We tend today to use the terms “ratio” and “proportion” interchangeably.  But, as Densmore points out, in the Euclidean/-Archimedean/Newtonian tradition, this is not correct.

 

She notes Euclid’s definition of “ratio”, Definition 3 from Book 5 of his The Elements: “A ratio is a sort of relation in respect of size between two [M.D.: geometric] magnitudes of the same kind.” [Densmore, p. xlvii, emphasis added by M.D.], and explains as follows: “For example, in [a] first ratio, one area may be twice as large as another area; in [a] second ratio, once force may be twice as large as [an]other force.”  [p. xlviii].

 

“We may call this relationship “same ratio” even if the like magnitudes of the second ratio are of a different kind from the like magnitudes of the first.”

 

“Two ratios that are the same constitute a proportion, and it is abbreviated as “::”.  For example, if a:b is the same ratio as c:d, [M.D.: e.g., for lines a, b, c & d], then [M.D.: one can rightly assert] a:b::c:d .” [p. xlviii].

 

This means that a “proportion” is a ‘meta-ratio’ – a ‘ratio of ratios’. 

 

In terms of the Seldonian first dialectical-categorial calculus, then, for the systematic/taxonomic categorial progression of these two categories, with ratios as its «arché»-category, we have –

 

ratios --) ‘ratios of ratios’ = ratios(ratios) = ~ratios = ratios²

 

=

 

(ratios + D(ratios))   =   (ratios + meta-ratios)

 

|-=

 

ratios + proportions, [--) q₁ + q₂ in the generic _NQ arithmetic – 

 

and such that the category named “ratios” is qualitatively different from, and qualitatively unequal to, the category named “proportions”.

 

Densmore notes, regarding Euclid’s definition of the ratios category, that ratios of qualitatively different kinds of magnitudes are scrupulously avoided by Euclid and Newton alike: “Note that the magnitudes must be of the same kind: areas may be in ratio to areas, lines to lines, velocities to velocities, forces to forces.  But Euclid doesn’t compare unlike magnitudes, and Newton follows him… and doesn’t try (for example) to put force in ratio to area.”

 

“Since ratio is a relation in respect of size, how would we know whether some area were larger, smaller, or equal to some force?” [p. xlviii].

 

In fact, in Seldonian terms, a finite force, of whatever magnitude, and a finite area, of whatever magnitude, are of qualitatively different kinds.  

Any finite force is not less than any finite area, and is not equal to any finite area, and is not greater than any finite area. 

 

Such a force, and such an area, are qualitatively unequal to one another.  

The relation that they form is one of non-quantitative, i.e., of qualitative inequality.

 

As we shall see with more specificity below, what makes them qualitatively different is not the quantifier of a force or of an area, e.g., the “2” in “2 Newtons of force”, or the “2” in “2 square kilometers of area”, which can be grasped as unqualified, hence homogeneous “pure quantities”. 

 

It is the metrological unit qualifier that makes such forces and areas heterogeneous – e.g., the “Newton” unit of measurement for a force is qualitatively different from, e.g., the “kilometer squared” unit of measurement for an area.  

 

Densmore continues:   

 

“They cannot be directly compared; they have no relation in respect of size.”

 

Regarding equations, on the contrary, Densmore notes –

 

“When we use algebraic equations, on the other hand, we treat what were ratios as fractions, that is, as quotients of numbers.  Then we are no longer thinking of the numerator and denominator as magnitudes with a kind.”

 

“Newton sometimes makes use of equations in this way; however, he [M.D.: usually] avoids doing so, because when we do that we no longer have a geometric, and thus visual and intuitive, picture of what we are doing.”

 

“The manipulation of numbers [M.D.: i.e., of the “pure”, ‘unqualifed quantifiers’ of the Q or R arithmetics] using algebra is convenient, but it’s a bag of tricks that loses for us the reality of the things behind it, at least for the duration of the transformations.” [p. xlviii].

 

Densmore notes that proportions can be “alternated, such that the antecedent [M.D.: e.g., the a in a:b] is taken in relation to the antecedent, and consequent [M.D.: e.g., the b in a:b] to consequent… .  One must be careful with this: if the second antecedent and consequent [M.D.: e.g., the c:d in a:b::c:d] are [of] a different kind of magnitude from the first [M.D.: e.g., if a:b::c:d is ‘area a: area b:: force c: force d’], alternation will produce illegitimate “ratios” between unlike magnitudes.” [p. xlix].

 

“Newton avoids doing this; when he must alternate unlike ratios he switches to algebraic equations.”

 

“If the ratios are not ratios of magnitudes but of numbers [M.D.: i.e., of metrologically-unqualified “pure quantities”], then we don’t have this problem.  Since all [M.D.: e.g., Q and R] numbers are homogeneous with one another, a proportion of numbers remains a proportion of numbers even after alternation.” [p. xlix].  The result remains

 numerical; remains a “pure quantity”. 

 

Whereas ratios of different kinds of magnitudes result, not in a “pure” quantity, but in a ‘qualo-quantitative’ expression, because the metrological units do not “cancel-out” in such cases.

 

Algebraic equations that are expressed in terms of the standard arithmetics and number-systems typically don’t even show the metrological units, the units of measure, that can lead to the illegitimate inhomogeneities of which Densmore warns us above.  

The metric units are not an explicit part of the standard equational algebra, hence are not visible in the [standard] mathematics.

 

However, beginning with the ‘Mu’, 

or ‘R m_’ 

arithmetics, the seventh of the Seldonian arithmetics for modeling dialectics in the longform version of the categorial progression for the Seldonian arithmetics for modeling dialectics, fully-mathematical expressions for metrological, unit-of-measurement qualifiers, as distinct from, but also as united with, their numerical quantifiers, emerge.  

In that arithmetic, the metrological unit qualifiers, the qualitative units of measure, are no longer invisible, or merely implicit.  They are part of the new mathematics.

 

This arithmetic can be used to make what Dana Densmore is explaining above more visible.

 

If we assign m°u°1 to the “sec.” syncopated metrical 

unit for “seconds”, and m°u°2 to the “gm.” syncopated 

metrical unit for “grams”, and m°u°3 to the “cm.” 

syncopated metrical unit for “centimeters”, then –

 

Areas are metrically qualified by the “cm.²” compound unit, or, in ‘Mu’ arithmetic terms, by the “purely”-qualitative unit ‘meta-numeral’ m°2u°3, whereas forces 

are metrically qualified by the compound unit for mass times acceleration, i.e., m°u°2 + u°3 – 2u°1.  

 

In the Seldonian dialectic arithmetics, whenever two ‘dialectical meta-numerals’ differ in their subscripts – even if their subscripts are “purely quantitative” rather than being ‘qualo-quantitative’ – then those two ‘meta-numerals’, and their corresponding ‘meta-numbers’, differ qualitatively.

 

Clearly then, finite forces, whatever their “pure” quantifiers, are not less than nor equal to nor greater than finite areas, whatever their “pure” quantifiers may be. 

 

Finite forces and finite areas are, on the contrary, qualitatively unequal; heterogeneous.

 

It is not that, in the Seldonian ‘Mu’ arithmetic, forces cannot be divided by areas.  It is that the result of such a division will not produce a “pure” quantity, and may produce a metrical, unit-of-measure qualifier that may not be meaningful or useful, e.g. –

 

m°u°2 + u°3 – 2u°1     ¸    m°2u°3  =   

 

m°u°2 + u°3 – 2u°1      ´   m°–2u°3 =

 

m°u°2 +  – 1u°3 – 2u°1

 

– i.e., equals grams ¸ (centimeters ´ seconds squared).

 

 

The key to the “purely” quantitative meanings resulting via Euclidean/-Newtonian magnitude ratios is their homogeneity; that antecedent and consequent are both of the same unit, the same metrological, unit-of-measure kind, e.g. –

‘area a:area b’, or,

 

(4)m°2u°3:(2)m°2u°3 =    (4/2)m° 2u°3 –  2u°3

 =   (2)m°u0   

=    2,

 

i.e., 4cm.²/2cm.²   =   2, and cm.²/cm.²   

=   1.

 

 

 

 

 

 

 

 

 

 

 

For more information regarding these Seldonian insights, and to read and/or download, free of charge, PDFs and/or JPGs of Foundation books, other texts, and images, please see:

 

www.dialectics.info

  

and

https://independent.academia.edu/KarlSeldon

 

 

 

 

 

 

 

For partially pictographical, ‘poster-ized’ visualizations of many of these Seldonian insights -- specimens of ‘dialectical art’ – as well as dialectically-illustrated books published by the F.E.D. Press, see –

 

https://www.etsy.com/shop/DialecticsMATH

 

 

 

 

 

 

 

 

 

 

 

¡ENJOY!

 

 

 

 

 

 

 

 

 

 

 

Regards,

 

Miguel Detonacciones,

 

Voting Member, Foundation Encyclopedia Dialectica [F.E.D.];

Elected Member, F.E.D. General Council;

Participant, F.E.D. Special Council for Public Liaison;

Officer, F.E.D. Office of Public Liaison.

 

 

 

 

 

 

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