Copyright © 2013Avello Publishing Journal

ISSN: 2049 - 498X

Issue 1 Volume 3: Principia Mathematica

Philosopiae Naturalis Principia Mathematica

Jason WakefieldUniversity of Cambridge

The analytic way to understand Sir Isaac Newton's natural philosophy through the useful werzeuge

(tools) of sprachspiele (language games) is to decipher the original Latin in his personal copies of

the Principia with Ludwig Wittgenstein's rare notebooks (in German) and hand-written letters (in

English) archived in the Wren Library, Trinity College and the University Library's Munby Rare

Books Room, Cambridge. This requires walking across the black and white tiles of the Wren

Library; past the book stacks arranged in rows perpendicular to the walls with limewood carvings

by Gibbons; past the marble bust standing on a plinth of Newton carved by Roubiliac; past the

display case containing Philosopiae Naturalis Principia Mathematica Classmark: NQ. 16. 200; past

the large statue of Byron sculpted by Thorvaldsen, to a desk at the south – end of the library next to

the light of a large stained - glass window showing an allegorical scene of Newton being presented

to King George III.1

Before my comparison of three of Newton's personal copies of Philosopiae Naturalis

Principia Mathematica (Classmarks: C.17.4 / NQ.17.34 / NQ. 16. 200) in the Wren Library; Ed

Potten, the Head of Rare Books at Cambridge University Library, gave me the rare honour of

inspecting the rarest treasure out of Newton's unique, personal copies of the Principia in the

Montaigne Office of the Munby Rare Books Room. There are four layers of libraries in the

University of Cambridge archive system and Ed Potten's Montaigne Office is at the nucleus of the

U.L building at the centre of our 114 libraries. The Montaigne Office's Philosopiae Naturalis

1 Christopher Wren designed the library in 1676. This window was designed much later by Giovanni Cipriani in 1771.

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Principia Mathematica (Classmark: Adv.b.39.1) is interleaved with blank pages so that Newton

had sufficient space for his hand – written annotations and corrections for his second edition. My

close examination of this key, fragile gem revealed marginilia by Newton's hand that included

crossings out, insertions, amendments, errata, omissions and equation alterations. Newton's hand –

written annotations on his personal copy of the first edition of the Principia can be used as

werzeuge (tools) to collapse the analytic / continental divide between the philosopher Ludwig

Wittgenstein and his contemporary, the etymologist Martin Heidegger; by refining how each thinker

clearly emphasise or sprach (signify) sorgfalt (carefulness). This should clarify my methodological

approach to analysing Newton's hand-written notes via the influence of Leibniz's correspondence

with Mencke and Oldenburg from July 1674 to July 1684.

Before Newton annotated this personal copy of his Philosopiae Naturalis Principia

Mathematica with new calculations, notes, revisions and answers to critics, fragments of the

Principia's where exchanged by Newton with others working on the laws of motion in letters to

Leibniz (through the intermediacy of Oldenburg), Flamsteed and Halley. My original contribution

to this over saturated area of mathematical scholarship, is a fresh reading of the Principia through

analysing Newton's marginal revisions as grammatical games of successful semantics where

redefinitions and repropositions involve necessitism, and its negation contingentism. Thus the vis

inertiae, or force of inactivity, in original Principia scholarship is forced out of its present state with

pressure from not only Wittgenstein but also from his colleague Ramsey's The Foundations of

Mathematics (1925)2 response to Russell & Whitehead's three - volume Principia Mathematica

(1910-13). The centripetal force of Newtonian Principia scholarship towards analysis through the

magnetism of Wittgenstein's rare notebooks may seem clearly far too obvious due to the original

manuscripts close proximity at the University of Cambridge. Heidegger's 1925-50 Nachlass

(written in German) slings Newton's personal copy of his Principia in to a different orbit; this is not

in to a continental orbit clearly defined as a contrary force in opposition to an analytical orbit; but

2 See Ramsey's original publication in Proceedings of the London Mathematical Society, ser. 2, 25, p. 1-36.

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another natural rectilinear way to analyse Newton's hand-written annotations or corrections for his

second edition.

Newton may have chosen to publish the first edition of his Principia in Latin because infinite

series, speculation about the ratios which exist between straight and geometrical curved lines, as

well as, parabola was in heavy circulation in the Latin edition of Geometria (1649) written by

Descartes. It is clear that Newton was not satisfied with the mathematics of the Geometria (1649)

nor Kepler's problems in Astronomia Nova Stellae Martis (1609) from his letter to Oldenburg dated

13th June 1676:

In priori Diagrammate primus terminus valoris ipsorum p, q, r in prima columna invenitur dividendo primum terminum Summae proximè superioris per coefficientum secundi termini ejusdem Summae: Et idem terminus eodem ferè modo invenitur in secundo Diagrammate. (Turnbull 2008: 24)

It is clear that Newton is responding to extracts from letters written by Leibniz to Oldenburg which

have been forwarded on to him. Newton reduces fractions to infinite series by division; and radical

quantities by extraction of the roots. The extractions of affected roots, of equations with several

literal terms, resemble in form their extractions in numbers, which Newton devises a method he

represents in a diagram display, where the right – hand column exhibits the results of substituting in

the middle column the values, for example, of y, p, q, r that are shown in the left – hand column.

This letter is critical evidence of the formation of the first edition of the Principia, as these

reductions of equations to infinite series can be used to determine the lengths of curves; the volumes

of solids; centres of gravity; mechanical curves; angular motion; elliptic sectors; the circuits of

ellipses adapted to hyperbola; arithmetical progressions and other numerical coefficients.

Metaphysicians would perhaps analyse this letter via Ramsey to reread Kepler's second law of

planetary motion or the values of solids in a continuation up to infinity by certain numerical series /

infinite equations - instead my focus is on F = ma.

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Figure I. Sir Isaac Newton's Philosopiae Naturalis Principia Mathematica 1686. page 26 (Newton's hand-written corrections). Cambridge University Library. Munby Rare Books Room. U.L Classmark: Adv.b.39.1.

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Newton's corrections here on page 26 are substantial. The full page of corrections is free from both

diagrams and equations, thus can be read using the same textual techniques as many of Newton's

letters to Oldenburg, without having to do things like draw a geometrical curve to pass through a

number of given points. We can analyse this page of corrections or additions by Newton to his own

mathematico – physicum, not by Halley's remarks on calculus but through Wittgenstein's

investigations in to language. Newton's hand-written additions are specifically about the innate

force of matter, vis insita, as a power of resisting, in Definition III and vis inertia, as a force of

inactivity. Wittgenstein scholars may object to a Heideggerian reading of these notes as a deviation

away from the rectilinear course of Trinity, Cambridge principles of mathematical analysis; my

counter-argument or case against this would be that the velocity of this mathematical motion does

not change with a continental deceleration. In the winter of 1925, Heidegger gave a lecture course

on logic as the question of truth called Logik: Die Frage Nach Der Wahrheit, starting with the

linguistics of Aristotle's λόγος; in the summer of 1928 he gave a different lecture course on the

metaphysical foundations of logic,3 where we find his most sustained attempt at elucidating

Leibniz's metaphysical landscape (without digressions); culminating with his lecture course in the

summer of 1934 on logic as the question concerning the essence of language. In all these lectures

Heidegger returns to the propositional λόγος and its other abbreviation or expression λογική. The

reason why Wren Library scholars are taught not to read the first edition of Newton's Principia

through these specific lecture courses by Heidegger, is that they suffer from obtuseness, lacking the

vivid clarity of Whitehead & Russell's Principia Mathematica (1910-3) or Ramsey's The

Foundations of Mathematics (1925). This is incorrect. These lectures have been significantly

ignored by Fellows of both King's and Trinity, Cambridge; in addition to suppressing the analytical

impact of Heidegger's Unterwegs zur Sprach (1959) because of the tensions between Russell and

Heidegger during World War II. This contempt for each others analytical philosophy between

England and Germany can be traced back to the hostile reception of Nietzsche's later literary output

3 Micheal Heim has translated these studies in to English.

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at the University of Cambridge during the British Empire's Boer Wars during 1880-1881 / 1899-

1902. The Peter Gast (Heinrich Kōselitz) – Friedrich Nietzsche correspondence written between

1876-1891 reveals that his books where ignored by Oxbridge, despite a French translation of his

Richard Wagner in Bayreuth being published (from the German) in 1877 and Georg Brandes

lecturing on his other works in Copenhagen during 1888. Russell continued dismissing Nietzsche

during the 20th Century in print, as a man of weak literature, rather than a strong academic in

chapter XXV of The History of Western Philosophy (1945). Russell also has a negative chapter on

Leibniz in this book, prefering to crown Locke as the apostle of philosophical revolutions around

1688. John Locke was writing his Essay Concerning Human Understanding at the same time John

Wallis published an essay (during January 1687) in Philosophical Transactions, six months earlier

than the publication of the Principia. Wallis wrote to Halley asking for an update on Newton's

progress in the formation of his treatise De Motu Corporum, that went on to form the nucleus of the

Principia. Newton and Wallis reached similar conclusions for linear motion under resistance.

Def. VI.Vis centripeta quantitas absoluta est mensura ajusdem major vel minorpro efficacia caufae eam propagantis acentro per regiones in circuitu.

Ut virtus Magnetics major in uno magnete, minor in alio.

Newton corrected by hand the third line of Definition VI to 'thus the magnetic force is greater in

one load-stone and less in another according to their sizes and strength of intensity.' This correction

faces the entire page of corrections as shown earlier in Figure I. Newton clarifies in very brief terms

that the 'absolute quantity of a centripetal force is the measure of the proportional to the efficacy of

the cause that propagates it from the centre, through the spaces round about.' The impact of this

specific proposition on Leibniz, has not been articulated clearly yet. There is no complete collection

of Leibniz's writing; his contributions on natural philosophy remains scattered in various academic

journals, unpublished manuscripts and several thousand letters on the principles of mathematics in

different languages such as Latin, French and German. In Principia studies, Leibniz's letters to

Oldenburg from Paris (written in Latin) during 1676 are perhaps the most analysed; however the

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Leibniz – Des Bossess correspondence written between 1706 – 1716 (also in Latin) have a rare

elegance that we can not ignore. The latter correspondence is available on microfilm from the

Bibliothéque Nationale, Paris, France and Hanover, Germany. During this period Newton had by

now entered his eighth decade. His letter writing became dominated with a dispute with leibniz over

calculus, which Newtonians such as Abbé Conti, Brook Taylor, Des Maizeaux, De Moivre and

Keill turned in to a controversy, demanding that both the draft letters, notes and personal revisions

of Newton as well as Leibniz be closely examined. This is the first example of a public rift between

British and Continental mathematicians. Very few Britons based in Cambridge and London did not

support Newton; whilst virtually all of Leibniz's friends in France and Germany joined in a vendetta

against Newton, claiming that they all acquired calculus solely from Leibniz alone. Despite this,

Abbé Conti was warmed by the Newtonians in London and expressed his support for the English

side in this extract from a letter dated March 1716 to Leibniz:

Monsieur,

J'ai differé jusqu'à cette heure de repondre à votre Lettre, parce que j'ai voulu accompagner ma Ré ponse de celle que M. Newton, vient de faire à l' Apostille que vous avez ajoutée. […] On attribute à ce grand homme bien des choses qu' il n' admet pas, comme il l'a fait voir à ces Messieurs François qui vinrent à Londres l' occasion de la grande Eclipse...

A Londres le de Mars 1716.

Conti searched the archives of the Royal Society for any letters sent to London from Cambridge or

Oxford to resolve the calculus dispute that started in 1713 from Bernoulli's private criticism of the

Principia Book II, Prop. X - whilst Newton was preparing its second edition. Bernoulli eventually

decided to publish his criticism in Acta Erud during February 1713 which sparked Leibniz to

publish Charta Volans on the 29th July 1713. Chamberlayne with Conti attempted several

reconciliations both privately and publicly between Britain and the Continent; however this did not

stop Leibniz from challenging Newton's competence in mathematical philosophy, demanding that

he produce hand-written corrections, in defence of charges of treason against the republic of letters!

Newton responded with Commercium Epistolicum (1712), that proved that he clearly had the

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concepts of calculus fluxions in 1676 at the latest, several years before Leibniz invented differential

calculus. It is clear that several mathematicians requested from Newton his hand-written additions

and changes in his notebooks on the calculus of fluxions. One urges scholars to source Newton's

original manuscripts used to compile the Principia, so that we can check together his revisions;

alternative equations; crossing outs of both typed and hand-written words; corrected typographical

errors; added numerical symbols and other marginalia.

The dispute over Newton's hand-written notes on the calculus of fluxions has not swayed me

in to having a bias towards British mathematicans over Continental mathematicians or vice versa.

Time did not heal this dispute, as even over a century later, during the time Hegel was lecturing on

the history of philosophy in 1825-6, apologetic defences from 1716 where still being rejected by

archivists at the University of Berlin, Germany. Analytical commentary on natural philosophy

written in German did not begin to be taken seriously at King's or Trinity College, Cambridge again

until the arrival of Wittgenstein. He impressed Anscombe, Keynes, Ramsey, Rhees, Russell, von

Wright and Whitehead greatly by writing approximately 20, 000 pages of primarily notebooks,

bound volumes and typescripts. What is interesting is how can we apply Wittgenstein's Germanic,

linguistic pluralism to the Principia's deletions, overwritings, interlinear insertions, marginal

remarks, substitutions, counterpositions, shorthand abbreviations, orthographic errors and slips of

the hand. We need to establish Newton's editors habits of (or not) combining insertions with

deletions to form substitutions between alternative expressions. Newton's revision work needs a

deep Wittgensteinian understanding of the underlying functions of the tools of editing. The

Principia contains several layers of text with intertextual links that we can use in language as a kind

of tool that does different, diverse things in different applications. Anscombe & Rhees editorial

pruning of Wittgenstein's notes in 1953 culminated in the publication of Philosophische

Untersuchungen as Wittgenstein's will of 29th January 1951 desired. He died on the 29th April 1951,

aged 62. 4Anscombe's translation of the Philosophische Untersuchungen from German in to English

4 Ludwig Wittgenstein was decorated a number of times as an officer on the frontline during World War I, from 1914

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in 1953 and the Russell – Einstein manifesto issued on July 9th 1955 (in the midst of the Cold War),

healed the rift slightly between British mathematicans and Continental mathematicians increased by

World War II. Newton's hand-written notes to the first edition of the Principia, when analysed

through Wittgenstein's puzzles in the fields of semantics, logic and mathematics as outlined in the

Philosophische Untersuchungen, can be read analytically, rather than as a magnitude of analysis

diminished in infinitum due to obscure, homologous German.

Russell was very interested in Einstein's Germanic description of Newton's law of universal

gravitation as early as 1905. He also paid close attention to Einstein's later experiments in

mathematics that showed a Newtonian connection as not integrable, from which Einstein deduced

that spacetime is curved and a geometric description of the derivative of the connection proving

how the modified acceleration of bodies in geometry is caused by the presence of mass. In

Newtonian gravity, the source is mass; in Einstein's equations of special relativity, mass includes

energy, momentum densities, pressure, shear and tensors written in abstract index notation. This

German convergence on Newton's law of universal gravitation by Einstein, was taken much more

seriously at the Royal Society of London and the University of Cambridge, then Leibniz's attack.

Einstein extended this expansion and correction of Newtonian gravity in to a parametrized post-

Newtonian (PPN) formalism by using a metric to represent the Newtonian limit whilst treating the

orbiting body as a test particle. The precession of apsides encoded in the non-linearity of Einstein's

equations can be seen to be partially inherited by those concerned with evolutionary equations

involving non-local terms; for example the Surface Quasi – Geostrophic equation (SQG). These

equations support the ontology in my monograph The Question of Non-Being? A Pragmatic

Methodology of Casino Contingency (2013) when discussed as a non-linear maximum principle for

linear non-local operators; however they provide a case against my work in astrophysics in 5D +

when applied to questions of regularity of solutions, long time dynamics and the absence of

anomalous dissipation in 2D SQG.

to 1918.

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We must not digress away from the first edition of the Principia (that Newton corrected by

hand) in to the mathematics of liquid crystals or mathematical modelling for analysis of complex

fluids and active media in evolving domains. Focus shall be directed straight on the Principia

rebound by Douglas Cockerell & Son in Grantchester, January, 1965; microfilmed and classmarked

Adv.b.39.1 by the rare books librarians at the U.L, Cambridge. Samuel Pepys was the President of

the Royal Society when Newton was making changes to his personal copies of the first edition of th

e Principia. Pepys' name is on the Latin title page, underneath a sub – heading showing Newton's

authorial position at Trinity, Cambridge, as Lucasian Professor of Mathematics. The Newton –

Pepys probability problem that arose out of the correspondence between Newton and Pepys

contains accurate gambling advice from Newton to Pepys, however the logical argumentation is

weak. Newton was particularly interested in having personal conversations at Trinity throughout

1686 about Galileo's dynamics and Galileo's mocking of Kepler over laws of universal gravitation.

The simplicity of the pioneering formulas for Newton, as expressed in the Principia, was that a few

natural laws apply to the whole universe.

Newton's representation of the reality of the universe can be juxtaposed with Newton's

corrections and marginalia as performing many different functions. 'I'll call these ingredients non-

representationalism and functional pluralism, respectively.' (Price 2011: 201). This is how Huw

Price, current Bertrand Russell Professor of Philosophy and Fellow of Trinity, Cambridge makes

sense of Wittgenstein's linguistic pluralism in the Philosophische Untersuchungen. Price does not

analyse assertion strictly through the Philosophische Untersuchungen alone, turning to Micheal

Dummet's chapter on 'Assertion' in Frege: Philosophy of Language and Robert Brandom's non-

representationalism. Before bringing the Principia up to our present date, let us re-read Ramsey.

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Figure II. Frank Plumpton Ramsey. Knowledge. September 1929. Original Manuscript. The page number and pencilled alterations were made by R.B Braithwaite after Ramsey's sudden, tragic death in 1930. Archived at the University of Cambridge.

If we read Newton's hand-written notes through the reliable process of analysing Ramsey's hand-

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written notes, then we can turn Newton's mathematical beliefs of logic in to true knowledge. To

clarify, the reliability of the Principia's hand-written alterations can guarantee truth, when grasped

with the truth-conditions of Ramsey's note on knowledge in Figure II, as opposed to an alternative,

unsatisfactory, unreliable, fallacious methodological apparatus. On a macro – level, Newton's

universals of law, can be read through Ramsey's hand-written notes as general axioms of a

deductive system of knowledge; on a micro – level, Ramsey's epistemology intersects with

Newton's semantics, as universals of fact deducible from particular facts. Counter to this

argumentation, are objections by Ramsey himself, in another hand-written note 'General

Propositions and Causality', also dated 1929. This note also features page numbers and pencilled

alterations by R. B Braithwaite. Here Newton's first edition of the Principia can be described as

unrestricted universal generalizations that are not genuine propositions. Ramsey's later view is a

change of mind from a few years earlier in his 'Facts and Propositions'. Elsewhere 'General

Propositions and Causality' also rebuts Ramsey's earlier article 'Universals of Law and of Fact'. Due

to the unstable nature of Ramsey's ideas of infinity that underly Wittgenstein's U – turning point in

1929 on his return to the University of Cambridge, we should clarify these shifts in thought before

using them to analyse the Principia.

Wittgenstein's Tractatus Logico – Philosophicus was first published in German in 1921, then

translated by Ogden with Ramsey's help in to English a year later. The book was developed out of

Wittgenstein's correspondence with Russell, Moore, Keynes and as a reaction to Frege's logic in

language. The Ogden / Ramsey translation has subtle differences in nuance from the Pears /

McGuinness translation, thus Tractatus Logico – Philosophicus is best read in the original German.

The misconception that Continental or German thought is not analytical philosophy is completely

uninformed. The finest examples of Cambridge University analysis, in the form of Wittgenstein and

Ramsey is an unstable product of a partial conversion of Continental philosophy in to British

analytical philosophy. The werkzeuge (tools) of sprachspiele (language games) as found in

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Heidegger's Unterwegs Zur Sprach as a way of reading Newton's hand-written notes to the

Principia is viable; as it does not stretch far beyond the Kripkensteinian conclusions of Saul

Kripke's Wittgenstein on Rules and Private Language (1982). This portmanteau was later rejected

by John McDowell's paper Wittgenstein on Following a Rule5 in an attempt to respond to Simon

Blackburn's Rule – Following and Moral Realism in Wittgenstein: To Follow a Rule. 6 Despite

McDowell's rejection of Kripkensteinian conclusions, the division between analytical, British

philosophy and Continental, German philosophy; is not a case of Ramsey versus Heidegger. The

sorgfalt (carefulness) of Heidegger can be applied to deciphering Newton's hand-writing on his

personal copy of the Principia to collapse the analytic / Continental divide. Germanic philosophers

of language are finer spun than the hands of crude Oxbridge logicians have the inkling of! Indeed

Ramsey might be in agreement with Einstein, that the theory of general judgements existentially is

the clue to the semantic foundations of the Principia's universe.

When comparing three of Newton's personal copies of Philosopiae Naturalis Principia

Mathematica (Classmarks: C.17.4 / NQ.17.34 / NQ. 16. 200) word by word in the Wren Library, it

quickly came to my attention that the difference in meaning derived from the slight variations in the

texts concerned are semantic differences – differences in truth conditions for mathematics; the

application of which, in physics, is a pragmatic difference, in virtue of their different forces. This is

the pragmatic use of the corrections or alterations of the Principia, in Heideggerian terminology, as

languages-and-language-users-in-a-world. The mathematical principles of Newton's natural

philosophy thus appeals to real use. It is interesting to see Newton making a mathematical

statement and withdrawing it later on the same page in his hand-written corrections, this is often

because Newton is drawing a subtle distinction between content-specifying and use-specifying

theories of propositions and lemmas presented geometrically.

Newton's methodology of revisions is through elaborating on and extending on geometric

5 Synthese: March 1984, Volume 58, Issue 3, page 325-363. 6 Wittgenstein: To Follow a Rule (1981) Edited by Leich & Holtzman. Routledge.

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interpretations of previous propositions and lemmas. A variable quantity such as A in the relevant

formula often has a small moment a or o added to it, changing the formula. After evalution, Newton

calls this genitam. Generating formulas like this later influenced A. R Forsyth, Sadlerian Professor

of Pure Mathematics, Cambridge in his Theory of Functions of a Complex Variable (1918). In this

text we see atomic / molecular primary propositions, laws of nature and processes inherited from

Newton by Riemann. Correction of the proof – sheets of Theory of Functions of a Complex Variable

(1918) was executed by Burnside, Professor of Mathematics at the Royal Naval College, Greenwich

and Taylor, Fellow of Trinity College, Cambridge. Substantial, external correction of these sheets

proves the importance of errata correction to lightening the labour of Trinity mathematicians

between 1893 and 1918. The rare (1900) edition of Theory of Functions of a Complex Variable that

once belonged to Charlie Dunbar Broad can be found in the Wren Library.

Now that we have determined the orbit of Newton's hand-written marginalia from Halley's

correspondence to Taylor's corrections of Forsyth's Theory of Functions of a Complex Variable

(1918); we should return to Wittgenstein's science of mechanics in the Tractatus Logico –

Philosophicus. Plochman, Potter and several others have already exhausted the mathematics in

Wittgenstein's Tractatus in several articles from June 1965 onwards in journals such as Philosophia

Mathematica. In the interests of originality, my analysis at this point will be acute and brief.

Newton's main hand-written alterations to the first edition of the Principia are to the second section

of the first book; the seventh section of the second book and the third book. To be precise, the

additions are on the determination of forces; the theory of the resistances of fluids; lunar theory; the

precession of equinoxes and the calculation of the orbits of comets are enlarged with greater

accuracy. Newton's marginal notes in his personal copies of the Principia reveal and illustrate

significant new experiments, this mirrors ideas in Wittgenstein's Tractatus that logic deals with

possibilities as its facts and the possibilities of the state of affairs must be written into the mechanics

of the thing itself. The use of Wittgenstein's Tractatus in analysing Newton's Principia is often

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relegated in importance to its affect on Russell in revising his second edition of Principia

Mathematica (1910) co-authored with Whitehead. This book has interesting formation rules for

substitution of strings for the symbols called variables and the formula:

(Ǝx). φx. = . ~(x) . ~φx

This formula is created by Russell and Whitehead to express clear logic in addition to applications

of symbolism in variables in a formula such as:

⊃x, ⊃y, ⊃x, y

The question of the non-being of the behaviour of fictitious objects has a peculiar notation in the

first edition of Russell & Whitehead's Principia Mathematica:

⊢: x ε ẑ(φz) .≡. (φx)

Gōdel, Ramsey and Wittgenstein7 all argued critically but sympathetically against these equations;

however Russell and Whitehead made very little alterations, instead prefering to add a new

appendix to a second edition that was published in 1927. Wittgenstein as a reader of Newton's

Principia's arithmetical generalization, can be suggested as a thinker deriving meaning from its

proof. This may not be restricted to arithmetical accounts of the Principia's notes but also to its

semantics. This may seem incompatible, thus a clarification is in order. There is a distinctness about

the Tractatus that allows the Principia to be analysed both mathematically and semantically with

success. The grasp of Frege in the Tractatus, allows for us to ascertain the meaning of Newton's

7 Russell mentions Wittgenstein's criticism of the Tractatus and Cantor's proof in the preface to the second edition of his Principia Mathematica.

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marginal sentences, whilst its grasp of Russell allows for us to ascertain Newton's additional

arithemetic. Conversely, Newtonian universal interpretations of gravity contained in the Principia,

may not need the Tractatus, as Carnap's semantics (in Meaning and Necessity) evaluates sentences

as holding / not holding state-descriptions, which Carnap parallels to Leibniz's possible worlds and

Wittgenstein's possible states of affairs.

My approach to analysing Newton's hand-written notes closely word by word in the Wren

Library, Cambridge, can be applied to those wishing to visit the Leibniz - Archive in Hanover,

Germany, to analyse the collection of manuscripts they have archived, which encompasses about

200, 000 sheets – filling approximately twelve shelving units. Some of these manuscripts, like

Newton's are fire – damaged, water – damaged, or yellowed with age over the centuries. Newton

and Leibniz where also similar in that they did disparate and unrelated occupations; Newton

enjoying the hedonism of London life (after the success of the Principia), Leibniz engaging in

theology (despite success in computational calculus). Leibniz's hand-written papers show the use of

alchemical and planetary symbols for algebraic variables that are missing from modern texts such as

Forsyth's Theory of Functions of a Complex Variable. Also we do not use overbars for roots today,

nor the notation he used for the monads of kinetic energy. These outdated monads of kinetic energy

have evolved in to what Ernest Rutherford later named protons when he split an atom in 1917 in a

nuclear reaction between nitrogen and alpha particles. Rutherford was promoted to Director of the

Cavendish Laboratory, Cambridge as a result of the experiment in 1919. Over the next few decades,

Rutherford's redefinition of Leibniz's monad caught the attention of several theoretical physicists

across the world, in particular those preparing for studies at Massachusetts Institute of Technology

and the University of Princeton. In 1942, Richard Feynman starting drawing diagrams or pictorial

representations of the mathematical behaviour of subatomic particles; which revolutionized nearly

every aspect of theoretical physics. Feynman had personal conversations with Einstein about

formulas and calculations of neutron equations; however his best formulas remain secret, classified,

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restricted data by the government of the United States. The Leibniz / Rutherford influenced formula

for calculating the yield of a fission bomb is called the Bethe – Feynman formula, by the military of

the United States. If a = internal energy per gram, b = growth rate, c = sphere radius then a = (bc) ² f

. We can then add a numerical coefficient to this formula to calculate the yield of a fission bomb.

This formula, in addition to Einstein's revisions we find to the calculations in the Principia, are the

mathematical principles of natural philosophy that defeated the innovations in physics being

developed in Berlin during World War II.

Wittgenstein's student Anscombe introduced the term consequentialism into the language of

analytical, moral philosophy in modernity in 1958. During this time, Anscombe was a Fellow at the

University of Oxford, and was very vocal in debates against atomic warfare, denouncing Truman as

a mass murderer for his use of atomic bombs at Hiroshima and Nagasaki. Anscombe was later

buried at the Ascension Parish burial ground in Cambridge, on the same plot as Wittgenstein. The

Principia and its relation to quantum / relativity theory and classical mechanics is currently being

taught at Trinity, Cambridge by Jeremy Butterfield. His Ph. D was supervised by Hugh Mellor

before he left to teach at All Souls, Oxford; University of Princeton; University of Pittsburgh and

the University of Sydney. During the course of these positions, Butterfield proposed a clarification

of stochastic Einstein locality, a resolution of Einstein's 1913 hole argument and clarified indexicals

with tense in the realm of semantics. Thus this chapter on the Philosopiae Naturalis Principia

Mathematica is not only very cohesive and analytically lucid but is also very relevant to what is

being debated at Oxbridge today.

Brandom, University of Pittsburgh, gave a lecture at UC Berkeley recently about

Nietzschean genealogies and Hegelian hermeneutics of magnanimity. Brandom is a philosopher

very similar to my self, who explicity promotes pragmatic logic, without negating the importance of

post – Davidsonian elements of Hegel's theory of agency. Brandom is an analytical philosopher who

is not afraid to articulate how analytical philosophy has failed cognitive science. My aim is to heal

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the rift between classical German mathematics, American pragmatism and English semantics, to

create an inclusive philosophy. Analytical pragmatism need not be devoid of the Continental

thought of Hegel, Heidegger, Kant or Leibniz. American help for my British quest for a Leibnizian

and Hegelian pragmatic social emancipation has been provided by not only Brandom but also

Dreyfus. Others need to join our pragmatik (pragmatic) enlightenment of pragmatismus

(pragmatisms). Brandom has done a lot of pragmatic work on Hegel and Heidegger, however there

needs to be a lot more analytic work on Leibniz and Nietzsche.

The remarks by our dear brother President Barack Obama at Pariser Platz, Brandenburg

Gate, Berlin, Germany on June 17th 2013 are an apt way to conclude. President Obama, stood next

to Chancellor Merkel, along the fault line where the city of Berlin was once divided, in a free and

united Germany. Obama remarked that Germany was renowned as a land of poets and thinkers,

among them Immanuel Kant, who taught us that freedom is the “unoriginated birthright of man, and it belongs to him by force of his humanity” […] Today, 60 years after they rose up against oppresison we remember the East German heroes of June 17th. When the wall finally came down, it was their dreams that were fulfilled. (Obama 2013: 1)

What our dear brother Obama's speech is telling the people of Germany, is that citizens not

governments can choose whether or not to be defined by a wall, or whether to tear it down. The

same applies to the division between Newtonian analytical philosophy and Leibnizian Continental

philosophy; we can tear down this wall. On a deeper, ethical and moral level, we must uphold our

universal, human rights and reject the nanotechnology (invented by Feynman) that is being misused

by governments to police citizen's private thoughts covertly. Regimes are suppressing our universal,

human rights, free will and suffocating our soul discretely, with nanoscience originally designed by

physicists. The dignity of man and woman kind is inviolable; no matter how sophisticated the nano-

engineering and the statistical dialogue systems motivated by neuronal data – driven frameworks

are. Newtonian laws of motion also function internally. Computational optimisation of nanobots,

including Bayesian models of uncertainty, is the link between internal biophysics and the forced

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speech that is manipulated from input speech components. This is how Kantian imperatives of

universal human dignity and freedom of thought as well as freedom of speech relate directly to the

Principia. Chancellor Merkel reminded President Obama that former President J.F.K once said –

'Ich bin ein Berliner.' Thus we, British mathematicians, should also show solidarity with the key

German, moral philosophers.

With all due respect to President Obama, we should remember the objections from pragmatist

philosopher Cornel West, to President Obama putting his hand on Martin Luther King's Bible, at the

George Washington University in January 2013. The black tradition that produced a man of such

high decency and dignity as Martin Luther King Jr. is offended that Obama used his Bible for just a

brief moment in a Presidential pageantry. The same can be said by serious scholars who are

attempting to redeem the divide between Newtonian mathematics and Kantian mathematics, who

listened to Obama's brief remark about Kant at the Brandenburg Gate, Berlin, Germany. Martin

Luther King Jr, like myself, proposes a challenge to all those in power, no matter what colour they

are. Martin Luther King Jr opposed the physicists involved in the carpet bombing of Vietnam who

where avid readers of Newton and Feynman. Despite the negative uses of Sir Isaac Newton's

Philosopiae Naturalis Principia Mathematica as werzeuge (tools) for atomic destruction; we can

agree with David Hume,8 that his hand-written corrections sprach (signify) a sorgfalt (carefulness)

that England can boast belongs to the greatest and rarest genius that ever arose for the instruction of

our species.

Bibliography

8 David Hume (1754-61) The History of England.(6 volumes) Indianapolis: Liberty Fund.

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Carnap, R. 1947. Meaning and Necessity: A Study in Semantics and Modal Logic. Chicago University Press.

Forysth, A. R. 1918. Theory of Functions of a Complex Variable. Cambridge University Press.

Turnbull, H.W. The 1661-1727 Correspondence of Isaac Newton (1959) Vol I-VII. The Royal Society. Cambridge University Press. 2008.

Heidegger, M. (1925-6) Logik: Die Frage Nach Der Wahrheit (Logic: The Question of Truth) tr. Sheehan, T. Indiana University Press. 2010.

Heidegger, M. 1959. Unterwegs Zur Sprach (On The Way To Language). tr. Hertz, P. D. San Francisco: Harper.

McDowell, J. 1984. Wittgenstein On Following A Rule. Synthese, Volume 58, Issue 3.

Newton, I. 1686. Philosopiae Naturalis Principia Mathematica 1686. (Newton's personal copy). Munby Rare Books Room. Cambridge University Library. U.L Classmark: Adv.b.39.1.

Newton, I. 1686. Philosopiae Naturalis Principia Mathematica (Classmarks: C.17.4 / NQ.17.34 / NQ. 16. 200) Wren Library. Cambridge University.

Obama, B. 2013. Remarks by President Obama at the Brandenburg Gate – Berlin, Germany. Washington: The White House. Office of the Press Secretary.

Price, H. 2011. Naturalism Without Mirrors. Oxford University Press.

Ramsey, F. Knowledge. September 1929. Original Manuscript. Archive: Cambridge University.

Russell, B. & Whitehead, A. N. 1910. Principia Mathematica. Cambridge University Press.

Wittgenstein, L. 2000. Nachlass. The Bergen Electronic Edition. Oxford University Press.

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