Essays On Education And Kindred Subjects (Fiscle Part- 11), Herbert Spencer [historical books to read .txt] 📗

Then, To Find The   One _Rational_ Order, Amongst

A Host Of    Possible Systems." ... "This Order Is Determined By The   Degree

Of Simplicity, Or, What Comes To The   Same Thing, Of    Generality Of    Their

Phenomena." And The   Arrangement He Deduces Runs Thus: _Mathematics_,

_Astronomy_, _Physics_, _Chemistry_, _Physiology_, _Social Physics_.

This He Asserts To Be "The True _Filiation_ Of    The   Sciences." He Asserts

Further, That The   Principle Of    Progression From A Greater To A Less

Degree Of    Generality, "Which Gives This Order To The   Whole Body Of

Science, Arranges The   Parts Of    Each Science." And, Finally, He Asserts

That The   Gradations Thus Established _À Priori_ Among The   Sciences, And

The Parts Of    Each Science, "Is In Essential Conformity With The   Order

Which Has Spontaneously Taken Place Among The   Branches Of    Natural

Philosophy;" Or, In Other Words--Corresponds With The   Order Of    Historic

Development.

 

 

 

Let Us Compare These Assertions With The   Facts. That There May Be

Perfect Fairness, Let Us Make No Choice, But Take As The   Field For Our

Comparison, The   Succeeding Section Treating Of    The   First

Science--Mathematics; And Let Us Use None But M. Comte's Own Facts, And

His Own Admissions. Confining Ourselves To This One Science, Of    Course

Our Comparisons Must Be Between Its Several Parts. M. Comte Says, That

The Parts Of    Each Science Must Be Arranged In The   Order Of    Their

Decreasing Generality; And That This Order Of    Decreasing Generality

Agrees With The   Order Of    Historical Development. Our Inquiry Must Be,

Then, Whether The   History Of    Mathematics Confirms This Statement.

 

 

 

Carrying Out His Principle, M. Comte Divides Mathematics Into "Abstract

Mathematics, Or The   Calculus (Taking The   Word In Its Most Extended

Sense) And Concrete Mathematics, Which Is Composed Of    General Geometry

And Of    Rational Mechanics." The   Subject-Matter Of    The   First Of    These Is

_Number_; The   Subject-Matter Of    The   Second Includes _Space_, _Time_,

_Motion_, _Force_. The   One Possesses The   Highest Possible Degree Of

Generality; For All Things Whatever Admit Of    Enumeration. The   Others Are

Less General; Seeing That There Are Endless Phenomena That Are Not

Cognisable Either By General Geometry Or Rational Mechanics. In

Conformity With The   Alleged Law, Therefore, The   Evolution Of    The

Calculus Must Throughout Have Preceded The   Evolution Of    The   Concrete

Sub-Sciences. Now Somewhat Awkwardly For Him, The   First Remark M. Comte

Makes Bearing Upon This Point Is, That "From An Historical Point Of

View, Mathematical Analysis _Appears To Have Risen Out Of_ The

Contemplation Of    Geometrical And Mechanical Facts." True, He Goes On To

Say That, "It Is Not The   Less Independent Of    These Sciences Logically

Speaking;" For That "Analytical Ideas Are, Above All Others, Universal,

Abstract, And Simple; And Geometrical Conceptions Are Necessarily

Founded On Them."

 

 

 

We Will Not Take Advantage Of    This Last Passage To Charge M. Comte With

Teaching, After The   Fashion Of    Hegel, That There Can Be Thought Without

Things Thought Of. We Are Content Simply To Compare The   Two Assertions,

That Analysis Arose Out Of    The   Contemplation Of    Geometrical And

Mechanical Facts, And That Geometrical Conceptions Are Founded Upon

Analytical Ones. Literally Interpreted They Exactly Cancel Each Other.

Interpreted, However, In A Liberal Sense, They Imply, What We Believe To

Be Demonstrable, That The   Two Had _A Simultaneous Origin_. The   Passage

Is Either Nonsense, Or It Is An Admission That Abstract And Concrete

Mathematics Are Coeval. Thus, At The   Very First Step, The   Alleged

Congruity Between The   Order Of    Generality And The   Order Of    Evolution

Does Not Hold Good.

 

 

 

But May It Not Be That Though Abstract And Concrete Mathematics Took

Their Rise At The   Same Time, The   One Afterwards Developed More Rapidly

Than The   Other; And Has Ever Since Remained In Advance Of    It? No: And

Again We Call M. Comte Himself As Witness. Fortunately For His Argument

He Has Said Nothing Respecting The   Early Stages Of    The   Concrete And

Abstract Divisions After Their Divergence From A Common Root; Otherwise

The Advent Of    Algebra Long After The   Greek Geometry Had Reached A High

Development, Would Have Been An Inconvenient Fact For Him To Deal With.

But Passing Over This, And Limiting Ourselves To His Own Statements, We

Find, At The   Opening Of    The   Next Chapter, The   Admission, That "The

Historical Development Of    The   Abstract Portion Of    Mathematical Science

Has, Since The   Time Of    Descartes, Been For The   Most Part _Determined_ By

That Of    The   Concrete." Further On We Read Respecting Algebraic Functions

That "Most Functions Were Concrete In Their Origin--Even Those Which Are

At Present The   Most Purely Abstract; And The   Ancients Discovered Only

Through Geometrical Definitions Elementary Algebraic Properties Of

Functions To Which A Numerical Value Was Not Attached Till Long

Afterwards, Rendering Abstract To Us What Was Concrete To The   Old

Geometers." How Do These Statements Tally With His Doctrine? Again,

Having Divided The   Calculus Into Algebraic And Arithmetical, M. Comte

Admits, As Perforce He Must, That The   Algebraic Is More General Than The

Arithmetical; Yet He Will Not Say That Algebra Preceded Arithmetic In

Point Of    Time. And Again, Having Divided The   Calculus Of    Functions Into

The Calculus Of    Direct Functions (Common Algebra) And The   Calculus Of

Indirect Functions (Transcendental Analysis), He Is Obliged To Speak Of

This Last As Possessing A Higher Generality Than The   First; Yet It Is

Far More Modern. Indeed, By Implication, M. Comte Himself Confesses This

Incongruity; For He Says:--"It Might Seem That The   Transcendental

Analysis Ought To Be Studied Before The   Ordinary, As It Provides The

Equations Which The   Other Has To Resolve; But Though The   Transcendental

_Is Logically Independent Of    The   Ordinary_, It Is Best To Follow The

Usual Method Of    Study, Taking The   Ordinary First." In All These Cases,

Then, As Well As At The   Close Of    The   Section Where He Predicts That

Mathematicians Will In Time "Create Procedures Of    _A Wider Generality_",

M. Comte Makes Admissions That Are Diametrically Opposed To The   Alleged

Law.

 

 

 

In The   Succeeding Chapters Treating Of    The   Concrete Department Of

Mathematics, We Find Similar Contradictions M. Comte Himself Names The

Geometry Of    The   Ancients _Special_ Geometry, And That Of    Moderns The

_General_ Geometry. He Admits That While "The Ancients Studied Geometry

With Reference To The   Bodies Under Notice, Or Specially; The   Moderns

Study It With Reference To The   _Phenomena_ To Be Considered, Or

Generally." He Admits That While "The Ancients Extracted All They Could

Out Of    One Line Or Surface Before Passing To Another," "The Moderns,

Since Descartes, Employ Themselves On Questions Which Relate To Any

Figure Whatever." These Facts Are The   Reverse Of    What, According To His

Part 2 Chapter 3 (On The Genesis Of Science) Pg 103

Theory, They Should Be. So, Too, In Mechanics. Before Dividing It Into

Statics And Dynamics, M. Comte Treats Of    The   Three Laws Of    _Motion_, And

Is Obliged To Do So; For Statics, The   More _General_ Of    The   Two

Divisions, Though It Does Not Involve Motion, Is Impossible As A Science

Until The   Laws Of    Motion Are Ascertained. Yet The   Laws Of    Motion Pertain

To Dynamics, The   More _Special_ Of    The   Divisions. Further On He Points

Out That After Archimedes, Who Discovered The   Law Of    Equilibrium Of    The

Lever, Statics Made No Progress Until The   Establishment Of    Dynamics

Enabled Us To Seek "The Conditions Of    Equilibrium Through The   Laws Of

The Composition Of    Forces." And He Adds--"At This Day _This Is The

Method Universally Employed_. At The   First Glance It Does Not Appear The

Most Rational--Dynamics Being More Complicated Than Statics, And

Precedence Being Natural To The   Simpler. It Would, In Fact, Be More

Philosophical To Refer Dynamics To Statics, As Has Since Been Done."

Sundry Discoveries Are Afterwards Detailed, Showing How Completely The

Development Of    Statics Has Been Achieved By Considering Its Problems

Dynamically; And Before The   Close Of    The   Section M. Comte Remarks That

"Before Hydrostatics Could Be Comprehended Under Statics, It Was

Necessary That The   Abstract Theory Of    Equilibrium Should Be Made So

General As To Apply Directly To Fluids As Well As Solids. This Was

Accomplished When Lagrange Supplied, As The   Basis Of    The   Whole Of

Rational Mechanics, The   Single Principle Of    Virtual Velocities." In

Which Statement We Have Two Facts Directly At Variance: With M. Comte's

Doctrine; First, That The   Simpler Science, Statics, Reached Its Present

Development Only By The   Aid Of    The   Principle Of    Virtual Velocities,

Which Belongs To The   More Complex Science, Dynamics; And That This

"Single Principle" Underlying All Rational Mechanics--This _Most

General Form_ Which Includes Alike The   Relations Of    Statical,

Hydro-Statical, And Dynamical Forces--Was Reached So Late As The   Time Of

Lagrange.

 

 

 

Thus It Is _Not_ True That The   Historical Succession Of    The   Divisions Of

Mathematics Has Corresponded With The   Order Of    Decreasing Generality. It

Is _Not_ True That Abstract Mathematics Was Evolved Antecedently To,

And Independently Of    Concrete Mathematics. It Is _Not_ True That Of    The

Subdivisions Of    Abstract Mathematics, The   More General Came Before The

More Special. And It Is _Not_ True That Concrete Mathematics, In Either

Of Its Two Sections, Began With The   Most Abstract And Advanced To The

Less Abstract Truths.

 

 

 

It May Be Well To Mention, Parenthetically, That In Defending His

Alleged Law Of    Progression From The   General To The   Special, M. Comte

Somewhere Comments Upon The   Two Meanings Of    The   Word _General_, And The

Resulting Liability To Confusion. Without Now Discussing Whether The

Asserted Distinction Can Be Maintained In Other Cases, It Is Manifest

That It Does Not Exist Here. In Sundry Of    The   Instances Above Quoted,

The Endeavours Made By M. Comte Himself To Disguise, Or To Explain Away,

The Precedence Of    The   Special Over The   General, Clearly Indicate That

The Generality Spoken Of    Is Of    The   Kind Meant By His Formula. And It

Needs But A Brief Consideration Of    The   Matter To Show That, Even Did He

Attempt It, He Could Not Distinguish This Generality, Which, As Above

Proved, Frequently Comes Last, From The   Generality Which He Says Always

Comes First. For What Is The   Nature Of    That Mental Process By Which

Objects, Dimensions, Weights, Times, And The   Rest, Are Found Capable Of

Having Their Relations Expressed Numerically? It Is The   Formation Of

Certain Abstract Conceptions Of    Unity, Duality And Multiplicity, Which

Are Applicable To All Things Alike. It Is The   Invention Of    General

Symbols Serving To Express The   Numerical Relations Of    Entities, Whatever

Be Their Special Characters. And What Is The   Nature Of    The   Mental

Process By Which Numbers Are Found Capable Of    Having Their Relations

Expressed Algebraically? It Is Just The   Same. It Is The   Formation Of

Certain Abstract Conceptions Of    Numerical Functions Which Are The   Same

Whatever Be The   Magnitudes Of    The   Numbers. It Is The   Invention Of

General Symbols Serving To Express The   Relations Between Numbers, As

Numbers Express The   Relations Between Things. And Transcendental

Analysis Stands To Algebra In The   Same Position That Algebra Stands In

To Arithmetic.

 

 

 

To Briefly Illustrate Their Respective Powers--Arithmetic Can Express In

One Formula The   Value Of    A _Particular_ Tangent To A _Particular_ Curve;

Algebra Can Express In One Formula The   Values Of    _All_ Tangents To A

_Particular_ Curve; Transcendental Analysis Can Express In One Formula

The Values Of    _All_ Tangents To _All_ Curves. Just As Arithmetic Deals

With The   Common Properties Of    Lines, Areas, Bulks, Forces, Periods; So

Does Algebra Deal With The   Common Properties Of    The   Numbers Which

Arithmetic Presents; So Does Transcendental Analysis Deal With The

Common Properties Of    The   Equations Exhibited By Algebra. Thus, The

Generality Of    The   Higher Branches Of    The   Calculus, When Compared With

The Lower, Is The   Same Kind Of    Generality As That Of    The   Lower Branches

When Compared With Geometry Or Mechanics. And On Examination It Will Be

Found That The   Like Relation Exists In The   Various Other Cases Above

Given.

 

 

 

Having Shown That M. Comte's Alleged Law Of    Progression Does Not Hold

Among The   Several Parts Of    The   Same Science, Let Us See How It Agrees

With The   Facts When Applied To Separate Sciences. "Astronomy," Says M.

Comte, At The   Opening Of    Book Iii., "Was A Positive Science, In Its

Geometrical Aspect, From The   Earliest Days Of    The   School Of    Alexandria;

But Physics, Which We Are Now To Consider, Had No Positive Character At

All Till Galileo Made His Great Discoveries On The   Fall Of    Heavy

Bodies." On This, Our Comment Is Simply That It Is A Misrepresentation

Based Upon An Arbitrary Misuse Of    Words--A Mere Verbal Artifice. By

Choosing To Exclude From Terrestrial Physics Those Laws Of    Magnitude,

Motion, And Position, Which He Includes In Celestial Physics, M. Comte

Makes It Appear That The   One