strong. john stuart mill, john herschel, and the probability of causes (article)

8/11/2019 Strong. John Stuart Mill, John Herschel, And the Probability of Causes (Article)

1/12

John Stuart Mill, John Herschel, and the 'Probability of Causes'Author(s): John V. Strong

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2/12

John

Stuart

Mill,

John

Herschel,

and the

'Probability

of

Causes'

John

V.

Strong

Boston

College

1. Introduction

Any discussion of

probability

theory

in

early

nineteenth-century

Bri-

tain

requires a

specially

vigorous

exercise

of the

historical

imagination.

A

century of

subsequent

thought

about

the

foundations

of

the

subject

has

made

both

historians and

philosophers

of

science

only

too

aware

of

howi

elusive

and

polyvalent

a

term

'probability'

is;

indeed,

since

Carnap,

it

has

scarcely

been

possible

to use the

word

without a

qualifying

adjective

or

subscript.

Before

the nineteenth

century,

on

the other

hand,

during

the

'emergence'

of formalized

probabilistic

thinking,

writers from Pascal

and

Fermat

onwards were

keenly

conscious that the

concept

they

used

was

often ill-defined

and

equivocally

employed.

There

is

a

sensation,

therefore,

of

sailing

unexpectedly

into a

patch

of

flat

calm when one turns

to such

early

Victorian

writers

(I

use

the

term in

its

common,

extended sense

to refer

to the

period

from

around

1830

till the

middle of the

century)

as

Lubbock,

Drinkwater-Bethune,

Galloway,

and,

above

all,

De

Morgan.2

For

them,

probability

is

Laplace's

Essai

phil-

osophigue [16]

and

Th4orie

analytique

[17]

with

such

semi-popularized

pre-

sentations

of

Laplace's

ideas

as

Lacroix's

Traite e"lementaire

[15]

and

the

writings of

Quetelet

([23

]; [24

],

English

translation

[25 ]).

Even a

mathematician

of

the

stature of

De

Morgan

devotes

virtually all

his ef-

forts in

this

field to

making

Laplace's

results

accessible

to

English-

speaking

readers.

Given

the

state of

British

mathematics

during

these

years,

this is

not

too

surprising.

Only a

short

while

had

passed

since

the

subject had

been

wakened

from

its

slumbers

by the

labors

of

the

Young

Turks

of the

Cambridge

'Analytical

Society':

John

Herschel,

George

Peacock,

Charles

Babbage and

their

contemporaries.

Much

of

the

best

British

mathematics was

still an

import from

the

Continent

(and

indeed

the

early

efforts

of the

Cambridge

group

had

gone

largely

into

translating

standard

French

treatises).

In

PSA 1978, Volume 1, pp. 31-41

Copyright

(

1978

by

the

Philosophy

of

Science

Association.

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3/12

32

probability

theory (as

in celestial

mechanics),

Laplace

towered

over

his

peers,

and

hence the obvious

task for

the British

innovators

was the

ex-

position

and

dissemination

of

his ideas

and, as their confidence

grew,

the further

development

and

application

of

them.

Both precedent (in

the

form of Laplace's ownworks, as well as those of earlier writers like Con-

dorcet3),

and

acquiescence

in Laplace's

serene

assurance

that

the theory

as a whole

was no

more

than "good

sense

reduced

to calculation,"

encour-

aged such

steps;

after

all, who would

be so perverse

as

to insist

that

his

field

of interest

required no

common

ense?

At this

point

events

took a

turn of

a kind familiar

in the history

of

ideas.

Just as

in the

seventeenth

century

a much-simplified

version

of

the philosophy

of

Bacon's

Novumorganum

was adopted

as 'Baconianism',

and

just

as

doctrines

about

nature and

scientific

method

far

less

nu-

anced than Newton's

own views

became

the 'Newtonianism'

of

the Enlighten-

ment;

so

Laplace's

richly

ambivalent

view of

the

subject matter

of the 'doc

trine of chances' became, in the nineteenth century, a muchmore univocal

'Laplacean'

interpretation

of probability.

"Chance,"

says De Morgan

in an

early

essay, "is

merely

an expression

of our ignorance

of the chain

of

events

which have

led to

any

particular

occurrence."([7],

p.

103).

In

modern parlance,

probabilities

are

rational

degrees

of belief,

and

any

question

of

probability

is

simply

"an inquiry

into

the number

of

ways in

which,

under given

conditions,

an event may

happen

or fail";

when

these

are enumerated

"the proportion

of these two

is that of

the

probabilities

for and against."(ibid.).

The ground

is thus

laid for

a

critic

dissatis-

fied with

the received

interpretation

of Laplace's

ideas,

to

question

whe-

ther Laplace

himself

was

a

'Laplacean'

in this

sense;

and as

we shall

see,

this is in fact what happens.

When

these

criticisms

did develop they

were (as might

be anticipated)

directed against

both the

presuppositions

of

the

Laplacean

doctrine and

its

applications.

That some objections

were based

simply

upon

misconceptions

or bias is clear

from

the fact

that writers

like

De

Morgan

took

pains

on

repeated

occasions

(see

[7], [8],

[9])

to

insist

that the

theory

was nei-

ther

irreligious

nor an inducement

to

gambling

Other criticisms

were

aimed chiefly

at

the

applications

of

the

theory:

for

instance,

whether

in

particular

cases

like

the calculations

of Condorcet

on

judicial

verdicts,

the

mathematics

had

not

run

away

from its

users,

giving

impressively

pre-

cise but totally unrealistic conclusions.4 One line of criticism, though--

that

which

challenged

the

legitimacy

of the so-called

'inverse'

use of

the

probability

calculus

to estimate

which

of

several

causes,

each

capable

of

producing

an observed

effect,

had in fact done

so--bore

much

more

directly

on

the fundamental assumptions

of the

Laplacean

interpretation.

As

I

have

argued

elsewhere

[29]

,

the attacks of

George

Boole (most

notably

in

his

The Laws of

Thought

[4])

and

of

John

Venn

(in

The

Logic

of

Chance

[31])

drastically

undermined

the

whole

project

of

merging

degree-of-belief

ac-

counts

of

probability

with the

method

of

hypothesis,

to

produce

a

probabil-

istic

logic

of

scientific

inference.

Indeed,

after

the

appearance

in

1874

of

that

curiously

anachronistic

work,

Jevons'

The

Principles

of

Science

[13] ,

such

efforts were almost

entirely

abandoned

for

nearly

half

a

cen-

tury, until revived (in a variety of very different guises) by Keynes [14],

Reichenbach [27],

Ramsey [26],

and

Carnap

[63.5



4/12

33

Though a

really

adequate

account

of

the

developments

noted in the

last

paragraph

has

yet to

be

written,

the

major events,

and

the

identities

of

the

principal

contributors,

seem

well

enough

established.6

What

I

should

like

to

argue

here is

that,

even

as

a

crude

sketch,

the

picture

I

have

drawn is flawed, omitting as it does (and as does every discussion ofthe

topic

of which

I am

aware),

one

of the most

interesting

treatments

of

probability, both 'pure'

and

applied,

to

be

found

in

the

nineteenth

cen-

tury.

I

refer

to the

following

root-and-branch

attack on

Laplace, which

begins with this

gloss on the

initial

definition of

probability

found

in

the

Essai

philosophique:

If his

[i.e.,

Laplace's]

unrivaled

command

ver the

means which

mathematics

supply for

calculating

the results

of

given data,

necessarily

implied an

equally

sure judgment

of

what the data

ought

to

be,

I

should

hardly

dare

give utterance to

my

conviction,

that

in

this

opinion he

is

entirely

wrong;

that his

foundation

is

altogether insufficient for the superstructure erected upon it;

and that

there

is

implied, in all

rational

calculation of

the

prob-

abilities of

events, an

essential

condition,

which

is

either over-

looked

in Laplace's

statement, or

so

vaguely

indicated as

neither

to

be suggested

to the

reader, nor

kept in

view by

the writer

him-

self. ([21],

pp.

1140-1141).

The

author of this

passage is John

Stuart

Mill; the

book

is the System

of

Logic. But

these

views

were

likely

not familiar

to

many of Mill's con-

temporaries,

since they appeared

only in the

first

(1843)

edition

of the

Logic,

being

replaced in

all

editions from

1846

onwards with

what

was (or

could at least easily be read as) a muchmore sympathetic appreciation of

the

standard

'Laplacean'

account

of

probability. What

I would

like to do

in

the

remainder

of this

paper, after

describing

briefly

Mill's

position

in the

first edition of

the Logic

(hereafter SLI),

is

suggest some

answers

to the

following

questions: (1)

Why s

Mill

talking

about

probability at

all in

the

Logic? (2) What is

the

source of

his

initial, very

negative

views

on the

received

interpretation?

(3) Whydid

these

views alter

so

markedly in the

interval between

1843

and 1846?

and (4) What

was

his final

position on the

matter?

Finally, I

shall

indicate why

the

whole subject

seems to

me not

only

interesting

in

itself,

but

also

significant

for under-

standing how British

philosophy

in

the

nineteenth

century tried to

come to

grips with the

alluring but

recalcitrant

'doctrine

of chances

2.

Probability

and the

Structure of

the System of

Logic

For those

accustomed

to think

of

the

relationship

between

probability

and

scientific

inference in

twentieth-century

terms, the fact

that Mill

devotes

three

chapters of the

System

of

Logic (in

all

editions) to the sub-

ject

of

probability

may

come as a

surprise.

Unlike,

say,

Jevons,

Mill al-

lows

only

a

minor

and

subsidiary

role

to

hypotheses in science. For

Mill,

scientific

explanation

is,

broadly

speaking,

subsumption

under causal

laws,

with the

latter

interpreted as

invariable sequences

of

phenomena

([21],

pp.

323-369);

the

logic of

hypotheses is a

set

of

procedures for

eliminating

all but one candidate among a set of laws, all of which purport to cover

the

same

phenomena.



5/12

34

Probability

does find

a natural

place in Mill's

discussion,

however.

Central

to Mill's treatment

of causality

(and

hence to the

argument

of

the Logic

as a

whole) is the

doctrine

of the plurality

of causes

([21],

pp. 434-453).

Events are commonly

he effect

of an

intersection

of

causal chains,

of a

'collocation'

of causes.

In order to

find if

there

is in any given case a deeper order underlying such collocations, we re-

quire

means for

deciding whether

observed

regularities

amongphenomena

are in fact mere

coincidences,

or

whether

they betray

the

presence of

hitherto

unsuspected

causes at work.

Now this

is an issue

which

had already been

raised by several

genera-

tions

of writers.

Originally discussed

in connection

with a kind

of ar-

gument

to

design in the

early

eighteenth

century

by John Arbuthnot

(see

[11],

pp.

166ff.), it emerged in

a

full-blown,

secularized

form in a

1735

essay

by Daniel

Bernoulli

on the origin

of the solar

system

([3]; cf.

r30],

pp. 222-224).

The

question,

used

as an

illustration

by

virtually

every writer on probability for the next hundred and fifty years, was this:

given

the fact

that

all the known

planets

revolve in

the same

sense about

the

sun,

and

that

their orbits

all lie more

or

less in a plane,

it seems

sensible to

ask whether

the

observed

order could

have

arisen simply

by

co-

incidence,

rather

than from

some single

cause.

In other

words, prior

to

theorizing

about the

mechanismwhich

might

account for

the

regularity,

one

ought

first to ask

whether

there

need

be any

mechanism.

This question

about

the

'probability

of causes'--in

Mill's

language,

the problem

of

"how to distinguish

coincidences

which are

casual

from

those

which

are the result

of law"

([211,

p. 1148)--is

clearly crucial

to

any account of scientific explanation of the sort Mill proposes. And in

fact Mill

accepts

the

validity

of

the calculation

done

by

Daniel

Bernoulli

(and, following

him in

essentials,

by Laplace).

What Mill

objects

to strong

ly

in

ELa

is the implication

that

somehow such

an assessment of

probabili-

ties

can

go beyond

what is

given

in

experience;

"that

there can be

a

mea-

surement

of probability

where

there

is no

experience."(

[21],

p.

1143).

For Mill,

Laplacean probability

calculations

are valid

so

far,

but

only

so

far,

as their foundations

are

empirical:

"Conclusions

respecting

the

prob-

ability

of

a fact

rest not

upon

a

different,

but

upon

the

very

same

basis,

as conclusions respecting

its

certainty;

namely,

not our

ignorance,

but

our

knowledge:

knowledge

obtained

by experience,

of

the

proportion

between

the cases

in

which

the

fact

occurs,

and

those

in which

it does

not occur.

Every calculation of chances is grounded upon an induction..." ([21], p.

1142; emphasis

added).

It

seems

quite

clear from

this

passage

(and

the

impression

is

further strengthened

when it

is

taken

in

context)

that

Mill

is

arguing

for a

frequency

interpretation

of

probability,

almost

a

quarter

century

before

Venn.

I think

the likeliest

reason

for

this,

and for

Mill's

stand

against

the

near-unanimous

acceptance

of

Laplaceanism

by

his

contempo-

raries,

is

implicit

in what

has

already

been

said

about the

depth

of

his

empiricist

commitments

though

'empiricism'

was

not for him an

attractive

label).

The received

version

of

probability

smacked

too much

of

apriorism,

of

a

device

whereby

truths about

the world could be

spun

out

of the

human

mind without

observation

or

experiment,

and

"by

a

system

of

operations

upon

numbers, our ignorance...coined into a science."(ibid.).8



6/12

35

Nonetheless,

I would like

to

suggest

one source

of

influence on

Mill's

ideas that

may

have

been

of

significance.

The writer to whomMill

seems

to have turned most regularly for

methodological inspiration

is the

Scot-

tish Common

ense philosopher Dugald Stewart.9 Stewart

has

relatively

little to say on probability, but he does devote one long note to the sub-

ject

in

his Philosophy

of the

Active and Moral

Powers of

Man

(1828).

A

large part of this note is

given

over to

quotations from

a

paper

of Pre-

vost

and L'Huillier

(1796)--characterized

by

Stewart

as

"very

able"-- in

which the authors assert that

our

judgments about 'Laplacean'

probabili-

ties, and those about observed

causal regularities among

natural events,

"dependent de deux ordres

de

facultes

differents":

in

particular, that

the latter

rely wholly on

"la

liason des

idees'"lfound

in

our experience

([28], VII,

pp.

116-118). Given

the paucity of criticism of

the founda-

tions of

probability prior to

Mill's time, and the accessibility of this

passage

to Mill while he was

writing

the

Logic,

the

arguments

of Prevost

and

L'Huillier could have

supported,

even if

they did not

inspire,

Mill's

misgivings about the Laplacean view.10

3. Herschel, Mill, and the Second Edition

of

the

System of

Logic (1846)

If the

above account

is even

roughly correct, especially

in its

empha-

sis on the ease

with

which

a

non-Laplacean, frequency

interpretation

of

probability finds a place in Mill's

philosophy of science, it becomes all

the more

surprising

to find him

apparently turning away sharply from

this

interpretation only three years

later, in the 1846 edition of the Logic

(?P2).

"The

theory

of

chances,"

he

writes

there,

"as conceived

by

Laplace

and by

mathematicians generally,

has not the fundamental

fallacy which

I

had ascribed to it."([211, p. 535). And on another point on which he had

taken issue with

Laplace

in

SLI

(the

rules for estimating the credibility

of

witnesses

testifying

to the

occurrence

of

an

improbable

coincidence),

Mill now

writes, "This argument of Laplace's, though I

formerly thought

it

fallacious,

is

irrefragable

in the

case which he supposes..."

([21],

p. 636).

Fortunately

for

the

historian,

it

is

possible to trace

in

great detail

the

stages

in

Mill's

change

of heart.ll

There is abundant

evidence that,

for

a writer

preparing

a

treatise

of

unprecedented scope

on the

'methods

of

scientific

investigation' (as

the

subtitle

of the

Logic

ran),

Mill

knew painfully little about the natural sciences of his day.12 Even after

the

publication of

?L1,

he

was

conscious

that

the

book might

well contain

errors

or

omissions that he

had

been

unable

to detect; and

accordingly,

when Sir

John

Herschel wrote

a

cordial letter of thanks for

a presentation

copy of

the Logic which Mill had

sent him, the latter made bold to ask if

Herschel

would suggest any corrections that seemed

necessary.13

The

choice

of

critic was

a

shrewd

one.

If

Mill knew surprisingly

lit-

tle

science, Herschel had

mastered an

astonishingly large

amount.

(His

on-

ly

rival

would

have been William

Whewell,

with whose

Philosophy

of

the In-

ductive

Sciences

Mill

had

dealt

quite sharply

in the

Logic.)

Even

more im-

pressive than Herschel's accomplishments

was his reputation;

as

one

recent

commentator has put it, for the early Victorians, to be scientific was to

be as

much

like

Herschel

as

possible ([5], p. 219). Though five

years

re-



7/12

36

turned

from

his epochal

survey of the southern

skies

at the Cape of Good

Hope,

Herschel

was still deep

in the monumental

task of preparing his

ob-

servations

for publication; but

he readily

turned his attention

to the

Logic when

Mill began serious

work on a new

edition in 1845. In

December

of that year, Herschel sent a long letter to Mill, detailing his objec-

tions to Mill's strictures

against

Laplace's statement

of the theory

of

probabilities, as presented

in

SL1. "With these objections

I can

no ways

agree,"

he wrote,

"and I

will

not conceal

from you that

I read themwith

great concern and an

earnest wish

that you would give

them a full

recon-

sideration."14 Granted,

says

Herschel, that Laplace

shares the

tendency

of many

mathematicians

sometimes

"to prefer

mathematical calculation

by

rule and formula to

the plain

exercise of common ense."

But this

is no

more than

a lapse on Laplace's

part.

Mill's

criticism misses the

point

that

Laplace

was aware that only

through experience could

the formal

cal-

culus of probabilities--which

is, as Herschel notes,

"simply the

theory of

combinations"--be linked with

the real world.

Probability is, to

be sure,

epistemic, "a matter of opinion & judgment" that certain events are equal-

ly possible

"as regards our limited

knowledge

or conceptions."

But the

basis of all such assessments

is experience, whether

in the form

of the di-

rect

evaluation

of

relative

frequencies,

or of

more generalized

and tacit

use

of observation and experiment.

"How do we know,"

asks Herschel,

"that

in a forest

of

trees

half

of

which are oaks

it

is equally possible

to stum-

ble on an

oak and another

tree?

Has anybody

ever tried it?" And

yet such

a judgment is

in no sense

a

priori:

"A

total absence

of all knowledge

of

the connexion

of

events--or

so to speak,

of

the mechanism

of the

events

is incompatible

with that state

of

mind which leads us

to assert an equal

probability...

.We must

see enough

of

the case

to

get

or make

for

ourselves

an impression (perhaps an erroneous one) that there does exist a similarity

of

circumstances.

In short

an

opinion

or a

judgment

to

be worth the name

must be grounded

on

something

not

merely negative."

The 1846

revision

of the

Logic incorporates

all these

criticisms.

After

summarizing

his

original

view

(with

an

amplified quotation

from

Laplace's

Essai

to

open

the chapter),

Mill

devotes several

pages

to a detailed

recan-

tation

of his

position

in

SLI.

Probability

is

indeed

"probability

to

us,"

that

is,

the

warranted

degree

of

expectation

we have that an event

will

occur

([21],

p. 535).

This

degree

of

expectation

can

be

calculated

by

the

"common

heory"

of

probability

even

in

those

cases in which

we are

unable

to appeal to any "special experience"; under such conditions,

the

"logical

ground

of the

process

[of calculation]

is our

knowledge,

such

knowledge

as

we

then

have,

of the laws

governing

the

frequency

of occurrence

of the

dif-

ferent

cases;

but

in

this

case

our

knowledge

is limited to that

which, being

universal

and

axiomatic,

does

not

require

reference

to

specific

experience,

or to

any

considerations arising

out of the

special

nature

of the

problem

under

discussion."([21],

p. 537).

4.

Conclusion

It

is

clear

that

one

wants

to ask at least

two

questions

about

Mill's

revised

estimate

of the

Laplacean (or perhaps

Laplacean-Herschelian)

ac-

count of probability: is it fairer to Laplace's own views, and is it a

valid

interpretation

of the doctrine

of

chances

in its own

right?

The



8/12


9/12

38

6In America

the project

was going

forward,

in a

quite

different

form,

in

the work

of C.S.

Peirce.

7Mill's

criticisms

in SL1

may

thus antedate

those of Robert

Leslie

El-

lis, whose brief, -difficult papers in the Transactions of the Cambridge

Philosophical

Society

for 1843 and

1856

also

argued for

the

frequency

interpretation.

8Hacking ([10],[11])

has

shown

that Laplace's

views

uneasily

embrace

both degree-of-belief

and

relative-frequency

accounts

of probability,

and that this

duality can

be traced

back

to the writings

of James

Bernoul-

li and Leibniz

on the

subject.

9See

Olson

([22],

esp. pp.

94-124)

for an

extensive

discussion

of

Stew-

art's views

and

of their

influence

on Victorian

philosophy

of science.

1OThe

importance

of this

Continental

tradition

for the

overall

develop-

ment of

methodology

in Britain

has

been stressed

in

a

recent paper

by

Lau-

dan [18 ]

llThis

process

has been

noted

by

J.M. Rob-,on

in his "Textual

Introduction

to

[21];

see

esp. pp.

lxxx-lxxxi.

12Mill

was engagingly

aware

of his

limitations

in

this

regard,

as

his

re-

marks

on the

matter

in

his Autobiography

([19],

pp.

133-134)

testify.

13Although I have concentrated here on what seem to me to be the chief

points

of Herschel's

criticisms

regarding

Mill's views

on

probability,

Herschel

also provided

a

number

of

corrections

on

other

topics;

see

Mill's

letter

to Herschel

of

28

February

1846

([20],

p.

695).

14Herschel

to

Mill,

22 December

1845;

manuscript

(apparently

a fair copy

byr

n

amanuensis)

at

the

Royal

Society,

London.

I am

grateful

for

permis-

sion

to

quote

from

this

letter,

to which the

Society

holds

copyright.



10/12

39

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strong. john stuart mill, john herschel, and the probability of causes (article)

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