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Joseph W Dauben

AbrahamRobinson

and

Nonstandard

Analysis:

History Philosophy and

Foundations ofMathematics

Mathematics

is the

subject

in

wh ich

we

don'tknow* wha t

we are talking about.

Ber t r and

Russel l

*

Don't care would

be

more

to the

point .

M a r t i n Davis

I never unders tood why logic should be reliable

everywhere

else, but not in mathematics .

A . Heyt ing

1. Infinitesimals

and the

History

of

Mathematics

H istorically, the du al concepts of infinitesimals and infinities have

alwaysbeen at the center of crises and fou nd ations in m athem atics, from

the first " foundat iona l crisis" that some, at least, have associated with

discovery

ofirrational n um bers (mo re properly speaking, incomm ensurable

magnitudes) by thepre-socartic Pythagoreans

1

,to thedebates that arecur-

rentlywag ed between intuitionistsand formalistbetween th edescendants

of Kronecke r

and

B r o u w e r

on the one

ha nd ,

and of

Cantor

and

Hilbert

on the other. Recently,a new crisis hasbeen identifiedby theconstruc-

tivist Erret Bishop:

There

is a

crisis

in

contemporary mathematics ,

and

anybody

who has

This paper

w asfirst

presented

as the second of two Harvard Lectures onRobinson and his

wo rk delivered at Yale Un iversity on 7 M ay 1982. In revised versions, it has been presented

to colleagues at the Boston Colloquim for the Philosophy of Science (27A pril 1982), th e

American Mathematical Society meeting

in

Chicago

(23

M arch 1985),

th e

Conference

on

History

and

Philosophy

of

Modern Mathematics held

at the

Universi ty

of

Minnesota (17-19

May 1985), and, most recently,

at the

Centre National

d e

Recherche Scientifique

in Paris

(4

June 1985)

and the

Depar tment

of

Mathematics

at the

Universi ty

o f

Strassbourg

(7

June

1985).

I am

gra te fu l

for

energetic

a nd

constructive discussions with many colleagues whose

comments

and

suggestions have served

to

develop

and

sharpen

th e

argum ents presented here.

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775

Joseph

W.

Dauben

not

noticed

it is

being willfully blind. The

crisis

is due to our

neglect

of

philosophical issues....

2

Bishop , too, relates

his

crisis

in

part

to the

subject

of the

infinite

and in-

finitesimals.

A rguing that formalists mistakenly concentrateon the truth

ra ther than

the meaning of a

mathematical s ta tement ,

he

criticizes

Abraham Robinson 's nonstandard ana lys is

as

"form al f inesse," ad ding

that "it is

difficult

to

believe

that

debasement

of

meaning could

be

car-

ried

so

far.

3

N ot all

m athematicians, h ow ever,

are

prepared

to

agree that

there is a crisis in modern mathemat ics , or that Robinson 's work con-

sti tutes

any

debasement

of

meaning

at

all.

K u r t Godel , fo r

example, bel ieved that Robinson more than anyone

else had

succeeded

in

bringing mathematics

and

logic together,

and he

praised Ro binso n 's creat ion of nonstand ard analysis for enl is t ing the

techniques

of

mod ern logic

to

provide rigorous foundations

for the

calculus

using

actual infinitesimals.

The new

theory

w as first

given wide public i ty

in

1961 when Robinson out l ined

the

basic idea

of his

nonstandard

analysis

in a

paper presented

a t a

jo int meet ing

of the

Amer ican

Mathemat ica l Society and the Mathematical Association of

Amer i ca .

4

Subse quently, im pressive applications of R obin son 's approach to

inf in-

i tesimals have con firm ed

h is

hopes that non stand ard analysis could

enrich

standard mathematics

in important ways .

As for his

success

ind efining

infinitesimals

in a

rigorously m athematical

wa y, Ro binson saw his w ork not only in the tradi t ion of o thers

like

Leib-

niz

and Cauchy before him, but even as v indicat ing and

just i fy ing

their

views. The

relation

of

the i r work , however ,

to

Robinson 's

ow n

research

is

equally significant,

as

R obinson himself real ized,

and

this

for

reasons

that are of particular interest to the historian of mathemat ics . Before

returning

to the

question

of a

"new" crisis

in

mathem atics

due to

Robin-

son's

work, i t is important to say something,

briefly,

about the history

of

infinitesim als,

a

history that Ro binson took w ith

th e

utm ost seriousness.

This is not the place to rehearse the long his tory of infin itesim als in

mathematics. There is one historical

f igure ,

however, who especially

interested Robinsonnamely,

Ca u c h y a n d

in wh at

follows

Cauchy pro-

vides a focus for considering the historiographic significance of Robin-

son 's

ow n

w o r k .

In

fact , fo l lowing R obinson 's lead, o thers l ike

J. P.

Cleave, Charles Ed w ards , Detlef Laug w itz,and W . A . J . Luxem burg have

used nonstandard analysis

to

rehabili tate

o r

vindicate earlier

infin-

i tesimalists.

5

Leibn iz ,

Euler , and

Cauchy

are

among

the

more

p romi-

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ABRAHAM ROBINSON AND NONSTANDARD A N A L Y S I S

179

nent mathematicians

w ho

have been rationa lly

reconstructed evento the

point

of

having had,

in the

views

of

some

com m entators,

Robinsonian

nonstandard infinitesimals

in

mind

from

the

beginning.

T he

most detailed

and

methodically sophisticated

of

such treatments

todateis

that provided

by

Imre Lakatos ;

in

what fol lows,

it is his

analysis

of

Cauchy that

is

emphasized.

2. Lakatos , Robinson, and Nonstandard Interpreta t ions of

Cauchy ' s

Infinitesimal Calculus

In

1966, Lakatos read

a

paper

that

prov oked considerable discussion

at the International Logic Colloquium meeting that year in Hannover.

Thepr imaryaim ofL akatos 'spaper w asmade clear in itstitle: Cauchy

and the C on t i nuum: The Significanceof Non-standard Analysis for the

History

and

Philosophy

ofMathematics.

6

Lakatos

acknowledged

his ex-

changes

w i th Robinson

on the

subject

of

nonstandard analysis

as he

con-

tinued to revisethew orking draf t of hispaper. Although Lakatos never

published the

article,

it

enjoyed

a

rath er w ide private circulation

and

even-

tually

appeared

after

Lakatos's death in volume 2 of his

Mathematics,

Science

and Epistemology.

Lakatos realized that tw o important thingshad happened withthe ap-

pearan ce of Ro bins on 's new theor y, indebted as i t was to the results and

techniques of m od ern m athem atical logic. He took i t above all as a sign

that metamathematics

w as

turning away from

its

original philosophical

beginnings

and w as grow ing into an im portant b ranch of mathematics.

7

This

view,

now

more than twenty years later, seems

fully

jus t i f ied .

The second claim that L akatos m ade, ho w eve r, is that non standa rd

analysis

revolutionizes

the

historian's picture

of the

history

of the

calculus.

The

g rounds

for

this assertion

are

less

c lear and in

fact ,

are

subject

to

question. Lakatos explained

his

interpre tation

of

Robinson's achievement

as

follows

at the

beginning

of his

paper:

Robinson's work.. .offers

a rational

reconstruction

of the

discredited

infinitesimal

theory w hich satisfies m od ern requ irements

o f

rigour

and

which

is no

weaker than W eierstrass's theory. This reconstruction makes

infinitesimal

theory an almost respectable ancestor of a fully-fledged,

powerfu l modern theory ,

lifts it

from

the

status

of

pre-scientific gib-

berish and

renew s interest

in its

partly fo rgo tten, partlyfalsified

history.

8

But consider the word

almost.

Robinson, says Lakatos, only makes

the achievements of earl ier infinitesimalists

almost

respectable. In fact ,

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180 Joseph

W .

Dauben

Ro binson's w ork in the twentieth century cannot vindicate Leibniz's w ork

in

the seventeenth century, Euler 's in the eighteenth century , or Cauchy 's

in thenineteenth century . There isnothingin the language orthought of

Leibniz, Euler,

or

Cauchy

(to

whom Lakatos devotes most

of his

atten-

tion) that would make them early Robinsonians. The

difficulties

of

Lakatos's rational reconstruction, however, are clearer in some of the

details

he offers.

For

example, consider

Lakatos 's

interpretation

of the

famous theorem

from Cau chy's Cours d 'analyse of 1821 , which purports to prove that the

limit of a sequence of continuous functions

s

n x) is con tinuous. T his is

what Lakatos,

in the

spirit

of

Robinson's

ow n

reading

of

Cauchy,

has

tosay:

In fact

Cauchy's theorem was true and his proof as correct as an

infor-

m alproof can be.

Following Robinson

. . .

Cauchy 's argum ent,

if not

interpreted

as a

proto-Weierstrassian argument

bu t as a

gen uine Leibniz-

Cauchy one,runs

as fo l lows: . . .

s

n

x )

should

be

defined

and

continuous

and

converge

n ot

only

at

stan-

dard Weierstrassian points

but atevery

point

of the denser

Cauchy

con t inuum,and. . .thesequences

n

(x ) shouldbedefinedfor

infinitely

large indices

n and

represent continuous functions

at

such

indices.

9

In one last sentence, this is all summarized in startling terms as follows:

Cauchy made absolutely

no

m istake,

he

only proved

a

completely dif-

ferent theorem , abo ut tran sfinite sequencesof

functions which

Cauchy-

converge on the

Leibniz

cont inuum.

1 0

But upon reading Cauchy's Cours d'analyseor either of his later

presentations of the theorem in his

Resumesanalytiques

of

1833

or in the

Comptes Rendues fo r 1853one

finds

no

hint

of transfinite

indices,

se-

quences,

or

Leibnizian continua made

denser

than standard intervals

byth eadd itionof infinitesimals.

C auchy, when referring

to

infinitely large

numbers

n ' > n,has

"very

l a rge"but f in i tenumbers in

mind ,

notac-

tually infinite C antorian-type transfinite num bers.11

This is unmistakably clear

from

another work Cauchy published in

1833

Sept lecons

dephysique generategivenat

Turin

in the

same year

he

again published the continuous sum theorem. In the

Sept lecons,

however,

Cau chy explicitly denies

the

existence

of

infinitely large numbers

for their allegedly contradictory

properties.

1 2

Moreover ,

if

Lakatos

w as

m istaken about C auchy 's position concern-

ing

the actually

infinite,

he was also wrong about Cauchy's continuum

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A B R A H A M R O B I N S O N A N D N O N S T A N D A R D A N A L Y S I S

1 81

being

one of

Leibnizian infinitesimals.

If, by

virtue

of

such infinitesimals,

Cauchy's original proof had been correct all along, w hywould he then

have issued

a

revisedversion

in 1853,

explicitly

to

improve

upon

the earlier

proofs? Instead, were Lakatos and Robinson correct in their rat ional

reconstructions, a ll Cauchy would need to have done w aspoint out the

nonstand ard meaning of his

infinitesimals explaining

how infinitely large

and

infinitely

small numbershad givenhim a correct theorem, aswellas

a p roof , all along.

Lak atos also draw s some rather remarkab le conclusions abo ut w hythe

Leibnizian version

of

nonstandard analysis failed:

The

downfa l l

of

Leibnizian theory

w as

then

not due to the

fact that

it wasinconsistent, but that it wascapable only of limited growth. It

was the heuristic potential of growthand explanatory powerof

W eierstrass's theo ry that brou ght about the dow nfall of infinitesimals.

1 3

This

rational reco nstruction m ay complement the overall

view

Lakatos

takes of the importance of research programs in the history of science,

but it

does

no

justice

to

Leibniz

or to the

subsequent history

of the

calculus

in

the eighteenth and early nineteenth centuries, which (contrary to

La ka tos) dem onstrates th at (i) in the eighteenth cen tury the (basically Leib-

nizian)

calculus constituted

a

theory

of

considerable power

in the

hands

ofth e

Bernoullis,

Euler, and

m any others;

and

(ii)

the

real stumbling block

to inf ini tes imalsw as their acknowledged inconsistency.

The first pointiseasily established byvirtueof the remarkable achieve-

mentsof eighteenth-century m athem aticians who used the calculus because

itw aspowerfu l it produ ced str iking resultsand wasindispensable in ap-

plications.

1 4

But it was also suspect

from

the beginning, and precisely

because of the question of the contradictory nature of infinitesimals.

This bringsus to th e second point: despite Lakatos 's dismissalof their

inconsistency, infinitesim als were perceived even

by

Newton

and

Leibniz,

and

certainly

by

their successors

in the

eighteenth century,

as

problem atic

preciselybecause of their contradictory qualities. Newton was specifical-

ly

concerned with

th e

fact

that

infinitesimals

did not

obey

th e

Archime-

dean axiom

and

therefore could

not be

accepted

as

par t

of

r igorous

mathemat ics .

1 5

L eibniz was similarly concerned abou t the logical accept-

ability of inf ini tes imals .The first public presentation of his differential

calculusin 1684was severely de term ined by his attemp t toavoid the logical

difficulties

connected with

the

infinitely small.

H is

article

in the

Acta

Eruditorum

on m axima and m inima, for example, presented the differen-

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182 Joseph W .Dauben

tial as a

finite line segment rather than

the

infinitely small quantity that

was used in

practice.

1 6

This con fusion b etween theoretical considerations and practical applica-

tions carried over

to

Leibniz's metaphysics

of the

infinite,

for he was

never

committed to any one view but made conflicting pronouncements.

Philosophically,

as

Robinson himself

has

argued, Leibniz

had to

assume

the reality of the

inf in i tethe

infinity of his monads, for exampleor

th e

reality

of

infinitesimals

not as

mathem atical

pointsbut as

substance-

or force-pointsnamely,

Leibniz's monads themselves.

1 7

That the eighteenth cen tury wa s concerned not w ith dou bts about the

potential of infinitesim als but p rim arily with fears about th eir logical con-

sistency

is clear from the proposal Lagrange drew up for a prize to be

awarded by the Berlin Academy for a rigorous theory of infinitesimals.

As the prize proposal put it:

It is well kn ow n that higher m athematics continually uses infinitely large

and infinitely

small quantities. Nevertheless, geometers,

and

even

th e

ancient analysts, have carefully avoided everything which approaches

the infinite; and some great modern analysts hold that the terms of the

expression

infinite magnitude

contradict

one

another.

The A cadem y ho pes, therefore , that i t can be explained how so many

true theorems have been deduced from a contradictory supposition,

and

that

a

principle

can be

delineated which

is

sure,

c lear in

a

word ,

t ruly

mathematicalwhich can appropriately be substituted for

the

infinite.

18

Lakatos seemsto appreciate all

th is and

even contradicts himselfon

th e

subject

of

Leibniz's theory

and the

significance

of its

perceived

in-

consistency. Recalling his earlier assertion that Leibniz's theory was not

ove rthrow n because of its inconsistency , consider the follow ing line, jus t

a few

pages later, wh ere La katos asserts that non stand ard analysis raises

the problem

of

"how

to

appraise

inconsistent

theories like Leibniz's

calculus, Frege's logic,

and

Dirac's delta function."

1 9

Lakatos apparently had not made up hismind as to thesignificance

ofth e

inconsistency

of

Leibniz's

theory,

which raises questions abou t

th e

historical value

and

appropriateness

of the

extreme sort

of

rational

reconstruction

that

he has

proposed

to vindicate th e work of earlier

generations. In fact, neither Leibniz nor Euler nor Cauchy succeeded in

giving

a

satisfactory foundation

for an

infinitesimal calculus that also

demonstrated its logical consistency. Basically, Cauchy's "epsilontics"

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ABRAHAM R O B I N S O N

AND

NONSTANDARD

A N A L Y S I S 183

wereameans ofavoiding infinitesand infinitesimals. Nowheredo Robin-

sonian infinitesimalsorjust i f icat ion s appear inCa uchy 's explanat ionsof

the

rigorous acceptabil i ty

of his work.

2 0

Wholly apart f rom what

Lakatos and

others like Robinson have

at-

tempted in

reinterpre ting earl ier resu lts

in

terms

of

nonstandard analysis,

it isstillimp or tant to understand Robinson's ow nreasons fordeveloping

his historical knowledge in as much detai land with as much scholar-

shipash e

did.

For

Robinson,

the

history

of

infinitesimals

w as

more than

an an tiqua rian interest; i t was not one that developed w ith advan cing age

or retiremen t, but was a simu ltaneous developm ent that began w ith his

discovery o f nonstandard analysisin theearly 1960s. Moreover, there seem

to hav e been serious reasons for R ob inso n's keen attention to the history

of

mathematics as part of his own "research program" concerned with

the future of non standard analysis.

3. Non standard A nalysis and the History of Mathematics

In

1965,

in a

paper titled

"On the

Theory

of

Normal

Families,

Robin-

son began

with

a short look at the history of mathematics.

2 1

He noted

that for about onehundredand fifty years

after

itsinceptionin theseven-

teenth cen tury , m athematical analysis developed vigoro usly on inadequate

foundations. Despite this inadequacy,

th e

precise, quantitative results pro-

duced

by the leading mathematicians of that period have stood the test

of t ime.

In the first half of the nineteenth century, however, the concept of the

limit,

advocated previously

by

Newton

and

d 'Alembert , gained ascendan-

cy . Cauchy, whose

influence

w as inst rumental in bringing about the

change, still based

his

arguments

on the

intuitive concept

of an

infinitely

small

n um ber as a variable tending to zero. At the same t ime, how ever,

he

se t the stage for the form al ly m ore sat isfactory theoryofW eierst rass ,

and

today deltas

and

epsilons

are the

everyday language

of the

calculus,

at least for m ost m athem aticians. I t was this precise approach that p aved

the wa y for thefo rm ula t iono fm ore generaland m ore abstract concepts.

Robinson used this history to explain the importance of compactness as

applied

to

functions

of a

complex variable, which

had led to the

theory

of normal families developed largely

b y

Paul Montel . Therefollowed

the

qualitativedevelopmentof comp lex variable theory , suchasPicard the ory ,

and, finally, against this background, more quantitative theories like those

developed

byRolf

Nevanlinnato

wh om Robinson's paper w asd edicated

as

part

of a Festschrift.

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184

Joseph

W .Dauben

Thehistorical notes to befoundat thebeginningo f Robinson's paper

were echoed again at the end, whenhe turned to askwhether theresults

he had achieved using nonstandard analysis couldn' t be achieved just

as well by standard methods. Although he admitted that because of the

transfer principle (developed

in hispaperof

1961,

"Non-Standard

A nal-

ysis")thiswasindeed possible,headded that such translations into stand-

ard term s usually complicated m atters considerably. As for non stand ard

analysis

and the use of

infinitesimals

it

permitted,

his

conclusion

w as

emphatic:

Nevertheless, w eventure to suggest that our approach has a certain

natural appeal, as shown by the fact that it waspreceded in history

by

a long line of attempts to introduce

infinitely

small and

infinitely

large numbers into Analysis.

22

A nd so the

reason

for thehistoricald igression was itsusefulnessinserv-

in g

a

mu ch broader purpose than merely introdu cing some rather rem ote

historical connections between Newton, Leibniz, Paul Montel, and Rolf

Nevanlinna. History could serve

th e

mathematician

as

propaganda.

Robin-

son was apparent ly concerned that m any m athematicians w ere prepared

to adopt a "so what atti tude toward nonstandard analysis becauseo f

the m ore fam iliar reduction that wa s alway s possible to classical fou nd a-

tions. There w ere several way s to ou tflan k those who chose to m inimize

nonstandardanalysis because, theoretically,

it

could

do

nothing that w asn't

equallypossible in stand ard analysis. A bove all, non stand ard analysis was

often simpler and more intuitive in a very direct, immediate way than

standard approaches. But ,asRobinson also beganto argue w ith increas-

ing

frequency

and ingreater detail,

historically

theconceptof infinitesimals

had always seemednaturala nd intuitively preferable to more convoluted

and less intuitive sorts of rigor. Now that nonstandard analysis showed

w hy

infinitesim als were

safe

for consumpt ion inm athematics, therewas

no reason not to exploit their natural advantages. The paper for Rolf

Nevanlinna w asmeant to exhibit both the technical applications and, at

leastinpart throu ghitsappeal tohistory,thenaturalnessofnonstandard

analysis in

developing

th e

theory

of

normal families.

4.Foundations

and

Philosophy

of

Mathematics

If Ro binson regarded the history of infinitesima ls as an aid to the

justification

in a very general way of nons tand ard analysis, w hat con tribu-

tion did it make, along with hisresults in model theory, to the founda-

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A B R A H A M R O B I N S O N A N D N O N S T A N D A R D A N A L Y S IS

185

t ions and philosophy of mathematics? Stephan K orner, w ho taught a

philosophy of mathematics course with Robinson at Yale in the fall of

1973, shortly before

Robinson's

death early

th e

following year,

w as

doubtless closest to Robinson's maturest viewson the

subject.

23

Basical-

ly ,K orn er sees Ro binson as a follow ero for at least w ork ing in the sam e

spirit

as Leibniz and Hilbert. Like Leibniz (and Kant

after

him), Robin-

sonrejectedanyem pirical basisforknowledge about th e

infinitewhether

in the formof infinitely largeo rin finitely small quan tities, sets, w ha teve r.

Leibniz is

f a mous

for his

view that infinitesimals

a re

useful

f ict ionsa

posit ion deplored

by

such critics

as

Nieuwentij t

or the

more f lamboyant

and popular Bishop Berkeley, whose condemnation of the Newtonian

calculus m ight equally well have applied to

Leibniz.

24

Leibniz adopted bo th

the infinitely large and the infinitely small inm athematics for pragmatic

reasons, aspermitt ing an economy of expression and an intuitive, sug-

gestive, heurist ic picture. Ultimately, there

w as

nothing

to

wor ry about

since

the m athematic ian could e liminate them from his

final

result

after

having infini tesimals and infini t ies to provide the m achinery and do the

work

of a

p roof .

Leibniz and Robinson shared a similar view of the ontological status

of

infinitiesand infinitesimals. Theyare not just fictions, butw el l-found

ones"fictiones

benefondatae, in thesensethattheir

ap plications

prove

useful

in

penetrating

the

complexity

of

natural phenomena

and

help

to

reveal relationships in naturethat purely empirical investigations would

never produce.

A s

Emil

Borel once said of Georg Cantor 's transfinite set theory (to

paraphrasenot too grossly): al thoug hheobjected to t ransf ini te num bers

or induct ions in the formal presentat ion of finished results, it wascer-

tainly perm issible to u se them to d iscover theore m s and create

proofs

again, w hateverworks .

2 5

I t w as only necessary to b e sure that in the

final

versionthey w ere elim inated, thus m aking

n o

official appearance. Robin-

son, however,

w as

interested

in

more, especially

in the

reasons

w hy the

mathematics worked

as it

did,

and in

particular

w hy

infinities

and in-

finitesimals weren ow admissible as rigorous entities despite centuries of

doubts and attempts to eradicate them entirely.

H ere R obinson succeeded w here Leibniz and his successors failed. Leib-

niz, for example, never demonstrated the consistent foundations of his

calculus, forw hichhisw o r kwassh arp ly criticizedbyNieuwentij t , among

others. Th roug hou t the eighteenth century , the trou bling found ations (real-

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186 Joseph W . Dauben

ly, lack of foundations) of theLeibnizian infinitesmal calculus continued

tobothe r mathem atic ians, unt i lthe

epsilon-de lta

m ethodsofCauchya nd

th e

arithmetic

rigor

of

W eierstrass reestablished analysis

o n

acceptably

finite terms. Because,

as

Kornerremarks, Leibniz's

approachw as

con-

sidered irremediably inconsistent, hardly anyefforts w ere made to improve

this del iminat ion."

26

Robinson

w as

clearly

not

convinced

of the

inconsistency

of infini-

tesimals,

and in

developing

th e

methods

of

Skolem (who

had

advanced

theidea ofnon standard arithm etic) he w as led toconsider th e possibility

of no nstand ard analysis . A t the same t im e, his w ork in model theory and

mathematical logic contributed

not

only

to his

creation

of

nonstandard

analysis, but to his views on the foundat ions of mathematics as well.

5. Robinson and Formalism 64

In the

1950s, w ork ing un der

th e influence of his

teacher Abraham

Fraenkel, Robinson seemsto have been satisfied witha fairly straightfor-

w ard philosophy ofPlatonicrealism . But by1964, Ro binson's ph ilosoph-

ical

views

had

undergone considerable change.

In a

paper titled simply

"Formalism 64," Ro binson emp hasized

tw o

factors

in

rejecting

his

earlier

Platonism in favor of a formalist position:

(i) Infinite totalities do not exist in any sense of the word (i.e., either

reallyor

ideally). M ore precisely,

any

m ent ion ,

or

purpor ted m ent ion,

of infinite totalities is, literally, meaningless.

(ii) N evertheless, w eshould continue th e business ofM athematics "as

usual,

i.e.

w e

should

act as / / inf in i te

totalities really

existed.

27

Georg Kreisel once com mented that ,

as he

read Robinson's Formalism

64, it was notclearto himwh ether Robinson meant 1864or7P64 Robin-

son, h ow ever, was clearly responding in hisviewson fo rm alism to research

that had made a start l ing impression upon mathematicians only in the

previous year namely,Paul Cohen'simportant wo rkin 1963on forcing

and the independence of the continuum hypothesis.

A s long as it appeared that the accepted axiomatic system s of set theory

(the Z ermelo-Fraenk el axiom atization, for example)wereable to copewith

all

set theoretical problems that wereofinterestto thewo rking m athemati-

cian, belief in the existence of a unique

"universe

of

sets

w as almost

una nim ous. H ow ever, this simpleviewof thesituationwasseverelyshaken

in

th e 1950s and early 1960s by two dist inct dev elopm ents. One of these

w as Cohen's proof of the independence of the cont inuum hypothesis ,

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ABRAHAM

R O B I N S O N

ANDNONSTANDARD A N A L Y S I S 757

which revealedag reat dispar i ty betweenthe scaleo f trans fini te ordinals

and the scaleof

cardinalsor

power sets. A s Robinson himself noted in

an article in

Dialectica,

the relation "is so flexible tha t it seems to b e qu ite

beyond control,

at

least

fornow.

28

The

second development

of

concern

to

Robinson

was the

emergence

of new and varied axioms of infini ty. Although the orthodox

Platonist

believes

that in the real world such axioms must either be t rueor false,

Robinson

found himself persuad ed otherwise. Despite

his n ew

approach

to foundat ionsin "Formalism 64," he was not dogmat ic ,butremained

flexible:

The development of meaningless infinitistic theories may at some

future date

become so unsatisfactory to methat I shall bewilling to

acknowledge

the

greater intellectual seriousness

of

some form

of

con-

structivism.

B ut I

cannot imaginethat

I

shall ever retu rn

to the

creed

of the true platonist, who sees the world of the actual infinite spread

out b efo re him and believes

that

he can comprehend the incomp rehen-

sible.

29

6. Erret Bishop: Meaning, Truth, and Nonstandard Analysis

Incomprehensible ,

however ,

is what someof Ro binson 's critics have

said, almost li terally,o f nonstandard analysis i tself . Of all Robinson's

opponents ,

a t least in public, none has been more vocalor m o r e

vehementthan Erret Bishop.

Inth e

s u m m e r

of

1974,

it w as

hoped that R obinson

and

B ishop would

actually have a chanceto discuss their views in a forum ofm athemat i-

cians

and

historians

and

philosophers

of

mathematics

w ho

w ere invited

to aspecial W orksh opon theE volution of Modern Mathematics held at

th eA merican AcademyofA rtsandSciencesinBoston. Garrett

Birkhoff ,

one of the workshop's organizers, had intended to feature Robinson as

the keynote speaker for the section of the Academy's program devoted

tofoundat ionsofm athematics ,butRo binson's unexpected death inA pr il

of 1974 ma de this imp ossible. Instead , Erret B ishop presentedthefeatured

paper

for the

section

on foundations.

Birkhoff

comparedRob inson'sideas

with those of Bishop in the

following

t e rms :

During

the past twenty years, significant contributions to the foun da-

tions of m athematics have been m ade by two opposing schools. O ne,

led

by

A braha m Ro binson, claims Leibnizian antecedents

for a

"non-

standardanalysis stemm ing from the

model

theory ofT arski.The

other (smaller) school, led by Errett Bishop, attempts to reinterpret

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188 Joseph W .Dauben

Brouwer 's intuitionism in terms of concepts of "constructive

analysis."

30

Birkhoff

went

on to

describe briefly

(in a

written report

of the

session)

th e spirited discussions following Bishop's talk, marked, as henoted, "b y

the

absence

of

po sitive reactions

to

Bishop's view."

31

Even

so,

Bishop's

paper raised a funda m ental question about the philosophy of mathem atics,

which

he put simply as follows: "As pu re m athem aticians, w e m ust decide

whetherw e are playing a game, or w heth er our theorem s describe an ex-

ternal reality."

32

If theseare the only choices, then one's response is ob-

viously

limited.

For

Robinson,

the

excluded m iddle w ould have

to

come

into play

here for he

viewed m athematics,

in

particular

the

striking results

hehad

achieved

in

model theory

and

nonstandard analysis,

as

constituting

much more than a meaningless game, although he eventually came to

believethat m athematics didnot necessarily describe any e xternal reality .

B ut more of Robinson's own metaphysics in a moment .

Bishop m ade his concerns over th e crisis he saw in contemporary

mathematics quite clear

in a

dramatic characterization

of

wha t

he

took

to be the

pernicious

efforts of

historians

and

philosophers alike.

N ot

only

isthere acrisisat the foundations of mathematics, according to Bishop,

but avery real danger(as he put it) in the role that historians seemed to

be

playing, along with non stand ard analysis itself,

in

fueling

th e

crisis:

I think

that

itshould be a fundamental concern to thehistorians that

what they are doing is potentially dangerous. The superficial danger

isthat

it will be and in

fact

has

been systematically distorted

in

order

to support the status quo. And there is a deeper danger: it is so easy

to accept the problems that have historically been regarded as

signifi-

cant as actually being

significant.

33

Interestingly,

in his own historical w riting, Robinson sometimes m ade

the

same point concerning

the

t r iumph ,

as

many historians (and math-

ematicians

as

well) have come

to see it, of the

success

of

Cauchy-Weier-

strassianepsilontics over infinitesimals

in

making

th e

calculus rigorous

in

the

course

of the

nineteenth century.

In

fact,

one of the

most impor-

tant achievements of Robinson's work in nonstandard analysis has been

his conclusive de m onstration of the po verty of this kind of

historicism

ofthe m athem atically W higgish interpreta tion of increasing rigor over the

mathematically unjustif iable "cholera

baccillus"

of infinitesimals, to use

Georg Cantor 's colorful description.

34

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1 8 9

A s

for nonstandard analysis , Bishop had this to say at the Boston

meeting:

A

more recent

at tempt

at

mathematics

by

formal finesse

is

nonstand-

ard analysis. I gather that it has met with some degree of success,

whether at the expense of giving significantly less meaningful proofs

I

do not k n o w . M y interest in nonstand ard analysis is tha t attempts

are being made to introduce it

into

calculus courses. It is difficult to

believe

that debasement

of

meaning could

be

carried

so far .

3 5

Two thing s deserve comm ent here. The first istha tB ishop (surprising-

ly ,

in light of some of his later comm ents abo ut nonstand ard analysis)

does not dismiss it as completely meaningless, but only asks whether its

proofs

are "significantly less meaningful" than constructivist proofs. Leav-

ingopen for the moment what Bishophas inmind herefo r

meaningless

in terms of proofs, is seems clear that by one useful indicator to which

Bishop refers, nonstandard analysis

is

year-by-year showing itself

to be

increasingly "meaningfu l . "

3 6

Consider, for example, the pragmatic value of nonstandard analysis

in terms of its application in

teaching

th e calculus. Here it is necessary

to

consider

the

success

of

Jerom e K eisler 's textbook Elementary Calculus:

An A pproach U sing Infinitesimals, w hich uses nonstandard an alysis to

explain in an

in t roductory course

th e

basic ideas

of

calculus.

The

issue

of its pedagogic valu e

will

also serve to reintrod uce, in a m om ent, the ques-

tion of meaning in a very direct way.

Bishop claims that the use of nonstandard analysisto teach thecalculus

is

wholly pernicious.

He

says this explicitly:

The

technical complications introduced

by

Keisler 's approach

are of

minor impor tance .

The

real damage lies

in his

obfuscation

and

devitalization of those

wonderfu l

ideas. No invocation of Newton and

Leibniz is going to just i fy developing calculus using [nonstandard

analysis] on the grounds tha t the usual definition of a limit is too

c o m p l i c a t e d . . .

Although

it

seems

to be

futile,

I

always tell

m y

calculus

students

that

m athem atics is not esoteric: i t is com m onsense. (Even the noto rious

e,dde finition of limit iscomm onsense, and moreover iscentral to the

important pract ical problems

of

approximation

and

estimation.) They

do not believe m e.

37

O ne

reason Bishop's students

may not

believe

him is

thatwhat

he

claims,

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190

Joseph W .

Dauben

in fact, does not seem to be true. There is another side to this aswell,

for one may also ask whether there is any truth to the assertions made

by Robinson (and emphatically

by

Keisler)

that

"the whole

point

of our

infinitesimal approach to calcu lus is that it is easier to de fine and explain

limitsusing

infinitesimals."

38

O f course, this claim also deserves examina-

tion, inpart because Bishop 's ow nat tempt to dismiss Ke isler 's m ethod s

as

being equivalent

to the

axiom

"0 = 1" is

simply

nonsense.

39

In

fact,

thereareconcrete indica tions that despitetheallegations madebyB ishop

about obfusc ation and the nonintuitiveness of basic ideas in non standa rd

terms, exactly the opposite is true.

N otlongago astudywasund er taken to assessthevalidityof theclaim

that "from this nonstandard approach, th ed efinitions of the basic con-

cepts [of the calculus] becom e simpler and the argu m ents more

intuitive."

40

Kathleen Sullivan

reported

th e results of her dissertation, written at the

University of Wisconsin and designed to determine the pedagogical

usefulness of nonstandard analysisin teaching calculus, in the

Am erican

Mathematical Monthly in 1976. This study , there fore, was presum ably

available to Bishop when his review of Keisler's book appeared in 1977,

in

which he attacked the pedagogical validity of nonstandard analysis.

What did Sullivan's study reveal? Basically, she set out to answer the

following questions:

Will the students acquire the basic calculus skills? Will they really

understand the fundamental concepts any differently? How difficult

willit be for them to m ake the transition into standard analysis courses

if

they want

to

study mo re mathematics?

Is the

nonstand ard approach

only suitable

for gifted

mathematics students?

4 1

To answer these questions, Sullivan studied classes at five schools in

th eChicago-Milwaukee

area

duringth eyears 1973-74. Four ofthem w ere

small private colleges,

the

fifth

a

public high school

in a

suburb

of

Milwaukee. The same instructors who had taught the course previously

agreed

to

teach

one

introductory course using Keisler 's book (the 1971

edition) aswellasanother introductory course usingastandard approach

(thus serving as a control group ) to thecalculus.Com parison of SA T scores

showed

that both

the

experimental (nonstandard) gro up

and the

standard

(control) group were com parable

in

abili ty befo re

the

cou rses began.

A t

the end of the course, acalculus test w asgivento both groups. Instruc-

torsteaching thecourses were interviewed, and aquestionnaire w asfilled

out by everyone who has used Keisler 's book withinthe last five years.

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A B R A H A M R O B I N S O N A N D N O N S T A N D A R D A N A L Y S I S

191

Th e single question that br ou gh t out the greatest

difference

be tween

the two g roups w as quest ion 3:

Define f(x)

by the

rule

Prove using the definit ion of l imit tha t

lim

f(x) = 4 .

Control Group Experimental Group

(68

students)

(6 8

students)

Did

not attempt 22 4

Standard arguments:

Satisfactory p roof

2 14

Correct

s ta tement , faulty proof

15

14

Incorrect a rgum ents 29 23

Nonstandard a rguments :

Satisfactory

proof

25

Incorrect a rgument s 2

Theresults,asshownin theaccompanying

tabulation,

seem to bestrik-

ing; bu t ,

as

Sull ivan cautions:

Seeking

to determ ine wheth er or not students really do perceive the basic

conceptsany

differently

is not s implyam atter oftabulating howmany

students

can

form ulate proper

mathematical

definitions. Most teachers

would

probably agree that this would

be a

very imperfec t ins t rumen t

for measur ing unde rs t and ing

in a

col lege freshman.

B ut

fur ther l ight

on this and other questions can be sought in the comments of the

instructors .

4 2

Here, too, the results are remarkable in their support of the heurist ic

value

of using no nstan da rd analysis in the classroom. I t would seem

tha t ,

contrary to Bishop'sviews,thetraditiona l

approach

to thecalculus m ay

be the

more pernicious. Instead,

the new

nonstandard approach

w as

praised

in

s t rong terms

by

those

w ho

actually used

it:

The

group

as a

whole responded

in a way

favorable

to the

experimen -

tal

method

on

every item:

the

students learned

the

basic concepts

of

the calculus more easily, proofs were easier to explain and closer to

intui t ion, and most felt that the students end up with a better under-

standing of the basic concepts of the calculus.

43

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192 Joseph W . Dauben

A s to

Bishop's claim

that th e d,e

method

is commonsense,

44

this

too is

open

to

question.

A s one

teacher hav ing

successfully

used Keisler's

book remarked,

"W hen

my most recent classes were presented with the

epsilon-delta

definition o f limit, they we re outraged by its obscurity com -

pared

to

what they

had

learned [via nonstandard analysis]."

4 5

B ut

as G. R . B lackley wa rned Keisler 's pub lishers (Prindle, W eber and

Schm idt) in a letter whe n he was asked to

review

the new textbook prior

to its

pu blication:

Such problem s as m ight arise

with

the book

willbe political.

It is revolu-

tionary. Revolutions are seldom welcomed by the established party,

although revolutionaries often

are.

46

Thepoint to all ofthisissimplytha t ,if onetake

meaning

as the stan-

dard ,

as

Bishop urges, rather than

truth,

then

it

seems clear that

by its

ow n

success nonstandard analysis

has

indeed proven

itself

meaningful at

the

most elementary level

at

which

it

could

be in t roduced namely ,

that

at

which calculus

is

taught

for the first

time.

B ut

there

is

also

a deeper

level

of meaning at which nonstandard analysis operatesone that also

touches

on

some

of

Bishop's criticisms. Here again Bishop's

views can

also be

questioned

and

shown

to be as

unfounded

as his

objections

to

nonstandard analysis pedagogically.

Recall that Bishop began

his

remarks

in

Boston

at the

American

Academy

of

Arts

and

Sciences workshop

in

1974

by

stressing

the

crisis

in contemporary mathematics that stemmed

from

what he perceived as

a

misplaced emphasis upon formal systems

and a

lack

of

distinction

be-

tween the

ideas

of truth and

meaning.

The

choice Bishop gave

in

Boston

was

between m athematics

as a

m eaningless game

o r as a

discipline

describing some objective reality. Leaving aside the question of whether

mathematics

actually

describes reality, in some objective sense, consider

Robinson 'sow n

hopes

fornonstandard

analysis, those beyond

the

pure-

ly

technical results

he

expected

the

theory

to

produce.

In the

preface

to

his

book on the subject, he hoped that

"some

branches of modern

Theoretical

Physics might benefit directly from th e application of non-

standard

analysis."

4 7

In

fact,

the

practical advantages

of

using nonstandard analysis

as a

branch

ofapplied

mathematics have been considerable. Although this

is

not theplaceto gointo detail ab out theincreasing num berofre sults aris-

in g

from nonstandard analysis

in

diverse contexts,

itsuffices

here

to

men-

tion impressive research using nons tand ard analysis in phy sics, especially

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A B R A H A M R O B I N S O N A N D N O N S T A N D A R D A N A L Y S IS 1 93

quantum theory andthermodynam ics, and in econom ics, w here studyof

exchange economies has been particularly amenable to nons tandard in-

terpretation.

4 8

7.

Conclusion

There is another purely theoretical context in which Robinson con-

sidered the importance of the history of mathematics that also warrants

consideration. In 1973, Robinson wrote an expository article tha t drew

its title

from

a famo us mon ograph writ ten in the nineteenth century by

Richard Dedekind: W assind und wassollen dieZahlen? This title w as

r o u g h l y t r a n s l a t e d o r t r a n s f o r m e d in R o b in s o n ' s

v e r s i o n a s

" N um ber s WhatA reThey and W hat A reThey GoodFor? A sRobin-

son put it:

"N um ber systems, like hair styles,

go in and out of fashion

i t ' s what 's underneath

tha t count s . "

4 9

This might

well

be taken as the leitmotiv of much of Robinson's

mathematical career, for his surpassing interest since the days of his disser-

tation written

at the

University

of

London

in the

late 1940s

w as

model

theo ry, and especially the way s in which mathematical logic could not on ly

illuminate mathematics, buthave very realanduseful applicationsw ithin

virtually

all of its

branches.

In

discussing number systems,

he

wanted

to

demonstrate, as he put it,

that

the collection of all number systems is not a finished totality whose

discoverywas

com plete

around

1600,

or

1700,

or 1800,but that it has

been and still is a growing and changing area, sometimes absorbing

new systems

and

sometimes discarding

old

ones,

or

relegating them

to

the

attic.

50

Robinson, of course, was leading up in his paper to the way in

which

nonstandard analysis had again broken the bounds of the traditional

Cantor-Dedekind unders tanding of the real numbers , especial ly as they

had been augmented by Cantorian transfinite ordinals and cardinals.

Tomakehispoint, R obinson turned m om entarilyto thenineteenth cen-

tury

and

noted that Hamilton

had

been

the

first

to

demonstrate that there

w as a larger arithmetical system than that of the complex numbers

namely, that represented

by his

qua ternions. These we re soon supplanted

by the system of vectors developed by Josiah Willard Gibbs of Yale and

eventually transformed into a vector calculus . This was a more useful

system,

one

more advantageous

in the

sorts

of

applications

for

which

quaternions had been invented.

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194 Joseph W. Dauben

Som ewhat later, another approach to theconceptofnumbe rw astaken

by Georg Cantor,

w ho

used

the

idea

of

equ inumeros i ty

in

te rm s

of

one-

to-one corresponden ces

to

de f ine num bers .

I n

fact ,

for

Cantor

a

cardinal

numbe r

was a

sym bol assigned

to a

set,

and the

same sym bol represented

all sets equivalent to the b ase set. The ad van tage of thisviewof the n ature

of numbers, of course, was that i t could be applied to infinite sets, pro-

ducing trans finite nu m be rs and eventually leading to an entire system of

transfinite ar i thmetic .

I ts

major d i sadvantage , however ,

w as

that

it led

Cantor

to

reject adamantly

any

mathematical concept

of infinitesimal.

51

A s

Robinso n points out , a l though

the

eventual

fate

of

Cantor ' s

theory

w asa success story , it w as not entirely so for its au tho r. D espite the clear

utility of Can tor ' s ideas, which arose in connection with his work on

trigonometric series (later applied with great success

by

Lebesgue

and

others

at the

turn

of the

century),

it was

highly criticized

by a

spectrum

of mathematic ians, including, among the most prominent , Kronecker ,

Frege, andPoincare. In

add ition

to thetraditionalobjection thatthe in-

finite

shou ld not be allow ed in rigoro us mathem atics, Cantor ' sw ork was

also questioned because of its

abstract character.

Ultimately, however,

Cantor ' sideas p revailed , despite cri t icism ,a nd today settheoryis a cor-

nerstone, if not the m ajor foundat ion, upon w hich m uch of modern

mathemat ics rests.

5 2

There was an impor tant lesson to be learned, Robinson believed, in

the even tual acceptance of new ideas of n um be r, despite their nove lty or

th e controversies they might provoke. Ultimately, utilitarianrealities could

not be overlooked or ignored forever. With an eye on the

future

of non-

standard

analysis, Ro binson w asimpressed by the fateofanother theory

devised late in the nineteenth century that also attempted, l ike those of

Hamil ton , Cantor,

and

Robinson,

to

develop

and

expand

the

frontiers

of n u m b e r .

In the

1890s, Kurt Hensel introduced

a

whole series

of new

numbe r

systems, his now

familiar p-adic numbers. Hensel realized that

he

could

use

h is

p-adic num bers

to

investigate properties

of the

integers

a nd

other

numbers . H e

also realized,

as did

others , that

the

same results could

be

obtained in other ways. Consequent ly, many mathematic ians came to

regard Hensel 's work

as a

pleasant game; but,

as

Robinson himself

ob-

served, Many

of

Hensel 's contemporaries werere luctant

to

acquire

the

techniques involved

in

handl ing

the new

numbers

and

tho ugh t they con-

stituted

an

unnecessary

burden.

53

The sam e might be said of n on stand ard analysis, par t icularly in l ight

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ABRAHAM R O B I N S O N

AND

NONSTANDARD A N A L Y S I S 195

of the transfer principle that demonstrates that theorems true in *R can

alsobe

proven

for R by

standard

methods .

M oreover, m any m athemati-

cians are clearly reluctant to m aster the logical m achinery o f m odel the ory

with which Robinson developed his original version of nonstandard

analysis. This problem has been resolved by Keisler and Luxemburg,

among

others,w ho

have presented nonstandardanalysis

in

ways accessi-

ble to mathematicians without their having to take up the difficulties of

mathematical logicas a

prerequisite.

54

But forthosewho see nonstandard

analysis

as a fad

that

may be a

currently pleasant gam e, likeHensel's p-adic

numbers , th e

later history

of

Hensel's ideas should give skeptics

an ex-

ampletoponder . For today, p-adic numbersareregarded ascoequal with

the

reals,

and

they have proven

a

fertile area

of

mathematical research.

The same has been demonstrated by nonstand ard analysis. I ts applica-

t ions in areas of analysis, the theory of com plex variables, math em atical

physics, economics, and a host of other fields have shown the uti l i ty of

Robinson's own extension of the number concept. Like Hensel 's p-adic

numbers, nonstandard analysis

can be

avoided, al though

to do so may

complicate proofs

and

render

the

basic features

of an

a rgument

less

intuitive.

W hat pleased R obinson as m uch about no nstan da rd analysis as the in-

terest it engendered

from

the beginning among mathematic ians was the

way itdem onstrated th e indispensability,as

well

as thepower ,oftechnical

logic:

It is

interesting

that

a

method which

had

been given

up as

untenable

has at

last turned

out to be

workable

and

that this development

in a

concrete

branch ofmathematics w asbrought about by the refined tools

made avai lable by modern mathematical logic .

55

Robinson

had

begun

his

career

as a

mathematic ian

by

s tudying

set

theo ry and axiomatics withA braham Fraenkel in Jerusalem, which even-

tuallyled to hisPh.D. from th eUn iversityofLondon in

1949.

56

H isearly

interest in logic w as later am ply repa id in his applications of logic to the

development of nonstandard analysis. As Simon Kochen once put i t in

assessing the

significance

of

Robinson 's cont r ibut ions

to

mathematical

logic

and model theory:

Robinson ,

via

model theory, wedded logic

to the

mainstreams

of

m a t h e m a t i c s . . . . A t

present, principally because

of the

wor k

of

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196

Joseph

W . Dauben

Abr aham

Ro binson, model theory

is

just that :

a

fully-fledged theory

with manifold interrelations with the rest of

mathematics .

57

Kurt

Godel

valued Ro binson 's achievement

for

sim ilar reasons:

it

suc-

ceeded inunitingm athemat icsand logicin anessen tial,fundamentalway .

That union

h as

proved

to be no t

only

one of

considerable m athematical

importance,

but ofsubstantial philosophicalandhistorical con tentas

well.

Notes

1. There is a considerable l i te ra ture on the subject of the supposed cr is is in mathe m atics

associatedwith

the

Pythagoreans.See,

for

example,(Hasse

and

Scholz 1928).

For a

recent

survey of this debate, see (Berggren 1984; Dauben 1984; Knorr 1975).

2.

(Bishop 1975, 507).

3. (Bishop 1975, 513-14).

4 .

Robinson first publ i shed

th e

idea

o f

nons tandard ana lys is

in a

paper submit ted

to

the Dutch Academy ofSciences (Robinso n 1961).

5.

(Cleave 1971; Edwards 1979; Laugwitz 1975, 1985; Luxemburg 1975) .

6. (Lakatos 1978).

7.

(Lakatos 1978, 43).

8.

(Lakatos 1978, 44).

9. (Lakatos 1978, 49).

10. (Lakatos 1978, 50). Emphasis in original.

11. Cauchy offers his def in i t ions of inf ini te ly large and small num be rs in severa l w ork s,

f irst in the

Cours

d'analyse,

subsequent ly

in

la ter versions withou t sub stantive chang es.

See

(Cauchy 1821, 19; 1823, 16; 1829, 265), as well as (Fisher 1978).

12.

(Ca uch y 1868).

13. (Lakatos 1978, 54).

14.

For

details

of the

successful development

of the

early calculus,

se e

(Boy er 1939;

Grattan-Guinness 1970, 1980; Grabiner 1981; Youshkev i tch 1959).

1 5. (New ton 1727, 39), w here he discusses the co ntrary nature of ind ivisibles as

demonstra ted by Euclid in Book X of theElements. For additional analysis of N e w t o n ' s

views on

inf in i te s imals ,

see

(Grabiner 1981, 32).

1 6.

See

(Leib niz 1 684).

For

details

and a

critical analysis

of

w h a t

i s

involved

in

Le ibniz ' s

presenta t ion and applica t ions of inf ini tesimals , see (Bos 1974-75; Engelsman 1984) .

17.

See

(Robinson 1967,

35 [in

Robinson 1979,

544]).

18. In (Lagran ge 1784, 12-13; Dugac 1980, 12).For details of theBer l in A cadem y's com-

petition, see (Grab iner 1981, 40-43; Y ou sh ke vitc h 1971, 149-68).

19. (Lakatos 1978, 59). Emphasis

added.

20. See (Grattan -Gu inne ss 1970, 55-56), w he re he discusses "limit-avo idanc e" and its

role

in

m a k i n g

th e

calculus r igorous.

21. (Robinson 1965b).

22.

(Robinson 1965b, 184); also

in

(Robinson 1979, vol.

2,

87).

23. I am

grateful

to

Stephan

Korner

and am

happy

to

acknowledge

his

help

in

ongoing

d iscuss ions we have had of R obinson and h is w ork .

24. For a

recent survey

of the

cont rover s ies sur rounding

th e

ea r ly deve lopment

of the

calculus, se e (H all 1980).

25. Borel in a letter to H a d a m a r d , in (Borel 1928, 158).

26. (KOrner 1979, xlii).Korner

notes,

however, that an exception to this generalization

is to be foun d in Hans Va ihing er 's general theory o f f ic t ions. Vaihing er t r ied to jus t i f y in-

finitesimalsby "a m e thod o f opposite mistakes," asolut ion that was too imprec ise , Ko rner

suggests , to have impressed mathematic ians. See (Vaihinger 1913, 51 I f f ) .

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A B R A H A M R O B I N S O NANDN O N S T A N D A R D A N A L Y S IS 197

27. (Robinson 1965a, 230; Robinson 1979, 507). Ne arly ten years later, R obinson re-

calledthemajorpoints of

Formalism

64 as follows: (i)

that

mathematical theories which,

allegedly, deal wi thinfinite totali t ies do not have any detailed meaning, i .e. reference, and

(ii)

that this has no bea ring on the question wh ether or not such theories should be developed

and

that , indeed, there

are

good reasons

w h y w e

should continue

to do

mathemat ics

in the

classical fashionnevertheless. Robinson added that nothing since1964had p rompted h im

to

change these views

andthat,in

fact, well-known recent developments

in set

theory repre-

sent evidence favoring these views." See (Robinson 1975, 557).

28. (Robinson 1970, 45-49).

29. (Robinson 1970, 45-49).

30. (Birkhoff 1975, 504).

31 .

(Birkhoff 1975, 504).

32.

(Bishop 1975, 507).

33.

(Bishop 1975, 508).

34.

For Ca ntor 's views, see his letter to the Ital ian mathem atician Vivanti in (M eschkow ski

1965, 505). A general analysis of Cantor's in terpretat ion of inf ini tes im als may be fo und

in

(Dauben

1979, 128-32, 233-38).

On the

question

of rigor, see

(Grab iner 1974).

35. (Bishop 1975, 514).

36. I t should also be noted, if only in passing, that Bishop has not bothered

himself ,

apparently, w ith

acareful

study

of

nonstandard analysis

or its

implications,

for he

offhandedly

admits that he only

gathers

that i t has met with some degree of success" (Bishop 1975,

514; emphasis added).

37.

(Bishop 1977, 208).

38.

(Keisler 1976, 298), emphasis

added;

quoted

in

(Bishop 1977, 207).

39. (Bishop 1976, 207).

40. (Sull ivan 1976, 370). N ote that Su ll ivan's study used the experim ental version of

Keisler 's book, issued

in

1971. Bishop reviewed

th e first

edit ion published

five

years later

by Prindle, W eber and S chm idt. See (Keisler 1971, 1976).

41 .

(Sullivan 1976, 371).

42. (Sullivan 1976, 373).

43. (Sullivan 1976, 383-84).

44. (Bishop 1977, 208).

45.

(Sullivan 1976, 373).

46. (Sullivan 1976, 375).

47. (Robinson 1966, 5).

48. See

especially (Robinson 1972a, 1972b, 1974, 1975),

as

well

as

(Dresden 1976)

and

(Voros

1973).

49.

(Robinson 1973, 14).

50.

(Ro binson 1973, 14).

51 .

For details , see (Dauben 1979).

52.

See (Dauben 1979).

53.

(Robinson 1973, 16).

54. (Luxemburg 1962, 1976;

Keisler

1971).

55 . (Robinson 1973, 16).

56. Robinson completed his disser tat ion,

TheMetamathematics of Algebraic Systems,

at

B irkbeck College,

Univers i ty

of

L o n d o n ,

in

1949.

I t was

published

tw o

years later;

see

(Robinson 1951).

57. (Kochen 1976, 313).

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