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Calculation - Thinking - Computational

Thinking

Seventeenth-Century Perspectives on

Computational ScienceMichael S. Mahoney

Princeton University

Published in Folkerts, Menso; Seising, Rudolf (Hg.):Form, Zahl, Ordnung. Studien zur Wissenschafts- und

Technikgeschichte. Ivo Schneider zum 65. Geburtstag(Boethius: Texte und Abhandlungen zur Geschichte der Mathematik

und der Naturwissenschaften), Stuttgart: Franz Steiner Verlag, 2003.

Festschriften allow one to begin on a personal note. Over thirty years ago Ihad the pleasure of daily conversations with Ivo Schneider during a year's

sabbatical at the Institut fr Geschichte der Naturwissenschaften in Munich.

We resumed those conversations two years later when Ivo spent a year as

visiting professor at Princeton. At the time, we shared a focus on

mathematics and the mathematical sciences in the 16th and 17th centuries,

and our long talks, often on the way to and from the coffee shop, gave me

an opportunity to sound out my ideas with a knowledgeable, imaginative,

and articulate colleague. Each of us has since moved off into other areas, in

my case to the history of computing since 1945. Yet those conversationsremain pertinent. For, the two subjects are not as far apart as they might

seem at first glance. They both involve the emergence of new disciplines, or

of new ways of thinking about old disciplines: in the 17th century, symbolic

algebra and the new mode of analytical reasoning that it fostered; more

recently, theoretical computer science as a mathematical discipline.

On the one hand, the two subjects display considerable continuity of theme.

Theoretical computer science has drawn its mathematical structure from

developments in abstract algebra that in turn exemplify many of the themes

that informed the first efforts in symbolic algebra in the 17th century. In that

sense, this essay completes a circle that I opened up during that year in

Munich with a lecture on "Die Anfnge der algebraischen Denkweise im 17.

Jahrhundert."(1) It may complete it in another sense as well, namely by

marking the end of algebraic thought as it was conceived in the 17 th century,

at least as far as it was thought to capture the relation of mathematics to the

world.

1. Published under that t itle

in the short-lived but

important journal founded

by Schneider and Eberhard

Schmauderer,Rete:

Strukturgeschichte der

Naturwissenschaften

1,1(1971), 15-31; English

trans., "The Beginnings of

Algebraic Thought in the

17th Century", in S.

Gaukroger (ed.),Descartes:Philosophy, Mathematics

and Physics (Sussex: The

Harvester Press/Totowa,

NJ: Barnes and Noble

Books, 1980), Chap.5.

Among the topics on our

daily walks was the

question of "s tructural

history" and Ivo's plans for

a journal devoted to it.

Let me start, then, with a brief account of the emergence of a new

mathematical world in the 17th century and then jump to the new world of

mathematics being created by means of the computer, of which we have

only limited mathematical understanding.
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I. Cutting the World Up and Putting it Back

Together with Mathematics

Toward the close of the 16th century, the French lawyer and mathematician

Franois Vite created a new symbolic algebra, or "logistic of species",

designed to extend the heuristic power of algebra from arithmetic to

mathematics in general and thereby to provide a new tool for carrying outthe form of mathematical reasoning which the Greeks had called "analysis".

He called his new algebra the "analytic art" (or "art of analysis").(2)

Essentially, it went beyond the old algebra by using symbols to denote both

knowns and unknowns (his convention was to use vowels for unknowns and

consonants for knowns; Descartes chose to work from opposite ends of the

alphabet) and thus to separate the form of the relationship among knowns

and unknowns from any particular values, indeed from any particular kind of

quantity, that they might represent. That is, an equation expressed a

relationship among things that could be added to one another, subtractedfrom one another, and so on. What interested Vite and his successors was

not so much the solution of an equation as the structure of the relation that it

expressed.

2. Franois Vite,In artem

analyticen isagoge (Tours,1591; republ. in Opera, ed.

F. van Schooten, Leiden,

1646)

The main task of his ars analytica, which distinguished it from all previous

algebras, was the investigation of the constitutio aequationum, i.e. the

structure of equations: how they are constituted and how they are related to

one another. In a series of treatises Vite set forth techniques for the analysis

and transformation of equations. He thus established the prototype, inessence if not in historical fact, for Book III of Descartes' Gomtrie, that

misnamed treatise on the theory of equations, where Descartes showed just

how an nth-degree polynomial is the product ofn linear binomials, how the

coefficients of the polynomial are the result of combinations of the roots, and

thus not only how terms can be removed by reducing or augmenting the

roots but also how one might imagine roots that do not correspond to real

numbers, although structurally they must be there.

xn

+a

1x

n-1

+... + a

n = (x

-r1)(

x - r2)...(

x - rn)

a1 = -(r1 + r2 + ... + rn), etc.

x3 - 1 = (x- 1)(x - a)(x - b); a = ?, b = ?

In short, the new symbolism and its associated techniques enabled one to

talk about equations and the number and nature of their solutions, even

without solving them. In retrospect and put anachronistically, Vite's analytic

art was a language of metamathematics as well as mathematics.

As such, it gave rise to three lines of development of interest to the matter at

hand. First, as already noted, it made symbolic algebra the study of the

abstract structures of mathematics. Second, in providing a language for


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talking about mathematical reasoning, it became a model for talking about

reasoning in general, that is, it suggested a form of symbolic logic by which

reasoning could be viewed as a kind of calculation. Third, it stimulated a

program of mathematical research that extended the heuristic power of

algebra into new realms and made the analytic art the language of

mathematical science. Let me begin with the last point and then return later

to the first two.

In his work, Vite set out an agenda which I have termed the "analytic

program" and which called for the application of his art to the works cited in

Book VII of Pappus of Alexandria'sMathematical Collection as

constituting the field of analysis, or as Newton happily phrased it, the

"Treasury of Analysis". The algebraic geometries devised by Descartes and

Fermat addressed this agenda, as did Fermat's new methods of maxima and

minima and of tangents. In the latter case, the application of the art carried

symbolic algebra into the realm of the indefinitely small, or infinitesimal, and

linked it, in ways not pertinent to the current discussion, to independent

efforts at recapturing the techniques of quadrature, or the determination of

the area of curved figures, associated with the name of Archimedes.

Methods of tangents and techniques of quadrature developed alongside one

another through much of the 17th century, in many cases without reference

to algebra.(3) As is well known, it was the signal achievement of Newton and

Leibniz to establish the inverse relationship between them. Both did so in the

language of symbolic algebra, and Leibniz in particular signaled the

importance of the symbolism. The 'd' was to be construed as an operation

on the quantity to which it was prefixed. Differentiation constituted a "certain

modification" (quaedam modificatio) of a quantity, and the rules governing

that modification gave rise to a new realm of structures to be analyzed.(4)

3. For the state of the art

just prior to the work of

Newton and Leibniz, see

my "Barrow's Mathematics:

Between Ancients and

Moderns", in M. Feingold

(ed.),Before Newton: The

Life and Times of Isaac

Barrow (Cambridge:

Cambridge UniversityPress, 1990), Chap. 3

4. For more extended

discussions on this and

what follows, see my

"Infinitesimals and

Transcendent Relations:

The Mathematics of

Motion in the Late

Seventeenth Century", in

D.C. Lindberg and R.S.

Wes tman (eds.),

Reappraisals of the

Scientific Revolution

(Cambridge: Cambridge

University Press , 1990),

Chap. 12, and "The

Mathematical Realm of

Nature", in D.E. Garber et

al. (eds.), Cambridge

History of Seventeenth-

Century Philosophy

(Cambridge: CambridgeUniversity Press , 1998),

Vol. I, pp. 702-55.

Although the calculus was not created for the sake of doing mechanics, it

was set to assume that role at the time Newton'sPrincipia appeared. It is,

of course, ironic that analytic mechanics was couched in the terms of

Leibniz's calculus rather than Newton's own fluxions, but the translation into

algebraic terms involved more than symbolism. In keeping with the heuristic

goals that had motivated Vite in the first place, Euler pointed to the difficulty

posed by Newton's geometrical original:

Newton'sMathematical Principles of Natural Philosophy,

by which the science of motion has gained its greatest

increases, is written in a style not much unlike [the synthetic

geometrical style of the Ancients]. But what obtains for all

writings that are composed without analysis holds most of all

for mechanics: even if the reader be convinced of the truth of

the things set forth, nevertheless he cannot attain a sufficiently

clear and distinct knowledge of them; so that, if the samequestions be the slightest bit changed, he may hardly be able to

resolve them on his own, unless he himself look to analysis and

evolve the same propositions by the analytical method.(5)
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By bringing out the essential structure of problems, algebraic analysis (Euler

would consider the phrase redundant) made clear how they and their

solutions were related to one another. One could not only do mathematics

but could see how the mathematics was done.5.Mechanica sive motus

scientia analytice exposita

(St. Petersburg, 1736),

Preface, [iv].

Euler's point concerned mathematics rather than mechanics, but the twowere so wrapped up in one another that in the 18th century analytic

mechanics was considered a branch of mathematics rather than of physics.

A few decades later, Lagrange took pride in the absence of diagrams from

hisMcanique analitique (1788):

No drawings are to be found in this work. The methods I set

out there require neither constructions nor geometric or

mechanical arguments, but only algebraic operations subject to

a regular and uniform process. Those who love analysis will

take pleasure in seeing mechanics become a new branch of it

and will be grateful to me for having thus extended its

domain.(6)

6. "On ne trouvera point de

Figures dans cet Ouvrage.

Les mthodes que j'y

expose ne demandent ni

constructions, ni

raisonnemens

gomtriques ou

mcaniques , mais

seulement des oprations

algbriques, assujetties

une marche rgulier et

uniforme. Ceux qui aiment

l'Analyse, verront avecplaisir la Mcanique en

devenir une nouvelle

branche, et me sauront gr

d'avoir tendu ainsi le

domaine." Avertissement.

The equations of the infinitesimal calculus had become the sole vehicle of

mechanics, the unchallenged means of mechanical thought. With the

Principia, and especially with its translation into the calculus, the

effectiveness of mechanics rested on that of mathematics. Proposition 41 of

Book I shows what that means. It is important to grasp the profound

implication of the condition in the statement of the problem (NB it is a

problem, not a theorem):

Assuming any

sort of centripetal

force, and

granting the

quadrature of

curvilinear

figures, required

are both the

trajectories in

which the bodies

move and the

times of motions

in the trajectories

found.(7)

7. Isaac Newton,

Philosophiae naturalis

principia mathematica

(London, 1687), 127.

[emphasis added]

In this proposition Newton maps the motion of an orbiting body on the left

onto a graph of motion at the "atomic" level on the right. The orbit and the

position of the body on it at any given time are thus captured mathematically,

provided that one can determine the area under the curves abzv and dcxw,
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i.e. that one can integrate the equations of motion. As Pierre Varignon put it,

after translating Newton's scheme into the two basic "rules", velocity v =

ds/dtand forcey = (ds/dx)(dds/dt2), wherex is measured along the axis

AC from A ands is measured along the curve VIK from V,

As to how these two rules are to be used, I say for now that,

being given any two of the seven curves noted above [curves

relating distance, time, force, and velocity in variouscombinations], that is to say, the equations of two taken at will,

one will always be able to find the five others,supposing the

required integrations and the solution of the equations that

may be encountered[emphasis added].(8)

8. Pierre Varignon, "Du

mouvement en gnral par

toutes sortes de courbes;

& des forces centrales, tantcentrifuges que

centreptes, ncessaires

aux corps qui les

dcrivent",Mmoires de

l'Acadmie Royale des

Sciences (1700), 86.

The condition linked the success of mathematical physics to that of the

calculus. It was the job of the calculus to secure those integrations and

solutions, and that is where its practitioners directed their efforts over thenext centuries. The intellectual satisfaction derived from reductionist

explanations depended on the capacity of the mathematics to carry out the

integration that provided the reduction, in the sense of showing that the

behavior at the reduced level did produce or correspond to the behavior at

the observable level.

The situation did not change with the shift from central-force physics to other

models of physical action. Once couched in the terms of the calculus, the

effectiveness of the physical model and its capacity to convey understanding

depended on the capacity of the calculus to provide a solution to thedifferential equations that resulted from analysis. In some cases, it was a

matter of calculating, as in expansion into series and term-by-term

integration. In other cases, it was a matter of exploiting the power of

algebraic analysis to explore structural relationships among problems and

thus to determine conditions of solvability or, in some cases, to prove

unsolvability, as in the case of the general quintic. Over the course of the

eighteenth century, what began as a search for the algorithms that made

integration as straightforward and mechanical as differentiation ended in a

theory that settled for analysis into families of curves reducible to canonicalforms.

Despite the successes of analysis, it became increasingly clear that in many

cases, for example the n-body problem, the move from differential equation

to finite form could be accomplished only by numerical calculation, that is by

reducing the analytical expressions to explicit summations iterated over small

intervals. One could do that by hand, but it was clearly a job suited more for

a machine. The story of the development of mechanical computing devices,

both analog and digital, during the nineteenth and early twentieth centuries

has been recounted many times, and I do not want to retrace that story

here.(9) What is important is that, as far as a mathematical understanding of

the world is concerned, the turn to mechanical calculation has from the

outset been a matter offaute de mieux. A numerical solution may produce

9. See, for example, William

Aspray (ed.), Computing

Before Computers (Ames:

Iowa State University

Press, 1990).
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from the basic relations specific values to be matched against measurements,

but it generally brings very little insight into how those values reflect the

working of the underlying relationships. One may, of course, experiment with

various initial values and try to discern how the outcome changes, but doing

so does not bring insights of the sort provided by

relating work to energy by way of force

and momentum. Numerical solutions do not reveal how the system works

because they hide precisely the intermediate (mediating) relationships that

lead from the behavior of the parts to that of the whole.

In the seventeenth century, the interactive development of algebra and

mechanics led to an analytic view of the world that characterized scientific

thought for the next three centuries. The invention of symbolic algebra and its

extension into the realm of the infinitesimal ultimately provided a powerful

mathematical tool for the study of the world as matter in motion. What made

the tool so powerful was that the algebra that lay at its foundation could beused not only to do mathematics but to talk about it as well. Not only could

one solve problems using algebra, but one could use the same algebra to

analyze questions of solvability. Algebra and the calculus not only captured

the world in mathematical structures but also provided the tools for analyzing

those structures mathematically.

More than simply a means of thinking about mathematics, symbolic algebra

was considered a means of thinking about thought itself. Descartes was only

the first of a line of thinkers down to the present who have pictured the

workings of the mind as a form of calculation, or as Hobbes put it, ofratiocination. Looking toward a universal characteristic, Leibniz foresaw a

time when matters of controversy could be resolved by sitting down and

calculating. From Boole'sAlgebra of Thought, through Frege's

Begriffsschriftand Russell's and Whitehead'sPrincipia mathematica (the

title no coincidence), algebra formed the link between mathematics and logic

and thus provided a means of thinking about thought itself.

From the outset, algebra was also associated with the notion of a mechanical

procedure. Algebra proceeded by straightforward rules, by what Leibniztermed "algorithms", thus giving new and fateful meaning to a word that had

been synonymous with reckoning since the 12th century.(10) That is what

made it appealing to him as a vehicle of logic: one could move from

premisses to conclusions by calculation.(11) Or rather, one could carry out

logical analysis in the way one did algebraic analysis: by following the rules of

algebra. That is what lay behind the notion of mechanizing logic; it is what lay

behind Turing's idea of capturing the notion of computability in an abstract

machine.(12)

10. "Algorithm" derived

from "algorismus", which

in turn was the Latin form

of al-Khwarizmi, the author

of the first Arabic treatise

on calculation with

"Indian" numbers, the

decimal place-valuesystem. Translated by

Robert of Chester in the

12th century, theLiber

algoritmi de numero

indorum became the basis

of a series of textbooks on

arithmetic, the most widely

read of which was John of

Holywood'sAlgorismus

vulgaris (ca. 1220). The

word acquired a 'th' in the

17th century by a back-

formation evidently based

on the assumption that the

word was originally Greek.

11. "... quando orientur

controvers iae, non magis

disputatione opus erit inter

duos philosophos, quam

inter duos Computistas.

Sufficiet enim calamos in

manus sumere sederequead abacos, et s ibi mutuo

(accito si placet amico)

dicere: calculemus."Die

philosophischen Schriften

von Gottfried Wilhelm

Leibniz, ed. C. J. Gerhardt

(Berlin, 1890, VII, 200. (I

thank Siegfried Probst for

locating this source and

pos ting it to the Historia

Mathematica list.)

To make the notion of "computable" as clear and simple as possible, Alan

Turing proposed in 1936 a mechanical model of what a human does when

12. On the mechanization of

logic, see Sybille Krmer,

Symbolische Maschinen:
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computing:

We may compare a man in the process of computing a real

number to a machine which is only capable of a finite number

of conditions q1, q2, ..., qRwhich will be called "m-

configurations". The machine is supplied with a "tape" (the

analogue of paper) running through it, and divided into sections

(called "squares") each capable of bearing a "symbol".(13)

Turing imagined, then, a tape divided into cells, each containing one of a

finite number of symbols. The tape passes through a machine that can read

the contents of a cell, write to it, and move the tape one cell in either

direction. What the machine does depends on its current state, which

includes a signal to read or write, a signal to move the tape right or left, and

a shift to the next state. The number of states is finite, and the set of states

corresponds to the computation. Since a state may be described in terms of

three symbols (read/write, shift right/left, next state), a computation may itselfbe expressed as a sequence of symbols, which can also be placed on the

tape, thus making possible a universal machine that can read a computation

and then carry it out by emulating the machine described by it.

Die Idee der

Formalisierung in

geschichtlichem Abri

(Darmstadt:

Wissenschaftliche

Buchgesellschaft, 1988)

and Martin Davis, The

Universal Machine: The

Road from Leibniz to

Turing(NY/London: W.W.Norton, 2000).

13. "On Computable

Numbers, with an

Application to the

Entscheidungsproblem",

Proceedings of the London

Mathematical Society,

ser.2, vol. 42(1936), 230-

265; at 231.

Turing's machine, or rather his monograph, belonged to the then current

agenda of mathematical logic. TheEntscheidungsproblem stemmed from

David Hilbert's program of formalizing mathematics; as stated in the

textbook he wrote with W. Ackermann,

TheEntscheidungsproblem is solved when one knows a

procedure by which one can decide in a finite number of

operations whether a given logical expression is generally valid

or is satisfiable. The solution of theEntscheidungsproblem is

of fundamental importance for the theory of all fields, the

theorems of which are at all capable of logical development

from finitely many axioms.(14)

14. D. Hilbert and W.

Ackermann, Grundzge

der theoretischen Logik

(Berlin: Springer, 1928), 73-

4: Das

Entscheidungsproblem ist

gelst, wenn man ein

Verfahren kennt, das bei

einem vorgelegten

logischen Ausdruck durch

endlich viele Operationen

die Entscheidung ber die

Allgemeingltigkeit bzw.

Erfllbarkeit erlaubt. Die

Lsung des

Entscheidungsproblems ist

fr die Theorie allerGebiete, deren Stze

berhaupt einer logischen

Entwickelbarkeit aus

endlich vielen Axiomen

fhig sind, von

grundstzlicher

Wichtigkeit.

Turing designed his machine to compute theEntscheidungsproblem, or

rather to show that it was uncomputable. Just as he was submitting his paper

to the London Mathematical Society, Alonzo Church published an articlewhich anticipated Turing's results by means of a different sort of logical

calculus, namely the lambda calculus, and an equivalent notion of

"computability", which Church called "effective calculability". With the help


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of W.H.A. Newman, for whose course Turing originally wrote his paper,

Turing went to study with Church at Princeton, where he subsequently

showed that his machine had the same power as Church's lambda calculus

or Stephen Kleene's recursive function theory for determining the range and

limitations of axiom systems for mathematics.(15) For the next several years,

all these schemes remained abstract devices of metamathematics. For quite

independent reasons, the war changed that.

15. Stephen C. Kleene,

"Origins of Recursive

Function Theory",Annals

of the History of

Computing3,1(1981), 52-67

II. Thinking Numerically - Computational

Thinking: The Computer

The Harvard Mark I and ENIAC marked the culmination of the

development of a mechanical calculator, and the latter marked the turning

point to electronic digital computation. John von Neumann first encountered

ENIAC in his role as mathematical physicist looking for means of rapid

numerical solution of non-linear partial differential equations.(16) But as he

talked with ENIAC's creators about the next version of their device, he

shifted his focus to a different sets of concerns. In the concept of the stored

program he not only saw a means of achieving a working device with the

capacity in principle to behave as a Turing machine, but he had a vision also

of what that capacity might mean for doing science. By analogy with

organisms viewed as natural automata, computers as artificial automata had

the potential to grow with the problems they were meant to solve. In

particular, he contemplated the conditions under which an automaton couldreplicate itself. While von Neumann imagined an actual machine floating in a

primeval sea of components, his colleague Stanislaw Ulam suggested instead

the model of a cellular automaton, that is a two-dimensional array of cells

each containing a finite automaton which changes its state as a function of the

states of the cells surrounding it. One could then ask about the possible

configurations of cells that would be capable of reproducing themselves in

the space of the cellular automaton.(17)

16. On von Neumann, see

William Aspray,John von

Neumann and the Origins

of Modern Computing

(Cambridge: MIT Press ,

1990).

17. Arthur Burks, "VonNeumann's Self-

Reproducing Automata", in

Papers of John von

Neumann on Computing

and Computer Theory, ed.

William Aspray and Arthur

Burks (Cambridge,

MA/London: MIT Press;

Los Angeles/San

Francisco: Tomash

Publishers, 1987), 491-552.

Von Neumann also pointed to a fundamental problem posed by the use of

the computer as a means of thinking about the world, and indeed about

thinking itself. To the extent that science seeks mathematical understanding,

that is understanding that has the certainty and analytical transparency of

mathematics, then one needed a mathematical understanding of the

computer. As of the early 1950s, no such mathematical theory of the

computer existed, and von Neumann could only vaguely discern its likely

shape:

There exists today a very elaborate system of formal logic, and,

specifically, of logic as applied to mathematics. This is a

discipline with many good sides, but also with certain serious
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weaknesses. This is not the occasion to enlarge upon the good

sides, which I certainly have no intention to belittle. About the

inadequacies, however, this may be said: Everybody who has

worked in formal logic will confirm that it is one of the

technically most refractory parts of mathematics. The reason

for this is that it deals with rigid, all-or-none concepts, and has

very little contact with the continuous concept of the real or of

the complex number, that is, with mathematical analysis. Yetanalysis is the technically most successful and best-elaborated

part of mathematics. Thus formal logic is, by the nature of its

approach, cut off from the best cultivated portions of

mathematics, and forced onto the most difficult part of the

mathematical terrain, into combinatorics.

The theory of automata, of the digital, all-or-none type, as

discussed up to now, is certainly a chapter in formal logic. It

will have to be, from the mathematical point of view,

combinatory rather than analytical.(18)

Neither here nor in later lectures did von Neumann elaborate on the nature

of that combinatory mathematics, nor suggest from what areas of current

mathematical research it might be drawn.

Over the two decades following von Neumann's work on automata,

researchers from a variety of disciplines converged on a mathematical theory

of computation, composed of three main branches: the theory of automata

and formal languages, the theory of algorithms and computational

complexity, and formal semantics.(19) The core of the first field came to lie in

the correlation between four classes of finite automata ranging from the

sequential circuit to the Turing machine and the four classes of phrase

structure grammars set forth by Noam Chomsky in his classic paper of

1959.(20) With each class goes a particular body of mathematical structures

and techniques.

18. John von Neumann,

"On a logical and general

theory of automata" in

Cerebral Mechanisms inBehavior--The Hixon

Symposium, ed. L.A.

Jeffries (New York: Wiley,

1951), 1-31; repr. in Papers,

391-431; at 406.

19. For more detail see my

"Computers and

Mathematics: The Search

for a Discipline of

Computer Science", in J.

Echeverra, A. Ibarra and T.Mormann (eds.), The Space

of Mathematics

(Berlin/New York: De

Gruyter, 1992), 347-61, and

"Computer Science: The

Search for a Mathematical

Theory", in John Krige and

Dominique Pestre (eds.),

Science in the 20th

Century (Amsterdam:

Harwood Academic

Publishers, 1997), Chap. 31.

20. Noam Chomsky, "On

certain formal properties of

grammars",Information

and Control2,2(1959), 137-

167.

Two features of the mathematics warrant particular attention. First, as the

study of sequences of symbols and of the transformations carried out on

them, theoretical computer science became a field of application for the most

abstract structures of modern algebra: semigroups, lattices, finite Boolean

algebras, -algebras, categories. Indeed, it soon gave rise to what otherwise

might have seemed the faintly contradictory notion of "applied abstract

algebra".(21) Second, as the computer became a point of convergence for a

variety of scientific interests, among them mathematics and logic, electrical

engineering, artificial intelligence, neurophysiology, linguistics, and computer

programming, algebra served to reveal the abstract structures common to

these enterprises. Once established, the mathematics of computation then

became a means of thinking about the sciences, in particular about questionsthat have resisted traditional reductionist approaches. Two examples of

particular importance to biology are Aristide Lindenmayer's L-systems, an

application of formal language theory to patterns of growth, and, more

21. See, for example, Garrett

Birkhoff and Thomas C.

Bartee,Modern AppliedAlgebra (New York:

McGraw-Hill, 1970) and

Rudolf Lidl and Gunter Pilz,

Applied Abstract Algebra

(NY: Springer, 1984).

22. Aristide Lindenmayer,

"Mathematical models for

cellular interactions in

development",Journal of

Theoretical Biology

18(1968), 280-99, 300-15. W.Fontana and Leo W. Buss,

"The barrier of objects:

From dynamical systems to

bounded organizations ", in
http://www.princeton.edu/~hos/Mahoney/articles/ivofest/17Cpersp.ivo.html#N_21_http://newwin%28%27http//www.princeton.edu/~hos/Mahoney/articles/20thcSci/20thcent.html')http://newwin%28%27http//www.princeton.edu/~hos/Mahoney/articles/compsci/sansebas.html')http://www.princeton.edu/~hos/Mahoney/articles/ivofest/17Cpersp.ivo.html#N_20_http://www.princeton.edu/~hos/Mahoney/articles/ivofest/17Cpersp.ivo.html#N_19_http://www.princeton.edu/~hos/Mahoney/articles/ivofest/17Cpersp.ivo.html#N_18_


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recently, Walter Fontana's and Leo Buss's theory of biological organization

based on the model of the lambda calculus.(22)

J. Casti and A. Karlqvist

(eds.),Boundaries and

Barriers (Reading, MA:

Addison-Wesley, 1996),

56-116.

III. Generating the World

The computer is essential to those new approaches to biology, as it is to the

application of cellular automata to a range of physical, biological, ecological,

and economic investigations.(23) It is not a matter of calculating numbers

where analytic solutions are not possible, but rather of defining the local

interactions of a large number of elements of a system and then letting the

system evolve computationally, because we have neither the time nor the

mental capacity to derive that system. For example, rather than seeking a

numerical approximation to the non-linear partial differential equations of

fluid flow, one models the interaction of neighboring particles and displays

the result graphically. Instead of a mathematical function, what emerges is a

picture of the evolving system; an analytical solution is replaced by the stages

of a time series.

23. For a general view, see

Gary William Flake, The

Computational Beauty of

Nature: Computer

Explorations of Fractals,

Chaos, Complex Systems,

and Adaptation

(Cambridge, Mass: MIT

Press, 1998).

In other applications, the results may include new elements or new forms of

interaction among them. In particular, the system as a whole may acquire

new properties, which emerge when the interactions among the elements

reach a certain level of complexity. Precisely because the properties are a

product of complexity, that is, of the system itself, they cannot be reduced

analytically to the properties of the constituent elements. The current state of

mathematics does not suffice to gain analytical insight into the structures of

such systems, and hence, although the computer by its nature is

mathematical, we do not have means of understanding its mathematics, or

rather the computation does not afford mathematical understanding, certainly

not in the sense of Newton'sPrincipia.

In a certain sense, the notion of complexity as an emergent property of

systems governed locally by simple relationships may have lain inherent in themechanistic world view set forth in the seventeenth century. What was real

was matter in motion. Matter had no essential properties other than mass or

bulk, by which pieces of it occupied space to the exclusion of other pieces.

Motion was a matter of change of place with respect to time, brought about

by pieces of matter pushing one another around. None of this was directly

observable; rather, it underlay observation itself. What the senses perceived,

each in its own way, was changing patterns of matter impinging on matter.

The complexity of the world lay in the complexity of those patterns of

interaction. For Descartes, the behavior of heavy bodies referred to as"gravity" emerged from the interaction of the particles of the vortex

surrounding the earth.

At one level, this remains the case. The ultimate particles may embody a


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different catalog of essential properties, the laws of interaction may take a

different form, but nothing of the "new" science challenges the role of those

particles as the ultimate building blocks of the physical world. Similarly,

nobody doubts that life as we know it is a chemical phenomenon, resting in

principle on the interaction of fundamental particles. People now speak of

"carbon-based" life, using the qualifier to suggest that there could be some

other form but in so doing also accepting and reinforcing the premiss that life

is a form of chemistry reflecting the potential inherent in the physicalproperties of carbon and hydrogen, which properties themselves emerge

from the different numbers and configurations of the electrons, protons, and

neutrons that constitute the atoms

What has changed is the attitude toward the means of expressing the

relationships among the fundamental particles and of transforming

expressions at one level into expressions at another level. In thePrincipia,

Newton could capture the basic relationship of bodies attracting one another

by the expression ma = mm'/r

2

, where a by definition is d

2

S/dt

2

. Movingfrom small particles to large bodies was facilitated by being able to show that

forces among the constituents of a body conjoined to act as a single force

concentrated at its center of mass. The equation relating the forces of two

bodies acting on one another over a distance proved mathematically

tractable, in the sense that one could solve it in closed analytic form.

Unfortunately, the equation for three or more bodies, needed for any precise

mathematical account of the motion of the planets and in particular for the

motion of the moon about the earth, did not yield so easily to the techniques

of the calculus.

The subsequent articulation of the mechanical model of the physical world

increasingly challenged the capacity of mathematics to transform descriptions

at the level of the fundamental elements into descriptions at the level of direct

experience. The main root of the modern computer leads directly from the

need to substitute numerical approximations for differential equations that

could not be solved in closed form. That was especially the case for systems

of non-linear partial differential equations. The complexity of the systems

they described did not lie in the equations but in their solutions, and it was

admittedly a complexity that could not be captured in closed form. Now themodels are not expressed in a general differential equation characterizing the

whole system, which equation is then solved analytically or calculated with

the aid of a computer. Rather, they are described at the local level by means

of interactions with the immediate neighborhood, and the result is then

generated. Thus the sciences seem to have given up on mathematical

explanation.(24)

24. For the most recent and

perhaps most extreme

argument against thecapacity of traditional

mathematics and

mathematical phys ics to

encompass the complexity

of the world, see Stephen

Wolfram,A New Kind of

Science (Champaign, IL:

Wolfram Media, Inc. 2002).

Not entirely, however, and not without a struggle. John Holland, a pioneer in

the application of cellular automata to biology and the creator of geneticalgorithms, shows that some in the new field are not yet ready to surrender

the insights of mathematical analysis. In the concluding chapter ofHidden

Order: How Adaptation Builds Complexity, Holland looks "Toward


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Theory" and "the general principles that will deepen our understanding ofall

complex adaptive systems [cas]". As a point of departure he insists that:

Mathematics is our sine qua non on this part of the journey.

Fortunately, we need not delve into the details to describe the

form of the mathematics and what it can contribute; the details

will probably change anyhow, as we close in on our

destination. Mathematics has a critical role because it alongenables us to formulate rigorous generalizations, or principles.

Neither physical experiments nor computer-based experiments,

on their own, can provide such generalizations. Physical

experiments usually are limited to supplying input and

constraints for rigorous models, because the experiments

themselves are rarely described in a language that permits

deductive exploration. Computer-based experiments have

rigorous descriptions, but they deal only in specifics. A well-

designed mathematical model, on the other hand, generalizes

the particulars revealed by physical experiments, computer-

based models, and interdisciplinary comparisons. Furthermore,

the tools of mathematics provide rigorous derivations and

predictions applicable to all cas. Only mathematics can take us

the full distance.(25)

25. John H. Holland,

Hidden Order: How

Adaptation Builds

Complexity (Reading, MA:

Addison-Wesley,

1995)161-2.

Details aside, Holland's goal, with which he associates his colleagues at the

Santa Fe Institute, reflects a vision of mathematics that he and they share

with mathematicians from Descartes to von Neumann.

As von Neumann insisted in 1948, the mathematics will be different. To

meet Holland's needs it "[will have to] depart from traditional approaches to

emphasize persistent features of the far-from-equilibrium evolutionary

trajectories generated by recombination."(26) Nonetheless, his sketch of the

specific form the mathematics might take suggests that it will depart from

traditional approaches along branches rather than across chasms, and that it

will be algebraic. As the most recent work of Fontana on the lambda

calculus applied to chemistry suggests, it will be a mathematics of adecidedly modern sort. That is to be expected. The Turing machine is a

modern concept, conceived by a thinker who was nothing if not a

reductionist. His 1936 paper sets on computation, and thus on computing

machines, limits that are no less firm and no less universally accepted than

the constraints of the laws of thermodynamics or of the constant speed of

light.

26.Ibid., 171-2.

Today we confront the question of whether the computer, the newest and

leading medium of scientific thought can be comprehended mathematically,

i.e. in some way algebraically or analytically. If so, then it will be viewed as

the newest chapter of a history that began in the 17th century with the

beginning of algebraic thought. If not, then perhaps fifty years from now


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someone will be giving a lecture on the topic of "The End of Algebraic

Thought in the 20th Century."

calculation thinking computational thinking

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