Abstract : This paper will discuss the works and influences of ...
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Abstract : This paper will discuss the works and influences of ...


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Norton 1 Jennifer Norton Jensen Mth 467 30 April 2009 Abstract: This paper will discuss the works and influences of Diophantus of Alexandria. Through his puzzling details of life, his surviving works, his advancement in algebraic notation, and his influence on mathematicians, such as Fermat, Diophantus made a great impression upon mathematics. The Life and Works of Diophantus of Alexandria Diophantus of Alexandria is known as the “father of algebra”.
  • sixteenth century through the eighteenth century
  • unknown quantity
  • diophantus
  • sum
  • algebra
  • life
  • numbers
  • problems
  • work



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Nombre de lectures 25
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Confronting Science’s
Logical Limits
The mathematical models now used in many scientific fields
may be fundamentally unable to answer certain questions about
the real world. Yet there may be ways around these problems
by John L. Casti
o anyone infected with the can answer all questions about numbers. this issue is to settle what we mean by
idea that the human mind is A few years later Alan M. Turing proved “scientific knowledge.” To cut throughTunlimited in its capacity to an- an equivalent assertion about computer this philosophical Gordian knot, let me
swer questions, a tour of 20th-century programs, which states that there is no adopt the perhaps moderately contro-
mathematics must be rather disturbing. systematic way to determine whether a versial position that a scientific way of
In 1931 Kurt Gödel set forth his incom- given program will ever halt when pro- answering a question takes the form of
pleteness theorem, which established cessing a set of data. More recently, a set of rules, or program. We simply
that no system of deductive inference Gregory J. Chaitin of IBM has found feed the question into the rules as input,
arithmetic propositions whose truth can turn the crank of logical deduction and
never be established by following any wait for the answer to appear.
deductive rules. Thinking of scientific knowledge as
These findings proscribe our ability to being generated by what amounts to a
know in the world of mathematics and computer program raises the issue of
logic. Are there similar limits to our abil- computational intractability. The diffi-
ity to answer questions about natural culty of solving the celebrated travel-
and human affairs? The first and per- ing-salesman problem, which involves
haps most vexing task in confronting finding the shortest route connecting a
large number of cities, is widely believed
to increase exponentially as the number
of destinations rises. For example, pin-
pointing the best itinerary for a sales-
man visiting 100 cities would require
examining 100 99 98 97 ... 1
possibilities—a task that would take
even the fastest computer billions of
years to complete.
But such a computation is possible—at
least in principle. Our focus is on ques-
tions for which there exists no program
at all that can produce an answer. What
would be needed for the world of phys-
ical phenomena to display the kind of
logical unanswerability seen in mathe-
matics? I contend that nature would
have to be either inconsistent or incom-
plete, in the following senses. Consis-
tency means that there are no true para-
the world’s fastest computer running for
billions of years to calculate the shortest
route between 100 destinations. Scientists
are now seeking ways to make such daunt-
ing problems more tractable.
102 Scientific American October 1996 Confronting Science’s Logical Limits
Copyright 1996 Scientific American, Inc.doxes in nature. In general, when we
encounter what appears to be such a
paradox—such as jets of gas that seemed
to be ejected from quasars at faster than
light speeds—subsequent investigation
has provided a resolution. (The “super-
luminal” jets turned out to be an opti-
cal illusion stemming from relativistic
Completeness of nature implies that a
physical state cannot arise for no rea- masses moving in accordance with New-
son whatsoever; in short, there is a cause ton’s law of gravitational attraction.
for every effect. Some analysts might One version of the problem addresses PROTEIN-FOLDING PROBLEM con-
siders how a string of amino acids (left)object that quantum theory contradicts whether two or more of these bodies
folds up almost instantaneously into anthe claim that nature is consistent and will collide or whether one will acquire
extraordinarily complex, three-dimension-complete. Actually, the equation gov- an arbitrarily high velocity in a finite
al protein (right). Biologists are now try-
erning the wave function of a quantum time. In his 1988 doctoral dissertation,
ing to unravel the biochemical “rules” that
phenomenon provides a causal expla- Zhihong ( Jeff) Xia of Northwestern proteins follow in accomplishing this feat.
nation for every observation (complete- University showed how a single body
ness) and is well defined at each instant moving back and forth between two bi-
in time (consistency). The notorious nary systems (for a total of five masses) plausible rules for protein folding would
127“paradoxes” of quantum mechanics could approach an arbitrarily high ve- need 10 years to find the final folded
arise because we insist on thinking of locity and be expelled from the system. form for even a very short sequence
the quantum object as a classical one. This result, which was based on a special consisting of just 100 amino acids. In
geometric configuration of the bodies, fact, in 1993 Aviezri S. Fraenkel of the
A Triad of Riddles says nothing about the specific case of University of Pennsylvania showed that
our solar system. But it does suggest that the mathematical formulation of the
t is my belief that nature is both con- perhaps the solar system might not be protein-folding problem is computation-Isistent and complete. On the other stable. More important, the finding of- ally “hard” in the same way that the
hand, science’s dependence on mathe- fers new tools with which to investigate traveling-salesman problem is hard.
matics and deduction hampers our abil- the matter. How does nature do it?
ity to answer certain questions about • Protein folding. The proteins mak- • Market efficiency. One of the pillars
the natural world. To bring this issue ing up every living organism are all on which the classical academic theory
into sharper focus, let us look at three formed as sequences of a large number of finance rests is the idea that financial
well-known problems from the areas of of amino acids, strung out like beads on markets are “efficient.” That is, the
physics, biology and economics. a necklace. Once the beads are put in market immediately processes all infor-
• Stability of the solar system. The the right sequence, the protein folds up mation affecting the price of a stock or
most famous question of classical me- rapidly into a highly specific three-di- commodity and incorporates it into the
chanics is the N-body problem. Broadly mensional structure that determines its current price of the security. Conse-
speaking, this problem looks at the be- function in the organism. It has been es- quently, prices should move in an un-
havior of a number, N, of point-size timated that a supercomputer applying predictable, essentially random fashion,
Confronting Science’s Logical Limits Scientific American October 1996 103
Copyright 1996 Scientific American, Inc.
ILLUSTRATIONS BY LAURIE GRACEdiscounting the effect of inflation. This, sured position of planets or the actual
in turn, means that trading schemes observed configuration of a protein.
based on any publicly available infor- Such observables generally constitute a
mation, such as price histories, should discrete set of measurements taking their
be useless; there can be no scheme that values in some finite set of numbers.
performs better than the market as a Moreover, such measurements are gen-
whole over a significant interval. But ac- erally not exact.
tual markets do not seem to pay much In the world of mathematics, on the
attention to academic theory. The fi- other hand, we have symbolic represen-
nance literature is filled with such mar- tations of such real-world observables,
ket “anomalies” as the low price–earn- where the symbols are often assumed
ings ratio effect, which states that the to belong to a continuum in both space
stocks of firms whose prices are low rel- and time. The mathematical symbols
ative to their earnings consistently out- representing attributes such as position involved, just plain English. On the oth-
perform the market overall. and speed usually have numerical values er hand, constructing a convincing caus-
that are integers, real numbers or com- al argument without recourse to mathe-
The Unreality of Mathematics plex numbers, all systems containing an matics may be a daunting task. In the
infinite number of elements. In mathe- case of the stability of the solar system,
ur examination of the three ques- matics the concept of choice for charac- for example, one must find compellingOtions posed above has yielded what terizing uncertainty is randomness. nonmathematical definitions of the
appear to be three answers: the solar Finally, there is the world of compu- planets and gravity.
system may not be stable, protein fold- tation, which occupies the curious posi- Given these difficulties, it seems wise
ing is computationally hard, and finan- tion of having one foot in the real world to consider approaches that mix the
cial markets are probably not complete- of physical devices and one foot in the worlds of nature and mathematics. If
ly efficient. But what each of these pu- world of abstract mathematical objects. we want to invoke the proof machinery
tative “answers” has in common is that If we think of computation as the exe- of mathematics to settle a particular real-
it involves a mathematical representa- cution of a set of rules, or algorithm, the world question, it is first necessary to
tion of the real-world question, not the process is a purely mathematical one “encode” the question as a statement in
question itself. For instance, Xia’s solu- belonging to the world of symbolic ob- some mathematical formalism, such as
tion of the N-body problem does not jects. But if we regard a computation as a differential equation, a graph or an
explain how real planetary bodies move the process of turning switches on or off N-person game. We settle the mathemat-
in accordance with real-world gravita- in the memory of an actual computing ical version of the question using the
tional forces. Similarly, Fraenkel’s con- machine, then it is a process firmly root- tools and techniques of this particular
clusion that protein folding is computa- ed in the world of physical observables. corner of the mathematical world, even-
tionally hard fails to address the issue One way to demonstrate whether a tually “decoding” the answer (if there is
of how real proteins manage to do their given question is logically impossible to one!) back into real-world terms. One
job in seconds rather than eons. And, answer by scientific means is to restrict challenge here is establishing that the
of course, canny Wall Street operators all discussion and arguments solely to mathematical version of the problem is
have thumbed their noses at the effi- the world of natural phenomena. If we a faithful representation of the question
cient-market hypothesis for decades. So follow this path, we are forbidden to as it arises in the real world. How do
to draw any conclusions about the in- translate a question such as “Is the so- we know that mathematical models of
ability of science to deal with these ques- lar system stable?” into a mathematical a natural system and the system itself
tions, we must either justify the mathe- statement and thereby to generate an bear any relation to each other? This is
matical model as a faithful representa- answer with the logical proof mecha- an old philosophical conundrum, en-
tion of the physical situation or abandon nism of mathematics. We then face the tailing the development of a theory of
the mathematics altogether. We consid- problem of finding a substitute in the models for its resolution. Moreover,
er both possibilities in what follows. physical world for the concept of math- mathematical arguments may be sub-
What these examples show is that if ematical proof. ject to the constraints revealed by Gö-
we want to look for scientifically unan- A good candidate is the notion of del, Turing and Chaitin; we do not know
swerable questions in the real world, causality. A question can be considered yet whether the real world is similarly
we must carefully distinguish between scientifically answerable, in principle, if constrained.
the world of natural and human phe- it is possible to produce a chain of causal
nomena and mathematical and compu- arguments whose final link is the answer The Noncomputational Mind
tational models of those worlds. The ob- to the question. A causal argument need
jects of the real world consist of directly not be expressed in mathematical terms. here may be ways to sidestep these
observable quantities, such as time and Tissues. The problems identified byFor example, the standard deductive ar-
position, or quantities, such as energy, gument “All men are mortal; Socrates is Gödel and others apply to number sys-
tems with infinite elements, such as thethat are derived from them. Thus, we a man; therefore, Socrates is mortal” is
consider parameters such as the mea- a causal chain. There is no mathematics set of all integers. But many real-world
104 Scientific American October 1996 Confronting Science’s Logical Limits
Copyright 1996 Scientific American, Inc.N-BODY SYSTEM consisting of a point
mass oscillating between two binary sys-
tems (left) is unstable, according to a the-
orem by Zhihong Xia of Northwestern
University. Such work may reveal wheth-
er the solar system will someday expel
one of its planets into deep space.
problems, such as the traveling-sales-
man problem, involve a finite number
of variables, each of which can take only mathematical physicist Roger Penrose
a finite number of possible values. of the University of Oxford, have ar-
Similarly, nondeductive modes of rea- gued that human cognitive activity is not
soning—induction, for instance, in which based on any known deductive rules and
we jump to a general conclusion on the is thus not subject to Gödelian limits.
basis of a finite number of specific ob- Recently this viewpoint has been bol-
servations—can take us beyond the stered by studies carried out under the
realm of logical undecidability. So if we aegis of the Institute for Future Studies
restrict our mathematical formalisms to in Stockholm by me, the psychologist
systems using finite sets of numbers or Margaret A. Boden of the University of
nondeductive logic, or both, every math- Sussex, the mathematician Donald G.
ematical question should be answer- Saari of Northwestern University, the tists may be able to solve some seem-
able; hence, we can expect the decoded economist Åke E. Andersson (the insti- ingly intractable problems.
real-world counterpart of such ques- tute’s director) and others. Our work Of course, science’s ability to plumb
tions to be answerable as well. strongly suggests that in the arts as well nature’s secrets is limited by many prac-
Studies of the human mind may re- as in the natural sciences and mathemat- tical considerations—such as measure-
veal other ways to bypass logical limits. ics, the human creative capacity is not ment error, length of computation, phys-
Some artificial-intelligence proponents subject to the rigid constraints of a com- ical and economic resources, political
have proposed that our brains are com- puter’s calculations. Penrose and other will and cultural values. But none of
puters, albeit extremely sophisticated theorists have conjectured that human these considerations bears on whether
ones, that perform calculations in the creativity stems from some still unknown there is a logical barrier to our answer-
same logical, step-by-step fashion that mechanisms or rules, perhaps related to ing a certain question about the natural
conventional computers (and even par- quantum mechanics. By uncovering world. My contention is that there is not.
allel processors and neural networks) these mechanisms and incorporating So a tour of 20th-century mathematics
do. But various theorists, notably the need not be so disturbing after all!them into the scientific method, scien-
The Author Further Reading
JOHN L. CASTI is a professor at the Technical University of Vi- Searching for Certainty. John L. Casti. William Morrow, 1991.
enna and at the Santa Fe Institute (casti@santafe.edu). He is grateful Randomness and Undecidability in Physics. K. Svozil. World
to Joseph F. Traub, Piet Hut, James B. Hartle and Åke E. Andersson Scientific, Singapore, 1994.
for stimulating discussions on these matters and to the Institute for Boundaries and Barriers. Edited by John L. Casti and A. Karl-
Future Studies in Stockholm for partial support of this research. qvist. Addison-Wesley, 1996.
Confronting Science’s Logical Limits Scientific American October 1996 105
Copyright 1996 Scientific American, Inc.

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