ISSN 1393-614X Minerva - An Internet Journal of
Philosophy Vol. 8 2004.
____________________________________________________
ADVANCE IN
MONTE CARLO SIMULATIONS AND ROBUSTNESS STUDY AND THEIR IMPLICATIONS FOR
THE DISPUTE IN PHILOSOPHY OF MATHEMATICS
Chong Ho Yu |
Abstract
Both Carnap
and Quine
made significant contributions to the philosophy of mathematics despite
their
diversed views. Carnap
endorsed the
dichotomy between analytic and synthetic knowledge and classified
certain
mathematical questions as internal questions appealing to logic and
convention.
On the contrary, Quine was opposed to the analytic-synthetic
distinction and
promoted a holistic view of scientific inquiry. The
purpose of this paper is to argue that in light of the recent
advancement of experimental mathematics such as Monte Carlo
simulations,
limiting mathematical inquiry to the domain of logic is unjustified.
Robustness
studies implemented in Monte Carlo Studies demonstrate that mathematics
is on
par with other experimental-based sciences.
Introduction
Carnap and Quine
made tremendous contributions to numerous areas of modern philosophy,
including
the philosophy of mathematics. Carnap endorsed the dichotomy between
analytic
and synthetic knowledge and classified certain mathematical questions
as internal
questions appealing to logic and convention. In addition, he regarded
the
ontological question about the reality of
mathematical objects as a pseudo-question. On the
contrary, Quine made an ontological commitment to mathematical entities
by
asserting that mathematical objects are on par with physical objects.
This
assertion is tied to his belief that there is no first philosophy prior
to
natural science. In addition, Quine was opposed to the
analytic-synthetic
distinction and promoted a holistic view of scientific inquiry. On one
hand,
Quine recognized that there are differences between logic/mathematics
and
physical sciences. On the other hand, Quine maintained that it is a
mistake to
hold a dualistic view. For Quine logic and mathematics are essentially
empirically-based and they are subject to revision according to new
evidence.
The purpose of this paper is to argue that in light of the recent
advancement
of experimental mathematics such as Monte Carlo simulations, limiting
mathematical inquiry to the domain of logic is unjustified. Robustness
studies
implemented in Monte Carlo Studies demonstrate that mathematics is on a
par
with other experimental-based sciences.
Quine (1966/1976)
wrote, “Carnap more than anyone else was the embodiment of logical
positivism,
logical empiricism, the Vienna circle” (p. 40). To discuss Carnap’s
philosophy
of mathematics, it is essential to illustrate the ideas of the Vienna
circle,
as well as how members of the Vienna circle adopted and rejected other
ideas.
In the following, the theories of Frege, Russell, Whitehead and Gödel
will be
briefly introduced. These are by no means the only ones who are related
to the
formulation of Carnap’s and the Vienna Circle’s notions. Nonetheless,
since
this article concentrates on the argument against the logical view of
mathematics endorsed by Carnap, discussion of Frege, Russell, Whitehead
and
Gödel is germane to the topic.
DIFFERENT VIEWS
ON THE PHILOSOPHY OF MATHEMATICS
The Vienna
Circle
Logical
positivism, which originated with the Vienna circle, embraced
verificationism
as the criterion for obtaining meaningful knowledge. The verification
criterion
is not just a demand for evidence. Verification does not mean that,
with other
things being equal, a proposition that can be verified is of vastly
greater
significance than one that cannot. Rather, the verification thesis is
much more
restrictive than the above. According to logical positivism, a
statement is
meaningless if verification is not possible or the criteria for
verification are
not clear (Ayer, 1936; Schlick, 1959). To be specific, the verification
principle is not an account of the relative importance of propositions,
but a
definition of meaning. Meaning and verifiability are almost
interchangeable
(Werkmeister, 1937). The principle of verification was used by the
Vienna
Circle as a tool to counteract metaphysics by enforcing adherence to
empiricism. However, one may then ask how we can substantiate
mathematical
knowledge when mathematics is considered by many to be a form of
knowledge that
cannot be verified by sensory input. Following the strict criterion of
verifictionism, the analytic philosopher Ayer (1946) has said that
mathematics
is nonsense. In his view, mathematics says nothing about the world.
What it can
accomplish is to enlighten us how to manipulate symbols.
Russell and
Whitehead
In order to make
sense out of mathematics, logical positivists adopted a view of
mathematics in
the Frege-Russell-Whitehead tradition. This tradition took care of
logic and mathematics,
and thus left a separate epistemological problem of non-logical and
non-mathematical discourse (Isaacson, 2000). According to Frege
(1884/1960),
logical and mathematical truths are true by virtue of the nature of
thought.
This notion is further expanded by Russell, and also by collaboration
between
Russell and Whitehead.
In Russell's view
(1919), in order to uncover the underlying structures of mathematical
objects,
mathematics should be reduced to a more basic element, namely, logic.
Thus, his
approach is termed logical atomism. Russell's philosophy of mathematics
is
mainly concerned with geometry. At the time of Russell, the existence
of
geometric objects and the epistemology of geometry could not be
answered by
empiricists. In geometry a line can be broken down infinitely to a
smaller
line. We can neither see nor feel a mathematical line or a mathematical
point.
Thus, it seems that geometric objects are not objects of empirical
perception
(sense experience). If this is true, how could conceptions of such
objects and
their properties be derived from experience as an empiricist would
require?
Russell's answer is that although geometric objects are theoretical
objects, we
can still understand geometric structures by applying logic to the
study of
relationships among those objects: "What matters in mathematics, and to
a
very great extent in physical science, is not the intrinsic nature of
our
terms, but the logical nature of their inter-relations" (1919, p.59).
Whitehead and
Russell’s work on “Principia Mathematica” (1910/1950) is a bold attempt
to
develop a fully self-sufficient mathematical system through logical
relationships. For Russell and Whitehead, mathematics is a purely
formal
science. The existence of mathematical objects is conditional upon
structures.
If a certain structure or system exists, then there also exist some
other
structures or systems whose existence follows logically from the
existence of
the former. In their view, mathematics could be reduced to logical
relationships within the logical system without external references.
The
Frege-Russell-Whitehead tradition is considered the logical approach to
mathematics. This approach is said to be a solution to infinite regress
or
circular proof.
However, the proposal
by Whitehead and Russell is seriously challenged by Gödel. Gödel (1944,
1961)
proposed that a complete and consistent mathematical system is
inherently
impossible, and within any consistent mathematical system there are
propositions that cannot be proved or disproved on the basis of the
axioms
within that system. Thus, the consequences drawn from mathematical
axioms have
meaning only in a hypothetical sense. In addition, mathematical
propositions
cannot be proved by using combinations of symbols without introducing
more
abstract elements. In Gödel’s sense, logicism in
mathematics does not solve the
problem of infinite regress or circular proof.
In rejecting the
logical approach, Gödel took an "intuitionistic" position to
mathematics. Unlike Russell, who
asserted mathematical structures exist in terms of relationships, Gödel
maintained that it is not a question of whether there are some real
objects
"out there". Rather, our sequences of acts construct our perceptions
of so-called "reality" (Tieszen, 1992). According to Gödel,
"despite their remoteness from sense experience, we do have something
like
a perception also of the objects of set theory… I don't see any reason
why we
should have less confidence in this kind of perception, i.e. in
mathematical
intuition, than in sense perception" (cited in Lindstrom, 2000,
p.123). Indeed, there are followers of
Gödel’s even in the late 20^{th} century. Jaffe and Quinn
(1993)
observed that there is “a trend toward basing mathematics on intuitive
reasoning without proof” (p.1).
Carnap
Carnap disliked
ontology and metaphysics. For Carnap intuition is a kind of mysterious
and
unreliable access to matters of independent fact. Creath (1990a, 1990b)
argued
that anti-intuition is one of the primary motives of Carnap’s
philosophy.
Carnap was firmly opposed to the Platonic tradition of accepting
"truths" based upon "supposed direct metaphysical insight or
grasp of objects or features of things independent of ourselves but
inaccessible to ordinary sensory observation." (p. 4) Creath (1900b)
pointed out,
Carnap's
proposal, then, is to treat the basic axioms of mathematics, of logic,
and of
the theory of knowledge itself, as well as the sundry other special
sentences,
as an implicit definition of the terms they contain. The upshot of this
is that
simultaneously the basic terms are understood with enough meaning for
the
purpose of mathematics, logic and so on, and the basic claims thereof
need no
further justification, for we have so chosen our language as to make
these particular
claims true… On Carnap’s proposal the basic claims are in some sense
truths of
their own making. It is not that we make objects and features thereof,
rather
we construct our language in such a way that those claims are true. (p.
6)
Following
Poincaré and Hilbert’s assertion that the axioms of mathematics can be
constructed as implicit definitions of the terms they contain, Carnap
viewed
numbers as logical objects and rejected the intuitionist
approach to mathematics. Although Gödel’s theorem brought arguably
insurmountable difficulties to the Russell-Whitehead project, Carnap
still
adopted Russell’s logico-analytic method of philosophy, including
philosophy of
mathematics. By working on logical syntax, Carnap attempted to make
philosophy
into a normal science in a logical, but not empirical, sense (Wang,
1986).
Carnap accepted Russell and Whitehead’s thesis that mathematics can be
reduced
to logic. Further, Carnap asserted that logic is based on convention
and thus
it is true by convention. In his essay entitled “Foundations of logic
and
mathematics” (1971, originally published in 1939), Carnap clearly
explained his
position on logic and convention:
It
is important to be aware of the conventional components in the
construction of
a language system. This view leads to an unprejudiced investigation of
the
various forms of new logical systems which differ more or less from the
customary form (e.g. the intuitionist logic constructed by Brouwer and
Heyting,
the systems of logic of modalities as constructed by Lewis and others,
the
systems of plurivalued logic as constructed by Lukasiewicz and Taski,
etc.),
and it encourages the construction of further new forms. The task is
not to
decide which of the different systems is the right logic, but to
examine their
formal properties and the possibilities for their interpretation and
application in science. (pp. 170-171)
The preceding
approach is called linguistic conventionalism, in which things can make
sense with
reference to particular linguistic frameworks. Once we learn the rules
of a
certain logical and mathematical framework, we have everything we need
for
knowledge of the required mathematical propositions. In this sense,
like the
Russell-Whitehead approach, a linguistic framework is a self-contained
system.
As mentioned
earlier, the verification criterion of logical positivism might face
certain
difficulties in the context of mathematical proof. Carnap supported a
distinction between synthetic and analytical knowledge as a way to
delimit the
range of application of the verification principle (Isaacson, 2000). To
be
specific, Carnap (1956) distinguished analytic knowledge from synthetic
knowledge, and also internal questions from external questions. An
external
question is concerned with the existence or reality of the system of
entities
as a whole. A typical example is, “Is there a white piece of paper on
my desk?”
This question can be answered by empirical investigation. A question
like “Do
normal distributions exist?” is also an external question, but for
Carnap, it
is a pseudo-question that cannot be meaningfully answered at all.
On the other
hand, an internal question is about the existence of certain entities
within a
given framework. Mathematical truths, such as 1+1=2, or a set theoretic
truth,
are tautology in the sense that they are verified by meanings within a
given
frame of reference; any revision may lead to a change of meanings. In
Carnap’s
view it is meaningful to ask a question like “Is there a theoretical
sampling
t-distribution in the Fisherian significance testing?” In other words,
to be
real in logic and mathematics is to be an element of the system. Logic
and
mathematics do not rely on empirical substantiation, because they are
empty of
empirical content.
Quine
Unlike Carnap,
Quine did not reject the ontological question of the realness of
mathematical
entities. Instead, for Quine the existence of mathematical entities
should be
justified in the way that one justifies the postulation of theoretical
entities
in physics (Quine, 1957). However, this notion is misunderstood by some
mathematicians such as Hersh (1997), and thus needs clarification.
Hersh argued
that physics depends on machines that accept only finite decimals. No
computer can
use real numbers that are written in infinite decimals; the
microprocessor
would be trapped in an infinite process. For example, pi (3.14159…)
exists
conceptually, but not physically and computationally. While electrons
and
protons are measurable and accessible, mathematical objects are not.
Thus,
Hersh was opposed to Quine’s ontological position. Hersh was confused here
because he was equating measurability and representation to existence.
In the
realist sense, the existence of an object does not require that it be
known and
measured by humans in an exact and precise manner. While the numeric
representation of pi does not exist, one could not assert that π also
does
not exist. Actually, the ontological commitment made by Quine, in which
mathematical objects are considered on par with physical objects, is
strongly
related to his holistic view of epistemology. While Quine asserted that
logic/mathematics and physical sciences are different in many aspects,
drawing
a sharp distinction between them, such as placing logic/mathematics in
the
analytic camp and putting physical science on the synthetic side, is
erroneous.
In his well-known paper “Two dogmas of empiricism,” Quine (1951)
bluntly
rejected not only this dualism, but also reductionism, which will be
discussed
next.
Quine (1966/1976,
originally published in 1936) challenged Carnap’s notion that
mathematics is
reduced to logic and that logic is true by convention. Quine asserted
that
logic cannot be reduced to convention, because to derive anything from
conventions
one still needs logic. Carnap viewed logical and mathematical syntax as
a
formalization of meaning, but for Quine a formal system must be a
formalization
of some already meaningful discourse. Moreover, in rejecting the
analytic-synthetic dichotomy, Quine rejected the notion that
mathematics and
logical truths are true definitions and we can construct a logical
language
through the selection of meaning. A definition is only a form of
notation to
express one term in form of others. Nothing of significance can follow
from a
definition itself. For example, in the regression equation, y=a+bx+e,
where y
is the outcome variable, x is the regressor/predictor, a is the slope, b is the beta weight, and e is
the error
term, these symbols cannot not help us to find truths; they are nothing
more
than a shorthand to express a wordy and complicated relationship. For
Quine,
meaning is a phenomenon of human agency. There is no meaning apart from
what we
can learn from interaction with the human community. In this sense,
logical
truths are not purely analytical; rather, constructing logic can be
viewed as a
type of empirical inquiry (Isaacson, 2000).
Quine (1951)
asserted that there are no purely internal questions. Our commitment to
a
certain framework is never absolute, and no issue is entirely isolated
from
pragmatic concerns about the possible effects of the revisions of the
framework. In Putnam’s (1995) interpretation, Quine’s doctrine implies
that
even so-called logical truths are subject to revision. This doctrine of
revisibility is strongly tied to the holistic theme in Quine’s
philosophy. To
be specific, the issue of what logic to accept is a matter of what
logic, as a
part of our actual science, fits the truth that we are establishing in
the
science that we engaged in (Isaacson, 2000). Logics are open to
revision in
light of new experience, background knowledge, and a web of theories.
According
to Quine’s holism, mathematics, like logic, has to be viewed not by
itself, but
as a part of all-embracing conceptual scheme. In this sense, even
so-called
mathematical truths are subject to revision, too.
It is essential
to further discuss two Quineian notions: revisability of terms and
holism,
because viewing these Quineian notions as opposition to Carnapian views
is a
mistake. According to Friedman (2002), criticism of
Carnap by Quine is based on Quine’s “misleading” assumption that
analytic
statements are said to be unrevisable. However, Carnap did not equate
analyiticity to unrevisability. It is true that in Carnap’s linguistic
conventionalism logical and mathematical principles play a constitutive
role.
Nevertheless, even if we stay within the same framework, terms can be
revised
but their meanings would be changed. Further, we could move from one
framework
to another, whicn contains a different set of principles. Consequently,
terms
are revised in the process of framework migration.
According to
Creath (1991), the holist view that Quine embraced in Quine’s earlier
career
might be called radical holism. In Quine’s view it is the totality of
our
beliefs which meets experience or not at all. French scientist Duhem
was cited
in defense of this holism, but Duhem’s argument was not that extreme.
In the
Duhemian thesis, scientists do not test a single theory; instead, the
test
involves a web of hypotheses such as auxiliary assumptions associated
with the
main hypothesis. On the other hand, radical holism states that in
theory
testing the matter is concerned with whether the totality of our
beliefs meets
the experience. Creath (1991) criticized that if that is the case, then
all our
beliefs are equally well confirmed by experience and also are equally
disposed
to give up as another.
In Quine’s later
career (1990/1992), he modified his holist position to a moderate one,
in which
we test theories againist a critical mass rather than a totality. A
critical
mass is a big enough subset of science to imply what to expect from
some
observation or experiment. The size of this critical mass will vary
from case
to case. According to Friedman (2002), Carnap explicitly embraced
certain
portions of holism such as the Duhemian thesis. For Carnap, a
linguistic
framework is wholly predicated on the idea that logical principles,
just like
empirical ones, can be revised in light of a web of empirical science.
In this
sense, the philosophies of Quine and Carnap share the common ground
based on
the Duhemian thesis.
According to Pyle
(1999), Quine viewed moderate holism as an answer to certain questions
in
philosophy of mathematics, which are central to Carnapian philosophy.
Carnap
asserted that mathematics is analytic and thus mathematics can be
meaningful
without empirical context. Moderate holism's answer is that mathematics
absorbs
the shared empirical content of the critical masses to which it
contributes. In
addition, Carnap’s analytic position to mathematics makes mathematical
truth
necessary rather than contingent. Moderate holism's answer is that when
a
critical mass of sentences jointly implies a false prediction, we could
choose
what component sentence to revoke. On the other hand, we employ a maxim
of
“minimum mutilation” (conversativism) to guide our revision, and this
accounts
for mathematical necessity. Nevertheless, Carnap might not have
objections to
this, because as mentioned before, Carnap accepted revision of beliefs
in light
of empirical science. Indeed, moderate holism, as the guiding principle
of
mathematical and other scientific inquiries, is more reasonable and
practical
than radical holism.
Discussion
Carnap’s views on
logic and mathematics, such as distinguishing between
analytic-synthetic
knowledge, reducing mathematics to logic and basing logic on
convention, are
problematic. Indeed, Quine has deeper insight than Carnap because he
asserted
that logic and mathematics are based on empirical input in the human
community;
and thus they are subject to revision.
Statistical
Theories and Empirical Evidence
There are many
examples of mathematical theories that have been substantively revised
in light
of new evidence. How the newer Item Response Theory amends Classical
True Score
Theory is a good example. In the article “New rules of measurement,”
prominent
statisticians Embretson and Reise (2000) explained why the conventional
rules
of measurement are inadequate and proposed another set of new rules,
which are
theoretically better and empirically substantiated. For example, the
conventional theory states that the standard error of measurement
applies to
all scores in a particular population, but Embretson found that the
standard
error of measurement differs across scores but generalizes across
populations.
In addition, R.
A. Fisher criticized Neyman’s statistical theory because Fisher
asserted that
mathematical abstraction to the neglect of scientific applications was
useless.
He mocked that Neyman was misled by algebraic symbolism (Howie, 2002).
Interestingly enough, on some occasion Fisher was also confined by
mathematical
abstraction and algebraic symbolism. In the theory of maximum
likelihood
estimation, Fisher suggested that as sample size increases, the
estimated
parameter gets closer to closer to the true parameter (Eliason, 1993).
But in
the actual world, the data quality may decrease as the sample size
increases.
To be specific, when measurement instruments are exposed to the public,
the
pass rate would rise regardless of the examinee’s ability. In this case
the
estimation might be farther away from the true parameter! Statisticians
could
not blindly trust the mathematical properties postulated in the
Fisherian
theorems.
Someone may argue
that the preceding examples have too much “application,” that they are
concerned with the relation between a measurement theory and
observations, not
a “pure” relation among mathematical entities. Nevertheless, on some
occasions,
even the functional relationship among mathematical entities is not
totally
immune from empirical influence. For example, the Logit function, by
definition, is the natural log of the odd ratio, which is the ratio
between the
success rate and the failure rate. However, in the context that the
rate of
failure is the focal interest of the model, the odd ratio can be
reversed.
Putting
statistical findings in the arena of “applied mathematics” seems to be
an
acceptable approach to dismissing the argument that mathematics is
subject to
revisions. Actually, the distinction between pure and applied
mathematics is
another form of dualism that attempts to place certain mathematics in
the
logical domain. In the following I argue that there is no sharp
demarcation
point between them, and mathematics, like the physical sciences, is
subject to
empirical verification. Empirically verifying mathematical theories
does not
mean using a mapping approach to draw correspondence between
mathematical and
physical objects. Counting two apples on the right hand side and two on
the
left is not a proof that 2+2=4. Instead, empirical verification in
mathematics
is implemented in computer-based Monte Carlo simulations, in which
“behaviors”
of numbers and equations are investigated.
Distinction
Between Pure and Applied Mathematics
Conventionally
speaking, mathematics is divided into pure mathematics and applied
mathematics.
There is a widespread belief that some branches of mathematics, such as
statistics, orient toward application and thus are considered applied
mathematics. Interestingly enough, in discussion of the philosophy of
mathematics, philosophers tend to cite examples from “theoretical
mathematics”
such as geometry and algebra, but not “applied” mathematics such as
statistics.
Although I hesitate to totally tear down the demarcation between pure
and
applied mathematics, I doubt whether being so-called “pure” or
“applied” is the
“property” or “essence” of the discipline. As a matter of fact,
geometry could
be applied to architecture and civil engineering, while statistics can
be
studied without any reference to empirical measurement. To be specific,
a
t-test can be asked in an applied manner, such as “Does the IQ mean of
Chinese
people in Phoenix significantly higher than that of Japanese people in
Phoenix?” However, a t-test-related
question can be reframed as “Is the mean of set A higher than that of
set B
given that the Alpha level is 0.5, the power level is 0.75, both sets
have
equal variances and numbers in each set are normally distributed?” A
research
question could be directed to the t-test itself: “Would the actual Type
I error
rate equal the assumed Type I error rate when the Welch’s t-test is
applied to
a non-normal sample of 30?”
A mathematician
can study the last two preceding questions without assigning numbers to
any
measurement scale or formulating a hypothesis related to mental
constructs,
social events, or physical objects. He/she could generate numbers in
computer
to conduct a mathematical experiment. There is another widespread
belief that
computer-based experimental mathematics is applied mathematics while
traditional mathematics is pure. A century ago our ancestors who had no
computers relied on paper and pencil to construct theorems, equations,
and
procedures. Afterwards, they plugged in some numbers for verification.
Today
these tasks are performed in a more precise and efficient fashion with
the aid
of computers. However, it is strange to say that mathematics using
pencil and
paper is pure mathematics while that employing computers is applied.
In brief, I argue
that the line between pure and applied mathematics is blurred.
Conventional
criteria for this demarcation are highly questionable; the subject
matter
(geometry or statistics) and the tool (pencil or computer) cannot
establish the
nature of mathematics (pure or applied). In the following discussion I
will
discuss how mathematicians use Monte Carlo simulations to support my
argument
that mathematics is not purely logical but rather has empirical
elements. Next,
I will use an example of a robustness study to demonstrate how
traditional
claims on certain statistical theories are revised by findings in
simulations.
With the
advancement of high-powered computers, computer simulation is often
employed by
mathematicians and statisticians as a research methodology. This school
is
termed "experimental mathematics" and a journal entitled “Journal of
Experimental Mathematics” is specifically devoted to this inquiry
(Bailey &
Borwein, 2001). Chaitin (1998), a supporter of experimental
mathematics,
asserted that it is a mistake to regard mathematical axioms as
self-evident
truths; rather the behaviors of numbers should be verified by
computer-based
experiments. It is important to differentiate the goal of controlled
experiments in psychology, sociology, and engineering from that of
experimental
mathematics. In the former, the objective is to draw conclusions about
mental
constructs and physical objects, such as the treatment effectiveness of
a
counseling program or the efficiency of a microprocessor. In these
inquiries,
mathematical theories are the frame of reference for making inferences.
But in
the latter, the research question is directed to the mathematical
theories
themselves.
Both of them are
considered “experiments” because conventional experimental criteria,
such as
random sampling, random assignment of group membership, manipulation of
experimental variables, and control of non-experimental variables, are
applied
(Cook & Campbell, 1979). Interestingly enough, in terms of the
degree of
fulfillment of these experimental criteria, experimental mathematics
has even
more experimental elements than controlled experiments in the social
sciences.
Consider random sampling first. In social sciences, it is difficult, if
not
impossible, to collect true random samples. Usually the sample obtained
by
social scientists is just a convenient sample. For example, a
researcher at
Arizona State University may recruit participants in the Greater
Phoenix area,
but he/she rarely obtains subjects from Los Angeles, New York, Dallas,
etc.,
let alone Hong Kong, Beijing, or Seoul. In terms of controlling
extraneous
variables or conditions that might have an impact on dependent
variables, again
social sciences face inherent limitations. Human subjects carry
multiple
dimensions such as personality, family background, religious beliefs,
cultural
context, etc. It is definitely impossible that the experimenter could
isolate
or control all other sources of influences outside the experimental
setting. On
the other hand, computer-based experiments achieve random sampling by
using a
random number generator. It is argued that some random number
generators are
not truly random, but the technology has become more and more
sophisticated.
Actually, even a slightly flawed random number generator could yield a
more
random sample than one collected in the human community. Also,
computer-based
experimental mathematics does not suffer the problem of lacking
experimental
control, because numbers and equations do not have psychological,
social,
political, religious or cultural dimensions. In brief, the preceding
argument
is to establish the notion that experimental mathematics is
experimental in
every traditional sense.
Monte Carlo
Simulations and Robustness Study
Traditional
parametric tests, such as t-test and ANOVA, require certain parametric
assumptions. Typical parametric assumptions are homogeneity of
variances, which
means the spread of distributions in each group do not significantly
differ
from each other, and normal distributions, which means the shape of the
sample
distribution is like a bell-curve. Traditional statistical theories
state that
the t-test is robust against mild violations of these assumptions; the
Satterthwaite t-test is even more resistant against assumption
violations; and
the F-test in ANOVA is also robust if the sample size is large (please
note
that in these theories the sample can be composed of observations from
humans
or a set of numbers without any measurement unit). The test of
homogeneity of
variance is one the preliminary tests for examining whether assumption
violations occur. Since conventional theories state that the preceding
tests
are robust, Box (1953) mocked the idea of testing the variances prior
to
applying an F-test: "To make a preliminary test on variances is rather
like putting to sea in a rowing boat to find out whether conditions are
sufficiently calm for an ocean liner to leave port!" (p.333).
However, in recent
years statisticians have been skeptical of the conventional theories.
Different
statisticians have proposed their own theories to counteract the
problem of
assumption violations (Yu, 2002). For instance,
(1) Some researchers construct
non-parametric procedures to
evade the problem of parametric test assumptions. As the name implies,
non-parametric tests do not require parametric assumptions because
interval
data are converted to rank-ordered data. Examples of non-parametric
tests are
the Wilcoxon signed rank test and the Mann-Whitney-Wilcoxon test. Some
version
of non-parametric method is known as order statistics for its focus on
using
rank-ordered data. A typical example of it is Cliff’s statistics
(Cliff, 1996).
(2) To address the violation
problem, some statisticians
introduce robust calculations such as Trimmed means and Winsorized
means. The
trimmed mean approach is to exclude outliers in the two tails of the
distribution while the Winsorized mean method “pulls” extreme cases
toward the
center of the distribution. The Winsorized method is based upon
Winsor's
principle: All observed distributions are Gaussian in the middle. Other
robust
procedures such as robust regression involve differential weighting to
different observations. In the trimmed mean approach outliers are given
a zero
weighting while robust regression may assign a “lighter” count, say
0.5, to
outliers. Cliff (1996), who endorsed order statistics, was skeptical of
the
differential weighting of robust procedures. He argued that data
analysis
should follow the principle of “one observation, one vote.” Mallows and
Tukey
(1982) also argued against Winsor's principle. In their view, since
this
approach pays too much attention to the very center of the
distribution, it is
highly misleading. Instead, Tukey (1986) strongly recommended using
data
re-expression procedures, which will be discussed next.
(3) In data re-expression, linear
or non-linear equations
are applied to the data. When the normality assumption is violated, the
distribution could be normalized through re-expression. If the
variances of two
groups are unequal, certain transformation techniques can be used to
stabilize
the variances. In the case of non-linearity, this technique can be
applied to
linearize the data. However, Cliff (1996) argued that data
transformation
confines the conclusion to the arbitrary version of the variables.
(4) Resampling techniques such as
the randomization exact
test, jackknife, and bootstrap are proposed by some other statisticians
as a
counter measure against parametric assumption violations (Diaconis
& Efron,
1983; Edgington, 1995; Efron & Tibshirani, 1993; Ludbrook &
Dudley,
1998). Robust procedures recognize the threat of parametric assumption
violations and make adjustments to work around the problem. Data
re-expression
converts data in order to conform to the parametric assumptions.
Resampling is
very different from the above remedies, for it is not under the
framework of
theoretical distributions imposed by classical parametric procedures.
For
example, in bootstrapping, the sample is duplicated many times and
treated as a
virtual population. Then samples are drawn from this virtual population
to
construct an empirical sampling distribution. In short, the resampling
school
replaces theoretical distributions with empirical distributions. In
reaction
against resampling, Stephen E. Fienberg criticized that "you're trying
to
get something for nothing. You use the same numbers over and over again
until you
get an answer that you can't get any other way. In order to do that,
you have
to assume something, and you may live to regret that hidden assumption
later
on" (cited in Peterson, 1991, p. 57).
It is obvious
that statisticians such as Winsor, Tukey, Cliff, and Fienberg do not
agree with
each other on the assumption violation and robustness reinforcement
issues. If
different mathematical systems, as Russell and Whitehead suggested, are
self-contained systems, and if mathematics, as Carnap maintained, is
reduced to
logic that is based on different conventions, these disputes would
never come
to a conclusive closure. Within the system of Winsor’s school, the
Gaussian
distribution is the ideal and all other associated theorems tend to
support
Winsor’s principle. Within the Tukey’s convention, the logic of
re-expression
fits well with the notions of distribution normalization, variance
stabilization, and trend linearization.
It is important
to note that these disputes are not about how well those statistical
theories
could be applied to particular subject matters such as psychology and
physics.
Rather, these statistical questions could be asked without reference to
measurement, and this is the core argument of the school of data
re-expression.
For example, researchers who argue against data re-expression complain
that it
would be absurd to obtain a measurement of people’s IQ and then
transform the
data like [new variable = 1/(square root of IQ)]. They argue that we
could
conclude that the average IQ of the Chinese people in Phoenix is
significantly
higher than that of the Japanese, but it makes no sense to say anything
about
the difference in terms of 1/(square root of IQ). However, researchers
supporting data re-expression argue that the so-called IQ is just a way
of
obtaining certain numbers, just like using meters or feet to express
height.
Numbers can be manipulated in their own right without being mapped onto
physical measurement units. In a sense non-parametric statistics and
order
statistics are forms of data re-expression. For example, when we obtain
a
vector of scores such as [15, 13, 11, 8, 6], we can order the scalars
within
the vector as [1, 2, 3, 4, 5]. This “transformation” no doubt alters
the
measurement and, indeed, loses the precision of the original
measurement.
Nevertheless, these examples demonstrate that statistical questions can
be
studied regardless of the measurement units, or even without any
measurements.
Monte Carlo simulation is a typical example of studying statistics
without
measurement.
As you may notice
in the section regarding bootstrapping, statisticians do not even need
empirical data obtained from observations to conduct a test; they could
“duplicate” data by manipulating existing observations. In
bootstrapping,
number generation is still based on empirical observations, whereas in
Monte
Carlo simulations all numbers could be generated in computer only. In
recent
years, robustness studies using Monte Carlo simulations have been
widely
employed to evaluate the soundness of
mathematical procedures in terms of
their departure from idealization and robustness against assumption
violations.
In Monte Carlo simulations, mathematicians make up strange data (e.g.
extremely
unequal variances, non-normality) to observe how well those
mathematical
procedures are robust against the violations. Box is right that we
cannot row a
boat to test the condition for an ocean liner. But using computers to
simulate
multi-million cases under hundreds scenarios is really the other way
around—now
we are testing the weather condition with an ocean liner to tell us
whether
rowing a boat is safe. Through computer simulations we learn that
traditional
claims concerning the robustness of certain procedures are either
invalid or
require additional constraints.
There are
numerous Monte Carlo studies in the field of statistics. A recent
thorough
Monte Carlo study (Thompson, Green, Stockford, Chen, & Lo, 2002;
Stockford,
Thompson, Lo, Chen, Green, & Yu, 2001) demonstrates how
experimental
mathematics could refute, or at least challenge, the conventional
claims in
statistical theories. This study investigates the Type I error rate and
statistical power of the various statistical procedures. The Type I
error rate
is the probability of falsely rejecting the null hypothesis, whereas
the
statistical power is the probability of correctly rejecting the null
hypothesis. In this study, statistical procedures under investigation
include
the conventional independent-samples t-test, the Satterthwaite
independent-samples t-test, the Mann-Whitney-Wilcoxon test
(non-parametric
test), the test for the difference in trimmed means (robust procedure),
and the
bootstrap test of the difference in trimmed means (resampling and
robust
methods). Four factors were manipulated to create 180 conditions: form
of the
population distribution, variance heterogeneity, sample size, and mean
differences. Manipulation of these factors is entirely under the
control of the
experimenters. No other non-experimental factors could sneak into the
computer
and affect the conditions. The researchers concluded that the
conventional
t-test, the Satterthwaite t-test, and the Mann-Whitney-Wilcoxon test
produce
either poor Type I error rates or loss of power when the assumptions
underlying
them are violated. The tests of trimmed means and the bootstrap test
appear to
have fewer difficulties across the range of conditions evaluated. This
experimental study indicates that the robustness claims by two versions
of the
t-test and one of the non-parametric procedures are invalid. On the
other hand,
one of the robust methods and one of the resampling methods are proved
to be
true in terms of robustness. Although the scope of this study is
narrowed to
one of each statistical school, the same approach can be applied to
various
versions of parametric tests, non-parametric tests, robust procedures,
data
re-expression methods, and resampling.
Conclusion
The above
findings are not achieved by the methods suggested by Russell and
Carnap, such
as the study of logical relationships, truth by definitions or truth by
convention. Rather, the claims result from experimental study. When
Quine
introduced his philosophical theory on logic and mathematics, computer
technology and the Monte Carlo method were not available. Nonetheless,
his
insight is highly compatible with recent development in experimental
mathematics. I strongly believe that if researchers put aside the
analytic-synthetic distinction by adopting Quine’s moderate holistic
view to
scientific inquiry, many disputes could come to a conclusive closure. Indeed, a holistic approach has been
beneficial to mathematical inquiry. Although R. A. Fisher was a
statistician,
he was also versed in biology and agriculture science, and indeed most
of his
theorems were derived from such empirical fields. Winsor’s principle is
based
on the Gaussian distribution, but Gauss discovered the Gaussian
distribution
through astronomical observations. Survival analysis or the hazard
model is the
fruit of medical and sociological research. As discussed before,
Embretson and
Reise, as psychologists, used the psychometric approach to revise
traditional
measurement theories. The example of robustness study demonstrates how
social
scientists employed Monte Carlo studies to challenge traditional claims
in
mathematics. As Quine’s holism proposed,
logic, mathematics, observation, and a web of scientific theories are
strongly
linked to each other.
References
Ayer, A. J.
(1936). ‘The Principle of Verifiability’. Mind
(New Series), 45, 199-203.
Ayer, A. J.
(1946). Language, Truth, and Logic (2^{nd}
ed). London: V. Gollancz.
Bailey, D. H.,
& Borwein, J. M. (2001). ‘Experimental Mathematics: Recent
Developments And
Future Outlook’. In B. Engquist, & W. Schmid, (Eds.). Mathematics
Unlimited: 2001 And Beyond (pp. 51-65). New York:
Springer.
Box, G. E. P.
(1953). Non-Normality And Tests On
Variances. Biometrika, 40, 318-335
Carnap, R.
(1971). ‘Foundations of Logic And Mathematics’. In O. Neurath, R.
Carnap, &
C. Morris, (Eds.). Foundations of The
Unity of Science, Toward An International Encyclopedia of Unified
Science
(pp. 139-212). Chicago: University of Chicago Press.
Carnap, R.
(1956). Meaning And Necessity: A Study In
Semantics And Modal. Chicago, IL: University of Chicago Press.
Cliff, N. (1996).
Ordinal Methods For Behavioral Data
Analysis. Mahwah, NJ: Erlbaum.
Chaitin, G. J.
(1998). The Limits Of Mathematics: A
Course On Information Theory And The Limits Of Formal Reasoning.
Singapore:
Springer-Verlag.
Cook, T. D.,
& Campbell, D. T. (1979). Quasi-Experimentation:
Design And Analysis Issues For Field Settings. Boston, MA: Houghton
Mifflin
Company.
Creath, R.
(1990a). Carnap, Quine, and The Rejection
of Intuition. In Robert B. Barrett & Roger F. Gibson (Eds.), Perspectives on Quine (pp. 55-66).
Cambridge, MA: Basil Blackwell.
Creath, R. (ed.).
(1990b). Dear Carnap, Dean Van: The
Quine-Carnap Correspondence and Related Work. Berkeley, CA:
University of
California Press.
Creath, R.
(1991). Every Dogma Has Its Day.
Erkenntnis, 35, 347-389.
Diaconis, P., and
B. Efron. (1983). Computer-Intensive
Methods in Statistics. Scientific American, May, 116-130.
Edgington, E. S.
(1995). Randomization Tests. New
York: M. Dekker.
Efron, B., &
Tibshirani, R. J. (1993). An Introduction
to The Bootstrap. New York: Chapman & Hall.
Eliason, S. R.
(1993). Maximum Likelihood Estimation:
Logic and Practice. Newbury Park: Sage.
Embretson, S. E.,
& Reise, S. (2000). Item Response
Theory For Psychologists. Mahwah, NJ: LEA.
Frege, G.
(1884/1960). The Foundations of
Arithmetic (2nd ed.). New York: Harper.
Friedman, M.
(2002). Kant, Kuhn, and The Rationality
Of Science. Philosophy of Science, 69, 171-190.
Gödel, K. (1944).
‘Russell’s Mathematical Logic’. In Paul A. Schilpp, (Ed.). The
Philosophy of Bertrand Russell (pp.125-153). Chicago: Northwestern
University.
Gödel, K. (1961).
Collected Works, Volume III. Oxford:
Oxford University Press.
Hersh, R. (1997).
What is Mathematics, Really? Oxford:
Oxford University Press.
Howie, D. (2002).
Interpreting Probability: Controversies
and Developments In The Early Twentieth Century. Cambridge, UK:
Cambridge
University Press.
Isaacson, D.
(2000). ‘Carnap, Quine, and Logical Truth’. In D. Follesdal (Ed.). Philosophy Of Quine: General, Reviews, and
Analytic/Synthetic (pp. 360-391). NewYork: Garland Publishing.
Jaffe, A., &
Quinn, F. (1993). “Theoretical
Mathematics”: Toward A Cultural Synthesis of Mathematics And
Theoretical
Physics. American Mathematics Society, 28, 1-13.
Ludbrook, J.
& Dudley, H. (1998). ‘Why Permutation Tests Are Superior To T And F
Tests
In Biomedical Research’. American
Statistician, 52, 127-132.
Lindstrom, P.
(2000). ‘Quasi-Realism in Mathematics’. Monist,
83, 122-149.
Mallows, C. L.,
& Tukey, J. W. (1982). ‘An Overview of Techniques of Data Analysis,
Emphasizing Its Exploratory Aspects’. In J. T. de Oliveira & B.
Epstein
(Eds.), Some Recent Advances in
Statistics (pp. 111-172). London: Academic Press.
Peterson, I.
(July 27, 1991). ‘Pick a Sample’. Science
News, 140, 56-58.
Putname, H.
(1995). ‘Mathematical Necessity Reconsidered’. In P. Leonardi & M.
Santambrogio, (Eds.). On Quine: New
Essays (pp. 267-282). Cambridge: Cambridge University Press.
Pyle, A. (ed.).
(1999). Key Philosophers in Conversation:
The Cogito Interviews. New York: Routledge.
Quine, W. V.
(1951). Two Dogmas of Empiricism.
Philosophical Review, 60, 20-43.
Quine, W. V.
(1957). ‘The Scope and Language of Science’. British
Journal for the Philosophical Science, 8, 1-17.
Quine, W. V.
(1966/1976). The Ways of Paradox, and
Other Essays. Cambridge, MA: Harvard University Press.
Quine, W. V.
(1990/1992) Pursuit of Truth (2^{nd}
ed.). Cambridge, MA: Harvard University Press.
Russell, B. (1919). Introduction to
Mathematical Philosophy.
London: Allen & Unwin.
Schlick, M.
(1959). ‘Positivism and Realism’. In A. J. Ayer (Ed.), Logical
Positivism (pp. 82-107). New York: Free Press.
Stockford, S.,
Thompson, M., Lo, W. J., Chen, Y. H., Green, S., & Yu, C. H. (2001
October). ‘Confronting The Statistical Assumptions: New Alternatives
For
Comparing Groups’. Paper presented at the Annual Meeting of Arizona
Educational
Researcher Organization, Tempe, AZ.
Thompson, M. S.,
Green, S. B., Stockford, S. M., Chen, Y., & Lo, W. (2002 April).
‘The .05
level: The probability that the independent-samples t test should be
applied?’
Paper presented at the Annual Meeting of the American Education
Researcher
Association, New Orleans, LA.
Tieszen, R.
(1992). ‘Kurt Gödel and Phenomenology’. Philosophy
of Science, 59, 176-194.
Tukey, J. W.
(1986). The Collected Works of John W.
Tukey (Volume IV): Philosophy and Principles Of Data Analysis 1965-1986.
Monterey, CA: Wadsworth & Brooks/Cole.
Wang, H. (1986). Beyond Analytic
Philosophy: Doing Justice To
What We Know. Cambridge, MA: MIT Press.
Werkmeister, W.
H. (1937). ‘Seven Theses Of Logical Positivism Critically Examined Part
I’. The Philosophical Review, 46, 276-297.
Whitehead, A. N.,
& Russell, B. (1910/1950). Principia
Mathematica (2^{nd} ed.). Cambridge, UK: Cambridge
University
Press.
Yu, C. H. (2002).
‘An Overview Of Remedial Tools For Violations Of Parametric Test
Assumptions in
The SAS System.’ Proceedings of 2002
Western Users of SAS Software Conference, 172-178.
Copyright © 2004 Minerva
All rights are reserved, but fair and good faith use with full
attribution may
be made of this work for educational or scholarly purposes.
Chong Ho Yu
has a Ph.D. in Measurement, Statistics, and Methodological Studies from
Arizona
State University, USA. Currently he is pursuing a second doctorate in
Philosophy at the same institution. He is also a Psychometrician
at Cisco
Ssystems/Aries Technology, USA. His research interests include
philosophical
foundations of research methodology, and relationship between science
and
religion.
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