Mathematical psychology : Prospects for the 21st century : A guest editorial

Journal of Mathematical Psychology 52 (2008) 269–280
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Journal of Mathematical Psychology
journal homepage: www.elsevier.com/locate/jmp
Mathematical psychology: Prospects for the 21st century: A guest editorial$
James T. Townsend 1
Indiana University, Department of Psychological & Brain Sciences, 1101 E. 10th Street, Bloomington, IN 47405-7007, United States
a r t i c l e
i n f o
a b s t r a c t
Article history:
The twenty-first century is certainly in progress by now, but hardly well underway. Therefore, I will take
Received 30 April 2007
that modest elasticity in concept as a frame for this essay. This frame will serve as background for some
Received in revised form
of my hopes and gripes about contemporary psychology and mathematical psychology’s place therein.
29 April 2008
It will also act as platform for earnest, if wistful thoughts about what might have (and perhaps can still)
Available online 18 July 2008
aid us in forwarding our agenda and what I see as some of the promising avenues for the future. I loosely
structure the essay into a section about mathematical psychology in the context of psychology at large
Keywords:
and then a section devoted to prospects within mathematical psychology proper. The essay can perhaps
Future
Mathematical psychology
be considered as in a similar spirit, although differing in content, to previous editorial-like reviews
Fields of mathematical psychology
of general or specific aspects of mathematical psychology such as [Estes, W. K. (1975). Some targets
History of mathematical psychology
for mathematical psychology. Journal of Mathematical Psychology, 12, 263–282; Falmagne, J. C. (2005).
Psychological science
Mathematical psychology: A perspective. Journal of Mathematical Psychology, 49, 436–439; Luce, R. D.
Clinical science and mathematical
(1997). Several unresolved conceptual problems of mathematical psychology. Journal of Mathematical
psychology
Psychology, 41, 79–87] that have appeared in this journal.
Neuroscience and mathematical
© 2008 Elsevier Inc. All rights reserved.
psychology
Mathematical psychology and other
quantitative fields
Computer science and mathematical
psychology
Physics and mathematical psychology
1. Psychology and mathematical psychology therein
of discourse. Ponderings about psychological topics, casual and
systematic, recede into recorded history. Yet we all know that
1.1. A glimpse of recent history
scientific psychology is little more than one-hundred twenty-
Psychology is a young science as opposed to a young field
five years old or so, dating by convention from 1879, the year
of establishment of Wundt’s famous laboratory in Leipzig; this
according to Boring’s (1957) calendar of experimental psychology.
$ A word is in order concerning the scope and style of this essay. There are
Without question, we have made rather startling progress by
many themes that could be selected for such a piece. The topic I have chosen
almost any measure, in that scant time. This progress has arguably
centers on the health and future of mathematical psychology. It does include a
accelerated since the 1940s due to the amazing new tools, so
necessarily very succinct history and a number of bordering issues along with some
attractive and appropriate for non-physical sciences, proffered by
suggestions for remediation and improvement. Due to space limitations, I have had
von Neumann, Wiener, Shannon, and others.2 I speak, of course, of
to limit mention and citations of a plethora of important names outside those of a
very few of the pioneers in the field. A vast number of important investigators es-
automata theory (von Neumann), utility theory (von Neumann &
pecially if they’re on the younger side are neglected. In particular, I had to virtually
Morgenstern), cybernetics/feedback control theory (Wiener), and
entirely omit the contributions, many overlapping with mathematical psychology,
information theory (Shannon) (see, e.g., Townsend and Kadlec
stemming from the field of psychometrics. The same goes for a voluminous set of
(1990)).
researches on categorization. I regret and apologize for these necessary omissions.
And, though an individual may have performed valuable research in several areas
A significant paradigm shift in psychology occurred when it
of mathematical psychology, I refrained from making these multiple mentions.
began to move away from the grand and perhaps overweening
I should also mention that I cannot guarantee that all citations within a set of
schools toward less capacious, but more articulated venues con-
references in the same location in the essay will be homogeneous (e.g., all reviews
taining more testable models and hypotheses. The grand schools
of their work, all at about the same date, etc.). It was already laborious simply
collecting these as they are. I also take license to occasionally mention important
were highly conspicuous in abnormal psychology, especially in the
names sans specific citation. The style is meant to be informal, even conversational,
so some notable individuals receive mention without specific citation.
E-mail address: jtownsen@indiana.edu.
1 President of Society for Mathematical Psychology 2004–2005.
2 In all cases of name listing, there is no significance to their order.
0022-2496/$ – see front matter © 2008 Elsevier Inc. All rights reserved.
doi:10.1016/j.jmp.2008.05.001

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J.T. Townsend / Journal of Mathematical Psychology 52 (2008) 269–280
various off-shoots of Freudian theory: for instance, Adler, Fromm,
of phenomena and interesting, if rarely conclusive, explanatory
Jung, Horney, and Reich, to name a few. But experimental psychol-
models or theories. A downside is that the careful and steady,
ogy also possessed broadly encompassing theories identified with
incremental growth of the science can be neglected, since the
individuals, as witness Thorndike, Lewin (also featured in social
obvious rewards, or at least the ‘grand prizes’ in the field are
psychology), Skinner, Tolman, Hull, and Guthrie.
accorded the discoveries of effects. But the interpretation of the
The 1950s and then the 1960s saw such quantitative areas
initial effect that stands unaltered in the face of replication and
as signal detection theory, mathematical learning theory, and
parameter variation is quite atypical. Often, follow-up experiments
foundational measurement theory aid and abet this transition to
require elaboration or complexification of the original rationale
a less global but more rigorous, science.3 Names we think of in the
and frequently, yet more experimentation. Thus, this approach
signal detection connection are Peterson, Birdsall, Fox (engineers),
can be productive if carefully pursued experimentally and
Swets, Tanner, Green and Egan. Pioneers in mathematical learning
theoretically. Otherwise, our priorities can be decidedly skewed
theory include Bush, Mosteller, Estes and again rather quickly,
toward unearthing of new effects.
Suppes, Atkinson, Bower, and Crothers, and Murdock.
In addition, just about the same time, psychologists and friends
1.4. Another portentous trade-off approach: Operationalism
began to work out new theories of measurement that included
structures more appropriate for the behavioral and biological
A close cousin of the ‘effects’ bias is the embodiment of one or
sciences than had the ‘classical’ approaches of Campbell and
more theoretical notions in an experimental paradigm, or set of
others. We note well-known contributors like Suppes, Krantz,
paradigms. The growth of ‘operationalism’ was associated with the
Luce, Tversky, and in somewhat different vein, Coombs, and
philosophical movement of logical positivism and the Vienna Circle
N. Anderson. These investigators also overlapped the growing field
(think Schick, Ayer, Carnap, Neurath, Hempel and influencers and
of decision theory oriented toward actual human behavior. Other
fellow travelers such as Russell, Wittengenstein, Popper) and found
notable names around the same time, in allied pursuits with these
a vent in psychology through efforts of Meehl & MacQuordale and
various topics were Restle, Greeno, LaBerge, Sternberg, and Theios.
the physicist Bergmann (it has been said that Bergmann influenced
Garner and Attneave and Laming helped bring information theory
the social sciences far more than he did physics). Again, this
into the fold of quantitative psychology. And, the rather neglected
approach possesses benefits, and aided psychology in shaking off
(admittedly not quite so much in psychometrics), but fundamental
the residue of impossible-to-answer philosophical conundrums
topic of geometry in psychology was given a considerable boost by
that still adhered to the field in the early twentieth century. The
Shepard, Kruskal and others.
loved and hated school of behaviorism came into existence and
With the foregoing eye-blink history as preface I move to a list
dominated experimental psychology for several decades.
of woes that afflict psychology, afflictions that can be somewhat
In any event, there were snags, snags which were proba-
ameliorated by the practice of mathematical psychology.
bly unanticipated by the founders of the ‘operational definition’
wherein philosophical claptrap was avoided by defining theoret-
1.2. The tortoise and the hare
ical entities by way of the empirical operations through which
observations were recorded. A substantial snag is the risk of cir-
It can prove a frustrating experience to compare psychology’s
cularity where a theoretical hypothesis points to an experimental
pace of advance with progress in the ‘hard’ sciences. Except
result and vice versa. The theoretical and phenomenal restrictions
perhaps in the flurry of studies one may witness on discovery
are evident. Or, if as sometimes occurs, there are several more-or-
of a dramatic ‘effect’ (see discussion immediately below), steps
less distinct operations relating to a phenomenon and presumed
in filling in data about a phenomenon not to mention testing
theoretical concept, these may turn out to be closely related (iden-
of major theoretical issues and models, seem to occur with all
tical occasionally, this is good), unrelated, or even contradictory,
the urgency of a glacier. One may wait years, before a modeler
depending on a subsequent theory developed to encompass the
picks up the scent of an intriguing theoretical problem and carries
paradigms associated with respective operational definitions.
it ahead. It is disheartening to contrast our situation with, say,
Mathematical psychology serves as a stiff antidote to the af-
that of microbiology. Perhaps the influx of quantitatively prepared
flictions of ‘effect-philia’ and cul-de-sacs of overindulged oper-
‘outsiders’ (look for later discussion) will helps speed things up.
ationalism. The necessity of providing a rigorous, economical,
accounting of concepts and empirical through a quantitative model
clearly combats the overly particular, and acts not only to ac-
1.3. The eternal pursuit of effects
commodate an entire set of phenomena, but assays the abil-
ity of diverse theoretical notions and experimental operations to
One of the ‘blessings and curses’ of modern psychology is
‘live together’ within the same theory.
the everlasting quest for ‘effects.’ Some of our most famous
investigators made their name by discovery of a novel and
1.5. Anti-replication and anti-null effects bias
enchanting effect. In some cases, these have led to a rich set
With regard to afflictions in psychology (and probably a number
of other sciences), I mention two which are not particularly
3 Somewhat truthfully and perhaps somewhat waggishly, the period of the 1960s
solvable through modeling, but I want to get them off my chest:
at Stanford has been referred to, at least by the erstwhile graduate students of
One is the heavy bias against replication. Too few studies are
the times, as “the golden days of mathematical psychology”. Frequent workshops
published that precisely replicate an earlier study. In fact, younger
with other math–psych bastions such as University of Michigan and University of
investigators in particular, are warned to include variations so that
Pennsylvania were held, the milieu was outstanding with a modeling faculty that
included William Estes, Richard Atkinson, Gordon Bower and Patrick Suppes. Many
editors and reviewers will not reject their ‘replication’ out of hand.
of the students during that period were to become leaders in various facets of
This is not an earmark of a mature science. Long ago, William K.
psychology. The pack in my cohort includes Michael Humphreys, Richard Shiffrin,
Estes spoke of this regrettable prejudice in a seminar during my
Don Hintzman, Robert Bjork, Stephen Link, Chiziko Izawa, William Batchelder,
graduate student days. It seems just as true now as it did then.
Donald Horst, Jack Yellott, Joseph Young, David Rumelhart, and Michael Levine. In
A cognate weakness that many writers have remarked on,
addition, many already or soon-to-become, notables visited in a post-doctoral or
visiting scientist capacity. With no apologies for the nostalgia, it was a fabulous time
using a variety of terms, is the bias against publication (or even
and place to be entering scientific psychology.
consideration for publication) of ‘null’ results. How on earth can

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J.T. Townsend / Journal of Mathematical Psychology 52 (2008) 269–280
271
psychology advance when only ‘positive’ findings with regard to a
citing even mathematical theorems that have been previously
hypothesis, model, etc., are made available to the scientific public?
published elsewhere. Mathematical psychology is occasionally
A number of suggestions have been put forth to circumvent this
prey to this tendency since we overlap such a broad sweep
obstacle, including an archive of negative findings, perhaps sorted
of mathematically oriented research venues. The writer has
according to topic. To my knowledge, this strategy has never been
witnessed several examples of these oversights. Economics,
seriously attempted. With the current revolution in electronic
industrial engineering, operations research, biophysics, many
publishing, it might be worth a try. With the increasing ability
areas of artificial intelligence (robotics, pattern recognition,
of information systems to implement content-addressable search,
problem solving . . . ) clearly provide for rich interaction among
even just the compilation of frequencies of positive vs. negative
investigators but are likewise prone to this type of abuse.
findings could be implemented in the drawing of inferences.
Aside from null results, recent years have seen a lively debate
2. Mathematical psychology: Rumination on its evolution and
about the value of null hypothesis testing, in addition to various
position
means of improving the strategy and avoiding its threatening
sloughs of despond.
2.1. Continued development of areas of mathematical psychology and
rumors of a demise
1.6. Methodology = Modeling plus statistics and psychometrics
As observed earlier, mathematical psychology emerged from
Another associated topic involves methodology in psychology.
embryo in the late fifties and early sixties of the twentieth century.
A multitude of writers have decried the methodological fetters
Already by the end of the sixties, some were pronouncing the
accompanying decades of over-reliance on standard statistics,
demise of mathematical psychology.5 Did it really die, despite the
statistics originally intended more for assessing grain production
ostensible continued existence of practitioners of that field? If not
under varying conditions than for an evolving systematic science.
an obvious corpse, is it in dire peril?
On the one hand, traditional statistics and its offspring have served
Let’s pause a moment to espy the fate of some of the major
as effective tools for much of the advancement of psychological
areas of early effort in mathematical psychology mentioned earlier.
science in the last century. On the other, an argument can be
Perhaps they may shed some light on this assertion. In addition,
made that we have become much too dependent on its fruits in
this little tour will allow us to discern where major branches of the
ways that have succored the tendency to ‘live-with’ loose, verbal
field have themselves perambulated.
theories. Done right, the field of mathematical psychology offers a
prescription for using quantitative theory to impel theory-driven
methodology.
2.2. Foundational measurement
Of course, there will always be a need for statistics. In addition
to testing various traditional hypotheses, providing for confidence
Several of the founders’ names were mentioned earlier but it
intervals, etc., it is highly useful to have in hand tools to test model
would be remiss not to cite several major seminal works in this
fits and, even better, to compare models against one another (see,
area: The three foundational volumes of Krantz, Luce, Suppes, and
e.g., the useful special issues of Journal Mathematical Psychology
Tversky (1971), Luce, Krantz, Suppes, and Tversky (1990a,b) and
(Myung, Forster, & Browne, 2000; Wagenmakers & Waldorp,
Suppes, Krantz, Luce, and Tversky (1989), the excellent pedagogical
2006)). This topic will reappear in later sections of the present
text of Roberts (1979), and the seminal topologically based volume
essay.
of Pfanzagl (1968).
Foundational measurement theory has continued to attract
highly skilled mathematicians and quantitatively oriented psy-
1.7. A regrettable neglect of proper scientific referencing
chologists from throughout the world and to advance knowledge,
especially on the critical topic of ‘meaningfulness’. Technical dis-
I will conclude this section with a little belly aching pertinent
cussion is beyond my scope here, but informally, ‘meaningfulness’
to the behavioral sciences in general and psychology in particular.
relates jointly to how a measurement scale represents qualitative
Obviously, we all know of areas in psychology that are amenable
to modeling and could clearly profit from it. Often, verbally based
aspects of the real world, and the degrees and types of invariance
“theories” may be underpinned (or even contradicted) through
that properties of a scale enjoy under permitted transformations of
previously published quantitative research but the verbal theorist
that scale (e.g., Krantz et al. (1971) and Roberts (1979); for a recent
fails to use or cite it. It may well be that the ‘errant’ investigator
thoughtful statement, see Narens (2003)).
thought up the approach independently in a verbal and qualitative
One impediment to more usage of foundational measurement
way, as psychologists often do. Or it may be that they simply don’t
theory has undoubtedly been the relative paucity of effort and re-
feel that anything accomplished mathematically has to be cited by
sults on an ‘error’ theory which could provide a ready implemen-
non-modelers, as noted above. In either case, it is bad science and
tation of statistical procedures with data. Groundbreaking work
insulting to the theorists and methodologists who labor mightily
continues on this challenge (for recent progress on stochastic ap-
to advance psychology as a rigorous science. I trust that a physicist
proaches to foundational measurement which subsume the tra-
would not fail to recognize mathematical results that formed a
ditional error theory, see, e.g., Niederée and Heyer (1997) and
sturdy basis under her theory or tools.4
Regenwetter and Marley (2001)).
A related gripe concerns the tendency of investigators in
I believe this field is and will continue to be, of interest not only
different quantitative, but overlapping, disciplines to ignore work
to the behavioral and biological sciences, but also to philosophy
in the areas of their research ‘cousins’. Sometimes this simply
of science and epistemology in physics, although at present the
takes the form of failing to utilize helpful material from other
substrata primarily relate to Newtonian rather than relativistic
research domains, but not infrequently, it may involve lapses in
physics. At any rate, this branch of mathematical psychology is
apparently not responsible for the reputed passing of the field.
4 I suspect a number of readers can think of instances of this phenomenon,
perhaps intersecting their own work. I rather cravenly fail to mention names in
5 As far as I can ascertain, these announcements have been confined to verbal
order to avoid litigation and other unpleasant retribution.
remarks. However, they have occasionally been uttered by renowned psychologists.

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J.T. Townsend / Journal of Mathematical Psychology 52 (2008) 269–280
2.3. Signal detection theory
In any event, there is no reason to think that signal detection
theory encouraged anyone to proclaim the decease of mathemati-
Signal detection theory, as most readers of this essay will
cal psychology.
know, emerged in the 1950s as a confluence of Neyman/Pearson
statistical decision theory and ideal detector theory of electrical
2.4. Decision theory
engineering (e.g., one of my early favorites is Peterson, Birdsall,
and Fox (1954)). In psychology, the Green and Swets (1966) book
What about the field of decision making? One branch of
has become a classic with other volumes such as Egan (1975)
effort primarily theoretical but impelled partly by a growing
following up on interesting byways. Early on, there were clear
literature of experimentation, finds its roots in the axiomatic
and continuing associations to be made with Thurstone’s popular
foundations laid down by von Neumann and Morgenstern (1953).
discriminal processes theory.
Psychologists dedicated to this tradition consisted partly of those
Signal detection theory went through a period during the sixties
also contributing to foundational measurement, and indeed, often
and early seventies in which psychologists proffered a number
similar tools are found in their theoretical arsenal (e.g., Luce,
of models that lay outside the engineer-oriented mathematical
Suppes, Krantz, and Tversky). Of course, this branch also included
communication theory (as exemplified by the central content of
those with statistics or economics backgrounds (e.g., Savage).
the popular (Green & Swets, 1966)), the latter also incorporating
A massive and influential development in the field came about
(but was not limited to) elementary statistical decision theory.
through the efforts of Tversky and Kahneman, who discovered
In addition to Luce’s adaptation of his choice theory to detection
a number of human choice situations in which people veer
and recognition situations (e.g., Luce (1959, 1963a)), a number of
drastically away from the classic (and even some newer) axiomatic
finite state models were put forth and evaluated (e.g., Atkinson
theories. Certain of these cases flow from their theoretical results,
and Kinchla (1965), Krantz (1969) and Luce (1963b)). Although
especially prospect theory (Kahneman & Tversky, 1979). As
these contained interesting capabilities, not always captured by
everyone should now know, this corpus of work earned Kahneman
the dominant theory (as exemplified by Green and Swets (1966)),
the Nobel Prize in economics and would undoubtedly been
especially learning effects (e.g., Kinchla, Townsend, Yellott, and
awarded simultaneously to Tversky were it not for his untimely
Atkinson (1966) and Luce (1959)), they currently see only
passing.
occasional employment.
The field of decision making has historically been somewhat
In contrast, the dominant theory is now ubiquitous in
separated into a set of quantitative theorists (mostly in the
psychology in general as well as in sensory sciences, usually
axiomatic or statistical framework) and a set of experimentalists,
employing the ubiquitous Gaussian distributions (nonetheless,
the latter largely made up of psychologists. Of course, there still
two-state models vie with the continuous theory in certain
are many who do both (e.g., One subdivision of the experimental
areas; see the Wixted reference below and citations therein).
group has inclined toward testing predictions made by the
With regard to the latter characteristic, it would seem that
axiomatic or statistically-based models (Birnbaum, 2004). Another
certain foundational results could be beneficially employed by
area has concentrated on continuing the quest for psychological
experimental researchers (e.g., Marley (1971)). For example,
behavior that seems at odds with various facets of the axiomatic
ideal detector theory is flourishing as are multidimensional
theories [especially utility theory]). Yet another has evolved
theories of signal detection. An example of the latter is general
non-quantitative models or theories that attempt to be heavily
recognition theory, which was originally developed in order to
real-world oriented (e.g., how do people make decisions in a
study interdimensional interactions (e.g., Ashby and Townsend
group-crisis?), and experimented thereon.
(1986), Kadlec and Townsend (1992) and Thomas (1999, 2003)).
Although various facets and extensions of utility theory per
One of the major contributions of signal detection theory was
se are still active research areas, a number of investigators have
the centrality of decision mechanisms, even in putative “purely
moved on to explore the preferential laws governing risk (Weber,
sensory” domains. This facet continues to offer fresh perspectives
Shafir, & Blais, 2004). In addition, the perhaps overdue appearance
as in the Wenger and Ingvalson (2002) study which found a
of hedonics in the consequences of decisions has occurred (Mellers,
powerful role of the decision process in the perception of holistic
2000). Gigerenzer and colleagues have been a powerful voice
vs. non-holistic object perception. It has been widely employed as
in plumping for models based on all-too-human limitations in
a theory of categorization (e.g., Ashby and Maddox (1990); many
processing capacity and sometimes rationality (e.g., Gigerenzer
of the models in Ashby (1992) are signal detection based).
and Todd (1999)). Wishing to avoid charges of excessive modesty,
Certainly, signal detection theory should be studied (usually
I hasten to mention work that seeks to be in the spirit of strong
it is covered rather cursorily, if at all, and in a non-quantitative
quantitative theory but heavily invested in psychological and
fashion–more is the pity; see below) even by students in their
biologically flavored knowledge (Busemeyer & Townsend, 1993).6
introductory courses. It stands as the prototypical theory-driven
In any event, I can locate little in the evolving field of decision
methodology, since it can be employed to discern decision and
making that should have precipitated augurs a terminal malady of
learning bias from ‘true’ sensory or sensitivity (e.g., signal-to-
mathematical psychology.
noise ratio) effects in such diverse fields as hypnotic phenomena,
to trial-witness memory, to laboratory psychophysics or learning
and cognition experiments e.g., Swets (1996); see Balakrishnan
2.5. Psychophysics
(1999) for an alternative method of analyzing sensory and bias
effects). For instance, Wixted (2007) argues for a ‘traditional’ type
I somewhat artificially separated signal detection from psy-
of signal detection model against less mathematized but process
chophysics and it might legitimately be argued that neither is
oriented, two-process kinds of models, in certain areas of cognition
strictly a subfield of mathematical psychology. Moreover, due to
(e.g., see Balota, Burgess, Cortese, and Adams (2002), Diana, Reder,
space and ‘psychological distance’ concerns, I must neglect the now
Arndt, and Park (2006), Heathcote (2003), Hockley and Cristi
sizeable field of sensory sciences, which of course, heavily over-
(1996), Malmberg, Zeelenberg, and Shiffrin (2004) and Yonelinas
laps both psychophysics and signal detection (and uses methods
(1994)). MacMillan and Creelman (2005) compile a worthy set
of signal detection-based methodologies provided in a tutorial
style. Wickens (2002) provides a basic introduction to many of the
6 I refrain from mentioning the likelihood of a pummeling about my head and
fundamental concepts.
shoulders by my colleague, Jerome Busemeyer.

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J.T. Townsend / Journal of Mathematical Psychology 52 (2008) 269–280
273
from both). Nonetheless, I think it is fair to say that in some senses
non-linear (typically multiplicative) differential equations, most
the psychophysics of Weber, Fechner (to a lesser extent Wundt),
often with strong systems-oriented interpretations afforded the
Helmholtz, and others was a progenitor, along with statistics, of
equational elements.
mathematical psychology. Also, modern mathematical psycholo-
About this time too, two investigators in the cognitive science
gists and psychophysicists have long been attracted to it, apply-
revolution, Minsky and Papert (1969), published a book innocently
ing concepts from measurement theory and functional analysis
entitled “Perceptrons”. As most readers of this essay are well
(e.g., see Falmagne (1985) and Luce, Bush, and Galanter (1963)) as
aware, perceptrons are basically linear, and usually deterministic,
well as more process models (e.g., Baird (1997) and Link (1992)).
pattern recognizers; members of the class of systems known as
From some viewpoints, Stevens (1951) should probably be con-
linear discriminant classifiers. These authors showed in a carefully
sidered as the twentieth century heir of classical psychophysics,
reasoned treatise, that perceptrons are incapable of seemingly
through his innovation of the necessity for a hierarchy of mea-
quite elementary topological distinctions.
surement scale types as well as his new experimental methods of
Needless to say, their exegesis did not have the effect of
scaling, the so-called direct methods. One influential investigator
furthering the general interest in perceptron theory or perhaps,
whose work could be listed in several of the present categories in-
even an encouragement with regard to neural modeling per se. In
cluding the present is that of Shepard (1964). Another is Anderson
fact, some have felt that it had a rather devastating consequence
(1981). A substantial and evolving theory of psychophysics with
on activity in neural modeling.
roots in Fechner’s original developments is found in the work of
During the years that interest in neural modeling paled,
Dzhafarov and Colonius (e.g., Dzhafarov (2002) and Dzhafarov and
Grossberg and a few other theorists, such as James Anderson,
Colonius (2007)). Nosofsky has contributed an influential modeling
kept the fires aflame (Anderson, 1995). Anderson’s models could
approach which links up psychophysics, psychometrics (via multi-
be viewed as more sophisticated upgrades (e.g., incorporating
dimensional scaling), and information processing (e.g., Nosofsky
stochastic noise, Hebbian learning devices, and non-linear decision
(1984)).
structures) of perceptronic tenets.7
An enduring challenge, not unrelated to the issue of multidi-
Then, in the 1980s, hurtled the connectionistic meteor, driven
mensional interactions mentioned earlier, is the inclusion of con-
in substantial part by the labors of Rumelhart and McClelland
text effects in psychophysical scaling (see, e.g., Helson (1964) and
(1986). Interestingly, Rumelhart had been a graduate student
Parducci (1956)). The drawing together of the psychophysical ap-
in the Stanford mathematical psychology training program in
proaches with process model approaches to context and inter-
the 1960s. Even though some specialists would prefer to offer
actions has begun (e.g., Chapter 16 in Baird (1997), Hughes and
distinct definitions for “connectionism” vs. say, “distributed
Townsend (1998) and Link (1992)) but is likely still in its infancy.
processing”, they are often used interchangeably to indicate what
Nonetheless, it appears that such a synthesis can be useful. For
we might call neuralistic modeling. “Neuralistic” might be a
instance, a coalition of psychophysical with process methodol-
reasonable neologism for this field since practitioners strive to
ogy has elucidated aspects and challenges of the Stroop effect not
let neurophysiology and neuro-anatomy guide their efforts while
previously visible (Melara & Algom, 2003). Sarris has made funda-
benignly neglecting less critical aspects of these disciplines. They
mental contributions to relational psychophysics in comparative-
are occasionally taken to task for transgressions in this regard,
developmental psychology (e.g., Sarris (2006)).
but theorization has always been a matter of emphasizing what
seems most vital and ignoring the rest. Certainly such criticisms
2.6. Neural modeling
apply to all but the most microscopic models of neural functioning;
and the latter are often of little interest to psychologists. It is also
One very significant player in quantitative theory in psychology
worth remarking, given our earlier discussion, that learning theory,
as well as cognitive science has been neuropsychological modeling.
always a key ingredient of neural modeling, made an impressive
It might have been expected to provide a major counter weight to
come back in the wake of the renaissance of the latter.
some of the less propitious forces facing mathematical psychology.
Although endeavors were made by those involved with the
Due to the potentially symbiotic linkages that could result when
Society for Mathematical Psychology to encourage affiliations and
both are applied in the spirit of reductionism (but see Uttal (1998)),
interactions (e.g., by appointing connectionistic associate editors
plus its almost startling resurgence in recent years, I take a little
to Journal of Mathematical Psychology; inviting keynote addresses
more space on this topic.
by leaders in connectionistic modeling, etc.), and although many
Like some of the other branches of research, an adequate
mathematical psychologists have labored from this perspective,
history of neural modeling would take up at least one volume.
the attempts to embrace this field perhaps have not been entirely
However, we can limn in some of the most evident sign-posts.
successful. There are notable exceptions, including the work of
Although neural modeling was certainly around at the time that
Kruschke (1992).
mathematical psychology received its formal impetus, and there
Neuropsychological modeling has certainly waxed and waned
have been significant intersections over the years, it is not routinely
over the past half century or so, and has proceeded more or less
thought of as a sub-field of mathematical psychology.
independently of mathematical psychology, but it doesn’t appear
Modern history of neural modeling goes back at least to
to have been responsible for the obituary of the latter at any point
Rashevsky and Ashby in the 1940s. The fifties saw emergence of
in time.
logic network based thinking (flowing undoubtedly from automata
theory of von Neumann, Turing, and others) by McCulloch and Pitt.
2.7. Information processing approach
The ultimate influence of Hebb’s well-known principles of synaptic
learning, though not so rigorously presented originally as some
Like the other topics discussed here, the information processing
of the other candidates for attention, would be difficult to over
approach possesses somewhat fuzzy boundaries, but perhaps
emphasize.
At the juncture where mathematical learning theory began to
fade in popularity (no causal implication intended here, merely a
7 In the discipline of neural modeling even more than others, space and the
time-marker), Grossberg (1969) began to publish his first neural
natural emphasis of this essay prohibit listing of a sizeable set of investigators
modeling work which, incidentally, included an emphasis on
primarily associated with other fields, which have made fundamental contributions
learning and motivation. That work was, and is, founded on
to neuro-psychological quantitative theorizing.

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even more so. A central tenet seems to be the representation
Thus, by this time, the budding field of cognitive science, with
of perceptual, cognitive, and/or motoric mechanisms, usually
both its theoretical content as well as implementation heavily
confined to a certain task setting, via a set of subsystems with
determined by digital computers and automata theory, was in-
the information flow (itself usually a fuzzy concept) depicted via
creasing enormously in popularity, with Simon and Newell per-
the so-called, and ubiquitous, flow diagram. Lachman, Lachman,
haps leading the charge in areas close to experimental psychology.
and Butterfield (1979) provided what I regard as a quintessential
Contributions began pouring in not only from experimental psy-
tutorial on the information processing approach, and one of the last
chologists but also philosophers, computer scientists, electrical en-
rigorous treatments of cognitive psychology for undergraduates. Of
gineers and applied physicists. Such innovations as production
the early textbooks in mathematical psychology, probably Laming
systems soon provided a powerful lure to psychologists seeking a
(1968) was the closest to the information processing approach.
richer milieu for concepts about mental operations. Likely, it is pri-
Undoubtedly spawned by computer science (e.g., automata
marily the confluence of cognitive science as a new and promising
theory), and aided and abetted by information theory and
field along with the perceived failure of mathematical learning the-
cybernetics, the generic conception was picked up just as fast by
ory to ‘pay off’ that caused a substantial hiatus, if not termination,
general experimental psychologists as by mathematical modelers,
of the latter. Into the bargain, many of the founders of mathemati-
sans the rigor of the latter. Yet, the seminal efforts of Sternberg,
cal learning theory were increasingly attracted to regions of study
Sperling, Estes, Falmagne, Atkinson, and others, helped attract
more closely allied with cognitive science, such as the broad ap-
modelers intrigued by the idea of analyzing a system down into
proach of human information processing. It seems fair to say that
its functional components, even if (or because) the properties
the well-known Atkinson and Shiffrin model of short-term mem-
of the system in action might reveal emergent properties. Later
ory and control processes (Atkinson & Shiffrin, 1968) epitomized
contributors of the modeling ilk include Nosofsky, Massaro, Link,
this movement. The subsequent launching of the ACT models by
Yellott, Ratcliff, D. Meyer, Colonius, Diederich, J. Miller, Vorberg,
J. R. Anderson and colleagues (e.g., Anderson and Bower (1973))
Bundesen, E. A. C. Thomas, Massaro, Logan, Schweickert, Dzhafarov
in some ways intersect both the information processing approach
and others.
and the emerging cognitive science paradigm (e.g., a la Simon &
Models of response time have been especially influential and
Newell).
productive in uncovering underlying processing mechanisms. The
The evidence seems to point to the recession of mathematical
most popular types of response time models, often including
learning theory in the late 1960s as one key marker that
accuracy predictions have been those based on random walks (e.g.,
convinced some investigators of the weak health, if not demise,
Link and Heath (1975)), diffusion processes (e.g., Ratcliff (1978)
of mathematical psychology. Moreover, a number of modelers
and Busemeyer and Townsend (1993)), and counting processes
who cut their teeth with the ‘classical’ learning models, were
(e.g., Smith and Van Zandt (2000)).
now moving wholesale into ever broader and more rigorous
I believe the essence of the information processing approach
models of memory. I refer in particular, to the works of Hintzman
provided a milieu that helped prompt investigations into issues
(1986), Izawa (1971), Murdock (1982), (e.g., Raaijmakers and
of model mimicking. This influence was even felt in modeling of
Shiffrin (1981)), (e.g., Humphreys, Bain, and Pike (1989)), which
learning and memory. Thus, Greeno and Steiner (1964) analyzed
investigators have over the past few decades made this field into a
model equivalence within Markov chain models of memory.
poster child of how behavioral research should be carried out.
Batchelder (1970) attacked what seemed to be a mimicking issue
Okay, so we could lay some of the blame for the perhaps pre-
regarding incremental vs. all-or-none learning and showed how
mature rumor of the death (but not interment!) of mathematical
to distinguish them. My own efforts on parallel vs. serial model
psychology on the adventitious events pertaining to learning the-
testing began shortly thereafter (e.g., Townsend (1969, 1971,
ory in the late 1960s. However, it has always seemed paradoxical
1972) and Townsend and Wenger (2004)). Of course, knowledge
to me how learning per se was shoved into the corner for sev-
of when and how models cannot be differentiated can assist in
eral decades, even though memory became supersedent as a le-
erecting a meta-theory or methodology that is capable of such
gitimate topic in cognitive science and mathematical psychology
assay (e.g., Townsend (1976a,b)). In my not-unbiased view, the
per se. How did these memories originally become instated? With
information processing approach is still alive and prospering,
few exceptions only in the relatively lonely (for awhile!) terrain of
although perhaps not always under that rubric.
neural modeling (e.g., Anderson (1973) and Grossberg (1969); and
shortly with a resurgence from the Anderson quarter e.g., Ander-
2.8. Mathematical learning theory
son (1990)), did learning theory persevere. While it is fair to say
that learning played some part in some of the burgeoning mem-
And then there is that other early pillar of mathematical
ory models, it was at best a minor role. Learning as a legitimate
psychology, mathematical learning theory. The germinating work
research topic is happily now back with us big time.
of Bush and Mosteller (1955) and Estes (1950) gave rise to a decade
In any event, it is time to abandon this historical, if somewhat
or so of fervent activity on mathematical models of learning,
whimsical, excursion and pose the query as to the state of health
including a wave of anthologies and monographs (e.g., Bush and
of the field today.
Mosteller (1955); later, Atkinson, Bower, and Crothers (1965); very
late, Restle and Greeno (1970)).
2.9. Trends, education, politics, and tenure
Even during the 1960s, mathematical learning theory was
beginning to move toward more emphasis on memory and less
One can attempt answers along many dimensions. First, the
on learning. By the mid-seventies publications of theoretical and
“glass is half full” perspective: The Society of Mathematical
experimental effort devoted to mathematical learning theory had
Psychology continues to serve many functions that are conducive
diminished precipitously. Furthermore, it could be (and was,
not only to the highest standards of research in mathematical
at least informally) argued that finite-state Markov learning
psychology, but to encourage the entering of young scientists
models, a mainstay in the field, were beginning to appear rather
into our field. Its members contribute expert reviewership to a
baroque, sometimes without convincing signals from the data that
broad spectrum of research areas vis-à-vis journals and scientific
such elaborations were required. Perhaps before self-corrective
grant proposals. They provide some of the top research, especially
measures could eventuate, other forces essentially moved in to
through model and theory building, in scientific psychology. Many
occupy the territory.
regularly bring in valuable grant resources to their universities,

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275
even in these extreme financially harsh times. Many members of
2.12. Undergraduate training: Following the trends in society
the Society have served on strategic committees and panels in
national institutes and other societies and federations.
Many surveys and studies have documented the steep decline
Nonetheless, there are serious “glass is half empty” concerns:
of scientific training and acumen in our youth, not to mention the
For instance, with reference to membership of Society of Math-
deterioration of scholarship in general (e.g., witness the lamentable
ematical Psychology, we find that unfortunately, numbers have,
pruning of elementary and high school courses in music, foreign
with some fluctuation, tended to decline in number since its for-
languages, and even physical education, starting in the 1970s).
mation in the late 1960s (following inauguration of Journal of Math-
Hence, it should come as no surprise to learn that scientific
ematical Psychology in 1964). We will see that there is widespread
education of US college students is woefully inadequate.
anxiety about quantitative training in psychology, not simply in-
With the possible exception of some well endowed private uni-
clusive of mathematical psychology per se. We can question ‘why’
versities and colleges, psychology departments serve as bountiful
cash cows for their institutions, even (or especially) in large, public
and ask about relationships to various movements in psychology
research-oriented universities. In addition to sizeable flocks of ma-
and cognate areas.
jors plus non-majors, they typically bring in large amounts of out-
side research monies, sometimes close to or exceeding the more
2.10. Fractionation vs. generality
established laboratory science departments. And, their classroom
lab facilities cost little to nothing in comparison with the latter.
One trend over the past century up to the present has been the
The debt to the devil in all of this is that there is immense
steady fractionation of science in general and certainly psychology
pressure, if usually implicit, against driving down enrollments by
in particular. This trend has been associated with many benefits,
increasing standards, for instance, by requiring majors to take
such as deepening knowledge and specialization within many
more physical and biological science and mathematics. Of course,
branches of study. Yet, it has arguably, if rather paradoxically, been
this influence sums with many others, including the aversion
at the expense of areas like mathematical psychology which seek to
certain sectors and individuals within psychology feel towards
encompass a wide sweep of individual domains, from learning and
mathematics and hard science.8 Other pernicious forces include
memory, to sensation and psychophysics (indeed to at least one
the seemingly perpetual inclination of publishers to persuade
society for each modality!), to models of social behavior, to neural
authors to ‘dumb down’ their textbooks and sometimes, even
theory. As these specialties have budded and flowered, generalistic
scientific monographs, and the well-documented grade inflation
groups such as SMP have often experienced hurdles in capturing
that has plagued higher education at least since the advent of the
time from over-extended researchers.
1970s.
With regard to undergraduate training, given the above
and other factors, expecting a sea change toward solid-science
2.11. Is psychology now ‘sold’ on mathematical psychology, having
education, even for most psychology majors, not to mention the
effectively absorbed it?
legions from other departments who take our courses, is akin to
belief in the tooth fairy. The only practical solution I can espy is
One argument heard occasionally, even by ‘friends’ of mathe-
for psychology departments to offer a true scientific psychology
matical psychology, is that now quantitative modeling has been
track, with mandatory courses in the sciences, mathematics and
absorbed by the field at large and hence a separate sub-discipline
statistics. It could, but need not, be incorporated into a true honors
devoted to this specialty is superfluous. For instance, serious ob-
program.9
stacles in the path of publishing mathematical models encouraged
The latter could include options for coursework in engineering,
the establishment of Journal of Mathematical Psychology. Support-
economics, ecological sciences, and more recently available,
ing evidence is proffered that many experimental journals now
informatics and biocomplexity, which would increase the chances
regularly accept quantitative modeling of the experimental data
of those who decide not to pursue postgraduate education, to find
within submitted articles. I agree that this is definite evidence of
employment. An added benefit to psychology graduate programs
progress. Yet, the acknowledgement of the fact does not lead inex-
throughout the land, would be a diminution in the usual scenario
orably to the consequent. For one thing, journal editors reveal an
every spring: thirty or so departments fighting over a pitiably
astonishingly high variance with regard to their attitudes toward
small handful of qualified students, the latter of which either
quantitative modeling. And, many if not most, outside of quantita-
come from other sciences or somehow manage to acquire decent
tive journals, pre-form some type of upper-bound on the degree of
background in the face of feeble departmental curricula and
abstraction they deem acceptable.
inadequate counseling.10 Certainly, the uninterrupted flow of milk
My view is that import alone should determine what sees
the light of day even in our more experimental or qualitative
organs, but with the stipulation that the editor should be free to
8 Naturally, much of this aversion is muted, especially overt statements of
request fairly considerable clarification and even tutorial material.
opposition to “hard science” in general. However, one could not get far, even
A rebuttal might be to the effect that “Well that just means more
in rigorous neuroscience, in the absence of a modicum of real mathematics
publications for the Journal of Mathematical Psychology”. A problem
(e.g., the calculus). In addition, shocking as it may seem, occasionally first-hand
anecdotes surface from individuals in major universities, of prominent research
with this reasoning is that if there is little or no overlap with
psychologists disparaging quantitative methodology and training (and this includes
the experimental journals, it is even easier for less quantitative
psychometrics and statistical methodology; not just mathematical modeling).
scientists to ignore our work.
9 It has appeared to me that many departments and universities began to
I think it can be fairly argued that SMP and Journal of
downgrade the standards associated with honors programs back in the 1960s and
Mathematical Psychology, in addition to sister organizations and
70s. The reasons may vary but at least in some cases, one motive was to avoid
damage to self esteem—a noble goal, but perhaps at odds with the definition of
journals such as Psychometric Society and Psychometrika, are
“honors”.
greatly needed by scientific psychology as “keepers of the flame”.
10 Who among us has not heard some version of the following undergraduate’s
We serve as upholders of the highest quantitative standards by
refrain (and for me across several universities where I’ve taught): “But the counselor
our multitudinous duties as reviewers of papers and grants and as
laughed when I asked about taking math and science, and asked why a psychologist
practitioners of modeling and methodological science. All this, in
would need something like that”. Of course, the one saving grace has been the
relatively continuous influx of quantitatively prepared students from abroad.
addition to our function in training of graduate and undergraduate
However, even this palliative may fade as countries begin to establish their own
psychology majors.
fine universities and research institutes.

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J.T. Townsend / Journal of Mathematical Psychology 52 (2008) 269–280
from the contented cow should gladden the hearts of deans and
to me that it is the very rare neophyte (i.e., newly minted
presidents.
quantitative assistant professor) who can publish at the rate
easily achieved by their cohort in other specialties of the field.
2.13. Graduate recruitment and training
Some departments, T&P committees and significant individuals
(especially pertinent is the chair or department head) do
Given this prelude, perhaps it should not come as a surprise
appear to ‘handicap’ according to sub-discipline when considering
that even graduate training in psychology has seen the devolution
promotions and salaries, but I think this is not common and when
of quantitative training; never that auspicious in the first place.
done, probably not to the appropriate degree.
Several prominent psychometricians reported and discussed a
Why, it might be countered, does the burgeoning field of
survey on quantitative training of psychologists in 1990 (Aiken,
cognitive neuroscience seem to have little difficulty in attracting
West, Sechrest, and Reno (1990); cf. Townsend (1994)). At that
apprentices? It is true that a huge range of technical ability
time outdated topics were a major concern with a consequent
and background is accommodated within that discipline but the
under representation of newer, powerful strategies.
same could be true in quantitative psychology. More persuasively,
Fifteen years later we have a reprise on a considerably more
neurophysiology has been a more entrenched part of psychology
frightening note, in an article by Clay (2005) written for the
— indeed, almost every department has an ‘area’ devoted to
APA Monitor: the shocking scarcity of new quantitatively-trained
neuroscience or neurocognition (the “in” terms for this region,
psychologists, and even of qualified programs to train them.11
‘physiological psychology”, “psychobiology”, and so on, seem to
Stephen West (Editor of Psychological Methods and Professor at
change every decade or so) — since the very inception of scientific
Arizona State University) is quoted “At lot of the major quantitative
psychology. There are also feeder sources and ancillary training
programs over the years have died. We’re one of three larger
posts for the physiologically inclined, like pre-med and biology
programs in psychology in the country, and we produced one Ph.D.
for which there is little concomitant in our area. Interestingly,
this year”.
the current movement toward neuroscience forms one of the few
Although mathematical psychology per se, probably never saw
trends in psychology toward hard science.
more than three or four formal programs of training in the US,
In any event, it can be cogently argued that the central
it is still arguable that even that limited presence has declined.
advantage psychology has over other fields in years past has been
Of course, the dearth of rigorously trained post-baccalaureate
the relatively heavy component of education in practical statistics
psychologists mentioned above, and the degradation of science in
and methodology and potentially, modeling. Hundreds of hours
general in the US, are undoubtedly contributing factors.
of arduous, often boring and occasionally exciting, labor in the
Now, the Clay article portrays the view from some methodol-
laboratory plus the subsequent data analysis and model testing,
ogists that quantitative psychologists are in great demand, with
put psychologists in a solid position not only for academic positions
some institutions finally ‘giving up’ when years go by without a
but also research and management in industry and government.
successful quantitative hire. That may be and if so, I applaud that
The pivotal role of experimental and methodological psychol-
there is now a palpable appreciation of our specialty, without, of
ogists in effectively leading medical research teams especially in
experimental design and data analysis, while typically serving un-
course, hearty clapping for the diminution in numbers. I must say
der the obligatory M.D principal investigator, is well known. And,
though, that in my opinion, years went by where superbly trained
clinical psychologists have neither the political clout, M.D. prestige,
mathematical and psychometric psychologists were not accorded
nor monetary recompense afforded psychiatrists. But, they have
the best job opportunities. Too often, the unspoken refrain seemed
in the past been able to contribute their knowledge and practice
to be something of the form “. . . well, everyone learns statistics at
of test theory and administration. These skills largely disappear
a sufficient level to teach our introductory courses [often the only
in the unfortunate trend toward so-called Psy.D. degrees, which
kind offered], so we might as well hire in one of our favorite con-
require little if any research experience, statistical knowledge or
tent areas and be doubly happy . . . ”. And, a rather dangerous pitfall
even training in psychological test theory. However, there are now
often awaited young quantitative psychologists, that of being ex-
forty-five member training institutions in clinical science, which
pected to provide ‘free’ consulting services to faculty and students
adhere to the principles of the Academy of Psychological Clinical
(who often didn’t bother to take the assistant professor’s classes
Science. The goal of these programs is to emphasize the rigorous
on the topic), yet receiving little credit for these activities at tenure
training of a core of clinical scientists (see, e.g., McFall (2006)) per-
and promotion time.
haps somewhat compensating for an avalanche of Psy.D. practi-
As other possible aversions to the beginning graduate student,
tioners over the past couple of decades. I don’t know if research
in addition to the sheer arduous technical preparation (years of
has been accomplished regarding the relative proficiency in ther-
mathematics plus the applied quantitative tools), it has appeared
apy of Psy.D. personnel vs. traditional Ph.D. clinical psychologists,
but there is no doubt about their respective statistical and research
skills. Although as one would expect, one of the major regions of
11 Partly to lodge support for this Monitor article and partly to stress the
cross disciplinary training for clinicians is in neuro-science, some
contributions of mathematical psychology to quantitative training in psychology,
departments, such as that at Indiana University, train some stu-
I co-authored with Richard Golden and Thomas Wallsten, a guest editorial on this
dents in mathematical modeling and collaborate with the quanti-
issue for the APA Science Directorate (Townsend, Golden, & Wallsten, 2005). The
tatively oriented faculty.
topic of quantitative training in psychology was raised several times in various
‘break-out’ sessions at the recent APA sponsored Science Leadership Conference.
Many of us have witnessed even people with Ph.D.s in such
Admittedly, APA has in the past been accused of short-changing the ‘science’ in
fields as physics, engineering, computer science, and mathematics,
favor of professional concerns, undoubtedly one of the several motivations for
making grave errors in experimental design when they ‘cross-
the establishment of American Psychological Society. Nonetheless, there is some
over’ into experimental psychology in the absence of collaboration
evidence that APA, in addition to its decided interest in professional matters, is
within the latter.12 The same is true, and more, with regard to
credibly moving into the game of promoting scientific psychology. For instance, I
was recently able to recommend ‘challenges in quantitative training in psychology’
as a theme for the next leadership conference. Steve Breckler, Executive Director for
Science, APA, appears to provide a healthy force in this direction (see Psychological
Science Agenda (see psaA@APA.ORG, for more on APA’s role in psychological
12 This is not a criti