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Three questions about mathematical psychology

This project investigates mathematical psychology's historical and philosophical foundations to clarify its distinguishing characteristics and relationships to adjacent fields. Through gathering primary sources, histories, and interviews with researchers, author Prof. Colin Allen - University of Pittsburgh [1, 2, 3] and his students  Osman Attah, Brendan Fleig-Goldstein, Mara McGuire, and Dzintra Ullis have identified three central questions: 

  1. What makes the use of mathematics in mathematical psychology reasonably effective, in contrast to other sciences like physics-inspired mathematical biology or symbolic cognitive science? 
  2. How does the mathematical approach in mathematical psychology differ from other branches of psychology, like psychophysics and psychometrics? 
  3. What is the appropriate relationship of mathematical psychology to cognitive science, given diverging perspectives on aligning with this field? 

Preliminary findings emphasize data-driven modeling, skepticism of cognitive science alignments, and early reliance on computation. They will further probe the interplay with cognitive neuroscience and contrast rational-analysis approaches. By elucidating the motivating perspectives and objectives of different eras in mathematical psychology's development, they aim to understand its past and inform constructive dialogue on its philosophical foundations and future directions. This project intends to provide a conceptual roadmap for the field through integrated history and philosophy of science.

This video was published on Prof. Colin Allen’s YouTube channel.

The text of the transcript video presentation with screenshots of presentation slides published with the permission of the Author, Prof. Colin Allen - University of Pittsburgh.


This is a pre-recorded presentation for the 2021 meeting of the Society for Mathematical Psychology. 

I'm a member of the Department of History and Philosophy of Science at the University of Pittsburgh, and here we're going to introduce you to a project that we have started with encouragement from several members of society to write a book-length treatment of the history and philosophy of mathematical psychology. 

And I'm going to be joined in this presentation by four wonderful students from my department Osman Attah, Brendan Fleig-Goldstein, Mara McGuire, and Dzintra Ullis. They will cover the three questions that are mentioned in the title and Dzintra will wrap up with some final thoughts and ideas about how you can contribute. 

But I'm going to start by giving you an introduction to the project, explaining a little bit about the nature of history and philosophy of science, give you some caveats about the project itself, where we are currently, suggest a timeline, and indicate some of the sources that we have been relying on.

The Project History & Philosophy of Mathematical Psychology

So The Project History & Philosophy of Mathematical Psychology to frame it in terms of the history and philosophy of science, it's useful to start with this quote by Norwood Hanson, who was one of the founding members of the History and Philosophy of Science at Indiana University, who wrote back in the 60s, with a nod to Kant, that the philosophy of science, without a history of science, is empty and the history of science, without philosophy of science, is blind.

And so the goal, from the beginning of history and philosophy of science, has been to try to find some middle ground between the particular attention to context that historians are fond of and the more general search for abstract ideas that philosophers focus on. 

Ordinary historians tend to focus on social and political context; historians of science tend to bring in a little bit more of the intellectual context, also known as the history of ideas. 

Similarly, philosophers of science look to science for ideas about epistemology and ontology - these more general abstract ideas that they're looking for, because they are philosophers, but with guidance from science itself. 

So, history and philosophy of science researchers tend to seek a middle ground between the particular and the abstract, with the goal of integrating insights from both directions. 

Now it's useful in thinking about what we're doing here, especially today, and especially in the context of mathematical psychology, to twist this statement that many of you will be familiar with about models into one about what we're doing: all histories are wrong, some are useful. 

So what we intend to indicate here is that not everybody has the same view of what happened and what was important, and what happened. 

But our goal is to try and identify some of the diversity in those views and thereby do something useful.

And it's very hard to tell the story of mathematical psychology without bringing Stanford into the very center of the story. So, on the right-hand side here, you see a photograph that was posted online recently by Stephen Link, who was looking for information about what was happening in Stanford's Institute for Mathematical Studies in the Social Sciences in the 1960s. 

And in that photograph, you will see, but perhaps not recognize, some of the people who are perhaps even present virtually at this talk, and we hope later at the live Q&A session that we have scheduled. 

This picture, I believe, was taken at Ventura Hall, which is pictured at the bottom left, and one of the key things that we've learned from our interviews is the idea that the psychologists in this group were set apart from the rest of psychologists was very important in building a certain kind of corps spirit. 

It's also possible, then, to identify some of the people in that picture and others as related to each other through a kind of academic lineage. 

So here, I tried to map out just a few important figures, putting a big blue blob around those who might self-identify as mathematical psychologists and using colors to try to code what we're, seeing as some of the more important institutions, at least initially. So Stanford is represented by this orange color, although many of the people at Stanford going back and forth have relationships to Indiana University, so I've used the crimson color for that and UC Irvine also looms large among this particular cluster of nodes, but there are many others that will eventually be brought into the story. 

We can do a similar thing and try to identify some of the key events, and I'm not going to take you through all of these at the moment. 

But the timeline is just one way of keeping track of what happened, but it doesn't necessarily tell us how it came to be and what was important, and how it laid the groundwork for the science that emerged and the continuation of this enterprise of mathematical psychology.

So some of the sources that are going into this presentation, are some interviews that are still ongoing, various retrospective and programmatic articles by, again, some of you who may be here, formal histories written in particular, in this recent some book of a much older period, but where mathematics is important in psychology, by Murray with contributions from Steve Link. 

And, of course, various online sources. And in the future, we're going to be doing more interviewing, and more investigation of various sources and material, but at this point I want to hand it over to Osman who's going to address this first question in our outline. 

How did psychology become mathematical?

Hi everyone, I'm Osman, and in this part of the presentation and I'll be talking about some of the historical and intellectual context around the mathematization of psychology. 

I will also highlight examples of integrated history and philosophy of science questions that these concepts raise and which are of interest to us, and, as the title suggests, I will be framed in this part of the presentation on the question of how psychology became mathematical. 

So we might begin with a very telescopic history by going all the way back to what is traditionally regarded as the beginning of modern science, namely the so-called Galilean revolution in which is argued there was a shift or transition from the dominantly qualitative mode of classical or Aristotelian science to the dominant quantitative mode, we now characterize with scientific work. 

The essence of this revolution as exemplified by Galileo's famous quote "The Book of nature is written in mathematical language" is what the historian & philosopher of psychology Joni Michell calls the quantitative imperative, namely the idea that science requires quantification and mathematics essentially. 

As the story goes the quantitative imperative and the tenets of the Galilean revolution became institutionalized in the scientific revolution. 

Works like Newton's Principia became a standard for what the new science was supposed to be, and there was almost a frenzy to build or develop sciences in the image set forth by works like Newton's Principia. This is sometimes, especially in later times called physics envy. 

Physics envy, of course, is a very pervasive attitude, and the quantitative imperative that comes along with it is very dominant also in the history of psychology. Indeed one of our interviewees remarked that all sciences move in the direction of becoming quantitative as time passes. 

Sort of an indication of the presence of the quantitative imperative in psychology, and this is of course a common sentiment, In the history of psychology we find several psychologists arguing that if psychology is to become scientific it must necessarily become mathematical. 

So there's always been a desideratum to develop a mathematical psychology, but the question is, how does one mathematize psychology? 

This is an important question because, while the examples in the physical sciences might provide hints about how we should go about doing this there are clearly some significant differences between the physical sciences and social and behavioral sciences such as psychology. 

For example, it's taken for granted that objects in the domain of inquiry in physics, for instance, are quantifiable. 

However, in the 1930s and 40s there was a very vigorous debate about whether or not, for instance, psychophysical and psychometric measurements were meaningful at all. 

This is because of a much more fundamental debate about whether psychological phenomena were quantifiable at all. 

For instance, the British Association for the Advancement of Science tasked its psychological arm with producing a report about this question, namely whether psychological phenomena are quantifiable. 

The infamous report released in 1940 controversially argued in the negative saying that psychometric and psychophysical approaches, for instance, were not really meaningfully quantifying psychological phenomena. 

The debate that ensued from the publishing of this report, of course, led to interesting developments in psychometrics, psychophysics, and eventually measurement theory. 

So there are differences like these, and these differences introduce complications to the project of mathematizing psychology. 

And every attempt to mathematize psychology has to deal with these complications. 

In so doing, they come to define how mathematics could and should be used in psychology -- namely the role or place of mathematics in psychology. 

Part of what we're charged to do in this project is to study historical episodes in which psychologists are negotiating the use of mathematical methods in a field that seems to almost resist quantification because of the nature of the object in the domain of inquiry. 

What we want to do is study these episodes to understand the role or place of mathematics in psychology. 

So to exemplify some of the questions, or this tension and some of the questions we are interested in, I'll use the example of the relationship between experimental data to formal models. 

So, this is a longstanding debate in the history of psychology going all the way back to the seminal work of Herbart in the handbook of psychology in which he tries to build a mathematical psychology. 

Herbart, of course, working in a sort of pre-Wundtian pre-experimental psychology period, believed that there was not really meaningful psychological experimentation and that as a result, the role of mathematics in psychology had to be very central, you had to put mathematics up front. 

if you're going to mathematize psychology. We wanted accuracy predictive accuracy, for instance, and if you wanted that we have to put the mathematics upfront and whatever empirical facts, we might know about the psychological domain would be secondary. 

Of course, Fechner in his work believed the opposite. He believed he argued against Herbart, arguing that there is meaningful experimentation in psychology and that, if we were going to build mathematical psychology these experimental data, or these experiments, the results produced from these experiments, had to be central, and at the forefront. 

And then the mathematics will be secondary to it, the mathematics would be attendant to the experimental data. 

The tension between Herbart and Fechner is of course a longstanding tension and contemporarily we sometimes refer to this as a tension between data-driven and theory-driven approaches to the mathematization of psychology. 

Some of the related, some of the more contemporary views about the relationship between experimental data to formal models include what we might call psychophysical and psychometrical approaches. Here, just like in Fechner's case, because of course this descended from the Fecnher line.

Here the experimental data is put up front. The mathematics is made secondary. Mathematics is usually in the form of classical statistics or the Bayesian statistics has been used contemporarily as well. 

And here the idea is to put the experimental data up front and make the mathematics secondary. 

Contrary to this is the measurement-theoretical approach to the mathematization of psychology. Here the mathematics is center stage and the experimental data is put second -- it's made secondary to it. And the mathematics here is largely axiomatic. 

And the relationship between the experimental data to the mathematical model is viewed in a representational sense. 

And I mean representational in the technical sense in that an isomorphism is found between some axiomatic system and the empirical structure of some domain. 

And, of course, you have to produce a representation theorem to show that there is this isomorphism between the empirical domain and the axiomatic system. 

Similar to this are computational approaches to the mathematization of psychology. Here again, just like in the measurement-theoretical approach, the mathematics is upfront. 

The data is somewhat secondary. However, unlike the measurement-theoretical approach, the relationship between the experimental data and the formal model, the mathematical model is not regarded in representational terms, at least not in the sense of the measurement-theoretical approach. 

Here the mathematics is supposed to characterize the causal-mechanistic background of the psychological phenomena of interest, so the data produced by, the data characterizing the psychological phenomena is measured, of course, against the mathematical, the formal model, but here the relationship is not viewed in representational terms, it's viewed as the mathematics is characterizing the causal-mechanistic background of the phenomena of interest. 

So these are three views about the role, or how we should view the relationship of mathematics to experimental data in the project of trying to mathematize psychology. 

But, so far, we have seen some of our interviewees and authors in the field have conceived of their work to be somewhat distinct from the sort of work done by psychophysicists, psychometricians, computationalists, and measurement theorists, suggesting a different model of how to mathematize psychology and the role of mathematics in psychology. 

Part of what we're trying to understand is this distinction and to what extent it is historically and contemporarily constitutive of the math psych community. 

If there is a central core that views its work as somewhat distinct from these different approaches, what is it in their use of mathematics, that is different? And this will be the subject of the next part of the presentation by Brendan. 

What makes math psych's use of mathematics different from other branches of psychology?

In the next two sections, we will discuss what makes mathematical psychologist use of mathematics different from other branches of psychology. 

I will be looking at this question, more generally, and I will offer four provisional answers. I'll then hand it off to Mara who will look at this question focusing more specifically on the relationship between mathpsych and cognitive science. 

So what makes mathematical psychology's use of mathematics different from other branches of psychology? 

As a first answer-- nothing. Mathpsych simply wanted psychology to embrace quantitative models. 

As a historical, sociological fact psychology has perennially resisted quantitative methods and models. 

The origins of mathpsych as a field were in part simply a movement advocating the use of quantitative methods and models. It is not part of mathpsych's essence that it advocates a particular type of mathematics. 

Nevertheless, the claim that the mind is appropriately studied with quantitative concepts is a bold and interesting claim as Osman discussed in the previous section. 

In 1995 Duncan Luce wrote that (quote) "Mathematics becomes relevant to science whenever we uncover structure in what we are studying. The existence of psychological structure, cannot be in doubt, but what that structure is another matter." 

In our interviews, we have heard that (quote) "all sciences move in the direction of becoming quantitative as time passes. We use quantitative modeling, to be precise, and to be able to communicate properly what we're doing." 

So in this sense, mathematical psychologists seem to advocate generally for a quantitative psychological science, more so than for any particular kind of mathematics. 

That said, there are some trends, there are many different dimensions along which the use of math and psychology can vary, and practitioners of mathematical psychology have expressed views to us that would locate mathematical psychology in specific areas of these spaces. 

So as a second answer to our question mathpsych's use of mathematics is distinctive in that in practice it is more likely to adopt tools for modeling from a wider range of areas in mathematics. 

Partly, this is a result of mathpsych valuing advanced training in mathematics beyond basic statistics. Greater knowledge of mathematics simply means more mathematics to draw from. 

In particular, more geometric math seem to appear more frequently, such as topology, group theory, and differential geometry. For example, considerations of geometry, group theory, and symmetry lay at the heart of Shepard's work on psychological spaces. 

These types of models may still incorporate probabilistic elements, but they nevertheless include mathematics that make these models distinct from more standard statistical models. 

Additionally, when it comes not to the math in models, but to the formal techniques used to evaluate models we have heard from our interviewees ways in which mathematical psychology stands out. In particular, we have heard how mathematical psychology brings together statistical inference and experimental design in a more principled way. 

Instead of experimental design and statistical analysis occurring as separate and consecutive stages of investigation, as is often the case in psychology mathematical psychologists, to a greater degree first establish which statistical tests will allow them to answer the question they're interested in and then design their experiments with such a statistical test in mind. 

In general, our interviewees have described mathematical psychology as having a tighter relationship between models and experiments than in other areas of psychology although people disagree on what this means exactly. 

One sense in which this is true, however, is that the norms for evaluating models, with a clearly specified mathematical form relative to data are more established than in the case of quote "numerical simulation models" -- that is quantitative computational models that may not have an apparent or specifiable mathematical form. 

Mathpsych has traditionally valued having models with a clearly specified mathematical form so that the models' properties transformations and evaluations can be better understood. 

This then leads us to our third answer-- theoretical math, analytic solutions, and hand-fitted models. 

Mathpsych is distinctive in part in that it has always included branches that focus on "pure" mathematical results such as those that came from measurement theory, and then mathpsych modelers of empirical data have expressed preferences for analytic models and models whose features are not derived from automated procedures, but are instead chosen by scientists. 
Many figures in mathpsych have expressed concern over the ease with which computers can help produce predictively accurate models. In 1975 Bill Estes wrote of the computer revolution (quote): 

"The fitting of the model to data is accomplished via computer programs with four, eight, twelve, virtually unlimited numbers of parameters being evaluated by computer search "of the parameter space, almost completely bypassing both the opportunities for ingenuity and the quantities of blood, sweat and tears that used to go into this enterprise. And it's become almost literally the case that the fitting of a model to a set of data is primarily a test of the technical competence of the investigator and has little to do with the adequacy of the model for interpreting empirical phenomena".
While mathpsych has embraced the use of computers to automate the fitting of models and the like, there's still a desire for models can be generated and evaluated in principled ways that go beyond the desire for mere predictive accuracy. 
Again, what this looks like is not agreed upon, but I will try to say a little bit more about what this might mean in our fourth and last answer. 

So our last answer-- aspirations for general abstract and well-supported cumulative theory. 

Mathematical psychologists frequently express a strong desire to attain well-evidenced, mathematically expressible psychological theory that generalizes beyond a particular experimental context. This is arguable, for example, what makes mathpsych different from psychometrics. 

This means models that go beyond descriptive summaries of data, go beyond merely establishing effects and capture underlying causal processes. Such a goal requires moving beyond goodness of fit for the evaluation of models. 

I will now briefly note two ideas that mathematical psychologists have discussed that can serve as indicators that some work in mathematical psychology is moving towards this sort of goal. 

The first is the idea of a model providing insight about how to lump and split data from diverse experiments and other contexts by capturing causal processes. 

Lumping data occurs when seemingly disparate mental phenomena can be shown to be subsumed under a single, general model. 

Again Shepard's laws provide a good example of this. Splitting data occurs when undifferentiated, seemingly undifferentiated behavioral data can be shown to be quite naturally carved up and shown then to be generated by distinct submodels of cognitive processes. 

Estes in 1975 gives the example in classical physics of the wave and particle models of light capturing different aspects of complicated optical phenomena. 

In psychology, he gives the example of Stevens' 1957 categorization of prothetic versus metathetic sensory dimensions in relation to scales of measurement. I invite you all to think of your own examples. 

Lumping and splitting are both valuable and they illustrate the goal of moving beyond mere predictive accuracy towards well-evidenced general theory. 

The second idea concerns the nature of cumulative theory. Duncan Luce wrote in 1995 (quote): 

"It's moderately rare to find a psychologist who, when confronted by a new set of data invokes already known mechanisms with parameters estimated from different situations. 
"When each model is unique to a particular experimental situation, all of the model's free parameters must be estimated from the data explained. Frequently the resulting numbers of parameters outrun the degrees of freedom in the data. This reflects a failure of the science to be cumulative, an unfortunate feature of psychology and social science that is widely criticized by natural scientists. I view it as one of the greatest weaknesses of modeling (and other theory) in our science." 

In many cases, a psychological theory that adequately predicts some body of data does not generalize to new data and so must be modified in order to adequately predict the new data. 

This is a weak form of cumulative theory in that it is more impressive when models of experimental findings are already constrained by previous experimental findings. 

In this case, the goal is to approach a new experimental context with a well-supported theory in hand such that the new data, together with the well-supported theory entail new psychological theoretical posits. 

In this sense, new psychological factors can be said to be established through a process of theory-mediated measurement. 

Townsend writes that signal detection theory (quote):
"stands as the prototypical theory-driven methodology, since it could be employed to discern decision and learning bias from true sensory sensitivity, e.g. signal-to-noise ratio effects in such diverse fields as hypnotic phenomena, to trial-witness memory, to laboratory psychophysics, or learning and cognition experiments". 

This process again illustrates a desire to go beyond goodness of fit. 

So while not unique to mathematical psychology, these aspirations that I have been discussing are voiced strongly by many mathematical psychologists. 

There's a sense of optimism about the possibility of attaining these goals and a belief that we should not lower our standards. 

Just as many psychologists eschew the use of math in psychology, many psychologists feel that we are not entitled to this sort of lofty scientific aspiration, but mathematical psychologists have expressed that we must attain this goal, if we wish for psychology to ever become a mature science. 

I now hand it off to Mara, who will discuss the question we've been looking at as it pertains more specifically to the relationship between mathematical psychology and cognitive science. 

How should we think about mathematical psychology in relation to cognitive psychology and cognitive science?

Hi, like Brendan said, my name is Mara and I'm going to be talking about the third and final question of today's talk which is really just a kind of a smaller question than the broader one that Brendan was discussing in the previous section. 

So here I've been looking at how we should think about the relationship of modern mathematical psychology, cognitive psychology, and cognitive science. 

Before I get into the details though, I do want to just acknowledge that this relationship could be explored from many different angles, right, we can look at socio-institutional relationships among them, conceptual, methodological, and eventually we do want to have all of these, as well as the interactions among them in view. 

At this point, though, the project is in its very early stages, and so what we have here is quite fragmented. 

Today, what I'm going to be doing though mostly is just sort of breaking this question down into a few smaller ones that we are exploring, and will continue to explore moving forward. 

And to this end I've broken it down into three questions in particular. So, the first is "Where do the institutional and intellectual roots of mathpsych, cognitive psychology and cognitive science overlap? Where do they diverge?" 

Second, "What makes mathematical psychology's understanding of the role and method of modeling distinct from that of cognitive psychology and cognitive science?"

And then third, "What is the role of computers and all of this?" 

So one way in which we're trying to address the first question is to map out the individuals and disciplines that were influential at the start of mathematical psychology. 

And all I've done here is take the image that Colin showed in the introduction and add a few names. Now, there are people's names who should be on here who aren't and I haven't even traced the lineages among these people. But here I just want to show the diversity of disciplines that were involved. 

So I've grouped people roughly into disciplines or areas of research, so we have people from engineering, with the development of signal detection theory and its application to human cognition, people with more of a background in formal measurement theory, and at least one philosopher with Pat Suppes there. People from psychometrics. 

Mathematical learning theory, with its roots in the behaviorism of both Skinner and Hull. Information processing. I've also included Chomsky here, next to Miller, both because many people seem to think that he did have some role intellectually in the development of mathpsych, and also because he and Miller were at one of the early Stanford summer schools. 

I just want to briefly compare this to one characterization of cognitive science, so the image here on the left, the hexagon, is a reconstruction of an image from 1978 when the Sloan foundation assembled an ad hoc committee to both characterize the state of the field and create a plan for action moving forward, one of the members was George Miller. 

The report was quite controversial, it was never published. But just note that there is some overlap in the disciplines that I mentioned on the last slide and the disciplines mentioned here. 

Also, it's the case that the way that the cognitive science society describes their field does still mention these six disciplines. They are found on their logo, in addition to education, which is the seventh. 

So, with this in mind, I think those two smaller questions that we're looking at. So both for cognitive science, but also, you know, for mathematical psychology in particular we have a number of different disciplines that were involved and what we want to look at is, what is the nature of the interaction among these disciplines? 

Was it the case that all of these disciplines retained their boundaries, but were perhaps using some sort of shared method? 

Or was it the case that they really did come together to create an entirely new field? I think it's very much an open question. 

The way that some people talk about mathpsych is as if it is a method. The way other people talk about it is as if it is a discipline. Another question that we're trying to address is, you know, despite the overlap of these fields, What is the core difference between them? 

And, you know, one thing that we are finding is the strong background in mathematical... sorry ... strong background in measurement theory that is found in mathpsych. 

This image here is just tracing the kind of area of research being published within the mathematical psychology journal, and you just see that work in measurement theory and mathematics, you know, is represented throughout the history of the journal, but it is worth noting that in our interviews and in our research people seem to have quite mixed views on the importance of measurement theory, with some people questioning the role it played in both the development of the field, more generally, and also in their own work. 

And so, turning to kind of a subset of these questions, I just want to briefly talk about differences between the way mathematical psychologists conceive of modeling in their field, at least the people we've talked to, and the way they conceive of it in cognitive science. So, a number of people that we've talked to have noted that the goal of modeling in mathematical psychology is not simply to simulate human performance or behavior. 

Herb Simon is included here because he's been singled out a couple of times as someone whose approach to modeling is sort of emblematic of that in cognitive science. 

Namely, as long as you get something that looks kind of roughly human, as one interviewee put it, that's enough. Right, so just simulating behavior, fitting you know the model to the data, that's sort of the end goal. 

Estes back in 1975 was characterizing some of the modeling in cognitive science in this way in the quote that I included here but that for mathematical psychology, you know, simulating human behavior is not actually the end goal. Rather what they're trying to do is locate mathematical structures within the data that may be given a cognitive or psychological interpretation, however that is to be cashed out. 

And importantly, you know, it's the experiment and experimental data that is playing a role both in the construction of the theory or model. I haven't really talked about the relationship between theory and model and I'm glossing over it here, but it is something we're going to explore. But anyway, so that the data is important in the construction of the model. 

But then predictions of the model are tested against the data and you revise the model based on the results. So there's this iterative back and forth between you know the model and the experiment with the model confronting the data at multiple stages. And so computers did play a role in all of this both for cognitive science and for mathpsych. 

But what we've been seeing is that the role of computers within these disciplines has varied. So the way in which you know computers, or one of the ways in which computers have been important for cognitive science is, of course, as a model of the mind itself. 

Howard Gardner back in 1980 was characterizing sort of the core commitments of cognitive science and he says that one of them is that "central to any understanding of the human mind is the electronic computer." 

That it is the most viable model of how the human mind functions. Now, not everyone in cognitive science, you know, adheres to this but it's undeniable that in cognitive science, the computer as a model of the mind has been an important theme. 

This doesn't seem to be the case in mathematical psychology it doesn't seem to be the case that the computer was the model. Rather the computer was a tool for modeling. And what we have here is an image of a PDP-1 computer which a couple of people have talked about, as you know, a very early computer that was used for mathematical psychology and modeling within it. 

So, as a positive thing, computers have been used as a tool within mathematical psychology for modeling. 

Right, both when you're working with highly complex models, or you are working with large amounts of data or both, computers seem to have been indispensable in terms of progress in this area of mathpsych. But this is accompanied by a more cautionary note that has shown up both in our interviews and in, you know, people within the field, their characterization or discussion of the role of computers in mathpsych. And that is that, you know, you shouldn't extend the use of computers too far, that computer can become a crutch for the development of mathematical psychology for a number of reasons. One may be because. when you're working with computers, right, fitting the model to the data comes quite cheap, so you think that is some sort of criteria for, you know, deciding between competing models or deciding if a model is a model that we should keep. 

It just no longer becomes, you know, a good criterion for doing this. And furthermore, when you're working with, you know, computational models in this way there's a risk of watering down our understanding of theory because we've no longer made explicit the underlying mathematical structure. 

And sort of this, both of these things, are sort of expressed in this quote here from an article by Estes from the 1970s. 
So these are three questions that we are considering at this point, and you know I'm sure more will come up in this area as well, but with that, I want to turn it over to Dzintra who will wrap things up, and talk about directions for future research. 

Conclusions and Future Directions

Thank you Mara for the introduction, my name is Dzintra, and I'll be presenting some conclusions and future directions. 

Here's a little outline of what I'll be doing today. So the first thing I'll be doing is presenting a couple of slides on our approach to the history and philosophy of mathematical psychology as was presented today. 

And then I'll have a couple of slides on a couple of the differences between mathematical psychology and other sub-disciplines. 

These are things that have been recurring throughout the presentation, I just wanted to highlight them and a couple of questions that arise from them. 

And then I'll conclude with a few philosophical questions that are still open questions and we would like your feedback on answering them. 

The first thing to highlight here is that our approach to this project requires a balancing act between the local immediate context and the more global, deeper historical context. 

Throughout this presentation, we highlighted tidbits of both local and global context, a number of key figures to focus on, and sets of contrasting ideas. 

We also spent some time delineating mathematical psychology and its approach from other fields and sub-disciplines. We started our history centering on the Stanford group. 

And then we continued by tracing academic lineages through other hubs of mathematical psychology such as Indiana and Irvine. 

And these might not be the only schools that are hubs for mathematical psychology. Other hubs might include the University of Pennsylvania and Michigan, and we would love your thoughts if we missed any centers or other hubs from our narrative thus far. We then trace lines of influence from these hubs backward in time. 

And in Osman's presentation, for example, we saw glimpses of this deeper global historical context in his discussion of the rise of quantification in psychology. 

And we start to see the birth of psychology as a science. During that time there was a transition from qualitative to quantitative models of science. 

And many philosophers and psychologists have thought, and still continue to think that the mind is subjective and private, making it hard, if not impossible to measure aspects of the mind in the same way that we would measure physical materials. 

This complex nature of psychological phenomena creates difficulties for quantification in psychology, as one can't simply apply the tools from physics to psychology, ... and off we go with mathematical psychology! 

But just as a quick disclaimer the history that we've started to sketch in this presentation is by no means complete and we welcome and encourage your feedback on any of the missing pieces. 

As is true in every discipline, each individual's position can't be understood in isolation, rather we must understand who and what influenced these groups and what they are reacting to. This applies to the position of each and every mathematical psychologist from the founding fathers up until contemporary mathematical psychologists. 

Throughout this presentation, a number of apparent contrasts have emerged in various degrees And this list that I'm providing, it's not an exhaustive one, but it's just an example of some of the contrasts that we found. The first being this contrast between quantitative and qualitative models of science, that we've seen with the rise of psychology as a science. 

And then we see a contrast between models and laws and causal processes versus statistical summaries of data, etc, etc. 

And we're still actively in the process of our historical investigation, so we want to know more from you guys, our audience, on how key figures came to take on these apparent contrasts. Who were their intellectual influences? 

As historians, we want to know the intellectual, social, and political context which drives these apparent contrasts. As philosophers, we want to know whether these contrasts are still useful for contemporary mathematical psychologists today. 

To what extent are these themes and contrasts that we have identified as historical artifacts? Are they still figuring, and going on in debates today? 

The themes and contrasts that we've raised may have been important in the early days of mathematical psychology. But we want to know how they're impacting current experimentation and modeling and theory building. 

In either case, the questions that mathematical psychologists ask, if they change over time, and the debates and the conceptual contrasts are going to inevitably seep into, and shape the modern concepts and debates. 

Moreover, we want to know, How are these diverse sets of contrasts that have been cropping up throughout our presentation reconciled within the umbrella of mathematical psychology. 

And lastly, from our perspective of integrated history and philosophy of science, we aim to further untangle the finer-grained distinctions between each of these contrasts and also within each of these contrasts. The hope is that we would get a more explicit set of foundations for the field of mathematical psychology and perhaps even a more deliberate strategy for future progress in the field. 

Now I'm going to go through a couple of slides that highlight some of the differences that we discussed, between mathematical psychology and other fields or subfields. 

So here we have a slide on the differences between psychometrics and psychophysics, on the one hand, and mathematical psychology, on the other. 

What I want to highlight on this slide is the claim that mathematical psychology seems to be theory-building. 

This is reflected in a couple of the sources that we've read and some of the people that we've heard from, but we want to invite your thoughts on the matter. 

What does it really mean to you, qua mathematical psychologist, for mathematical psychology to be theory building? 

Are other disciplines and sub-disciplines equally theory-building? Or do they emphasize theory building less? Or is your approach somehow markedly different from that of other sub-disciplines? 

And how does theory building relate to identifying laws or models that generalize beyond a particular context? 

And just one other thing to note on this slide is that the points under these slides may be a little bit misleading because the distinctions may not be as cut and dried as they appear, so this is especially the case in contemporary debates. Some psychometricians, for example, have actually begun to integrate theory and the search for causal processes into their research programs. 

Now we're seeing another slide on some differences between mathematical psychology, and here the contrast is between computation and cognitive sciences. 

The one thing that I want to highlight here is the last point under mathematical psychology. 

Namely, it construes modeling as an iterative process. In what sense is modeling an iterative process? 

It may be that modeling and experimentation are tightly linked as we've seen before, and that there's a bidirectional influence between these two things. If so, if that's correct then I invite you guys to let us know, What do you think the nature is of this bidirectional relationship? 

I now turn to our last set of slides. A couple of philosophical questions arose from our presentation and from the work that we've done so far in this project. It's still in our early. stages and we hope to get some answers to these questions, and hopefully you guys can help us figure out some of these answers as well. So the first thing is that throughout this presentation we've talked about a major aim of using quantitative modeling in mathematical psychology. 

And this has been to identify and describe the underlying psychological structure in terms of causal processes. 

So we've discussed how psychological phenomena are subjective. This makes it hard to measure and it has a complicated and complex nature. 

A couple of mathematical psychologists have stated that models don't aim at being true, they don't admit being accurate, and they don't aim that being correct. 

So the question here is, why is that the case? Is the reason due to the complexity of the phenomena itself, or is there another reason that is independent of the nature of the phenomena under study? 

Second, what are the goals of modeling? Is there one goal, or are there many goals? My guess or my inclination would be to think there are many goals and that goals might be disagreed on between different mathematical psychologists. So some of the goals that we've seen crop up are explainability, or increases in explainability, increases in understanding, increases in predictive power, etc, etc, but this list is definitely not exhaustive and it most likely does not adequately represent the diversity and ideas that mathematical psychologists hold, so I would like to invite your feedback once again on this idea and on this question. 

I now want to turn to just a couple of questions regarding the status of models and their relationship to both pluralism and realism. So a number of mathematical psychologists seem to think that plurality of models is a strength. 

The thought is that it's better to have multiple models capturing a single phenomenon, than less. 

So, is this true? Is this a generally held tenet across mathematical psychologists? If so, what benefits are gaining from this view? And second, what can be done when two models are incompatible, but they explain the same phenomenon, or at least they target the same phenomenon? 

Presumably, one could use mathematical methods to ensure that two models are mathematically distinct, and then one could design experiments to uncover those differences empirically. 

But we would like to hear more about what you guys think in terms of these two questions and get your feedback on these questions. 

So both realism and pluralism about models are hot issues in the philosophy of science, and as historians and philosophers of science, we now have, we might have now, one example of contemporary scientists -- you guys -- who adopt a pluralistic stance. 

And last but not least, I just wanted to invite your thoughts on the progress of the field. 

So, for one, do you think that the advancements in quantitative methods will lead to more accurate or true models? Or do you think that mathematical psychology will perennially remain distanced from truth and accuracy as the goals of modeling? Secondly, how do you see the future of the field? 


Colin Allen
Prof. University of Pittsburgh.

Faculty member in the Department of History & Philosophy of Science at the University of Pittsburgh, specializing in Cognitive Science, Animal Minds, Cognitive Evolution, Machine Morality, and Computational Humanities
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Hi there! Let me ask you, do you already familiar with Spiral Dynamics?