The Science of the Soul

Now that cognitive scientists and neuroscientists are learning more and more about the brain, and are able to point to the parts of the brain whose function is typically thought of to be part of the mind or soul, can a scientist (or even a religious person) still believe in the soul? This is the question that is posed by this article from the New York Times. I just want to offer some thoughts I have on the article.

The article begins with Popes Pius XII and John Paul II and later has quotes from two Catholic academics (one a theologian- John F. Haught, one a biologist- Kenneth R. Miller) that make clear that Catholics can believe in evolution and the existence of a created world where each human has a soul. (N.B. It is up to the individual Catholic whether or not she wants to believe in evolution. It is not an article of faith one way or another.) This is important for a number of reasons. First, people whose religion is some form of secularism (e.g. Dawkins) lump all religious people- and all Christians- together, so it is good to have an article to show that there is nuances in belief among Christians. Second, it demonstrates that the Catholic faith is amenable to the progress of science, but at the same time it cannot contradict the most basic tenets of the faith such as the created world by God and the existence of a soul, which leads to the last paragraph of the article.

“What do you say as a scientist about the soul?” His [Dr. Miller's] answer, he said, is always the same: “As a scientist, I have nothing to say about the soul. It’s not a scientific idea.”

I think this is precisely right. And this is why Intelligent Design is such a dangerous idea. Not only it is bad science, it is bad theology. It is bad theology because Intelligent Design (ID) is stating that some aspect of the Creator can be quantified and scientifically explained, whereas the point of faith is to believe in something that cannot be scientifically demonstrated. (In the Creed, we say "I believe in one God, the Father almighty, maker of heaven and earth"- not "I believe in Intelligent Design that demonstrated that God is the creator of earth, but we can't quite use ID to show that God created heaven....".) All religious believers must make that Kierkegaardian leap of faith (preferably after critical thought), and Intelligent Design denies that thereby making religion nothing other than a bad science.

And the reason that Intelligent Design is bad science is since it has absolutely no explanatory power. The reason that evolution is undeniable true is that evolutionary theory and in particular, evolutionary dynamics answers numerous questions in biology, anthropology, economics, physics, linguistics, psychology (cognitive science), neuroscience, etc. There are hundreds of papers each month that are published each month that use evolutionary theory, but the numbers that use ID is 1 in all of 2006 and none so far in 2007. So a basic tenant of any worthwhile philosophy of science is that the model must have explanatory power. And preferably the model can be applied to many situations, has empirical evidence, and can be formalized (which for evolution is through various differential equations like the replicator model and evolutionary Game Theory). ID has none of these things.

To bring the conversation back to the soul. Intelligent Design claims, essentially, that God can be quantified. This is a various dangerous proposition for a religious person to make for it opens her up to allowing the soul to be quantified as well. Stating that science can prove that God exists, God is the creator, or humans have a soul, is stating that faith is refutable. So as Miller said, the scientist has nothing to say about the soul, but he has a lot to say about the brain.

The article states that the recent advances in brain science have refuted Descates Cogito ergo sum (I think, therefore I am) with respect to animals. I don't think that is quite right. While Descartes' Cogito can be refuted a number of ways, I don't see how advances in brain science would do so. Descartes' claim is fundamentally an a priori claim to refute skepticism, not a scientific claim about the soul. And what Descartes considered to be thought, cannot be done by animals. (Now Descartes, contra Aristotle and St Thomas Aquinas, did not believe that animals had souls.) But it seems that if brain science refutes any fundamental part of Cartesian philosophy, it would be dualism as a scientific philosophy. All religious believers are de facto dualists in the sense that they believe in the material body and the immaterial soul. But a believer like myself is not a dualist in the sense that I believe that only the material body can be scientifically and philosophically investigated.

That seems enough for today, perhaps more on another day.

The On-Line Encylopedia of Integer Sequences

So I am finally getting into my summer routine, which means that I will be able to blog at least every 2-3 days, but hopefully more often...

Regardless, while doing some combinatorics on trees I was reminded of the most excellent website brought to you by N. J. A. Sloane at AT&T Research. (Who knew phone companies still had labs?) Anyway, pretty much any sequence you can think of is already up there and it tells you many of the known applications of the sequence. And if you know of a sequence that is not up there or of an application of a sequence that is already in the database, then you can e-mail the webmaster and get it on there.

For example, I have been dealing with a sequence that I call the "phylogenetic numbers" since they show up in the study of phylogeny, i.e. evolutionary trees. So I put the first part of the sequence into the database (1, 3, 15, 105, 945) and it returned this, which are (as it turns out) the double factorial numbers of 2n-1, i.e. (2n-1)!!. And then they give about a dozen applications of this sequence.

I have to warn you, that this site can consume hours and hours of your time. Just for fun, you should look at the Catalan numbers. Speaking of number sequences and combinatorics, I suggest that you get familiar with the two-volume opus by Prof Stanley on Enumerative Combinatorics.

Addendum
I realized that I should state briefly what the double-factorial numbers are. Two references are here and here. The gist of the double factorial numbers is the following.

n!!=(n-2)(n-4)(n-6)...kn; where kn is 1 if n is odd and 2 if n is even. E.g., 8!!= 8*6*4*2 and 9!!=9*7*5*3*1

Teaching gap between American & Asian math teachers

This is from the UCI website since it was done by a UCI and two UCLA researchers. (The full article was published in May 25th issue of Science.)

The upshot is the following: while both Asian and American math instructors use analogies to teach mathematics, the Asian teachers do so in a way that encourages reasoning in their students; in particular, this means that the Asian teachers "reduce the processing demands on their students." In other words, the analogies that Americans teachers give are too abstract in order to be helpful.

These findings are very interesting to me for two reasons (excluding that fact that something worthwhile came out of Education Departments), one pedagogical and the other psychological. First, as someone who teaches introductory college mathematics (from Pre-Algebra to College Algebra and statistics) this let's me know that when I use analogies (which I do), I need to make sure that I use a lot of mental and visual imagery in conjunction with arm gestures to get the analogy across. (I think I do this, at least to an extent, but I am sure that I can be better at it.) Secondly, I am investigating (on a side-research project) the different way that Asians and Westerners think. Prof Richard Nisbett has shown that East Asians are better at seeing relationships among events that Westerners. So if Nisbett is correct, which I preliminarly I believe he is at least correct to an extent, then this would match up with the aforementioned article.

The social psychological make-up of Asians allows them to think about relationships in a way that is not natural (in the sociological sense, not the biological) for Westerners and this allows them to teach mathematics in a more efficient way. It appears from the article that the instructors were teaching arithmetic, algebra, and calculus, but not geometry. If Nisbett is correct that Asians have a better intuition of arithmetic and algebra and Westerners with geometry, it might be the case that there would be different results for this study if geometry was the topics being taught.

Regardless, this is an interesting little article which has both practical and theoretical import. Although it is about mathematics and psychology (but not mathematical psychology), I thought that it would be good to talk about on this blog.

Back, sort of...

I finished my last requirements for the quarter and I have begun teaching a College Algebra class for the summer session at a local community college. I have finally begun to settle in, so I plan to post something of substance either tonight or tomorrow.

Finishing last paper of the school year...

I am taking a class this quarter on Cognitive Modeling and I have a paper due on Monday. As a result, until I finish it, I am not able to post much. Basically the paper is running some of my advisor's old data using Bayesian graphical models.

But more on Bayesian graphical models another day. Actually, various Bayesian tools will be the subject of many future posts.

UCLA's IPAM: Graduate Summer School

Proof that I am not the only person who thinks that mathematical behavioral science is important; UCLA's Institute for Pure & Applied Mathematics agrees.

FYI (from the website).

"Probabilistic Models of Cognition: The Mathematics of Mind” will involve leaders from Cognitive Science and experts from Computer Science, Mathematics and Statistics, who are interested in making bridges to Cognitive Science. The goal is to develop a common mathematical framework for all aspects of cognition, and review how it explains empirical phenomena in the major areas of cognitive science - including vision, memory, reasoning, learning, planning, and language. The summer school is motivated by recent advances which offer the promise of modeling human cognition mathematically. These advances have occurred largely because the mathematical and computational tools developed for designing artificial systems are beginning to make an impact on theoretical and empirical work in Cognitive Science. In turn, Cognitive Science offers an enormous range of complex problems which challenge and test these theories.

Sadly I won't be able to go (teaching obligations), but if you are at all interested in this sort of thing, you should check it out. (Although, I am not sure that they still have room.) That being said, the graduate program that I am in allows one to study any or all of these areas extensively.

The Mathematics of Behavior

I have bumped into another book, which I will begin to selectively review in the near future.

This book is The Mathematics of Behavior by Earl Hunt, which by its title is obviously very relevant to this blog.

Describing full binary trees with a context-free language

Today I describe a type of graph (full binary tree) using formal language theory, and in particular, a context-free language. The purpose of this exercise is to use the typical mathematical trick of moving from one area of mathematics to another in order to solve a problem. The problem in this case is how to describe a tree-based statistical model that is used in cognitive science. (But we will save talking about those models for another day.)

Let's recall some graph theory. A binary tree is a directed acyclic graph (DAG) where each node has no more than two children. A full (or proper) binary tree is a tree where each non-terminal node has two children. In other words, each node either has 0 (the terminals or leafs) or two children. The node which is not a child of any other node is referred to as the root. Above you can see an example of a full binary tree that was created in Matlab.

Another notion that we need that comes from graph theory (and like binary trees is used a lot in data structures in computer science) is the preorder traversal. A traversal is a particular way of visiting the nodes of a tree. And the preorder traversal is when one begins with the root of the tree and then moves downward from left-to-right. A sample implementation of this is:

preorder(node)
print node.value
if node.left  ≠ null then preorder(node.left)
if node.right ≠ null then preorder(node.right)

But this can be clearly seen in the following figure.

And now let's recall some formal language theory. A formal language is set of strings (finite or infinite) of symbols with precise rules (often recursive) for generating those strings. A context-free language is a particular type of formal language. Formally, a context-free language (CFG) is a 4-tuple (V, Σ, R, S) where:

1. V is a finite set called the variables ;
2. Σ is a finite set, V ∩ Σ = ∅, called the terminals;
3. R is a finite set of productions (or rules), with each production being composed of a variable and a string of variables and terminals, i.e. α → A, where A ∈ (V∪Σ)*, where '*' is Kleene (or star) operation: for any set A, A* = {x1x2…xk| k≥0 ∧ ∀i≤k xi∈A} ;
4. S ∈ V is the start variable.

Now we can let the grammar for full binary trees defined by the preorder traversal be defined as follows:

GFBT={{α},{L, R}, R, α},

where the productions R are given by (where '|' represents the disjunction) and ε represents the empty string:

α → LαR|LRα|LαRα|ε.

Following Chomsky and Miller ["Introduction to the formal analysis of natural languages", Handbook of Mathematical Psychology: Volume II, 1963, pp. 290, 293] we can refer to the first production as left-recursive, the second as right-recursive, and the third as self-embedding. Naturally we can think of the 'L' symbol as representing left and the 'R' symbol representing right.

In order to see how this works, let's use this grammar's productions to generate the string the represents the above tree (the same one we used to demonstrate the preorder traversal). Note that since the language is context-free, we don't have to replace alpha with the same string each time:

α → LαRα →LLRRα →LLRRLαRα→LLRRLLRRLR.

And so now we can use this language to generate a string that represents any full binary tree with the preorder traversal.

At a later point I will tie this into a particular application in mathematical psychology, viz. with multinomial processing tree models.

Real Analysis with Economic Applications

Ok, E.A. (2007). Real Analysis with Economic Applications. Princeton: Princeton University Press.

This is book that I picked up today. I spent about two hours going through its 800 or so pages, and it seems excellent. I plan to write up a "selective" review on within the next week or so. (Naturally I will not read the whole book, but I will pick and chose some topics and review what Prof Ok has to say about them.) But if you bump into it at your bookstore, make sure to take a look.

Addendum

So I think what I am going to do is talk about a section or two every week. I think the first topic that I am going to cover is order relations and social choice theory (Part I, Chapter A, Section A.1, Sub-section A.1.4).

5 Best O' Math Psych Books

The Wall Street Journal has gotten in the habit of publishing a list of the 5 books that are the best for whatever subject. I am going to give my thoughts on the five best (i.e., classic and mandatory) books for the mathematical psychologist. The thing that makes many of the articles in these volumes amazing compared to reading Frege's Begriffsschrift or the Principia (actually, either of the two Principias- Newton's or Whitehead & Russell's) is that you can still learn from the science contained within, they are not merely of historical or philosophical interest. I am not saying that the context of the aforementioned texts are wrong, but if you want to learn logic you pick up a modern logic textbook and similarly for physics. (Though I think that mere historical and philosophical interest is more often that not more than sufficient reason to study something, which is why I own copies of Newton's Principia and W&R's up to *56.) Perhaps, later on, I will do the same for some related fields such as mathematical logic, philosophical logic, and game theory.

Mandatory books for the mathematical psychologist who specializes in applications of logic and computation to cognition (i.e., this list is naturally prejudiced toward my research interests) and in no particular order:

1. Anderson, J. & Rosenfield, E. (Eds.) (1999). Neurocomputing: Foundations of Research. Cambridge: MIT Press.

This has many of the classic articles in the field, to include the most classic of them all, viz. McCulloch & Pitts (1943)- the seminal paper on neural networks, Rosenblatt (1958) that introduces the perceptron, Minsky & Papert (1969) where they express the limitations of the perceptron, Hopfield (1982) where he brought neural nets to the popular scientific front and combined many disparate ideas elegantly, and many more- 43 papers (or excerpts from books) in all.

2. Shannon, C.E. & McCarthy, J. (1956). "Automata Studies," Annals of Mathematical Studies, Number 34. Princton: Princeton University Press.

An early collection of works in automata from the incomparable Annals of Mathematical Studies from Princeton University. The papers come in three topics: Finite Automata, Turing Machines, and Synthesis of Automata. The first section has exceptional papers on logic, neural nets, and automata by Kleene, von Neumann, Minsky, and Moore. These (along with the McCulloch & Pitts paper) are the founding papers on finite automata. The next two sections also have splendid papers by Davis, de Leeuwe, et al, MacKay, and a pair of papers by Uttley.

3. Luce, R.D., Bush, R.R., Galanter, E. (1963). Handbook of Mathematical Psychology, Vol. II, Chapters 9-14. New York: John Wiley & Sons.

There are six exceptional papers in this volume, where any one would be worth purchasing the book. The first two papers are on stochastic learning theory (Sternberg) and stimulus sampling theory (Atkinson & Estes) while the last paper is on mathematical models of social interaction (Rapoport). All are great introductions to their respective fields and still relevant. However, in the middle there is a trinity of papers by Chomsky: Chomsky & Miller's "Introduction to the formal analysis of natural languages," Chomsky's "Formal properties of grammars," and Miller's & Chomsky's "Finitary Models of Language Users." I cannot adequately express how wonderful these papers are, whatever you feel about Chomsky's politics (for good or ill), just ignore all of that and enjoy the pure genius of he and Miller combining Chomsky's groundbreaking work in linguistics in the 1950s with automata theory. So freakin' brilliant. These papers are on par with Frege's Foundations of Arithmetic. Really. They are that good.

4. Minsky, M.L. (1967). Computation: Finite and Infinite Machines. Englewood Cliffs: Prentice-Hall.

One of the first (if not the first) textbooks that took the ideas that are scattered in the pre-1965 articles in (1) and the articles in (2) and synthesized them clearly using contemporary logical notation. Still a lot to learn from the book. His proof of the equivalence of neural nets and finite automata is still one of my favorite proofs.

5. Krantz, Luce, Suppes, Tversky (1971). Foundations of Measurement, Vol I: Additive and Polynomial Representations. Mineola: Dover.

While the second of the three volumes, which is on Geometrical, Threshold, and Probabilistic Relations, is more clearly linked to my research, the first volume is essential to read. It is essential since it lays down the foundations of what measurement theory is. Measurement theorists examine how the physical world (both nature and humans) can be measured quantitatively.

Bonus. Luce, R.D. (1964). "The Mathematics Used in Mathematical Psychology," The American Mathematical Monthly, Vol. 71, No. 4.

Since this is an article, not a book, I have included it as a bonus; this article explains the position of mathematics and psychology and lays out the challenges that are still facing the mathematical psychologist today. It also includes a nice synopsis of measurement theory. The moral of the story: mathematical psychologists are getting a 400 year late (if not a three thousand year late) start than mathematical physicists and dealing with a much more complex subject, viz. humans, and so new mathematical techniques will have to be invented in order to solve some of the open problems in mathematical psychology.

Late July Conferences

These two conferences run back-to-back and are about 48 miles apart. I will be at both.

The Society for Mathematical Psychology is having its 40th Meeting in Costa Mesa, CA (near UCI, who is the sponsor) from July 25 to July 28. Incidentally, I will be giving a talk there.

Then at UCLA there is the bi-annual conference on the Mathematics of Language, MoL 10, that is from July 28-July 30.

Institute for Mathematical Behavioral Sciences

The Institute for Mathematical Behavioral Sciences (IMBS) at UC Irvine is an inter-disciplinary Institute within the School of Social Sciences. It was founded by Duncan Luce in 1989, who served as director from then until 1999. Bill Batchelder served from 1999-2003. And since then, Don Saari has been the director. Luce is a mathematical psychologist and economist. Batchelder is a mathematical psychologist who has also done work in mathematical sociology and mathematical anthropology. Saari is a mathematician and economist. From this one gets a good flavor of what the Institute is about. The Institute has over 60 affiliated faculty and a small, but superb graduate program. (Disclaimer: Of which I am a member and Batchelder is my advisor.)

The five main stands of research of the members of the Institute (both faculty and grad students) are the following:

• Measurement Theory, Foundational Issues, and Scaling Models
• Statistical Modeling
• Cognitive; Economic; Anthropological/Sociological
  
• Individual Decision Making
• Perception and Psychophysics
• Vision; Psychophysics and Response time
• Social and economic phenomena
• Economics and Game Theory; Public Choice; Social Networks; Social Dynamics and Complexity
  

I am presently working in foundational issues and cognitive statistical modeling. Game Theory and Social Dynamics is an interest of mine, but I have not yet done any serious research in that area. I do hope to remedy that soon.

To say a few more words about the graduate program. While there are only six full-time students and one part-time student currently in the program, due to the unique nature of the program we have contact with students from various disciplines all of the time, e.g. this conference. This allows us to gain insight into the research of students outside are particular field of study. For example, I have learned a fair amount about research into vision due to my inter-action with fellow students- something that would never occur in a normal graduate setting. I should also re-iterate that IMBS has the greatest collection of mathematical behavioral scientists in the world and that in the program we have contact with all of these people.

Science does good...

This would make Aristotle happy, since he claimed that all, to include science, is naturally directed toward the good.

From Science:

Scientists this week reported major advances toward a central goal of stem cell research: directly reprogramming fetal mouse cells so that they are indistinguishable from embryonic stem (ES) cells. The technique, which they say should also work on adult cells, could one day enable researchers to generate cell lines tailored to individual patients without the use of eggs or embryos.

Thus, while experimenting on ES make me and many others very uncomfortable, we can hope that the controversy surrounding research on ES cells with become a non-issue. So, hurray for scientists!

Hard Analysis vs. Soft Analysis

Terence Tao, the UCLA Math Professor and Fields Medalist, has a blog. In a recent posting, he discussed the differences and similarities between hard analysis and soft analysis. I will not re-iterate what Dr Tao writes on his page since that would just be ridiculous. However, I would like to state how this is related to the titular mathematical behavioral sciences. Tao notes that analysts generally specialize in either hard or soft analysis.

At first glance, the two types of analysis look very different; they deal with different types of objects, ask different types of questions, and seem to use different techniques in their proofs. They even use^[2] different axioms of mathematics; the axiom of infinity, the axiom of choice, and the Dedekind completeness axiom for the real numbers are often invoked in soft analysis, but rarely in hard analysis. (As a consequence, there are occasionally some finitary results that can be proven easily by soft analysis but are in fact impossible to prove via hard analysis methods; the Paris-Harrington theorem gives a famous example.) Because of all these differences, it is common for analysts to specialise in only one of the two types of analysis. For instance, as a general rule (and with notable exceptions), discrete mathematicians, computer scientists, real-variable harmonic analysts, and analytic number theorists tend to rely on “hard analysis” tools, whereas functional analysts, operator algebraists, abstract harmonic analysts, and ergodic theorists tend to rely on “soft analysis” tools. (PDE is an interesting intermediate case in which both types of analysis are popular and useful, though many practitioners of PDE still prefer to primarily use just one of the two types.

Now one reason the mathematical behavioral sciences are so difficult is that one often has to use the tools of both soft and hard analysis. For example, in my work on researching the formal properties of models, I use hard analysis (e.g., mathematical logic and algorithms) and I use the tools of soft analysis when I investigate the dynamics of a system (e.g., dynamics and ergodic theory).

Thus, this posting by Tao ends up being an excellent exposition on how a mathematician can move from one domain of mathematics to another by going through a particular example, viz. the (finite) convergence principle. Being able to do this is important for many mathematicians, but this sort of situation arises for us in the mathematical behavioral sciences all of the time.

Third time is a charm...

So, this is my third attempt at blogging. Hopefully, it works better than the previous times. This blog, as the name suggests, will primarily be about mathematical behavioral sciences. I will also cover topics that are related to that field of study, but technically fall outside of it. For more details on mathematical behavioral sciences (MBS), see my personal website.