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December 13, 2009

Can Five-Year-Olds Compute Coproducts?

Posted by John Baez

Charlie Clingen points out the following article:

The abstract explains what the authors are up to:

Transitive inference, class inclusion and a variety of other inferential abilities have strikingly similar developmental profiles—all are acquired around the age of five. Yet, little is known about the reasons for this correspondence. Category theory was invented as a formal means of establishing commonalities between various mathematical structures. We use category theory to show that transitive inference and class inclusion involve dual mathematical structures, called product and coproduct. Other inferential tasks with similar developmental profiles, including matrix completion, cardinality, dimensional changed card sorting, balance-scale (weight-distance integration), and Theory of Mind also involve these structures. By contrast, (co)products are not involved in the behaviours exhibited by younger children on these tasks, or simplified versions that are within their ability. These results point to a fundamental cognitive principle under development during childhood that is the capacity to compute (co)products in the categorical sense.

A lot of the paper is taken up with explaining coproducts, which suggests a variant of the old Groucho Marx joke:

Coproducts are so simple, any five-year-old child can understand them. Quick, someone find me a five-year-old child!

But mathematicians who already understand coproducts might find this the most interesting part:

Children acquire various reasoning skills over remarkably similar periods of development. Transitive Inference and Class Inclusion are two behaviours among a suite of inferential abilities that have strikingly similar developmental profiles—all are acquired around the age of five years. For example, older children can infer that if John is taller than Mary, and Mary is taller than Sue, then John is taller than Sue. This form of reasoning is called Transitive Inference. Older children also understand that a grocery store will contain more fruit than apples. That is, the number of items belonging to the superclass is greater than the number of items in any one of its subclasses. This form of reasoning is called Class Inclusion. These two types of inference appear to have little in common. Transitive Inference typically involves physical relationships between objects, while Class Inclusion involves abstract relative sizes of object classes. Nonetheless, explicit tests of these and other inferences for a range of age groups revealed that success was attained from about the median age of five years.

Since Piaget, decades of research have revealed important clues regarding the development of inference, yet little is known about the reasons underlying these correspondences. A common theme in two recent proposals is the computing of relational information. In regard to Relational Complexity theory, the correspondence between commonly acquired cognitive behaviours is based on the maximum arity of relations that must be processed (e.g., tasks acquired after age five involve ternary relations, i.e., relations between three items). In regard to Cognitive Complexity and Control theory, the correspondence is based on the common depth of relation hierarchies. Although a relational approach to cognitive behaviour has a formal basis in relational algebra, certain assumptions must be made about the units of analysis. For tasks as diverse in procedure and content as Transitive Inference and Class Inclusion, it is difficult to see how the analysis of one task leads naturally to the other. For Relational Complexity theory, Transitive Inference is considered to involve the integration of two binary relations between task elements into an ordered triple, or ternary relation; whereas Class Inclusion is regarded as the integration of three binary relations between three sets of elements (one complement and two containments) into a ternary relation. For Cognitive Complexity and Control theory, Transitive Inference involves relations over items; whereas Class Inclusion involves relations over sets of items.

This theoretical difficulty is symptomatic of the general problem in cognitive science where the basic components of cognition are unknown. In the absence of such detailed knowledge, cognitive modelers have been forced to assume a particular representational format (e.g., symbolic, or subsymbolic). This approach, however, does not lend itself to the current problem, because the elements of Transitive Inference and Class Inclusion tasks (i.e., objects and classes of objects) do not share a common basis. Understandably, then, these sorts of behaviours have tended to be studied in detailed isolation, narrowing the scope for identifying general principles.

Category theory was born out of a desire to establish formal commonalities between various mathematical structures, and has since been applied to the analysis of computational structures in computer science. The seminal insight was a shift from objects as the primary focus of analysis to their transformations. Contrast, for instance, sets defined in terms of (the properties of) the objects they contain—Set Theory—against sets defined in terms of the morphisms that map to or from them—Category Theory. This insight motivates our categorical approach to the analysis of inference, and our way around the current impasse. In cognitive science, several authors have used category theory for a conceptual analysis of space and time, though we know of only one other application that has modeled empirical data. Since our application of category theory to cognitive behaviour is novel, we first introduce the basic category theory constructs needed for our subsequent analysis of Transitive Inference, Class Inclusion, and other paradigms. The analysis begins with a brief introduction of the sort of data our approach is intended to explain, which primarily concerns contrasts between younger and older children relative to age five, and correlations across paradigms. Finally, we extend our categorical approach to more complex levels of inference. Our main point is that, despite the apparent lack of resemblance, all these tasks are formally connected via the categorical (co)product, to be defined below. The significance of this result is that it opens the door to an entirely new (empirical) approach to identifying general principles, particularly in regard to the development of inferential abilities, that are less likely to be revealed by standard modeling methods.

How convincingly do the authors argue that products and coproducts are the unifying theme of the reasoning skills that develop around the age of 5? I’m not sure… they tackle this later in the paper, but I haven’t read it carefully yet.

Posted at December 13, 2009 5:27 AM UTC

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Re: Can Five-Year-Olds Compute Coproducts?

The basic point seems to me to be that the more difficult tasks (the ones that children older than about 5 years are able to perform while children younger than about 5 years are not) involve analysing two pieces of data at once, while the simpler tasks (that even younger children are able to perform) involve analysing only one piece of data at a time (and other pieces are irrelevant even if present). Of course, this is just the sort of thing that may be modelled categorially with products.

Why use category theory at all? One answer may be to get other researchers to focus on the relevant points. While the others are worrying about the level of complexity of each piece of data (such whether it refers to individuals or to sets of individuals), the authors want to consider the relationships between the data.

From the introduction:

This theoretical difficulty is symptomatic of the general problem in cognitive science where the basic components of cognition are unknown. In the absence of such detailed knowledge, cognitive modelers have been forced to assume a particular representational format (e.g., symbolic [6], or subsymbolic [7]). This approach, however, does not lend itself to the current problem, because the elements of Transitive Inference and Class Inclusion tasks (i.e., objects and classes of objects) do not share a common basis. Understandably, then, these sorts of behaviours have tended to be studied in detailed isolation, narrowing the scope for identifying general principles.

Posted by: Toby Bartels on December 13, 2009 6:52 AM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

What a coincidence. Just over the past few days Todd and I were discussing Piaget and cognitive development , keeping in mind Piaget’s consideration of category theory (however superficial) in his theory of genetic epistemology, lo and behold, you brought up this paper! This should make some interesting reading.

For what its worth (and perhaps others may have thought along these lines), I have always suspected that there is a good reason why CT is able to unify so many diverse areas of mathematics, that reason being we are perhaps genetically endowed with cognitive structures that help us reason categorically with varying degrees of success. To me, a promising line of research in cognitive neuroscience would be one that verifies (perhaps elucidates) such structures. Of course, such structures are not meant to be located in specific parts of the brain; rather, they must exist as dynamic representations in the mind/brain.

Posted by: Vishal Lama on December 13, 2009 11:21 AM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

There are a range of related studies brought up in this thread. Also some nice papers in The Logical foundations of cognition by John Macnamara and Gonzalo E. Reyes.

Posted by: David Corfield on December 13, 2009 11:48 AM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

Had a look a some of the papers in “The Logical foundations of cognition” since the prospect that category theory could be of use in psychology made me curious.
But I have to admit that I’m confused by the use of some of the math nomenclature. Is it just me?
If you take a look at the introduction of chapter 5 “Category Theory as a Conceptual Tool in the Study of Cognition” you’ll find:

“2. Category theory provides means to circumscribe and study
what is universal in mathematics and other scientific disciplines.”

Here “universal” is meant to denote something that several scientific disciplines have in common - somewhere else “universal” is explicitly used to denote some structure that all natural languages have in commen, yet it is also introduced to denote what is called a “universal object resp. property” in category theory. This is off putting.

Do they do this on purpose? Has my math education spoiled me?

Posted by: Tim vB on December 14, 2009 1:45 PM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

Tim wrote:

Has my math education spoiled me?

Yes. You are no longer comfortable with ordinary language.

I think the sentence you quote is using ‘universal’ in the ordinary sense, not the technical one. We can’t expect people in cognitive psychology to avoid using all words that have been borrowed by mathematicians and given technical meanings. Of course this can get a bit confusing when applying category theory to cognitive psychology. But I think we should be forgiving, because these applications are potentially important — and the subject may advance more quickly if some mathematicians contribute. Of course one thing mathematicians are good at doing is detecting and rejecting ambiguity and baloney, and this is one way to contribute. But if an effort seems sincere, and has interesting ideas behind it, it seems unfortunate to focus on the fact that the authors don’t write exactly the way mathematicians would.

Are there interesting ideas in the chapter you read?

Posted by: John Baez on December 14, 2009 3:52 PM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

I’ll report back the moment I understand something…
Some moments after my post it occured to me too that a psychologist might even strive to use “universal” both in the ordinary sense and in the technical sense (one the same page!) in order to MOTIVATE the abstract term to his intended audience.

As a mathematician I instinctively avoid that (just think about “The function of an operator is to function as a function of functions, while a function that operates on operators is a function with operators functioning as argument.”)

But I’ll try to adapt :-)

Posted by: Tim vB on December 14, 2009 4:47 PM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

I’ll try to explain my understanding why category theory is of interest to cognitive science at all:
Cognitive science as understood here tries to describe processes of learning that are universial in the sense that they do not depend on time, culture or other environmental influences. (The concurrent use of universal in this sense and in the sense of category theory - vexing to me on the first reading - seems to be deliberate and is intended to point out the analogy of the two concepts).

If you try to understand how a child learns to count, you need a theoretical model of counting. You may use set theory, use finite sets (or rather the isomorphism classes of finite sets, isomorph = having equal numbers of elements) to model numbers and set-theoretic constructions to model arithmetic operations.
Here’s the problem: To model addition, you need a concept of disjoint union of finite sets. This concept is not “universial”, because there is no “natural = canonical way” to define it.

If you buy yourself a category, you get a “universial” definition (in the sense of canonical) of disjoint union of sets for free, you substitute it with the definition of the product of two objects (which is a universal object in the sense of category theory).

Thus I think I’m not mistaken to state that the motivation to get interested in category theory is very close to the reasons that mathematicians are thinking about it.

(I’ll stop here, but there are also nice papers about children learning nouns from adults, by being shown objects, “look, there is a dog!” It’s fun to compare these to your personal experience, if you happen to have children. My nephew had already aquired the use of the noun “dog” at a very young age, when we met a dog and it’s owner during a walk. The owner pointed at her dog and said to my nephew “look, there is a wuff wuff” - which he instantly mimicked - his mother then told him “stop the bullshit, it’s a dog, for god’s sake!”).

Posted by: Tim vB on December 15, 2009 2:30 PM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

To model addition, you need a concept of disjoint union of finite sets. This concept is not “universial”, because there is no “natural = canonical way” to define it.

Wait, hold up.. I missed something huge here, because last time I checked disjoint union of finite sets was the coproduct in the category of finite sets.

Posted by: John Armstrong on December 15, 2009 4:11 PM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

First, the meta-question: My problem with these papers is that they mix trivial mathematics with subtle points about their vision of cognition science, which makes it hard to know if you understand what they are going for or not. (The math is trivial from my point of view, but, considering that the authors aren’t mathematicians but psychologists - with the exception of Lavwere - I hold them in high regard for this).
So, yes, “am I missing something huge here?” is my constant companion, too.

The point that Frangois Magnan and Gonzalo E. Reyes try to illustrate in chapter 5 with this example is IMHO this: They define the disjoint union of two sets A and B as union of A x {0} and B x {1}. See, the choice of 0 and 1 is arbitrary. The definition of a (co)product however does only use the very basic parts that a category is made of (objects, morphisms, composition of morphisms, existence of identity morphisms and associativity of composition). No arbitrary choices enter the scene after you have settled for the concept of category.

They do have other examples that justify the use of categories, I tried to pick the easiest one.

Posted by: Tim vB on December 15, 2009 4:40 PM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

They define the disjoint union of two sets A and B as union of A x {0} and B x {1}. See, the choice of 0 and 1 is arbitrary. […] They do have other examples that justify the use of categories, I tried to pick the easiest one.

How about product then? (For multiplication of numbers.) Even after the arbitrary choice of 0 and 1, you've still got an arbitrary choice of how to define the cartesian product. (Another arbitrary choice: whether to use A x {0} or {0} x A.)

Posted by: Toby Bartels on December 15, 2009 7:00 PM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

(Another arbitrary choice: whether to use A x {0} or {0} x A.)

(Actually, this is balanced by a corresponding choice in the definition of cartesian product; we shouldn't count it twice.)

It seems that what they really need here is structural set theory instead of material set theory.

Posted by: Toby Bartels on December 15, 2009 7:07 PM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

Well yes, sure, you pick some definition of disjoint sets. Within set theory you also pick some definition of numerals (Church, von Neumann, etc, etc) as we’ve discussed many times in structuralism conversations here.

So it seems that if their objection is the one you cite, then they shouldn’t come anywhere near addition-as-disjoint-union before it crops up. They should note that counting numbers themselves are “non-canonical”.

Posted by: John Armstrong on December 15, 2009 7:40 PM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

I'm guessing that they haven't swallowed the material Kool-Aid so much as to believe that everything is a set. (Really, I think that only mathematicians are ever that far gone.) So if I'm right, then they don't care about any method of representing numbers as sets as such.

However, they may still think of numbers as being cardinal numbers of sets (themselves sets of, say, blocks that children manipulate). And then once they get sets into the picture, then they worry about defining operations on them.

Posted by: Toby Bartels on December 15, 2009 8:09 PM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

Toby wrote:

I’m guessing that they haven’t swallowed the material Kool-Aid so much as to believe that everything is a set. (Really, I think that only mathematicians are ever that far gone.)

I’m not sure which ‘they’ you mean.

If you mean the authors of Category theory as a conceptual tool in the study of cognition, namely Frangois Magnan and Gonzalo E. Reyes, it’s worth noting that Gonzalo Reyes is a mathematician.

But, it’s also worth noting that he’s an excellent category theorist! He wrote a great paper called ‘The history of categorical logic 1963-1977’ with Jean-Pierre Marquis, which unfortunately Marquis was unable to put on the arXiv. And he’s also written lots of papers on synthetic differential geometry — i.e., doing differential geometry in topoi that allow for infinitesimals.

So, he has certainly not swallowed the Kool-Aid of material set theory.

But what’s more interesting to me, ultimately, is what the 5-year-olds think about all this!

Posted by: John Baez on December 15, 2009 8:41 PM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

I’m not sure which ‘they’ you mean.

I was thinking of the authors of the paper cited in the original post, but I guess that I should have been talking about Magnan & Reyes. And of course Reyes is an expert in category-theoretic logic, so no fears about him.

Posted by: Toby Bartels on December 15, 2009 9:22 PM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

Well, I don’t have any five year olds around here and now, but my own experience with children who are virtually grown was that they could very easily deal with abstractions. I never attempted to teach them category theory — my bad. But I certainly taught my boys stuff that their peers did not know. They never feared negative numbers, became quasi-comfortable with modular arithmetic, and learned various algebraic facts were true via arithmetic experimentations.

I believe that the biggest flaw in education is to assume that young people cannot grapple with abstraction at an early age. The whole world is an category of abstractions to children. The problem with teaching children the neat mathematical stuff that is there in the world is that there are far too few educated adults to re-enforce the ideas.

Posted by: Scott Carter on December 16, 2009 1:19 AM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

What would be the question to ask a five year old to find out what (s)he thinks about coproducts?

If you try to find out what abstraction the human mind uses when it learns counting, transitive inference etc. you should look into dyscalculia.

There is much to learn from children who fail to learn certain abstractions. Here are two examples:

  • Some children don’t understand the order relation of natural numbers (the way we do), i.e. why 1 < 2? As far as I now, we don’t yet understand the reason - but from my personal experience it could be something like “2apples are smaller than 1 pumpkin, so sometimes 2 is smaller than 1”.

  • Some children don’t understand equality (the way we do): A = A is obviously false, because one of the A’s is on the left of the equality sign, one is on the right, they are obviously not the same. Or this: A = a is right, because both sides sound the same.

Posted by: Tim vB on December 16, 2009 9:37 AM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

I do have a five-year old around so if anyone does come up with a good question to test whether or not five-year olds can compute coproducts, I’d be happy to try them out.

If you want comparative data, then I can also try them out on a four-year old, and if I ask the questions now and then wait a couple of weeks and ask them again I can get data for a six-year old, too.

(I guess there’s not a lot of point in asking the two-year old.)

For comparison, this particular five-year old is fine with limits: a common game in our house is for the children to cling on tightly to something (usually an adult’s leg) and declare “I’m a limit!”[1].

More seriously, he did make the leap from the statements “I’m the youngest in my class” and “I’m the eldest at home” to figuring out a group in which he was both the youngest and eldest. That’s a limit, right?

[1] s/(?<=m)i/pe/

Posted by: Andrew Stacey on December 16, 2009 11:13 AM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

aha! no 2 things can be equal
because then there wouldn’t be 2 of them

Posted by: jim stasheff on December 16, 2009 1:11 PM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

The notion of equality is subtle, which I learned from a child that had serious problems to understand it, which was quite an experience because the notion is so selfevident even to most little children!

I was reminded of this because one topic of the book “Category Theory as a Conceptual Tool in the Study of Cognition” is “reference”: If you point to a dog and tell a child “this is a dog”, how does it know what “dog” means? After all, it could be the proper name of that specific individual (like “Freddy”), it could refer to a part of the dog (“it is grey” or “look, the fur”) etc.

If you say A = A is true, you are using a list of abstract properties to identify both objects, but which are they? Let’s have a little test:

Is A = A true? (yes, but why do I think that?)

How about

  • a = A ? (no, but why not?)
  • A = A ?
  • A = A ?
  • A (imagine this letter to be red, is there a way to change the color of text here?) = A ?

In order to learn the count noun “dog”, a child has to be able to understand the list of properties that identify an object as a “dog”, and I think this list is not constant, but depends on the context. “dog” may not be a good example of this (this is the one the authors use throughout the book), but try to list the properties that identify an object as a “table”.

(Remark: If you know little children you may have noticed that this learning process sometimes does not work and e.g. a child, for some time, calls every dog “Freddie”).

Posted by: Tim vB on December 16, 2009 3:05 PM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

Maybe you already advanced the discussion far beyond the point the authors tried to make, the paper I talked about is an introduction to the subject and tries to illustrate and motivate the concept of a category to an audience that is unfamiliar with it.

This excerpt from page 61 could help to clarify this:

“The set-theoretical operations underlying the arithmetical ones are so natural that they are taken for granted when counting, which creates the impression that they are not involved in this process. We believe that these operations constitute universal “blueprints” of the mind and
we may ask what distinguishes them from others. Set theory does not give us too much help with this problem. Indeed, from a purely set-theoretical point of view, nothing seems to distinguish these operations from countless others that may be defined starting from two sets. To
make things worse, the definition of disjoint union is rather artificial. The main point is to transform the original sets so as to make them disjoint and this could be achieved in a variety of ways. How can one decide which, if any, is the right operation? It is at this point that category theory comes to our rescue with a new idea: instead of seeing a cartesian product as a set, let us view it as
a structure consisting of a set together with the relations between this set and the original ones. This structure satisfies a universal property which essentially characterizes it.”

Posted by: Tim vB on December 16, 2009 8:53 AM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

As I recall, Levi-Strauss states in no uncertain terms in several places that the sense of ‘structure’ that he uses is pretty much the same as in mathematics. I think Piaget’s introductory book on structuralism has group theory in it somewhere, for example. At a superficial level, structuralists were looking for (approximate) isomorphisms between various concrete structures (e.g. families). Perhaps one could say they tried to model a concrete collection of objects as a single category, with varying degrees of success.

Since category theory, on the other hand, is interested in functors between different kinds of structures, one might have suspected it to be a more suitably flexible framework for making the kind of comparisons anthropologists or cognitive scientists are interested in. So I had assumed that someone had already done something of this sort, convincingly or otherwise.

Posted by: Minhyong Kim on December 13, 2009 12:53 PM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

Do you know the paper by Morava – On the canonical formula of C. Levi-Strauss:

The anthropologist Claude Levi-Strauss has formulated a theory of the structure of myths using a formalism borrowed from mathematics, which has been difficult to interpret, and is somewhat controversial. Nevertheless, Levi-Strauss’s old school chum Andre Weil took his work seriously, and in this note I propose an interpretation of Levi-Strauss’s ‘canonical formula’ in terms of an anti-automorphism of the quaternion group of order eight.

Posted by: David Corfield on December 14, 2009 5:37 PM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

Re the title…

I am 38 years old and cannot compute coproducts :)

PS: Greetings from HK :)

Posted by: Eric on December 15, 2009 11:25 PM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

Yes you can!

Posted by: Dan Piponi on December 16, 2009 1:27 AM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

Who let Bob the Builder in here?

Posted by: Andrew Stacey on December 18, 2009 9:20 AM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

You’ve computed coproducts at least since the age of 5, Eric. For example:

Problem: Suppose Fred starts with an empty bucket. He puts an onion in bucket. Then he puts in a tomato. What’s in the bucket?

The hard part is not computing coproducts. The hard part is learning what ‘coproduct’ means.

PS - Lisa just got back from Hong Kong today! I really miss that place. Luckily I’ll get to visit it when we’re in Singapore next year.

Posted by: John Baez on December 16, 2009 3:03 AM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

Suppose Fred starts with an empty bucket. He puts an onion in bucket. Then he puts in a tomato. What’s in the bucket?

That is a very basic example of having two think about two things at once. I can imagine that there is some age at which a child will answer simply ‘a tomato’. But surely that age is younger than four years?

Posted by: Toby Bartels on December 16, 2009 3:39 AM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

I’ve been trying to analyse why I don’t like this example. I think it’s the bucket.

For me, the key step about colimits would be creating the colimit out of “nothing”: building it up from the pieces rather than starting with a container and putting all the pieces safely inside it and then working purely within that container. Of course, there always is a container but if it’s big enough then it’s not obvious.

So perhaps Lego is a better example! What’s the colimit of a collection of bricks? The Millennium Falcon, of course.

A slightly more serious example is the question: “I have 10 kroner, you have 5 kroner, how much money is there?”. This might still be too simplistic, but it springs to mind because of Calvin’s answer (that’s Calvin as in Calvin and Hobbes) which was (paraphrased): “10 kroner: that’s _my_ 5 kroner there and you’re not having any of it.”.

But I’m still not completely happy with this example so I shan’t test it out on my 5-year old. A better one would be where there’s a non-trivial intersection between the two sets. Maybe, “I’ve got all the pieces to build the plane and all the pieces to build the rocket but not all the pieces to build both at the same time.”

Posted by: Andrew Stacey on December 18, 2009 9:28 AM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

Hehe :)

Yeah, I know, I’ve computed things that are “coproducts”, but to recognize it as a coproduct requires that you know what a category is and what a colimit is, etc. Looking at colimit diagrams still makes my brain hurt (although they are pretty).

PS: I don’t know if you heard (I mentioned it here and there), but I am in HK permanently. I moved here last month. Finally! :) So definitely let me know next time you’re in the neighborhood.

Posted by: Eric on December 16, 2009 3:43 AM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

Eric wrote:

Yeah, I know, I’ve computed things that are “coproducts”, but to recognize it as a coproduct requires that you know what a category is and what a colimit is, etc.

Right. This is related to the distinction raised by Rajesh Kasturirangan.

Looking at colimit diagrams still makes my brain hurt (although they are pretty).

I’m convinced this ‘brain ache’ feeling is related to the growth of new neurons, which is now known to occur during learning. If so, it’s a good thing, not to be avoided — just like the ‘muscle ache’ feeling you get when you exercise.

I don’t know if you heard (I mentioned it here and there), but I am in HK permanently. I moved here last month. Finally! :) So definitely let me know next time you’re in the neighborhood.

You didn’t tell me, but you linked to your blog, and you explained it there. Congratulations!

Like you, I’m fleeing the sinking United States for Asia… but unfortunately, only for a year. It’s looking 99.9% certain that I’ll be spending a year at the Centre for Quantum Technologies in Singapore. And my friend Jiang-Hua Lu, who works on Poisson geometry, Lie groups, and Hopf algebras, has invited me to visit the University of Hong Kong, so I’ll almost certainly see you.

Posted by: John Baez on December 16, 2009 6:13 PM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

At the risk of sounding pedantic, I think it is worth distinguishing between:

Hypothesis A: Children compute coproducts (starting at age X or so)

Hypothesis B: Children perform certain tasks (such as transitive inference) that are best explained by cognitive scientists as evidence for the possession of category theoretic structures in their minds.

In cognitive science the distinction between hypothesis A and hypothesis B is quite important - what Chomsky has called the difference between competence and performance. For example, if A is true, we would be entitled to look for brain mechanisms that implement coproducts, while if B is the better hypothesis, it would be premature if not false to look for brain mechanisms.

In my admittedly biased view, good cognitive science has mostly happened when research has proceeded along directions that resemble hypothesis-B, resulting in competence theories. Unfortunately, the authors of this paper have not clarified their views on the matter, though it seems as if they have a hypothesis-A like view.

What I find interesting about the tasks that they test in children is that all of them involve a comparison between one kind of structure represented in one mental state and another structure represented in another mental state (such as the appearance/reality false belief test in the theory of mind segment). A strong case could be made that some category theory like formalism is needed to perform cross structural comparisons, especially if the comparison abilities are systematic and productive. However, arguing for category theory on grounds of systematicity is a competence argument, not a performance argument and the authors lose out on an opportunity as a result of their confusion between the two categories.

To give a linguistic analogy, English speakers are able to perform wh-movements (replace an assertion by a question, for example). Generative grammarians have tried to describe children’s knowledge of wh-movements using constraints on tree structures that represent the grammatical structure of a sentence. However, Chomsky and other generative grammarians never claimed that children actually compute those trees in their heads or that those tree operations are implemented somewhere in our brains.

Another analogy: we can clearly predict the motion of planets using differential equations, but no one believes that planets compute the differential calculus. (I should say ‘almost no one’, for there are people like Ed Fredkin, Norm Morgolus and Steve Wolfram who believe the universe IS a computer).

I find it strange that views that are demonstrably false and/or problematic in physics are so easily accepted when it comes to cognitive science and the philosophy of mind.

Posted by: RajeshKasturirangan on December 16, 2009 3:24 PM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

At the risk of sounding pedantic, I think it is worth distinguishing between: Hypothesis A: Children compute coproducts (starting at age X or so) Hypothesis B: Children perform certain tasks (such as transitive inference) that are best explained by cognitive scientists as evidence for the possession of category theoretic structures in their minds.

The Phillips/Wilson/Halford paper does not mention this distinction - or I missed it, too. But John Macnamara and Gonzalo E. Reyes do it in their paper “Foundational Issues in the Learning of Proper Names, Count Nouns and Mass Nouns”. Maybe it goes without saying from the point of view of most scientists in the field?

Excerpt from page 146:

“Since representations of the learner’s knowledge are at the core of our theory, we must clarify our notion of representation. A representation need not resemble what it represents in the way a map or a photograph do. For example, in dynamics the trajectory of a projectile is represented by a system of equations, which enable one to calculate height, velocity and direction at each point in the trajectory. There is no suggestion that the projectile itself is computing any of these values. Similarly, there is no suggestion in our representations that children express the knowledge in precisely the form we offer. Rather, we claim that they must represent the same information in some manner. In particular, we do not expect that children can tell other people the information they have learned. We do, however, expect them to reveal their knowledge indirectly, in their utterances and actions.”

RajeshKasturirangan said:

I find it strange that views that are demonstrably false and/or problematic in physics are so easily accepted when it comes to cognitive science and the philosophy of mind.

I grant others the benefit of the doubt, after all the methods used to aquire knowledge that work for mathematicians and for physicists are not universally applicable :-)

Posted by: Tim vB on December 16, 2009 4:27 PM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

You quote the paper saying:

For example, in dynamics the trajectory of a projectile is represented by a system of equations, which enable one to calculate height, velocity and direction at each point in the trajectory. There is no suggestion that the projectile itself is computing any of these values. Similarly, there is no suggestion in our representations that children express the knowledge in precisely the form we offer.

It can be put even more clearly by anyone who has tried to tutor an NCAA star basketball player in calculus.

Yes, we can express the problem of specifying a launch angle and force that will make the ball go through the hoop in terms of a calculus problem. And yes, he makes some absurd percentage of his free throws. But he most certainly isn’t thinking of it in terms of calculus problems, or he’d do a lot better on these exams!

Posted by: John Armstrong on December 16, 2009 7:08 PM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

I had an interesting chat the other day with an ex-colleague now at the engineering department in Cambridge. He contrasted two ways to have a machine move a cart along a track so as to balance a rod upright which is attached to it. Either you work out the equations of motion for the rod and use control theory, or else you use machine learning techniques and some learning experience with feedback of how well its doing.

It appears that the latter method works very well with little ‘prior knowledge’ encoded in the algorithm. This also worked for balancing a jointed rod, but as yet only in a simulation.

There’s an interesting challenge now to see which technique works to allow a robot to ride a unicycle. It would not surprise me if the machine learning method worked best - clearly we don’t know the equations of motion for a bicycle when we learn to ride.

Posted by: David Corfield on December 17, 2009 9:35 AM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

If we compare different machine learning algorithms to humans, maybe chess could be of interest: What do chess masters think when contemplating a specific situation and is it possible to get computers to do the same?

The wikipedia article (chess psychology) does not mention this, unfortunatly, here is a book about what I mean:

Chess Players’ Thinking: A Cognitive Psychological Approach (amazon)

The book I read is available in German only, it would seem:

Reinhard-Munzert (amazon)

One point of interest: Munzert describes chess masters that record their stream of conciousness, which revealed that they don’t dig deeper than the average player, but instinctivly/unconciously pick the best moves and spend their time thinking about those. The average player wastes his time thinking about other moves.

For some time developers of chess programs tried to understand this and to implement it - via heuristic algorithms - in computer programs. These programs lost to the brutal force approach (compute every possibility until you run out of time) somewhere during the 1990ies, I don’t know if anyone still tries to advance in this direction.

Posted by: Tim vB on December 17, 2009 10:54 AM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

Rodney Brooks style robotics is based on the idea that knowing-how (to ride a bicycle, for example) doesn’t require detailed descriptions of the underlying dynamics. In contrast, the equations of motion that capture the laws of classical mechanics are a form of knowing-that.

To reverse the robot riding a bicycle challenge, I would be very curious to find out what happens when you train the best machine learning algorithms on the kind of data that Kepler and Copernicus used to study planetary motion. Are they as good as the classical Newtonian solutions? Can they handle 3-body problems?

In his classic paper “Intelligence Without Representation” Brooks gives good arguments as to why we do not need internalized equations of motion to solve robotic navigation etc. Of course, Brooks style subsumption architecture doesnt run on machine learning algortithms, but Tommy Poggio made a similar argument on the basis of his work on Support Vector Machines (see Poggio’s “How the Brain Works”).

Other cognitive scientists - J. J. Gibson, Francisco Varela etc - have also made a similar argument. There is now a pretty thriving field in cognitive science called enactive cognition that is trying to shift the focus of cognitive epistemology from knowing-that to knowing-how.

Posted by: RajeshKasturirangan on December 17, 2009 2:37 PM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

Yes, in highschool I was very frustrated that I could calculate how I should throw the ball, but could not do it.

But aren’t we working towards another necessary refinement? We already distinguish if humans use categories (Hypothesis A) or if categories are a useful tool to describe how humans manage their knowledge (Hypothesis B).
If you look at Hypothesis A you can further distinguish if humans know that they use categories (or calculus) or if they don’t.
Your star basketball player could have a robotic arm with a microcontroller that has a program performing (approximatly) calculus, he neither needs to know that nor to understand how it works :-)

The passage I quoted seems to state that the authors don’t claim that Hypothesis A is true, nor that it can be tested by asking children how they manage their knowledge, because even if Hypothesis A is true, you cannot expect them to be able to know and explain it.

Posted by: Tim vB on December 17, 2009 9:48 AM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

Rajesh Kasturirangan wrote:

At the risk of sounding pedantic, I think it is worth distinguishing between:

Hypothesis A: Children compute coproducts (starting at age X or so)

Hypothesis B: Children perform certain tasks (such as transitive inference) that are best explained by cognitive scientists as evidence for the possession of category theoretic structures in their minds.

I agree that this is a crucial distinction! Of course when it comes to 5-year-olds, I only think Hypothesis B is interesting. My title ‘Can Five-Year-Olds Compute Coproducts?’ was merely meant to be eye-catching, amusing and terse.

I assume that any cognitive psychologist who hasn’t been asleep for many decades is aware of this distinction. Like Tim vB, I am willing to grant them the benefit of the doubt instead of demanding that they demonstrate their awareness of it.

Posted by: John Baez on December 18, 2009 4:29 AM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

As a mathematician turned cognitive scientist, my experience is that psychologists are aware of the distinction between hypotheses A and B but they often slip when doing their work and writing their papers as can be seen in the PLOS paper that started this discussion, and unlike the Macnamara-Reyes paper Tim vB cited (which I am certain is because of Macnamara’s familiarity with Chomskian linguistics; Macnamara is a particularly insightful cognitive psychologist).

Chomsky made the distinction between competence and performance precisely because he wanted to address the concerns of psychologists and others who were critiquing his work on Hyp A grounds when the claims were being made on Hyp B grounds.

Posted by: Rajesh Kasturirangan on December 18, 2009 5:56 AM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

My colleague Chris Athorne sent me the following by email, which I’m reposting here with his permission.

I still maintain that I taught Thomas how to “count” before he could speak by giving him a set of cards with numbers of dots marked on them and showing him that he should sort them into correspondence classes, which he was able to do. Cards of the same … er, Card … were not geometrically congruent so he did not have any patterns to offer clues. I told the child psychologist this and she gave me a look over her glasses as though I’d just claimed there were fairies at the bottom of my garden. Which, of course, there are but I thought it prudent not to go there!

Posted by: Tom Leinster on December 17, 2009 8:00 PM | Permalink | Reply to this

Your left fusiform gyrus Mileage May Vary; Re: Can Five-Year-Olds Compute Coproducts?

I believe Tom Leinster. My father was dubious about my claim that my son, younger than 18 months, knew the alphabet flawlessly. When my (Harvard Cum Laude) father visited, he tested my son with alphabet blocks. “Show me the “X.”
“What is the name of this one?”

In less than a year, my son was beating my Dad at chess.

I kept notes of his language acquisition. He jumped in one month from 3-word sentences (i.e. “It fall down.”) to this one, which stunned my father when my boy said so, still much younger than 2:

“Mommy, I want you to cut the tag from the back of Daddy’s pajama shirt.”

What we are born able to do, or predetermined to be able to acquire easily, varies widely. I observe this in Mathematics, in Reading (via left fusiform gyrus – which is necessary for normal, rapid understanding of the meaning of written text as well as correct word spelling – per Tsapkini et al. The orthography-specific functions of the left fusiform gyrus: Evidence of modality and category specificity. Cortex, 2009), and in Learning generally.

See also:
Education: Learning Styles Debunked
Journal References:

1. Rohrer et al. Increasing Retention Without Increasing Study Time. Current Directions in Psychological Science, 2007; 16 (4): 183 DOI: 10.1111/j.1467-8721.2007.00500.x
2. McDaniel et al. The Read-Recite-Review Study Strategy: Effective and Portable. Psychological Science, 2009; 20 (4): 516 DOI: 10.1111/j.1467-9280.2009.02325.x
3. Roediger et al. Test-Enhanced Learning. Taking Memory Tests Improves Long-Term Retention. Psychological Science, 2006; 17 (3): 249 DOI: 10.1111/j.1467-9280.2006.01693.x
4. Hal Pashler et al. Learning Styles: Concepts and Evidence. Psychological Science in the Public Interest, (in press)

Association for Psychological Science (2009, December 17). Education: Learning styles debunked. ScienceDaily. Retrieved December 17, 2009, from http://www.sciencedaily.com­ /releases/2009/12/091216162356.htm

Posted by: Jonathan Vos Post on December 17, 2009 10:46 PM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

Why was the child psychologist sceptical?

As a child psychologist she was surely aware that humans differ considerably in their development during childhood, and that an abstract concept like counting does not rely on language (see the introductory remarks about nonverbal thinking in this essay about human intelligence: Norimasa Kobayashi: An Essay on Human Intelligence ).

Posted by: Tim vB on December 18, 2009 12:45 PM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

I don’t know about psychologists, but from my personal experiences with physicians, although they may recognize on an intellectual level that such variation exists, many seem to react skeptically (or assume evidence of some problem) when actually presented with individuals who differ significantly from the median. I’d guess many psychologists are similar.

Posted by: Mark Meckes on December 20, 2009 12:52 AM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

Well, yes, I know this from my personal experience, too…but what was the reason in this case?

We know that even animals can count.
And of course we know of many Wunderkinder that were able to do amazing tasks at young ages. (I do not say that being able to count before you are able to speak proves that you are a Wunderkind).

One of my favorite stories is about Samuel Reshevsky who could beat world class players at chess when he was only a few years old. Asked how he was able to do it he used to say “I don’t now, it just happens, like breathing”.
As a child he toured the world. When he visited Vienna, people called Milan Vidmar, then one of the world’s top players, because no one else could hold a candle to him (this anecdote is mentioned briefly here: game agains Milan Vidmar). First, Vidmar did not take him seriously and lost, then gave his best and won, which caused Sammy to burst into tears. Vidmar later recalled that he never regretted to win a chess game more than on this occasion.

Posted by: Tim vB on December 20, 2009 3:28 PM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

Conc. ‘abstract’ thinking and language, I find archeological hints about language-free precursors of modern humans interesting. On tool using skills , here is an interesting article.

Posted by: Thomas on December 20, 2009 12:39 PM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

I think we have to distinguish between subitizing, and counting, the former is hard-wired into our brains, while the latter is acquired at a later age. Thus lot of animals can be demonstrated to be counting, but instead they are just using their brains hard-wired pattern recognition algorithm, while not understanding the underlying correspondence between numbers, and quantities.

Posted by: Németh Lukács on December 26, 2009 1:12 PM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

I think we have to distinguish between subitizing, and counting

Ah, so there may be a language barrier; what psychologists mean by ‘counting’ is more specific that what mathematicians mean by that term.

But if Tom used cards with large enough Card, then the psychologists should still accept it as counting.

Posted by: Toby Bartels on December 26, 2009 2:14 PM | Permalink | Reply to this

Re: Can Five-Year-Olds Compute Coproducts?

Interesting! I did not know that. Follow up question would be if there is a clear way to distinguish the two, but one way that does not seem very promising would be to search for an upper bound of items to count for the subitizing ability to still work.

Why not?

Well, it would seem that even ants can “count” (i.e. subitize):

Do ants count their steps to find their way home?

Posted by: Tim vB on January 2, 2010 12:55 PM | Permalink | Reply to this

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