# Source for analysis of identification of structures in learner's mind and mathematical structures?

Concerning the structure of the learner's mind, psychologist Piaget claimed that

There exists, as a function of the development of intelligence as a whole, a spontaneous and gradual construction of elementary logico-mathematical structures and that these 'natural' ('natural' the way that one speaks of the 'natural' numbers) structures are much closer to those being used in 'modern' mathematics than to those being used in traditional mathematics. (p. 79 in Piaget 1973).

Piaget appears to postulate an affinity between, on the one hand, the structures of the mind and, on the other, the structures of modern mathematics (mainly following Bourbaki). The essay in question is

Piaget, J. "Comments on Mathematical Education," in A. G. Howson, ed., Developments in Mathematical Education: Proceedings of the Second International Conference on Mathematical Education, 79--87, Cambridge: Cambridge University Press, 1973.

Piaget's postulated affinity has apparently been challenged by some scholars in the context of the New Math controversy.

Question. Is there a source that provides a detailed analysis of such a postulation of a connection between the structures in the learner's mind on the one hand, and Bourbaki-style structures in the foundations of mathematics, on the other?

(Note that I am not looking for general sources on the New Math/Modern Math controversy, nor am I particularly interested in Piaget's work in general, but rather for an analysis of this particular identification of Piaget's psychological structures and Bourbaki's mathematical structures).

I just came across a book that might be relevant:

All Positive Action Starts with Criticism: Hans Freudenthal and the Didactics of Mathematics. By Sacha la Bastide-van Gemert. Springer, 16 Jan 2015

Here the author quotes Freudenthal as follows on page 211:

It thus did not begin with the Sputnik shock. It had already begun in the early 1950s. They had even managed to convince Piaget, who did not understand anything of it except for the fact that the word "structure" appealed to him. With Piaget's name on the billboard they felt confident of the support of psychology. What now, psychology! Mathematics is ruled by a logical order and he who teaches mathematics is easily seduced to sacrifice the psychological, the educational order to the logical order. I have done my utmost to avoid this and in my 'fragment Rekendidactiek' of 1942, if not earlier, I wanted to warn others. But what was now happening before my very eyes? A logical order brought to ecstasy, a systematic of mathematics as a whole--that is how mathematics should be taught.

It is clear from this that Freudenthal was sceptical of these developments but unfortunately he does not elaborate the details of his objections.

• Crossposted to matheducators.SE: matheducators.stackexchange.com/questions/12507/… Jun 26, 2017 at 5:27
• "Piaget's postulated affinity has apparently been challenged by some scholars in the context of the New Math controversy." To which scholars are you referring? (And RE: already given answer, yes, Dubinsky -- as I understand matters -- is extending the work of Piaget to higher st/ages of learners. Cf. many posts on MESE, e.g., 12337.) Jun 26, 2017 at 18:19
• However fascinating it may be, I don't feel comfortable with this discussion as fitting for MO. There is a Cognitive Sciences Stack Exchange platform which I imagine would take such questions in psychology. Jul 2, 2017 at 15:30
• But it sure looks like this is a question about Piaget's research and empirical evidence for it. Jul 2, 2017 at 16:02
• You should, at the very least, make the question you are asking more visible. Right now, it is well-hidden in the middle of the wall of text. Jul 2, 2017 at 23:03

You say that Freudenthal does not elaborate his objections, but in a sense he did just that in his book Mathematics as an educational task (1973), Appendix I, starting: “The somewhat summary criticism I administered on several occasions to Piaget's work demands more detailed argument.” However, this addresses egregious mathematical misconceptions and methodological flaws (ill-chosen questions) more than the (“ontogeny parallels phylogeny”?) postulation you ask about.

Against that, his argument seems much shorter:

(p. 46): Bourbaki. How convincing this organization of mathematics is! So convincing that Piaget could rediscover Bourbaki's system in developmental psychology. Poor Piaget! He did not fare much better than Kant, who had barely consecrated Euclidean space as "a pure intuition" when non-Euclidean geometry was discovered! (...) Mathematics is never finished – anyone who worships a certain system of mathematics should take heed of this advice.

(p. 192): Impressed by Cantor's analysis he turned to studying the development of the number concept under this aspect. (...) Piaget believed that the concept of natural number could be entirely derived from potencies. (...) This may have seduced him to believe that it is also psychologically true; it was one of his ideas to trace in developmental psychology the system of mathematics he happened to be acquainted with.

Or, as reported by F. Goffree in The legacy of Hans Freudenthal (1993):

(p. 36): Piaget thought that the cognitive development in children took place from poor to rich structures whereas HF thought it was the other way round. Geometry showed something similar. In the Erlanger Program Klein had given a hierarchy of geometrical structures: topological, projective, affine and euclidean, in HF's terms from poor to rich. Children start by drawing irregular circles, anyone can see that. This was sufficient reason for Piaget to presume that their geometric development began with the topological structure. HF remarked condescendingly that those very same children were quite capable of distinguishing between correctly drawn circles and other figures.

(Note added: These last remarks, not sourced by Goffree, are in Revisiting mathematics education (1991), p. 27. So far as I can tell, the criticized “postulation” is most detailed in Épistémologie des mathématiques. Partie II. Chap. 8 & 11 (1961, transl. 1966).)

A much-cited attempt to analyze Piaget's ideas and carry them further has been given by Ed Dubinsky in Reflective abstraction in advanced mathematical thinking (1991).

Reflective abstraction is a concept introduced by Piaget to describe the construction of logico–mathematical structures by an individual during the course of cognitive development. Two impor- tant observations that Piaget made are first that reflective abstraction has no absolute beginning but is present at the very earliest ages in the coordination of sensori–motor structures and second, that it continues on up through higher mathematics to the extent that the entire history of the development of mathematics from antiquity to the present day may be considered as an example of the process of reflective abstraction.

In the majority of his own work, however, Piaget concentrated on the development of mathematical knowledge at the early ages, rarely going beyond adolescence. What we feel is exciting is that, as he suggested, this same approach can be extended to more advanced topics going into undergraduate mathematics and beyond. It seems that it is possible not only to discuss and conjecture, but to provide evidence suggesting, that concepts such as mathematical induction, propositional and predicate calculus, functions as processes and objects, linear independence, topological spaces, duality of vector spaces, duality of topological vector spaces, and even category theory can be analyzed in terms of extensions of the same notions that Piaget used to describe children’s construction of concepts such as arithmetic, proportion, and simple measurement.

• Now if someone would kindly explain to me the difference between this and something that programs like SCIgen produce in fractions of a second . . . Jun 25, 2017 at 19:26
• @FranzLemmermeyer The language here is consistent but obscure, while in the programs you mentioned, it is transparent but inconsistent. On the other hand, I agree with you that the insertion of a few thereof, henceforth, and hitherto at random places would bridge the gap (at least for me). It is most unfortunate because what the philosophers speak of makes sense sometimes while the way they present it is always totally incomprehensible (see the original texts by Hegel, for instance). They make the same mistake as lawyers equating the sophistication of the language with that of the thought. Jun 25, 2017 at 20:10
• @fedja, who is the philosopher here? Jun 26, 2017 at 8:18
• @MikhailKatz Anybody who talks about a craft without practicing the craft himself. The craft in this case is mathematics. The problem with Piaget statement is that it refers to the things that are very hard to define even for a practicing mathematician (logico-mathematical structures) and makes a distinction between modern and traditional mathematics (there are $\ge$ 3 timelines in math.: temporal (Theorem X was proved in year Y), historical (The work of mathematician X was influenced by that of Y) and logical (X implies Y). They are tangled too much to make such words meaningful (continued) Jun 26, 2017 at 11:51
• You might be interested to know that among the bunch of texts that Grothendieck wrote during his years of isolation there exists one entitled "Structure de la psyché". I have no idea of its availability whatsoever. Jun 26, 2017 at 20:47

Daniel Everett described in his book Dont sleep there are snakes the language of an Amazonian tribe (Piraha) which has no words for numbers. This tribe also uses "immediacy of experience" for believing other people. The author also argues that this language is a counter-example to the idea of a language instinct as proposed by Whorf and Chomsky.

Caleb Everett in Numbers and the Making of Us: Counting and the Course of Human Cultures argues, based on the language of the Piraha, that counting is a cultural achievement and not innately laid down in the structure of the brain. On page 274, in "Notes" to the section "6. Quantities in the Minds of Young Children", Caleb E. mentions Piaget as follows:

1. Jacques Mehler and Thomas Bever, “Cognitive Capacity of Very Young Children,” Science 3797 (1967): 141–142. See also the enlightening discussion on this topic in Dehaene, The Number Sense: How the Mind Creates Mathematics, particularly as it relates to the work of Piaget. I should mention, however, that an insightful reviewer notes that there have been issues replicating the results of Mehler and Bever with very young children.

A review of Caleb Everett's book is in the New York Times.

In the second part of his book, Caleb E. summarizes empirical evidence that people with anumerate languages and human babies have an innate sense to recognize quantities up to 3 precisely, and another innate sense to recognize the larger quantity between two aproximately (like: 16 is more than 8). To connect the two and to start to count needs exposure to language with number words and is a slow process. The approximate number sense is also there in rats, etc.

On page 168 he writes:

• As Elizabeth Brannon and Joonkoo Park, two animal cognition specialists, recently suggested: “it is challenging to understand how such a primitive system that is not capable of representing exact large numbers could give rise to the formal mathematics that is uniquely human.”20 The quantitative thought with which we and other species are innately equipped is orders of magnitude removed from the kinds of quantitative thought that most humans eventually possess. This suggests that biological explanations of such thought are inherently limited. Most of our numerical cognition owes relatively little to our neurobiological equipment and owes much more to the ways we manipulate that equipment. This manipulation can only be possible if there are external tools interacting with our innate mechanisms for quantity differentiation. The principal external tools in question are numbers, symbolic representations of quantities that are linguistically reified and used in culturally variant ways. The existence of numbers explains the gulf between actual human numerical thought and the numerical thought to which we are innately predisposed.
• Peter, thanks. But innateness of language structures (proposed by Chomsky and others), whether true or not, is a different issue from the innateness of mathematical structures as proposed by Dieudonne and others; see this answer on an amazing Piaget-Dieudonne exchange concerning this point. Jul 2, 2017 at 7:44
• Mikhail, the mention of the language instinct was an aside. I am astonished that even counting seems to be cultural, as argued in Caleb Everett's book. And this is based on empirical evidence and not introspection. Jul 2, 2017 at 7:52
• So your main point is from Caleb Everett and not from Daniel Everett? Do Caleb Everett's book or the NYT review of it mention either Piaget or Dieudonne &CO @Bourbaki and/or new math? Jul 2, 2017 at 7:55
• Daniel E. found that there are no words for numbers and that the Piraha do not count (besides one and many). Caleb E.'s book takes this as a starting point and adds many more arguments. Jul 2, 2017 at 7:58
• Do Caleb Everett's book or the NYT review of it mention either Piaget or Dieudonne &CO @Bourbaki and/or new math? Jul 2, 2017 at 8:01

Grothendieck wrote a text entitled "Structure de la psyché" that might help shed a light on the issues you consider. As a former member of Bourbaki, it seems highly plausible that he tried doing so to describe how the brain works mathematically, perhaps from a categorical point of view. I don't know whether this text is available online, but imho there's no doubt it should be quite instructive. Maybe the book by Philippe Douroux contains some information on this, but I haven't read it yet.

• Thanks! Do you have some details on Douroux's book? Jun 28, 2017 at 7:27
• The title must be "Alexandre Grothendieck : Itinéraire du dernier génie des (en ?) mathématiques". I have no idea whether it has been translated into English or not. Jun 28, 2017 at 8:51
• here I read that "Structure de la psyché" is about the combinatorics of oriented graphs, which does not seem to be related to the OP. There is also another manuscript entitled "Psyché et structures" which may be the one you are referring to? Both these manuscripts are in the Lasserre collection. Jun 28, 2017 at 10:21
• @CarloBeenakker Could you please provide a link? I have hard time finding this online. Jul 2, 2017 at 7:26
• @მამუკაჯიბლაძე --- all I have is the inventory of the Lasserre-griboillis that I linked to in my previous comment, which says: 2. Structure de la psyché. (12/10/1992-28/09/1993) 3600 pages This one is about some combinatorics of oriented graphs with extra-structure (part of the structure are successor and predecessor operators on the set of arrows). 3. Psyché et structures (26/03/93-20/06/93) 700 pages. This one is non-mathematical. None of this material is online. Jul 2, 2017 at 11:45