Wednesday, 1 June 2011

Intuitions Regarding Geometry Are Universal, Study Suggests

Are geometric intuitions universal? A study reported in Science Daily suggests:
All human beings may have the ability to understand elementary geometry, independently of their culture or their level of education. This is the conclusion of a study carried out by CNRS, Inserm, CEA, the Collège de France, Harvard University and Paris Descartes, Paris-Sud 11 and Paris 8 universities. It was conducted on Amazonian Indians living in an isolated area, who had not studied geometry at school and whose language contains little geometric vocabulary. Their intuitive understanding of elementary geometric concepts was compared with that of populations who, on the contrary, had been taught geometry at school. The researchers were able to demonstrate that all human beings may have the ability of demonstrating geometric intuition. This ability may however only emerge from the age of 6-7 years. It could be innate or instead acquired at an early age when children become aware of the space that surrounds them. This work is published in the PNAS.
The article is:
Véronique Izard, Pierre Pica, Elizabeth S. Spelke, and Stanislas Dehaene. 2011. Flexible intuitions of Euclidean geometry in an Amazonian indigene group. Proceedings of the National Academy of Sciences, 23 May 2011.


  1. I tend to believe anything this group of authors say (in particular, I'm a huge fan of Dehaene), so if they say so, it must be true! But incidentally, they are also the authors of the famous 'Log or Linear?' paper, which argues that the same people (the Amazonian tribes that Pica studies) do NOT have a liner intuition of numbers, but rather a logarithmic one.

    So Kant is partially vindicated: no innate intuition of numbers as linear, but probably an innate intuition of space that looks like 'our' geometry. Numbers would then be more of a 'cultural invention' than geometrical concepts.

  2. Numbers a cultural invention? Crazy talk!

    (Just kidding - sort of!)

  3. Go read the stuff and we can talk about it! :) Here is a summary of the main results:

    Abtract of 'Log or Linear?' (Science, 2008, same authors in different order)

    The mapping of numbers onto space is fundamental to measurement and to mathematics. Is this mapping a cultural invention or a universal intuition shared by all humans regardless of culture and education? We probed number-space mappings in the Mundurucu, an Amazonian indigene group with a reduced numerical lexicon and little or no formal education. At all ages, the Mundurucu mapped symbolic and nonsymbolic numbers onto a logarithmic scale, whereas Western adults used linear mapping with small or symbolic numbers and logarithmic mapping when numbers were presented nonsymbolically under conditions that discouraged counting. This indicates that the mapping of numbers onto space is a universal intuition and that this initial intuition of number is logarithmic. The concept of a linear number line appears to be a cultural invention that fails to develop in the absence of formal education.

  4. Hi Catarina, I've read Dehaene's book The Number Sense, and some of this sort of work. The problem is that they're making certain claims that don't make much mathematical sense. E.g.,

    "The mapping of numbers onto space is fundamental to measurement and to mathematics."

    1. What mapping?
    2. What do "the" "mapping", "numbers" and "space" mean?
    3. How do they define "mapping"?
    4. Is there a unique mapping? If they say so, how do they prove this? What assumptions are needed?
    5. By "numbers", do they mean the structure (R, <)? Some other structure? A proper initial segment of the positive reals R? Some metric structure on R? A topological structure?
    6. Let R* be the positive reals. log_{10} is a function f: R* -> R. What does "logarithmic" mean in this context?
    7. What is a "symbolic number"? (seems like a use/mention confusion).
    8. By "space" do they mean standard models of Euclidean geometry? Non-standard models? Or physical space? Something else? (They show no knowledge of the standard literature on representation theorems.)

    "This indicates that the mapping of numbers onto space is a universal intuition and that this initial intuition of number is logarithmic."

    9. What does "the" mean?
    10. What does "mapping" mean?
    11. What does "onto" mean? (They probably mean functions from initial segments of (R*, <) to a line segment in 1-dimensional models, but they don't say so.)

    "The concept of a linear number line appears to be a cultural invention that fails to develop in the absence of formal education."

    12. What is a "concept"? What is the mass or velocity of a concept?
    13. If a concept is massless and velocityless, how is it "invented"? What does that mean?
    14. What does "linear" mean?
    15. What does "number line" mean?

  5. Jeff, I know what you mean, I have a similar kind of reaction when I read what the psychologists studying deductive reasoning say about logic and deduction. Still, in this case I think it's slightly different, as Dehaene does have a degree in math (from the Ecole Normale Superieure, no trivial feat), so he knows his math (probably much better than I know mine, with a modest minor).

    Psychologists and cognitive scientists often seem to think that we philosophers are being 'pedantic' when we ask questions such as 'what is a "concept"?', but at least some of these questions do have an impact on the research they do. I'd be curious to hear what you think when you read the whole article (if you haven't yet), as it's hard to judge just from the abstract how solid the research is from a philosophical point of view.

    Generally, and again wrt my contact with the literature on the psychology of reasoning, my attitude is: even if the conceptual framework is not as clear as it should be, it doesn't mean that the empirical data are all to be discredited. The hard part is to filter the good stuff through!

  6. Hi Catarina, actually, I think the empirical results of these studies are generally very interesting - plus other work by authors in cognitive & developmental psychology, like Elizabeth Spelke, Alison Gopnik, etc., going back to Piaget. It's the philosophy that bothers me - "crazy talk", as Roy says.

    The term of art here is "concept" and the underlying subjectivism or psychologism. I don't know what concepts are, and maybe in some sense, concepts are "constructed". In another sense (e.g., Fregean functions), concepts aren't constructed. Are concepts subjective patterns of associations? Can two different minds both "grasp" the same concept? Are concepts applicable to things? Should we say,
    (1) The classical concept of negation is different from the intuitionistic concept of negation.
    (2) The concept of classical negation is different from the concept of intuitionistic negation.

    The first (1) encourages the view that there are many *concepts* of negation. (2) encourages the view that there are many *negation operations*, all related somehow (by "concept of classical negation" we mean "classical theory that governs negation"). I don't have any good answers to these questions, but I'd like to know how the mind "grasps" concepts.

  7. "I don't have any good answers to these questions, but I'd like to know how the mind "grasps" concepts."

    Well, I guess that's pretty much what we would all like to know, psychologists in particular :) Do we, philosophers, have truly robust theories of what concepts are? I guess we don't. But if this means that, in the absence of such a conceptualization, all psychological research will be 'crazy talk', then that would be a problem too, I think.

    I don't mean to say that all uses of the term 'concept' and related terminology by psychologists should be tolerated, but just that these considerations seem to me to introduce a significant risk of foundational skepticism. *That* we humans cognize is beyond doubt, but the question of *how* we cognize is daunting, and we seem to be forced to use the term 'concept' as a blanket term to refer to the objects of cognition.

    And just to clarify, what term would you prefer to use to describe the idea (concept?) of the series of the natural numbers as evenly-spaced and linear?

  8. Hi Catarina, my fault - I'm being very very unclear! Two separate things:
    (i) the subjectivism/psychologism (Roy's "crazy talk");
    (ii) the unclarity about the mathematics.

    On (i), it's not the talk of the concepts (though what concepts are is very moot). I happily talk of concepts, notions, ideas, etc. The trouble is the subjectivism or psychologism: the psychologistic assumption that certain mathematical entities *are* (mental) concepts, or intuitions, or inventions. For example, they write:

    "Is this mapping a cultural invention or a universal intuition?"

    Usually, a mapping f from X to Y is a set of ordered pairs (a subset of the Cartesian product X x Y). Can a function be an "invention"? How can a function be an "intuition"?

    Cf., would someone say: "Is the millisecond pulsar an invention or an intuition?" Only an idealist about physical entities would say such a thing.

    "what term would you prefer to use to describe the idea (concept?) of the series of the natural numbers as evenly-spaced and linear?"

    Quite happy with the word "concept" (and similar, like "notion", etc.)! Not happy with the psychologism about mathematical objects. But your question focuses on the interesting claim by Dehaene et al about mathematical cognition, which seems interesting but is difficult to follow.

    So, I don't quite get the phrase "the concept of the series of natural numbers as evenly-spaced and linear". Actually, I think they mean "beliefs about the numbers as being ..." or "manner of visualizing" (and not "the concept" - or maybe they simply have a rather holistic concept of concepts).

    Here "linear" doesn't mean a property of orderings (an order (X, <) is linear if < is transitive, irreflexive and trichotomous). I think it means "linear" as a property of functions on the real numbers R (i.e., functions f : R -> R of the form f(x) = ax +b.). Also, it's not clear what "evenly spaced" means, for the linear order (N, <) of the natural numbers isn't "evenly spaced" (though it is discrete). There is no metric defined on it. But they must mean some *metric* structure? (The definition of "evenly-spaced" discrete point set I think would be that the points all lie on a line, and the smallest steps are congruent.)

    When they say "space" I'm still confused. So, let's assume they mean the ordered real numbers (R, <). But they might mean Euclidean 3-space, or maybe the 1-dimensional real line construed some other way.

    It may be that people's minds do, for some reason, attribute extra properties to, say, (N, <). For example, it might be "visualized" as embedded in (R, <) by inclusion, along with the natural metric on (R, <), which induces a metric back onto N. Namely, d(x,y) = |x-y|. Then, under an embedding of (N, <) into (R, <) with this metric, the "distance" between n and n+1 is always the same (d(n, n+1) = 1, for all n). But perhaps people think of (N, <) as being embedded into (R, <) not by inclusion, but by some other function, call it j: (N, <) -> (R, <). Then, if I understand what they're saying: j(n) is approximately a log(n) + b. This seems eminently plausible.