The origins of abstract concepts:  Natural number


By Elizabeth Spelke (Harvard)


January 2016


During the week of 19 January 2016, Harvard cognitive psychologist Elizabeth Spelke gave lectures and seminars at the departments of psychology at Uppsala University and Karolinska Institute, focusing on the origins of knowledge of number. She also gave a lecture at YBC gymnasium.


Liz Spelke is an internationally recognized cognitive scientist, particularly well-known for her research on infants and young children, and in recent years she has had an important role in the debate on cognitive differences between men and women. She defends the position that there is no scientific evidence of any significant disparity in the intellectual facilities of males and females. Liz is the author of over 200 hundred scientific papers on how children learn about objects, numbers and how they interact with others.




YBC Gymnasium, Nacka.


Karolinska Institute, Department of Clinical Neuroscience, Nobels väg 11 (Rockefeller).


Uppsala University, Department of Psychology.


At Uppsala University on 21 January, prior to her talk, the department staff gave Liz a lab-tour that included many of the projects they run in the lab. There she interacted with 10-15 members of the Uppsala Child and Babylab. Following her talk there were meetings with key projects that overlap with her interests. She met with two projects funded by the Wallenberg Foundation, ”The Social Foundation of Cognition” and ”Infants' sense of number - Investigating Cognitive Mechanisms and Training Interventions for Learning Mathematics Early in Life”. Both meetings included a wide range of participating researchers, including assistants, PhD students and permanent research staff.


Lectures Topic at Karolinska Institute (20 Jan) and Uppsala University (21 Jan) was:


The origins of abstract concepts:  Natural number


Liz Spelke was lecturing to a full audience, consisting of approximately 80 undergraduates, graduate students, and faculty. Her talk on the development of mathematical thinking and on the possibilities of training the basis for such, provided both students and faculty with an excellent background and introduction to this area, but also on the latest finding in the research field. A specific focus in this talk was on the plasticity of mathematical thinking, in which she described the preliminary findings of a large-scale training study in which the approximate number system was in focus for the intervention. The presentation of both positive findings and limitations of the training program provided interesting insights into the transition from basic research to applied value. Liz Spelke’s talk kept the audience fully attended.




The natural numbers may be our simplest system of abstract concepts: two axioms, together with some logic, suffice to generate all of them. Natural number concepts also are extremely useful: it is hard to think of any product of our culture, from measurement to money to mathematics, that does not depend on them.


Moreover, natural number concepts have been richly studied over the last century, beginning with the pioneering research of Piaget and accelerating with every successive decade. Nevertheless, basic questions concerning the origins and nature of these concepts continue to be debated. Is the system of natural number part of our innate endowment, or do we construct it as children? Are these concepts unique to humans or shared by other animals? Are they universal across all human cultures, or accessible only to human groups whose ancestors discovered them and wove them into the groups' contemporary language and cultural practices?


Today, theories of the development of natural number tend to cluster around two sets of answers to these questions. On one view, the system of natural number is innate in humans, shared by other animals, and universal across cultures. On a second view, the system is learned by children, unique to humans, and variable across cultures. Here I argue for a third view. Spelke suggests that natural number concepts emerge over the course of human development and are unique to humans: in this sense, they are not innate. Nevertheless, natural number concepts also are universal across humans and depend on three innate, early emerging cognitive systems.




Spelke’s talk focused on these three systems. First, humans and other animals have a core system of number: an innate sense of approximate numerical magnitudes. With this system, we can compare sets of objects or events on the basis of their relative numerosity, we can match numbers of visible objects to approximately equivalent numbers of sounds, and we can add, subtract, multiply and divide numbers with approximate precision.


Second, humans and other animals have a core system of naive physics, by which we can track up to three objects in parallel and follow their interactions. This system serves to represent objects whether they are visible or hidden, to establish relations of one-to-one correspondence between two small sets of objects, and to make sense of events in which a single object joins or is removed from such a set.


Third, humans have as a species the unique capacity to learn a productively combinatorial natural language. As children learn their native language, they master the meanings of the smallest number words by combining information from the first two systems. Then they learn to combine number words so as to formulate expressions that designate larger numbers. Because the combinatorial rules of natural language are productive, they can serve to generate the infinite series of natural numbers.


In her talk, I Spelke discussed research that bears on this view and its two more popular rivals. She also considered future research directions that might serve to distinguish between them definitively.


Lecture at YBC Gymnasium, Nacka


19 January 2016, 09.00 to 10.30


Topic: Origins of Knowledge


Human capacities to sense the world and move about within it are similar to those of many other animals, but our capacities to understand and transform our surroundings are strikingly different. Our knowledge of the world is organized around abstract concepts: physical concepts like gravity and mass; social concepts like rights, beliefs and intentions; mathematical concepts like parallel and pi. Our most useful abstract concepts develop over time, replacing earlier concepts that are less powerful. As far as we know, we are the only species that has these concepts.


These phenomena raise three general questions:


(1) What makes humans so much smarter than other animals?


(2) Where do our abstract ideas come from?


(3) How do we gain new concepts that change our picture of the world?


Philosophers and scientists have pondered these questions for millennia, and they continue to be debated. Spelke suggested that answers to all three questions lie within our collective reach, and will come from research with two properties. First, the research will be interdisciplinary: Converging, systematic studies, bridging the fields of cognitive science, neuroscience, and computer science, will shed light on the nature of human intelligence. Second, the research will focus on the origins of knowledge, probing the cognitive capacities of human minds, human brains, and intelligent machines as they first begin to learn about the world.


Spelke’s own contribution to this interdisciplinary work focuses on the minds of human infants. In her talk, she considerd infants' developing knowledge of objects and their mechanical interactions, of people and their intentions and desires, and of number and geometry. She also discussed how cognitive scientists gain insight into what infants know and  reviewed some key findings that shed light on the nature of infant minds.

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