13. Number

What this lecture covers

The lecture asks how humans and animals represent number and quantity, and how that ability is implemented in the brain. It sits in a course on the human brain, following earlier lectures on perception and spatial cognition, and leans heavily on behavioral evidence before turning to neuroscience. After watching, you can explain what the approximate number system is, why it follows Weber's law, and what evidence links it to the parietal lobe in both humans and animals.

The lecture opens by asking why number matters: for foraging, group comparisons, and abstract reasoning in animals, and for engineering, science, and everyday tasks in humans. It builds from this behavioral case toward brain data collected from patients, human neuroimaging, and single-neuron recordings in monkeys.

Key ideas

• Number sense: the ability to represent large numerical magnitudes approximately, without verbal counting, shared by humans, infants, and many animal species.
• Weber's law: the finding that the ability to discriminate two quantities depends on their ratio, not their absolute difference; a fundamental property of perception in general.
• Approximate number system (ANS): the term for this ratio-dependent, abstract sense of quantity, measurable behaviorally with tasks such as comparing dot arrays.
• Abstractness: number representations generalize across sensory modality (dots versus tones) and across space and time, and can support approximate addition and subtraction without counting.
• Developmental dyscalculia: a specific deficit in number sense, independent of IQ or other abilities, analogous to developmental prosopagnosia for faces.
• Predictive value: early speed and precision at approximate number tasks in kindergarten predicts arithmetic ability years later, independent of general IQ measures like Raven's matrices.
• Intraparietal sulcus (IPS): a brain region, particularly its horizontal segment, repeatedly implicated in number and magnitude processing, though the lecture argues it is not exclusively devoted to number.

Walkthrough

Why number matters, in animals and humans (32)

The lecture surveys everyday and evolutionary uses of number: humans use it for basic tasks and to build engineering and science, while animals use quantity judgments for foraging, joining larger groups for safety, assessing rival group sizes, and even in male frog calls that escalate by adding sound components. Stanislas Dehaene's claim that number sense reflects an innate, biologically determined, domain-specific system is introduced as a strong hypothesis the lecture will later qualify.

Demonstrating number sense and Weber's law (454)

Using live in-class demonstrations with dot arrays, the lecture shows that people can judge which of two arrays has more items without counting, but performance degrades as the two quantities get closer together. Plotting accuracy against the ratio of the two numbers (rather than their difference) collapses the data onto a single curve, demonstrating Weber's law: discriminability depends on ratio, a property shared with judgments of brightness, weight, and loudness. The lecture notes that number is normally confounded with area and density in dot displays, and that careful experiments vary these separately across trials to rule them out as the basis for judgments.

Individual differences and predictive value (454)

Measured with the Weber fraction, number acuity shows large individual differences, improves until about age 30, and dissociates from other abilities in developmental dyscalculia, much as developmental prosopagnosia dissociates face recognition from general intelligence. A kindergarten dot-estimation task is shown to predict arithmetic ability measured years later, while not predicting non-numerical measures like Raven's matrices, suggesting the early measure taps something number-specific.

Symbolic numbers still invoke continuous magnitude (1866)

A class demonstration comparing single numerals to 65 shows that people respond more slowly as the numeral gets numerically closer to 65, even though this is a purely symbolic, trained task. This is used as evidence that trained adults still rely on an underlying continuous magnitude representation even for exact, symbolic numbers.

Cross-modal and operational abstractness (2020)

Experiments comparing dot arrays to sequences of tones show similar accuracy to within-modality comparisons, indicating the underlying number representation is not tied to a particular sense. Further demonstrations show people can approximately add or subtract these quantities, including across modalities, without literal counting, arguing that the system supports operations, not just comparison.

Evidence from infants and animals (2390)

Four-day-old infants look longer at visual arrays that match the number of sounds they just heard, showing cross-modal number matching from birth, though only for large ratio differences. The lecture then reviews animal evidence: a macaw and a chimpanzee ordering symbols, honeybees learning to add or subtract one from a remembered quantity and apparently recognizing zero, and rats that spontaneously combine two light flashes and two tones to choose a lever trained for four, without any prior training on that combination.

Brain damage, imaging, and the intraparietal sulcus (2799)

Two patients with acalculia show a double dissociation: one, with left parietal damage, is impaired at approximation and subtraction but not multiplication, while the other, with left temporal damage, shows the opposite pattern, suggesting rote memorized facts (like multiplication tables) and approximate magnitude representations depend on different brain systems. Neuroimaging work from Dehaene's lab identifies the horizontal segment of the intraparietal sulcus (HIPS) as active during calculation, but the lecture argues this region also responds to non-numerical magnitude and spatial tasks, including eye movements, citing a study that trained a classifier on leftward versus rightward saccades and found it could also distinguish subtraction from addition in the same region. A TMS study disrupting the left intraparietal sulcus impaired both symbolic and non-symbolic number judgments but not a control shape task matched for difficulty.

Single-neuron evidence in monkeys (2970)

Andreas Nieder's recordings from monkey parietal and frontal cortex identify neurons tuned to specific numerosities, such as a neuron that responds maximally to two dots and generalizes with a broad gradient to nearby quantities. These number neurons generalize across spatial arrangement, across time (sequences rather than static arrays), and across modality (tones as well as dots), and similar neurons have been found in monkeys before any training, suggesting the system is not purely learned.

Before you watch

• Review earlier lectures on functional specificity and domain-specific brain systems, since the lecture explicitly connects number sense to that framework.
• Recall the concept of double dissociation from patient studies covered earlier in the course, since it is used to interpret the two acalculia cases.

Check your understanding

1. What does Weber's law say about how number discrimination depends on the ratio versus the difference of two quantities?
2. Why does the lecture treat the double dissociation between two acalculia patients as informative about the brain organization of arithmetic?
3. What evidence suggests that trained adults still rely on an approximate, continuous representation even when comparing exact symbolic numbers?
4. How did researchers rule out simple explanations like "choose the larger option" in the honeybee addition and subtraction experiments?
5. Why does the lecture argue that the intraparietal sulcus is not a region exclusively dedicated to number?