Lecture 5 & 6: Numbers fintb

breakdown of key concepts and areas to focus on:

I. Core Concepts of Numbers

  • What is a number? Understand that a number isn't an inherent property of physical objects themselves, but rather a property of a set of objects. The definition of the set is crucial for determining the number.

  • Sets: A set is a collection of things (dogs, ideas, etc.). Any part of the physical world can be described by infinite sets.

  • Human Perception: Humans selectively group individuals together mentally, which then allows us to apply a specific number.

II. Origins of Our Sense of Numbers

  • Two Competing Theories:

    • Empiricism (John Stuart Mill): We learn about numbers through observation of the world.

    • Nativism (Immanuel Kant): Our sense of numbers is innate.

  • Types of Evidence to Test These Theories: Be familiar with these categories

    • Developmental studies

    • Neuropsychological studies

    • Cross-cultural studies

    • Cross-species (comparative) studies

III. Cross-Cultural Findings

  • Counting Systems:

    • Almost all cultures have them.

    • Defined by:

      • Labels used in a stable order

      • One-to-one correspondence

      • The last label represents the total

    • Example: Oksapmin tribe in Papua New Guinea (know generally what their system involves).

  • Magnitude Representation:

    • The Pirahã tribe:

      • Lack a counting system and number words.

      • Have words that sort of mean "one," "two," and "many."

      • Have difficulty with exact numbers above 2 or 3.

      • Conclusion: They have an approximate sense of numerical magnitudes even without linguistic number terms.

  • Summary of Cross-Cultural Studies:

    • Most cultures have counting systems for exact numbers.

    • Without language, humans have a sense of approximate numerical magnitudes.

    • Speculation: Language with a counting system may enable us to represent exact numbers.

IV. Animal Studies

  • Two Main Questions:

    • Can animals represent numbers (exact or approximate)?

    • Can animals compute over numbers?

  • Key Findings:

    • Rats (Tunnel Experiment): Can learn to go to a specific tunnel (e.g., the 5th tunnel). They can generalize this to new mazes, suggesting a sense of ordinal position.

    • Rats & Pigeons (Key Pressing):

      • Must press a key N times for food.

      • They tend to "play it safe" and press more times than required.

      • The "insurance factor" is proportional to N.

      • Conclusion: Representing approximate value, not exact number.

      • Insurance Factor Influences: The cost of being wrong (bigger cost = more insurance).

    • Rats & Pigeons (Discrimination):

      • Must tell if the number of stimuli (X) is the same as or smaller than a trained number (Y).

      • Performance depends on the ratio of X to Y, not the absolute difference.

      • Conclusion: Representing approximate numerical magnitudes, not exact numbers (like humans discriminating magnitudes).

    • Honeybees: Use landmarks to find food (right place, wrong number of landmarks vs. wrong place, right number of landmarks).

    • Alex the Parrot: Could answer questions about the number and color of objects.

    • Dogs (Addition): Violation-of-expectation experiments suggest they notice incorrect outcomes of simple addition (1+1=1 or 1+1=3).

    • Chicks (Addition/Subtraction): Can find a larger group of "broodmates" hidden behind a barrier.

    • Monkeys (Addition): Looking-time experiments suggest they represent the approximate sum.

    • Ducks: Can quickly distribute themselves to locations with different amounts of food, demonstrating an ability to compute ratios.

  • How Animals Represent Numbers of Things:

    • Approximate magnitudes

    • Object Tracking:

      • Visual system represents individual objects in working memory.

      • Limited capacity (adults = ~4).

      • Each object gets its own "object file."

  • Magnitudes vs. Object Tracking: Understand the difference.

    • Magnitudes: "This many bones is behind the screen"

    • Object Tracking: Relies on individual object representation.

V. Key Distinctions

  • Approximate vs. Exact Number Representation: This is a recurring theme.

  • Magnitude Representation: Understanding "more," "less," and proportionate differences.

  • Object Tracking: Limited to a small number of visually perceived objects.

VI. Overall Conclusions

  • Humans and animals can represent approximate numbers using a magnitude system.

  • Humans (and possibly some other species) can represent small, exact numbers of physical objects using object tracking.

  • Only humans (as far as we know) can represent exact numbers of any kind of thing, using a counting system.

I. Key Studies & Findings (Babies & Brains)

  • Wynn (1992): Addition/Subtraction with Small Numbers:

    • Violation-of-expectation experiments (e.g., Mickey Mouse doll).

    • Showed that infants seem to have some understanding of basic addition and subtraction with small numbers.

    • Important Point: Consider the counter-argument: are babies thinking "Mickey & Other Mickey" or "Two-ish Mickeys?"

  • Xu & Spelke (2000): Discriminating Larger Numbers:

    • 6-month-olds.

    • Key finding: Babies can discriminate between large numbers, but only when the ratio is high enough (2:1 works, 3:2 does not).

    • Example: They can distinguish 8 from 16, or 16 from 32, but not 8 from 12, or 16 from 24.

    • Ratio is Key: Performance depends on the ratio between the numbers being compared, suggesting an approximate number system.

  • Izard et al. (2009): Are Babies' Larger Numbers Abstract?

    • Studied newborns.

    • Focus on abstract representation of number.

  • McCrink & Wynn (2007): Addition/Subtraction with Larger Numbers:

    • Violation-of-expectation.

    • Infants look longer at unexpected outcomes (e.g., 5 + 5 = 5 or 10 - 5 = 10).

    • Since the numbers are larger than the object-tracking limit, this suggests approximate representations of number.

  • Feigenson et al. (2002): Cracker Choice Task:

    • Can babies discriminate small numbers of items?

    • Focus on small number discrimination abilities.

  • Feigenson & Carey (2003):

    • Follow-up on small number addition/subtraction (hiding objects in boxes).

    • Expected Empty vs. Remaining Trials.

II. Summaries of Baby Abilities

  • What Babies CAN Do:

    • Represent approximate large numbers (magnitudes).

      • Objects, sounds, events, actions.

      • Add, subtract, compute ratios.

    • Represent small, exact numbers (object files).

      • Objects they can see/hear.

      • Add, subtract.

      • Discriminate if high ratios (1:2 is okay, 2:3 is not).

      • Limited to small numbers (<4).

  • What Babies CANNOT Do:

    • Represent and compute with large exact numbers (e.g., 9, 72, 1239).

    • Knowledge about numbers (even/odd, prime numbers).

    • More advanced operations (multiplication, division, etc.).

    • Non-natural numbers (negatives, fractions, irrationals).

III. Where Does the "Rest of Math" Come From?

  • Hypothesis: Language

    • Counting system.

    • Number words.

    • Verbally stored number facts.

  • Evidence 1: Bilingual Speakers: Do mental math in the language in which they learned math in school.

  • Evidence 2: Neuropsychological evidence (brain lesions).

IV. Neuropsychological Findings (Brain Lesions)

  • N.A.U. (Dehaene & Cohen, 1991): Left Frontal Lobe Damage:

    • Can answer simple math questions correctly (2+2=4).

    • But cannot answer complex math questions, or provide any number knowledge.

    • Key finding: Retains approximate number knowledge.

    • Conclusion: Left Frontal Lobe holds exact number info, but not approximate number knowledge.

  • Mr. M (Dehaene, 1997): Parietal Lobe Damage:

    • Cannot say which of two numbers is larger/smaller.

    • Cannot say what lies midway between two numbers.

    • Key finding: Can still state things learned in school (times tables, days in a year). Possesses exact number knowledge, but lacks approximate number sense.

    • Conclusion: Approximate number system is in the Parietal Lobes!

V. Big Summary

  • All cultures have a sense of approximate magnitudes.

  • Animals, infants, and adults have an approximate number sense that supports some sophisticated computations (addition, subtraction, ratios).

  • Specialized brain circuitry for the approximate sense (shared across species and cultures).

  • Separate, language-based brain circuitry for precision math (which depends on language).

VI. Three Cognitive Systems that Represent Number

  1. Language:

    • Labels of our counting system.

    • Represents exact numbers.

    • Small and large numbers.

    • Number knowledge we've learned (times tables, etc.).

  2. Magnitude Representations ("Accumulator" mechanism):

    • Represents approximate values.

    • Small and large numbers.

    • Of objects, actions, sounds, events, etc.

  3. Visual Object Tracking:

    • Represents exact number.

    • Only for small numbers (up to 3 or 4 for adults).

    • Only represents numbers of objects.

VII. Key Takeaways for the Test

  • Distinguish between approximate number representation (magnitudes) and exact number representation (object tracking and language-based systems).

  • Understand the limitations of each system.

  • Know the key studies (Wynn, Xu & Spelke, Dehaene & Cohen, etc.) and their major findings.

  • Consider the "extraordinary claims require extraordinary evidence" argument in the context of infant number abilities.

  • Understand the role of language in developing more advanced numerical abilities.

  • Know the brain regions associated with approximate vs. exact number processing (parietal vs. left frontal).