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COGS 152

Cognitive Foundations of Mathematics

eπi + 1 = 0

Some questions …

!  What is Mathematics? !  What is its nature? !  What is so special about mathematics? !  Can we study these questions

scientifically?

Mathematics: the Queen of the Sciences (… but, is it?)

"  Some (implicit) principles of modern science: "  “The book of nature is written in mathematics” (Galileo) "  Mathematics: the “universal language” "  If you can put it in mathematical terms, do it "  Ultimate argument or proof is stronger when formulated

mathematically "  The more a sc. discipline formulates facts and theories

mathematically, the better, … “Hard sciences” "  Education, evaluation: mathematics, always there

Mathematics: the Queen of the Sciences (… but, is it?)

Plaque carried by Pioneer 10

What is so special about Mathematics?

Mathematics: An amazing conceptual system! "  Extremely precise "  Highly effective "  Symbolizable "  Generalizable "  Stable across time and communities

"  (Temptation to believe that it is shared with “extraterrestrials”!) "  Intrinsically abstract

"  Its very subject matter lies beyond experience (e.g., Euclid’s point, Actual infinity—infinitesimals, points at infinity, transfinite numbers, etc.)

"  Verification/falsification is not empirical (proof!) "  A belief (conjecture) must be proved (accepted body of knowledge)

Mathematics, an amazing(!) conceptual system

… which also has many (apparently) arbitrary facts (i.e., not grounded in nature)

"  Why the multiplication of negative numbers yields positive numbers?

"  Why is the empty set a subset of every set? "  Why is 0.9999… actually equal to 1? "  Why is 0! = 1? "  … etc.

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Usually answers in (pure) Mathematics are dogmatic

"  formal definitions "  axioms "  rules, algorithms "  etc.

! The validation of those answers is provided by proof, not by meaning ! Proof determines that something is true but not necessarily why it is true

Benjamin Peirce on eπi + 1 = 0

« Gentlemen, that is surely true, it is �absolutely paradoxical; we cannot�understand it, and we don’t know �what it means. But we have proved�it, and therefore we know it must�be truth. »

Harvard mathematician, astronomer (1809-1880)

What is Mathematics? The question of the nature of mathematics has been studied … mainly philosophically, and mathematically (formal logic, metamathematics)

"  Platonism "  Formalism "  Logicism "  …

"  Naturalism

What is Mathematics?

"  Dedekind "  Cantor "  Hilbert "  Poincaré "  Weyl "  ...

Limitations: "  subject matter "  introspection

Moral: "  purely philosophical

inquiry and introspection give a limited picture

The question of the nature of mathematics has been studied … mainly philosophically, and mathematically (formal logic, metamathematics)

Our approach: The question(s)

"  Cognitive Sciences "  Conceptual systems "  Cognitive semantics "  Psycholinguistics "  … Neuroscience

"  What is the nature of the human conceptual system called Mathematics?

"  What makes Mathematics possible?

"  A scientific, but ‘carefully’ reductionistic approach

Concepts, meaning, idealization, abstraction, and the human animal

"  Naturalizing the problem "  Living systems: Resources are limited

"  “recycling” "  New stuff with old parts

"  Concepts, ideas, meaning, minds are the result of biologically viable living cognition "  Mathematical concepts/ideas are not an exception "  NOT TAKING ANY MATH FACTS for granted

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A common picture in science …

Dynamical Systems

Differential Equations

Chaos Theory

Bifurcation Theory

Trigonometry Atractors

Probabilities Instability

Physics, Chemistry, etc.

Taking as given: Equations, numbers, operations, axioms, …

… Formalization and Quantification

A common picture in science …

Dynamical Systems

Differential Equations

Chaos Theory

Bifurcation Theory

Trigonometry Atractors

Probabilities Instability

Mind, Cognitive Phenomena

Taking as given: Equations, numbers, operations, axioms, …

… Formalization and Quantification

Dynamical Systems

Differential Equations

Chaos Theory

Bifurcation Theory

Trigonometry Atractors

Probabilities Instability

Mind, Cognitive Phenomena

Taking as given: Equations, numbers, operations, axioms, …

… Formalization and Quantification

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A common picture in science … The Cognitive Sciences OF Mathematics

Science and “the (human) mind”

Philosophy Neuroscience

Linguistics

Education

Psychology

Anthropology Computer Science

Memory

Language

Attention

Perception Learning

“Mind”

Thought

Reasoning Communication

Concepts