Where Mathematics Comes From Seminar:Cognitive mechanisms of mathematic cognition Stefan Schneider April 18, 2011 Lakoff, G., & Núñez, R. E. (2000). Where Mathematics Comes From. New York: Basic Books, Inc.
George Lakoffhttp://georgelakoff.com/writings/books/Lakoff, G., & Johnsen, M. (2003). Metaphors welive by. London: The University of Chicago Press.● linking linguistics & cognitive science.● metaphors (“love is a partnership”). Have you ever had the idea yourself that most abstract concepts can be understood in terms of very basic intuitions?
Rafael Núñezhttp://www.cogsci.ucsd.edu/~nunez/web/links.htmlNúñez, R. E., & Freeman, W. J. (Eds.).(1999). Reclaiming Cognition: The Primacyof Action, Intention and Emotion. Thorverton,U.K. Imprint Academic.● for Embodiment (and against AI)● developmental psychology influence from Jean Piaget; educated in Switzerland
Indeed, where does math come from?platonic mathmathematical objectsand structures existindependent fromhumans
Indeed, where does math come from?platonic math non-platonic mathmathematical objectsand structures exist mathematics is aindependent from pragmatic humanhumans invention
The romance of mathematics● transcendence - existence of mathematics independent of humans, structuring the universe● mathematical truth as the gateway to transcendental truth● reasoning is logical, therefore mathematical● logic is transcendent, independent of humans, “disembodied”: therefore AI is possible
L & N in contrast● Theorems that human beings prove are within a human mathematical conceptual system.● All the mathematical knowledge that we have or can have is knowledge within human mathematics.● There is no way to know whether theorems proved by human mathematicians have any objective truth, external to human beings or any other beings.
TODO: Show how math cognitively develops(a project somewhat analogous to the axiomatizationof math, but searching for basic cognitive structuresand for mechanisms that develop more complicatedconcepts)-> Embodiment
Innate math● subitizing (up to 4)● innate arithmetic (up to 3)● estimate numerosity (size of collections)● similar in animals - argument that this really is possible without conceptual capabilities● Neural evidence (to which LN often refer to)
Ordinary cognition● it is not all about conscious reflection, but works to a large part independent from it● abstract “fancy” math rooted in normal cognition● image schemas; aspectual schemas; conceptual metaphor; conceptual blend
Image schemas● a conceptual primitive that appears to be universal e.g. “the book is on the table”: “on” is composed of orientational, topological and force-dynamic schemas● forms a gestalt● “Image schemas have a special cognitive function: They are both perceptual and conceptual in nature.” (31)● “complex image schemas like In have built-in spatial logics “ (31) (“self-evident”)● arguments that the visual system does conceptual processing
Aspectual schemas● the dynamic side, operations - “the structure of events”● e.g. the “source-path-goal schema” - “the principal schema concerned with motion”● has also internal spatial logic and built-in inferences● metaphorically - “fictive motion”: “The road runs through the woods”, “The fence goes up the hill”, and “two lines meeting at a point”, “a function graph reaching a minimum at zero”
conceptual composition[INTO and OUT-OF schema]
conceptual Metaphor● a central process in everyday thought” - remember L to be linguist● “abstract concepts are typically understood, via metaphor, in terms of more concrete concepts” (39)● “Many arise naturally from correlations in our commonplace experience, especially our experience as children.” (41)● neural argument: conflation, simultaneous activation, linking through strengthening of association
structure of metaphors“Each such conceptual metaphor has the samestructure.”A is BorB A[Example: STATES-ARE-LOCATIONS]
image schema inferences inherited“the logic of Container schemas is an embodied spatial logic thatarises from the neural characterization of Container schemas [sinceit] preserves the inferential structure of the source domain.” (44)“folk Boolean logic”, “which is conceptual, arises from a perceptualmechanism - the capacity for perceiving the world in terms ofcontained structures” (45)“From the perspective of the embodied mind, spatial logic is primaryand the abstract logic of categories is secondarily derived from it viaconceptual metaphor. This, of course, is the very opposite of whatformal mathematical logic suggests. It should not be surprising,therefore, that embodied mathematics will look very different fromdisembodied formal mathematics.” (45)
Metaphors introduce elements[love is a partnership]
Conceptual blends● “conceptual combination with fixed correspondences between source and target domain” [boat house / house boat]● a blend “has entailments that follow from these correspondences, together with the inferential structure of both domains” (49) - Gestalts again!
Abstractioncontinuous building through metaphor mechanismeventually makes college mathsL&Ns approach from the book→ metaphorical decomposition(exemplified on Eulers formula)
Wrap up● Innate math● Image & aspectual schemas● Metaphors ● A is B (B can be very basic) ● A inherits built-in logic of B● Blends● metaphorical decomposition
Questions / Discussion / Critique● Embodiment? ● What does that really mean, and how is it realized? Can we understand how it is realized in a functional way? (cf. Searles Chinese Room) ● What about the omnipresent tables of LN - they appear very formal.● Built-in inferences ● How are these computed?● Basic structures ● What mechanism generates such a basic structure as e.g. “modus ponens” in the CATEGORIES-ARE-CONTAINERS metaphor?