1. 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.

2. Where Mathematics Comes From

3. George Lakoff
http://georgelakoff.com/writings/books/
Lakoff, G., & Johnsen, M. (2003). Metaphors we
live 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?

4. Rafael Núñez
http://www.cogsci.ucsd.edu/~nunez/web/li
nks.html
Núñez, R. E., & Freeman, W. J. (Eds.).
(1999). Reclaiming Cognition: The Primacy
of Action, Intention and Emotion. Thorverton,
U.K. Imprint Academic.
● for Embodiment (and against AI)
● developmental psychology influence
from Jean Piaget; educated in
Switzerland

5. Indeed,
where does math come from?

6. Indeed,
where does math come from?
platonic math
mathematical objects
and structures exist
independent from
humans

7. Indeed,
where does math come from?
platonic math
non-platonic math
mathematical objects
and structures exist mathematics is a
independent from pragmatic human
humans invention

8. 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

9. 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.

10. TODO:
Show how math cognitively develops
(a project somewhat analogous to the axiomatization
of math, but searching for basic cognitive structures
and for mechanisms that develop more complicated
concepts)
-> Embodiment

11. Outline
● Innate mathematical capabilities
● Cognitive unconscious & ordinary cognition
Concept development:
● Image schemas & Aspect schemas
● Conceptual composition
● Conceptual metaphor
● Conceptual blends

12. 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)

13. 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

14. 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

15. 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”

16. conceptual composition
[INTO and OUT-OF schema]

17. 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

18. structure of metaphors
“Each such conceptual metaphor has the same
structure.”
A is B
or
B A
[Example: STATES-ARE-LOCATIONS]

19. image schema inferences inherited
[CATEGORIES-ARE-CONTAINERS]

20. image schema inferences inherited
“the logic of Container schemas is an embodied spatial logic that
arises from the neural characterization of Container schemas [since
it] preserves the inferential structure of the source domain.” (44)
“folk Boolean logic”, “which is conceptual, arises from a perceptual
mechanism - the capacity for perceiving the world in terms of
contained structures” (45)
“From the perspective of the embodied mind, spatial logic is primary
and the abstract logic of categories is secondarily derived from it via
conceptual metaphor. This, of course, is the very opposite of what
formal mathematical logic suggests. It should not be surprising,
therefore, that embodied mathematics will look very different from
disembodied formal mathematics.” (45)

21. Metaphors introduce elements
[love is a partnership]

22. 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!

23. Abstraction
continuous building through metaphor mechanism
eventually makes college maths
L&Ns approach from the book
→ metaphorical decomposition
(exemplified on Eulers formula)

24. 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

25. Thanks !

26. 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. Searle's 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?