Presentation given for Foundations of Neuroscience course. University of West Georgia, Fall 2013

1. The Cognitive Science of
Mathematics
Ron Hopkins

2. Background
 The central thesis of this presentation is taken from Where Mathematics
Comes From, written by cognitive scientists George Lakoff and Rafael E.
Núñez.
 For this presentation I will be presenting a brief introduction into the
cognitive science of mathematics. I will be focusing on the author’s central
thesis regarding neuroscience and the biological origins of mathematical
ideas.
 Note: Their book was written in 2000 when research into the biological origins and
mechanisms of ‘numerical reasoning’ was in its infancy. So in addition to their original
cited sources from their text, I have referenced later research which supports their central
thesis.

3. WHAT IS THE ORIGIN OF
MATHEMATICAL IDEAS?
At the time of this book, there was no
discipline for the (cognitive) analysis of
mathematical ideas. The book was written to
propose the creation of such a discipline: the
cognitive science of mathematics.
 Mathematics had been previously
viewed as the ‘epitome of precision.’
 We employ symbols in calculations which
then allow us to create proofs.
 These proofs essentially claims to valid
and logical conclusions.
 However, symbols are not inherently
meaningful. They are basically
‘signposts’ to ideas.

4. HOW DO WE STUDY
MATHEMATICAL IDEAS?
Thesis: If mathematics involves the
manipulation of symbols then the intellectual
content of mathematics is not found in the
application of mathematics. Instead, it will be
found in the realm of human ideas.
“Mathematics may be defined as the subject where we
never know what we are talking about, nor whether
what we are saying is true.” – Bertrand Russell, prominent
philosopher of mathematics
 By itself, mathematics cannot
empirically study human ideas.
 Human cognition is a distinct and
separate field of study from
Mathematics.

Therefore, to understand mathematical ideas
we must apply the science of mind.
 Previously, mathematics was defined as
‘that which mathematicians do.’
 Accordingly, mathematical ideas were
simply the ideas that mathematicians
have consciously taken them to be.
 But to the cognitive scientist, human
ideas are not so simple.

5. 
HOW DO WE STUDY ‘IDEAS’?
According to the authors of this
book, ‘human ideas are to a significant
extant, ground in sensory-motor
experience.’

This theory is consistent with embodied
cognition, a contemporary and popular theory
regarding multiple sources of input contributing
to human cognition.

Embodied Cognition: “Cognition is embodied
A simple way of understanding this theory is
this: cognition involves not only the brain but
also information provided by the central
nervous system.
when it is deeply dependent upon features of
the physical body of an agent, that is, when
aspects of the agent's body beyond the brain
play a significant causal or physically
constitutive role in cognitive processing.”
(Stanford Encyclopedia of Philosophy)

Abstract human ideas make use of cognitive
mechanisms such as conceptual metaphors
that import modes of reasoning from
sensory motor experience.

Therefore, the nature of human ideas are always
the subject of empirical questions. Having
establish this, the authors contend that
mathematics as know it arises from the nature
of our brains and our embodied experience.

6. Empirical Investigation into Mathematical Ideas
The Brain’s Innate Arithmetic

7. INNATE MATHEMATICAL
ABILITIES
Research strongly suggests that we are born
with the ability to perform rudimentary
arithmetic. (In this context, arithmetic is
defined as calculations concerning simple
quantities)
For those curious as to how this was done…
 At 3-4 days, babies can discriminate
between collections of two and three
items.
 By 4 and a half months, a baby can
tell that one added to one is two and
two minus one is one.
 Infants can later tell that two plus one
is three and that three minus one is
two.

8. INNATE MATHEMATICAL
ABILITIES
 The key principle of their developing
theory involved studies which
suggested that:

Babies appear to use mechanisms more
abstract than mere object location.

Number or quantity of objects was more
important to the babies than object identity.
 In sum, the mechanism for
mathematical ideas and reasoning
allow for the consideration of abstract
concepts rather than merely
responding to objects. This is where the
critical concept of conceptual
metaphors becomes important.

9. 
The supramarginal gyrus and the angular
gyrus are found in the posterior and
superior side of the temporal lobe. Adjacent
to Wernicke’s area, this section of the brain
is believed to be a possible source for our
mathematical ability. Consider the following:

The supramarginal gyrus involved in
phonological and articulatory processing of
words.

The angular gyrus is involved in semantic
processing.

The neurons in this area are very well
positioned to process the phonological and
semantic aspects of language which allows
for the identification and categorization of
objects.
SUPRAMARGINAL GYRUS AND
ANGULAR GYRUS

10. 
Later research after 2000:



Numerosity and Symbolic Thinking demonstrated in the
inferior parietal cortex (Coolidge and Overmann, 2012)

INFERIOR PARIETAL CORTEX
Neural correlates of relational reasoning and symbolic
distance effects in the parietal cortex (Hinton, Dymond, von
Hecker, Evans; 2010)
Developmental Changes in Mental Arithmetic correlates with
increased functional specialiization in the left interior parietal
cortex (Rivera, Reiss, Eckert, and Menon; 2005)
At the time of this book’s publication the amount of
evidence supporting the theory of mathematical
ability originating in the inferior parietal cortex was
minimal.

Cited was an example of patients with epilepsia arithmetices.
This is a rare form of epilepsy in which seizures are occur
when an individual attempts to perform mathematical
calculations. EEG studies on these patients showed abnormal
rhythmic discharges in the inferior parietal cortex.

Also, patients with lesions in the inferior parietal cortex
demonstrate knowledge of other sequence based knowledge
(i.e, the alphabet and the days of the week) but could not
recall missing numbers in a consecutive series.

11. 
The inferior parietal cortex is located
anatomically near the connections for
auditory, visual, and touch come together.
It’s a highly associative area so why would
this be where mathematical cognition is
found?

Or to quote Dehaene: “What is the
relationship between
numbers, writing, fingers, and space?”
INFERIOR PARIETAL CORTEX

Numbers are connected to fingers because
children learn to count on their fingers.

Numbers are connected to writing because they
are symbolized by written numerals.

Numbers are related to space in various ways
such as considering how many objects will be
able to occupy an empty space.
(Dehaene, 1997)

12. 
MATHEMATICAL COGNITION
Accepting the theory that mathematical
ability originates within the inferior parietal
cortex, then this suggests that we have a
part of the “brain specialized for a sense of
quantity.” (Dehaene, 1997)

The authors propose that the involvement of
the inferior parietal cortex in mathematical
cognition implies that the conceptual
mechanisms involved in mathematical reasoning
are embodied conceptual mechanisms.
Simply stated, conceptual mechanisms can be
reduced to primitive image schemas. These
schemas provide a special cognitive function of
being both perceptual and conceptual at the same
time.
“As such, they provide a bridge between language
and reasoning on the one hand and vision on the
other.” (Lakoff and Nunez, 2000)

13. 
NEURAL MODELS
Cited in the text is the work of Terry
Regier, who proposed the following neural
model of conceptual metaphors:


2. A visual mechanism fills in from the ‘outside world’ to
the ‘inside map’ we create. The topological properties of
the Container schema are created.

3. Orientation sensitive cell assemblies found in the
visual cortex are employed by orientation schemas.

“Ideas do not float abstractly in the world.
Ideas can be created only by, and instantiated
only by in, brains.” (Lakoff and Nunez, 2000)
1. Topographic Maps of the visual field are needed in
order to link cognition to vision.
4. Map comparisons, requiring neural connections
across maps, are needed.
The authors contend that ideas originate
within the brain and are therefore only
understandable via empirical methods.

All of the above are considered to be the
biological necessities for the creation of an
image schema.

Why is this important? Because according to the
authors, the embodied mind model of mathematical
ideas utilizes conceptual metaphors of containment.

14. 
THE EVOLUTION OF THE
MATHEMATICAL MIND
The neural circuitry involved in
mathematical reasoning is also involved in
other evolved processes. It appears to be
something which is not exclusive to other
processes but rather makes use of our
adaptive capacities.

So, mathematical ideas are produced by
cognitive processes which have evolved just like
any other processes.

For example, the prefrontal cortex is involved in
complex arithmetic calculation as well as
abstract thinking and thought. When some
patients have lesions in this area they become
incapable of performing complex sequential
operations such as multiplying two numbers
together. This is notable for multiplication of the
variety used in basic multiplication tables does
not utilize rote memory!

This brings up a very interesting point…

15.  Mathematical abilities are not
exclusive to one particular region of
the brain.
THE MATHEMATICAL BRAIN
 Basic arithmetic is separate from rote
memorization of addition and
multiplication tables.
Here is a two question math test:
1. 7 x 2 =
2. (2x - 4) = ¼ (1x+15) [Solve for x]
 In answering those questions you would
be using completely different parts of your
brain.
Question 1 involves rote memorization which appears
to be subcortical and associated with the basal
ganglia.
Question 2 involves conceptual thinking and would
most likely involve the inferior parietal cortex.

16. So What?
This presentation has concerned a small portion of Lakoff and Nunez’s proposal for a
cognitive science of mathematics. This is essentially a form of mathematical idea
analysis which utilizes empirical research methods in order to understand the
biological origins of mathematical ideas.
Why does this matter?
In terms of mathematics education, a thorough understanding of how we think and
reason with numbers may provide us with an entirely new means by which to facilitate
better instruction in the classroom.
In terms of general use, quantitative reasoning is a critical skill not just in our modern
society but in evolved beings. Some animals species have demonstrated a capacity to
reason in regards to quantity and this suggests that numerosity is a necessary and
evolved mental capacity. (In fact, some have theorized that our ability to reason with
numbers preceded our ability to use language!) Therefore, it is an important skill with
survival value. The ability to think and reason with numbers is not merely a trivial skill
or an optional mode of thought only for those engaged in careers which utilize
numbers. It is an innate ability for us all.