The Cognitive Science of MathematicsRon Hopkins

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.

WHAT IS THE ORIGIN OF MATHEMATICAL IDEAS?

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.

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.

HOW DO WE STUDY MATHEMATICAL IDEAS?

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.

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

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.

A simple way of understanding this theory is this: cognition involves not only the brain but also information provided by the central nervous system.

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.

Embodied Cognition: “Cognition is embodied

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)

The Brain’s Innate ArithmeticEmpirical Investigation into Mathematical Ideas

INNATE MATHEMATICAL ABILITIES

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.

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…

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.

SUPRAMARGINAL GYRUS AND ANGULAR GYRUS

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.

INFERIOR PARIETAL CORTEX

Later research after 2000: Neural correlates of relational reasoning and symbolic distance

effects in the parietal cortex (Hinton, Dymond, von Hecker, Evans; 2010)

Numerosity and Symbolic Thinking demonstrated in the inferior parietal cortex (Coolidge and Overmann, 2012)

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.

INFERIOR PARIETAL CORTEX

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?” 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)

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)

NEURAL MODELS

Cited in the text is the work of Terry Regier, who proposed the following neural model of conceptual metaphors:

1. Topographic Maps of the visual field are needed in order to link cognition to vision.

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.

4. Map comparisons, requiring neural connections across maps, are needed.

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.

“Ideas do not float abstractly in the world. Ideas can be created only by, and instantiated only by in, brains.” (Lakoff and Nunez, 2000)

The authors contend that ideas originate

within the brain and are therefore only

understandable via empirical methods.

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…

THE MATHEMATICAL BRAIN

Mathematical abilities are not exclusive to one particular region of the 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.

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.