The nature of mathematics

Why does it matter & cognitive approach

Why nature of mathematics matters

• Where do laws of arithmetic come from?
• Would aliens have lambda calculus?
• Why can mathematics explain the world?
• How do we make mathematics accessible?

Cognitive science of mathematics

The only mathematics we know or can know is
a brain-and-mind-based mathematics.

It is up to cognitive science and the neurosciences to (...)
apply the science of mind to human mathematical ideas.

Lakoff, Nunez (p1, xi)

Components of the analysis

Innate arithmetic
Babies have some mathematical capacities

Conceptual metaphors
Links concepts via neural conflations

Layering metaphors
Explain more abstract mathematical concepts

Innate arithmetic experiments

Metaphors

Cognitive reconstruction of mathematics

Metaphors are central to thought

One of the principal results in cognitive science is that abstract concepts are typically understood, via metaphor, in terms of more concrete concepts.

Lakoff, Nunez (p39)

Everyday mathematical understanding

Mathematical ideas (...) are often grounded in everyday experience.

Many mathematical ideas are ways of mathematicizing ordinary ideas, as when the idea of a derivative mathematicizes the ordinary idea of instantaneous change.

Lakoff, Nunez (p29)

Analysing mathematical metaphors

Grounding metaphors
Sets are like physical containers

Linking metaphors
Numbers as sets, i.e. \(\emptyset, \{\emptyset\}, \{\emptyset, \{\emptyset\}\}\), ...

Introduction of elements
Metaphors introduce concepts into target domain

Arithmetic

Metaphors for arithmetic

Metaphors for arithmetic

Arithmetic is like...

Arithmetic laws...
Come from physical experience!

Arithmetic is object collection

Linguistic examples
Add onions and carrots to the soup
Which is bigger, 5 or 7?

Equational properties
Adding A to B gives the same result as
adding B to A for object collections

Limitations of the metaphor
Zero in terms of collections?

Arithmetic is movement along a path

Linguistic examples
4.9 is near 5, result is around 42

Equational properties
Moving from A by B gives the same
result as moving from B by A.

Nice features
Explains zero and fractions well

Beyond arithmetic

Infinity, Booleans, sets and \(e^{\pi i} + 1 = 0\)

Other mathematical metaphors

Basic metaphor of infinity
Infinity as the end of iterative process

Algebra
Folk theory of essences, substance and forms

\(e^{\pi i} + 1 = 0\)
\(\pi i\) as rotation, \(e^x\) turns multiplication into addition

Basic metaphor of infinity

Adding concepts to target domain
Metaphor adds actual infinity

Explaining actual infinity
\(\infty\) as the end of an iterative process

Implications

What can we learn from cognitive science?

The Romance of Mathematics

• Mathematics is an objective feature of the universe
• It has absolute truths about any possible universe
• It characterizes the nature of rationality
• Mathematical truths are universal and absolute

The Romance of Mathematics

The Romance of Mathematics makes a wonderful story (...). It perpetuates the mystique of the Mathematician [as someone who is] more rational, more probing, deeper, visionary. (...) But sadly, for the most part, it is not a true story.

Lakoff, Nunez (p341)

Is mathematics independent of culture?

Everything in the universe has an essence (...). [S]ince Euclid (...)
essence can be given by a small number of obviously true postulates.

[The idea] that theories, like buildings, must have secure, solid, permanent foundations on which all else is built is at least as old as Aristotle.

It has been governing metaphor behind [Western] theories that
pretend to give an account of certain and absolute knowledge.

Lakoff, Nunez (p355-358)

Metaphors in computer science

Metaphors matter
The choice of metaphors affects what we can think

What metaphors we use?
What cognitive metaphors lead to \(\lambda\)-calculus, monads, etc.?

Summary

Would aliens understand \(\lambda\)-calculus?

Would aliens understand \(\lambda\)-calculus?

Metaphors for logic
Logic is derived from container schema.

Would aliens have containers?
Imagine gaseous universe that does not have "in".

Where mathematics comes from

The nature of mathematics
Human-based, not Platonic ideals

Constructed via metaphors
Grounding (physical experience) and linking

Important consequences
Metaphors change how we think & teach

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