This is a really key text, referenced by many. It’s written by a linguist and a psychologist, interested in launching a cognitive science of mathematics to answer a question about how human beings comprehend infinity. I’ve been recommended a few papers criticising it – de Freitas and Sinclair have their own reservations – but I’ll come to that later.
The preface comments amusingly on “The Romance of Mathematics” – as having an objective yet disembodied existence, transcendent, above and beyond human mathematics, present in all of the universe and so shared by any intelligent beings.
The first chapter argues that the question of the existence of a platonic mathematics cannot be scientifically answered, and that mathematics known and used by a human mind must be a human mathematics, which must therefore be explored using cognitive science. They point to important developments: the embodiment of mind, cognitive unconscious and metaphorical thought, and outline some of the structure and aims/conclusions of the book.
Part 1, Chapter 1. A whole bunch of studies showing number perception in babies, subitising, very basic adding etc.
2 is a brief introduction to the cognitive science of the embodied mind, image schemas, (good bit about structures being topological ie. can be distorted), flow diagram of motion control used to reason about events/actions = aspect schema (source-path-goal schema → graph). Then they talk about conceptual metaphors, and looks at how the category schema gets its logics from the container schema – folk Boolean logic is conceptual arising from a perceptual mechanism. Metaphors can introduce new elements into the target domain, things can be conceptual blends.
…much of the “abstraction” of higher mathematics is a consequence of the systematic layering of metaphor upon metaphor, often over the course of centuries.
we are also interested in the all-too-common conception that mathematics is about calculation and about formal proofs from formal axioms and definitions and not about ideas and understanding. From the perspective of embodied mathematics, ideas and understanding are what mathematics is centrally about.
3 examines what arithmetic is – precise, consistent, symbolisable, calculable, etc, and says that the cognitive capacities needed in order to count up to four are grouping, ordering, pairing, memory, exhaustion-detection, cardinal-number assignment and independent-order capacities, beyond four also combinatoral-grouping capacity and symbolising capacity, and to characterise arithmetic operations and properties also metaphorising capacity and conceptual-blending capacity. They articulate the difference between grounding metaphors and linking metaphors, the former of which are discussed in this chapter, starting with arithmetic as object collection and the mapping providing commutativity, transitivity, linearity and all sorts of other stuff. With addition the two (numbers and collections) can be left separate but multiplication requires metaphoric blending, doing things to collections a certain number of times etc. The “zero collection” is an entity-creating metaphor. Arithmetic as object construction has another set of implications, ditto arithmetic as a physical segment (all of these are noticeable in natural language) and these can be used to define fractions. Physical segments are a bit more interesting (see measuring stick metaphor) because they are unidimensional and continuous, and the blend of source and target dimensions goes beyond the uni-directional mapping to create irrational numbers. Nice! Then there’s arithmetic as motion along a path and algebra as metonymy allows general proofs. There’s a great bit about zero (lack, destruction, smallness, origin) and one (individuality, unity, a standard, a beginning).
4 gives the implications of these four grounding metaphors of arithmetic. There are structural relationships between them, isomorphisms. The metaphor numbers are things in the world creates the Platonic realm idea and closure, that the end product of an operation would also be a number, which led to irrational and imaginary numbers and all sorts. Every natural number can be considered polynomial, calculation in different numerals would be different. Equivalent result frame gives the “axiom” of associativity. Motion along a path can help via rotation to make a blend that works for multiplying by negative numbers, further stretching is discussed. Arithmetic is summarised as coming from subitizing plus, primary experiences, conflations thereof, laws from entailments of grounding metaphors, and the equivalent result frames. It fits the world because that’s what it was made of, and thats why we think numbers “exist” and imaginary ones are OK, too. Positional notation lightens memory load. More advanced maths will also need linking metaphors, arithmetisation metaphors and foundational metaphors.
Part 2, Chapter 5. The folk theory of essences produced Greek axiomatic maths – essence is form.
What makes algebra a central discipline in mathematics is its relationship to other branches of mathematics, in which algebraic structures are conceptualised as the essences of other mathematical structures in other mathematical domains.
The axioms are thought of as causal. Algebra is conceptually mapped onto other domains using algebraic essence metaphors.
6 talks about classes and symbolic logic, beginning with the classes are containers metaphor and then the linking metaphor between classes and arithmetic that created Boolean algebra. These were then given symbolisation by a mapping between classes and symbols. Boole did not achieve the rigorous calculus that he thought he had or that he’s said to have. The propositional logic metaphor conceptualises propositions in terms of classes and the blend creates symbolic logic. They give four inferential laws of logic and show their source in container schemas.
7 is sets and hypersets (ooh). The sets are objects metaphor allows sets to be members of sets, the ordered pair metaphor allows sets to define relations, functions, etc, and the natural numbers as sets metaphor happens. Cantor’s metaphor (a redefinitional metaphor) uses the one-to-one correspondence thing to make all those infinities. Axiomatic Set Theory is that really formalised one, we get into ZFC here, and the axioms don’t particularly assume a container metaphor, but it’s often conceptualised that way, which would rule out self-membership (which can make a hot mess), which was ruled out by an extra axiom that von Neumann proposed. Hypersets map between Accessible Pointed Graphs and sets, and may or may not contain loops.
Part 3, Chapter 8 – infinity and beyond! (sorry)
the aspectual system is those event concepts, and some (like breathing) are not conceptualised as having completions. “Said it over and over” or “flew on and on” are from indefinite continuous processes are iterative processes metaphor. The target domain of the basic metaphor of infinity is the domain of processes without end – it’s the metaphor in which processes that go on indefinitely are conceptualised as having an end and an ultimate result. This final resultant state is unique and after all other states. The BMI produces a whole bunch of infinities. Processes can be static objects like series or can exist in time. BMI for enumeration has infinity be the highest number. BMI for mathematical induction makes a whole way of proving things.
9 is real numbers and limits – the BMI maps to numerals for natural numbers, same with decimal points, infinite polynomials map with infinite decimals, so it takes a metaphor (BMI) to make the real numbers (lol). Limits of infinite series (which seem to approach a number) have BMI and also a spatial thing going on, which work together. General limits, sums of infinite series, limits of functions, least upper bounds and infinite intersections of nested intervals are all discussed.
10 Transfinite numbers! Transfinite ordinals are mapped with the natural numbers, then one-to-one mappings for sequences are mapped with ordinal numbers.
11, infinitesimals. Infinitesimals interact with real numbers, so Monads are clusters of infinitesimals around each real number. Monads have no least upper bounds. Guess what, you can get infinitesimals by doing something cool with the BMI. Multiplication by an infinitesimal is zooming in is a brilliant metaphor for granular numbers. Granular numbers mean that calculus can become arithmetic, you don’t even need limits. Hyperreals. AN IMPORTANT POINT:
The study of infinitesimals teaches us somehting extremely deep and important about mathematics – namely, that ignoring certain differences is absolutely vital to mathematics! This idea goes against the view of mathematics as the supreme exact science, the science where precision is absolute and differences, no matter how small, should never be ignored.
Berkeley rejected it back in the day. People don’t like infinitesimals.
To be continued.