Visual Thinking in Mathematics: An Epistemological Study by Marcus Giaquinto (2007)

This book assesses the extent to which the popular belief that visual thinking in mathematics is useful only in a psychological, rather than epistemological way. Giaquinto examines instances where visual thinking has proved unhelpful such as peano’s space-filling curves and continuous nowhere-differentiable functions, and briefly discusses conventionalism, holistic empiricism and intuitionism as justifications for purely axiomatic systems. Hilbert once wrote “A theorem is only proved when the proof is completely independent of the diagram” in some notes on geometry, but then Hilbert was off on one for quite a while.

In the second chapter the author explores shape perception and concept-making wrt simple geometric shapes. The third chapter begins with the statement that

in having geometrical concepts we have certain general belief-forming dispositions that can be triggered by visual experiences; and if that happens in the right circumstances, the beliefs we acquire constitute knowledge.

The fourth chapter concerns geometric discovery by visualising, and takes an example similar to one used by Plato, a visual proof that a square a whose side is the length of the diagonal of a square b will have twice the area of that square, exploring how that belief could be arrived at through visualisation. This may be complemented by prior beliefs and inferential dispositions. It suggests that visual experience can be an a priori means of acquiring belief rather than simply evidential. The diagrams are not superfluous, as it is they that activate the relevant inferential dispositions in the reader.

Chapter five looks at diagrams in geometric proofs, moving from discovery to justification and investigating the view that diagrams, though often presented alongside proofs to aid comprehension, would make the argument vulnerable if it were a part of the argument itself (an interesting difference in priority here – understanding vs. rigour). The author examines the dangers in generalising from an example, and considers diagrams that work in parallel with symbolic expressions in a proof, or a generalising argument.

Number lines are the subject of chapter six, and looks at various psychological experiments and perception of magnitude. Evidence is found of a mental left-to-right number line, an interaction between cultural and innate endowments.

Visual aspects of calculation – this chapter looks at the evidential role of finger-counting and visual imagery in calculation. General Theorems From Specific Images use some dot pictures of indeterminate size to discover arithmetical truths and discusses the dangers of unintended exclusions in generalisations. The author argues that a spatial form can play the role of a variable to allow a specific image to bear general significance.

The creative heart of the discovery process lies in viewing a form in two ways at once.

The ninth chapter is on Visual Thinking in Basic Analysis, the most complex mathematics discussed, and gives some examples of discovery through diagrams.

Next is Symbol Manipulation, visual reasoning using numerals or variables, their relationships being visually assessed. If sufficiently formal the steps in a computation can be purely syntactic, and with a semantic justification, the performance thereof can deliver knowledge. This is potentially very interesting – the relationship of symbols to one another, how this aids understanding... Chapter eleven is about cognition of structure and extrapolation to infinite structures.

This guy hangs out with Stewart Shapiro, the one who talks about philosophy.

Giaquinto concludes by claiming that the algebraic-geometric contrast is actually more of a spectrum.  The epistemic value of visualisation is its ability to bring to attention new information.

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