Epistemology of Visual Thinking in Elementary Real Analysis
Bibliographic Data
| ID | 8397694 |
|---|---|
| Authors | Marcus Giaquinto (University College London, corresponding author) |
| Year | 1994 |
| Volume | 45 |
| Issue | 3 |
| Pages | 789-813 |
| Publication date | 1994-09-01 |
| Peer Reviewed | Yes |
| Open Access | No |
| Type | ARTICLE |
| Venue | The British Journal for the Philosophy of Science (JOURNAL) |
| Journal identifiers | ISSN: 0007-0882 • E-ISSN: 1464-3537 |
| Publisher | Oxford University Press (PUBLISHER • GB) |
| DOI | 10.1093/bjps/45.3.789 |
| OpenAlex | W2070331304 |
| Language | EN |
| Citations received | 4 |
| References cited | 4 |
Can visual thinking be a means of discovery in elementary analysis, as well as a means of illustration and a stimulus to discovery? The answer to the corresponding question for geometry and arithmetic seems to be ‘yes’ (Giaquinto [1992], [1993]), and so a positive answer might be expected for elementary analysis too. But I argue here that only in a severely restricted range of cases can visual thinking be a means of discovery in analysis. Examination of persuasive visual routes to two simple theorems (Rolle, Bolzano) shows that they are not ways of discovering the theorems; the type of visual thinking involved can never be used to discover analytic theorems of a certain generality. The hypothesis that visual thinking is never a means of discovering the existence or nature of the limit of some infinite process is considered, and a likely counter-example is set out. It is still possible that restricted theorems can be discovered visually: an example from Littlewood is examined in detail and not found wanting. Even when visual thinking is not a means of discovery it can provide the idea for a proof in a direct way; an example is presented (Intermediate Value Theorem). In conclusion: it may be possible to discover theorems in elementary real analysis by visual means, but only theorems of a restricted kind; however, visual thinking in analysis can be very useful in other ways.
Discrete mathematics · Epistemology · Generality · Mathematics education · Range (aeronautics) · Real analysis · Visual thinking · Aesthetic Perception and Analysis · Computer Science · Data Visualization and Analytics · History and Theory of Mathematics · Mathematics · Philosophy · Psychology
| Unique citing works | 4 |
|---|---|
| Citations per year | 0,14 |
| Citation span | 1997 - 2012 (16) |
| Citation velocity | historical |
| Highly cited | No |
| Citation types | Neutral: 2 |