Abstract
Diagrams are frequently used in mathematics, not only in geometry but also in many other branches such as analysis or graph theory. However, the distinctive cognitive and methodological characteristics of mathematical practice with diagrams, as well as mathematical knowledge acquired using diagrams, raise some philosophical issues – in particular, issues that relate to the empiricism-realism debate in the philosophy of mathematics. On the one hand, it has namely been argued that some aspects of diagrammatic reasoning are at odds with the often assumed a priori nature of mathematical knowledge and with other aspects of the realist position in philosophy of mathematics. On the other hand, one can claim that diagrammatic reasoning is consistent with the realist epistemology of mathematics. Both approaches will be analyzed, referring to the use of diagrams in geometry as well as in other branches of mathematics.
M. Sochański—Independent scholar.
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Notes
- 1.
- 2.
It has to be noted that realism is typically opposed to anti-realism, in that the latter claims that mathematical objects do not exist. The dichotomy empiricism-rationalism, on the other hand, concerns the source and nature of mathematical knowledge. However, every realist is an apriorist in that he claims that we come to know mathematical objects independently of physical experience. In that sense it is possible to oppose realism and empiricism.
- 3.
It should be noted that R1-R3 are independent of each other (although clearly strongly related) and so any combination of them can be held within one position.
- 4.
I will intentionally omit the two other big families of views on mathematics – varieties of neo-Kantian positions and logical positivism, in order to focus my attention on the empiricism-realism debate. Both claim that mathematics is a priori, but just as the followers of Kant hold that the apriority rests on in-born structure of our minds, the followers of logical positivism defend the view that they rest on the analyticity of mathematics.
- 5.
As far as ontology is concerned, it is has been claimed that the subject-matter of geometry is extension, empirical space or shape. John Stuart Mill, in turn, argued that the subject matter of geometry are the actual physical diagrams.
- 6.
“These things themselves that they mold and draw, of which there are shadows and images in water, they now use as images, seeking to see those things themselves, that one can see in no other way than with thought” [7].
- 7.
Plato names five aspects of a circle: “The first is the name, the, second the definition, the third. the image, and the fourth the (…) knowledge, intelligence and right opinion” [8], stressing that each of them is a necessary step in forming our knowledge of the true geometrical objects.
- 8.
By a “reasoning” I will understand the whole psychological process that starts with assumptions and a given diagram (or diagrams) and results in appearing of a belief state which is the conclusion of the reasoning.
- 9.
Visual characteristics typically fall into two categories: topological and metric. Metric properties refer, e.g., to the length of lines or size of angles. Topological properties in turn, include betweenness, incidence and inclusion. It has been claimed that only topological properties have been used in Euclid’s Elements.
- 10.
According to Coliva we can distinguish between perceptual concepts which correspond to deliverances of the senses and geometrical concepts which are of purely mathematical nature and for example used when we take a straight line that is not perfectly drawn, to represent a straight line. A similar distinction has been made and analyzed by Marcus Giaquinto who has also argues that geometrical arguments that use diagrams in a significant way are a priori [3].
- 11.
Mark Greaves notes for example, that according to Helmholtz, “kinematic properties are not given a priori, but rather are derived empirically from the subject’s own perceptions of bodily movements, and thus that there is a necessary linkage between the truths of geometry and the laws of mechanics”[4].
- 12.
An exception could be e.g. topology, which studies properties of shape in a more general and abstract way that geometry.
- 13.
In a similar vein, realism allows the use of Axiom of Choice or impredicative definitions.
- 14.
This point has been made by David Sherry, who claims that the Platonist cannot explain “use of the same diagram in proving theorems about radically incompatible figures” [9].
- 15.
To give just one example, Felix Klein used the notion of “refined intuition” which made use of sensual experience, but was “armed’ with precise mathematical concepts.
- 16.
One exception was Proclus, who adds another cognitive faculty to the ones postulated by Plato, which he calls imagination (phantasia). In words of Dmitri Nikulin, imagination is “intermediate between sense perception and discursive reason: with the former, imagination shares the capacity to represent geometrical figures as extended; with the latter, it shares the capacity to represent its object as unchangeable according to its properties” [6].
- 17.
However, it seems that the character of diagrammatic reasoning has little relevance to theses (R2) and (R3), which relate to the objective and necessary nature of mathematical knowledge.
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Sochański, M. (2018). Interpreting Diagrammatic Reasoning – Between Empiricism and Realism. In: Chapman, P., Stapleton, G., Moktefi, A., Perez-Kriz, S., Bellucci, F. (eds) Diagrammatic Representation and Inference. Diagrams 2018. Lecture Notes in Computer Science(), vol 10871. Springer, Cham. https://doi.org/10.1007/978-3-319-91376-6_18
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