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Proving With Pictures?
The Limited Role of Diagrams in Mathematical Argumentation

Patrick Scotto di Luzio
Stanford University
June 2001

In my dissertation, I assess the extent to which non-linguistic ways of representing information (such as using diagrams, pictures, and maps, as opposed to sentences and formulas) can be used appropriately in mathematical proofs. It is hoped that such a study will contribute to a broader and more refined understanding of how mathematics is developed, taught, and studied, which in turn may help to make its methods and results accessible to a wider audience.

I begin my investigation by arguing that mathematical proofs have a distinctive style, resulting from the measures mathematicians take to ensure that their proofs are interpreted in exactly the way they intended. For example, mathematicians try to use a notation which is precise and unambiguous. Furthermore, they tend to be quite explicit about what axioms or assumptions they are appealing to in the course of the proof, and also indicate how each step in the argument follows from previous ones. I proceed to show, first, that the adoption of such an argumentative style is crucial for the development of mathematics, and second, that linguistic notation systems are better suited than diagrammatic ones for achieving this style.

This result may seem surprising at first, since in everyday life we often find that diagrams are much better at representing information than language. Imagine, for example, how long it would take to write down all the information present in a road map (and how useless the result). Also, it is plausible to think that the effectiveness of diagrams vs. language would depend on who is using them, since some people claim to be more "visual" and thus less able to assimilate linguistic information than others. However, I argue that mathematical proof requires a special discipline which overrides such factors. Much of my work, then, involves articulating why mathematical proof is distinctive in this way.

My work appeals to two different traditions in the philosophy of mathematics, and offers a bridge between them. The first tradition, the "logical" one, tends to see mathematics as a clean, formal (almost computational) discipline. The second tradition, the "sociological" one, sees mathematics as an (often messy and complex) activity involving fits of discovery, imagination, and intuition. Unfortunately, there is very little dialogue and interaction between the two traditions, resulting in a fragmented account of the nature of mathematics. However, by looking at whether diagrams can legitimately be used in proof, one is thereby compelled to seek unification.

The logical tradition, for instance, is very much preoccupied with the nature of mathematical proof and has developed a sophisticated model for describing the proofs that are offered in mathematics journals and classrooms. This model employs what are called "formal systems," which are precisely specified languages along with rules for reasoning correctly when using such languages. For example, these rules only allow pieces of reasoning as obviously correct as, "If John is a student, then there exists a student." Notably, these formal systems almost never employ diagrams, but rather only use sentences. As a result, philosophers and mathematicians tend to think that "ideal" proofs consist only of linguistic statements.

And yet, the sociology of mathematics shows us that diagrams have considerable real-life utility. Mathematicians often rely on visual images when presenting their work to colleagues (or teaching it to students) in order to help develop an intuitive "picture" of the mathematical domain they are talking about. These images are often integral to their own understanding of the material as well, perhaps prompting an insight which leads to the discovery of a proof.

Putting the logical and sociological perspectives together, we get the following view diagrams are very useful for much that SURROUNDS the development and communication of some proof, but they ought not to be used IN the proof itself. Any proof which relies on a diagram for its content is generally thought to be flawed and deficient. At best, the diagrams can be used as optional illustrations, but are not of any theoretical necessity.

Over the past decade or so, a handful of philosophers have developed formal systems which appear to challenge this stance. In these newer formal systems, reasoning rules are applied directly upon diagrams instead of sentences. It turns out that these systems share many of the important properties possessed by the more mainstream, linguistic ones. In particular, the reasoning rules of these systems are often "sound" (they never allow someone to derive a conclusion that is false from premises that are true), "complete" (all valid arguments can be obtained using just those rules), and "decidable" (there is an effective method for checking whether the rules were applied correctly). From these results, it has been argued that diagrams deserve to be in proofs just as much as linguistic statements do, since there is no discernible "logical" difference between formal systems which use diagrams and those which do not.

I claim, however, that these results are not sufficient to show that diagrams can be used legitimately in proof. First, I argue that, while these results do indeed show that it is possible to reason correctly with diagrams, there is in fact much more to writing a proof than merely saying correct things. A proof also has to be a DEMONSTRATION of how it is that a conclusion follows from some premises. Traditional formal systems model such demonstrations in a particular way. The linguistic reasoning rules ensure that they are written in a certain style, so that one is forced to produce very precise and explicit proofs. Such precision and explicitness is crucial for the establishment of mathematical knowledge; without these features, it is doubtful that mathematics would have the character of objective, incontrovertible truth. Thus, there is an important feature of mathematical proof which, though modeled by linguistic formal systems, is not captured by the common "logical" notions of soundness, completeness, and decidedness. To point out that the diagrammatic, formal systems are sound, complete, and decidable is not to show that they are "just like" linguistic ones in all relevant respects.

I go on to argue that it is more difficult (though perhaps not impossible) to achieve the same style of proof with diagrams. Certain intrinsic properties of diagrams limit the extent to which it is possible to be precise and explicit when using them, and thus language appears to be an inherently superior tool for achieving the particular style associated with proving. Such a finding, of course, does not preclude there being important tasks for which diagrams are better suited. An important next step would be to positively characterize what those tasks might be, yielding a full account of all the phases that are involved in the discovery and development of mathematics.