On Mathematical Understanding


What is mathematical understanding?

First, I will try of set out some questions concerning mathematical understanding.

  • If we take the view that mathematical understanding is a process of acquiring mathematical knowledge, then what kind of process is it? What are its attributes? Is it necessarily gradual, or are there genuine moments when the realization of something crucial hits you? How much of mathematical understanding is just matter of imitation and repetition?

  • In Poincaré on Understanding Mathematics, Gerhard Heinzmann writes

    Now, since Aristotle, understanding is connected with learning. And it is well known that, according to Poincaré, in order to teach and to learn mathematics there must be appeal to intuition and reasoning by analogy. The same capacities are also necessary to create new mathematics.

Heinzmann, G. (1999), Poincaré on Understanding Mathematics, Philosophia Scientiae, 3: 43‐ 60

There is a philosophical technicality I like to expand on it here. The modes of “knowledge” has been an important subject in the history of philosophy and it at least goes back to ancient Greeks particularly to Plato and Aristotle who gave a substantial account of it for the first time. The notions concerning “knowing” we encounter in the work of Plato and Aristotle are epistêmê (knowledge), technê (craft or art), gnôsis (understanding, theory), epistasthai (knowing how to do, practice), etc. Broadly speaking (and this is really an oversimplification) one can divide up the notions into following categories.

  • epistêmê which deals with knowing essences, i.e. to understand things as they are.
  • technê which is the technical knowledge to manufacture things and perform kind of physical tasks. This is often translated as art or craft. The word epistasthai (knowing how to do) is usually associated with technê.

A word of warning: the above translation of Greek words must be taken with more than the proverbial pinch of salt as intended meaning of those words by their authors reflect a world picture corresponding to their time, and we should be absolutely conscious of our misjudgements that may come from our contemporary assumptions about the state of things. For instance some people have translated the word epistêmê in Plato and more so in Aristotle as scientific knowledge. To be frank, I do not like this translation; it enforces on the reader a contemporary understanding of science which certainly has experimentation as one of its essential components, which is of course a much later development in history of human thoughts, notably pursued by philosopher and scientist Ibn al-Hatham and subsequently and much more systematically and robustly by Francis Bacon, Galileo Galilei, and Antoine Lavoisier, and others. For this reason, I propose the term “exact knowledge” for the rest of this article.

I must add that although there is a fine distinction between epistêmê and technê in Plato, their relationship is not of being separated but rather it is of tension. Perhaps the following paragraph from SEP aticle on Episteme and Techne makes this tension more clear:

As the concept of technê develops, the role of reflective knowledge is emphasized. Whereas technê is associated with knowing how to do (epistasthai) certain activities, epistêmê sometimes indicates a theoretical component of technê. Then it is associated with understanding (gnôsis). On the one hand, the physician knows how to care for the sick (Rep. 342d), to prescribe a regimen (Rep. 407d), to provide for the advantage of the body (Rep. 341e), to make someone healthy (Charm. 174c), to make someone vomit (Laws 933b). On the other, the physician knows or recognizes (gignôskein) health by medical knowledge (epistêmê) (Charm. 170c). Since health is the goal of the medical craft, the physician understands the goal of the craft. Plato emphasizes this knowledge as a distinct aspect of the craftsman’s skill. Sometimes this aspect is theoretical in the root sense of theôria — looking.

But then enters Aristotle. In his Book VI of the Nicomachean Ethics, he sharply distinguishes technê from both epistêmê and aretê (virtue). An archetypal example of epistêmê (exact knowledge) for Aristotle is geometry. What renders a kind of knowledge to be an epistêmê (exact science) is its necessity (as opposed to contingency) and the unchangeability of objects which this science studies. The way Aristotle thinks about epistêmê here is that of an exact knowledge presented as a deductive system in which the relations among terms are invariable and necessary.

The full account of epistêmê in the strict sense is found in Posterior Analytics, where Aristotle says that we think we know something without qualification (epistasthai…haplôs) when we think we know (gignôskein) the cause by which the thing is, that it is the cause of the thing, and that this cannot be otherwise (71b10-15).

A remark on the notion of the casuse in Aristotle is in order. First, let us disclose the root of the word “cause”. The word derives from the Latin noun causa, which stems from the verb cadere, meaning ‘to fall’. The Romans conceived that a result ‘falls’ from a previous event. For the Greek, on the other hand, cause (αἴτιον) signified debt. In Physics II 3 and Metaphysics V 2, Aristotle systematizes causality. Aristotle’s account enumerates four types of causes: material cause, efficient cause, formal cause, and final cause. Here is an example explaining all these type of causes. Consider the production of an artefact like a silver chalice 1. the silver (and its transformed states in the process of production) from which it was made is the material cause; the silver smith who made it (even more importantly, for Aristotle, the art rather than the artisan, the craft rather than the craftsman) is the efficient cause; the shape and idea (or to use a more contemporary word “model”.) of chalice or ‘chalice-ness’ that makes it the type of thing it is the chalice’s formal cause ; and to the ends or purposes that a chalice serves its final cause.

According to SEP article Aristotle on Causality

There is no doubt that the art of bronze-casting resides in an individual artisan who is responsible for the production of the statue. But, according to Aristotle, all the artisan does in the production of the statue is the manifestation of specific knowledge. This knowledge, not the artisan who has mastered it, is the salient explanatory factor that one should pick as the most accurate specification of the efficient cause (Phys. 195 b 21–25). By picking the art, not the artisan, Aristotle is not just trying to provide an explanation of the production of the statue that is not dependent upon the desires, beliefs and intentions of the individual artisan; he is trying to offer an entirely different type of explanation; an explanation that does not make a reference, implicit or explicit, to these desires, beliefs and intentions.

We must mention here that although Aristotle’s account has been dominant throughout the history of philosophy, not everybody agrees with him. The most critical voice is that of the pastoral philosopher Martin Heidegger. You can read more about what he has to say in his work The Question Concerning Technology.

What then distinguishes the spheres of exact knowledge and of craft is the classic division between the purely theoretical and the purely practical. Scientific knowledge concerns itself with the world of necessary truths, which stands apart from the world of everyday contingencies, the province of craft.

However the distinction between technê and epistêmê fades away in his other works such as Physics and Metaphysics. However, the contrast becomes between technê and epistêmê on one side and empeiria (experience) on the other. At the beginning of Metaphysics,

Aristotle contrasts the person of experience (empeiria) with someone who has technê or epistêmê. The former knows that, when Callias had such and such disease, thus and such helped him, and the same for Socrates and many others. However, the person who has a technê goes beyond experience to a universal judgment. This judgment is that this remedy helped all individuals of this type, with this disease. Examples of the types of individuals are the phlegmatic and the bilious, when afflicted with a burning fever (981a5-15). However, it is important to note that the universals cited — phlegmatic and bilious — have a role to play in explaining a fever and, thus, a role to play in the account of a cure. As Aristotle says, the master craftsman (technitês) is wiser than the person of experience because he knows the cause, the reasons that things are to be done. The mere artisan (cheirotechnês) acts without this knowledge (981a30-b5). Aristotle goes on to say that in general the sign of knowing or not knowing is being able to teach. Because technê can be taught, we think it, rather than experience, is epistêmê ( 981b10).

There is a close parallel between our discussion of technê or epistêmê and Heidegger’s distinction between Zuhandenheit (ready-to-hand) versus Vorhandenheit (present-for-hand). This is worth more elaborating. See this Wiki articel.

Also, concerning what constitutes understanding “Turing Test” and Searle’s Chinese Room thought expriment are noteworthy.

  • To what degree can we really relate understanding and learning even in the theoretical domain of pure mathematics?

[1] The example of silver chalice is actually from Heidegger’s lecture “The Question Concerning Technology”

[2] Goldfarb, W. (1988), Poincaré Against the Logicists, in Kitcher and Aspray, pp. 61–81.

[3] Gray, J. (2012) Henri Poincaré: a scientific biography. Princeton University Press 2012.