What are the standards for ‘giving an account’? This is obviously crucial for the question of whether you really know, that is, understand something. Minimally, of course, you have to be able to keep your end up in an argument and show that your position is consistent. But something more positive than this seems required too. One major strand of ancient epistemology consists of exploring the requirements for ‘giving an account’, providing the reasoned basis necessary if you are to have understanding of a field of knowledge.
In many of Plato’s dialogues Socrates suggests that you have to be able to provide a satisfactory answer to the questioner who wants to know what virtue is, or courage, or friendship, or the like. This is obviously not provided by trivially appealing to the meaning of words; it has to express the nature of virtue, or courage, in such a way that the person to whom it is successfully conveyed will be able not only to recognize examples of virtue, or courage, but to explain why they are examples of the virtue in question, relating them to its nature. This exploration is sometimes called a search for ‘Socratic definitions’, although ‘definition’ is an unhappy term here.
One standing puzzle about these dialogues is the following. Socrates is ambitiously searching for understanding of difficult concepts like virtue and courage. But his approach is always to question others, starting only from shared premisses. This kind of ad hominem arguing relies only on what the opponent accepts and what it produces, time after time, are conclusions as to what virtue, courage, friendship and so on are not. Some self-styled expert makes a claim as to what virtue, etc. are, and Socrates shows that this cannot be the right answer. This does not, however, seem to move us towards understanding what virtue, courage and so on are. Socrates shows that others lack understanding, but not in a way that seems to be cumulative towards obtaining understanding of his own. There appears to be a mismatch between the goal and the methods. There are many ways of resolving this puzzle, and philosophers are divided over it.
Plato has a variety of concerns with knowledge, to some of which we shall return. Some of his most famous passages, however, show the dominance of what we can call the expertise model for knowledge. What is taken to matter for knowledge is whether you can, as an expert can, grasp the relevant items in a way that relates them to one another and to the field as a whole, and can give a reasoned account of this, one which explains the particular judgements you make and relates them to your unified grasp of the whole. And in some places Plato rethinks the crucial idea of giving a reasoned account, taking mathematics as his model.
In the Republic, Plato develops possibly the most ambitious model for knowledge that any philosopher has put forward. Now the aspiring knower has to complete an apprenticeship of many years’ mathematical studies. Mathematics – by which he is primarily thinking of the kind of systematized geometry of which we have a later example in Euclid – is remarkable for its rigour, system and clarity. It struck Plato as a perfect example of the kind of structured body of knowledge that had been presupposed all along by the expertise model. Moreover, all the features of the expertise model seem to fit mathematics in a clear and impressive way. Mathematics is not a heap of isolated results; particular theorems can clearly be seen to depend on other results which are proved in turn. The whole system begins from a clear and limited set of concepts and postulates. The way in which we get from these first principles to particular results is also lucid and rigorous. It is easy to see why Plato might see in an earlier version of Euclid a splendid model of knowledge as a structured and unified system, one where it is absolutely clear what the knower knows and how she knows it, how the system holds together and what it is to give a reasoned account of what you know – namely, a proof.
Mathematics as a model for knowledge also introduces two new notes. One is that mathematical results are peculiarly unassailable; we do not waste time arguing that Pythagoras’ theorem is wrong. We have seen that certainty and justification of what is known are not prominent in the cluster of issues that are the focus of the expertise model, where what matters is understanding that can be applied in practice. But Plato is clearly at some points attracted by the idea of a body of knowledge that is not open to serious questioning.
The other point flows from the fact that mathematics provides us with a body of firm knowledge which does not seem in any plausible way to have as its object the world that we experience, in an everyday way, through the senses. Pythagoras’ theorem was not discovered by measuring actual drawn triangles and their angles, and irregularities in these are obviously irrelevant to it. Plato is attracted to the view that a body of knowledge can exist to which our access is solely by using our minds and reasoning. He is not the last philosopher to be tempted by the view that the powers of philosophical reason are more developed versions of our ability to reason mathematically.
In the Republic’s central books, we find that to have knowledge requires mastery of a systematic field whose contents are structured as rigorously as the axioms and theorems in Euclid, and linked by chains of proof. Moreover, Plato goes one better than the mathematicians in claiming that philosophy actually proves its own first principles – something which mathematicians fail to do – and so does not begin from assumptions, but shows how everything flows from a first principle which is proved and not assumed. (Here matters get obscure, particularly since Plato makes everything depend on what he calls the Form of the Good.) As with mathematics, what is known is a formal system – what Plato famously regards as the world of Forms – and not the world of experience revealed to us by the senses; indeed, Plato goes out of his way to stress the extent to which the person thinking in abstract mathematical terms will come to conclusions at odds with experience.
This is, of course, an ideal, something emphasized by the way that the only people who, Plato thinks, have a chance of attaining it are those who are exceptionally talented by nature and have been brought up in ideal cultural circumstances. This warns us against thinking that we can find any actual example of knowledge. The expertise model on its own seemed to hold out the chance at least that knowledge was attainable. But when the requirements are made as formal and demanding as they become when mathematics is the model, the conditions for knowledge get set so high as to be unattainable by us.
Aristotle, in this as in many matters Plato’s greatest pupil, takes over the Republic model, but with important modifications which make it philosophically far more fruitful. He develops the idea in his work on the structure of a completed body of knowledge, the unfortunately titled Posterior Analytics. (It is so called because it follows his treatise on logic, the Prior Analytics.)
For Aristotle, Plato goes wrong in thinking that all knowledge hangs together in a unified structure. This makes the mistake of thinking that all the objects of knowledge together make up a single system, and can be known as such. But, Aristotle thinks, there is no such single system; different branches of knowledge employ fundamentally different methods, and do so because their subject-matters are fundamentally different. Aristotle does not disagree that something like Euclid’s geometry is a reasonable model for knowledge; like Plato, he is willing to appeal to mathematics to beef up the idea of expertise. But there is no such thing as knowledge as a whole, only the different kinds or branches of knowledge – or, as we are tempted to say, sciences. (The Greek word for knowledge, episteme, forms a plural, but we cannot say ‘knowledges’, and have to make do either with ‘branches of knowledge’ or ‘sciences’. This can obscure the way that, for example, Aristotle’s notion of a science is a restriction of Plato’s conception of knowledge.)