Let’s return to the case of Plato.  Philosophy has often taken mathematics as its model of knowledge.  “Let none enter here who have no knowledge of geometry!” it says over the doors of the academy.  The mathematical presents us with the clarity of the pure concept—one itself and not one orange or person or star –and its claims are deductively demonstrated without remainder (at least until Gödel came along).  Math, as Serres points out in Geometry, attains true universality overcoming the indeterminancy of difference and interpretation, and attains certainty.  It doesn’t matter whether you’re Greek or Persian, you cannot fail to come to the same conclusions when confronted with the Pythagorean Theorem.  It’s not that the content of geometry is particularly important to the philosopher, but rather that geometry gives the philosopher the ideal of knowledge and the method of how to attain that knowledge.  We see much later the fruition of this ideal in texts like Spinoza’s Ethics.

In the open to Plato’s Euthyphro, Socrates asks what piety is an Euthyphro responds, “piety is doing what I’m doing now; prosecuting the wrongdoer.”  Socrates responds that that’s not what he asked.  Euthyphro is confused.  “Well suppose, Euthyphro, I asked you what oikonomia is?  And you gave me a list of things like knowledge of how to cook, maintain a home, maintain finances, etc.  That would not answer the question, because I didn’t ask for a list, but for that common feature that all of these things share in common.”  Perhaps we could call this the function that haunts all of the instances of oikonomia, in the sense of the mathematical function.  This is the concept of oikonomia.  Cooking, managing a home, and household finances are all very different from each other, but they must nonetheless share a common concept (form) or essence that makes them what they are.  5, 7, 9, 11, 13, 15…  All of these numbers are very different from each other.  A list tells us very little.  Yet when we know the concept or function behind the list—f(x) = 2x + 3 –we know what is common to them all and how to proceed to find the next in the series.  We even have a rule that allows us to determine whether the new thing that we haven’t encountered before belongs to the series or not.  This is the mathematical ideal in philosophy.

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It is not at all unusual to hear Socrates’s ventroliquized voice deployed when theorists appeal to examples.  “I did not ask for an example of x, but for x itself!”  The giving of an instance or case becomes prohibited.  One is commanded to work at the level of the pure concept without any examples as Hegel strives to do in his magnificent Science of Logic.  “Being, pure being, and nothing but!”  However, what if we are not able to think pure concepts in this way?  What if behind what seems to be the pure concept, there is always a veiled particular that secretly governs the thinking of that thing?  Here we have a sort of inverted Platonism.  It is not the concept, universal, or function that governs the particular, but the particular that secretly governs our formulations of the universal.  Would we not then have a responsibility lay our examples bare so that we can evaluate whether or not these examples are good examples?

Much to our astonishment, Plato seems to say as much later in Book VI of the Republic.  In the Analogy of the Divided Line we are surprised to discover that mathematics, discursive reasoning, is not listed as the highest form of knowledge.  After all, we’ve daily read what is over the doors of the academy.  Plato’s reason for not placing mathematics at the highest level is interesting and instructive:  mathematical reasoning relies on diagrams and symbols; images or examples.  The geometer draws a triangle to help her think about the pure concept of triangleness.  The theorist of arithmetic uses roman numerals to think about arithmetic.  From the lowest level of the divided line, we had already learned that all images, no matter how faithful they are, distort what they represent because they can never capture the true essence of the thing.  Try as she might, the geometer cannot draw triangleness.  She always draws a right triangle or a scalene triangle or an equilateral triangle or an isosceles triangle.  Her diagram or example therefore carries the risk of leading her to overlook other relations that belong to the essence of triangleness.  Roman numerals get in the way of thinking operations on numbers that can be carried out in arithmetic (perhaps this is why the Romans made little in the way of contributions to mathematics).  The mathematician, says Plato, falls prey to what Hegel will much later disdainfully call “picture-thinking”, thinking in terms of images and examples.

The ideal, by contrast, would be noesis:  a thinking of pure essences, concepts, or functions without the use of any images or examples.  But what if this is an impossibility for hairless chimpanzees such as ourselves?  What if our thought is always secretly animated by a case or an example?  If this is the case, no matter how pure our thought on ethics, morality, or justice might be, there would always be a secret paradigmatic example or prototype of what is moral or just that governs our thinking.  The paradigmatic example would then function as the function that determines our thinking of all other cases and whether they belong to the class.  Something similar would happen in our thoughts about the nature of knowledge or being.  In thinking of the being of objects, we take a paradigmatic case of what an object is such as rocks.  Even if we don’t explicitly talk about rocks in our meditations on objects, we nonetheless use the case of rocks to determine whether or not other beings fall into the category of class of objects.  Likewise, when Kant evokes 7 + 5 = 12, has he given us a good or significant example or mathematical thinking, or a trite one?  What would happen if we chose a very different mathematical example as our paradigmatic case?  Would we come to the same conclusions about time and space that Kant reaches?  What of the education reformer whose paradigmatic example of knowledge and learning is “In 1492, Columbus sailed the ocean blue.”  Is the ability to recite this truth a good example of what it means to learn and of how to evaluate whether or not learning has taken place?  Is representational knowledge—the ability to repeat such referential statements about the world –a good model of what knowledge is at all?  Yet these cases nonetheless secretly govern these discourses, determining the entire destiny of education policies and practices.

If it is true that noesis, while an ideal, is nonetheless impossible for us, if it is true that hairless chimpanzees like us always think in terms of paradigmatic cases, we have a responsibility to places our paradigms on the table so that we might evaluate whether or not they are good examples and so that we might also bear responsibility to the examples we use.  In this last instance, bearing witness to examples and responsibility entails raising the question of whether we’ve been true to our own cases, or whether we’ve betrayed and distorted them.  Yet all too often we bury the cases in a crypt, quickly moving on from them, engaging in an idle chatter that seems to be saying something without really saying something at all.  Can we really speak of Lacanian psychoanalysis without speaking of the clinic and specific cases in the clinic?  I don’t think so.  I don’t think we know anything of desire or objet a or the real or the symbolic without seeing these things truly at work in the lives of analysands.  Otherwise we’re just building castles in the sky.

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