Cognition, Riemann and Plato

I’ve been spending a lot of time reading about the significance of Riemann’s Habilitation Dissertation and, today, a little bit of looking into the pervasive human desire to generalize led me yet again to Plato.  I keep thinking that a closer look at what Plato actually said is consistent with even the most brain-based thoughts on how we come to know anything.  The words that first got me going today were from William Kingdon Clifford’s translation of Riemann’s lecture (#20 on this list of Riemann papers has a pdf of the translation):

Researches starting from general notions, like the investigation we have just made, can only be useful in preventing this work from being hampered by too narrow
views, and progress in knowledge of the interdependence of things from being checked by traditional prejudices.

The key words for me are general notions, knowledge of the interdependence of things, and traditional prejudices. Riemann’s thoughts are carefully considered generalizations, directed at the notions of space, geometry and measurement, that greatly affected the course of modern mathematics.  And I will likely address them again in future posts.  But my point for this post centers more on epistemological ideas and hence another reference to Plato.

In Book 6 of the Republic, Plato describes what has been called the simile of the sun.  In the dialogue he writes, he makes the following statements:

The old story, that there is many a beautiful and many a good, and so of other things which we describe and define; to all of them the term “many” is implied.

And there is an absolute beauty and an absolute good, and of other things to which the term “many” is applied there is an absolute; for they may be brought under a single idea, which is called the essence of each.

The many, as we say, are seen but not known, and the ideas are known but not seen.

He goes on to talk about sight which he says, unlike the other senses, is bonded to something else – visibility, or light, or the sun itself.  While each of the following is formed as a question, he makes these observations:

And the power which the eye possesses is a sort of effluence which is dispensed from the sun

Then the sun is not sight, but the author of sight who is recognized by sight

And the soul is like the eye: when resting upon that on which truth and being shine, the soul perceives and understands, and is radiant with intelligence; but when turned toward the twilight of becoming and perishing, then she has opinion only, and goes blinking about, and is first of one opinion and then of another, and seems to have no intelligence

About students of geometry and arithmetic and the “kindred sciences” he says (or asks!):

And do you not know also that although they make use of the visible forms and reason about them, they are thinking not of these, but of the ideals which they resemble; not of the figures which they draw, but of the absolute square and the absolute diameter, and so on — the forms which they draw or make, and which have shadows and reflections in water of their own, are converted by them into images, but they are really seeking to behold the things themselves, which can only be seen with the eye of the mind

Among many there is a reluctance to accept as ‘a world,’ the world of forms.  There is material that gives rise to ‘the mind,’ and it is the body alive in its world but what the mind gives rise to seems to be material-less.  Yet clearly there is structure and meaning to ideas.  These structures can be explored as vigorously as one might explore any number of materials.  And, in fact, there will always be some relationship between thought and material.  The prejudice we have is believing that ideas are ‘contained’ in us while physical things are external to us.  But it is worth remembering that the images constructed by what neuroscientists call the visual brain are largely internal events.  I think Plato’s simile of the sun contains more than just an interesting metaphor.  I think it is taking note of how human worlds (both physical ones and conceptual ones) can only be brought into existence in relationship, leaving us to wonder again about the source of the conceptual ones.  Our habitual distinction between internal and external experience may yet give way.

Mathematics, I believe, uniquely demonstrates where an extraordinarily careful consideration of concepts can lead and assigns indisputable value to the research of form.

By way of an ode to Riemann I’ll end with a quote from Labyrinth of Thought by Jose Ferreiros Dominguez:

…Riemann transgressed the limits of the traditional conception of mathematics, turning it into a discipline of unlimited extent and applicability, since it embraced all possible objects.

3 comments to Cognition, Riemann and Plato

  • Joselle

    I’m glad you asked these questions. Our reading of Riemann’s own words can be a slippery affair because we all have inherited some habits of meaning that may not be consistent with his. For example, I would say that Riemann is downgrading empirical certainty when he says that Euclid’s matters of fact are not necessary, but ONLY of empirical certainty. In other words, they are only validated by what we think we see (like the way parallel lines look like they will never meet). In the next sentence he proposes to inquire about their (the lines) extension ‘beyond the limits of observation.’

    One of my favorite math historians (José Ferreirós) has written about Riemann fairly often. In one of his articles (Dogmas and the Changing Image of Foundations) he proposes this as Riemann’s view of knowledge:

    “…it begins in everyday experiences and proceeds to propose conceptual systems which aim to clarify experience going beyond the surface of appearances.”

    The conceptual systems are the hypotheses which can be improved by “spotting conceptual or theoretical inadequacies, striving for greater generality, and eliminating traditional prejudices.” Which brings me to your question about Lyndon Larouche. I watched part of a video presentation of Riemann’s lecture on his website and, as far as I saw, the speaker was getting the mathematics right. But I don’t think it’s possible to pull a political or economic views out of this. A philosophical one, perhaps, but it would be fairly broad. One of the points Ferreirós makes about Riemann is that he rejects the idea that applying reason or logic to experience gives us truth. Instead, Riemann recognizes the multifaceted interplay of experience(broadly conceived) and reflection or what Ferreirós calls “reconceiving.”

    I do agree, however, that Riemann’s perpsective does explore the mind itself (but I often make that argument about all of mathematics).

  • Donald King

    On a less serious point, what do you think of Lyndon Larouche’s stuff about Riemann? Does he understand it properly or not?

  • Donald King

    Riemann said in ‘On the Hypotheses which lie at the Bases of Geometry’ that his problem was ‘to discover the simplest matters of fact from which the measure-relations of space may be determined.'(1) Fair enough, but he then comments, ‘These matters of fact are – like all matters of fact – not necessary, but only of empirical certainty; they are hypotheses.'(1-2)

    When he says that ALL matters of fact are not ‘necessary,’ I take it that he means that nothing at all we call facts are ‘given’ i.e. accepted ‘a priori.’ Nothing is a ‘fact,’ and everything is dependent upon its possession (or not) of an ’empirical certainty.’ And from thence Riemann embarks on his habilitation dissertation.

    But this (to me post-modern!) assertion itself that nothing is ‘necessary’ (and everything a hypothesis seeking an empirical certainty) is accepted by Riemann himself without any argument. In a circular way, this assertion itself is a ‘necessary’ assumption. How can I seek an empirical certainty for the suggestion that everything (does he mean everything ELSE?) requires… an empirical certainty.

    His declaration that all facts are only hypotheses is an incredible one. It may help us to investigate ‘the measure-relations of space,’ but ‘all facts’? Is God too an ‘hypothesis’ demanding ’empirical certainty’?

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