Visualization in philosophy

While spring-cleaning my archive of weblinks, I rediscovered Walter Whiteley on Visualization in Mathematics: Claims and Questions towards a Research Program. I'm merely going to quote a few passages from that paper, on the principle that a link to the wise does not sate, but only stimulates the appetite.

There is a connection here with my readings of Luhmann, Nassehi and other German sociologists/philosophers of the "radical constructivist" type. I find that their ideas are easy to understand when understood as schematic, visualizable scripts. But more on that mysterious remark in later posts. Whiteley's suggestions are straightforward enough:

Context:
I am a research mathematician, working in discrete applied geometry. My own practice of mathematics is deeply visual: the problems I pose; the methods I use; the ways I find solutions; the way I communicate my results. The visual is central to mathematics as I experience it. It is not central to mathematics as many teachers present it nor as students witness it. This contrast is striking.

I also work with future and in-service teachers of mathematics: elementary, secondary and post-secondary. They are surprised to learn that modern abstract and applied mathematics can be intensely visual, combining a very high level of reasoning with a solid grounding in the senses. They wonder how good visual work in elementary school connects to their own experiences with algebraic and formula centric (but visually meager) presentations of mathematics in courses. They wonder how any of these approaches connect to students’ future work that might use mathematics. They suspect that visual and hands-on work is not the ‘real math’ but is a crutch or bridge to be left behind as one matures. They are surprised at distinct and varied forms of visual reasoning within mathematics. They are surprised that what their students see is not what they see. We cannot just show a visual and say ‘behold’. Learning to effectively use visuals takes as much teaching and time as algebraic and symbolic reasoning [7,31]. The challenge is that visual representation and thinking skills can be as important to students’ futures as the symbolic and language based reasoning.

...

Claim 5:
We create what we see. Visual reasoning or ‘seeing to think’ is learned. It can also be taught and it is important to teach it.

Comments:
As cognitive science reports, we learn to see [13,21,23,24,26]. We learn what to notice and what to ignore, and how to interpret ambiguous cues. We work with images in the brain, as wholes and as parts, with symmetry, and with transformations at many levels [15]. In mathematics, what the expert sees and does with an image is not what the novice sees, even with the same diagrams. What the teacher sees is not what the students see. What one student sees is not what their neighbor sees. All of these differences impact our classroom work with diagrams and visuals. Since we create what we see, we can change what we see. Consciousness of alternative ways to see, and of the value of seeing differently, is one step. It takes ongoing guidance (cognitive apprenticeship), practice and evolving imaging (and imagination) to ‘learn to see like a mathematician’[7,31]. A well-known book on learning to draw says: “I will change how you see and the rest will be easy” [9]. Something analogous can be true for the learning of mathematics.

...

Claim 6:
Visual and diagrammatic reasoning is cognitively distinct from verbal reasoning.

Comments:
Brain imaging, neuroscience, and anecdotal evidence confirm this distinction, in the brain and in functional problem solving. Imaging suggests connections of mathematical reasoning with brain areas for eye-hand coordination, and an association of visual and kinesthetic reasoning. For example, we do proportional reasoning in this area of the brain, appearing to use a logarithmic number line associated with eye-hand coordination [1,5,16,17]. Studies of the brain during problem solving show distinct paths and forms for visual reasoning and verbal reasoning. Symbolic reasoning appears to be a distinct amalgam of these, with parallel paths dependent on parallel representations. To quote C.S. Pierce: "Diagrammatic reasoning is the only really fertile reasoning."

Peek-a-boo

Sig on Mars reports that certain passages quoted from a NY Times book review discussed in my last post here have been deleted from the newspaper's site.

I have on various occasions noticed that happening at newspaper sites, not just that of the NY Times. In the present case, I assume it was because a number of people wrote to the editors complaining that the passages made them "uncomfortable". So these were removed, allowing everyone to settle down in their couches and pull the blanket back up over their knees.

Intelligent life-forms are always torn between comfort and curiosity - it's part of the definition. Techniques have been found to meet these conflicting demands, but they are not equally fine-tunable. Since text as a medium is technically more primitive than film, it is not easy to prick a text into a teasing shape, concealing offensive passages in a way that still allows a prurient glimpse. You pretty much have to castrate it.

Of course the old practice of replacing letters by "*" is still available. However, due to the deterioration of reading skills in many of the formerly industrialized countries of the West, America in particular, to use "*" would make it difficult for readers to get that prurient glimpse. To reconstruct an outrageous original text from asterisks, you must know how to spell.

The film medium is more flexible - software can blur parts of the image, or overlay them with rectangular black areas just that little bit too small. The basic approach is that of the pastie, that 100% American juridical invention: it is legal to show female breasts in all their gory, on condition that form-fitting, light-reflecting pieces of paper have been pasted over the nipples.

The same peek-a-boo techniques are used in films, for body parts as well as for advertising. In a sequel of the American TV series "Jackass", which is currently being re-run here in Germany on one of the prole channels, I noticed that even naked male butts are overlaid with see-through blur boxes. So now buttilation is on the Index, in addition to titillation.

In MTV reports on famous whatevers, advertising slogans on the t-shirts worn by the whatevers are so blurred that you can't guess what is being flogged. I assume MTV would restore the focus if they could get the advertising revenue. Unfortunately there is no money to be made flogging male butts on TV, so you never get to see much. Male nipples may be shown without any restrictions, because the monitors of morals apparently do not know the naughty things that can be done with them.

Today's lesson was: cringe-making puns are more injurious to comfort than is curiosity.

Boredom, spuds and charity

In this post I will be discussing cats, concepts, Mr. Potato Head and "being difficult".

In an email discussion recently, the possible merits of a new book by Stephen Pinker were being weighed on the basis of a review entitled "Is Violence History ?". In discussing the book the reviewer, Peter Singer, employs notions such as reason, ethics, justice, repression, mind, morality, cognitive and emotional faculties, hunter-gatherers and so on. He comes to the conclusion that Pinker's book is "supremely important". I have quoted some passages at the end of this post.

My reaction to this review was very different from that of the other participants in the discussion. So different, in fact, that I thought I had better not say it straight out, but only remarked: "This is kaleidoscope thinking: round and round go the same old notions, flashing and rattling. Morality, reason, violence ... I got tired of reading such stuff decades ago." Having thought over all this later, including my decision to hold back, I identified boredom and charity as essential components.

1. Boredom

It was around the age of 15 or so that I first encountered books and articles about reason, ethics and so on. I was particularly taken by Gilbert Ryle's The Concept of Mind. This led me to the British analytic school of moral philosophy and R.M. Hare, all of whose books I read that I could get my hands on in Texas (I see in the Wipe article that the reviewer, Peter Singer, studied under Hare). For a long time I read with particular interest the TLS reviews of books of philosophy.

Over the last forty years, however, I have lived in Germany, reading Leibniz, Schopenhauer, Dilthey, Nietzsche, Plessner, Gadamer, Sloterdijk, Deleuze, Rheinberger, Rorty, Atlan, Morin, Bachelard ... As a result I seem to have lost my ability to make much sense of those TLS reviews (even the fretfully humorous ones by Fodor !) or the Singer review. This is where the trouble starts with my American and British acquaintances - where I find myself being charged with being difficult, cynical, insecure ...

The image that sprang immediately to mind when I read the Singer review was Mr. Potato Head. The words "reason", "ethics" etc. seemed to be bits of plastic being rearranged yet again to produce a striking result. Now whether or not an Anglophone person has read any of the German and French writers I mentioned above, surely - I feel - he/she must have tired of all those inconclusive but insistent Anglophone discussions over the years: reason is a component of morality (or not), violence is natured (or nurtured), mind is a neural epiphenomenon (or not), justice is positive (or natural), cognition and emotion are separate (or bound together in our hunter-gatherer natures) ...

The fact that I find all this worthless - does it imply cynicism, a belief that "there is no such thing as reason and justice" ? It does not. It implies that I believe (at the very least) that the words, the conceptual cut-lines, are worthless. Exercise: try to formulate what you want to say without using the words "reason", "justice" and the rest of 'em. Do you find it difficult to break that habit ? It is rather reminiscent of addiction, isn't it ? And that potato - is it not "the neutral subject" surveying the world of ideas without a clue as to how it constructs that world, and is constructed by it ?

I am not trying to create the impression that I know-it-all and that these are easy-peasy issues - far from it. What I am trying to do is bring attention to how difficult they are. Glibly, skillfully shuffling traditional words around won't hack it. Boredom, though, is the mother of invention, a state of mind conducive to finding different approaches.

2. Charity

Dogs easily learn to look in the direction a person points with his index finger. Cats never learn this, but always look at the finger. I myself will look in the direction pointed when there's something there worth looking at. Otherwise I tend to stare at the finger, trying to figure out the point of pointing at such a pointless thing.

At first glance, it seems that I am being rather uncharitable, to put it mildly, to dismiss a serious discussion of Singer's/Pinker's ideas as a game of Mr. Potato Head. I'm sure the participants would not thank me for suggesting in this way that they are wasting their time. But do I find it uncharitable to be dismissed as "cynical" ? No, I find it frustrating, since my interventions are equally serious.

Maybe the problem is the analogy with Mr. Potato Head. This image may seem offensive, but I found it apposite and funny. And this is, I think, the core of the problem: I tend to be serious and make jokes at the same time. Many people interpret that as cynical frivolity, whereas I expect it to be taken as a token of self-deprecation and open-mindedness.

It seems there is more seriousness in the world than I had expected, and charity is not going to fix it. Thank God for books and blogsites, where one can speak one's mind without immediate reprisals.

Quotes from the review:

When you heard that a gunman had slaughtered scores of Norwegian teenagers on a holiday island earlier this summer, did you think that here was another symptom of our sick and violent world? So did I, until I read Steven Pinker's brilliant, mind-altering book about the decline of violence.

The real fascination of this book is how we got from being a species that enjoyed the spectacle of roasting each other alive to one that believes child-killers have the same rights as everyone else. As Pinker shows, it is both a long story and a relatively recent one. The first thing that had to happen was the move from a nomadic, hunter-gatherer existence (where your chances of meeting a violent end could be as high as 50:50) to settled communities. The trouble was that early governments showed themselves at least as capable of cruelty as anyone else: most of the truly horrific instruments of torture Pinker describes were designed and employed by servants of the state.

To readers familiar with the literature in evolutionary psychology and its tendency to denigrate the role reason plays in human behavior, the most striking aspect of Pinker’s account is that the last of his “better angels” is reason. Here he draws on a metaphor I used in my 1981 book “The Expanding Circle.” To indicate that reason can take us to places that we might not expect to reach, I wrote of an “escalator of reason” that can take us to a vantage point from which we see that our own interests are similar to, and from the point of view of the universe do not matter more than, the interests of others. Pinker quotes this passage, and then goes on to develop the argument much more thoroughly than I ever did. (Disclosure: Pinker wrote an endorsement for a recent reissue of “The Expanding Circle.”)
[The escalator of reason appears to be a cautious intellectual version of the Rapture]