Monday, December 26, 2011

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:
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.

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.

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."

Friday, December 23, 2011


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.

Sunday, November 6, 2011

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]

Saturday, September 17, 2011

More to the point

I have renamed this blog to "concertnotes" because it gives a better sense of the kind of posts I do. It used to be called "grumbleby", but grumbling is only an amusing sideline. My basic disposition is that of a spectator at concerts that do not seize his attention. As a result, I find myself examining the ceiling, riffling the program notes and in general not so much watching the show as considering the circumstances of its production.

This is what Luhmann calls "second-order observation". To be clear: engaging in this does not grant "objectivity", or privileged access to "the truth". It is merely a technique of attention to attentiveness - looking at the finger instead of what it's pointing to, studying the window display without buying anything (possibly for lack of money). The Little Match Girl was a second-order observer, and so was Wilde.

Sunday, April 24, 2011

No more ugly makeup !

John Cowan has pointed out a "bookmark resource" for tarting down the websites I complained about in my last post. As he explains here:
The Readability bookmarklet helps a lot with pages like that. Go to this webpage and drag the "Read Now" button to your browser's bookmarks bar. Then click on it whenever you want to make a page more easily readable. It deals with bad colors, annoying formats, etc.
People should take a look at Readability's FAQ, which answers the questions "What is Readability?" and "What happened to the free version of Readability?".

In short: the bookmark resource is "still free", and this is what it's all about:
We’re turning Readability into a monthly subscription service with a unique twist: the great majority of your fees (70%) will go directly to the writers and publishers you enjoy. We’re tethering a small, passive transaction to the reading decisions you make through the platform. You can even publicly share the top domains you’re enjoying through Readability. It’s a new type of badge: “I support these writers & publishers.”
In other words, this is another scheme to keep track of where you go with your browser - but only when you actually use the bookmark. Clicking it merely takes you to another website (Readability), and there is not even a cookie involved. Exactly the same happens when you point to any link. You'll have to decide if you want to live with that. It seems reasonable enough to me.

The make-it-more-readable feature appears to work as follows (I've inspected the page sources, but am not a browser/javascript expert):

1. Let's say you are positioned on some web page "" in your browser. That is, the "page frame" showing the contents of that page is in the foreground of your browser.

2. You click on the "Read now" bookmark. This is a "widget" containing javascript activated by your click.

3. The bookmark javascript is called with the "" address as parameter, causing the browser to call another piece of javascript at This other javascript technically reads the HTML page at the parameter value "" (the stuff which your browser had rendered in the original, hard-to-read form). It converts that original page into a "more readable" HTML page at, which is then the final page that is actually rendered in your browser (no longer the one at "").

Saturday, April 23, 2011

Relatively speaking

Recently, a commenter on a blog post at another site complained bitterly that he could not read the yellow-mauve-and-white-on-black presentation of a web page to which the post had linked. I myself could make out the text, but reading it was hard on the eyes. There are a lot of websites in the internet that are tarted up in this way.

One could speculate about the motives of the authors of such websites, and whether they are at all aware that there are "accessibility" aspects to web design. But over time I have found it more efficient and productive to take unreadable things at face value, and simply not read them. Given the amount of text in the world that is clamoring for attention, I rejoice at every badly designed website I encounter, and every badly written book - in each case one less thing to deal with !

It is an extravagance to posit that there must be substance behind appearances. The principle I apply here is: if less is more, than nothing is most to be desired. In terms of biological evolution, rejection is just the flip side of selection, but it's algorithmically simpler. How to weigh the criteria for selection from a large set of alternatives is a difficult problem, and requires goal-directed intelligence. In contrast, all you need to reject something is a garbage can (in case you need it later on after all).