Notation And Thinking at Dick Lipton’s Blog

December 3, 2010 at 8:16 am 8 comments

My colleague Dick Lipton writes wonderful blog posts, and he just wrote one this last week on the role of notation in our thinking, a mathematical/computational take on the Whorfian Hypothesis.  The below quote (attributed to Alfred North Whitehead) is actually from the commentary on the post, and captures the issues succinctly.

An important issue that Dick left unconsidered in his piece (but is raised in the commentary) is the up-take cost of a new notation.  There is a learning cost in developing the abstraction and learning the mapping that this symbol stands for that.  Much of Dick’s post is on APL which is a wonderful example of exactly this point.  APL is very compact and powerful — but it uses an unusual symbol set, and requires the newbie to learn vector operations as a way of describing computations.  That’s a valuable perspective to learn, but it comes with a cost.  In the end, all educational decisions are economic: Is the power and brevity in notation worth the cost to learn?

It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilisation advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle — they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.

via Notation And Thinking « Gödel’s Lost Letter and P=NP.

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8 Comments Add your own

  • 1. Darrin Thompson  |  December 3, 2010 at 8:56 am

    May I please have that quote printed and framed, 36″ X 50″?

  • 2. Alan Kay  |  December 3, 2010 at 11:24 am

    Hi Mark,

    Marvin Minsky’s quip about the “New Math” of the 60s was “The trouble with new math is that you have to understand it every time you use it!”

    However, many people chose to interpret this quite wrongly, as Marvin sanctioning teaching arithmetic as pattern skills without understanding.

    What he (and Seymour Papert) actually wanted was to have the children really understand what was going on via one kind of thinking and doing, and to also be super-fluent in “operating” using another kind of thinking and doing.

    This is what Whitehead actually meant also, as is readily apparent by reading his books about thought and education. So many users of this quote misrepresent his actual meaning.

    To me, the most thoughtful comments on Dick’s blog were the ones about Cartan’s p-notation on the one hand, and especially about notations that reduce names and symbols as much as possible.

    (I gave an invited talk last year that is to this same point for the Institute of General Semantics on the idea of “how to not use words when using words”.)

    For example, “trig” can be taught quite readily to 4th and 5th graders if “what it is” is employed directly as purely geometric relations with no additional terminology.

    Similarly, the trick (even more than a trick) of learning a language like APL is to avoid the symbols as much as possible (they clog up the 7+-2 chunks and distract from what is powerful and good about APL). One could even imagine a graphics preprocessor for a dataflow notation like G or Dan Ingalls’ Fabrik to get started. The symbols are useful once the operations have been internalized. (This is generally the case for most good notational schemes.)

    It’s both the ideas and the outlook of APL that make it so valuable both pragmatically but also philosophically. (And which lead readily to better designs for this kind of thinking.)

    Best wishes,


    • 3. John "Z-Bo" Zabroski  |  December 4, 2010 at 5:45 pm


      @For example, “trig” can be taught quite readily to 4th and 5th graders if “what it is” is employed directly as purely geometric relations with no additional terminology.

      A couple of thoughts here… First, why doesn’t eToys have tools like the Spherical Easel applet [1] and the NonEuclid applet [2]. These are currently considered some of the fancier ways that US K-12 teachers have for showing students non-euclidean and hyperbolic geometries. They’re entirely constructive and the tools themselves could be made much better. (By the wya, as I understand it, Arizona is the US state that leads all others in geometry teaching.)

      It seems like we hit a barrier with most learning tools, where they are either closed source (like those applets) or they don’t allow a learning slope to occur, where students can progressively immerse themselves more and more in a topic.

      Finally, have you read Wildberger’s Divine Proportions? I mentioned it in the comments here recently in another post by Mark. The basic idea is to use purely geometric relations and get rid of transcendental functions, because they basically have nothing to do with the essence of geometry.

      BTW, to Mark: That quote comes from Whitehead’s Introduction to Mathematics. In that book, Whitehead goes over various notational advancements such as for arithmetic and then vectors. Its worth checking out. Its sad that we reduce quotes to just the quote and don’t engage people in the source. — I don’t think that is why Whitehead put the effort into writing a slim, novel formatted 96 page book. (I also really hate programming books that start off a chapter with an obtuse quote…. usually that just kills trees and shows the author’s lack of ability to write poetically by involving the quote inline with the lesson.)



      • 4. Alan Kay  |  December 5, 2010 at 9:37 am

        Hi John,

        Both the original Etoys and the version currently in use were aimed at a very narrow age group (the latter about 9-11) and at being a vehicle for helping to learn “powerful ideas” (mostly in the world of science).

        There are many things in the realm of mathematics that would be useful even for this narrow age group that we had in mind but didn’t do.

        However, the general success of Etoys made a sequel of much wider range in many dimensions quite exciting to think about. We are starting to think about that, and welcome constructive ideas about what and how.

        It’s interesting to think about “geometry” for K-6 and how one would go about helping children learn. I don’t think I would touch the computer for quite a while — the world is right there and should be used.

        I don’t think I would worry at all about either Euclid or non-Euclid, and if I did, I would just use physical objects, such as basketballs and globes of the Earth and real saddles.

        Re: Whitehead. I was very into Whitehead and Russell when I was a teenager and first saw “the quote” in a Scientific American article in the 50s, and thought “this doesn’t sound like anything Whitehead would say or mean”. So I chased it down, and discovered that the actual context of the quote is quite different than it is usually used.



    • 5. Mark Guzdial  |  December 6, 2010 at 3:26 pm

      Hi Alan,

      I’m intrigued by the comment: “The symbols are useful once the operations have been internalized.” That makes sense, and is a form of transfer that what we know about cognitive science suggests should work. It’s not about moving knowledge from one context to another, but instead is about coming up with a “name” (or symbol) for something already learned. I can imagine a study where we explicitly teach operations, test that students have developed fluency at them, then teach symbols for these operations, test for learning the symbols, then finally, test to see if students can use the symbols as they did the operations. Such a study would provide some useful insights into CS education — can we teach semantics before syntax, and then introduce syntax later?


      • 6. Alan Kay  |  December 6, 2010 at 4:47 pm

        Hi Mark

        I am certainly prejudiced along these lines, and in that order.

        I think this is especially key for younger learners.



  • 7. Catherine Lathwell  |  December 21, 2010 at 10:30 am

    First I have to confess. I went to Art School. My degree is in Fine Art and English Literature.

    My second confession. I learned APL as a child.

    When I hear COMPUTER Scientists complain that APL is too hard, and that it can’t be learned by children, I truly truly deeply honest completely wonder, why on earth NOT?

  • 8. Winter tributes to APL | Chasing Men Who Stare at Arrays  |  January 9, 2011 at 7:43 pm

    […] In addition, thanks to a tip from my Twitter buddy @kaleidic, I learned that Allan Kay defends APL when Lipton’s colleague at Georgia Tech, Mark Guzdial, chimed in with what boils down to “APL is too hard“. […]


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