[Humanist] 31.218 on mathematics: mixed and manipulatory

Humanist Discussion Group willard.mccarty at mccarty.org.uk
Mon Jul 31 07:22:10 CEST 2017

                 Humanist Discussion Group, Vol. 31, No. 218.
            Department of Digital Humanities, King's College London
                Submit to: humanist at lists.digitalhumanities.org

        Date: Sun, 30 Jul 2017 09:23:31 -0400
        From: "William L. Benzon" <bbenzon at mindspring.com>
        Subject: Re:  31.217 on mathematics: mixed and manipulatory
        In-Reply-To: <20170730073702.452F36909 at digitalhumanities.org>

Yes, Willard, there’s something worth developing.

As you’ve mentioned, when we first learn to count, we do so through a physical process: we enumerate collections of small objects while uttering number words. And then we’ll be given sheets of paper where we see depicts of small collections and we’re asked to list the number of objects in the collection. That is, we are to write the numeral that corresponds to the appropriate number word (‘1’ for ‘one’) etc. Then when we are taught calculation we are taught to write the numbers in a certain format. As exactitude is important we drill on this for hours and this particular kind of exactitude does not come "naturally.”

More abstractly, think about the Turing machine. Turing asked us to image a paper tape moving over a head which can read symbols from the type and write to it:

> ...an unlimited memory capacity obtained in the form of an infinite
> tape marked out into squares, on each of which a symbol could be
> printed. At any moment there is one symbol in the machine; it is
> called the scanned symbol. The machine can alter the scanned symbol,
> and its behavior is in part determined by that symbol, but the
> symbols on the tape elsewhere do not affect the behavior of the
> machine. However, the tape can be moved back and forth through the
> machine, this being one of the elementary operations of the machine.
> Any symbol on the tape may therefore eventually have an innings.[17]
> <https://en.wikipedia.org/wiki/Turing_machine#cite_note-17> (Turing
> 1948, p. 3[18]
> <https://en.wikipedia.org/wiki/Turing_machine#cite_note-18>)
From Wikipedia entry: https://en.wikipedia.org/wiki/Turing_machine <https://en.wikipedia.org/wiki/Turing_machine>

You might also want to take a look at the cognitive metaphor work by George Lakoff and Raphael Núñez. There’s their 1997 article, The Metaphorical Structure of Mathematics: Sketching Out Cognitive Foundations for a Mind-Based Mathematics, which you can download here: http://escholarship.org/uc/item/5qq7q51z  http://escholarship.org/uc/item/5qq7q51z

On page 37 you will find this:

> Arithmetic Is Motion
> Numbers Are Locations on a Path.
> The Mathematical Agent is a Traveler along that path.
> Arithmetic Operations Are Acts of Moving along the path.
> The Result of an arithmetic operation Is A Location on the path.

And so forth. They’ve devoted a book to the topic: Where Mathematics Comes From (Basic Books 2000), which has a Wikipedia page (including critical comments): https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From <https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From>

I should note that I’m skeptical of the cognitive metaphor program for various reasons. Over the years Lakoff & Co. have turned it into something of a Theory of Everything, which automatically invites either uncritical acceptance or suspicion. But there IS something at the core of this enterprise that is worth attending to. 

The Wikipedia article on Numerical Cognition may also be of interest: https://en.wikipedia.org/wiki/Numerical_cognition <https://en.wikipedia.org/wiki/Numerical_cognition>

Piaget has some remarks on the biological origins of mathematics in his Biology and Knowledge (1974). Alas, my copy is in storage so I cannot find the relevant passage.


> On Jul 30, 2017, at 3:37 AM, Humanist Discussion Group <willard.mccarty at mccarty.org.uk> wrote:
>                 Humanist Discussion Group, Vol. 31, No. 217.
>            Department of Digital Humanities, King's College London
>                       www.digitalhumanities.org/humanist
>                Submit to: humanist at lists.digitalhumanities.org
>        Date: Sun, 30 Jul 2017 08:10:21 +0100
>        From: Willard McCarty <willard.mccarty at mccarty.org.uk>
>        Subject: mixed and manipulatory
> Rather than talk exclusively about 'applied' mathematics, which suggests 
> taking a tool developed elsewhere, then applying it in a new 
> situation, I'd like to hear more about what was once called mixed 
> mathematics (Oki, Historia Scientarum 23.2, 2013), also known as 
> physico-mathematics (Schuster, Synthese 185, 2012) or sub-scientific 
> (Høyrup, Hist. of Science 28.1, 1990), and in anthropology, 
> ethnomathematics (Ascher, Mathematics Elsewhere, 2002). I know, I am 
> blurring over some distinctions here, but my basic interest is in 
> getting at manipulatory, combinatoric operations in which some kind of 
> mathematics, even if by proxy, accompanies or arises from 
> kinaesthesis, as when you count with your fingers or move calculi 
> (little stones) around, i.e. compute with an abacus. How about (to
> shift to geometry), when a South Pacific islander moves a boat 
> according to a memorised schematic of the sea currents? 
> It seems to me that the term 'mixed mathematics' would do us well, 
> at least as a starting point, but then I am conditioned by my history.
> I learned programming first from assembler language (e.g., in English, 
> 'load accumulator with contents of memory location X, shift left 
> accumulator 1 bit, store result in memory location Y') and similar. 
> So I have a penchant for thinking of what goes on computationally 
> in such physical terms. But what about those not so long in the tooth? 
> Does computing have kinaesthetic appeal or meaning? Is there
> anything here worth developing?
> Suggestions? Comments?
> Yours,
> WM
> -- 
> Willard McCarty (www.mccarty.org.uk/), Professor, Department of Digital
> Humanities, King's College London; Adjunct Professor, Western Sydney
> University and North Carolina State University; Editor,
> Interdisciplinary Science Reviews (www.tandfonline.com/loi/yisr20)

Bill Benzon
bbenzon at mindspring.com


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