Mathematics, Jewishness, and Direction

When I was nearly 18 I was part of the British team to the International Mathematical Olympiad (IMO) in Bucharest (see my account of the experience).  The US team were Jewish1, all eight of them.  While this was noteworthy, it was not surprising. There does seem to be a remarkable number of high achieving Jewish mathematicians, including nearly a quarter of Fields Medal recipients (the Maths equivalent of the Nobel Prize) and half of the mathematics members of the US National Academy of Sciences2.

Is this culture or genes, nature or nurture?

As with most things, I’d guess the answer is a mix.  But, if of culture, what? There is a tradition of Biblical numerology, but hardly widespread enough to make the substantial effects. Is it to do with the discipline of learning Hebrew, maybe just discipline, or perhaps is it that mathematics is one of the fields where there has been less prejudice in academic appointments3.

I have just read a paper, “Disembodying Cognition” by Anjan Chatterjee, that may shed a little light on this.  The paper is an excellent overview of current neuroscience research  on embodiment and also its limits (hence ‘disembodying’).  One positive embodiment result related to representations of actions, such as someone kicking a ball, which are often depicted with the agent on the left and the acted upon object on the right.  However, when these experiments are repeated for Arab participants, the direction effects are reversed (p.102).  Chaterjee surmises that this is due to the right-to-left reading direction in Arabic.

In mathematics an equation is strictly symmetrical, simply stating that two thinsg are equal.  However, we typically see equations such as:

y = 3x + 7

where the declarative reading may well be:

y is the same as “3x + 7”

but the more procedural ‘arithmatic’ reading is:

take x, multiple by three, add seven and this gives y

In programming languages this is of course the normal semantics … and can give rise to confusion in statements such as:

x = x + 1

This is both confusing if read as an equation (why some programming languages have := read as “becomes equal to”), but also conflicts with the left-to-right reading of English and European languages.

COBOL which was designed for business use, used English-like syntax, which did read left to right:

ADD Tiree-Total TO Coll-Total GIVING Overall-Total.

Returning to Jewish mathematicians, does the right-to-left reading of Hebrew help in early understanding of algebra?  But if so then surely there should be many more contemporary Arab mathematicians also.  This is clearly not the full story, but maybe it is one contributory factor.

And, at the risk of confusing all of us brought up with the ‘conventional’ way of writing equations, would it be easier for English-speaking children if they were introduced to the mathematically equivalent, but linguistically more comprehensible:

3x + 7 = y

  1. Although they did have to ‘forget’ while they were there otherwise they would have starved on the all-pork cuisine[back]
  2. Source “Jews in Mathematics“.[back]
  3. The Russians did not send a team to the IMO in 1978.  There were three explanations of this (i) because it was in Romania, (ii) because the Romanians had invited a Chinese team and (iii), because the Russian national mathematical Olympiad had also produced an all Jewish team and the major Moscow university that always admitted the team did not want that many Jewish students.  Whether the last explanation is true or not, it certainly is consonant with the levels of explicit discrimination in the USSR at the time. [back]

tales from/for Berlin – appropriation, adoption and physicality

A few weeks ago I had a short visit to Berlin as a guest of Prometei, a PhD training program at the University of Technology of Berlin focused on “prospective engineering of human-technology-interaction”. While there I gave an evening talk on “Designing for adoption and designing for appropriation” and spent a very pleasant afternoon seminar with the students on “Physicality and Interaction”.

I said I would send some links, so this is both a short report on the visit and also a few links to appropriation and adoption and a big long list of links to physicality!

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matterealities and the physical embodiment of code

Last Tuesday morning I had the pleasure of entertaining a group of attendees to the Matterealities workshop @ lancaster. Hans and I had organised a series of demos in the dept. during the morning (physiological gaming, Firefly (intelligent fairylights), VoodooIO, something to do with keyboards) … but as computer scientists are nocturnal the demos did not start until 10am, and so I got to talk with them for around an hour beforehand :-/

The people there included someone who studied people coding about DNA, someone interested in text, anthropologosts, artists and an ex-AI man. We talked about embodied computation1, the human body as part of computation, the physical nature of code, the role of the social and physical environment in computation … and briefly over lunch I even strayed onto the modeling of regret … but actually a little off topic.

Alan driving

physicality – Played a little with sticks and stones while talking about properties of physical objects: locality of effect, simplicity of state, proportionality and continuity of effect2.

physical interaction – Also talked about the DEPtH project and previous work with Masitah on natural interaction. Based on the piccie I may have acted out driving when talking about natural inverse actions

ubiquity of computation – I asked the question I often do “How many computers do you have in your house” … one person admitted to over 10 … and she meant real computers3. However, as soon as you count the computer in the TV and HiFi, the washing machine and microwave, central heating and sewing machine the count gets bigger and bigger. Then there is the number you carry with you: mobile phone, camera, USB memory stick, car keys (security codes), chips on credit cards.

FireFly on a Christmas treeHowever at the Firefly demo later in the morning they got to see what may be the greatest concentration of computers in the UK … and all on a Christmas Tree. Behind each tiny light (over 1000 of them) is a tiny computer, each as powerful as the first PC I owned allowing them to act together as a single three dimensional display.

embodiment of computation – Real computation always happens in the physical world: electrons zipping across circuit boards and transistors routing signals in silicon. For computation to happen the code (the instruction of what needs to happen) and the data (what it needs to happen with and to) need to be physically together.

The Turing Machine, Alan Turing’s thought experiment, is a lovely example of this. Traditionally the tape in the Turing machine is thought of as being dragged across a read-write head on the little machine itself.

However … if you were really to build one … the tape would get harder and harder to move as you used longer and longer tapes. In fact it makes much more sense to think of the little machine as moving over the tape … the Turing machine is really a touring machine (ouch!). Whichever way it goes, the machine that knows what to do and the tape that it must do it to are brought physically together4.

This is also of crucial importance in real computers and one of the major limits on fast computers is the length of the copper tracks on circuit boards – the data must come to the processor, and the longer the track the longer it takes … 10 cm of PCB is a long distance for an electron in a hurry.
Alanbrain as a computer – We talked about the way each age reinvents humanity in terms of its own technology: Pygmalion in stone, clockwork figures, pneumatic theories of the nervous system, steam robots, electricity in Shelley’s Frankenstein and now seeing all life through the lens of computation.

This withstanding … I did sort of mention the weird fact (or is it a factoid) that the human brain has similar memory capacity to the web5 … this is always a good point to start discussion 😉

While on the topic I did just sort of mention the socio-organisational Church-Turing hyphothesis … but that is another story

more … I recall counting the number of pairs of people and the number of seat orderings to see quadratic (n squared) and exponential effects, the importance of interpretation, why computers are more than and less than numbers, the Java Virtual Machine, and more, more, more, … it was very full hour

AlanLcoblo - artefactsAlan

  1. I just found notes I’d made for web page in embodied computation 5 years ago … so have put the notes online[back]
  2. see preface to Physicality 2006 proceedings[back]
  3. I just found an online survey on How many computers in your house[back]
  4. Yep I know that Universal Turing machine has the code on the tape, but there the ‘instructions’ to be executed are basically temporarily encoded into the UTM’s state while it zips off to the data part of the tape.[back]
  5. A. Dix (2005). the brain and the web – a quick backup in case of accidents. Interfaces, 65, pp. 6-7. Winter 2005.[back]

Single-track minds – centralised thinking and the evidence of bad models

Another post related to Clark’s “Being there” (see previous post on this). The central thesis of Clark’s book is that we should look at people as reactive creatures acting in the environment, not as disembodied minds acting on it. I agree wholeheartedly with this non-dualist view of mind/body, but every so often Clark’s enthusiasm leads a little too far – but then this forces reflection on just what is too far.

In this case the issue is the distributed nature of cognition within the brain and the inadequacy of central executive models. In support of this, Clark (p.39) cites Mitchel Resnick at length and I’ll reproduce the quote:

“people tend to look for the cause, the reason, the driving force, the deciding factor. When people observe patterns and structures in the world (for example, the flocking patterns of birds or foraging patterns of ants), they often assume centralized causes where none exist. And when people try to create patterns or structure in the world (for example, new organizations or new machines), they often impose centralized control where none is needed.” (Resnick 1994, p.124)1

The take home message is that we tend to think in terms of centralised causes, but the world is not like that. Therefore:

(i) the way we normally think is wrong

(ii) in particular we should expect non-centralised understanding of cognition

However, if our normal ways of thinking are so bad, why is it that we have survived as a species so long? The very fact that we have this tendency to think and design in terms of centralised causes, even when it is a poor model of the world, suggests some advantage to this way of thinking.

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  1. Mitchel Resnik (1994). Turtles Termites and Traffic Jams: Explorations in Massively Parallel Microworlds. MIT Press.[back]

multiple representations – many chairs in the mind

I have just started reading Andy Clark’s “Being There”1 (maybe more on that later), but early on he reflects on the MIT COG project, which is a human-like robot torso with decentralised computation – coherent action emerging through interactions not central control.

This reminded me of results of brain scans (sadly, I can’t recall the source), which showed that the areas in the brain where you store concepts like ‘chair’ are different from those where you store the sound of the word – and also I’m sure the spelling of it also.

This makes sense of the “tip of the tongue” phenomenon, you know that there is a word for something, but can’t find the exact word. Even more remarkable is that of you know words in different languages you can know this separately for each language.

So, musing on this, there seem to be very good reasons why, even within our own mind, we hold multiple representations for the “same” thing, such as chair, which are connected, but loosely coupled.

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  1. Andy Clark. Being There. MIT Press. 1997. ISBN 0-262-53156-9. book@MIT[back]