Machine Math?

Calculators are an often used example in the philosophy of mind.  Sometimes they’re used analogously, to show how computational algorithms can be implemented in a variety of mediums (say, the very different circuitries of the calculator and the human brain).  Other times, they’re used metaphorically, as objects that we can attribute intentional states: the calculator ‘knows’ how to add and ‘believes’ that 2+2=4.  But how appropriate are comparisons between calculators and humans?  Is it a matter of implementing the same (or nearly the same) algorithm?  Or is the comparison a mere metaphor?   Stanislas Dehaene is the champion of the surprising view that neither of these (caricatured) approaches can be right: calculation is neither a matter of merely attributing intentional states, nor do humans and calculators implement algorithms in the same way.

(Apologies if this topic seems old hat to any – if you are a person already familiar with Dehaene, ‘cultural re-mapping’, number sensing, and the like, the payoff to re-reading this extremely cool and interesting stuff about human mathematical capabilities, is some very exciting and interesting new advances in brain localization and machine-learning)

Dehaene’s view is that our mathematical abilities result from the mixture of two evolved mechanisms, and, importantly, a sprinkling of language.  The first of these evolved mechanisms is a capacity to distinguish a certain amount of discrete quantities, or numerosity: the ability to tell apart one, two, three, and maybe four and five.  Then, there is the capacity to distinguish differences in quantity: that six is bigger than one, or that twenty is less than sixty.  Both of these abilities can be found in animals, and, yes, human children.  And it’s easy to understand why such mechanisms might persist over time*:  as an organism, it is very handy to have a capacity to determine between alternatives; whether option (a) was better than (b) because more nutrients, or less competition, or what have you.

What is gained when language is thrown in the mix is an ability to systematize quantity, to give every number a label that is easy to organize and remember.  This process builds upon the evolved mechanisms mentioned above, and extends them, particularly, our capacity in naming and utilizing discrete quantities (Andy Clark, in particular, has a good account of how this might work).  Eventually, through individual learning from teachers and parents, an infant gain the ability to determine absolute difference between number.  This replaces the kind of logarithmic guesstimation of our biological heritage.  (Radiolab has a fantastic podcast focussing on this aspect of Dehaene’s theory) –And from this, our ability to do mathematics follows (more or less).

So now, let us return to the calculator. Can it still serve as an image, a guiding picture for an understanding of how humans do math?  Well, maybe.  Whatever it does – and it certainly does calculate – a calculator certainly doesn’t have the same mathematical upbringing that we do.  All this means is that, whatever analogical perspicacity a calculator might bring to the mathematical abilities of full-grown, mathematically-capable humans, it won’t shed any light on how these mathematically-capable humans come to have their (near) algorithmic capacities.  If we really want to understand human-style math, we’ll need a new, genetic way of understanding such capacities.

Luckily though, we are beginning to create such a genetic account.

Marzo Zorzi, of the University of Padua has led a team that have created a neural network with the power of comparing estimated quantities.  Unfortunately while the article doesn’t link this research to Dehaene, it’s clear to see that this neural network mirrors the evolved mechanism he (that is, Dehaene) invokes to explain human numerosity.  This is exciting stuff!  Not only might this research help us situate such a pattern in the brain (perhaps via this new groundbreaking method of localization), but it may eventually help us to understand the way in which such a circuit gets recruited by language to allow for our distinctively human way of implementing mathematics.

Of course, there are always caveats.  This neural network has to ‘learn’ the trick of comparing estimated quantities, whereas it seems that evolution has made this ability in some way ‘innate’.  And further, we seem to be a ways away from any ‘hybrid’ approach in machine learning (indeed, language comprehension is a huge, if not the problem for artificial intelligence).  Never the less, I’m please and excited that we are slowly phasing out the more simplistic calculator-cum-human-calculation image, for more realistic comparisons and metaphors for our distinctively human ability to use math.


*It’s a particular bugbear of mine when scientists use the phrase ‘why such a (mechanism/organ/function) evolved’ – rather than this more innocuous and, in my view, correct phrase – thus reading a certain amount of teleology into the particular mechanism/organ/function that isn’t there.



Klein, Colin – Philosophical Issues in Neuroimaging

Clark, Robin – Generalized Quantifiers and Number Sense

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