Sunday, 24 April 2011

2 Become 1

One occasionally reads that arithmetic doesn't apply "exactly" because, for example, if you physically aggregate two drops of water the result is one drop of water. An example of this generic kind of thought is:
If one adds a litre of water to a reservoir a billion times one would certainly not end up with exactly a billion litres of water, because litres of water cannot be measured that accurately. In scientific experiments measurements of this type (should) always have error bounds, which quantify the degree to which simple arithmetic fails. (E.B. Davies, 2005: "Some Remarks on the Foundations of Quantum Theory", Brit. J. Phil. Sci. 56, p. 530.)
From the true premise that "adding 1 litre" a billion times does not yield a quantity of water of one billion litres, the conclusion is drawn that "simple arithmetic fails". How is this inference justified?

There are two mistakes here concerning how arithmetic is applied in ordinary reasoning. First, about what the bearers of cardinalities are. As Frege, Russell and others pointed out, cardinalities are born by sets (or classes, or concepts), not concrete lumps: for what would the cardinality of, say, Karl Kautsky be? Second, about the meaning of the addition symbol $``+"$ in such applications. Write $``n = c(X)"$ to mean "$n$ is the cardinality of the set $X$". Then addition, $``n + k = p"$, is defined as:
$\exists X, Y, Z(n = c(X) \wedge k = c(Y) \wedge p = c(Z) \wedge (X \cap Y = \emptyset) \wedge (Z = X \cup Y))$
So, addition $+$ of cardinal numbers "represents" set-theoretic union $\cup$ of disjoint sets, and not physical juxtaposition(or aggregation) of concrete lumps.

[There is different notion of addition defined for ordinals, and this does involve "concatenation" of sequences: that's how a Turing machine adds numbers. But for finite cardinals and ordinals, the addition structures are isomorphic.]

So, in Davies's argument above that "simple arithmetic fails", there was an implicit assumption that adding one litre somehow corresponds to the successor operation $S$ on $\mathbb{N}$. But this is not the case. For the successor $S$ operation, if $X$ is a finite set, then adding a distinct element $e$ to $X$ yields a set of cardinality $S(c(X))$. But this quite different from aggregating or fusing a new concrete thing with some other.

Of course, two concrete things can be fused to form one concrete thing. But indeed the union of two sets is one thing. This is just functional application: $f(A,B)$ is just one thing, by definition, despite the operation taking two arguments.

On this topic, a Spice Girls video: "2 Become 1",

[Update, 25th April: I edited the post, with an example.]


  1. The Spice Girls video, that was a cheap shot! :)

    But seriously now, on the notion of 'bearers of cardinality', a Wittgenstein quote that I like:

    This is how our children learn sums; for one makes them put down three beans and then another three beans and then count what is there. If the result at one time were 5, at another 7 (say because, as we should now say, one sometimes got added, and one sometimes vanished of itself), then the first thing we said would be that beans were no good for teaching sums. But if the same thing happened with sticks, fingers, lines and most other things, that would be the end of all sums.
    “But shouldn’t we then still have 2 + 2 = 4?” – This sentence would have become unusable. (RFM, § 37)

  2. Hehe - I have to think of lame mathematical excuses for these Hendrix, Jacksons, Spice Girls, etc., videos!

    Yes, it'd definitely be harder to learn arithmetic in W's world, if the concrete things that mundane count nouns refer to keep vanishing, and fusing mysteriously - concrete implementations of computations would be unreliable in Ws world.
    If things were that bad, the same would apply to tokens of syntax too: a token of "thank you" might transform into a token of "goodbye", and communication would be impossible.
    Since most predicates have a built-in time parameter, the set S(t) of Fs at time t, = {x : x is an F at t}, will have a time-varying cardinality n_F(t).
    In W's world, n_{bean}(t) is an unpredictable function of t.

    But even so, n_F(t) is a property of the set of Fs at t, and not a property of any F, or even its aggregate (since, as Frege pointed pointed, an aggregate of n Fs can be an aggregate of k Gs). So, it'd still be true that if there are n Fs, k Gs, and no F&Gs, there are (n+k) (FvG)s.

    Against W, though, I think that so long as expression *tokens* didn't fusing/disappear, etc, in the way W suggests, the sentence "2 + 2 = 4" would be usable; one could teach arithmetic using expression tokens: e.g., I say, "yeah - yeah - yeah" as a representation of 3. But count nouns whose referents behaved unstably, by fusing/disappearing, such as "bean", would be hard to use.

  3. Jeff, have you heard of populations in the Amazon and in the Australian outback who have few words for numbers? They've been quite extensively studied recently (e.g. the famous 'Log or Linear?' paper in Science, 2008). At least for some of these populations, the practice of counting is entirely alien.

    The reason why I think this is significant is that a world where objects behave in a sufficiently stable way (at least *some* objects) would appear to be a necessary but not *sufficient* condition for the practice of counting to emerge. I'm not sure how this observation would fit into your general story, but I find this material fascinating and philosophically important (still trying to figure out how exactly!).

  4. Catarina - yes, you probably know too of th recent controversy concerning the Piraha, a tiny isolated tribe in the Amazon, discussed on Language Log a few times. The empirical claim is that their language lacks recursion, which means even the weakest assumption in the Chomskyan research program is threatened.

    On the linguistics side, I'm not sure what to conclude, as I'm too ignorant of linguistic theory. But it's a further inference that the minds of the language users don't use (as opposed to have reflective capacity for meta-representation) basic computational operations.
    What I think is that mental processes in all humans involve concatenation and substitution - these are sufficient to generate all computable functions (concatenation gives +, while substitution gives x). So,
    (i) "||" concatenated with "|||" gives "|||||". (2 + 3 = 5)
    (ii) substituting "|||" (3) for each occurrence of "|" in "||||" (4) gives "||||||||||||". (3 x 4 = 12)
    But while these processes account for language competence, they needn't be meta-represented in the language itself. Possibly, social/cultural conditions might make the explicit implementation of that further capacity irrelevant.
    So, I think experiments involve concatenation and substitution need to be performed. The idea would be to see if subjects can perform certain (language-independent) tasks on tokens: e.g., substitute "aa" for all occurrences of "a" in "bababa". Also, one could test higher primates to see if they can perform substitution tasks.

    On the stability issue, I agree, yes. But my reply to W is that the point would equally apply to the mental tokens (whatever they are!) that are involved in mental reasoning. The mind has to store tokens of bits of information as it proceeds. If these mental tokens lacked stability (e.g., short term memory was very unreliable), thinking would be impossible too! So, *token* performances of counting, speaking and thinking do require stability conditions on their tokens (whether linguistic, mental or bundles of sticks, etc.).

    If you'd like to join us on the blog, Catarina - I can send you an invite!

  5. Hi Jeff,

    These are two slightly different debates, both fascinating, and as it turns out they concern the same linguistic groups (the Piraha in particular, but there is another tribe, the Mundukuru). But the people investigating their numerical conceptions (Pica, Dehaene etc.) are not really focusing on the recursion debate, but rather on the emergence (or non-emergence) of the conception of the natural numbers as a linear series. Take a look at this article, if you are interested:

    And thanks for the invite! If nothing else, I could be the token woman among the contributors of this blog :) But seriously, obviously I am very much enjoying all the posts here, and I guess it might be nice to cross-post here the more philosophical/logical posts that I write at New APPS (*very* different readership!). But remember that I hardly ever write symbols these days :)

  6. Hi again, just to clarify that I would be delighted and honored to become a contributor to this blog! I realized afterward that my 'token woman'joke may have been infelicitous; as it turns out, I am a feminist with a weird sense of humor...

  7. Cool - it'd be great! After I signed up the Center staff (all blokes, indeed...), I invited a few other friends, three men and three women too, but the women couldn't make it - hopefully one may join later.
    Can you send me an email (to jeffrey.ketland at

    Thanks for the link. I know Dehaene's book "The Number Sense", but it was mid-90s - so long ago I'd forgotten about it.