Two kinds of arithmetic.

Drawings of mathematical problems predict their resolution was a University of Geneva press release picked up by on March 7, 2024. It concerns research published in Memory & Cognition on February 14. That article, by Hippolyte Gros, Jean-Pierre Thibaut, and Emmanuel Sander, focuses on the role mental representations play in how people solve word problems involving whole numbers.

As the authors encapsulated their findings, “What we count determines how we count.” According to them, there are two different mental models for arithmetic, with different problem-solving strategies, and the one we choose is often determined by the context of the problem. They call the first model the cardinal representation. In a cardinal representation, a number stands for how many elements are in a certain set. The other model is an ordinal representation, in which a number refers to a certain position on the number line. (This second usage is related to, but different from, the way the term “ordinal” usually occurs in mathematics.)

Here is an example of a cardinal problem, taken from the list used in their experiment:

A bag of pears weighs 8 kilograms. It is weighed with a whole cheese. In total, the weighing scale indicates 12 kilograms. The same cheese is weighed with a milk carton. The milk carton weighs 3 kilograms less than the bag of pears. How much does the weighing scale indicate?

In an earlier paper the authors had established that subjects faced with a cardinal problem of this type, and asked to use as few operations as possible, would get to the solution in three steps. Using the terms from this example,

  1. The cheese weighs 12 – 8 = 4kg.
  2. The milk carton weighs 8 – 3 = 5kg.
  3. The weighing scale will indicate 4 + 5 = 9 kg.

Now an example of an ordinal problem, with the same arithmetic structure:

Obelix’s statue is 8 meters tall. It is placed on a pedestal. Once on the pedestal, it reaches 12 meters. Asterix’s statue is placed on the same pedestal as Obelix’s. Asterix’s statue is 3 meters shorter than Obelix’s. What height does Asterix’s statue reach when placed on the pedestal?

(Asterix and Obelix are cartoon characters, as familiar to the French as Tom and Jerry are here.)

Subjects with the ordinal problem and the same instructions would not be distracted by the equivalent of calculating the weight of the cheese (which would be the 4m height of the pedestal), and solve the problem in one step:

  1. Asterix’s statue reaches 12 – 3 = 9 meters.

As they describe it, problems involving weight, price and collection tend to lead to a cardinal analysis, “due to these quantities usually describing unordered entities.” On the other hand, problems involving height and duration are more often treated as ordinal problems “due to daily-life knowledge underlining the intrinsic order of the entities they mention.”

In this new paper, the authors investigate what mental activity leads to these different strategies. To this end, they asked subjects to give a solution to the problem, as well as a diagram of the problem. The authors worked with 111 subjects (59 adults and 52 fifth graders). Each was given a booklet of 12 problems, a mix of 6 cardinal and 6 ordinal (the $x, y, z$ in these tables were substituted with numbers in the range $z < 4 < x < y < 15$ in the tests).

Two of the diagrams submitted for the prompts quoted above. Top: the bag of pears, the cheese, and the carton of milk (LAIT) are identified on two sides of a scale and labeled with their weights. Bottom: the statues and the pedestal (SOCLE) are symbolized by rectangles; a vertical graduated axis registers height (taille) in meters. Images courtesy of Hippolyte Gros.

The authors then developed a protocol for evaluating the extent to which a diagram represented cardinal or ordinal thought. Identifiable clusters of objects and graphically rendered set inclusion were cues to the presence of cardinal thought, while ordinal thought would manifest itself as axes, graduations, and intervals. Analyzing the results, the authors found that the cardinality of the diagrams predicted the use of the three-step strategy, and the ordinality of the diagrams predicted the one-step, regardless of whether the problem itself was of cardinal or ordinal type. This held for both children and adults.

Abel Prize for a probabilist.

The awarding of the 2024 Abel Prize (sometimes described as the Nobel Prize of mathematics) to the French probability theorist Michel Talagrand was reported by Kenneth Chang in the New York Times, March 20, 2024. Chang largely followed the Abel Committee announcement in citing three different areas of Talagrand’s work.

  • Stochastic processes. These are phenomena subject to random variation. Chang mentions the water level in a river—let’s call it $L$. Characteristics of such a process are its average value $\mu$ and its standard deviation $\sigma$. The standard deviation is a measurement of how often and how much $L$ deviates from $\mu$. A simple example of Talagrand’s work is an estimate for the maximum of the water level $L$ over a long period of time. The estimate calculates the probability $P$ of $L$ exceeding its mean by a certain amount; it has the form (I’m following Bodhisattva Sen’s 2022 lecture notes at Columbia, p. 101)
    $$P[L\geq \mu + f(\mu, \sigma, x)] \leq e^{-x}$$
    where $f$ is a specific function devised by Talagrand.
  • Concentration of measure. As Chang describes it, in this second area Talagrand “helped show that there is a measure of predictability within random processes.” Here we have an expository lecture, “A new look at independence,” by the master himself. He describes the phenomenon as follows: “A random variable that depends (in a ‘smooth’ way) on the influence of many independent variables (but not too much on any of them) is essentially constant.” The setting for his estimate is much more general, but he starts his lecture with tossing a coin $N$ times, counting $+1$ for heads and $-1$ for tails and tallying the sum $S_N$. He estimates, for any $t\geq 0$, the probability that $S_N$ is greater than $t$ in absolute value:
    $$P[|S_N| \geq t] \leq 2\exp\left(-\frac{t^2}{2N}\right)$$
    for any $t\geq 0$. So the probability of $t = 550$ or more heads or tails in $N=1000$ tosses is $P[|S_{1000}|\geq 100]\leq .014$, and the probability that the number of heads is between 450 and 550 is at least $1-.014=.986$. (Besides the point for this exposition, a much sharper estimate is possible using the Chernoff inequality. This number, $.997$, is the one reported by Chang, presumably working from Talagrand’s 2019 Shaw Prize citation.)
  • Spin glasses. A spin glass is a mathematical model for a special kind of matter in which atoms are arranged “amorphically” (not regularly as in a crystal) and in which their interactions are not confined to nearest-neighbors (unlike, for example, the Ising Model.) To investigate what configurations give local minima for energy, physicists have made mathematical models such as the Sherrington-Kirkpatrick (SK) model. This model dates back to 1975 and is described allegorically in this 2014 article by Dmitry Panchenko, in terms of the Dean’s problem.
    Left, POS HALL, labeled $\sigma = +1$. Right, NEG HALL, labeled $\sigma = -1$. Between are cells with links between them labeled $g_{ij}$.
    The Dean’s problem. The Dean must assign each of $N$ students to one of two dorms, as copacetically as possible. Assignment of student $i$ to Pos Hall is recorded as $\sigma_i=+1$, and to Neg Hall as $\sigma_i=-1$. Students $i$ and $j$ may like each other ($g_{ij}>0$) or dislike each other ($g_{ij}<0$) with $|g_{ij}|$ telling us how much. Image by Tony Phillips.

    With this notation, in a perfect solution the product $g_{ij}\sigma_i\sigma_j$ is positive for every pair $i,j$: if students $i$ and $j$ like each other ($g_{ij}>0$), they get assigned to the same dorm ($\sigma_i\sigma_j =1$) and if they don’t ($g_{ij}<0$) they will be in different dorms ($\sigma_i\sigma_j =-1$). This is often too much to hope for—say 2 is BFFs with 4 and 5 while 4 and 5 cannot stand each other. But the Dean can make everybody as happy as possible by maximizing the sum $\sum_{i,j}g_{ij}\sigma_i\sigma_j$. Formally, this is the same problem as minimizing the energy in the SK model of a spin glass.

    When $N$ is large, solving the problem is very hard. Around 1982, the Italian physicist Giorgio Parisi found a way to package the configurations to make the computation possible. He derived his equation using his powerful physical intuition. This work, in part, earned him the 2021 Nobel Prize. But mathematically speaking, it remained a conjecture. Chang quotes Talagrand: “For a mathematician, this doesn’t make any sense whatsoever.” Nevertheless, Talagrand developed the mathematical machinery necessary for a rigorous proof in 2006.

Michel Talagrand’s Abel Prize was also covered by Davide Castelvecchi in Nature on March 20. Both reporters mention Talagrand’s “unconventional” career among French mathematicians: he did not attend the École Normale Supérieure in Paris, their traditional spawning ground, and went instead through the regular university system, in Lyon. But he was not some nobody from the provinces. At age 22 he took the Agrégation examination in mathematics, the same national test that all the math normaliens have to pass, and came in first.

—Tony Phillips, Stony Brook University

Read other recent Tony’s Take columns.