How long is a piece of string? This proverbial question is not really intended to have an answer. But what about asking which of two strings is longer? That question is well-defined and has a clear answer. In the laboratory we probably would not use strings but we might present lines next to each other on a computer screen and ask participants to pick the longer one by pressing a key.

Suppose you didn’t know the outcome already, can you guess which of the 3 pairs of lines in the figure below would give rise to the speediest decision? And which would be slowest?

Intuition suggests the pair in the center panel would be easiest to discriminate, and the pair on the right would be most difficult—and would therefore be expected to take the longest.

In this instance, our intuition is borne out by the data: In countless experiments it has been shown that on virtually all perceptual tasks, responding slows and becomes more error-prone if stimuli are more similar to each other. This is perhaps unsurprising but the impact of these results should not be under-estimated: after all, it is compatible with the idea that people’s representation of the stimuli is analog in some manner. That is, people encode the lengths of the lines in a manner that retains some of their spatial properties. With the initial assumption that this representation is noisy, and the noise increases with magnitude, an analog representation can explain the observed data very nicely.

While it may appear obvious, it is not at all inevitable: In PowerPoint, for example, the lines in the rightmost panel are associated with precise numbers that characterize their lengths (2.65 cm and 2.61 cm, respectively, for left and right). If those two lengths were to be compared in a computer program (e.g., by something along the lines of “if(L1>L2) {print(“L1 is longer)}”), that comparison would be as quick as it would be for the panel on the left, where the two lines are of lengths 1.5 cm and 2.61 cm. Abstract numeric representations in a computer lose some of the (spatial) properties of the objects that are being represented.

Surely, numbers similarly lose their “spatial” properties when people encode them and represent them in memory? After all, why would the digit “3” be smaller (shorter? lighter?) than the digit “9” when we bring it to mind from memory or when we look at it on the screen? How can it be meaningful to talk about a “physical” aspect of numbers?

If your intuitions about the mental representation of numbers are captured by some of the preceding statements, then they turn out to be incompatible with much existing research. It appears that our representations of numbers, and how we process them, differ considerably from the processes inside a digital computer.

To illustrate, consider the figure below, which dates back to 1880 and was provided by Sir Francis Galton in the *Journal of the Anthropological Institute**:*

As the legend suggests, the drawings represent the mental representation of numbers provided by various correspondents. Each drawing is an attempt to visualize how that person represented numbers in their heads and is accompanied by a textual explanation. For example, a Colonel Yule explained that “I used to see them in gradations of colour, but I cannot fix these now with truth. I can only remember that 30 and up to 40 were of a subdued sunny colour; a division of the shade took place at 12.”

The mental life of numbers is therefore far from boring or colorless.

The introspections of Galton’s correspondents may appear amusing to us today, but modern experimental research has confirmed that people appear to represent numbers in ways that are remarkably similar to our representation of perceptual stimuli.

In a classic study in 1967, Moyer and Landauer showed that when people were asked to compare single-digit numbers, performance is worse when the difference between the two numbers is relatively small (e.g., 3 vs. 2 as opposed to 2 vs. 8) or when the numbers are relatively large (e.g., 7 vs. 9 as opposed to 2 vs. 4). The former effect is known as the numerical distance effect and the latter as the numerical magnitude effect. Both effects can be explained by assuming that people translate numbers into analog representations along a “mental number line.”

The mental life of numbers is much like that of perceptual stimuli such as lines of varying lengths.

Or is it?

A recent article in the *Psychonomic Bulletin & Review* examined the issue of number representation further and sought to tease apart the distance and magnitude effects. Researcher Attila Krajcsi presented participants with a large number of pairwise digit-comparison trials, similar to the task pioneered by Moyer and Laundauer.

In another condition, participants performed a very similar comparison task, except that on those trials the stimuli were presented as dot patterns rather than digits. The rationale for this comparison was that if numeric magnitude is represented in analog fashion, perhaps along a “mental number line”, then it should not matter whether the representation is tapped by symbols (i.e., digits) or by a graphical display (i.e., a corresponding number of random dots).

The results suggest that there is a symbolic component in our representation of numbers that differs from the representation that is evoked by dot patterns.

Specifically, Krajcsi first computed the slopes that related each participant’s response latency to the numeric distance between stimuli and their magnitude, respectively. In replication of much previous research, those slopes were found to be negative for numeric distance—that is, people slowed down as the difference between magnitudes decreased—and positive for numeric magnitudes—larger digits (or more dots) were slower overall than smaller digits (dots), irrespective of the distance between them.

The intriguing new result involves the correlations between those slope values: Is the magnitude effect associated with the distance effect? If a person is highly sensitive to numeric distance, are they also highly sensitive to magnitude (and vice versa)?

The figure below shows the main results for one of the studies reported by Krajcsi.

The left-hand panel shows the correlation for the dot task. Each dot represents a different participant. It is clear that participants who showed a greater distance effect (i.e., a more negative effect of distance) also exhibited a greater magnitude effect (i.e., the slope for magnitude was more positive). This is exactly as one would expect if the same (analog) representation is involved in both tasks. Indeed, the strength of the association is quite striking.

The right-hand panel shows the data for the digit task. This pattern is quite different: although participants show a numeric distance and a magnitude effect, there is no correlation between those two measures. This result is in conflict with the classic analog model of number representation.

Krajcsi proposes an alternative model to explain the data which invokes a discrete semantic

System (DSS) as driving the observed effects. This model is sketched in the figure below in comparison to the conventional analog model (here called ANS):

The DSS suggests that the distance and size effects rely on different mechanisms. The distance effect is ascribed to a semantic representation that encodes digits by their proximity in the same way that some semantic-memory models are postulating a network of concepts, in which words such as ‘cat’ and ‘dog’ are more closely related than ‘dog’ and ‘milk.’ The magnitude effect, by contrast, is ascribed to the frequency of the symbols—that is, the smaller digits are encountered more frequently in our lives than larger digits. (A three-course meal is more common than a nine-course meal.) The greater frequency of those digits imbues them with a processing advantage.

The results of this study cannot confirm the applicability of the DSS (although in another paper Krajcsi has reported further support for the model), but they do suggest that the mental life of numbers is not limited to a single number line.

Perhaps Galton’s correspondents were introspecting about their DSS when they mused about the color of the numbers in their head.

*Article focused on in this post:*

Krajcsi, A. (2016). Numerical distance and size effects dissociate in Indo-Arabic number comparison. *Psychonomic Bulletin & Review*. DOI: 10.3758/s13423-016-1175-6