Comments on “The parable of the pebbles”

counting, rationality and Homo sapiens

Ed Giniger 2019-03-04

From the perspective of the evolution of rationality, or perhaps its potential universality, it is interesting to note that recognizing numerical correspondence is not an exclusively human, or even primate, capability. Crows have demonstrated numerosity. They have the intrinsic capability to compare numerical quantities and recognize even rather abstract correspondences (eg., matching n sequential sounds to n visual objects). Whether the mechanisms used by crows and primates reflect evolutionary convergence or orthology is unclear at this point, to the best of my knowledge. In the experimental setting, ambiguities are generally minimized, this could easily be modulated. Either way, recordings from single neurons makes it clear that there is intrinsic nebulosity in the neural representation of numerical quantities - in crows and in primates - and that the neural circuitry employs error functions in making comparisons.


David Chapman 2019-03-05

Hi Ed, nice to see you here, thanks for the comment!

Corvids are amazing all-around and I love them. I’ve read popular accounts of behavioral studies on their ability to count (among other feats of reasoning). I didn’t realize that this had been studied in the lab as well.

I’d be interested to hear more about the error correction functionality!

reference on crows; also about reproducibility in biology

Ed Giniger 2019-03-06

The best place to start to see what’s been done on numerosity in crows is to look up Andreas Nieder, in Tubingen. The short version is that individual neurons in the relevant region of the brain are tuned for specific numbers of inputs. It does not seem that the crow “counts” the way your shepherd does so much as that it estimates a count, which can be either from a simultaneous viewing of objects or a sequential interaction (depending on the percept). Thus, larger numbers light up cells at a glance, as do small numbers; there is no extra delayfor “counting”. The “error function” in a comparison is therefore comes from the usual sort of analog synaptic interactions among neurons tuned to different values. There is a certain amount known about the neurophysiological details, but I’d have to look it up to remember them. The error metric is linear on a log scale (which I think is also true in human psychophysics, but don’t quote me). There are also cells tuned to zero, but this seems to act as a numerical value, not a categorical one - it is closer to “1”, further from “2”, etc on the same scaling as any other small number. Categorical evaluations can be made, however, for example for >, <, = and not = .
On a different point, I was reading through “the eggplant” and noticed your comment alluding to the reproducibility crisis in science. I don’t believe this actually exists. There are individual subjects, notably human psychology, where it is very real and has largely to do with the fact that the observations are so astronomically distant from underlying mechanism, connected only by uncountable layers of speculative philosophizing. But in basic biology, for example cell and molecular biology, where such a noise has been made about it, what people have taken to be a reproducibility crisis is really just a progressive denial of what you dub nebulosity. (Thanks for the word, by the way; it gives me a perfect way of talking about a concept that ought to be obvious to anyone who has ever done biology, but increasingly is disallowed.) It seems that the more our observations show us the inherent fuzziness of categories, and the chaotic (sensu stricto) connection between initial conditions and final observations in biology, the more that people try to pretend to a pseudo-Newtonian determinism that just doesn’t fit biological reality. One can come up with various sorts of sociological babel to account for this, but it comes back at us as a belief that science is getting less “rigorous”, when in fact, increasingly we are simply denying the intrinsic messiness of reality.

Crows and replication

David Chapman 2019-03-08

Thanks for the pointer to Nieder! First thing I found was this good popular article, which said:

Specific NCL neurons reacted to specific numbers, and activity in the NCL reflected the crows’ behavior. The birds made more mistakes, for example, when comparing neighboring numbers, especially when the numbers were large (for instance, four versus five). This could potentially be explained by the pattern of neural activity Nieder saw in the NCL: Neurons that reacted strongly to higher numbers were also more likely to be activated by neighboring numbers than those that reacted to small numbers. Perhaps most intriguingly, the same behavioral and neural results were previously found in the primate PFC.

I hadn’t known this about even the primate case.

what people have taken to be a reproducibility crisis is really just a progressive denial of what you dub nebulosity… people try to pretend to a pseudo-Newtonian determinism that just doesn’t fit biological reality. … it comes back at us as a belief that science is getting less “rigorous”, when in fact, increasingly we are simply denying the intrinsic messiness of reality.

Ah, very interesting! I need to think about this more (which is part of why it took me a couple days to reply).

As a first cut, though… it seems that there’s two axes here. One is whether or not it is possible (even in principle) to find law-like patterns in some areas; the other is whether the claims of a study hold up when re-tested. There could be a replication crisis in an area that could support exact claims, if many studies were done badly. There could be high replicability in a field that is inherently messy, so long as the studies were done carefully and the conclusions drawn were suitably hedged or vague. (What do you think?)

I didn’t know that basic CMB was experiencing a (perceived) replication crisis. (Medical research, yes, but there’s some special factors there.) I did know that neuroscience is. In particular, my outsider impression is that fMRI kinda mostly doesn’t work.

Cellular and molecular bio replication

David Chapman 2019-03-10

[Ed had trouble posting the following comment because I recently installed a new anti-spam package that is overzealous. I’m posting it for him.]

Your interpretation is correct, of course. Reproducibility problems in biology come in many flavors. There are subjects that are irreducibly ambiguous (at least given current intellectual perspectives), like much of social psychology. Clinical psych as well, where I (and many others) would argue that the current, symptomatic classification of diseases is skew to the underlying biological nature of those diseases. fMRI has had its own issues, partly due to unthinking application of canned software packages by investigators who don’t actually understand, or think about, the underlying statistical issues. There was quite a to-do about this a couple of years ago when someone took the trouble to point out a bug in the common application of a common package (due partly to default values in the software that applied only under particular conditions) and showed that it called into question a lot of published work. That particular bug is now well understood; whether other such things exist I can’t tell you. In principle, however, done properly, there is nothing intrinsically wrong with fMRI if done and analyzed correctly. And, as you imply, some classes of medical research, particularly drug trials paid for by pharma, has the dual problem of reporting bias (negative results get buried) and bias introduced by the influence of lawyers and money.

Regarding cell and molecular biol, there has also been much “reproducibility” hullabaloo in the last few years. Partly, CMB has just been tarred by the same brush applied to other forms of research. Partly there really has been an increase in bad research that manages to get published, driven in particular by the proliferation of non-reputable open access journals that are little more than vanity presses (not to be confused with the proliferation of perfectly respectable open access journals that are trying to expand the availability of research findings). The relevant non-crisis, however, has to do with an increasing pretense that CMB should be “quantitative” in the way the mesoscale Newtonian mechanics is quantitative. A huge fuss was made a couple of years ago about a report that an effort had been made to reproduce a hefty number of seminal findings in cancer biology by farming them out to contract labs and asking them to recreate exactly the original experiments. People were shocked, just shocked, that in some cases, experiments that had required years of work by exceptionally talented trainees in the best labs in the world could not be gotten to work the same way by random technicians in unnamed contract labs working to a constrained timeline. In quite a number of other cases, the reproducing labs actually did get the same basic answer as the original finding but the actual, numerical values of things like effect sizes were different from those originally reported, and that was classified as “failure to reproduce”. Of course, exactly why anyone who has ever actually worked with real biological samples would expect subtle cellular experiments to give the same numerical values in unrelated experimental settings is a mystery to me, given the near-infinitude of unknown and (currently) unknowable differences in boundary conditions and details of handling. I’d classify this last kind of failure to reproduce as the one that is most relevant to our discussion, and most reflective of simple physics envy.

Link to “Using a rational framework" goes to

Benjamin Taylor 2020-11-03

Broken link

David Chapman 2020-11-06

Er, um, … The part of the outline that points into is nebulous enough that I can’t even attach a page name to it; so unfortunately I can’t fix it yet.

Thank you very much for pointing it out, nonetheless!

Origins of the parable

pozorvlak 2024-02-27

I can’t find an older exact match for the parable, but discussing the origins of mathematics in terms of pebbles goes back at least as far as John Stuart Mill’s 1843 “A System of Logic, Ratiocinative and Inductive” (, and people have been criticising it since at least Frege’s 1884 book “The Foundations of Arithmetic” (which makes fun of Mill’s “gingerbread and pebble” arithmetic, before discussing some cases in which natural number arithmetic initially seems to apply but gives wrong conclusions).

Prototypically countable

David Chapman 2024-02-27

Interesting, thank you!

Maybe pebbles are the prototypically countable objects—durable, easily manipulable, readily available…