Why the Human Brain Perceives Small Numbers Better

For what it's worth, the upper limit for subitization varies quite a bit.

The neurotypical limit is usually four, but can be raised with practice. Children with Down's Syndrome may not be able to subitize at all. Autistic savants can subitize up to thirty or more, with unconfirmed cases up to several hundred.

This was exactly my undergrad research topic! My knowledge is a bit out of date (perhaps this is in TFA), but the running theory is that there are two number systems. The first operates noiselessly and immediately for numbers under 4. The second works in the exact way you are describing, where the magnitude of errors are linear in the size of the set (the “analog magnitude” system). As I recall, there is substantial evidence supporting the idea of two separate systems rather than one.

I don’t really follow your point about integers. We know the brain does object detection at a low level, so it makes sense that the number systems will operate over countable elements.

I think this is a near-universal experience among people who’ve played a lot of Risk, but did everyone else collect their little dudes into units of 5, and then collect those up into further collections of five five-unit-squads? After that, 125 units is too many to really be necessary at once.

There are arena in the brain which encode integers up to <small number I forgot>. Alex’s Adventures In Numberland was a fascinating book.

It even kinda works for something like tar.

tar czvf output files

The czvf is really four arguments I guess, but it expresses a single concept in some sense.

Perhaps a fish or an insect might benefit more from knowing an accurate approximation of their or another school/swarm, than it would from knowing if there are 2 or 4 of them somewhere (since there seldom are so few).

Brain systems do handle order of magnitude changes. E.g. going from dimly lit room [1-3] to bright sunlight. [1000-2000]

For me it's four, my brain just instantly recognizes that there are four things, that instant pairing of pairings: oooo

Five is a bit more iffy, I can obviously tell quickly that there are five things, but then it feels like it's my 3 and 2 pattern recognition kicking in plus some addition math: ooo oo. If you spread the objects out in a weird way where I don't see that 3+2 grouping then I have to count em.

Many years ago at a LAN party at my friend's house his father came in with a news article saying studies showed that gamers could develop a better sense for instantaneous assessment of count, up to 6 I vaguely recall. Anyhow I do remember reconsidering my aptitude, since for me it gets confusing with >= 5.

For what it's worth, the upper limit for subitization varies quite a bit.

The neurotypical limit is usually four, but can be raised with practice. Children with Down's Syndrome may not be able to subitize at all. Autistic savants can subitize up to thirty or more, with unconfirmed cases up to several hundred.

It's a visual count. The visual system probably isn't optimized for counting. It identifies and tracks objects, but not many. I'm not surprised 4 is a common boundary, and I also wouldn't be surprised if it were somewhat hard-coded. It could well be limited by a "lower" layer.

The experiment is artificial, so it might not represent counting in a more natural setting, where e.g. the objects might have features that distinguish them better. The article brushes over that.

There's a lot of evidence from other studies of being able to track about 4 simultaneous objects. It looks like (but we don't have concrete evidence quite yet) that somewhere (maybe higher level parts of the visual cortex, maybe something higher level like state scratch pads, maybe something else we haven't thought of) there's a four wide logical pathway. When we overload it, it seems that we sort of time slice whatever it is, jumping between the objects at roughly four at a time. In addition to performance tests, there's also the data point that there's a surprising amount of languages in pre agricultural societies that have words for 'one' through 'four', but anything else is some degree of 'many'.

And yeah, you can train your brain to see larger sets if they're in a pattern that the middle parts of the visual cortex can train on, but the limit does seem to be around 4 visual symbols/tokens concurrently.

Another place you see an interesting bandwidth limitation is that pretty much all languages have a pretty fixed semantic bits per secondnwhen spoken. Specifically around 39 bits per second. We can listen and hear faster but rates, but it's very difficult to speak faster. Similar to the above discussion the cute hack to go faster is to essentially give a single internal symbol to a larger sequence of syllables through training, not casting off the limitation but sort of hacking around it. https://www.science.org/doi/10.1126/sciadv.aaw2594

Not sure if it's hard coded, but the brain is taking a lot of visual shortcuts because of the limited visual bandwidth you have.

https://youtu.be/HU6LfXNeQM4?si=PRdZP_hdwdyJQkdP&t=521

Your Brain: Perception Deception

This section is interesting on how little we see in high definition, and how much mental processing it takes.

Except here in the metric land when we want to buy a pound of something (or thereabouts) we ask for half a kilo. And the second can easily be 3/4 liters.

I don’t know if you intended this as a joke, but this is true and a good UX insight.

For day to day activities, the US customary units are more humane. Foot, pound, cup, are simply common sizes that one encounters. Foot and cup break into convenient units (powers of two in small scale volume, convenient rationals like a half, two thirds, etc for both). A standard stair riser is 2/3 of a foot — 8 inches — for example. The US system is better in this way than the imperial system as the former’s pint, quart etc are more conveniently sized with better fractions.

And, to your point: all this breaks down when the sizes get bigger (or tinier!) than you can physically “handle”. A yard/meter is pretty big (people don’t usually measure areas for rooms etc in yards) and beyond that metric becomes as good and most of the time easier. If only it hadn’t been designed around powers of 10!

But this principle extends to other special case units. For example while I prefer to use metric screws technical reasons, the fractions of 2 in drill threads are much more intuitive.

And you can see this principle breaks down for non-“intuitive” units: temperature is abstract, foot-pounds are not better than newton-meters etc.

But use the right tool for the job. I mostly write Lisp and C++ code but it doesn’t mean I won’t sometimes reach for Python.

Note: I was born in a (then) non-metric country and currently live in the remaining one, but have lived in metric land for a lot of my life and use them both all day long. So no bias.