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How the brain 'approximates' without counting (medicalxpress.com)
109 points by dnetesn 52 days ago | hide | past | web | favorite | 45 comments

Tangentially related, there was an interesting Reddit post on this a few years ago: https://www.reddit.com/r/askscience/comments/2ayhkp/if_someo...

Cool. It kind of reminds me of how problems are often solved in an FPGA.

You can only have so much combinatorial logic before you need to break the problem up into multiple stages. The subitizing thing seems like solving the problem in one step with combinatorial logic, whereas counting is like breaking the problem into multiple stages.

It makes sense that the brain can only do so much combinatorial computation before the problem needs to be broken up into smaller pieces, incorporating memory.

Thanks, that's golden!

Something related that I've always been curious/uncomfortable about:

Can you count without using a language?

Try this: Clap your hands or tap on something an arbitrary number of times. Can you tell how many times you did it without "saying" one, two, three in your head?

Even if you pay attention to it, it seems impossible to count without language.

At least not a stream of sensory inputs; small clusters of around 10 or fewer things seem easily countable from just looking at a "snapshot."

I can't turn it off, but I know I can very well do math without words or numbers in my head.

Geometry is the best example, anythig beyond the basics has no good common names and still you can do it in your head.

If you slice through a pair of carrots, how many pieces of carrots are there afterwards? (do this simple thing in your head an than read on)

I slice them imaginary and than start counting what I'm left with. The part where I go from 2 carrots to 4 carrot parts happens without words. Anyone doing it differently? You of course have to be self-observing enough to say more than "I just know" :).

That might as well be a shortcut, as in your brain has done 2*2 so many times that you have an immediate path for that fixed answer... if this is the case there's no actual calculation performed, as in you can't do the same thing for "what if you slice 73 carrots in half"...

> if this is the case there's no actual calculation performed

There is no conscious calculation performed. The result might be fetched from memory, but there might also be subconscious processes involved. And that seems to be what the argument is about. Grandparent claims we cannot count without language while the article claims we have estimation hardware builtin (that's count-ish, imo). That's not really conflicting, since GP implies a specific counting process that's usually done consciously. Do the clapping, ask someone unsuspecting how often you have clapped, and he'll probably be able to tell the number without concisely counting. Even for a stream of sensory inputs.

But yeah, consciously thinking about something usually involves words for many of us, as that the most common/effective way for our thoughts to interact with the world.

I'm kinda shure I do it visually :)

But yea, get's more difficult with 73. But even there I might determine "visually" that cutting across a row of carrots halfs each of them. So the 2 in 73*2 ist still determined without words.

It depends what you mean by counting I suppose. You can carry out algorithms that do counting for you without language.

For instance you can use tokens to do counting. If you want to count the number of sheep in a herd you could put a small rock in a bag for every sheep. This bag of rocks will then "represent" the herd. And you can use it to reason about properties of the herd.

If lets say you want to count that all sheep are present you can move the rocks from one bag to another. If you want to split the herd between two peeople you can make two piles roughly the same size and take as many sheep.

You could have another type of object representing months in a year. Then you could assign the rocks to months to ration food.

When you start grouping your tokens in well defined shapes, or introduce tokens of different denominations you are on the slippery slope towards language.

I seem to remember Feynman talking about how he "said" numbers in his head as he counted and ran across someone who saw numbers tick by visually.

Of course if you count symbols ticking by visually as "language" then yes, at the root the brain is going to use some sort of symbol to represent increasing quantity.

I think there's one way to count without using a language: music, or rhythm. When listening to music, I seem to be able to tell how many beats pass away, without actually keeping count without imagining any words in my head.

The book 'Teach your baby math' suggests that toddlers can learn to recognise how many dots are shown to them, up to at least 200.

I haven't fully explored this, but made a simple web app to implement part of the training system in the book. It shows you (or your child) a number of dots, and says the number out loud.


Very few humans can directly perceive more than maybe 10 individual items in a group without counting. If you're telling me you can show a toddler a group of 157 dots and have them say "157", I don't believe you.

(Maybe they can learn to recognize a particular grouping of dots in a book, but that's a different question.)

I have read reports online of people succeeding with this method using paper flashcards.

However, like you, I wonder:

- are these reports true?

- even if they are true, has the child just memorised a small section of each particular grouping?

Of course, this trick wouldn't work if a parent is using my online tool, as the patterns of dots are pseudo-random, so the same number is unlikely to have the same representation on subsequent views.

I mean, we don't really know the limits of human neuroplasticity. It is certainly a stretch, but I wouldn't rule it out for young children.

A crazier example, to me, is the Moken children in Thailand who can see clearly underwater (http://www.bbc.com/future/story/20160229-the-sea-nomad-child...).

Would be neat to know what the margin of error is. Is it less than 10%? ie given 157 dots, how difficult is it to estimate 150-170 dots?

I think part of this is a lack of practice. I've only counted how many macoroni noodles were in my bowl once

Who is the author of that book?

This is among other things is what the book Thinking, Fast and Slow by Daniel Kahneman is about.

Totally recommended read.

Many of the studies cited in that book have been toppled by the replication crisis in psychology.

Is there a consolidated source explaining this? Say, if I wanted to read the book, but avoid learning false factoids?

Here are a few examples: https://jasoncollins.blog/2016/06/29/re-reading-kahnemans-th...

I would just avoid factoids from psychology altogether for the next few years while the researchers in that field sort everything out. https://www.theatlantic.com/science/archive/2018/11/psycholo...

Worse, some of the famous studies aren't merely underpowered or based on nonrepresentative samples (like college kids) or sloppily executed — some famous studies are outright fraudulent. https://www.vox.com/2018/6/13/17449118/stanford-prison-exper...

I believe the bit about priming was refuted in some repro attempts.

If you're short on time, watch Kahneman's talk at Google, which basically summarizes the first chapter or so. https://www.youtube.com/watch?v=CjVQJdIrDJ0

I haven't yet read this and won't until after work but I did click through and notice it didn't mention subitizing.

Subitizing is pretty cool and it's the name for how you can look at items and instantly know how many there are if it's less than some small number (approximately 5 iirc).

Here's a wiki article on it for anyone interested: https://en.wikipedia.org/wiki/Subitizing

Interesting! I was having a conversation last weekend with a friend when I asked "How many objects can you count without counting, you just know the answer". I didn't know the name for it. For me the answer is three, four if they are in a good arraignment, five is almost always a group of three and two.

Yeah, I also have thought about this and didn't have a word for it. For me the number is definitely 5 most of the time, 6 if they're in a perfect rectangle.

I suspect this is why tally marks[1] are so effective and therefore common.

[1] https://en.wikipedia.org/wiki/Tally_marks

If I see a five-pointed star, or a five spoked wheel, I don't have to count to know how many. Probably same goes for six, or a hexagon. Maybe seven.

I believe that's recognizing a shape and knowing an attribute.

How does this perception mechanism fit in with how people conceptualize or register numbers?

A friend was telling me that when she thinks of a number, a particular visualization of it automatically comes into her mind. From how she described it, I recall thinking it sounded like an accidental approximation of something like a logarithmic scale, but less direct and less precise than it had to be. I knew her pretty well, and, other than this visual, she didn't appear atypical (was social, smart and educated, but no savant superpowers, nor any unusually low limits).

I was wondering whether she had a normal human conception of numbers, and somehow just had a little extra introspection on that. Or maybe this visual was an independent mechanism (perhaps learned in childhood). Or maybe she conceived of numbers differently than most people.

> A friend was telling me that when she thinks of a number, a particular visualization of it automatically comes into her mind. From how she described it, I recall thinking it sounded like an accidental approximation of something like a logarithmic scale

Sounds like spacial-sequence synesthesia. When I think of numbers I see them at specific points on a logarithmic(-ish) scale as well. I don't think it gives me any advantage in mental math / counting / estimating, but I can not disable it and try to experience numbers any other way. I have only learned it's not common in my early adulthood, until then I have just assumed that's the way everyone's mind works.

I wonder if they take into account the idea that you have some prior reference for expected object counts.

I felt like this is an important mechanism at play when I read the classroom example. If I saw a picture of a classroom for 2 seconds and am then asked to estimate the chair count, in my head it goes something like: mean #chairs in classrooms in my experience * fullness of the classroom in picture (e.g. .5-2 range) = count estimate.

But if I show you a bunch of made up widgets you never saw before, you’d still be able to guess.

reading articles on scientific paper is such as waste of time.

This is describing the introduction of the paper, i'd guess.

Since in cogsci the Gestalt effect/theory is widely accepted. What i believe the paper did (but i can't ready it) is to 'reverse engineer' the effect in some part of the brain and came up with an estimate of the number the process will 'estimate' based on your vision focal point into the collection of items.

...but who knows.


Here's an abstract for you. This link was provided in the article. Reading articles on scientific papers can be helpful if you like...read...and check the links/references. You know, the same way you might want to check the annotations in a scientific paper.

Thanks! couldn't find it at first.

I think it's more than just estimating the count of objects.

We also "approximate" for example a spatial distribution - think about how when running through rough terrain, you instantly know which path to choose to encounter fewer rocks. You are certainly not focusing and counting each individual rock.

The image in the article is really an example of a spatial distribution and fits very well. There's really no way to count which color there is more of in some area of the image, but you can still estimate which one there is more of. I'm not even sure how we could numerically assess that without guesstimation.

Eh? That is one of the easiest things you can do with parallel/analog/fpga computing. You simply sum (and this does not mean compute summing, it just happens) the signals that respond in a certain way. It doesn't require much.

Sure, with computing, but if you didn't have access to such technology?

I mean, just looking at our eyes: it's a field of evenly spaced measurement devices that send a direct signal to out brain. Ie we have a "parallel transmission of pixels".

You then just check what colour has the strongest (most) signals.

Or do you mean something else? Like how to do that consciously or something and not like, how it's possible?

Our rods and cones are no where near uniformly distributed, even after accounting for the blood vessel running around the retina.

If so, just add another layer that biases the result based on actual distribution.

So if I understand it correctly, the tl;dr is that we don't like "counting" items in our peripheral vision, so we approximate by mentally tallying how many "snapshots" it takes to see all the items in question?

I wonder what is the implication for this for people on the ASD spectrum.

How does that affect this?

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