We and many other creatures share a mechanism for what you might call extracting numerosity, or the number of
things, from the environment. It's really very important for all sorts of creatures and for all sorts of reasons.
- (Brady: Extracting numerosity from) (the environment, what does that mean?) What I mean by this is that a fish needs
to know how many other fish are in a shoal so it can join the larger shoal, so the numerosity of fish in shoals is important. Lions need to know how many invading lions into their territory there are; usually this is done through sound, they can hear them coming, so they can decide whether to flee or to fight. So this is just extracting the number of things from the environment. Of course they have to decide what's relevant in the environment; so the lions are paying attention to the number of roars that they can hear and the number of distinct roarers. The fish are paying attention, not to the the seaweed, but just to other fish that are swimming around. So they
have to focus on a particular set of objects. So this is a mechanism that fish and lions and and we share; at least we think we share them, we certainly have similar mechanisms. Whether all these mechanisms come from a common ancestor or not I don't know, and this requires genetic research - at the moment that's not available. (Brady: Professor do animals need to
know the) (number or they just need to know) (a greater than, equal sort of thing? Do) (they know, okay there's more on the left) (than the right; they don't need to) (extract an actual number do they?) Sometimes that's enough, having a you know a rough sense of how many there are may be sufficient. So if you've got a lot of fish swimming around in a shoal, let's say 100 fish over here versus 50 fish over there, you're not going to be able to enumerate them but you've got to have a way of extracting, at least approximate numerosity from that. The way in which that's done is probably something like this: you kind of extract the approximate density of the fish, then you see how big an area these fish cover, so you can; it's basically density times area, then fish, fish can do that. Sometimes you need to do it exactly. So if there are three lions defending against four lions they need to know that there's just one more of them, then they have a decision
to make. If we're four and they're three then we attack, we defend our territory. So sometimes you need to know it exactly, sometimes approximate will do. One proposal came out in the early 80s is that we all possess an accumulator. And so the more things that you experience the fuller the accumulator
gets. Let me give you a helpful analogy. Supposing I've got a jug, let's say with marks on it, and for each thing I see I have a cup of water and I pour it into the jug. So for each thing, however big the thing is, it's the same cup of water. Or wherever the thing is it's the same cup of water. So if I see three things then the jug will be filled to there, if I see four things it'll be to there, five things to there - up to about ten things. This is a very simple mechanism. There's evidence now, at least in frogs, that there's a particular neuron that does this. In monkeys there's evidence that there's a mechan- there's single neurons that seem to behave like one of these accumulators; that is the more things that the monkey sees the
more this particular neuron fires. So there's- I know this is Numberphile so I can say this; a monotonic relationship between the number and the firing rate. In the case of other animals it may not be firing rate, it might be done in some other way, we don't know. It may be done within a neuron or it may be done in terms of connections between neurons. Let me give you an example from
frogs, now, most people think frogs are pretty stupid but here they perhaps tell us humans something that we ought to have known all along. So imagine these, this particular species of frogs have been studied, they're in a swamp. And they want to mate and the male wants to mate with a female, so it has to attract a female. The female can't see it and he can't see the female; so it's- mating is done by, by sound. So what the male frog has to do is- (Brady: The choice of a mate at
least, not the mating itself) Not the- that's true, not the mating itself but the choice of a mate uh depends on sound. What the male has to do, it has to make it appear that it is the most- the fittest male in the swamp. And it does this by making a series of noises called chucks. [Frog noises] Okay I'm not very good at frog noises. The female will be attracted by the number of chucks in a phrase. Because the more chucks you can make more fit, more substantial you are. Now number's important. It's better than volume because volume will depend upon how far away you are from the male, but number doesn't depend on how far away you are from
the male. And it turns out that frogs compete with each other. So if I make five chucks, my neighbouring male might make five plus one chunks. And this might escalate until you reach the limit of how much breath you can have to make a number of chunks in a phrase. (Brady: It's like an auction)
- That- it is, it's a bit like an auction. Now why does- I don't know if it applies to humans that if you make a lot of noise, or more noises, you're more likely to attract a mate; but anyway that's the way it works in frogs. Now what's interesting is that Gary Rose at the university of Utah, he showed there's a neuron in the brain of the frog that seems to act like an accumulator. So it it hears the number of chucks
made by another frog and it thinks, 'ello he's made five, I have to make an extra one. So then it will generate an extra chuck to make it six chunks.
- (Brady: So it's not only counting the) (number that the other guy made, it
has to) (count how many it makes?) That's right. It has to- yeah I've only been making four so I better I better up my game. So it looks as though there's there is uh something rather like an accumulator in the brain of the frog. Now we don't know much about it for other- the brains of other creatures and we don't know whether there's an equivalent mechanism in our brain; but there's no reason why they shouldn't be. So for example when we think about timing there's a gene which you can find in the fruit fly, which we're only very very distantly related to, called TIM, which is a timing gene which responds to a particular interval. Now it turns out this TIM gene is conserved in us. Now we use it in a rather different way from
the the fruit fly; and it combines with other genes and other mechanisms to make our timing mechanism more general and more detailed than the fruit fly, but you know if there's a gene in the fruit fly for
time there could be a gene in the frog or even in insects, who can also count, for counting which we share with them; and which builds an equivalent type of mechanism. (Brady: You use the analogy of
like a measuring) (cylinder or a vessel being) (filled, in an actual brain what is being) (filled? Like where is, where's
this stuff) (being stored? What's being filled up to) (accumulate into?) (I can imagine the pulses from the) (neurons but how are they stored?) Yes. That's that's a very very good question, and it's one that is extremely controversial - you've hit on a very raw, or should I say nerve, um here. So one argument, which has been made
by Randy Gallistel at Rutgers University in in- he was also that he was one of the co-organisers of this meeting; he argues that things like time and number are actually stored in polynucleotides, in in the neuron itself. And this is going to have an effect on the way in which the neuron connects with other neurons. Another idea is that it's got to do with the connections between neurons. There have been some modelling exercises, usually neural networks, where either the network learns to recognise a particular numerosity, like fiveness, because - if it's fiveness it goes into particular state and the virtual neurons connect up in a particular way and if it's not five then they connect up in a different way. Or it can just be some kind of internal mechanism which, for example Tahan has proposed, whereby a network of neurons can organise itself into say twoness or threeness or fourness or fiveness, roughly speaking.
- (Brady: So does that mean, as) (each accumulation happens) (the neural network says, 'okay let's move) (into fourness position?) (Ooh there's been another one let's move) (into fiveness posit- oh now we're) (going to go into sixness position'?) Yeah that's that's what the neural network people would say, and it's not entirely clear how that's going to work. That's one of the the the current controversies. But in a simple accumulator mechanism it will be a single neuron that responds to a particular numerosity of whatever it happens to be tuned to. If it's, let's say tuned to the number of sounds it hears, that's what it will respond to so it'll respond more to fiveness than to fourness. And in fact there is some evidence that in monkey cortex, and indeed in a kind of homologous part of bird brain, there are individual neurons that respond preferentially to particular numerosities. So if the monkey sees an array of five this neuron will respond more to five than to four and so on to sixes. And similarly in the bird brain. There is evidence for single neurons acting as an accumulator. In fact in the monkey brain the number neurons seem to
be in a cleft in the parietal lobe. So the parietal lobe's round there, and there's a kind of a fold. At the bottom of the fold, technically the fundus, there are these number neurons. If the number neurons are down there, on the lateral surface here there are going to be accumulator neurons - sometimes called summation
neurons - which fire more when they experience more things. And it's believed that- it's not been yet demonstrated - that there's a connection between these neurons, these accumulator neurons, and the number neurons. So if this accumulator neuron gets to fiveness then it will activate down here the neuron which preferentially responds to fiveness. So that's the way people think it works at the moment. (Brady: If we've seen this
accumulator perhaps) (working in the frog, which is very
exciting,) (why haven't we been able to see it in) (humans that we probably study even more) (than frogs?) (Why has it not shown itself yet?)
- Ah, well that's an interesting question and that's because we can use very fine electrodes to probe the brains of frogs, and at the moment we're not allowed to do that with humans; unless for example it's in the course of surgery. So, for example, in intractable epilepsy the surgeon would like to remove the epileptic focus, but the surgeon would like to do that without damaging the parts of the brain that the patient really needs. So you have to find out which bits of the brain are really important to the patient and what you do is you open up the skull, you put a grid of electrodes on the exposed brain, and then you see with various tests which parts of the brain, rather selectively now, just a few thousand neurons, respond let's say to speech or to particular words; or in this case to particular numbers. And so what, for example, Josef Parvizi
in Stanford has done is he's found neurons, or perhaps neuron clusters, that respond to particular digits. So in part of the brain down there, which is where you do some of your visual processing, there seem to be little clusters of neurons that respond to the digit five, though not to fiveness, okay, not to the numerosity five. And now there's some other work, done by Carlo Samenza in Italy at the University of Padua. And what he does is something slightly different. With epileptic patients what he does is he stimulates the brain in such a way that it temporarily disables the bit that's being stimulated. So he's been looking at calculation; so there are particular bits of the brain which, if you stimulate them, stops you doing addition but not multiplication. And another bit that stops doing multiplication but not addition. So it looks as though, although addition and multiplication are spread over several different sites, each particular site, just a few thousand neurons, is doing a particular arithmetical operation. (Brady: These people have to be
conscious for) (them to know?)
- Yes they have to be conscious, that's right. So we open up the- well not me personally - but the brain is opened up and the person has to be
conscious; or else that the surgeon doesn't know what the different bits of the brain are doing.
- (Brady: Who are the people that really) (want to know) (these- the inner workings of how our) (brain's dealing with numbers?) One of the problems is that not everybody is very good at learning arithmetic. And there are
many different branches of mathematics, and some people are not very good at arithmetic but are very good at geometry; we've tested some of that. Some people are not very good at geometry, but are
very good at arithmetic, and we've tested some of those. So we're talking really here about arithmetic because this is the closest we get to what animals can do. So there's some people who, despite every opportunity, don't seem to be able to develop arithmetic in the normal way. Now if they've inherited a mechanism for rep- extracting numerosity from the environment, sorry if I have to use that expression again, and then representing it in there in their minds; if that's not working very well then this is a problem. Because this seems to be the foundation for learning arithmetic. Understanding what the numerosity of a set is is critical to learning arithmetic. A set has a definite number of members; if you add another member it changes the numerosity, if you take a member away it changes the numerosity, so it's very exact very specific. And it's the difference between a set of five and a set of six, which has just got one more member. So if you're not very good at representing sets and their numerosities - both of which by the way are abstract things. I mean a set is abstract, and the numerosity of
the set, property of the set - also abstract. That's a whole other issue that's plagued philosophers for at least two and a half millennia. The point here is that these
people are not very good at the foundations. Not very efficient at the foundations for arithmetic. So they don't, for example, know that fiveness is just one more than fourness. They can count okay but their internal representations not very good. We, we might speculate how- their accumulator doesn't work very efficiently. So if it doesn't work very efficiently they're not very- they don't have a very reliable representation, let's say a fiveness. And that means that the relationship between their representation of fiveness and the digit five, or the word five, is not very good. And this means that they are really bad at learning arithmetic, they need extra help. Rather in the way that a dyslexic who has a problem relating lesser
strings to words has a problem and needs special
help. What we've been trying to do is trying to create digital environments in which kids can play around with sets, combine sets together - which is like addition - or separate sets, which is like subtraction, and relate sets to digits and and to words. We feel this might help overcome this problem which is usually called dyscalculia. Now dyscalculia is actually quite a big problem. I mean it probably affects something like 5% of the population. And it's been calculated that this costs
us at least two and a half billion pounds a year in the UK. So worldwide if it's 5% of the worldwide population it's a serious and expensive business; um because people who've got dyscalculia have all sorts of consequential life problems. Like they're not very good with money, they're not very good at work; so it's known to be a serious problem in everyday life. One of the first dyscalculics we ever saw was in prison for shoplifting. And the reason he shoplifted was he was too embarrassed to go to the counter with his purchases because he didn't know how much money to give, he didn't know if he was getting the right change. And of course kids in school, you know the 5% of kids in school who have this problem get this every - well five days a week because they have maths lessons five days a week. So one of the things we're trying to do is to help these kids overcome that problem, for which you need specialised help. Now one of the things I'm trying to persuade teachers, educational professionals like educational psychologists, and ultimately the government is that the problem with these dyscalculics is that this mechanism that we've seen operating in a whole range of different
creatures might be the same in humans; that Johnny is not really as good as Billy or on these kinds of tasks. We're trying to show that this is an innate problem, rather like colour blindness. A small mutation has affected the way in which
this accumulator mechanism in the brain has been built, and this is this has got long long term consequences. Unfortunately the government doesn't recognise it at the moment. And other bodies that should be taking it seriously are not recognising it either. However, Singapore - which is a country that always
comes top or close to top on all international mathematical comparisons - they take it seriously. They say we've got 10% of kids who need learning support in mathematics, we don't know why they're having this problem, but we'd like to try and help them. At least- maybe half of those will be dyscalculics. So we're we're collaborating with the Singapore government, and the national institute of education there, to try and develop some digital environments in the first place to help these kids overcome their
problems. (Brady: Dyslexia doesn't have
this problem of) (recognition and help does it?) No it doesn't. One of the reasons: it's thought that if you can't- if you're finding reading difficult or spelling difficult, this is a real handicap - and it is a real handicap. Also a lot of celebrities have come out as dyslexic, showing that it's possible to be successful even though you're dyslexic. However it's also the case that if you have- if you've got poor literacy you've also got worse life chances. But it also turns out that if you've got poor numeracy you've got even worse life chances than having poor literacy. (Brady: But why isn't that coming out?) Because people think, oh well it's it's okay to say I'm terrible at maths; Where it's not okay to say I'm terrible at language or I'm terrible at reading
or I'm terrible at spelling - some might say
that. So I think attitudes are a bit different. Also, the other thing is that vastly more money has been thrown at the dyslexia problem than has been thrown at the dyscalculia problem - vastly more money. Both here, in America and Europe - but of course this is part of what I call the virtuous or the vicious triangle. Because there's no recognition there's no money, because there's no money there's no research, because
there's no research there's no recognition; and so on. Whereas if we could just intervene in one of those nodes in the triangle maybe we could make a big difference. If you've got better research or more research or more spectacular research we get more recognition; you get parents who are complaining to their local MP that their children are not getting the right type of help because they're dyscalculic, then government might change their mind. But even though there's a- the British dyslexia association, and there was dyslexia action; and of course there's autistic societies and ADHD societies; there isn't a dyscalculia association 'the British dyscalculia association'. So parents at least bear some of the responsibility here; but of course they don't know that the reason that that Johnny is not learning arithmetic is because he's dyscalculic - they think- either think that Johnny is lazy or Johnny is stupid; but in fact he's not he's just just
dyscalculic. [Extras] Okay so this is a
digital environment that we've created to help dyscalculics. And the idea is that they understand about the relationship between digits and sets and they understand about what sets are like...
Hopefully this will explain why I wrote 2 + 3 = 6 in an exam once.
Yes. Quite different from the other Numberphile videos. I like it. It opens a whole new area I didn't know about.
Very interesting video. I would've liked it if he could talk about the leap from counting to numbers. Obviously we don't have accumulator neurons operating on values in the tens of thousands and much higher. There must be some bridge of abstraction that connects the general concept of number (perhaps encoded visually or linguistically as a decimal) and a counting neuron. And how about continuous quantities? Does anybody know about this?
It's nice to see them do math-adjacent videos, but I wish they'd go back to just making interesting videos about interesting numbers.
Periodic Videos has a video about every element, there's no excuse for Numberphile not having a video about every number.
I suspect I might have discalculia, I learned to add and subtract by memorizing every instance of those operations involving single digits, something which I'm told is not how most people do it
This was a que so didn’t even know needed answering lol
Such simple questions I never thought about. E.g., how do lions determine if they are outnumbered? Great video.
Awesome video. Thanks for sharing!
Mine uses my fingers, sometimes a calculator.