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BI 137 Brian Butterworth: Can Fish Count?
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Brian Butterworth is Emeritus Professor of Cognitive Neuropsychology at University College London. In his book, Can Fish Count?: What Animals Reveal About Our Uniquely Mathematical Minds, he describes the counting and numerical abilities across many different species, suggesting our ability to count is evolutionarily very old (since many diverse species can count). We discuss many of the examples in his book, the mathematical disability dyscalculia and its relation to dyslexia, how to test counting abilities in various species, how counting may happen in brains, the promise of creating artificial networks that can do math, and many more topics.
Brian's website: The Mathematical Brain
Twitter: @b_butterworth
Can Fish Count?: What Animals Reveal About Our Uniquely Mathematical Minds
3:19 – Why Counting?
5:31 – Dyscalculia
12:06 – Dyslexia
19:12 – Counting
26:37 – Origins of counting vs. language
34:48 – Counting vs. higher math
46:46 – Counting some things and not others
53:33 – How to test counting
1:03:30 – How does the brain count?
1:13:10 – Are numbers real?
Brian 00:00:03 What underlies the numerical ability of, uh, ants or birds or fish also underpins our own numerical ability. My guess is that no Greek, no ancient Roman was a prodigious calculator because their notation was inappropriate for calculation. Self recognition is difficult when you can't do what everybody else in the class can. Do you think of yourself as being stupid? And I've had lots of adults come to me late in life, even in their eighties and say, at last I understand why I couldn't arithmetic. It's not,
Speaker 0 00:00:58 This is brain inspired.
Paul 00:01:11 Hey, it's Paul. My guest today is Brian Butterworth. Brian is emeritus professor of cognitive neuropsychology at university college, London he's written previous books like the mathematical brain and dyscalculia and his newest book. Can fish count what animals reveal about our uniquely mathematical minds, which is the main subject of our conversation today. The book like the subtitle suggests explores the counting and numerical abilities of many species, why it's important for organisms to count how evolution has shaped. What's important to count for individual species, the experiments to test counting abilities, how our brains might implement our numerical abilities and other related topics. Today we discussed dyscalculia for a while and its relation to dyslexia. Something Brian knows a lot about. Then we wander through a few of the topics. I just mentioned using examples like fish lions, birds, and bees, and a handful of others. Although the book has many more examples than we get to. So I link to the book in the show notes at brain inspired.co/podcast/ 137. If you like the topics discussed in general on this podcast and want to learn more about the intersection of neuroscience and AI, you may be interested in my free video series, open questions in AI and neuroscience in it. You'll learn about some key unanswered questions in both fields and the emerging field of neuro AI, which can help move those fields forward, go to brain inspired.co/open to get those videos. All right, here's my discussion with Brian Butterworth.
Paul 00:02:54 So Brian, the book is called can fish count, but, uh, really it's about the counting and numerical abilities of many species. It's a, a broad survey of which animals and organisms can, uh, count and how we tell how they can count, whether they can count. And of course, um, it's about, you know, the, the mechanisms of counting and so on. First of all, why are you interested in counting and why fish in particular?
Brian 00:03:25 Well, like much of science it's down to luck. So it was by luck. I, I became a neuropsychologist. I was just a regular psycho linguist. Then I got called in to do some analysis of, of phasic patients. And, uh, so I actually became quite well known for doing research on here. Then I, um, I had a, a pipeline, uh, to a university in Italy, uh, the university of, uh, Padua, uh, and, uh, under the European community, as it was, then I would send them students and they would send new students. And one of the students who came to work with me for her PhD wanted, uh, initially wanted to work Onia because that's what I did. And when she got London, she said, don't wanna work Onia I something that nobody works on.
Brian 00:04:25 So past and ed, I was interested in the foundations of mathematics and thought, why don't we, uh, look at, uh, mathematical abilities? And so we did a, a series of, of individual patient studies who had selective deficits initially of, uh, the mathematical abilities where previously there'd been okay at maths. And we also did a few patients who had, um, preserved mathematical abilities when other cognitive abilities like memory and language were shock. And what we discover or I say rediscovered was, was a particular the, which if damaged, um, led to difficulties with mathematics and if intact led to preserve mathematics, even though other parts of the brain might affect language and memory. And so on, then I started to ask why this part of the brain, and that's really, uh, the beginning of the story about how I gotta fish.
Paul 00:05:32 Ah, okay. So, so you're talking about dyscalculia dyscalculia.
Brian 00:05:37 Yes. <laugh> dyscalculia. Yeah. Um, that, that, uh, that, that came a little bit later. Um, so, um, one of the, one of the themes in neuropsychology, uh, back in the, uh, nineties was whether dissociations that you found in patients could be found developmentally. So a, a range of different disabilities in reading or in language had been identified in neurological patients. The question is could some kind of developmental disorder lead to a similar kinda pattern. So I wondered if this was the case with, uh, uh, numerical abilities. Um, if for example, uh, children have, uh, something anomalous about their, their paral, this lead to them having problems with, uh, at least with numerical tasks. And that's why I started thinking about, uh, um, developmental disco. And I think we were the first actually, um, to discover that, uh, children, in this case, they were years, 0 9, 9 years old, um, who were really bad at very, very simple tasks.
Brian 00:06:58 Like say how many dots there are in a display. Were there only one to 10 dots? Um, these were the kids who were, who had real trouble doing arithmetic. And, uh, since that original study, which I did with, um, a visiting, a brilliant visiting scientist from Austria called Karen land and an equally brilliant student of mine called Anna Bevin, um, uh, that was really the start of, uh, I would say modern investigations into developmental disco. We showed that if you're bad at enumerating dots, then you're gonna have trouble learning arithmetic. And so we began to think that maybe there's some kind of innate system for, um, doing very simple number tasks for extracting Nu American information from the environment. And, uh, that if, if like other inherited capacities, something goes wrong in the course of inheritance, then, um, you know, you could end up having trouble with things that other people find perfectly easy.
Paul 00:08:07 Well, you make the point, I want to ask about the relation to language in a moment, but you make the point that dyslexia is widely recognized and treated. And in fact, I don't know if it's just, was just underdiagnosed when I was younger, but it seems like half of the children, uh, that my, my kids go to school with have been diagnosed with dyslexia and are getting treatment for that. But, um, but you have made the point that, um, dyscalculia is under recognized and undertreated.
Brian 00:08:41 Yeah, that's, that's absolutely right. There was, uh, a major UK government report published in 2008, which compared the effects on life chances of, of dyscalculia and dyslexia. And what they found was that in terms both of lifetime's earnings and in terms of, uh, educational opportunities, having dyscalculia was more of a handicap than dyslexia. Mm-hmm <affirmative>, uh, now why dyslexia should be better recognized than dyscalculia is a deep historical problem? Um, I think one of the reasons is because it seems to be okay to say to a friend or a teacher, well, um, or a parent, you know, I'm just not very good at that, but you can't really say, well, you know, I'm not very good at reading or I'm not very good at, you know, hearing language,
Brian 00:09:38 Right? Yeah. That that's the cause of most, most, most dyslexic. Um, and, uh, and, and the other thing, there was a, a wonderful English actress called Susan Hampshire, who many years ago, um, collected famous people who were dyslexic, she herself as an actress, was dyslexic and found it very, very hard oh yeah. To, uh, learn her lines. Um, and she wanted, if there were other people who were somehow made it in that, in their own world, uh, who were also dyslexic. So for example, racing driver, Jackie Stewart was one of those who was, um, who came out if you will, as dyslexic. Uh, we had a deputy prime minister, Michael Hesel who nearly became prime minister, fortunate. He didn't, um, and he admitted to being dyslexic, uh, and there are many of them. So in our current parliament, there are quite a few people who to being dyslexic, but none, so far who admit to in disco. Well, but I, I, I put, I put out, I put out a, a request. Any of, any of you famous people who are disco QIC, or think you might be disco QIC, please come forward and we'll test
Paul 00:10:57 You think you might be, is the key there <laugh>. Yeah. Cuz I'm curious when people are dyslexic and or dyscalcu leic, uh, how, how easy it is to self recognize that
Brian 00:11:08 Self recognition is difficult, uh, as it was for dyslexia originally, when you can't do what everybody else in the class can, do you think of yourself as being stupid? And I've had lots of adults come to me late in life, even in their eighties and say at last I understand why I couldn't learn arithmetic. It's not. Cause I was stupid. I thought, you know, all this time I thought I was just stupid cause I couldn't do arithmetic. And uh, and now I know that, you know, it could be something quite specific, um, nothing to do with stupidity at all. And it's a bit like color blind. The did, uh, professor Butterworth, you know, some people are color blind, some people aren't and it's not to do with their intelligence. It's just that something's gone wrong with the inheritance. So, um, so I feel that that being able to identify it, uh, relieves people of the burden of thinking they're, they're just stupid. Yeah.
Paul 00:12:07 This is a total aside, but I was thinking the, um, so just going back to dyslexia again, I, I don't know that there's a, as much of a stigma now to dyslexia and in fact it seems like at least a portion of the population, like some mental, um, phenomena, uh, dyslexia is almost kinda like celebrated as something special now. Uh, I, I, do you have that sense or am I misreading society in that regard?
Brian 00:12:38 No, that that's absolutely right. I mean, I I've even got a book somewhere, um, called the gift of dyslexia. Right. Okay. Um, which has well dyslexics just think differently and, and, and this can be very, very creative. Um, I dunno, there've been any proper studies here reviewed studies of, uh, whether dyslexics are more creative than, than non dyslexics. So, you know, um, I'm agnostic about that. Um, but of course dyslexic a serious problem and yeah, unless, unless you, you get the right sort of help early on, you are never gonna be able to read efficiently. And uh, if you do get the right help early on, uh, then you can then to read efficiently, you won't necessarily read in the same way as non dyslexics. Um, but you know, you can, you can get by in fact, the first dyslexic that we ever tested, uh, a young woman who was in my, uh, seminar group at university college London, which by the way, is an extremely hard university to get into.
Brian 00:13:48 You have to be really good to get into it. And so all the students there are really good and, you know, she, she was reading out her essay one day in the seminar group. And every time I think it was an essay on neurotransmitters. So these are, are words that you, you come across in text, but you don't often hear spoken. And also by authors that, um, you see in text, but don't hear spoken. So she read out a perfectly good essay, neurotransmitters, and every time she came to the name of a neurotransmitter or an author, she wanted to cite, she turned to the student next to her and said, um, how do you say that? And he would say whatever it was, for example, something really. And next time she came across G a, B a, she would say GABA. So each word that she knew how to pronounce, she had to learn as a separate item.
Brian 00:14:49 Interesting. So to get, to, to get to the psychology course at university college, she had to do a lot of very, very hard work. So she had to learn all those words separately. And, uh, we tested her formally after, after this. Oh, this is interesting. And so we gave a really simple letter strength that she'd never seen before and asked her to pronounce them and she couldn't. Hmm. Uh, but once we told her how to pronounce them, she was okay. And that was a particular, rather specialized case of dyslexia. I mean, there are cases like that and her own cases, which are rather different from that. Um, but she formed a pattern and, um, you know, we, we thought that this was actually worth letting the world know about. And we, we wrote some papers about it and curiously, we found a Japanese boy, uh, well, we didn't find him.
Brian 00:15:43 His father found us and this Japanese boy was exactly like this, this, uh, student of mine in that he was really, really bad at reading English. Mm-hmm <affirmative>, he could only pronounce words whose pronunciation he already knew. Um, but his Japanese reading was very superior. Hmm. So, um, we thought this was a very interesting, uh, observation. And we actually went to Japan, tested all his classmates, tested him again, tested his father, tested his mother, tested his siblings. And it turns out that his classmate, I mean, if you, if you spoke to him, you wouldn't know that he wasn't fluent English speaker. He wasn't fluent English speaker as good as me, if not better. Um, but his classmates of course would, you know, just learn English at school. Um, so they weren't very good, but their reading of English was better than his, his, uh, siblings, uh, were normal English readers, his father who was a famous journalist. Well, I won't say who,
Paul 00:16:59 Why can't you say who?
Brian 00:17:02 Um, <laugh> I, well, I, I, I don't wanna say who. Okay. Um, and, um, the famous English journalist who worked for a famous English publication then for other things, uh, he, uh, he was an excellent, an excellent reader, but he didn't read normally. And we think he was DYS dyslexic as well. And when we asked him, he said, yeah, well, I had trouble reading at school. So I said, well, why you become a journalist in that journalist the dyslexia <affirmative> so dyslexia, we, we we've worked on dyslexia. We, we, we know quite a bit about it, but dyscalcu is different because there are varieties of dyslexia, but there's only really one type of developmental to
Paul 00:17:51 Is, is, does that have to do with the, um, accumulate, like the kind of fundamental, simple accumulator mechanism that underlies our counting and then therefore numerical ability.
Brian 00:18:04 That's the claim of the book. Yes. And, um, so we are the only species that really, that has proper language. I creatures communicate extent and there isn't really model of language. Whereas as I point out in the book, there are lots of creatures, maybe all creatures have some degree of numerical ability. And so there are animal models for numerical ability. And the claim in the book is that what underlies the numerical ability of, uh, ants or birds or fish also underpins our own numerical ability. And of course we have lots of other things that help us. Like we have counting words and mm-hmm <affirmative>, we have, uh, symbols, uh, and we have education, which, um, fish, even though they, they live in schools, um, don't actually have any education.
Paul 00:19:14 So when I, um, I guess let's talk a little bit about, um, the relation between language and, uh, counting. I've had this, I don't know if this is a common thing you you've, you can tell me, uh, I will often be walking and then notice, um, around some, you know, sometimes it's in the thirties, sometimes it's in the twenties that I've subconsciously been counting my steps. It happens a lot, especially on stairs for some reason. Um, but those are counting words. Uh, so that's my, uh, language ability on top of some counting ability. So what's going on there and <laugh>, uh, but the, you, you've already alluded to the fact that our counting ability, that very basic mechanism is distinct from many, if not all of our other higher cognitive functions like language, although in the case I just described, that's an interaction between language and counting.
Brian 00:20:08 Yes. Well, one of the things that, uh, modern humans learn in the course of growing up are counting words. Now, it's not clear when humans started to have counting words. Now we know that, you know, 30,000 years ago, humans counted because we've got marks on bones and stones and cave walls, which are in my view correctly, interpreted as counting. Now whether they actually have counting words like 1, 2, 3 to go with those, uh, marks is, is not certain. Now there is a, a school of thought, uh, some modeling that's been done, uh, particularly by mark Paal and his colleagues at bedding university, uh, which suggests that you can actually get a sense of when humans started to use counting words. So if you've got a, you can see how words change over time, over historical time, because they're written down. So you've got a record and a time stamp for quite a lot of words.
Brian 00:21:20 And you can tell how quickly, or how slowly, uh, words change. And one of the things that, uh, Pagan's colleagues discovered is that in three of the largest language families, Australian, um, African languages and in the European languages counted words are among the slowest changing. And so if you kind of, uh, retro or predict back from the oldest examples, you've got, you could say, well, you know, maybe even 30,000 years ago, um, humans had counting words, but we dunno certain of course, but that that's plausible now, but even before then, you know, with Neanderthals, uh, they also probably counted looking at the kind of artifacts they made and the marks that they made on cave wars. And they probably counted as well. And, um, well, they were certainly did according to me. Um, but whether they had counting words to do it, it's not clear cause we don't really know whether they had something like modern language as, as such, uh, now why you should count your steps.
Brian 00:22:33 Um, the question I would then be tempted to ask is when you were with your parents, did they count steps with you? I know that with my kids, you know, we would walk along the road, we'd swing them along between, you know, the mother and the father and go 1, 2, 3 up when we'd count the steps up the, up the stairs. Um, so they got quite a lot of experience of counting words when they were very young and that maybe stick in, in, in the mind. So later on it gets, uh, if you like, uh, uh, remembered or, or reconstructed and it it's commonly, it's commonly assumed. And there's a lot of evidence with this there, no proper research that when you do, uh, if you are multilingual, are you multilingual?
Paul 00:23:25 No, I'm not fluent in any other languages.
Brian 00:23:28 Okay. Um, so if, if for example, you brought up, let's say just bilingual, say English and French. Then if you were taught counting in English, you would count in English, preferentially your whole life. Um, we did a bit of work in Spain, but with, um, looking at, uh, kids brought up speaking in, in Catalonia, kids speaking Spanish and speaking Catalan, and some of them went to Catalan schools. Some of them went to Spanish schools and on the whole, the ones who went to Catalan schools counted in Catalan. Those in Spanish schools counted Spanish in Spanish. And we know lots of very, very educated, um, Europeans, um, uh, say brought up in, in Italy, speak perfect English. They counted Italian. Um, and, uh, something that, you know, you get it into children's brains early enough. It kinda stays there.
Paul 00:24:26 What seems like such a useless thing, such a useless thing. Every time I notice that I'm, you know, in the thirties or something, I think now I think because having read your book, I think I'm not an aunt, so I don't need to count my steps. So what, you know, what a waste of brain energy essentially
Brian 00:24:42 <laugh> yes. Yeah. I'm sure. I'm sure. I'm sure there's no point to it. Um, but you never know. I mean, you might be, you might like the ants be making an estimate of how far you've gone by the numbers steps you've counted, which is what one species of aunt aunt does. Um, yeah. Um,
Paul 00:25:03 So blame blame. My parents is what you're saying.
Brian 00:25:06 Well <laugh> yeah, for so many, for so many things, we should blame my parents. Right. Um, but uh, I mean, the other thing is that we we've, uh, I think I reported briefly about this in the book we've actually done studies and other people have as well that you can, you count unconsciously, uh, without words mm-hmm <affirmative> um, so you, you can present an array of, uh, dots and you ask them very quickly after their presentation. So if you are, if you ask the subject, uh, how many dots said you see they, they can't answer cause it hasn't kind of registered in their consciousness, but nevertheless, we know that it has an effect on something that they do next. So if you, if they see four dots and then, uh, they can't report how many there are, and then you ask them to, uh, count some dots that they can see.
Brian 00:25:56 Then if they've, um, if, for example, the number of dots is the same in their, uh, their unconscious perception, then they'll be quicker. Um, and, uh, uh, it was, we had some kind of strange results there, but lots of people have found that, uh, you, you, you can count unconsciously, uh, and you can't of course ask people, uh, what, how many things you saw cause they're unconscious, but you can see the effect of having made that, uh, uh, Nuer assessment on, on later behavior. So you don't have to do it with words. You can do it in other ways,
Paul 00:26:37 Sticking with words and, and language. Um, just for another minute, I took a class that was, uh, interesting about the origins of writing, um, not the origins of language per se, but, um, the professor focused a lot on early QE form and marks on clay tablets back in the Fri Crescent and how the need to keep track of how many sheep you get and how many I get and things like that. And those marks that you were talking about on the clay tablets, uh, probably I don't know what the current, uh, hypothesis is, but her idea was that that sort of thing led to written language. And of course that's not spoken language, but so thinking about the counting abilities of us and what seems to be just across all species so far that have been tested, it seems, um, would you, and however, we were just alluding to the fact that language seems to be separate from this counting mechanism. So in your, in your thoughts, uh, counting and numerical ability must have preceded language, perhaps, but do you think that it had anything to do with development of language in the human species?
Brian 00:27:50 Well, according to choky and his followers, our ability to count is really consequential upon our ability, right. To do syntax linguistic syntax. And the reason for this is cause we can count as high as we like, and at least we can with words. And so, okay. Um, if you're gonna count as how you like, without words gonna be bit difficult. Yeah. Um, so he thinks that the recursive mechanism that you need for syntax, uh, underpins our ability to count. Well, it certainly, um, is related to our ability to count large numbers precisely. That's true, but you know, ants don't have language, at least these particular desert aunts don't have language. They won claim that they do, but yet they can do some quite good, I would say recursive counting, um, um, fish too. So I don't think he's right about that, but obviously, uh, you know, you, you can't counter, you know, let's say 213 precisely, unless you've got the linguistic needs to do it.
Brian 00:29:01 I mean, you might be an estimate of, you know, around two bit more than two, a bit less than two 50, even without the language, but you can't get two, I would say without the language. Um, so, you know, language, I'm not saying language isn't important. Um, and, uh, you know, it's, um, I's mathematician Whitehead said, you know, um, a good notation, uh, makes life a whole lot easier and we do have, well, we now have a good notation for numbers and actually one of the things that, um, uh, I've just been writing about. Um, and I, I mentioned it briefly in the book is that if you look at people who are exceptional calculators, I mean, not like me, but you know, really, really good likes. Yeah. Well like, but they're not, I mean, you know, professional mathematicians, some of them mm-hmm <affirmative>, um, and, and all engineers, you know, anybody who's really been very good.
Brian 00:30:04 Uh, I see. And some of them have even, you know, been in, been formally tested and been in brain scans and so on. Um, I mean, one of the things that, uh, that that is, is clear from, from these people, they've got an enormous, uh, memory store of facts that, that they've for whatever reason, a fact about numbers, hundreds of digits of pie, the square of every number between one and hundred, you know, these, these guys are, are really good. And, um, yeah, they, that, I mean, absolutely not, you know, the classic Rainman type of, uh, uh, I mean, a lot of them are just kids. Who've gone to a onto Abacas training after school, in, in India or Japan or China or Taiwan, um, who just learned to do, uh, a medical calculations really, really quickly. Now, I don't think they're using words by the way.
Brian 00:31:11 Um, I mean, the ones I've spoken to about this, the one I've observed, they have this, they have a, a, um, a calculating, they have the Abacus in their head, so they're not doing worse. They're just moving these imaginary digs around in their head. So that's, um, why some of them, you know, just have been to Abacas Juku, um, are really good. But the other thing I would say is that all the other prodigious calculators who haven't been to Abacas school, they depend on our positional notation. And my guess is that no Greek, no ancient Roman was a prodigious calculator because their notation was inappropriate for calculation mm-hmm <affirmative>. And we know this, it was inappropriate because when people were writing Roman numerals back in the day, they, if they were doing calculations, they used accounting board, uh, with, uh, units, tens and hundreds on the counting board.
Brian 00:32:19 And they'd put, uh, BES on, on, on each of these, um, columns to indicate, uh, how many, uh, uh, how many tens, how many units, how many hundreds there were. Um, so you needed a proper notation if you're gonna be really good at this. Um, now there were other ways, I'm sorry, I've gone into this, but no, please, you tempted, you tempted me down this path. <laugh> um, there were also finger counting methods and some of them were really complicated and you could do really complicated calculations on your fingers and whether you were using, uh, counting words or, or, and this was before you had position rotation, uh, whether you use counting words as well. I dunno, it's not reported, uh, but there are a lot of kinda many evil manuscript, like from the venerable bead about how you could use each, uh, each part of your finger to represent a different number. And so on
Paul 00:33:18 The, the, the different segments of your fingers of each individual finger.
Brian 00:33:22 Yeah, yeah. That's right. Um, and so you, and you, you work out ways of, you know, doing it,
Paul 00:33:29 Huh?
Brian 00:33:29 Skillfully. Now don't ask me to do it cause I can't right. I never been taught, but some people could anyway, but the, just the final point about Lang language and numbers is this is if you like crude different parts of the brain. And this wasn't discovered by us, it was discovered by, um, a Swedish neurologist called he who did a case study. And he had a case series of over a hundred patients, maybe, maybe near 200 patients. And what he observed was you could have disturbed, uh, numerical abilities. If your paral load was dis was damaged, but preserved language, or you could have, um, impaired language, if your frontal lo was, uh, what we now call broke area was damaged, but your paral was intact. So he was the first really to identify this, but both areas are damage and you, you have trouble with average and numbers. And of course you, you often get that. And we, we recapitulated some of those started in a bit more detail and showed that, you know, he was basically right.
Paul 00:34:48 Uh, a lot of what you were just talking about with different calculation abilities is higher mathematically than the fundamental counting mechanism. So I, I would, I wonder, you know, what, what the relation is between our higher mathematical abilities and this kind of tally, uh, counter the accumulator mechanism that you describe, and it might be worth just, um, describing, you know, that fundamental mechanism. And, and just to throw one more question at you. So what you're talking about as important for counting is in our paral cortex. Uh, but if our, if counting, if the accounting ability is so fundamental and common across species, many species don't have a cortex, for instance. So, um, and, and I know that we don't know where in the brain, it happens in many other organisms, uh, but might there be like a, a more basal, uh, brain mechanism that underlies counting, even in us that gets elaborated in the paral cortex. Sorry. So many questions all at once, but there you go.
Brian 00:35:53 Oh, absolutely. Excellent questions. What we know about, um, counting mechanism in other animals is basically in, in, uh, primates like, uh, monkeys, monkeys, particularly, and in this is some brilliant work done by a, a German neuroscientist called Andrea. Uh, he did basically the same experiments with, uh, um, monkeys and, uh, birds Crow, and, uh, what he found in the, um, in the crows brain, which doesn't have a cortex, was that there were individual neurons in what's what's called the, of the, the crows brain, um, which look very much like the individual neurons in the brain, except in the brain they're in the cortex thes brain. They're in the, we Don know anything about at the moment about the fish brain. Hmm. Um,
Paul 00:36:54 Okay. Wow. So what is that a teaser?
Brian 00:36:57 <laugh>, that's a teaser. So come back next year. We may have an answer for that
Paul 00:37:01 One. Is it zebra fish that you're working on or the, uh,
Brian 00:37:03 Yeah, we're we, yeah, we're working on zebra fish. Okay. Um, um, I'm tell you what we're doing, but it's, uh, it's it's, uh, but we don't have the, we don't have the data yet. Um, great. Uh, so, uh, so yeah, I mean, this is a really good question. And when you've got, you know, really interesting counting abilities in, in ants and BS who don't even have a, um, then we're, we're, we're talking, you know, uh, deep mysteries, but then again, uh, the analogy here, at least for me is timing mechanisms. Oh yeah. So you can find timing mechanisms in, in all species, in, from fruit flies where the probably best described right up to humans. So in humans, they've got timing mechanisms in cortical structures, and of course, fruit flies don't have cortical structures, but they have that we, we share genes with them or of the same genes. And so somehow these genes build, um, timing mechanisms, if you like, where they can now, look, this is getting way above my pay scale. So, um, um, I'm not sure I can, I can say much more than this. Um, but when it, when it comes to fish, uh, we're, we're gonna, we're looking at now, we're gonna, we're looking at brains of, of fish and we're gonna doing some more experiments on the brains of fish.
Paul 00:38:30 Uh, are you doing like whole brain optogenetic imaging? Is that what's going on
Brian 00:38:34 Or, yeah, optogenetic exactly, yeah. We're doing optogenetics with microscopes cause um, we have to use laal sever and, um, because they're transparent optogenetics. So when, um, when bits of the brain are active, they light up in this, in these particular, uh, preparations. And so you can see the, where the micros go. Uh, and so, um, we know that even laal zebrafish will orient towards more objects than fewer objects. And so, you know, we just wanna know how these brains light up, but as I say, come back in a year and, um,
Paul 00:39:15 You're not gonna give us any preliminary results, even
Brian 00:39:18 Pre preference don't we don't have a pre now we haven't. No, no. I mean, this is, this is really difficult technical work. I mean, something that's, you know, beyond my particular competence and it's, it's the team I work with who know how to do this. And so far we haven't got any data. I mean, we know how to do the, I think we know how to do the experiment. We just, um, have got some other experiments that we have to do first. And, and then we can get onto this really exciting.
Paul 00:39:50 You, you mentioned the, uh, conservation of words for numbers in our language. So I, I just, it made me think, and then you were talking about timing mechanisms. I don't know if we wanna come back to the relation between timing and counting since there's, uh, overlap there. But, but, uh, I remember I had, uh, Dean guano on the podcast a long time ago and he makes the point that the word, uh, time is the most common noun in the English language. I can't speak to its conservation as a word, but, uh, but the actual word time. So, uh, an interesting overlap there as well, but, um, let's talk about Phish a little bit more in, in a, in a moment, but, but I, I still wanna know B because, uh, their counting ability is much more, let's say fundamental or, or lower level, but because you were talking about calculations and our ability to, um, manipulate numbers and perform higher mathematical calculations and operations, what is the relation between, you know, that fundamental accumulator counting mechanism and our higher mathematical abilities?
Brian 00:40:58 Well, uh, there's some basic arithmetic. So one of the things that, uh, my, my, my fish experimental colleagues have done is they've, uh, arranged an experiment in which the fish can only see one other fish at a time. So the fundamental or the basic paradigm is will the fish go to a larger should than a smaller should when it can only see, um, each show one fish at a time. So it can see, let's say three fish over on the left and four fish over on the right, but it's swimming around in between these two S so it's having to, uh, actually count or add one to one to one and on this side, and then one to one to one to one on the other side, um, maybe it's doing what one over here, one over there. Uh, it's not clear exactly how the fish does it, but that's adding mm-hmm <affirmative>.
Brian 00:41:59 Um, uh, what about fish, subtraction? I know about that. Um, there's been some work with bees that suggest that bees can subtract. Um, so that's where the, you know, and you just, just step the accumulator down one in order to subtract mm-hmm <affirmative> that that's, that's not, not so hard. So the basic accumulator mechanism is a bit like a tally counter. This is, this is a little machine you can buy on Amazon without, you know, a couple of dollars where you, you, you, you press a key every time. There's something that you want to count. So, for example, if you want to, the example, I'd give my book, which is rather biblical is counting sheep, but not goats. Uh, so for every sheep, you see you, you, you press the key and the counter accumulates another sheep. And then at the end of your counter, you look at the accumulation, it says 13 sheep.
Brian 00:42:57 So, uh, the difficult thing here, uh, is not doing the counting, which is very simple, just, uh, press it's actually deciding what object you are seeing is a sheep, the selector, and not a goat and the select, what I call the selector. And that's, that's kind of expensive. And in, in your terms, you know, and, and maybe in experiential terms, you have to learn what a sheep is, what a goes, is it doesn't, doesn't necess doesn't come naturally. Um, it may well be that the critical categories for bees, for example, uh, are landmarks. So the be just identifies landmarks and, and counts them up. It might also count up pedals on a flat to know which valve are really going to be rich in pollen. Um, now what we would do in those cases, counting their marks and counting, uh, pedals is we could say, do they have the same number or not?
Brian 00:44:00 So we can, uh, we can abstract away from the, um, the, the, the landmarks and the pedals. And so, yeah, they've got, you know, there are four pedals here. I've seen four landmarks that can the B do that dunno. Um, that's for B researchers to, uh, find out I've, I've tried to prompt green researchers to look into this. Um, anyway, um, we, we can do it, no problem, but, uh, other animals tend to only count things that are, are useful relevant for them. Now, of course, you could say that when bees are counting landmarks, they're not all counting the same landmarks every time they're counting, you know, uh, you know, three U trees here, and three Oak trees there, they're not the same objects. So there has to be a degree of abstraction in landmark counting as well. Now we don't know the extent to which, uh, be can abstract away from landmarks.
Brian 00:45:02 We know that they do, uh, because in the original experiment, by last Chica, he, he had large yellow tent. So it's always the same sort of landmark, but they weren't always in the same place. So the question he asked initially was, is the B when it goes from the hive to the food source, using a measure of distance, which doesn't involve landmarks, or is it actually counting the landmarks in order to make an assessment of, of distance, so he could move the tents farther apart, or bring them closer together and see whether the, the be was using the number of landmarks to locate the food source or whether it was using the distance, uh, from the hive to, to the original food source body found was that they were counting landlords. Um, and, uh, I mean that he could have tried doing it with red land red tens, or yeah, green tens or tents in a different shape, but he didn't do that originally. So it'd be interesting to see the extent to which, uh, bees can abstract from, uh, the set that they're enumerating. Uh,
Paul 00:46:11 But yeah, I just had a, a B expert on recently. Sereni Sereni Vaan and, um, at point also, yeah.
Brian 00:46:17 Device work on that. Yeah.
Paul 00:46:19 Yeah. Well, he, a lot of his work is with like navigation and, and flight control, but, um, in a nice review, he wrote about B cognition. Um, he talks about that, that bees can discern zero as well, which is an abstract, pretty abstract concept. But it's fascinating to me that something that you talk about in your book also, of course, you know, there's a question why do we need to count? And that, so that, that opens the whole evolutionary questions. Uh, but the idea that an organism can count some things, but not other things is kind of hard to fathom given our, what we would consider SU superior abstraction ability that we can apply nuity to anything, no matter what. Um, but it makes me wonder what are we, are there things that we can't count because they're not useful to us, or, you know, do we have, <laugh> such abstraction that we are free to count anything, you know, but it's an interesting fact,
Brian 00:47:16 Uh, well, uh, yeah, I, uh, I think we can count anything. Um, not everybody can count everything. I mean, some right. Some things you need to be trained to count. Uh, so there was a, you know, a famous 19th century calculator who counted the, uh, the, the number of hairs on the tail of a horse. Now he didn't need to do that. No <laugh> and he just wanted to do it now. I, and I don't actually know how he did it, whether he did it by kind of pulling the, the, uh, the tail apart so he could count each one, I dunno. Um, and there was another famous 19th century calculator. Uh, again, people are very low educational mobility, um, very little education at all, who, who counted the grains of, uh, corn that he gave to his chickens over the course of some large number of years. Um, I think it was 40 odd thousand. Um, wow. Uh, so you know, what you choose to count is really, uh, say up to you, um, you can count your steps if, if you really want
Paul 00:48:24 So useless. Yep.
Brian 00:48:26 <laugh> which may or may not be useful. Right. Um, uh, but, um, yeah. Uh, but nav, navigation's a very interesting issue and I, I deal with it a bit in the book. And, um, one of the, I was gonna say, well, the claims I make, maybe that's not the right way to put it. One of the, um, what should I say, explanatory ideas that I try to use here is that, uh, particularly for animals that migrate, most of them do to a greater or lesser extent, they know they go from their hive, looking for food, looking for a mate, and then coming back. Um, so how do they, how do they calculate distance? How do they calculate direction? How do they do what sailers would call dead reckoning, where you actually calculate each step of your, uh, of your, uh, uh, path, uh, then figure out where you got to, and then where you'd have to go next, where you'd have to go back to.
Brian 00:49:32 Um, and the analogy that, that I I thought about was that maybe animals have something like Google maps in their heads. So Google maps is just numbers. And, uh, when you ask Google map to say, as it, um, just before we came on, um, what's the shortest way from my house to the bridge theater, basically what the, what Google maps is doing. It's doing a computation over numbers, uh, complicated computation over quite a lot of numbers. And so one possibility for navigation is that animals have something like Google maps in their heads, all which is just numbers. Uh, now, you know, on Google maps, you can have additional information. You can ask it to tell you where restaurants are, where stations are, and maybe on a, a long journey, like the Bartel Godwin from which flies from Alaska to New Zealand every year. And then back again, uh, it needs to know where the restaurants are on, on route.
Brian 00:50:50 Actually. I dunno whether it does stop offer for a slack, but anyway, if it did, uh, that's what it would be information to have. And again, you know, restaurants or gas stations are just numbers in the Google database, but better than the goo Google maps in some respects is that these migrating birds need to have a three dimensional map because they're not flying flat, they're flying up in the air. They need to know about a wind speeds, wind directions, where it's best to fly, where normally you get, you know, a, a good following wind where you get a, a bad, um, crosswind. They need to have a, a three dimensional map that includes information about winds. Um, so maybe they don't have everything that you find in Google maps, or they have something like, I would say something bit like Google maps, uh, with at least the relevant information coded there somewhere in their brain. Now, you know, birds have small brains cause they they've got small aerodynamic heads. But one of the things we've learned recently is that the, the number of neurons in the bird's brain, um, could be greater than some of the neuro uh, the number of neurons in even some primates, um, because the neurons are packed more firstly together, uh, they have to be cuz you've got these small neurodynamic heads. Um, so,
Paul 00:52:20 Um, well, and, and birds count just about as well as primates or if not better in some cases, right?
Brian 00:52:26 In, in some cases they might do better. In fact, they've done the, uh, um, this brand and a team have done exactly, uh, did an experiment with, uh, with monkeys and, uh, another took all her stimuli and did it with, with birds and got almost identical results. So birds can, birds can be really pretty good at, uh, uh, numericals last, maybe not all birds, um, uh, but certainly CS, crows and Ravens. And so on, we've known for nearly a hundred years are very good at counting. And, um, also Parro can be pretty good gray parrot, like the famous gray par uh, uh, Alex who could count and actually tell you in words how many heat can do. Um, so they, they can be pretty good and pirates have been retested again. So yeah, you know, these are, so the term bird brain is, um, could even be considered a compliment
Paul 00:53:28 Like dyslexia now <laugh>
Brian 00:53:31 <laugh>
Paul 00:53:33 You, you also talk about, uh, clever Hans. You tell that story in the book where, um, you know, this was a horse that seemed to have great numerical ability, but was, uh, cleverly using other queues that were, were subtle. Um, and, and this is, you know, you also talk about how it's difficult to actually assess whether something is counting or whether it's using density or size or some other metric to accomplish tasks. W so I, maybe, I, I, maybe we should bring this back to Phish again, since that's, uh, your, uh, the main focus and kind of what you end up with in the book, but, well, yeah. So why do Phish need to count? Why do they count? And then, um, maybe we could talk about how, how to tell whether they're actually counting or using some other, uh, cues.
Brian 00:54:23 Well, uh, small fish, uh, like to live in S um, because being in a show reduces the risk of ation. Uh, so if you are, you know, little Fred fish in a big show, um, when the wise mouth comes around, wants to eat some, uh, your, um, your friends, um, it might well miss you because it's already eaten your friends. If it's a small, if it's a very small show that might eat your friends and you, because, uh, you can't, you can't escape from it. Um, so being able to choose the larger should is important for, for small fish. Uh, it also helps them, uh, forage, uh, particularly, you know, if what they're looking for, uh, our are, are small particles of the water. So many eyes do a better chop than just two eyes. The other question you ask is how do you know they're really responding toity rather than, uh, to other, uh, features of the environment?
Brian 00:55:34 Uh, it's very difficult in, uh, real life, uh, to separate out these two things cause they normally go together. So, uh, a show with more fish has more fishy stuff in it. Um, uh, so it could be choosing on the basis of fishy stuff. Um, and I, I think the, the short answer here is that in real life, uh, the animal will choose whatever is the most efficient way of deciding, which is more, um, so it won't necessarily choose the, the show with the most fish, but it will choose the show with the most fishiness about it, if you will. Uh, particularly if, if they're fish in different sizes and they're remember, they're all swimming around. So it's actually difficult to count anyway.
Paul 00:56:23 Yeah, yeah.
Brian 00:56:24 In the lab, you can control this, you can control the size of the fish, how much they swim around and so on. And you can show that, uh, under laboratory circumstances, the fish doing deep respond toity, so you can get rid of the, the other cues that it might be attending to like, uh, the amount of fishiness or which is, uh, be a combination of the actual area, vision area covered by the fish as it swims and the fishes as they around. Uh, so you can do it in the lab. It's harder to do it in, in real life. I mean, there are examples where you can do it in real life. So for example, in these wonderful experiments, by Karen, with lions, uh, she used, um, a playback technique. So she could tell how many intruders there were meant to be to lion a prides territory because she would place say, uh, three loud speakers at the edge of the, uh, the prides territory.
Brian 00:57:33 And so it seemed as though there were three vs going to attack the pride and its territory. Um, and then, uh, you could tell whether the defending lions would attack the apparent intruders would approach the land speakers or whether they would retreat. And this depended upon the calculation that the defending lines made about how many of themselves there were, and how many of the intruders there seemed to be. And this, by the way is also quite abstract because what they're doing is they're hearing intruders and they're seeing themselves, uh, so they're making a comparison across modalities, uh, the number of, uh, raw wars and the number of, um, uh, and the number of themselves, which they probably assessed by vision. Uh, so yeah, you can tell they're count into those circumstances.
Paul 00:58:33 You know, there's, there's a lot of, um, discussion about the need for ecological validity in testing of animals, cognition, whatever the cognitive abilities are. Um, and in a sense, um, uh, in a sense, you know, with these tests, you might not be necessarily addressing, uh, the fish, the fishes, or the lions or whatever organisms they're, they're VE right. They're normal, ecologically, valid, uh, perceptual perception and, and interaction with the world. On the other hand, uh, I don't know how much mathematical ability is innate in humans, right? Because without any education, none of us would go very far, uh, generally in a, in a mathematical sense. So in, in some sense, you know, studying our mathematical abilities is studying the, what we have our potential, right. For cognition, which is an interesting question still, is that how you view, like when you're testing fish, numerical abilities, not necessarily what they are doing or what they normally do, but their potential, what they can do?
Brian 00:59:41 Uh, yeah. Basically you are, you are saying, can they count? Um,
Paul 00:59:45 Well, do count, well, it goes back to the question of when, when might they count versus when might they use other queues and, you know, so, so we're like pushing their counting ability essentially. Right? Yeah.
Brian 00:59:55 Um, so you, you can look at things like, um, uh, whether they're using, uh, density of mm-hmm <affirmative>, uh, of, of, of two SHS or whether they're using nuity and, uh, you know, if, if they've got density and nuity, they'll use them both, um, cuz it's sensible, um, you know, if you've got, uh, lots of fish they're gonna, and they're in a small area, they're going to look dense. So you'll use that as a queue. Um, you can also manipulate it so that you can take the same number of fish and um, make them more or less dense by changing the way which the tank is arranged. And you're under those circumstances when you have the same luminosity, but uh, a denser group they'll go for the denser group. I mean, they might well have gone for the more spread out group. They might think the more spread out group is, um, there's more, but actually they go for the denser group, in least in some studies.
Brian 01:00:53 Um, so you know, the in, in, in real life or ecologically valid situations, I mean they'll use what they can. And it's interesting to note that two of the leading, uh, experimenters in this area, um, O who operated virtually before the second world war and pioneered many of the methods we still use, he thought that, um, animals didn't count in the wild, but could learn to count in the lab. So they had this capacity, which for some reason that I didn't fully understand, he thought they didn't use in the wild, um, uh, a more recent, uh, student, uh, uh, Hank Davis, um, Canadian, uh, scientist. He thought, well, they, they, they can use, uh, accounting in the wild, but only as a last resort when things like density and right, uh, area, uh, moving around, um, can't be used to make that judgment. Um, on the other hand in the lab, um, Elizabeth bran, uh, showed that actually animals use given a choice between, um, area density and number animals like to use number more than they like to use these other queues.
Brian 01:02:23 So that's an experiment in the lab, but, uh, it seems to me that you know, what you do in the lab, um, seems applicable elsewhere. So for example, in the lab, you find that most animals will go for more food rather than less food. Um, well, you know, in the wild, given the choice, they'll go for more food rather than less food. And you could do proper experiments in with free ranging monkeys, for example, um, about whether free ranging monkeys will go for more pieces of apple than fewer pieces of apple and surprise. Surprise. Yes. They do go for more pieces of apple. Interesting. <laugh> <laugh> yeah, I think it's a free ranging monkey, not Mon monkeys in the lab. You have to, you wait for them to come up to, you know, your testing area and then you show them bits of apple or fruit or whatever, going into one bucket, rather fewer into another bucket and stand back and see which one they go for. Mm-hmm <affirmative> well, you'll not be surprised to know they go for more pieces of apple or whatever fruit seats most to them at the time,
Paul 01:03:30 Stepping back to the brain for a moment. So you're, you're you're for a moment. So your, your, your friend, Randy Galle, um, has this, uh, idea that numbers are symbols and they must be, uh, stored in internally within the cell in something more stable like RNA. Um, first of all, I want to ask what you think of, of those ideas, cuz I know that you and he have had many discussions, but then, well, I'll ask you that first and then we'll move on to like different mechanisms in the brain.
Brian 01:03:59 Um,
Paul 01:04:00 <laugh> <laugh>
Brian 01:04:02 Yeah. Um, <laugh> well, I mean the, Randy has some quite good evidence, um, yeah, about this. I mean, it's not, I mean he has some, he has some evidence and he has some very, very good theoretical arguments about why it has to be something like that within the cell. Now, again, this is, this is a topic that is, uh, a bit above my pay scale and I'm not really competent to, to make a, an evaluation on that, but I don't see why it shouldn't be true when you have, uh, other ways of trying to represent pities. For example, in neural networks, mm-hmm, <affirmative>, it don't really work terribly well. I mean, we've done some of this stuff ourselves and it really, it only really works if you kind of set up your neural network where you like some, um, innate properties. And I did this, uh, many years ago with, uh, uh, was then the student mark DOK.
Brian 01:05:09 Now course <inaudible> distinguished professor running one of the best labs in Italy on the numerical. Um, and I don't think he really believes this anymore, but, um, what, what we did is we, we, we tried to set up these neural networks, uh, ones which, uh, simply learned to distinguish different human and others where you kind of built in a bit of innate structure, not actual numbers, but name, structure, the ones with the R RNA structures seem to work much better. They, they model the human hu various types of human behavior better. So I, I like now whether this is within a cell or whether it's, um, you know, connections between urines, I don't know. Um, I mean, I, I think, you know, time will tell and there's no reason why it shouldn't be both. There's no reason why it shouldn't have numbers within cells and some other kind of numerical representations, uh, in, in the, the strength of the connections between cells. I don't see these are exclusive. Yes. Uh, so, and that, that would be my, I think that would be my position. You can have both, but I's not really been, that will model so far, I would say,
Paul 01:06:25 Um, in neural networks, you mean, or as
Brian 01:06:27 A in, in neural, in neural networks. I know. I mean, I I've had arguments with some of, you know, the great theorists of this, like Jeff Hinton, Uhhuh, um, who thinks it can all be done in your networks. And, uh, and Randy who thinks actually have to do otherwise. Yeah, well, you know, but then you have to get the stuff outta the sale and do calculations with it. Right. So, uh, so I, I, I think, um, maybe invite me back in five years time and maybe
Paul 01:07:00 One year for results for
Brian 01:07:04 The theory are always more difficult,
Paul 01:07:08 But you describe in the book too. I mean, there are individual neurons that have activity correlating with a certain numerosity, for instance. Uh, and then there, you know, there are obviously these accumulator type neurons as well. Yeah, yeah. Uh, but, uh, you know, part of the thing with neurons, whether it's neural networks or individual neurons is stochasticity right. They're, uh, somewhat random. And in cases, when let's say in, in a fish trying to count, uh, the numbers in a show, you need to be pretty precise. So how do we reconcile the stochastic nature of neural networks? And let's say, you know, just thinking of, uh, the individual neurons that respond to the number five or something, right. The, the numeral five, um, then we also have a grandmother cell problem where if we kill that neuron, the organism would, would, would not then be able to, um, represent five, unless it trained up another neuron through the accumulator. Uh, so how do we, how do we recognize, uh, reconcile the stochastic nature and then that grandmother cell-like problem with, uh, the precision needed,
Brian 01:08:16 Uh, right. Well, I think there's, there's a, a prior question here, which is that we have the, the, the neuro cells that for example, uh, has identified in, in bird, uh, brains. Um, so what is it about those cells that makes them respond preferentially to ness? He doesn't say maybe we dunno yet. So it might, it might be some, uh, Gallian, uh, thing going on within the cell, which makes AIAN particularly <laugh>, which makes them particularly responsive to five. Um, and, um, and the, the, the question is, and I, I, I think then some of the stuff that, that some of the, um, pav that need is published there, aren't, there isn't just one, uh, five, five nurse sell there, quite a lot of five, the cell they're quite lot five the cells. So it's not quite the grandmother cell problem. Mm-hmm <affirmative>, um, and, uh, you know, a cell's fairly cheap, so there's no reason why you can't have lots of five cells, then they have to be connected up in some way so that you, I mean, there, there are <affirmative> so, I mean, there, there are complicated modeling problems here.
Brian 01:09:31 Yeah. Um, so that's, I think, I think that that's, uh, that's one issue. The, the other is you say about the stochastic nature. Well, you know, it's been known for a long time that, uh, um, that with, uh, say accumulated type, uh, neurons or accumulated type mechanisms of any sort, you get scale of variability, that is the amount of variance goes up with magnitude. Um, and I mean, as long as you know, that as the brain knows that it can take into account the amount of variability, uh, that, um, that might be going on. And, uh, there is, there is a bit of evidence that, that, that some, uh, some animals do in fact take account of, um, the amount of, uh, of variability that, uh, that some particular mechanism is, is showing and making a, a judgment on the basis of well, um, you know, we're, we, it's very likely to be five, but, you know, it could be something else. Um, so I'm only gonna commit so much resource to, um, the, the guest that it's five given the amount of variability. Sometimes it looks there isn't very much variability. And so, you know, you're more confident in, in the way in which you, uh, you make that, uh, you choose that action.
Paul 01:10:54 So, Brian, we have just a couple minutes here left, and I have, uh, as usual, a thousand more questions to ask you. So I will have to bring you back on in a year and we can discuss more fish, uh, in particular, and then, uh, you know, reach out when you have those results or, or you're excited to talk about it. I will, but maybe, maybe two, two final questions before we, um, say goodbye one, do we know of any organisms that cannot count you because you describe a ton of organisms that can whe whether they show evidence of being able to have we discovered any that can't
Brian 01:11:26 Well, uh, the question is, have we show that they can't count or have we just failed to show that they can <laugh>. So there's some very nice, uh, experiments, uh, by a colleague of mine pet, uh, who, who worked, did some work with lizards. And, um, so she found these Liz couldn't count and she different experiments and found that they, they could. So, uh, you, you've gotta, first of all, you've gotta have a well designed experiment then going back to your point about ecological, valid validity, you know, it's, um, it depends on, you know, what the lizard actually likes and, um, and you have to find both the kinda the rigorous experimental design and an understanding of Elit psychology. Um, so you need to experts as well. So if you, if you, so just going back to the point, if you fail to find evidence of counting, maybe you've done the wrong experiment, uh, on the other, maybe they can't count.
Paul 01:12:28 Hmm. So, but, but we have no definitive organisms yet. How, how about bacteria?
Brian 01:12:34 Ah, yeah, well, um, well there are <laugh>. Yeah, well, um, I don't, I personally don't think bacteria can count, but, um, I mean, there, there, there are bacteria that will light up if there's enough of them together, right?
Paul 01:12:50 Yeah. You described that in the book as
Brian 01:12:52 Well. Yeah. I describe that in the book, but there have to be millions of them, and I doubt that the single south organism can count to a million. Um, so I could be wrong about that, but, uh, I
Paul 01:13:05 Would maybe using some other signal besides, besides counting. They
Brian 01:13:07 Could be, they could be, yeah,
Paul 01:13:10 I lied. I have one more question for you, our numbers real what's the ontological status of, uh, mathematical entities.
Brian 01:13:17 Oh, right. Um, yeah. Okay. You, you, you leave the easiest and last one. Yeah.
Paul 01:13:23 That's the
Brian 01:13:23 Simplest one. <laugh> I know. Um, so this is what, uh, gall, so calls, um, going down the rabbit hole. Um, so, uh, I, I think you have to go down the rabbit hole. Well, um, my view is a kind of rather simple minded view. Um, I mean, I know, I know about the history of, uh, of, uh, foundation of mathematics. I, I worked on that for a while. Um, but, but my view is that are of course, uh, abstract, uh, that abstract objects can have no cause or properties. Um, course they're abstract. Uh, they're often in the world they're abstract. Um, but we experience instances of these abstract, um, concepts. And, uh, these instances do have cause and properties in the world. And, um, so we can generalize or abstract from instances of these, uh, um, these instances in order to get a, an abstract conception of numbers.
Brian 01:14:32 So, you know, I'm not, I'm not a plagiarist in that sense. Um, there's a, a, one of my colleagues, uh, Marcus, Quintos a philosopher at university London who takes, who takes this kind of view. I'm sure he would say that my presentation of it is, is grossly unsophisticated, but, um, it seems to me to make sense, um, and there are lots of lots of things that, where, which are abstract that, um, we only know through their instances, even letters, you know, there's lots of mm-hmm, <affirmative> different fonts for the letter. A, um, you've got handwriting, uh, for a, um, so a as a concept at the letter a as a concept is actually rather abstract. And what we know about it are instances of it and the same is true for lots of like words. You can hear words in the high voice, go voice your voice, my voice, some person's voice, but they're, they're all are instances of, if you like be abstract, uh, uh, word itself, uh, or, or music, you know, you can hear, you know, Beethoven's fifth symphony played on a Kazu, but it's still Beethoven's, it's still the tune, isn't it?
Brian 01:15:56 The tunes abstract. So anyway, I don't think it's a mystery that, um, we have and have knowledge of lots of abstract concepts and numbers are just one of them.
Paul 01:16:07 All right, Brian. Well, what's actually real is I have to go, uh, see the year end school play, which is, I, I suppose not real, it's a real play, but not about real. Anyway, now I'm getting just confused, but I enjoy learning about, uh, the counting and numerical abilities, uh, of many of the animals that you describe. Um, most of which we didn't even get to during our discussion, but, um, but enjoy talking with you. Thanks for being with me.


Approved a resolution to withdraw from the Henderson County Shared Services Arrangement and begin operation of the district's own special education program beginning with the 2011-2012 school year.
Proposed a total tax rate of $1.2763. This tax rate is composed of an M & O tax rate of $1.04 and an I & S tax rate of $0.2363.
The next regular meeting of the EISD Board of Trustees will be held on August 17, 2010.
Tonight the Eustace ISD Board of Trustees will hold their regular monthly meeting. The meeting will start at 7:00 PM and be held in the HS library. Everyone is encouraged to attend tonight's meeting.
Consider resolution to withdraw from the Henderson County Shared Services Arrangement and start the district's own special education program for the 11-12 school year.
Consider Code of Conduct, Student Handbook, and EIC(Local) Policy for the 10-11 school year.
1A and 2A schools will advance three teams to the playoffs in football. The district champion will have a bye in the first round.
Allow a school or a school district to designate start and end dates for summer strength and conditioning in grades 7-12 not to exceed a six-week period during the summer.
Require all coaches to complete an UIL Rules Compliance Program as a replacement for the COPE program.
Allow the selection of one additional individual and/or relay team in each event to advance from the regional meet to state.
Litigation against UIL from the sports officials continue.
There has been a temporary restraining order issued against sports officials registering with the UIL.
UIL plan for sports officials included having them pass an open book rules test with at least a 70 to officiate, having playoff officials score at least an 80 on the exam, and to require background checks on all officials.
Enrollment Figures – how should schools count their enrollment for reclassification purposes?
Conferences – How many should there be?
Secrecy – Can UIL be more open about the reclassification and realignment issue?
Split Conferences – Should these continue?
Playoff Teams – How many should advance?
Soccer – Is 3A ready for soccer? Currently there are 90 3A districts playing soccer.
As I mentioned yesterday, the vast majority of this year's summer conference focused on school finance. We got to hear Lisa Dawn-Fisher, deputy associate commissioner for school finance at TEA, Lynn Moak of Moak, Casey & Associates, and Omar Garcia, director of statewide school finance from ESC Region 13. Each of these people gave presentations about different aspects of the current school finance situation and what is likely to be coming up for the next biennium. We also heard from Catherine Clark, associate executive director, Texas Association of School Boards, and David Thompson, the attorney that has handled lawsuits against the state over school finance.
How did we get to our current financial situation?
Omar Garcia – Omar is the person that develops the template that estimates how much funding a school district is earning.
For the next three years (at a minimum) we will have to live with the same amount of funding we received in 09-10.
There will be a shortfall in funding for this current school year.
Will have to "borrow" from the 10-11 school year.
This will definitely cause a shortfall for the 10-11 funding year.
Legislature will have to appropriate money to cover this shortfall.
If legislature does not appropriate this money, then proration would occur.
As you can see, the economics are not great for additional funding opportunities. As always, as soon as I learn additional details, I will pass them on to you. If you have any questions about any of the above information, please do not hesitate to contact me at cholcombe@eustaceisd.net.
Last week, Mrs. Beasley and I attended the TASA Summer Conference in Austin. This is an annual conference that brings administrators together to hear topics of interest going into the new school year. The theme at many of the sessions this year was school finance. Over the next few days, I will be summarizing some of the sessions that I attended.
For the coming year, there will be a loss of about 300,000 teacher jobs across the country due to cutbacks in spending.
At the same time, the majority of the states have increased standards for students.
Efficiencies come first, but they should be focused and selective.
Dr. Poston's presentation was timely and necessary during these times of cutbacks. For the 2010-2011 school year, EISD used a version of program budgeting to examine expenditures in all program areas, to provide opportunities for collaborative decision making with regards to the budget, and to prioritize spending.
Tomorrow, we will talk about the current school funding situation in Texas.


Moms love Little Splashes of Color!
Explore our many colorful collections of nursery wall art created with you and your child in mind.
Gritty trucks, delicate unicorns, farm tractors, animals and LOTS more nursery prints for boys' and girls' rooms.
You'll love the quality of the fine art giclee prints and your kids will be delighted to be surrounded by their favorite things!


HANDMADE CHRISTMAS GIFTS ARE THE PERFECT PRESENTS FOR EVERYONE ON YOUR LIST. LEARN HOW TO MAKE SEVERAL HOMEMADE CHRISTMAS GIFTS THAT ARE ALL-NATURAL, PRACTICAL, AND THOUGHTFUL.
I can't believe I'm already thinking about Christmas! I mean it's definitely time, I just can't believe it's already been another year. Time can slow down now!
This time of year is suppose to be joyful, but often turns into stressful. With all the hustle, gatherings, shopping, and gift wrapping it can be overwhelming. Since getting into essential oils I have found that I can whip up something nice for basically everyone on my list.
You can make several of these items and put together a gift basket or make one to add personal touch to any gift.
Each of these tutorials have written instructions and a video. They are everything you need to put together a beautiful gift basket for Christmas.
Each individual tutorial has a free printable label but I decided to compile all the labels needed for this gift basket on one page to make it easier for you!
Head over to my online resource library for your free printable labels and gift tag.
The items above would be a great gift for your mom, aunt, grandmother, or cousin. If you are needing a gift for a baby or child check out these homemade items.
Looking for a gift for your man? I have you covered here too! I made this gift basket for my husband for Faher's Day last year and he loved it! Check out these simple tutorial.
Get all the free printables for this basket on my online resource library.
I hope you enjoy these tutorials and have fun putting toether some homemade Christmas gifts this year!


Grosgrain: Vivi Reborn GUEST GIVEAWAY!!!! CLOSED.
Vivi Reborn GUEST GIVEAWAY!!!! CLOSED.
I love pairing a white gauzy short sleeved blouse with a unique eye catching necklace. Vivi Reborn is one of those shops that carry pieces that look great with this look.
This giveaway is for a $50 gift certificate to their online shop.
My favorite is featured to the left. This necklace is FABULOUS! So romantic and feminine with a focal clasp at back.
Five 19mm incredibly patinaed vintage brass beads with cream colored vintage lace woven through in a serpentine pattern - all tied with a sweet bow. Set on 18 inches of antiqued brass chain.
The necklace featured to the right is simply AMAZING! Would look great in a photo shoot with a hazy white aura.
At Vivi Reborn Rebekah Lyons and Shannon Mescher create unique, hand-made pieces that embolden the sophisticated woman and intrigue those they engage.
The root meaning of the word Vivi is "to breathe new life". This inspires their work to recover vintage pieces, and style them for today's trends. They believe each vintage piece we restore has a story that will continue to be told through its next recipient.
A sustainable lifestyle is not only responsible, but essential. Every item they touch is repurposed from selected vintage medals, jewels, and stones that inspire their creativity and femininity.
When they recover them, they take special efforts to recreate them for women who possess discriminating taste. An eye that allows them to manage the complex relationship between aged beauty and fashion trend.
immersed in something deep and true, and they will be reminded that they are all part of a story that continues.
Vivi Reborn also believes in giving back. "Together, we can help women who are entrepreneurs, but don't have the same opportunities we do to launch their own shops. Our friend Jessica started a micro finance loan company called Kiva. Through them, we give 10% of all Vivi Reborn sales as loans to women around the world to launch their own creative design businesses. Every woman deserves the opportunity to pursue something that they are passionate about. With every Vivi purchase, you are helping them too."
Helen is 33 years old and married with a child. She sells jewelry in front of her house. She has been in this business for the past 3 years. Click here to read more of her story.
Each Vivi Reborn design comes beautifully packaged in a hand-stamped glassine envelope with vintage lace tie and hand-stamped dangle. Additional gift packaging options are available for a minimal upcharge. Please inquire should you be interested.
simply creating a post on your blog and including this link.
it'd be difficult to choose!!!
Oh my...I'd LOVE to win this!
I linked this post to my blog! Those are absolutely adorable!! I would ♥ to win one of those!!! I ♥♥♥ the second one! But I would be glad to win any of them!
I love the first one with white ribbon!!! Just perfect and sooooo me.
Oh, I have a link on my blog.
Wow, your stuff is beautiful!! I hope I can win something=)!!!
That first necklace is my fave too.
I created a link on my sidebar.
Gorgeous jewelry - I really like the style. I've created a link on my blog!
Oh, good golly, how would I ever choose! I posted a link on my blog, HoboMama.com.
I posted on my blog DIETS!!!
Im so excited I love VIVI REBORN!
I posted on my blog, nickandbridget.blogspot.com, but can you tell me when the giveaway is happening?
i love love love your grosgrain necklace and love the idea of wearing it with a plain white tee but don't have a blog... i'm more of a reader - can i stil enter?


What is Environmental Control Unit?
Environmental control unit (ECU) permits remote control of electronic devices for the individuals surroundings such as their home temperature and entertainment systems. This allows a person to turn on or off lights, a radio or television, and use a phone and or unlock a door from a remote or other room location. Any part of the person's environment can be controlled depending upon the system's complexity.
Glen is showing how to use the Environmental Control options from his Augmentative and Alternative Communication device.


It's the song EVERYONE is talking about, and Conor just #owned it.
Conor Maynard is like clockwork with his INCREDIBLE YouTube covers these days and this week is no different - apart from the fact he's taking on the AMAZING number one hit from Drake!
WATCH: Conor Maynard Just Turned Rihanna's 'Work' Into A HAUNTING Pop Anthem!
Yes, Conor's now tried his hand at Drake's latest hit single 'One Dance' - the number one track on the Vodafone Big Top 40 right now - and it's fair to say he absolutely SMASHES it.
Roping in a female vocalist for support, Conor strips back the incredibly popular 'One Dance' and gives it a bit of Maynard flavour, and we reckon he TOTALLY owns it. More of the same please, Con!
WATCH: Conor Maynard's Covered Nick Jonas' 'Close'… As A Mash-Up With Years & Years' 'King'!


text:
LAW Econometrics
Policy Practicum: Empirical Study of Patent Troll Litigation
Biochemistry I (BIO 288, CHEM 181, CHEMENG 181, CHEMENG 281)
Biochemistry II (BIO 289, CHEM 183, CHEMENG 183, CHEMENG 283)
Mechanisms of Neuron Death
Patently-O
Intellectual Property and Information Wealth: Issues and Practices in the Digital Age
Intellectual Property in the New Technological Age (5th ed.)
Intellectual Property: Patents, Trademarks, and Copyright in a Nutshell (4th ed.)
Ma+h You Can't Use: Patents, Copyright, and Software
No Law: Intellectual Property in the Image of an Absolute First Amendment
Symposium: Creators vs. Consumers: the Rhetoric, Reality and Reformation of Intellectual Property Law and Policy
[ Regulatory & Policy ] Intellectual Property: Patent Law
[ Regulatory & Policy ] as a Related Elective for those interested in Statistics : Drawing an inference from quantitative evidence lies at the heart of many legal and policy decisions. Students interested in policy work should consider taking one or more classes on statistics and statistical inference. This course develops statistical and empirical research expertise that is useful in a wide range of legal settings. The course is designed to be accessible to students with widely varying backgrounds in statistics or econometrics.
(Same as PUBLPOL 303B and IPS 205B). After developing/reviewing some statistical background material, the course focuses on the most essential and widely used statistical research tools for public policy analysis, including multiple regression analysis, multilevel modeling and Bayesian analysis. These tools provide vital background for a wide variety of professional pursuits - not only engaging in policy analysis in government, research institutes, and academic settings, but also handling complex empirical issues in litigation, investment banking, consulting or finance. Topics include hypothesis testing, regression specification, logistic regression, probit, heteroscedasticity, serial correlation, errors in the variables, instrumental variables, simultaneous equations, generalized linear models, simulation, model checking, causal inference, and missing data imputation. Extensive hands-on experience with empirical analysis via computer exercises using popular statistical packages is a key component of the course. The skills developed are critical for becoming an intelligent consumer or producer of empirical research. The course is designed to be accessible to students with widely varying backgrounds in statistics or econometrics. Students who desire a more mathematical treatment of the topics may take the two unit "Econometrics: Mathematical Methods" supplement (Law 367) concurrently. A knowledge of basic statistics at the level of QM: Statistical Inference is a prerequisite.
Course Style: A Substantive/Quantitative course is one that teaches the rules and theory of a quantitative subject area that relates to the practice of law.

redpajama_set_name: RedPajamaCommonCrawl


text:
Thomas P. DiNapoli
Born: 10-Feb-1954
Birthplace: Rockville Centre, NY
Occupation: Politician
Executive summary: Comptroller, State of New York
University: BA History, Hofstra University (1976)
University: MA Human Resources Management, New School University (1988)
Professor: Adjunct, Long Island University
New York State Official Comptroller (2007-)
New York State Assembly 16th (1987-2007)
AT&T New York Telephone:Management (1984-87)
Congressional Staff to Rep. Robert J. Mrazek (1983-85)
New York State Official Mineola Board of Education (1972-82)
Big Brothers Big Sisters of America Nassau County Board (1980-86)
Nassau County Democratic County Committee Chairman (1999-2001)
Sons of Italy
Italian Ancestry
http://www.osc.state.ny.us/
NNDB MAPPER Create a map starting with Thomas P. DiNapoli


text:
use crate::middle::region;
use crate::mir;
use crate::ty;
use rustc_data_structures::fingerprint::Fingerprint;
use rustc_data_structures::fx::FxHashMap;
use rustc_data_structures::stable_hasher::{HashStable, StableHasher, ToStableHashKey};
use rustc_query_system::ich::StableHashingContext;
use std::cell::RefCell;
use std::mem;

impl<'a, 'tcx, T> HashStable<StableHashingContext<'a>> for &'tcx ty::List<T>
where
    T: HashStable<StableHashingContext<'a>>,
{
    fn hash_stable(&self, hcx: &mut StableHashingContext<'a>, hasher: &mut StableHasher) {
        thread_local! {
            static CACHE: RefCell<FxHashMap<(usize, usize), Fingerprint>> =
                RefCell::new(Default::default());
        }

        let hash = CACHE.with(|cache| {
            let key = (self.as_ptr() as usize, self.len());
            if let Some(&hash) = cache.borrow().get(&key) {
                return hash;
            }

            let mut hasher = StableHasher::new();
            (&self[..]).hash_stable(hcx, &mut hasher);

            let hash: Fingerprint = hasher.finish();
            cache.borrow_mut().insert(key, hash);
            hash
        });

        hash.hash_stable(hcx, hasher);
    }
}

impl<'a, 'tcx, T> ToStableHashKey<StableHashingContext<'a>> for &'tcx ty::List<T>
where
    T: HashStable<StableHashingContext<'a>>,
{
    type KeyType = Fingerprint;

    #[inline]
    fn to_stable_hash_key(&self, hcx: &StableHashingContext<'a>) -> Fingerprint {
        let mut hasher = StableHasher::new();
        let mut hcx: StableHashingContext<'a> = hcx.clone();
        self.hash_stable(&mut hcx, &mut hasher);
        hasher.finish()
    }
}

impl<'a, 'tcx> HashStable<StableHashingContext<'a>> for ty::subst::GenericArg<'tcx> {
    fn hash_stable(&self, hcx: &mut StableHashingContext<'a>, hasher: &mut StableHasher) {
        self.unpack().hash_stable(hcx, hasher);
    }
}

impl<'a> HashStable<StableHashingContext<'a>> for ty::RegionKind {
    fn hash_stable(&self, hcx: &mut StableHashingContext<'a>, hasher: &mut StableHasher) {
        mem::discriminant(self).hash_stable(hcx, hasher);
        match *self {
            ty::ReErased | ty::ReStatic => {
                // No variant fields to hash for these ...
            }
            ty::ReEmpty(universe) => {
                universe.hash_stable(hcx, hasher);
            }
            ty::ReLateBound(db, ty::BoundRegion { kind: ty::BrAnon(i), .. }) => {
                db.hash_stable(hcx, hasher);
                i.hash_stable(hcx, hasher);
            }
            ty::ReLateBound(db, ty::BoundRegion { kind: ty::BrNamed(def_id, name), .. }) => {
                db.hash_stable(hcx, hasher);
                def_id.hash_stable(hcx, hasher);
                name.hash_stable(hcx, hasher);
            }
            ty::ReLateBound(db, ty::BoundRegion { kind: ty::BrEnv, .. }) => {
                db.hash_stable(hcx, hasher);
            }
            ty::ReEarlyBound(ty::EarlyBoundRegion { def_id, index, name }) => {
                def_id.hash_stable(hcx, hasher);
                index.hash_stable(hcx, hasher);
                name.hash_stable(hcx, hasher);
            }
            ty::ReFree(ref free_region) => {
                free_region.hash_stable(hcx, hasher);
            }
            ty::RePlaceholder(p) => {
                p.hash_stable(hcx, hasher);
            }
            ty::ReVar(..) => {
                bug!("StableHasher: unexpected region {:?}", *self)
            }
        }
    }
}

impl<'a> HashStable<StableHashingContext<'a>> for ty::RegionVid {
    #[inline]
    fn hash_stable(&self, hcx: &mut StableHashingContext<'a>, hasher: &mut StableHasher) {
        self.index().hash_stable(hcx, hasher);
    }
}

impl<'a, 'tcx> HashStable<StableHashingContext<'a>> for ty::ConstVid<'tcx> {
    #[inline]
    fn hash_stable(&self, hcx: &mut StableHashingContext<'a>, hasher: &mut StableHasher) {
        self.index.hash_stable(hcx, hasher);
    }
}

impl<'tcx> HashStable<StableHashingContext<'tcx>> for ty::BoundVar {
    #[inline]
    fn hash_stable(&self, hcx: &mut StableHashingContext<'tcx>, hasher: &mut StableHasher) {
        self.index().hash_stable(hcx, hasher);
    }
}

impl<'a, 'tcx, T> HashStable<StableHashingContext<'a>> for ty::Binder<'tcx, T>
where
    T: HashStable<StableHashingContext<'a>>,
{
    fn hash_stable(&self, hcx: &mut StableHashingContext<'a>, hasher: &mut StableHasher) {
        self.as_ref().skip_binder().hash_stable(hcx, hasher);
        self.bound_vars().hash_stable(hcx, hasher);
    }
}

// AllocIds get resolved to whatever they point to (to be stable)
impl<'a> HashStable<StableHashingContext<'a>> for mir::interpret::AllocId {
    fn hash_stable(&self, hcx: &mut StableHashingContext<'a>, hasher: &mut StableHasher) {
        ty::tls::with_opt(|tcx| {
            trace!("hashing {:?}", *self);
            let tcx = tcx.expect("can't hash AllocIds during hir lowering");
            tcx.get_global_alloc(*self).hash_stable(hcx, hasher);
        });
    }
}

// `Relocations` with default type parameters is a sorted map.
impl<'a, Tag> HashStable<StableHashingContext<'a>> for mir::interpret::Relocations<Tag>
where
    Tag: HashStable<StableHashingContext<'a>>,
{
    fn hash_stable(&self, hcx: &mut StableHashingContext<'a>, hasher: &mut StableHasher) {
        self.len().hash_stable(hcx, hasher);
        for reloc in self.iter() {
            reloc.hash_stable(hcx, hasher);
        }
    }
}

impl<'a> ToStableHashKey<StableHashingContext<'a>> for region::Scope {
    type KeyType = region::Scope;

    #[inline]
    fn to_stable_hash_key(&self, _: &StableHashingContext<'a>) -> region::Scope {
        *self
    }
}

redpajama_set_name: RedPajamaGithub