WEBVTT 00:00:00.990 --> 00:00:02.475 [SQUEAKING] 00:00:02.475 --> 00:00:03.960 [RUSTLING] 00:00:03.960 --> 00:00:10.500 [CLICKING] 00:00:10.500 --> 00:00:12.000 NANCY KANWISHER: All right, so we're 00:00:12.000 --> 00:00:13.587 going to talk about number. 00:00:13.587 --> 00:00:16.170 I got a little carried away with the behavioral work on number 00:00:16.170 --> 00:00:18.623 because I just think it is so awesome. 00:00:18.623 --> 00:00:20.790 And I think it's, frankly, a little more interesting 00:00:20.790 --> 00:00:22.040 than a lot of the neural work. 00:00:22.040 --> 00:00:25.740 So this is going to be sort of a behaviorally heavy lecture. 00:00:25.740 --> 00:00:30.390 But let's start by thinking about why we have number 00:00:30.390 --> 00:00:32.009 and what we use it for. 00:00:32.009 --> 00:00:33.900 And the first thing to realize is 00:00:33.900 --> 00:00:38.700 that we use concepts of number and quantity like all the time. 00:00:38.700 --> 00:00:41.182 Most obviously, if you're, say, getting change at a store. 00:00:41.182 --> 00:00:43.390 I guess that doesn't really happen very much anymore. 00:00:43.390 --> 00:00:45.510 People are going to forget how to subtract because they just 00:00:45.510 --> 00:00:47.370 put their credit card or bump their phone 00:00:47.370 --> 00:00:48.250 or whatever they do. 00:00:48.250 --> 00:00:50.610 But anyway, it used to be that you handed 00:00:50.610 --> 00:00:53.190 over this stuff called money and that coins came back 00:00:53.190 --> 00:00:56.280 and that was the subtraction involved. 00:00:56.280 --> 00:00:59.970 We use it to tell time or to fail to tell time, 00:00:59.970 --> 00:01:02.310 as in my case this morning. 00:01:02.310 --> 00:01:04.410 To choose the larger of two objects, 00:01:04.410 --> 00:01:06.250 that's a continuous idea of quantity, 00:01:06.250 --> 00:01:09.030 not a discrete idea of number. 00:01:09.030 --> 00:01:11.880 To choose the shortest line at a grocery store, right, 00:01:11.880 --> 00:01:16.110 and all of those kinds of things are comparing quantities. 00:01:16.110 --> 00:01:20.130 And we also take these basic ideas of number and quantity 00:01:20.130 --> 00:01:22.140 and we build on them in modern societies 00:01:22.140 --> 00:01:26.940 to do all kinds of amazing things like engineering. 00:01:26.940 --> 00:01:29.760 Like all of modern science is highly quantitative, 00:01:29.760 --> 00:01:32.440 like all of computer science. 00:01:32.440 --> 00:01:35.670 And so these are really fundamental ideas. 00:01:35.670 --> 00:01:40.770 And animals, it turns out, are capable of mastering 00:01:40.770 --> 00:01:45.030 very simple but sophisticated understandings of number 00:01:45.030 --> 00:01:49.650 and even arithmetic computations. 00:01:49.650 --> 00:01:54.540 They can learn about order and number and quantity. 00:01:54.540 --> 00:01:59.410 OK, and they need to for lots of reasons. 00:01:59.410 --> 00:02:03.210 So here, just a brief overview of some of the situations 00:02:03.210 --> 00:02:05.970 where animals need concepts of number and quantity 00:02:05.970 --> 00:02:07.770 in the wild. 00:02:07.770 --> 00:02:10.080 Foraging, right, so animals spend a lot of time 00:02:10.080 --> 00:02:11.922 rooting around for food over here, 00:02:11.922 --> 00:02:13.380 rooting around for food over there, 00:02:13.380 --> 00:02:16.650 deciding when to keep rooting around here despite diminishing 00:02:16.650 --> 00:02:20.910 returns and go somewhere else where there's unknown pay off, 00:02:20.910 --> 00:02:21.960 unknown amounts of food. 00:02:21.960 --> 00:02:25.460 So that's a whole math of foraging behavior. 00:02:25.460 --> 00:02:28.400 OK, so that deals with the rate of return 00:02:28.400 --> 00:02:31.490 of the food at each location and the amount and the quality. 00:02:31.490 --> 00:02:36.180 And you can imagine a whole math to optimize the amount of food. 00:02:36.180 --> 00:02:38.960 They also need to know about number and quantity 00:02:38.960 --> 00:02:42.620 when they form teams, which many animals across taxa 00:02:42.620 --> 00:02:44.550 do in different ways. 00:02:44.550 --> 00:02:48.050 So schooling fish can quickly pick out 00:02:48.050 --> 00:02:52.160 the more numerous school of fish to join. 00:02:52.160 --> 00:02:54.987 And that's what they want to do because your statistics are 00:02:54.987 --> 00:02:56.570 better if they're a predator if you're 00:02:56.570 --> 00:02:59.600 in the larger school than the smaller school, right? 00:03:02.170 --> 00:03:05.200 So your chance of getting eaten is reduced just 00:03:05.200 --> 00:03:07.910 dividing by the number of options. 00:03:07.910 --> 00:03:09.670 And then there's all kinds of animals 00:03:09.670 --> 00:03:13.630 that take into account the size of groups of their own species 00:03:13.630 --> 00:03:19.870 or other species when making decisions about how far to run 00:03:19.870 --> 00:03:23.380 or who to chase or who to predate on 00:03:23.380 --> 00:03:27.280 or who's at risk of predating upon you. 00:03:27.280 --> 00:03:30.160 So lions hunt in teams. 00:03:30.160 --> 00:03:31.460 And they have to work together. 00:03:31.460 --> 00:03:34.060 They have actually whole strategic situation 00:03:34.060 --> 00:03:36.700 where different lions play different parts like a football 00:03:36.700 --> 00:03:38.260 game. 00:03:38.260 --> 00:03:41.500 And they have to decide which groups of predators 00:03:41.500 --> 00:03:45.520 to take on based on numerical advantage. 00:03:45.520 --> 00:03:49.390 And my favorite is the n plus one frog, the Tungara 00:03:49.390 --> 00:03:54.890 frog that lives in the rainforest in Puerto Rico. 00:03:54.890 --> 00:03:58.420 And it literally one ups other frogs, the males 00:03:58.420 --> 00:04:01.210 do in trying to impress the females. 00:04:01.210 --> 00:04:05.320 And so what happens is that one frog will start calling out. 00:04:05.320 --> 00:04:07.570 One male frog will start calling out trying 00:04:07.570 --> 00:04:10.090 to sound all hot to the gals. 00:04:10.090 --> 00:04:13.690 And then another frog will one up him by doing that call 00:04:13.690 --> 00:04:17.050 but elaborating on it by adding an extra call 00:04:17.050 --> 00:04:18.970 or an extra component. 00:04:18.970 --> 00:04:21.955 So for example-- 00:04:21.955 --> 00:04:27.180 [FROG CALL] OK, so that's one dude calling out. 00:04:27.180 --> 00:04:32.766 And not to be outdone, the next guy calls back. 00:04:32.766 --> 00:04:37.780 [FROG CALL] And apparently, if you follow these guys, they 00:04:37.780 --> 00:04:42.760 pretty systematically add one to the previous frog's call, 00:04:42.760 --> 00:04:45.220 right, up to a point. 00:04:45.220 --> 00:04:47.800 The point being approximately four. 00:04:47.800 --> 00:04:51.010 So it's not like 100 and 101. 00:04:51.010 --> 00:04:53.950 But it happens. 00:04:53.950 --> 00:04:56.680 OK, so that's just a broad overview of some of the cases 00:04:56.680 --> 00:05:00.700 that understandings of number and quantity 00:05:00.700 --> 00:05:04.370 arise in natural environments without training. 00:05:04.370 --> 00:05:07.270 So we want to know how is all this computed in the mind 00:05:07.270 --> 00:05:09.020 and brain. 00:05:09.020 --> 00:05:12.430 And so what are the foremost thinkers on this topic 00:05:12.430 --> 00:05:15.260 is Stan Dehaene, shown here. 00:05:15.260 --> 00:05:20.620 And he wrote in a very widely cited book, actually 00:05:20.620 --> 00:05:24.400 review article and book quite a while ago, 20 years ago, 00:05:24.400 --> 00:05:28.210 he said, animals, young infants, and adult humans 00:05:28.210 --> 00:05:31.450 possess a biologically determined, 00:05:31.450 --> 00:05:35.240 domain-specific representation of number. 00:05:35.240 --> 00:05:38.488 So this is a very kind of extreme, hardcore claim. 00:05:38.488 --> 00:05:40.030 We will see at the end of the lecture 00:05:40.030 --> 00:05:42.700 that he has backed off that claim. 00:05:42.700 --> 00:05:46.090 OK, so a couple of things, biologically determined, 00:05:46.090 --> 00:05:49.720 he's kind of implying innate, right? 00:05:49.720 --> 00:05:52.690 Domain-specific, I've avoided this phrase, 00:05:52.690 --> 00:05:54.880 for the most part, because it's kind of like jargon 00:05:54.880 --> 00:05:55.600 gobbledygook. 00:05:55.600 --> 00:05:58.282 But it's actually so entrenched in our field 00:05:58.282 --> 00:05:59.740 that it's worth knowing what it is. 00:05:59.740 --> 00:06:02.860 Domain-specific is just this idea of functional specificity 00:06:02.860 --> 00:06:04.120 that I've been talking about. 00:06:04.120 --> 00:06:06.940 But you can apply it to not just a piece of brain like, 00:06:06.940 --> 00:06:09.550 does this patch of brain process only faces? 00:06:09.550 --> 00:06:12.160 You can also apply it to a mental process 00:06:12.160 --> 00:06:14.500 even if you don't know what its actual brain basis is. 00:06:14.500 --> 00:06:17.350 So do we have special mental machinery 00:06:17.350 --> 00:06:18.820 for thinking about numbers that's 00:06:18.820 --> 00:06:22.420 distinct from our machinery for face recognition or navigation 00:06:22.420 --> 00:06:24.100 or language or whatever else? 00:06:24.100 --> 00:06:26.410 OK, so that's what domain-specific means. 00:06:26.410 --> 00:06:27.970 And it's worth knowing because you'll 00:06:27.970 --> 00:06:30.430 encounter it in other contexts. 00:06:30.430 --> 00:06:35.410 OK, so in more detail, Stan says, 00:06:35.410 --> 00:06:38.050 a specific neural substrate, located 00:06:38.050 --> 00:06:40.900 in the left intraparietal area, is 00:06:40.900 --> 00:06:43.810 associated with knowledge of numbers and their relations, 00:06:43.810 --> 00:06:46.090 which he defines as number sense. 00:06:46.090 --> 00:06:47.950 The number domain is a prime example 00:06:47.950 --> 00:06:51.190 where strong evidence points to an evolutionary endowment 00:06:51.190 --> 00:06:54.250 of abstract, domain-specific knowledge in the brain 00:06:54.250 --> 00:06:56.620 because there are parallels between number processing 00:06:56.620 --> 00:06:58.720 in animals and humans. 00:06:58.720 --> 00:07:00.790 Again, kind of hardcore claims. 00:07:00.790 --> 00:07:04.120 Not just is there this so he doesn't quite say innate, 00:07:04.120 --> 00:07:05.770 but he's strongly implying innate. 00:07:05.770 --> 00:07:08.380 I mean, that's evolutionary endowment, that 00:07:08.380 --> 00:07:10.180 basically means innate, right? 00:07:10.180 --> 00:07:13.240 It's an evolved ability that lives in a particular part 00:07:13.240 --> 00:07:14.200 of the brain. 00:07:14.200 --> 00:07:14.980 OK? 00:07:14.980 --> 00:07:16.660 So who would a thunk, right? 00:07:16.660 --> 00:07:17.243 Number, right? 00:07:17.243 --> 00:07:19.660 You think of number as something you get taught in school. 00:07:19.660 --> 00:07:21.220 But no, he's saying it's really part 00:07:21.220 --> 00:07:23.000 of your biological endowment. 00:07:23.000 --> 00:07:25.480 It has a particular brain region. 00:07:25.480 --> 00:07:28.990 And all of that may be if not completely independent, 00:07:28.990 --> 00:07:32.370 it may exist without explicit training. 00:07:32.370 --> 00:07:32.870 OK? 00:07:32.870 --> 00:07:34.670 So that's quite a claim. 00:07:34.670 --> 00:07:38.200 So what does number sense mean exactly? 00:07:38.200 --> 00:07:41.910 Well, what Stan and others in the field mean by number sense, 00:07:41.910 --> 00:07:43.180 it's a bunch of things. 00:07:43.180 --> 00:07:46.085 First of all, the idea that for human adults 00:07:46.085 --> 00:07:48.210 to have number sense, that means they can represent 00:07:48.210 --> 00:07:52.350 large numerical magnitudes without verbal counting, right? 00:07:52.350 --> 00:07:53.850 So counting is an interesting thing. 00:07:53.850 --> 00:07:55.892 But we're going to leave it aside for the moment. 00:07:55.892 --> 00:07:58.200 Number sense is a more general idea 00:07:58.200 --> 00:08:01.200 that's going to apply to animals and infants 00:08:01.200 --> 00:08:02.815 without explicit counting. 00:08:02.815 --> 00:08:05.190 OK, so you can have some way of representing that there's 00:08:05.190 --> 00:08:06.065 a lot of things here. 00:08:06.065 --> 00:08:08.970 And there's fewer things there. 00:08:08.970 --> 00:08:10.470 Second of all, these representations 00:08:10.470 --> 00:08:12.870 are approximate. 00:08:12.870 --> 00:08:15.510 And the ability to discriminate two of them 00:08:15.510 --> 00:08:17.910 depends on the ratio of those two, 00:08:17.910 --> 00:08:20.160 not the absolute difference. 00:08:20.160 --> 00:08:22.620 OK, and I'll show you in more detail what I mean by that. 00:08:22.620 --> 00:08:24.450 It's a deep fact about number sense 00:08:24.450 --> 00:08:28.200 and actually all of perception, pretty much. 00:08:28.200 --> 00:08:31.800 Further, the idea is that these representations are abstract. 00:08:31.800 --> 00:08:35.100 They're not just, say, a particular visual form. 00:08:35.100 --> 00:08:38.169 Like approximately 13 looks like this. 00:08:38.169 --> 00:08:38.669 No. 00:08:38.669 --> 00:08:41.640 They're going to generalize across modality, OK, 00:08:41.640 --> 00:08:44.620 and space and time. 00:08:44.620 --> 00:08:50.100 Next, these mental representations of number 00:08:50.100 --> 00:08:51.870 can be used in operations. 00:08:51.870 --> 00:08:55.260 Even without counting and being explicitly informed, 00:08:55.260 --> 00:08:57.150 you can add approximate numbers. 00:08:57.150 --> 00:08:59.400 You may be thinking, what the hell am I talking about? 00:08:59.400 --> 00:09:02.050 But I'll show you in a second. 00:09:02.050 --> 00:09:07.650 So for example, I'm going to show you two sets of dots next. 00:09:07.650 --> 00:09:09.660 And you're just going to shout out first 00:09:09.660 --> 00:09:12.360 if the first set of dots had more, 00:09:12.360 --> 00:09:14.310 if there were more dots in the first array 00:09:14.310 --> 00:09:16.930 and second if the second array had more dots. 00:09:16.930 --> 00:09:17.430 OK, ready? 00:09:17.430 --> 00:09:18.750 Here we go. 00:09:18.750 --> 00:09:19.880 Boom. 00:09:19.880 --> 00:09:22.800 Boom. 00:09:22.800 --> 00:09:23.400 Second. 00:09:23.400 --> 00:09:23.910 Duh. 00:09:23.910 --> 00:09:24.993 OK, let's try another one. 00:09:28.980 --> 00:09:29.693 Duh. 00:09:29.693 --> 00:09:30.360 And another one. 00:09:33.162 --> 00:09:33.990 Uh huh. 00:09:33.990 --> 00:09:34.590 Another one. 00:09:40.930 --> 00:09:43.270 I noticed the volume decreasing. 00:09:43.270 --> 00:09:44.920 And I noticed some hesitancy. 00:09:48.950 --> 00:09:51.840 Actually, I'm not sure about that one. 00:09:51.840 --> 00:09:56.190 OK, so how did you do that? 00:09:56.190 --> 00:09:57.410 What did you do? 00:09:57.410 --> 00:10:00.620 Did you go 1, 2, 3, 4, 5? 00:10:00.620 --> 00:10:01.120 No. 00:10:01.120 --> 00:10:02.860 I tried to do it, so there wasn't time to do that. 00:10:02.860 --> 00:10:03.597 How'd you do it? 00:10:03.597 --> 00:10:05.680 AUDIENCE: I kind of tried to see like the density, 00:10:05.680 --> 00:10:07.110 like how close all the dots were. 00:10:07.110 --> 00:10:08.110 NANCY KANWISHER: Mm-hmm. 00:10:08.110 --> 00:10:08.590 Mm-hmm. 00:10:08.590 --> 00:10:09.673 And did that work for you? 00:10:09.673 --> 00:10:11.610 Did that work OK? 00:10:11.610 --> 00:10:13.015 AUDIENCE: It seems to be OK. 00:10:13.015 --> 00:10:14.890 NANCY KANWISHER: OK, what Jack is pointing to 00:10:14.890 --> 00:10:16.515 is a really important thing in thinking 00:10:16.515 --> 00:10:18.730 about number, which is that number 00:10:18.730 --> 00:10:21.040 is confounded with area extent. 00:10:21.040 --> 00:10:23.410 How much total yellow stuff is on the screen? 00:10:23.410 --> 00:10:25.460 And it's confounded with density. 00:10:25.460 --> 00:10:28.090 And this is a big problem in people who 00:10:28.090 --> 00:10:29.500 want to do research on number. 00:10:29.500 --> 00:10:31.500 And so what they usually do is you can't totally 00:10:31.500 --> 00:10:32.560 unconfound those things. 00:10:32.560 --> 00:10:34.850 But you can unconfound them one at a time. 00:10:34.850 --> 00:10:37.090 So you can vary the size of the objects. 00:10:37.090 --> 00:10:39.230 And you can vary the density across trials. 00:10:39.230 --> 00:10:42.343 So no one of those cues will enable you to do it fully. 00:10:42.343 --> 00:10:44.260 This example isn't great that way because they 00:10:44.260 --> 00:10:47.750 were all the same size, right? 00:10:47.750 --> 00:10:50.990 OK, but so the point is, without explicitly counting, 00:10:50.990 --> 00:10:52.950 and God knows what you do it, how you do it, 00:10:52.950 --> 00:10:55.400 you just feel like you have a sense of roughly how many. 00:10:55.400 --> 00:10:56.960 Everybody got that sense? 00:10:56.960 --> 00:10:59.150 OK, so that's what we mean by number sense 00:10:59.150 --> 00:11:01.310 is that sense that you can just look at something 00:11:01.310 --> 00:11:03.290 and have a sense of roughly how many. 00:11:03.290 --> 00:11:05.420 Like you don't know if it's 19 or 18, 00:11:05.420 --> 00:11:08.720 but you know it's not 13, right? 00:11:08.720 --> 00:11:12.080 OK, right. 00:11:12.080 --> 00:11:16.400 Oh, and you guys all got quieter when the numbers 00:11:16.400 --> 00:11:17.580 got closer together. 00:11:17.580 --> 00:11:18.080 OK? 00:11:18.080 --> 00:11:20.930 It gets harder when the numbers are closer together. 00:11:20.930 --> 00:11:24.657 OK, so in experiments that have quantified this, lots of people 00:11:24.657 --> 00:11:25.490 have looked at this. 00:11:25.490 --> 00:11:27.450 Here's one that I was involved in way back. 00:11:27.450 --> 00:11:30.590 Just like you did, this is the task here 00:11:30.590 --> 00:11:32.558 that you guys just did. 00:11:32.558 --> 00:11:34.100 And here are some of the data we got. 00:11:34.100 --> 00:11:35.433 So let me walk you through this. 00:11:35.433 --> 00:11:38.450 This is accuracy on a bunch of different comparisons. 00:11:38.450 --> 00:11:43.220 16 dots versus 32 dots, people are pretty much 100% correct. 00:11:43.220 --> 00:11:44.060 OK? 00:11:44.060 --> 00:11:46.970 This is just normal human adults. 00:11:46.970 --> 00:11:48.980 16 versus 24, great. 00:11:48.980 --> 00:11:51.020 16 versus 20, pretty good. 00:11:51.020 --> 00:11:54.110 16 versus 18, now we're really dropping. 00:11:54.110 --> 00:11:56.030 16 versus 17, forget it. 00:11:56.030 --> 00:11:57.000 Can't do it. 00:11:57.000 --> 00:11:57.500 OK? 00:11:57.500 --> 00:12:01.790 So performance falls off as the numbers get closer together. 00:12:01.790 --> 00:12:02.300 OK? 00:12:02.300 --> 00:12:04.310 So that's sort of intuitive. 00:12:04.310 --> 00:12:11.540 But now let's consider these are all comparing to 16. 00:12:11.540 --> 00:12:13.000 Here, we compare to eight. 00:12:13.000 --> 00:12:14.630 Eight versus 16. 00:12:14.630 --> 00:12:15.830 Eight versus 12. 00:12:15.830 --> 00:12:16.580 Eight versus 10. 00:12:16.580 --> 00:12:17.330 Eight versus nine. 00:12:17.330 --> 00:12:20.900 You see the same fall off as the numbers get closer together. 00:12:20.900 --> 00:12:21.920 OK? 00:12:21.920 --> 00:12:23.490 So far, so good. 00:12:23.490 --> 00:12:26.690 But now we can ask, what determines that fall off? 00:12:26.690 --> 00:12:30.090 Is it the absolute difference or the ratio? 00:12:30.090 --> 00:12:36.110 And the way we tell is we plot the ratio of those two curves, 00:12:36.110 --> 00:12:37.640 and we look at performance. 00:12:37.640 --> 00:12:41.090 And we see they are spot on top of each other. 00:12:41.090 --> 00:12:43.670 That tells us that it is not the absolute difference that 00:12:43.670 --> 00:12:45.440 determines your ability to do this 00:12:45.440 --> 00:12:48.030 but the ratio of the numbers of dots. 00:12:48.030 --> 00:12:48.530 OK? 00:12:48.530 --> 00:12:50.690 It's sort of intuitive, right? 00:12:50.690 --> 00:12:53.510 But it's amazing how clear the result is. 00:12:53.510 --> 00:12:55.040 Everybody get that? 00:12:55.040 --> 00:12:58.670 OK, so this is a really deep fundamental fact 00:12:58.670 --> 00:13:02.480 about perceiving approximate number. 00:13:02.480 --> 00:13:04.130 And it's actually, more generally, 00:13:04.130 --> 00:13:05.690 a fact about perception. 00:13:05.690 --> 00:13:07.130 It's called Weber's law. 00:13:07.130 --> 00:13:09.980 And it just means that the discriminability 00:13:09.980 --> 00:13:12.470 of, in this case, two numbers, two numerosities, 00:13:12.470 --> 00:13:15.530 depends on their ratio, not their absolute difference. 00:13:15.530 --> 00:13:20.240 The exact same thing holds for evaluating which of two stimuli 00:13:20.240 --> 00:13:23.280 is brighter, which of two objects is heavier, 00:13:23.280 --> 00:13:24.980 which of two sounds is louder. 00:13:24.980 --> 00:13:26.810 They all follow. 00:13:26.810 --> 00:13:28.430 The ability to do that is a function 00:13:28.430 --> 00:13:31.490 of the ratio of the, two not the absolute difference. 00:13:31.490 --> 00:13:32.550 Yeah? 00:13:32.550 --> 00:13:38.385 AUDIENCE: [INAUDIBLE] with the size of the dots? 00:13:38.385 --> 00:13:40.010 NANCY KANWISHER: So in this experiment, 00:13:40.010 --> 00:13:44.510 we varied the sizes every which way and the density. 00:13:44.510 --> 00:13:47.300 As I mentioned before, you can't completely 00:13:47.300 --> 00:13:51.710 unconfound both size and density within each trial. 00:13:51.710 --> 00:13:53.620 But across trials, you can muck them up. 00:13:53.620 --> 00:13:55.370 So you can ask whether people are doing it 00:13:55.370 --> 00:13:57.470 by size or by density. 00:13:57.470 --> 00:13:58.340 OK? 00:13:58.340 --> 00:14:00.690 And we did all that here. 00:14:00.690 --> 00:14:02.880 OK, so this is not shocking yet. 00:14:02.880 --> 00:14:06.600 It's just kind of a basic, deep, clear fact about whatever 00:14:06.600 --> 00:14:08.160 our mental representation of number 00:14:08.160 --> 00:14:10.650 is, that it's this approximate thing. 00:14:10.650 --> 00:14:11.730 It's pretty good. 00:14:11.730 --> 00:14:16.950 And its precision scales with the magnitude. 00:14:16.950 --> 00:14:22.560 OK, all right, so this has been quantified in lots and lots 00:14:22.560 --> 00:14:23.820 of experiments. 00:14:23.820 --> 00:14:25.560 And this is called the Approximate Number 00:14:25.560 --> 00:14:28.080 System, or ANS. 00:14:28.080 --> 00:14:30.570 And the standard test that's been used in lots of studies 00:14:30.570 --> 00:14:34.470 to measure people's kind of number acuity 00:14:34.470 --> 00:14:36.603 is a lot like what I just showed you. 00:14:36.603 --> 00:14:37.770 You show an array like this. 00:14:37.770 --> 00:14:41.340 And you say, are there more yellow dots or blue dots? 00:14:41.340 --> 00:14:43.710 And people very quickly say yellow, in this case. 00:14:43.710 --> 00:14:45.720 And then you ask for a case like this. 00:14:45.720 --> 00:14:47.940 And they're a little slower, right? 00:14:47.940 --> 00:14:50.550 And here, you can see that the sizes have changed 00:14:50.550 --> 00:14:53.160 and have been orthogonalized. 00:14:53.160 --> 00:14:57.510 OK, so the ratio of yellow to blue dots 00:14:57.510 --> 00:14:59.610 is called the Weber fraction, right? 00:14:59.610 --> 00:15:02.970 This is this idea of Weber's law that determines your accuracy 00:15:02.970 --> 00:15:05.700 from just that ratio. 00:15:05.700 --> 00:15:08.810 And so you can measure people's Weber fraction, their ability 00:15:08.810 --> 00:15:11.582 to do this task, their kind of number precision. 00:15:11.582 --> 00:15:13.040 And what you find is, first of all, 00:15:13.040 --> 00:15:15.800 that there's very big individual differences. 00:15:15.800 --> 00:15:16.790 OK? 00:15:16.790 --> 00:15:17.870 Now, this is interesting. 00:15:17.870 --> 00:15:20.180 It's like things that we've seen in other domains. 00:15:20.180 --> 00:15:22.160 There are very big individual differences 00:15:22.160 --> 00:15:24.040 in navigational ability. 00:15:24.040 --> 00:15:25.910 There are very big individual differences 00:15:25.910 --> 00:15:28.610 in face recognition ability. 00:15:28.610 --> 00:15:32.790 And in both of those cases as well, 00:15:32.790 --> 00:15:37.100 there are people who are just so bad at it, from an early age, 00:15:37.100 --> 00:15:38.390 that it's like a syndrome. 00:15:38.390 --> 00:15:42.170 In this case, it's called developmental dyscalculia. 00:15:42.170 --> 00:15:44.420 I think I didn't fit it into the navigation lectures. 00:15:44.420 --> 00:15:48.200 But there's a whole kind of developmental disability 00:15:48.200 --> 00:15:51.440 in navigation that's called developmental topographic 00:15:51.440 --> 00:15:52.040 agnosia. 00:15:52.040 --> 00:15:55.550 People were just always really awful at knowing 00:15:55.550 --> 00:15:57.320 where they are, right? 00:15:57.320 --> 00:16:00.560 And I did mention developmental prosopagnosia. 00:16:00.560 --> 00:16:03.380 People were just always awful at face recognition. 00:16:03.380 --> 00:16:07.040 In each of these cases, in the apparent lack 00:16:07.040 --> 00:16:11.900 of any evidence of brain damage and in the absence 00:16:11.900 --> 00:16:14.780 of differences in IQ or other abilities. 00:16:14.780 --> 00:16:16.910 So it seems like each of those abilities 00:16:16.910 --> 00:16:18.500 has a very broad range. 00:16:18.500 --> 00:16:20.030 At the bottom end of the range, it's 00:16:20.030 --> 00:16:23.390 really kind of affects your life you're so bad. 00:16:23.390 --> 00:16:25.028 And it's unrelated to other abilities. 00:16:25.028 --> 00:16:26.570 And I think that's pretty interesting 00:16:26.570 --> 00:16:28.340 because it goes along with the idea 00:16:28.340 --> 00:16:32.090 that those mental abilities are really distinct parts of mind 00:16:32.090 --> 00:16:32.930 and brain. 00:16:32.930 --> 00:16:34.820 You can have a crappy number sense, 00:16:34.820 --> 00:16:38.090 and it doesn't mean that you're bad at other things. 00:16:38.090 --> 00:16:39.770 You just have a crappy number sense. 00:16:39.770 --> 00:16:42.120 It's a separate system, right? 00:16:42.120 --> 00:16:42.620 OK. 00:16:45.800 --> 00:16:48.470 Approximate number sense develops slowly. 00:16:48.470 --> 00:16:50.210 It's best at age 30. 00:16:50.210 --> 00:16:52.010 You guys are still on the upswing. 00:16:52.010 --> 00:16:54.860 We won't talk about me. 00:16:54.860 --> 00:16:57.560 This is what do we have here? 00:16:57.560 --> 00:16:58.950 This is Weber fraction. 00:16:58.950 --> 00:17:01.910 So the Weber fraction is what that ratio 00:17:01.910 --> 00:17:05.640 needs to be for you to be fairly accurate on whatever criteria 00:17:05.640 --> 00:17:06.140 they chose. 00:17:06.140 --> 00:17:08.810 And so a small fraction means you're better. 00:17:08.810 --> 00:17:10.380 And so it goes down. 00:17:10.380 --> 00:17:13.520 And this is age here, best at 30. 00:17:13.520 --> 00:17:17.599 And this is reaction time, which goes up for everything. 00:17:17.599 --> 00:17:19.420 What a bummer. 00:17:19.420 --> 00:17:20.300 Anyway. 00:17:20.300 --> 00:17:24.290 Interestingly, early ability with approximate number 00:17:24.290 --> 00:17:27.710 on this kind of a test predicts later math ability 00:17:27.710 --> 00:17:32.150 with very different kinds of organized math 00:17:32.150 --> 00:17:34.560 that you learn in school. 00:17:34.560 --> 00:17:36.960 So here's a study that looked at that. 00:17:36.960 --> 00:17:39.690 They asked whether this early approximate number 00:17:39.690 --> 00:17:42.960 sense is predictive of later arithmetic ability. 00:17:42.960 --> 00:17:46.107 And so in this case, they did a task like this. 00:17:46.107 --> 00:17:48.690 And their measure, they didn't use the task I just showed you. 00:17:48.690 --> 00:17:50.773 This is another thing you can do with little kids. 00:17:50.773 --> 00:17:51.990 You just flash this up. 00:17:51.990 --> 00:17:54.120 And you just ask them, how many dots are there? 00:17:54.120 --> 00:17:55.800 And they have to say four, right? 00:17:55.800 --> 00:17:57.420 And you just measure reaction time. 00:17:57.420 --> 00:17:58.710 It's pretty basic. 00:17:58.710 --> 00:18:00.540 OK? 00:18:00.540 --> 00:18:04.080 And so then what you do is you run this on kindergarteners. 00:18:04.080 --> 00:18:06.810 And you define groups that are slow, medium, 00:18:06.810 --> 00:18:08.820 or fast at this task. 00:18:08.820 --> 00:18:10.120 OK? 00:18:10.120 --> 00:18:12.360 So then you follow them. 00:18:12.360 --> 00:18:14.430 And you look at them later, in this case, 00:18:14.430 --> 00:18:18.880 at age nine and six years. 00:18:18.880 --> 00:18:23.310 And what you see is, even these older kids, who 00:18:23.310 --> 00:18:28.330 are defined by the slow, medium, or fast group in kindergarten, 00:18:28.330 --> 00:18:31.470 this is now their accuracy at arithmetic tasks 00:18:31.470 --> 00:18:33.740 four years later. 00:18:33.740 --> 00:18:35.180 Yeah? 00:18:35.180 --> 00:18:38.570 So it's not just some weird little task 00:18:38.570 --> 00:18:42.140 that psychophysicists made up to measure God knows what. 00:18:42.140 --> 00:18:46.200 It's predictive of your later arithmetic ability. 00:18:46.200 --> 00:18:46.920 OK? 00:18:46.920 --> 00:18:48.910 So it matters. 00:18:48.910 --> 00:18:53.190 So the speed of this dot estimation task at kindergarten 00:18:53.190 --> 00:18:56.730 is not associated with later abilities of other kinds, 00:18:56.730 --> 00:18:59.760 like Raven matrices, which is one of the standard measures 00:18:59.760 --> 00:19:01.290 in an IQ test, right? 00:19:01.290 --> 00:19:05.970 It's a nonverbal and non-number kind of task. 00:19:05.970 --> 00:19:08.640 Or ability to name digits or letters or other things 00:19:08.640 --> 00:19:14.670 that you can test kids on in however old 00:19:14.670 --> 00:19:16.140 they are, nine years. 00:19:16.140 --> 00:19:16.860 OK? 00:19:16.860 --> 00:19:20.730 So it's specifically predictive of later arithmetic ability. 00:19:20.730 --> 00:19:21.900 Everybody with me? 00:19:21.900 --> 00:19:23.490 So it matters. 00:19:23.490 --> 00:19:28.230 All right, and that suggests that there would be ways 00:19:28.230 --> 00:19:30.000 to intervene in dyscalculia. 00:19:30.000 --> 00:19:31.890 Potentially, you could catch the kids early 00:19:31.890 --> 00:19:33.420 who are destined to have a hard time 00:19:33.420 --> 00:19:35.580 and maybe figure out what you could do about it. 00:19:35.580 --> 00:19:37.680 And there are efforts underway to do that. 00:19:37.680 --> 00:19:39.280 OK? 00:19:39.280 --> 00:19:39.780 OK. 00:19:42.830 --> 00:19:45.980 OK, so I'm going to show you. 00:19:45.980 --> 00:19:48.418 We're exploring these various number abilities. 00:19:48.418 --> 00:19:50.210 I'm going to show you something interesting 00:19:50.210 --> 00:19:51.885 about symbolic numbers. 00:19:51.885 --> 00:19:54.260 So far, we've been telling you about nonsymbolic numbers. 00:19:54.260 --> 00:19:55.823 That means just dot arrays. 00:19:55.823 --> 00:19:57.740 Now we're going to deal with symbolic numbers. 00:19:57.740 --> 00:19:59.810 I'm going to flash up a bunch of numbers. 00:19:59.810 --> 00:20:03.230 And you're just going to say bigger if it's bigger than 65 00:20:03.230 --> 00:20:06.470 or smaller if it's smaller than 65. 00:20:06.470 --> 00:20:07.260 Really easy. 00:20:07.260 --> 00:20:09.260 But you're going to shout it out loud and clear. 00:20:09.260 --> 00:20:10.460 Ready? 00:20:10.460 --> 00:20:12.124 Here we go. 00:20:12.124 --> 00:20:13.070 AUDIENCE: Smaller. 00:20:13.070 --> 00:20:13.994 NANCY KANWISHER: Good. 00:20:13.994 --> 00:20:14.702 AUDIENCE: Bigger. 00:20:14.702 --> 00:20:15.684 NANCY KANWISHER: Good. 00:20:15.684 --> 00:20:17.650 AUDIENCE: Smaller. 00:20:17.650 --> 00:20:19.540 Bigger. 00:20:19.540 --> 00:20:21.330 Smaller. 00:20:21.330 --> 00:20:23.320 Bigger. 00:20:23.320 --> 00:20:25.520 Smaller. 00:20:25.520 --> 00:20:27.083 Bigger. 00:20:27.083 --> 00:20:29.500 NANCY KANWISHER: OK, did you guys see what happened there? 00:20:29.500 --> 00:20:30.960 Did you feel what happened? 00:20:30.960 --> 00:20:34.900 When the numbers get closer to 65, you're slower. 00:20:34.900 --> 00:20:37.570 Now you think about it, why the hell is that, right? 00:20:37.570 --> 00:20:39.310 If you run this in Matlab, it's not 00:20:39.310 --> 00:20:43.720 going to take longer to tell you that 63 is smaller than 65 00:20:43.720 --> 00:20:48.100 than it takes to tell you that eight is smaller than 65, 00:20:48.100 --> 00:20:48.610 right? 00:20:48.610 --> 00:20:49.378 I assume. 00:20:49.378 --> 00:20:50.170 I haven't tried it. 00:20:50.170 --> 00:20:52.100 But I doubt it. 00:20:52.100 --> 00:20:53.720 So what does that mean? 00:20:53.720 --> 00:20:55.450 That means that even when you are dealing 00:20:55.450 --> 00:20:57.880 with symbolic numbers, numbers that you 00:20:57.880 --> 00:21:00.550 have this whole elaborate edifice you've been trained 00:21:00.550 --> 00:21:05.470 on how to operate with these guys, especially you guys, 00:21:05.470 --> 00:21:09.130 you are still invoking some kind of notion 00:21:09.130 --> 00:21:10.690 of the continuous quantity. 00:21:10.690 --> 00:21:12.940 You haven't totally left that idea behind 00:21:12.940 --> 00:21:15.770 and moved off into some abstract space. 00:21:15.770 --> 00:21:20.410 You're still, even in doing this very literal, exact 00:21:20.410 --> 00:21:25.050 symbolic number task, you find it easier 00:21:25.050 --> 00:21:27.550 when the numbers are farther apart than when they're closer. 00:21:27.550 --> 00:21:28.663 Yeah, Talia? 00:21:28.663 --> 00:21:31.790 AUDIENCE: Could it be because of the number you chose? 00:21:31.790 --> 00:21:35.530 So if you chose the numbers 60, let's say, 00:21:35.530 --> 00:21:37.840 I feel like we read left to right. 00:21:37.840 --> 00:21:39.880 And they maybe have a good concept 00:21:39.880 --> 00:21:41.660 for the number of digits that we see. 00:21:41.660 --> 00:21:44.590 So when we see a number like 62, we 00:21:44.590 --> 00:21:47.195 have to read both the digits instead of just the one. 00:21:47.195 --> 00:21:48.820 NANCY KANWISHER: Yeah, but all the ones 00:21:48.820 --> 00:21:50.895 I showed were at least two digits. 00:21:50.895 --> 00:21:51.520 AUDIENCE: Yeah. 00:21:51.520 --> 00:21:56.200 But when you read, like when you see a number like 25, 00:21:56.200 --> 00:21:57.580 you see the two. 00:21:57.580 --> 00:21:59.440 And then you automatically like know that. 00:21:59.440 --> 00:22:00.815 NANCY KANWISHER: OK, fair enough. 00:22:00.815 --> 00:22:03.040 OK, that's a good counter explanation. 00:22:03.040 --> 00:22:07.150 But you guys were slow even with 58. 00:22:07.150 --> 00:22:08.590 I think, right? 00:22:08.590 --> 00:22:09.730 We could test that. 00:22:09.730 --> 00:22:12.147 I'm pretty sure all this has been tested pretty carefully. 00:22:12.147 --> 00:22:15.310 I don't know this literature totally thoroughly. 00:22:15.310 --> 00:22:17.320 But I doubt-- it's a good alternative account. 00:22:17.320 --> 00:22:18.612 And there might be some effect. 00:22:18.612 --> 00:22:19.450 But I think it's-- 00:22:19.450 --> 00:22:22.060 oh, in fact, in fact, actually, there is, yeah, 00:22:22.060 --> 00:22:23.680 I have data coming up next. 00:22:23.680 --> 00:22:24.550 But right. 00:22:24.550 --> 00:22:25.270 Blah, blah. 00:22:25.270 --> 00:22:27.250 OK, here's the data. 00:22:27.250 --> 00:22:28.720 OK? 00:22:28.720 --> 00:22:30.640 It's pretty continuous. 00:22:30.640 --> 00:22:34.660 So I think your good, plausible alternative doesn't seem 00:22:34.660 --> 00:22:36.230 to capture very much of it. 00:22:36.230 --> 00:22:36.730 OK? 00:22:39.250 --> 00:22:41.000 So yeah, this is what you guys just did. 00:22:41.000 --> 00:22:43.583 And does everybody get how this kind of reveals that even when 00:22:43.583 --> 00:22:45.370 you think you're doing this kind of more 00:22:45.370 --> 00:22:48.010 symbolic abstract thing, you're still 00:22:48.010 --> 00:22:51.280 tapping into some kind of continuous notion? 00:22:51.280 --> 00:22:52.120 Yeah? 00:22:52.120 --> 00:22:53.320 OK. 00:22:53.320 --> 00:22:55.630 So that says not only does your ability 00:22:55.630 --> 00:22:58.390 to do that in kindergarten predict your ability 00:22:58.390 --> 00:23:01.900 to do arithmetic later, it says, even now as highly trained 00:23:01.900 --> 00:23:04.570 MIT students who do all kinds of much more 00:23:04.570 --> 00:23:06.280 sophisticated math than this, you're 00:23:06.280 --> 00:23:09.190 still invoking that same kind of continuous sense 00:23:09.190 --> 00:23:11.560 of approximate number or something like it. 00:23:11.560 --> 00:23:13.550 OK. 00:23:13.550 --> 00:23:15.147 All right, so where have we gotten? 00:23:15.147 --> 00:23:16.730 We started with this checklist of what 00:23:16.730 --> 00:23:18.200 number sense might mean. 00:23:18.200 --> 00:23:22.010 And I've argued that you adults can represent 00:23:22.010 --> 00:23:25.640 large numerical magnitudes without verbal counting, 00:23:25.640 --> 00:23:27.140 that these numbers are approximate, 00:23:27.140 --> 00:23:28.970 and that your ability to discriminate them 00:23:28.970 --> 00:23:32.220 depends on the ratio, not the difference. 00:23:32.220 --> 00:23:35.540 And I've sort of loosely told you 00:23:35.540 --> 00:23:37.040 that these experiments are generally 00:23:37.040 --> 00:23:40.610 done unconfounded from things like area 00:23:40.610 --> 00:23:43.850 and that they refer to the discrete number. 00:23:43.850 --> 00:23:45.860 OK, what about these other questions here? 00:23:45.860 --> 00:23:49.160 I haven't really shown you how abstract they are 00:23:49.160 --> 00:23:52.610 or whether you can actually use them in arithmetic operations. 00:23:52.610 --> 00:23:54.410 OK, so how would we tell that? 00:23:54.410 --> 00:23:56.690 Well, here's an experiment that we did way back. 00:23:56.690 --> 00:23:58.460 We did the very same task I did on you 00:23:58.460 --> 00:24:01.730 guys before, which has more, except the first thing was 00:24:01.730 --> 00:24:02.750 an array of dots. 00:24:02.750 --> 00:24:06.200 And the second thing was a series of tones. 00:24:06.200 --> 00:24:07.220 OK? 00:24:07.220 --> 00:24:10.090 Series of tones presented faster than you could count. 00:24:10.090 --> 00:24:13.070 Beep, beep, beep, beep, beep, like that, right? 00:24:13.070 --> 00:24:14.090 OK. 00:24:14.090 --> 00:24:15.800 And so you might think that if people 00:24:15.800 --> 00:24:18.260 are doing some literal perceptual thing that this 00:24:18.260 --> 00:24:21.980 would be just like freaking impossible, right? 00:24:21.980 --> 00:24:23.240 But it's not. 00:24:23.240 --> 00:24:25.280 Accuracy is just about the same, maybe 00:24:25.280 --> 00:24:27.230 a hair lower, but almost the same 00:24:27.230 --> 00:24:29.960 with the cross-modal comparison of which has more 00:24:29.960 --> 00:24:33.380 than with the within modality one, visual dots to dots 00:24:33.380 --> 00:24:34.443 or tones to tones. 00:24:34.443 --> 00:24:36.110 This is dots to dots and tones to tones. 00:24:36.110 --> 00:24:37.596 And that's across. 00:24:37.596 --> 00:24:39.215 It's a little bit surprising. 00:24:39.215 --> 00:24:41.340 So that shows you that whatever you're tapping into 00:24:41.340 --> 00:24:43.020 is a pretty abstract representation. 00:24:43.020 --> 00:24:44.370 It's not tied to vision. 00:24:44.370 --> 00:24:45.840 It's not tied to hearing. 00:24:45.840 --> 00:24:47.850 And it also completely eliminates 00:24:47.850 --> 00:24:50.100 worries about density or area or stuff 00:24:50.100 --> 00:24:52.140 like that because that doesn't work here at all. 00:24:52.140 --> 00:24:53.140 OK? 00:24:53.140 --> 00:24:53.640 All right. 00:24:56.220 --> 00:24:58.750 OK, can you do operations on these? 00:24:58.750 --> 00:24:59.250 Sure. 00:24:59.250 --> 00:25:00.000 Why not? 00:25:00.000 --> 00:25:02.820 You can give people a dot array and a dot array 00:25:02.820 --> 00:25:05.400 and then tell them to add and ask whether the sum of those 00:25:05.400 --> 00:25:07.450 is greater or less than that. 00:25:07.450 --> 00:25:10.450 Let's try it. 00:25:10.450 --> 00:25:11.080 OK, here we go. 00:25:11.080 --> 00:25:12.870 Everyone ready? 00:25:12.870 --> 00:25:19.210 Consider, is the sum of this plus this greater or less 00:25:19.210 --> 00:25:21.660 than this. 00:25:21.660 --> 00:25:22.410 AUDIENCE: Greater. 00:25:22.410 --> 00:25:24.000 NANCY KANWISHER: Yeah. 00:25:24.000 --> 00:25:24.960 OK? 00:25:24.960 --> 00:25:27.060 And I really didn't leave you time to count. 00:25:27.060 --> 00:25:28.830 And so whatever you were doing in adding, 00:25:28.830 --> 00:25:30.930 you weren't adding symbolic numbers. 00:25:30.930 --> 00:25:33.240 You were adding these approximate amounts. 00:25:33.240 --> 00:25:33.900 OK? 00:25:33.900 --> 00:25:36.550 Well done. 00:25:36.550 --> 00:25:39.880 And then we could go crazy and do it across modalities. 00:25:39.880 --> 00:25:42.160 I'm going to ask you to add dots to tones 00:25:42.160 --> 00:25:45.130 and ask whether the sum is greater or less than that. 00:25:45.130 --> 00:25:46.270 We won't do it. 00:25:46.270 --> 00:25:50.760 But it turns out, people are just as good at that. 00:25:50.760 --> 00:25:52.560 Amazing, huh? 00:25:52.560 --> 00:25:54.240 So where has this gotten us? 00:25:54.240 --> 00:25:56.820 This told us that whatever this approximate number sense 00:25:56.820 --> 00:26:00.060 that we all have, it's damned abstract. 00:26:00.060 --> 00:26:02.610 You can compare it across sensory modalities pretty much 00:26:02.610 --> 00:26:03.990 as well as within. 00:26:03.990 --> 00:26:05.820 And you can perform operations with it. 00:26:05.820 --> 00:26:06.720 You can do addition. 00:26:06.720 --> 00:26:09.690 And you can also do subtraction just as straightforwardly. 00:26:09.690 --> 00:26:11.330 OK? 00:26:11.330 --> 00:26:14.120 So that's pretty cool. 00:26:14.120 --> 00:26:18.080 But in all of these studies and the demos with you guys, 00:26:18.080 --> 00:26:20.900 these are done on people with years and years of training 00:26:20.900 --> 00:26:22.980 in arithmetic. 00:26:22.980 --> 00:26:25.460 And so we really want to know, are these things-- 00:26:25.460 --> 00:26:27.710 is any aspect of this system innate? 00:26:27.710 --> 00:26:29.450 Is it present in very young infants? 00:26:29.450 --> 00:26:32.420 And to what extent do animals have these abilities? 00:26:32.420 --> 00:26:33.268 OK? 00:26:33.268 --> 00:26:35.060 Well, how would we find out whether they're 00:26:35.060 --> 00:26:37.230 present in infants? 00:26:37.230 --> 00:26:38.480 Well, there's a bunch of ways. 00:26:38.480 --> 00:26:41.300 But looking direction and looking time 00:26:41.300 --> 00:26:44.180 are the key cues you have with newborn infants. 00:26:44.180 --> 00:26:49.160 And so here's a study that was done on four-day-old infants. 00:26:49.160 --> 00:26:52.920 And what they did was they presented-- 00:26:52.920 --> 00:26:56.090 they had a familiarization phase. 00:26:56.090 --> 00:26:58.130 This is done cross modally. 00:26:58.130 --> 00:26:58.730 OK? 00:26:58.730 --> 00:27:02.780 So they present either sets of 12 sounds, 00:27:02.780 --> 00:27:07.490 to, to, to, to, 12, right, of those, or ra, ra, ra, ra, 00:27:07.490 --> 00:27:09.590 present a bunch of those to infants. 00:27:09.590 --> 00:27:13.580 Or they present sets of four taking the same total duration, 00:27:13.580 --> 00:27:15.728 to, to. 00:27:15.728 --> 00:27:17.270 It's just a coincidence that it's to. 00:27:17.270 --> 00:27:19.070 This is, I think, done in French. 00:27:19.070 --> 00:27:23.960 So anyway, the infants won't be confused by the sound to. 00:27:23.960 --> 00:27:30.350 So during that, you then show the infants these arrays. 00:27:30.350 --> 00:27:34.230 And you ask what they look at more. 00:27:34.230 --> 00:27:35.840 OK? 00:27:35.840 --> 00:27:37.052 They're not told the task. 00:27:37.052 --> 00:27:38.510 There's no way to tell them a task. 00:27:38.510 --> 00:27:40.520 It's just something they do. 00:27:40.520 --> 00:27:46.250 And what you find is, in the four versus 12 case 00:27:46.250 --> 00:27:49.100 like here, that's four versus 12, 00:27:49.100 --> 00:27:53.700 the infants look more at the congruent number then 00:27:53.700 --> 00:27:56.110 the incongruent number. 00:27:56.110 --> 00:27:56.610 OK? 00:27:56.610 --> 00:27:59.010 So again, they're comparing across modality. 00:27:59.010 --> 00:28:02.670 They're hearing some number of syllables. 00:28:02.670 --> 00:28:06.060 And they're selectively looking at the corresponding number 00:28:06.060 --> 00:28:08.400 of visual forms. 00:28:08.400 --> 00:28:09.210 No instruction. 00:28:09.210 --> 00:28:09.960 No nothing. 00:28:09.960 --> 00:28:11.070 Four days old. 00:28:11.070 --> 00:28:12.180 Amazing. 00:28:12.180 --> 00:28:13.140 OK? 00:28:13.140 --> 00:28:16.320 So they can do that if the comparison is four versus 12. 00:28:16.320 --> 00:28:19.650 They can do it if it's six versus 18. 00:28:19.650 --> 00:28:21.330 But they kind of can't do it very well. 00:28:21.330 --> 00:28:23.340 I mean, it's significant, but it's not very good 00:28:23.340 --> 00:28:25.480 if it's four versus eight. 00:28:25.480 --> 00:28:25.980 OK? 00:28:25.980 --> 00:28:28.350 So they have some sense of number. 00:28:28.350 --> 00:28:29.920 But it's very approximate. 00:28:29.920 --> 00:28:30.420 Yeah? 00:28:30.420 --> 00:28:31.020 AUDIENCE: Did you say they looked 00:28:31.020 --> 00:28:32.860 at the one that matches the number, 00:28:32.860 --> 00:28:34.568 or they hear the sound that goes with it? 00:28:34.568 --> 00:28:35.610 NANCY KANWISHER: Matches. 00:28:35.610 --> 00:28:37.260 That's congruent means match. 00:28:37.260 --> 00:28:40.450 Looking time on congruent versus incongruent. 00:28:40.450 --> 00:28:42.471 AUDIENCE: Isn't that kind of different from-- 00:28:42.471 --> 00:28:43.015 NANCY KANWISHER: From adaptation. 00:28:43.015 --> 00:28:43.690 AUDIENCE: Yeah. 00:28:43.690 --> 00:28:44.590 NANCY KANWISHER: It is. 00:28:44.590 --> 00:28:46.257 It is totally different from adaptation. 00:28:46.257 --> 00:28:49.870 And herein lies a classic annoyance 00:28:49.870 --> 00:28:51.580 for developmental psychologists. 00:28:51.580 --> 00:28:54.250 Because sometimes kids match. 00:28:54.250 --> 00:28:56.530 And sometimes they show adaptation. 00:28:56.530 --> 00:28:59.920 And you kind of don't know. 00:28:59.920 --> 00:29:03.940 Sometimes you don't know which way it's going to go. 00:29:03.940 --> 00:29:04.575 I don't know. 00:29:04.575 --> 00:29:06.700 Heather, do we have any insights about how you know 00:29:06.700 --> 00:29:07.825 which way it's going to go? 00:29:07.825 --> 00:29:11.740 Or you just try and experiment and you find out and yeah? 00:29:11.740 --> 00:29:12.280 Yeah. 00:29:12.280 --> 00:29:12.780 Yeah. 00:29:12.780 --> 00:29:14.458 It does mean you have to be careful. 00:29:14.458 --> 00:29:16.000 Because if you run a whole experiment 00:29:16.000 --> 00:29:17.560 on a smallish number of infants-- 00:29:17.560 --> 00:29:19.720 and it's usually hard to get enough because people 00:29:19.720 --> 00:29:21.310 have to drive in with their kids. 00:29:21.310 --> 00:29:22.610 And this, how do you find them? 00:29:22.610 --> 00:29:24.850 And there's other developmental labs who have all the kids. 00:29:24.850 --> 00:29:26.808 And it's like you're always running experiments 00:29:26.808 --> 00:29:29.470 with barely enough kids, right? 00:29:29.470 --> 00:29:31.330 And so that means there's a problem here. 00:29:31.330 --> 00:29:34.750 Because if you would take the result in either direction, 00:29:34.750 --> 00:29:36.460 that's a statistical problem. 00:29:36.460 --> 00:29:38.530 You gave yourself two shots at it, right? 00:29:38.530 --> 00:29:42.490 And so you have to statistically discount your finding 00:29:42.490 --> 00:29:44.380 because it could have gone either way. 00:29:44.380 --> 00:29:45.863 That is if your prior hypothesis is 00:29:45.863 --> 00:29:48.280 it has to go in one direction, you're on stronger footing. 00:29:48.280 --> 00:29:51.580 But you just suck it up and run a few more kids. 00:29:51.580 --> 00:29:52.990 Yeah? 00:29:52.990 --> 00:29:56.230 OK, so good. 00:29:56.230 --> 00:30:00.595 So this also shows that ratio dependence, right? 00:30:00.595 --> 00:30:02.470 They're better at it with the big differences 00:30:02.470 --> 00:30:03.890 than the small differences. 00:30:03.890 --> 00:30:04.390 OK? 00:30:07.190 --> 00:30:11.240 OK, so that's infants having this very, very early, at least 00:30:11.240 --> 00:30:13.740 in very crude form. 00:30:13.740 --> 00:30:15.170 What about animals? 00:30:15.170 --> 00:30:18.710 OK, so let's meet Mercury the macaw. 00:30:18.710 --> 00:30:20.105 Here's Mercury the macaw. 00:30:20.105 --> 00:30:20.772 [VIDEO PLAYBACK] 00:30:20.772 --> 00:30:22.550 - To a human, the order of the symbols 00:30:22.550 --> 00:30:24.487 shown on the above screen are obvious. 00:30:24.487 --> 00:30:26.945 We have all learned from a young age which of these symbols 00:30:26.945 --> 00:30:27.170 represented-- 00:30:27.170 --> 00:30:27.290 [END PLAYBACK] 00:30:27.290 --> 00:30:27.980 NANCY KANWISHER: Oh, what a good birdie. 00:30:27.980 --> 00:30:28.647 [VIDEO PLAYBACK] 00:30:28.647 --> 00:30:31.280 - --the lowest number and which the highest. 00:30:31.280 --> 00:30:35.360 However, for Mercury, the blue-headed macaw we see here, 00:30:35.360 --> 00:30:37.430 he has had to learn by trial and error 00:30:37.430 --> 00:30:39.680 the specific order to press these symbols to get 00:30:39.680 --> 00:30:41.240 a piece of food. 00:30:41.240 --> 00:30:44.480 It took him quite a long time. 00:30:44.480 --> 00:30:47.810 Mercury's brother Mars can do a bit better than that. 00:30:47.810 --> 00:30:50.180 He has begun to learn the more general concept. 00:30:50.180 --> 00:30:53.640 That is the symbols will always have an order. 00:30:53.640 --> 00:30:55.640 So when presented with a new list, 00:30:55.640 --> 00:30:58.240 he was able to rapidly decipher the order 00:30:58.240 --> 00:31:00.710 of new symbols, in this case kingfisher, 00:31:00.710 --> 00:31:04.040 warhead, hawk, hummingbird. 00:31:04.040 --> 00:31:05.990 Pressing randomly on the screen would 00:31:05.990 --> 00:31:08.870 have led to him receiving the correct answer less than 1% 00:31:08.870 --> 00:31:09.890 of the time. 00:31:09.890 --> 00:31:12.050 He's clearly doing better than that. 00:31:12.050 --> 00:31:15.080 This is interesting, as it shows the very basic aspects 00:31:15.080 --> 00:31:17.390 of cognition related to numbers are 00:31:17.390 --> 00:31:19.610 present in an animal that is very distantly 00:31:19.610 --> 00:31:21.694 related to humans. 00:31:21.694 --> 00:31:22.580 [END PLAYBACK] 00:31:22.580 --> 00:31:23.780 NANCY KANWISHER: OK, mostly, I just showed 00:31:23.780 --> 00:31:24.738 that because it's cute. 00:31:24.738 --> 00:31:26.780 But it's impressive ordering. 00:31:26.780 --> 00:31:27.290 OK? 00:31:27.290 --> 00:31:29.120 Still, he's kind of slow. 00:31:29.120 --> 00:31:32.180 I think it only goes up to four things. 00:31:32.180 --> 00:31:35.930 OK, so now we're going to meet the chimp Ayumu, 00:31:35.930 --> 00:31:38.600 who lives in Kyoto and who's the son of a very 00:31:38.600 --> 00:31:42.020 famous chimp named Ai, who was like a number wiz. 00:31:42.020 --> 00:31:44.105 But anyway, here's Ayumu. 00:31:44.105 --> 00:31:46.750 [VIDEO PLAYBACK] 00:31:53.347 --> 00:31:53.930 [END PLAYBACK] 00:31:53.930 --> 00:31:54.930 NANCY KANWISHER: I know. 00:31:54.930 --> 00:31:56.420 I can only catch the first three. 00:31:56.420 --> 00:31:58.628 And then it's like I can't even tell if he's correct, 00:31:58.628 --> 00:32:01.460 except from the tone. 00:32:01.460 --> 00:32:02.120 Pretty good. 00:32:02.120 --> 00:32:05.578 [VIDEO PLAYBACK] 00:32:13.790 --> 00:32:14.470 [END PLAYBACK] 00:32:14.470 --> 00:32:15.928 NANCY KANWISHER: Oh, got one wrong. 00:32:20.818 --> 00:32:22.110 Anyway, mostly gets them right. 00:32:22.110 --> 00:32:24.210 Pretty impressive, huh? 00:32:24.210 --> 00:32:25.680 OK, so that's cool. 00:32:25.680 --> 00:32:27.720 And order is clearly relevant. 00:32:27.720 --> 00:32:28.740 It's part of the space. 00:32:28.740 --> 00:32:32.190 But it's not the same as quantity or number, right? 00:32:32.190 --> 00:32:35.700 OK, so now we're going to skip to the honeybee, 00:32:35.700 --> 00:32:37.650 just for kicks because this paper just 00:32:37.650 --> 00:32:38.890 came out a month ago. 00:32:38.890 --> 00:32:41.320 And I think it's awesome. 00:32:41.320 --> 00:32:43.837 Honeybees have 1 million neurons. 00:32:43.837 --> 00:32:45.670 And if you're impressed, don't be impressed. 00:32:45.670 --> 00:32:48.600 Remember like a mouse has 100 million. 00:32:48.600 --> 00:32:51.370 And we have 100 billion. 00:32:51.370 --> 00:32:51.960 OK? 00:32:51.960 --> 00:32:53.460 Six orders of magnitude. 00:32:53.460 --> 00:32:56.670 OK, so 1 million is like not-- no. 00:32:56.670 --> 00:32:57.910 eight orders of magnitude. 00:32:57.910 --> 00:33:01.020 So 1 million is not that many, right? 00:33:01.020 --> 00:33:03.000 OK, and further, these guys branched off 00:33:03.000 --> 00:33:05.800 from us, evolutionarily, a very long time ago, 00:33:05.800 --> 00:33:07.240 600 million years ago. 00:33:07.240 --> 00:33:09.120 So they're tiny little guys, not very 00:33:09.120 --> 00:33:11.220 many neurons, totally different kind of thing. 00:33:11.220 --> 00:33:14.430 Who would think they have any kind of numerical abilities? 00:33:14.430 --> 00:33:16.360 Of course, they wouldn't, right? 00:33:16.360 --> 00:33:20.130 Oh, and yet, they can do arithmetic. 00:33:20.130 --> 00:33:22.750 OK, so here's the design. 00:33:22.750 --> 00:33:24.840 So here's what these guys did, this wonderful lab 00:33:24.840 --> 00:33:25.440 in Australia. 00:33:25.440 --> 00:33:26.220 I love this stuff. 00:33:26.220 --> 00:33:28.680 OK, so they trained these honeybees. 00:33:28.680 --> 00:33:31.710 This was a chamber like this. 00:33:31.710 --> 00:33:33.390 Honeybees fly into the chamber. 00:33:33.390 --> 00:33:36.360 And they see a number in a color right here. 00:33:36.360 --> 00:33:37.090 It's blue. 00:33:37.090 --> 00:33:37.920 And it's two. 00:33:37.920 --> 00:33:38.580 OK? 00:33:38.580 --> 00:33:40.410 And then there's a little entry hole. 00:33:40.410 --> 00:33:42.900 And they can choose to play or not play. 00:33:42.900 --> 00:33:46.050 If they go into the chamber, then they're 00:33:46.050 --> 00:33:47.940 in this interior space, where they 00:33:47.940 --> 00:33:51.480 get to make a choice between that pattern and that pattern. 00:33:51.480 --> 00:33:52.380 OK? 00:33:52.380 --> 00:33:55.360 And there's a little pole underneath each pattern. 00:33:55.360 --> 00:33:57.690 And if they light and they land on the pole, 00:33:57.690 --> 00:33:59.500 they can get some liquid. 00:33:59.500 --> 00:34:00.000 OK? 00:34:00.000 --> 00:34:04.500 So in the blue case, they're rewarded over trials. 00:34:04.500 --> 00:34:06.330 That if it's blue, that means they 00:34:06.330 --> 00:34:08.406 should add one to this number. 00:34:08.406 --> 00:34:10.239 And hence, that would be the correct answer. 00:34:10.239 --> 00:34:12.460 And that's the incorrect answer. 00:34:12.460 --> 00:34:13.180 OK? 00:34:13.180 --> 00:34:14.290 That would be amazing. 00:34:14.290 --> 00:34:15.520 Yeah? 00:34:15.520 --> 00:34:17.020 And if they choose the wrong number, 00:34:17.020 --> 00:34:19.540 they get some nasty quinine. 00:34:19.540 --> 00:34:20.770 OK? 00:34:20.770 --> 00:34:26.020 All right, in contrast, if the shape out front is yellow, 00:34:26.020 --> 00:34:28.090 then they have to subtract. 00:34:28.090 --> 00:34:30.310 So that means they have to keep track of this number 00:34:30.310 --> 00:34:34.005 and go in there and choose that number minus one. 00:34:34.005 --> 00:34:35.949 All right? 00:34:35.949 --> 00:34:40.150 OK, so keep in mind, oh, so they balance the total surface area. 00:34:40.150 --> 00:34:41.650 It doesn't look like in this figure. 00:34:41.650 --> 00:34:43.659 But it says in the method section they did. 00:34:43.659 --> 00:34:45.699 I believe them. 00:34:45.699 --> 00:34:49.480 And further, realize that when the bee is in here, 00:34:49.480 --> 00:34:53.980 he has to be holding that number in memory and adding one to it 00:34:53.980 --> 00:34:56.960 or subtracting one to it to figure out what to choose here. 00:34:56.960 --> 00:34:58.600 So this is pretty sophisticated. 00:34:58.600 --> 00:35:01.780 It's not like they're side by side, right? 00:35:01.780 --> 00:35:05.740 OK, and yet, they're pretty good at it. 00:35:05.740 --> 00:35:07.900 Here's accuracy over training trials. 00:35:07.900 --> 00:35:12.320 By 100 trials, they're over 80% correct. 00:35:12.320 --> 00:35:14.270 Pretty amazing, isn't it? 00:35:14.270 --> 00:35:19.580 OK, so then, in any good animal or infant cognition study, 00:35:19.580 --> 00:35:21.530 you want to show whether it generalizes. 00:35:21.530 --> 00:35:24.740 So then they test the same ability with new numbers. 00:35:24.740 --> 00:35:26.120 I forget what this range was. 00:35:26.120 --> 00:35:28.200 But it went one to four or something like that. 00:35:28.200 --> 00:35:32.210 And then they go to five or six, just to generalize the numbers, 00:35:32.210 --> 00:35:35.330 and different shapes than were used in the training trial. 00:35:35.330 --> 00:35:39.080 And the accuracy is around mid 60s. 00:35:39.080 --> 00:35:40.438 It's not quite as good. 00:35:40.438 --> 00:35:41.480 But it's still very good. 00:35:41.480 --> 00:35:42.897 They're not being reinforced here. 00:35:42.897 --> 00:35:44.720 And they're still doing the task. 00:35:44.720 --> 00:35:46.520 Now, what are the pink and blue bars? 00:35:46.520 --> 00:35:50.420 OK, so you might think, well, is a bee just 00:35:50.420 --> 00:35:52.170 going to the one that has more or less? 00:35:52.170 --> 00:35:54.140 So instead of learning add one, he's 00:35:54.140 --> 00:35:57.200 learned go to the larger number, larger than the one 00:35:57.200 --> 00:35:59.060 that you saw at the entry chamber, 00:35:59.060 --> 00:36:00.590 or go to the smaller number. 00:36:00.590 --> 00:36:02.870 But no, that's not what they're doing. 00:36:02.870 --> 00:36:06.800 Because the pink bars show the performance 00:36:06.800 --> 00:36:09.980 when both of the options are in the same direction, right? 00:36:09.980 --> 00:36:13.160 So the thing is blue. 00:36:13.160 --> 00:36:15.140 So he's doing addition. 00:36:15.140 --> 00:36:16.250 And he sees a two. 00:36:16.250 --> 00:36:19.520 And he goes in, he has a choice between three or four. 00:36:19.520 --> 00:36:21.980 He can only do that if he knows the difference 00:36:21.980 --> 00:36:24.050 between adding one and just taking 00:36:24.050 --> 00:36:26.360 the thing that has more, right? 00:36:26.360 --> 00:36:30.200 And he's well above chance in the pink bars. 00:36:30.200 --> 00:36:32.750 OK, so he's not just saying, choose 00:36:32.750 --> 00:36:34.790 the one that has more or the one that has less. 00:36:34.790 --> 00:36:37.650 He's adding one, pretty accurately, 00:36:37.650 --> 00:36:38.870 I mean sort of accurately. 00:36:38.870 --> 00:36:40.510 Better than chance. 00:36:40.510 --> 00:36:42.540 OK? 00:36:42.540 --> 00:36:45.660 All right, now that's pretty cool. 00:36:45.660 --> 00:36:49.950 But adding one, subtracting one, it's cool. 00:36:49.950 --> 00:36:54.060 But do they really have abstract concepts? 00:36:54.060 --> 00:36:58.470 Do they understand the concept of zero? 00:36:58.470 --> 00:37:00.428 OK, so paper was published last year 00:37:00.428 --> 00:37:02.220 arguing that they have the concept of zero. 00:37:02.220 --> 00:37:03.840 Here's how it goes. 00:37:03.840 --> 00:37:06.390 Same lab trains them, in this case, 00:37:06.390 --> 00:37:08.950 just on greater than or less than. 00:37:08.950 --> 00:37:11.880 So the bees are given a choice like this. 00:37:11.880 --> 00:37:14.580 And one set of bees is trained on greater than 00:37:14.580 --> 00:37:15.940 and one is trained on less than. 00:37:15.940 --> 00:37:19.110 So this set of bees trained on greater than chooses 00:37:19.110 --> 00:37:22.650 this one and then this one and then this one and so on. 00:37:22.650 --> 00:37:24.060 OK? 00:37:24.060 --> 00:37:26.790 Another set of bees is trained to do the opposite. 00:37:26.790 --> 00:37:27.750 All right? 00:37:27.750 --> 00:37:29.910 OK, so that's the training phase. 00:37:29.910 --> 00:37:33.820 Then we want to test in a generalized situation. 00:37:33.820 --> 00:37:36.930 So now they're tested with different shapes 00:37:36.930 --> 00:37:41.547 and different numbers, so threes and fours were. 00:37:41.547 --> 00:37:42.630 Maybe threes weren't used. 00:37:42.630 --> 00:37:42.930 I forget. 00:37:42.930 --> 00:37:45.222 There's some numbers in here that were not used before. 00:37:45.222 --> 00:37:48.360 OK, so you test them with new shapes. 00:37:48.360 --> 00:37:52.200 And here is accuracy for less than or greater than. 00:37:52.200 --> 00:37:53.400 Chance is 50%. 00:37:53.400 --> 00:37:55.410 And they're 75%. 00:37:55.410 --> 00:37:56.790 Not bad. 00:37:56.790 --> 00:37:57.630 OK? 00:37:57.630 --> 00:38:00.120 So they get more than or less than. 00:38:00.120 --> 00:38:03.260 OK, now we want to test the generalization. 00:38:03.260 --> 00:38:04.220 OK, oh, yes, sorry. 00:38:04.220 --> 00:38:06.270 This is where they changed the range of numbers. 00:38:06.270 --> 00:38:11.013 So the bees had not dealt with sixes before. 00:38:11.013 --> 00:38:13.055 So now they still have to do greater than or less 00:38:13.055 --> 00:38:16.880 than with a new numerical range. 00:38:16.880 --> 00:38:19.430 And they're still well above chance. 00:38:19.430 --> 00:38:21.530 OK? 00:38:21.530 --> 00:38:25.330 So then finally, they test zero. 00:38:25.330 --> 00:38:26.350 OK? 00:38:26.350 --> 00:38:30.070 So the bees that have to do less than 00:38:30.070 --> 00:38:33.610 have to say which of those is correct, all right? 00:38:33.610 --> 00:38:35.800 And you can see-- 00:38:35.800 --> 00:38:38.980 where did it go? 00:38:38.980 --> 00:38:39.910 Where's the zero one? 00:38:39.910 --> 00:38:40.667 Right here. 00:38:40.667 --> 00:38:42.250 And they're well above chance for both 00:38:42.250 --> 00:38:44.440 less than and greater than. 00:38:44.440 --> 00:38:45.250 OK? 00:38:45.250 --> 00:38:46.960 So we could quibble about whether that's 00:38:46.960 --> 00:38:48.400 a concept of zero. 00:38:48.400 --> 00:38:50.860 But the cool thing is these bees had not been tested 00:38:50.860 --> 00:38:52.840 with a blank card before. 00:38:52.840 --> 00:38:58.510 And they spontaneously get the idea that that is less than one 00:38:58.510 --> 00:39:00.400 or two or three or anything else. 00:39:00.400 --> 00:39:01.970 Yeah? 00:39:01.970 --> 00:39:05.060 So arguably, they have a concept of zero 00:39:05.060 --> 00:39:09.390 with no training and only 100 million neurons. 00:39:09.390 --> 00:39:12.810 OK, so all of that is in trained animals. 00:39:12.810 --> 00:39:16.500 And we can see some of these kinds of abilities 00:39:16.500 --> 00:39:18.210 even with untrained animals. 00:39:18.210 --> 00:39:21.300 And I will tell you just one more animal experiment 00:39:21.300 --> 00:39:23.077 because it's my all-time favorite ever 00:39:23.077 --> 00:39:24.660 and the simplest one in the whole set. 00:39:24.660 --> 00:39:27.140 This was done a long time ago by Church and Meck. 00:39:27.140 --> 00:39:28.140 So here's what they did. 00:39:28.140 --> 00:39:29.310 This is done in rats. 00:39:29.310 --> 00:39:30.900 They have a training phase, where 00:39:30.900 --> 00:39:33.510 they train the rats to press the two 00:39:33.510 --> 00:39:37.830 lever if they see two light flashes or hear two sounds. 00:39:37.830 --> 00:39:40.110 And they press another lever, the four lever 00:39:40.110 --> 00:39:42.910 if they see four lights or hear four sounds. 00:39:42.910 --> 00:39:43.410 OK? 00:39:43.410 --> 00:39:44.957 That's kind of basic animal training. 00:39:44.957 --> 00:39:45.540 It's a rodent. 00:39:45.540 --> 00:39:46.415 They're good at this. 00:39:46.415 --> 00:39:47.470 No big deal. 00:39:47.470 --> 00:39:49.680 But then after the animals have learned this, 00:39:49.680 --> 00:39:53.190 they spontaneously throw, in the testing phase, 00:39:53.190 --> 00:39:56.720 a trial with two lights and two sounds. 00:39:56.720 --> 00:39:59.540 And the rats press the four lever, first time. 00:39:59.540 --> 00:40:00.290 No training. 00:40:00.290 --> 00:40:01.160 No nothing. 00:40:01.160 --> 00:40:02.600 Spontaneous addition. 00:40:02.600 --> 00:40:07.430 Spontaneous abstraction across tones and lights. 00:40:07.430 --> 00:40:10.200 Pretty awesome, huh? 00:40:10.200 --> 00:40:13.110 So it's not just that you can reveal these abilities 00:40:13.110 --> 00:40:15.580 with elaborate training. 00:40:15.580 --> 00:40:19.030 OK, so we have all of these different kinds of evidence 00:40:19.030 --> 00:40:20.860 of an abstract number sense. 00:40:20.860 --> 00:40:23.380 And they're present in newborn infants. 00:40:23.380 --> 00:40:25.150 And they're present in animals. 00:40:25.150 --> 00:40:28.210 And they just seem to be part of our basic cognitive machinery, 00:40:28.210 --> 00:40:31.040 machinery that we share with animals. 00:40:31.040 --> 00:40:34.600 So how are they implemented in the brain? 00:40:34.600 --> 00:40:38.440 OK, so a little neuroanatomy reminder of some basics. 00:40:38.440 --> 00:40:40.060 This is a weird angle of a brain. 00:40:40.060 --> 00:40:42.490 It's kind of like this, kind of back of the head, 00:40:42.490 --> 00:40:44.048 front of the head, temporal lobe, 00:40:44.048 --> 00:40:45.340 frontal lobe around the corner. 00:40:45.340 --> 00:40:46.900 Everybody oriented? 00:40:46.900 --> 00:40:49.690 There is one of the longest sulci in the brain that 00:40:49.690 --> 00:40:50.748 starts about here. 00:40:50.748 --> 00:40:51.790 On me, it goes like this. 00:40:51.790 --> 00:40:53.290 And it curves around like that. 00:40:53.290 --> 00:40:53.915 It's back here. 00:40:53.915 --> 00:40:54.700 It goes up. 00:40:54.700 --> 00:40:55.930 And it curves over. 00:40:55.930 --> 00:40:56.650 OK? 00:40:56.650 --> 00:40:58.660 It's called the intraparietal sulcus. 00:40:58.660 --> 00:41:00.310 And I mention that just because it's 00:41:00.310 --> 00:41:02.050 in a lot of the number literature. 00:41:02.050 --> 00:41:05.650 You saw it in the paper you guys read for last night. 00:41:05.650 --> 00:41:09.130 And above it is the superior parietal lobule. 00:41:09.130 --> 00:41:11.320 And below it is the inferior parietal lobule. 00:41:11.320 --> 00:41:14.110 And none of that matters other than that a lot of the action 00:41:14.110 --> 00:41:16.480 is in the parietal lobe, particularly up 00:41:16.480 --> 00:41:18.490 here around the intraparietal sulcus. 00:41:18.490 --> 00:41:19.780 OK? 00:41:19.780 --> 00:41:22.780 All right, so studies that have looked at this 00:41:22.780 --> 00:41:27.460 includes some classical studies of patients with brain damage 00:41:27.460 --> 00:41:29.950 and something called acalculia. 00:41:29.950 --> 00:41:32.650 That means loss of ability to calculate. 00:41:32.650 --> 00:41:33.430 OK? 00:41:33.430 --> 00:41:38.140 And so there's two basic kinds of acalculia 00:41:38.140 --> 00:41:40.330 that are really interestingly different. 00:41:40.330 --> 00:41:43.690 So there's one acalculic patient who 00:41:43.690 --> 00:41:46.210 has left parietal lobe damage, that same region I just 00:41:46.210 --> 00:41:47.990 talked about. 00:41:47.990 --> 00:41:49.970 And this person is bad at approximation. 00:41:49.970 --> 00:41:53.980 So the kinds of dot array tasks that I gave you guys, 00:41:53.980 --> 00:41:55.900 this guy, after brain damage right here, 00:41:55.900 --> 00:41:58.150 is really bad at that kind of stuff. 00:41:58.150 --> 00:42:01.960 And interestingly, he's more impaired on subtraction 00:42:01.960 --> 00:42:03.830 than multiplication. 00:42:03.830 --> 00:42:08.620 So for example, hes worse at, what is seven minus five 00:42:08.620 --> 00:42:11.745 than what is seven times five? 00:42:11.745 --> 00:42:13.120 So think about that for a moment. 00:42:13.120 --> 00:42:15.100 And think about what that might mean, 00:42:15.100 --> 00:42:19.160 especially in light of another acalculic patient who has 00:42:19.160 --> 00:42:20.410 a very different presentation. 00:42:20.410 --> 00:42:22.720 He's got left temporal damage. 00:42:22.720 --> 00:42:24.400 His approximation is fine. 00:42:24.400 --> 00:42:27.310 So all those kind of dot array kind of tasks and tone tasks 00:42:27.310 --> 00:42:30.250 that I told you about, he's good at. 00:42:30.250 --> 00:42:32.110 This guy shows the opposite. 00:42:32.110 --> 00:42:33.850 He's more impaired at multiplication 00:42:33.850 --> 00:42:36.910 than subtraction. 00:42:36.910 --> 00:42:39.720 So do you guys have any-- oh, so first of all, 00:42:39.720 --> 00:42:44.130 you put these two patients together, and what do you have? 00:42:44.130 --> 00:42:45.980 AUDIENCE: Double dissociation. 00:42:45.980 --> 00:42:46.240 NANCY KANWISHER: Yeah? 00:42:46.240 --> 00:42:46.480 What? 00:42:46.480 --> 00:42:47.140 AUDIENCE: Double dissociation. 00:42:47.140 --> 00:42:48.190 NANCY KANWISHER: Double dissociation. 00:42:48.190 --> 00:42:48.690 Right. 00:42:48.690 --> 00:42:52.060 Two patients with opposite patterns of deficit, right? 00:42:52.060 --> 00:42:54.523 If we just had one, then we could maybe tell a story. 00:42:54.523 --> 00:42:55.690 But it wouldn't really know. 00:42:55.690 --> 00:42:58.013 But we have two, and they have opposite patterns. 00:42:58.013 --> 00:43:00.430 And now that really kind of constrains the interpretation. 00:43:00.430 --> 00:43:00.930 David. 00:43:00.930 --> 00:43:03.700 AUDIENCE: Can the first person add fine? 00:43:03.700 --> 00:43:05.440 NANCY KANWISHER: Good question. 00:43:05.440 --> 00:43:06.908 He's not very good at adding. 00:43:06.908 --> 00:43:07.450 AUDIENCE: Oh. 00:43:10.207 --> 00:43:11.290 NANCY KANWISHER: Thoughts? 00:43:11.290 --> 00:43:13.050 What do you think it means? 00:43:13.050 --> 00:43:18.430 AUDIENCE: It might mean that the addition and subtraction 00:43:18.430 --> 00:43:22.345 [INAUDIBLE] use the same like-- 00:43:22.345 --> 00:43:23.470 NANCY KANWISHER: Used what? 00:43:23.470 --> 00:43:25.283 AUDIENCE: Like they use the same area. 00:43:25.283 --> 00:43:26.200 NANCY KANWISHER: Yeah. 00:43:26.200 --> 00:43:29.140 So one hypothesis is that addition and subtraction 00:43:29.140 --> 00:43:31.945 are just a different beast than multiplication. 00:43:31.945 --> 00:43:33.820 Different parts of the brain do those things. 00:43:33.820 --> 00:43:35.260 Totally possible. 00:43:35.260 --> 00:43:37.750 But there's a kind of more intuitive interpretation. 00:43:37.750 --> 00:43:40.450 AUDIENCE: Well, I think people tend to memorize times tables. 00:43:40.450 --> 00:43:41.680 NANCY KANWISHER: Bingo. 00:43:41.680 --> 00:43:42.200 Bingo. 00:43:42.200 --> 00:43:44.200 Often, like the right answer is something that's 00:43:44.200 --> 00:43:45.010 like right in front of you. 00:43:45.010 --> 00:43:46.885 Just think about, what is it like to do that? 00:43:46.885 --> 00:43:48.400 How do you do seven times five? 00:43:48.400 --> 00:43:50.560 You don't think about the meanings of the numbers. 00:43:50.560 --> 00:43:52.750 You just blurt out 35. 00:43:52.750 --> 00:43:54.020 Right? 00:43:54.020 --> 00:43:54.520 Right? 00:43:54.520 --> 00:43:57.640 It's not a very rich number task. 00:43:57.640 --> 00:43:58.810 I mean, it's a number task. 00:43:58.810 --> 00:44:02.680 But it's a concrete, rote, verbally memorized thing. 00:44:02.680 --> 00:44:03.880 Right? 00:44:03.880 --> 00:44:11.200 And so the idea is that those verbalized concrete number 00:44:11.200 --> 00:44:13.960 facts are in one domain. 00:44:13.960 --> 00:44:16.060 One set of brain damage would impair those. 00:44:16.060 --> 00:44:17.800 And it's a different thing to impair 00:44:17.800 --> 00:44:20.770 the actual representation of numerosity. 00:44:20.770 --> 00:44:25.090 And the idea is that this person is 00:44:25.090 --> 00:44:28.460 the one with the real damage to the approximate number system. 00:44:28.460 --> 00:44:28.960 Right? 00:44:28.960 --> 00:44:29.740 Yeah? 00:44:29.740 --> 00:44:32.920 AUDIENCE: Does that mean that patient can be it 00:44:32.920 --> 00:44:36.970 is a problem doing the seven times five normally. 00:44:36.970 --> 00:44:40.730 But when they ask for summing seven for five times, 00:44:40.730 --> 00:44:42.113 they're not very good. 00:44:42.113 --> 00:44:43.030 NANCY KANWISHER: Yeah. 00:44:43.030 --> 00:44:44.822 Well, I think the approximate number system 00:44:44.822 --> 00:44:49.750 might have a tough time dealing with summing seven five times. 00:44:49.750 --> 00:44:51.610 So yeah, it has limits, right? 00:44:51.610 --> 00:44:54.550 It can deal with it can add two approximate things. 00:44:54.550 --> 00:44:56.183 But you might really lose your mind 00:44:56.183 --> 00:44:57.850 if you tried to do a whole string of it. 00:44:57.850 --> 00:44:58.370 Yeah? 00:44:58.370 --> 00:44:59.762 Yeah? 00:44:59.762 --> 00:45:01.720 AUDIENCE: If he was working on the same digits, 00:45:01.720 --> 00:45:05.260 like maybe seven plus seven or seven minus seven, 00:45:05.260 --> 00:45:08.860 expect him to maybe do that fairly easily 00:45:08.860 --> 00:45:12.010 if that's the case, right? 00:45:12.010 --> 00:45:13.390 NANCY KANWISHER: Say more. 00:45:13.390 --> 00:45:16.240 AUDIENCE: If it's a case that his approximate-- 00:45:16.240 --> 00:45:17.510 NANCY KANWISHER: Yeah, yeah. 00:45:17.510 --> 00:45:18.010 Yeah. 00:45:18.010 --> 00:45:19.600 AUDIENCE: He should be able to do seven minus seven 00:45:19.600 --> 00:45:20.100 fairly easy. 00:45:20.100 --> 00:45:23.268 Because you know that when you subtract the same things, 00:45:23.268 --> 00:45:24.310 you're going to get zero. 00:45:24.310 --> 00:45:24.760 NANCY KANWISHER: Yes. 00:45:24.760 --> 00:45:26.170 But it's an interesting question, actually, 00:45:26.170 --> 00:45:28.000 whether that would be part of that system 00:45:28.000 --> 00:45:31.700 or whether that's kind of more abstract formal thing you 00:45:31.700 --> 00:45:32.200 learn. 00:45:32.200 --> 00:45:34.607 So I think it depends how you do it, right? 00:45:34.607 --> 00:45:35.440 So one of the ways-- 00:45:35.440 --> 00:45:36.482 I didn't talk about this. 00:45:36.482 --> 00:45:39.940 But those same experiments adding, say, 00:45:39.940 --> 00:45:43.660 adding dots to dots, those were also done with little kids. 00:45:43.660 --> 00:45:45.622 And there, what you do is you show-- 00:45:45.622 --> 00:45:47.080 I don't really remember what it is. 00:45:47.080 --> 00:45:48.880 But you show some array of things, 00:45:48.880 --> 00:45:50.320 and you hide it behind a screen. 00:45:50.320 --> 00:45:52.270 And then you show another array and hide it behind the screen. 00:45:52.270 --> 00:45:53.562 And then you reveal the screen. 00:45:53.562 --> 00:45:55.750 Like how many things are there? 00:45:55.750 --> 00:45:59.800 That kind of stuff works spontaneously. 00:45:59.800 --> 00:46:02.175 So it might tap into that system. 00:46:02.175 --> 00:46:03.800 I think that's an interesting question. 00:46:03.800 --> 00:46:06.250 I'm not totally sure how it would go. 00:46:06.250 --> 00:46:06.750 Yeah? 00:46:06.750 --> 00:46:08.125 AUDIENCE: So the second person is 00:46:08.125 --> 00:46:09.450 bad at recall across the board? 00:46:09.450 --> 00:46:10.710 Or is it just with numbers? 00:46:10.710 --> 00:46:11.790 NANCY KANWISHER: Just with numbers. 00:46:11.790 --> 00:46:12.180 Yeah. 00:46:12.180 --> 00:46:13.590 I mean, there's always a little bit messy. 00:46:13.590 --> 00:46:15.990 The patient literature is always like some other random stuff. 00:46:15.990 --> 00:46:17.470 And how do you account for that? 00:46:17.470 --> 00:46:19.110 And there's lesions in other places. 00:46:19.110 --> 00:46:21.450 But to a first approximation, these 00:46:21.450 --> 00:46:24.585 are reasonably number-specific deficits. 00:46:24.585 --> 00:46:26.190 All right? 00:46:26.190 --> 00:46:29.098 OK, so that's a bit of a hint from 00:46:29.098 --> 00:46:30.390 the neuropsychology literature. 00:46:30.390 --> 00:46:33.600 But there's mainly these two patients 00:46:33.600 --> 00:46:36.420 and some other like messier ones. 00:46:36.420 --> 00:46:40.710 And so one wants to use neuroimaging 00:46:40.710 --> 00:46:42.140 to get a better picture of it. 00:46:42.140 --> 00:46:44.140 Of course, that's been going on for a long time. 00:46:44.140 --> 00:46:47.520 And so here's one of the early papers from Stan Dehaene's lab. 00:46:47.520 --> 00:46:50.070 This is a top view of the brain. 00:46:50.070 --> 00:46:51.480 So this is this parietal zone. 00:46:51.480 --> 00:46:54.090 And this is what is often referred 00:46:54.090 --> 00:46:58.200 to as the horizontal segment of the intraparietal sulcus. 00:46:58.200 --> 00:46:59.402 hIPS to its friends. 00:46:59.402 --> 00:47:01.860 And it's that sulcus I talked about that goes up like this. 00:47:01.860 --> 00:47:02.970 It kind of curves over. 00:47:02.970 --> 00:47:04.650 And it's like this bit right there. 00:47:04.650 --> 00:47:05.490 OK? 00:47:05.490 --> 00:47:07.035 That little orange strip. 00:47:07.035 --> 00:47:09.660 And so what he's saying in this review article from a long time 00:47:09.660 --> 00:47:12.090 ago is that that region is activated only 00:47:12.090 --> 00:47:13.470 when you do calculation. 00:47:13.470 --> 00:47:16.170 He means basic arithmetic in this case. 00:47:16.170 --> 00:47:20.060 Not when you do all these other things. 00:47:20.060 --> 00:47:23.990 But when this paper came out, I'm like, yeah, right. 00:47:23.990 --> 00:47:25.850 I don't think so. 00:47:25.850 --> 00:47:28.910 I can't tell you how many experiments I've run and seen 00:47:28.910 --> 00:47:32.510 big ass activations right there on tasks that have nothing 00:47:32.510 --> 00:47:33.470 to do with numbers. 00:47:33.470 --> 00:47:34.280 So looks good. 00:47:34.280 --> 00:47:35.300 Sounded good. 00:47:35.300 --> 00:47:37.070 He got away with it for a while. 00:47:37.070 --> 00:47:38.110 And it's not true. 00:47:38.110 --> 00:47:38.873 Yeah? 00:47:38.873 --> 00:47:40.790 AUDIENCE: So is the reason sort of high enough 00:47:40.790 --> 00:47:41.623 that you can zap it? 00:47:41.623 --> 00:47:42.770 NANCY KANWISHER: Terrible. 00:47:42.770 --> 00:47:44.240 Being filmed too. 00:47:44.240 --> 00:47:45.680 He's a really smart, nice guy. 00:47:45.680 --> 00:47:47.972 I just like when people are a little bit fast and loose 00:47:47.972 --> 00:47:50.960 and make a big claim, which you can tell at the time 00:47:50.960 --> 00:47:51.785 isn't quite right. 00:47:51.785 --> 00:47:52.910 It's a little bit annoying. 00:47:52.910 --> 00:47:53.210 Anyway. 00:47:53.210 --> 00:47:53.710 Sorry. 00:47:53.710 --> 00:47:54.230 Go ahead. 00:47:54.230 --> 00:47:54.500 AUDIENCE: Yeah. 00:47:54.500 --> 00:47:56.577 Is the region high enough that you can zap it? 00:47:56.577 --> 00:47:57.410 NANCY KANWISHER: Ah. 00:47:57.410 --> 00:47:58.243 We're getting there. 00:47:58.243 --> 00:48:00.410 Yes, indeed, you can. 00:48:00.410 --> 00:48:02.330 But let's do a little more basic stuff first. 00:48:02.330 --> 00:48:05.630 OK, so the claim is that this hIPS thing 00:48:05.630 --> 00:48:07.850 is the locus of the approximate number system. 00:48:07.850 --> 00:48:09.120 That was the early claim. 00:48:09.120 --> 00:48:09.620 OK. 00:48:12.380 --> 00:48:15.800 And for further, the claim implicit in this article 00:48:15.800 --> 00:48:17.810 in this figure is that it's involved 00:48:17.810 --> 00:48:20.150 in numerical representations only, 00:48:20.150 --> 00:48:24.290 not any of these other things, grasping tasks, manual tasks, 00:48:24.290 --> 00:48:27.230 eye-movement tasks, et cetera, et cetera, et cetera. 00:48:27.230 --> 00:48:31.070 OK, really? 00:48:31.070 --> 00:48:33.680 And as I mentioned, like me and lots of other people 00:48:33.680 --> 00:48:36.780 had seen it looks like the same regions activated 00:48:36.780 --> 00:48:38.930 in all kinds of other situations, especially 00:48:38.930 --> 00:48:42.620 those involving reasoning about spatial location. 00:48:42.620 --> 00:48:45.823 You guys got short shrift six weeks ago. 00:48:45.823 --> 00:48:47.990 I meant to talk about the parietal lobe and its role 00:48:47.990 --> 00:48:48.885 in high-level vision. 00:48:48.885 --> 00:48:50.510 And it just somehow went by the boards. 00:48:50.510 --> 00:48:54.110 But all this stuff is involved in aspects 00:48:54.110 --> 00:48:58.040 of vision, particularly spatial vision, knowing what is where. 00:48:58.040 --> 00:48:59.690 OK? 00:48:59.690 --> 00:49:02.190 And so there's an alternate view, 00:49:02.190 --> 00:49:06.140 which is that there's no specific brain region that's 00:49:06.140 --> 00:49:10.760 specifically all only involved in discrete number per se. 00:49:10.760 --> 00:49:15.410 Instead, there's a common region for processing magnitude 00:49:15.410 --> 00:49:18.560 of almost any dimension, whether discrete or continuous, 00:49:18.560 --> 00:49:21.470 right, that approximate number system or your exact number 00:49:21.470 --> 00:49:25.570 system, and that it builds on previous representations 00:49:25.570 --> 00:49:26.920 of space. 00:49:26.920 --> 00:49:28.260 OK? 00:49:28.260 --> 00:49:31.380 For example, the number line, right? 00:49:31.380 --> 00:49:34.950 So you guys read this article for last night. 00:49:34.950 --> 00:49:36.990 And just to review what the key point was, 00:49:36.990 --> 00:49:40.740 this is, again, the kind of aerial view 00:49:40.740 --> 00:49:42.120 with the parietal lobe here. 00:49:42.120 --> 00:49:47.820 And that's the hIPS region, yeah, 00:49:47.820 --> 00:49:49.410 that was in the previous slide. 00:49:49.410 --> 00:49:51.870 And you can see it's this horizontal part of that sulcus 00:49:51.870 --> 00:49:53.430 way up in the parietal lobe. 00:49:53.430 --> 00:49:56.100 And the yellow and green means that there's 00:49:56.100 --> 00:49:59.370 overlapping activation for both symbolic calculation, that's 00:49:59.370 --> 00:50:02.430 like with symbols, and for nonsymbolic calculation. 00:50:02.430 --> 00:50:04.870 That's like dot arrays stuff like that, right? 00:50:04.870 --> 00:50:08.370 And so it's activated for both of those. 00:50:08.370 --> 00:50:13.950 And the point of this paper is, first of all, 00:50:13.950 --> 00:50:18.900 that there's also overlap with the eye-movement system, right? 00:50:18.900 --> 00:50:21.000 And so here, they're really asking, 00:50:21.000 --> 00:50:23.340 is this spatial representation kind 00:50:23.340 --> 00:50:26.310 of co-opted in your representation of number 00:50:26.310 --> 00:50:28.350 using a kind of spatial number line, right? 00:50:28.350 --> 00:50:29.700 It makes perfect sense. 00:50:29.700 --> 00:50:31.650 Animals need a representation of space. 00:50:31.650 --> 00:50:33.780 It's like extremely basic, right? 00:50:33.780 --> 00:50:37.080 And once you have that, you can co-opt it and represent numbers 00:50:37.080 --> 00:50:39.660 in that same spatial code. 00:50:39.660 --> 00:50:43.330 And as you guys all read, the cool result from that paper, 00:50:43.330 --> 00:50:47.250 which is also from Stan Dehaene's lab, 00:50:47.250 --> 00:50:51.570 is that when you take that region right in there, 00:50:51.570 --> 00:50:54.390 you take those voxels in there, and you train them 00:50:54.390 --> 00:50:57.840 on making leftward versus rightward saccades. 00:50:57.840 --> 00:50:59.700 So now you have a classifier that 00:50:59.700 --> 00:51:01.560 looks at the pattern of activation there, 00:51:01.560 --> 00:51:02.940 can distinguish a leftward versus 00:51:02.940 --> 00:51:04.050 a rightward versus saccade. 00:51:04.050 --> 00:51:05.050 I'm just reviewing this. 00:51:05.050 --> 00:51:07.110 Hopefully it was clear enough. 00:51:07.110 --> 00:51:10.200 That same classifier can then distinguish 00:51:10.200 --> 00:51:12.510 subtraction versus addition. 00:51:12.510 --> 00:51:14.430 Did you guys all get that from the paper? 00:51:14.430 --> 00:51:14.930 Yeah? 00:51:14.930 --> 00:51:17.170 It's pretty cool, isn't it? 00:51:17.170 --> 00:51:18.990 Anyway, so that's kind of nice evidence 00:51:18.990 --> 00:51:20.790 that the same spatial system that's 00:51:20.790 --> 00:51:23.070 used in spatial attention and eye movements 00:51:23.070 --> 00:51:28.850 has been co-opted to represent numbers as well. 00:51:28.850 --> 00:51:31.700 OK. 00:51:31.700 --> 00:51:35.300 All right, so I just wanted to incorporate that. 00:51:35.300 --> 00:51:37.520 In case anybody missed what the paper was about, 00:51:37.520 --> 00:51:39.800 those were the key points. 00:51:39.800 --> 00:51:42.710 Other early studies have asked more directly 00:51:42.710 --> 00:51:45.890 this question of whether different kinds of magnitude 00:51:45.890 --> 00:51:48.170 are all represented together in the brain. 00:51:48.170 --> 00:51:51.350 And this study is quite clever. 00:51:51.350 --> 00:51:55.280 They used a variant of the fact that I showed you guys before. 00:51:55.280 --> 00:51:57.230 Remember when saying whether the number is 00:51:57.230 --> 00:51:58.940 greater or less than 65, it's harder 00:51:58.940 --> 00:52:01.970 when it's closer to 65 than when it's farther from 65. 00:52:01.970 --> 00:52:04.310 OK, even though I was showing you symbols, 00:52:04.310 --> 00:52:05.780 that was key thing, right? 00:52:05.780 --> 00:52:08.570 So that's called the distance effect, right? 00:52:08.570 --> 00:52:10.160 And that's true for all comparisons. 00:52:10.160 --> 00:52:12.980 And so this study exploits that distance effect. 00:52:12.980 --> 00:52:15.560 And they use stimuli like this. 00:52:15.560 --> 00:52:19.020 And they ask, which one is larger? 00:52:19.020 --> 00:52:21.440 And it could be larger in absolute size. 00:52:21.440 --> 00:52:23.360 Like the two is larger here. 00:52:23.360 --> 00:52:25.370 Or it can be larger in number meaning 00:52:25.370 --> 00:52:26.990 like the seven is larger. 00:52:26.990 --> 00:52:29.030 OK, so in different blocks, you're saying, 00:52:29.030 --> 00:52:30.590 which one is physically larger? 00:52:30.590 --> 00:52:32.990 Which one is numerically larger? 00:52:32.990 --> 00:52:34.490 Which one is brighter? 00:52:34.490 --> 00:52:36.470 That would be this one here. 00:52:36.470 --> 00:52:41.120 And then they just have a control with letters. 00:52:41.120 --> 00:52:42.170 OK? 00:52:42.170 --> 00:52:43.580 And so then-- sorry. 00:52:43.580 --> 00:52:45.110 The design is slightly complicated. 00:52:45.110 --> 00:52:48.080 So there's these three main tasks and a control task. 00:52:48.080 --> 00:52:50.900 But then within each, they have the difficult version 00:52:50.900 --> 00:52:52.300 and the easy version. 00:52:52.300 --> 00:52:54.800 And the difficult version is when the comparisons are close, 00:52:54.800 --> 00:52:57.140 two similar brightnesses, two similar numbers, 00:52:57.140 --> 00:53:00.030 two similar sizes versus two larger ones. 00:53:00.030 --> 00:53:00.530 OK? 00:53:00.530 --> 00:53:02.990 So that's what all this garbage shows. 00:53:02.990 --> 00:53:05.047 OK, so then you do that subtraction. 00:53:05.047 --> 00:53:07.130 You look, and you say, OK, what parts of the brain 00:53:07.130 --> 00:53:11.000 are more active when you do the difficult versus easy number 00:53:11.000 --> 00:53:12.540 comparison? 00:53:12.540 --> 00:53:14.730 Like saying, which is larger? 00:53:14.730 --> 00:53:17.640 It's not that difficult. But two versus three 00:53:17.640 --> 00:53:19.860 versus two versus seven. 00:53:19.860 --> 00:53:21.780 OK? 00:53:21.780 --> 00:53:26.160 And so what they find is that similar regions of the brain 00:53:26.160 --> 00:53:29.680 are active for all three of those kinds of comparisons. 00:53:29.680 --> 00:53:30.180 OK? 00:53:30.180 --> 00:53:34.530 So it's not like you get just one for symbolic number 00:53:34.530 --> 00:53:38.370 or for the two magnitude tasks. 00:53:38.370 --> 00:53:40.140 All of those different kinds of magnitude 00:53:40.140 --> 00:53:43.440 activate the same regions. 00:53:43.440 --> 00:53:51.750 And so the conclusion is that number and size and brightness 00:53:51.750 --> 00:53:55.890 engage a common parietal spatial code, OK, an overlapping 00:53:55.890 --> 00:53:57.390 region for all of these. 00:53:57.390 --> 00:53:59.860 Does that make sense? 00:53:59.860 --> 00:54:01.830 OK. 00:54:01.830 --> 00:54:05.670 And so that shows, in this case, that it's not just 00:54:05.670 --> 00:54:10.030 symbolic number but also magnitude. 00:54:10.030 --> 00:54:11.910 Which one is bigger, right? 00:54:11.910 --> 00:54:14.910 It's kind of continuous magnitude idea. 00:54:14.910 --> 00:54:17.130 OK? 00:54:17.130 --> 00:54:19.480 OK. 00:54:19.480 --> 00:54:19.980 Right. 00:54:19.980 --> 00:54:23.190 So one worry is that, in each of these cases, 00:54:23.190 --> 00:54:27.590 they're comparing a difficult condition to an easy condition. 00:54:27.590 --> 00:54:30.020 And so maybe the regions they got 00:54:30.020 --> 00:54:34.030 are just engaged in any kind of task difficulty. 00:54:34.030 --> 00:54:39.070 Maybe if they had done a syntactic task 00:54:39.070 --> 00:54:41.978 on language stimuli that was difficult versus easy, 00:54:41.978 --> 00:54:43.270 they would get the same things. 00:54:43.270 --> 00:54:44.950 From this experiment, we don't know. 00:54:44.950 --> 00:54:47.080 We'll talk more about that in a couple weeks when 00:54:47.080 --> 00:54:50.650 we talk about language, right? 00:54:50.650 --> 00:54:54.070 But here's at least one control that deals with that sort of 00:54:54.070 --> 00:54:57.850 and which does a TMS experiment, as you suggested a while back. 00:54:57.850 --> 00:54:59.530 OK, so this is kind of cool experiment. 00:54:59.530 --> 00:55:01.780 I mean, it's weird, but sort of cool. 00:55:01.780 --> 00:55:02.830 OK, so what do they do? 00:55:02.830 --> 00:55:06.190 They use-- OK, so they have, again, 00:55:06.190 --> 00:55:09.100 an easy task and a hard task. 00:55:09.100 --> 00:55:12.700 Again, it's the thing greater or less than 65. 00:55:12.700 --> 00:55:14.593 Not very hard, right? 00:55:14.593 --> 00:55:15.760 The hard one it's that hard. 00:55:15.760 --> 00:55:19.030 But so is it greater or less than 65? 00:55:19.030 --> 00:55:23.230 And it's either a symbolic number, or it's a dot array. 00:55:23.230 --> 00:55:24.730 You can't really see it, but there's 00:55:24.730 --> 00:55:27.220 a bunch of teeny dots in there. 00:55:27.220 --> 00:55:29.020 Or in the other condition, they have 00:55:29.020 --> 00:55:34.270 to say whether that ellipse is more horizontal or vertical. 00:55:34.270 --> 00:55:35.347 OK? 00:55:35.347 --> 00:55:37.930 And so you spend a lot of time, before you run the experiment, 00:55:37.930 --> 00:55:42.610 measuring reaction time and accuracy to balance difficulty 00:55:42.610 --> 00:55:44.920 within the easy conditions and balance difficulty 00:55:44.920 --> 00:55:46.660 within the hard conditions. 00:55:46.660 --> 00:55:48.350 OK? 00:55:48.350 --> 00:55:53.260 So then what they do is they do something called offline TMS. 00:55:53.260 --> 00:55:54.040 OK? 00:55:54.040 --> 00:55:57.820 Offline TMS, I didn't talk about this much before. 00:55:57.820 --> 00:56:00.923 The standard kinds of TMS, you stick the coil right 00:56:00.923 --> 00:56:01.840 on the subject's head. 00:56:01.840 --> 00:56:03.910 There's a subject doing a task on a monitor. 00:56:03.910 --> 00:56:05.910 And somebody is standing there holding the coil. 00:56:05.910 --> 00:56:08.170 It's really kind of rudimentary. 00:56:08.170 --> 00:56:11.230 And right at a key point of the trial, 00:56:11.230 --> 00:56:13.850 you deliver a zap to disrupt that part of the brain. 00:56:13.850 --> 00:56:16.160 And you find out how much that interferes 00:56:16.160 --> 00:56:17.410 with performance on that task. 00:56:17.410 --> 00:56:20.620 That's the standard online kind of TMS thing. 00:56:20.620 --> 00:56:23.260 But there's also offline TMS, where 00:56:23.260 --> 00:56:28.420 you zap people at a slow rate for like 10 minutes. 00:56:28.420 --> 00:56:31.150 And then the idea is that you've kind of generally 00:56:31.150 --> 00:56:35.740 disrupted that piece of brain for, say, another 10 minutes. 00:56:35.740 --> 00:56:37.030 It's a little bit scarier. 00:56:37.030 --> 00:56:40.060 But it's just like 10 minutes, right? 00:56:40.060 --> 00:56:41.620 OK, and so that way, you don't have 00:56:41.620 --> 00:56:43.810 to be quite so fancy about the precise timing. 00:56:43.810 --> 00:56:46.480 You can just kind of reduce its effectiveness 00:56:46.480 --> 00:56:47.590 for a whole 10 minutes. 00:56:47.590 --> 00:56:50.650 OK, so that's what they did here, offline TMS. 00:56:50.650 --> 00:56:53.830 So you sit there and get zapped for 10 minutes slowly here. 00:56:53.830 --> 00:56:56.260 And then you do some math tasks. 00:56:56.260 --> 00:56:57.280 OK. 00:56:57.280 --> 00:57:01.000 OK, so what they find is that zapping 00:57:01.000 --> 00:57:06.070 the left intraparietal sulcus disrupts the magnitude tasks 00:57:06.070 --> 00:57:08.560 on both numbers and dots. 00:57:08.560 --> 00:57:11.410 But it doesn't mess up the shape tasks with the ellipses, 00:57:11.410 --> 00:57:14.720 even though the ellipses are balanced for difficulty. 00:57:14.720 --> 00:57:15.680 OK? 00:57:15.680 --> 00:57:17.630 So that's at least a little bit of an argument 00:57:17.630 --> 00:57:20.180 that it's not just about generic difficulty, 00:57:20.180 --> 00:57:22.230 at least in this experiment. 00:57:22.230 --> 00:57:24.240 OK? 00:57:24.240 --> 00:57:27.340 All right, I think that's what I just said. 00:57:27.340 --> 00:57:30.220 So that's some evidence for a role 00:57:30.220 --> 00:57:32.380 of at least the left intraparietal sulcus 00:57:32.380 --> 00:57:34.692 in both symbolic and nonsymbolic number. 00:57:34.692 --> 00:57:36.400 Again, nonsymbolic number just means dots 00:57:36.400 --> 00:57:40.120 without Arabic numbers, not just any difficulty. 00:57:40.120 --> 00:57:42.280 All right, so that's all very nice. 00:57:42.280 --> 00:57:44.050 But it's crude as hell, right? 00:57:44.050 --> 00:57:46.030 We found these big, blurry chunks of brain 00:57:46.030 --> 00:57:46.960 that are implicated. 00:57:46.960 --> 00:57:49.060 And we zapped a big chunk of brain 00:57:49.060 --> 00:57:50.980 and slightly reduced performance. 00:57:50.980 --> 00:57:52.450 It's like, OK, better than nothing. 00:57:52.450 --> 00:57:54.100 But it's not very impressive. 00:57:54.100 --> 00:57:57.260 What are the actual neurons doing in the brain? 00:57:57.260 --> 00:57:59.800 Well, now it becomes really important and useful 00:57:59.800 --> 00:58:02.710 that this approximate number system 00:58:02.710 --> 00:58:06.160 that I've been talking about is also present in animals. 00:58:06.160 --> 00:58:08.410 And that means we can use animal models. 00:58:08.410 --> 00:58:10.990 And we can record from individual neurons 00:58:10.990 --> 00:58:13.000 in the parietal lobes of monkeys when 00:58:13.000 --> 00:58:16.630 they do number tasks to find out what actual neurons are doing. 00:58:16.630 --> 00:58:17.500 OK? 00:58:17.500 --> 00:58:19.960 And so there's a guy named Andreas Nieder, who's been 00:58:19.960 --> 00:58:22.370 doing this for a long time. 00:58:22.370 --> 00:58:24.800 And he has some pretty remarkable data. 00:58:24.800 --> 00:58:29.900 And so he starts by training monkeys to do a number task. 00:58:29.900 --> 00:58:31.510 So here's what the monkey sees. 00:58:31.510 --> 00:58:35.320 Monkey sees a sample, some number of dots. 00:58:35.320 --> 00:58:38.440 And then there's a memory delay, in this case, one second. 00:58:38.440 --> 00:58:40.000 And then he has to do a matching task 00:58:40.000 --> 00:58:42.770 and choose that array, not this array. 00:58:42.770 --> 00:58:43.270 OK? 00:58:43.270 --> 00:58:45.228 So he's got to remember that there's three dots 00:58:45.228 --> 00:58:46.570 and choose the right three. 00:58:46.570 --> 00:58:49.090 And notice that the sizes and configuration of the dots 00:58:49.090 --> 00:58:49.640 have changed. 00:58:49.640 --> 00:58:51.940 So we have to do something more like remember 00:58:51.940 --> 00:58:56.950 three in whatever mental monkey E's version of three exists. 00:58:56.950 --> 00:58:58.060 OK? 00:58:58.060 --> 00:59:00.670 OK, simple matching tasks. 00:59:00.670 --> 00:59:03.580 Then he records from neurons in the parietal and frontal cortex 00:59:03.580 --> 00:59:05.660 in monkeys. 00:59:05.660 --> 00:59:10.040 And he finds neurons that are sort of specific for number. 00:59:10.040 --> 00:59:13.380 OK, so here's time in that task. 00:59:13.380 --> 00:59:17.570 This is the time that the sample is presented right here. 00:59:17.570 --> 00:59:21.680 And here is the response of a single neuron that likes 00:59:21.680 --> 00:59:26.400 two more than anything else. 00:59:26.400 --> 00:59:27.390 OK? 00:59:27.390 --> 00:59:30.540 And that too, notice, is all different kinds 00:59:30.540 --> 00:59:33.780 of spatial arrangements and sizes of the dots. 00:59:33.780 --> 00:59:37.770 What's common about all of them is that it's two. 00:59:37.770 --> 00:59:40.570 Next best, it likes four. 00:59:40.570 --> 00:59:41.070 OK? 00:59:41.070 --> 00:59:43.180 And it generalizes across number from there. 00:59:43.180 --> 00:59:44.520 So it's approximate. 00:59:44.520 --> 00:59:47.640 It's not like high for two and zero for everything else. 00:59:47.640 --> 00:59:50.550 It's got a kind of generalization gradient. 00:59:50.550 --> 00:59:52.060 But it prefers two. 00:59:52.060 --> 00:59:52.560 OK? 00:59:52.560 --> 00:59:54.630 So that's a number neuron. 00:59:54.630 --> 00:59:56.100 Yeah? 00:59:56.100 --> 01:00:01.270 OK, here's a six neuron. 01:00:01.270 --> 01:00:03.790 This neuron likes six. 01:00:03.790 --> 01:00:07.840 Here it is same task during presentation of trials here. 01:00:07.840 --> 01:00:09.766 Red is six. 01:00:09.766 --> 01:00:13.390 Next closest is like eight and maybe 10. 01:00:13.390 --> 01:00:16.360 So it also generalizes as well, but it responds more to six 01:00:16.360 --> 01:00:17.650 than anything else. 01:00:17.650 --> 01:00:18.550 Pretty awesome. 01:00:18.550 --> 01:00:19.940 Huh? 01:00:19.940 --> 01:00:25.370 OK, now that doesn't tell us how it was computed, right? 01:00:25.370 --> 01:00:28.970 So finding a single neuron that does something spectacular 01:00:28.970 --> 01:00:29.660 is thrilling. 01:00:29.660 --> 01:00:30.380 We all love it. 01:00:30.380 --> 01:00:32.270 It's great fun. 01:00:32.270 --> 01:00:34.400 And we're closer to the neural circuit 01:00:34.400 --> 01:00:37.160 because we found a neuron that seems to be part of the action. 01:00:37.160 --> 01:00:40.340 But notice it doesn't tell us how that neuron made 01:00:40.340 --> 01:00:42.620 that computation, right? 01:00:42.620 --> 01:00:45.290 What are the circuits that led into it, that enabled it 01:00:45.290 --> 01:00:48.285 to be specific to six or two? 01:00:48.285 --> 01:00:50.450 But it's still cool. 01:00:50.450 --> 01:00:53.850 OK, but next, we want to know, how abstract are those neurons? 01:00:53.850 --> 01:00:56.210 This is just dot arrays. 01:00:56.210 --> 01:00:57.840 OK? 01:00:57.840 --> 01:01:01.110 And they're just presented in one array. 01:01:01.110 --> 01:01:04.740 So next, Andreas Nieder trains his monkeys 01:01:04.740 --> 01:01:08.670 to keep track of the number of things that happen over time. 01:01:08.670 --> 01:01:10.290 It's not a spatial array. 01:01:10.290 --> 01:01:12.600 It's a temporal sequence. 01:01:12.600 --> 01:01:13.230 OK? 01:01:13.230 --> 01:01:16.870 So we have to see that there's four things coming in here 01:01:16.870 --> 01:01:20.460 and then choose the array that matches with four. 01:01:20.460 --> 01:01:22.800 OK? 01:01:22.800 --> 01:01:25.660 See how this is the way to ask how abstract those number 01:01:25.660 --> 01:01:26.280 neurons are. 01:01:26.280 --> 01:01:29.505 Are they really representing the abstract magnitude of two 01:01:29.505 --> 01:01:30.700 or six or whatever it is. 01:01:30.700 --> 01:01:32.117 Or are they representing something 01:01:32.117 --> 01:01:35.410 about the shape of a two-type array or a six-type array. 01:01:35.410 --> 01:01:40.540 OK, and they can also test over different modalities. 01:01:40.540 --> 01:01:43.530 So now they present four different tones. 01:01:43.530 --> 01:01:46.780 And the monkey has to choose the four dots. 01:01:46.780 --> 01:01:47.280 OK? 01:01:47.280 --> 01:01:51.690 Now it's both over time and over sensory modality. 01:01:51.690 --> 01:01:54.210 So how abstract are those number neurons? 01:01:54.210 --> 01:01:56.650 OK, they're pretty abstract. 01:01:56.650 --> 01:01:58.270 So here are a few number neurons. 01:01:58.270 --> 01:02:00.300 Cell one is in the blue colors. 01:02:00.300 --> 01:02:03.750 And here is its response in light blue to dots, 01:02:03.750 --> 01:02:06.660 one dot, two dots, three dots, four dots. 01:02:06.660 --> 01:02:12.120 And here is the same cell responding to sounds. 01:02:12.120 --> 01:02:16.605 It's specific to one, both for sounds and dot arrays. 01:02:16.605 --> 01:02:18.000 Isn't that cool? 01:02:18.000 --> 01:02:22.410 And you see the green cell is selected for two, 01:02:22.410 --> 01:02:26.440 whether in dots or sounds, and so forth. 01:02:26.440 --> 01:02:27.940 Pretty cool, huh? 01:02:27.940 --> 01:02:31.706 So these are very abstract number neurons. 01:02:31.706 --> 01:02:33.350 Does that makes sense? 01:02:33.350 --> 01:02:35.420 OK. 01:02:35.420 --> 01:02:38.110 OK. 01:02:38.110 --> 01:02:41.590 OK, now these monkeys are trained on number tasks. 01:02:41.590 --> 01:02:44.230 So you might think that these kinds of abstract number 01:02:44.230 --> 01:02:46.480 neurons-- and they're trained to do the generalization 01:02:46.480 --> 01:02:47.950 from tones to arrays. 01:02:47.950 --> 01:02:50.950 So maybe those neurons wouldn't live in their brains 01:02:50.950 --> 01:02:53.712 if they hadn't been trained to do that. 01:02:53.712 --> 01:02:55.670 But I don't have time to show you all the data. 01:02:55.670 --> 01:02:59.650 But in subsequent work, the same team 01:02:59.650 --> 01:03:02.260 has recorded from monkeys before any training. 01:03:02.260 --> 01:03:06.200 And you find similar number of neurons. 01:03:06.200 --> 01:03:10.570 So it does seem like these are things that exist in-- 01:03:10.570 --> 01:03:13.010 and remember that's consistent with what I said before, 01:03:13.010 --> 01:03:17.710 which is that a lot of these number abilities 01:03:17.710 --> 01:03:20.830 are present in animals without any training and in newborns. 01:03:20.830 --> 01:03:22.900 And so it makes sense that some of those neurons 01:03:22.900 --> 01:03:25.300 would be around even in advance of any training. 01:03:25.300 --> 01:03:27.550 AUDIENCE: How many neurons did they have to look at it 01:03:27.550 --> 01:03:28.330 to find? 01:03:28.330 --> 01:03:29.410 NANCY KANWISHER: Oh, that's a good question. 01:03:29.410 --> 01:03:30.760 I forget what percent it is. 01:03:30.760 --> 01:03:32.830 We could look it up in the Nieder paper. 01:03:32.830 --> 01:03:33.700 Yeah. 01:03:33.700 --> 01:03:35.830 It's not like you record from thousands, 01:03:35.830 --> 01:03:38.890 and you find 10, right? 01:03:38.890 --> 01:03:42.910 Remember they know where to look from, first, the human lesion 01:03:42.910 --> 01:03:45.940 literature and then the human functional imaging literature. 01:03:45.940 --> 01:03:48.940 And then there's also monkey neuroimaging literature 01:03:48.940 --> 01:03:50.860 where you can have monkeys doing dot tasks. 01:03:50.860 --> 01:03:52.647 So you can know where to look. 01:03:52.647 --> 01:03:53.980 Because the brain's a big place. 01:03:53.980 --> 01:03:55.813 If you're just sticking electrodes all over, 01:03:55.813 --> 01:03:56.650 God help you, right? 01:03:56.650 --> 01:03:59.597 So they know to go up in that parietal lobe 01:03:59.597 --> 01:04:01.930 if that region is homologous between humans and monkeys. 01:04:01.930 --> 01:04:03.520 And there's a lot of other evidence 01:04:03.520 --> 01:04:06.130 that that region is homologous. 01:04:06.130 --> 01:04:07.930 So they know how to get in the right zone. 01:04:07.930 --> 01:04:09.472 And I'm sure, once in the right zone, 01:04:09.472 --> 01:04:10.840 they're not all number neurons. 01:04:10.840 --> 01:04:12.730 I'm sure it's a relatively small percent. 01:04:12.730 --> 01:04:15.384 But it's not a trivial percent. 01:04:15.384 --> 01:04:16.638 Yeah? 01:04:16.638 --> 01:04:19.180 AUDIENCE: Do we have sense for how fractions are represented? 01:04:19.180 --> 01:04:21.105 Because all of these seem to be discrete. 01:04:23.710 --> 01:04:26.003 Or [INAUDIBLE], any ideas? 01:04:26.003 --> 01:04:26.920 NANCY KANWISHER: Yeah. 01:04:26.920 --> 01:04:30.220 Well, it's tricky because, certainly, at the single unit 01:04:30.220 --> 01:04:33.490 level, you'd have to either find some natural version where 01:04:33.490 --> 01:04:36.640 monkeys think about fractions naturally 01:04:36.640 --> 01:04:39.310 or teach them about fractions, which would be really hard. 01:04:39.310 --> 01:04:41.740 Because, for some reason, fractions are just really hard. 01:04:41.740 --> 01:04:44.800 Like all the people who study math education, 01:04:44.800 --> 01:04:46.862 it's like the key problem is, how do you get 01:04:46.862 --> 01:04:48.070 kids to understand fractions? 01:04:48.070 --> 01:04:50.950 I don't know why they're such a tough thing. 01:04:50.950 --> 01:04:53.710 But apparently, it's like a real dividing line, 01:04:53.710 --> 01:04:56.980 the kids who get fractions and the kids who don't. 01:04:56.980 --> 01:04:59.310 So I'd have to think. 01:04:59.310 --> 01:05:04.500 But, occasionally, there are patients with electrodes 01:05:04.500 --> 01:05:05.340 in their brains. 01:05:05.340 --> 01:05:07.230 And one could look at that. 01:05:07.230 --> 01:05:09.750 Actually, I took this slide out, but there's 01:05:09.750 --> 01:05:12.570 a paper that came out last year where they found number 01:05:12.570 --> 01:05:14.335 neurons in humans as well. 01:05:14.335 --> 01:05:15.960 I took it out because I didn't know how 01:05:15.960 --> 01:05:16.920 to integrate it in the lecture. 01:05:16.920 --> 01:05:18.480 Because the number neurons are deep 01:05:18.480 --> 01:05:20.910 in the medial temporal lobe, far from the parietal lobe. 01:05:20.910 --> 01:05:22.800 And it's like, I don't know how that fits. 01:05:22.800 --> 01:05:24.790 I don't know if that's the same thing or something else. 01:05:24.790 --> 01:05:26.340 But anyway, there are at least some number neurons 01:05:26.340 --> 01:05:27.820 that have been found in humans. 01:05:27.820 --> 01:05:31.050 And you could, in principle, look for number neurons 01:05:31.050 --> 01:05:33.420 up in the parietal lobe. 01:05:33.420 --> 01:05:38.125 In fact, I have a guy I'm trying to collaborate with. 01:05:38.125 --> 01:05:39.750 I'm begging him to collaborate with me. 01:05:39.750 --> 01:05:42.870 He's got two people who have arrays of electrodes 01:05:42.870 --> 01:05:46.410 chronically implanted right up in this region 01:05:46.410 --> 01:05:49.500 here because they are paralyzed. 01:05:49.500 --> 01:05:51.000 They had spinal damage. 01:05:51.000 --> 01:05:52.920 And like Michael Cohen's lecture, 01:05:52.920 --> 01:05:55.170 he's got arrays of electrodes where 01:05:55.170 --> 01:05:57.300 he's trying to use the neural responses there 01:05:57.300 --> 01:05:59.040 to direct robot arms. 01:05:59.040 --> 01:06:00.570 And so there's two of these people 01:06:00.570 --> 01:06:02.180 who have these chronically implanted things. 01:06:02.180 --> 01:06:03.722 I'm like, oh, please, please, please, 01:06:03.722 --> 01:06:05.760 can I collaborate with you and get responses 01:06:05.760 --> 01:06:07.680 from your patients' neurons? 01:06:07.680 --> 01:06:09.740 Was there a question over here a moment ago? 01:06:09.740 --> 01:06:10.550 Sorry. 01:06:10.550 --> 01:06:12.800 I thought I saw a hand go up. 01:06:12.800 --> 01:06:14.990 OK, so let me wrap up. 01:06:14.990 --> 01:06:17.690 So I've been arguing that this approximate number 01:06:17.690 --> 01:06:21.740 system is shared with animals and newborns. 01:06:21.740 --> 01:06:25.460 It's a pretty basic system that lots of animals have. 01:06:25.460 --> 01:06:28.100 It follows Weber's law, which you should remember. 01:06:28.100 --> 01:06:30.230 I don't like testing you guys on esoteric facts. 01:06:30.230 --> 01:06:32.780 But Weber's law is a very fundamental fact. 01:06:32.780 --> 01:06:36.110 And you should know it about perception and, in particular, 01:06:36.110 --> 01:06:36.930 about number. 01:06:36.930 --> 01:06:39.350 It tells you that the ability to discriminate two numbers 01:06:39.350 --> 01:06:44.360 goes as the ratio, not as the difference of those numbers. 01:06:44.360 --> 01:06:48.290 And that these approximate magnitude representations 01:06:48.290 --> 01:06:52.040 measured both behaviorally and neurally in humans and animals 01:06:52.040 --> 01:06:54.740 are very abstract to the particular objects, 01:06:54.740 --> 01:06:56.330 to the modality, to whether they come 01:06:56.330 --> 01:06:59.120 in over space or time, et cetera, 01:06:59.120 --> 01:07:03.140 whether they're represented in symbols or arrays of items. 01:07:03.140 --> 01:07:04.502 OK? 01:07:04.502 --> 01:07:06.710 I mentioned that there are big individual differences 01:07:06.710 --> 01:07:11.210 in humans in the precision of the approximate number system. 01:07:11.210 --> 01:07:15.080 And that is predictive of later arithmetic abilities 01:07:15.080 --> 01:07:18.170 independent of IQ. 01:07:18.170 --> 01:07:21.230 And we talked about the horizontal segment 01:07:21.230 --> 01:07:24.110 of the intraparietal sulcus as a key locus 01:07:24.110 --> 01:07:29.120 for the approximate number system in humans, 01:07:29.120 --> 01:07:32.330 including number-specific neurons. 01:07:32.330 --> 01:07:34.370 And we also talked about, both in some 01:07:34.370 --> 01:07:37.430 of the papers I mentioned and the paper you guys read, 01:07:37.430 --> 01:07:40.910 that there seems to be that that approximate number system up 01:07:40.910 --> 01:07:44.090 here in the parietal lobe, so far, doesn't seem 01:07:44.090 --> 01:07:48.710 to be one of these extremely specialized systems like faces 01:07:48.710 --> 01:07:51.680 and motion and navigation, which may turn out 01:07:51.680 --> 01:07:53.880 to be less specialized later with pending more data. 01:07:53.880 --> 01:07:55.670 But at the moment, we can already 01:07:55.670 --> 01:07:58.910 see that these number representations overlap a lot 01:07:58.910 --> 01:08:02.510 with representations of space, shown perhaps most dramatically 01:08:02.510 --> 01:08:03.980 in the paper you guys read showing 01:08:03.980 --> 01:08:07.730 cross decoding between eye-movement direction 01:08:07.730 --> 01:08:10.310 and arithmetic operations. 01:08:10.310 --> 01:08:12.845 OK, hang on. 01:08:12.845 --> 01:08:13.970 I'm almost done summing up. 01:08:13.970 --> 01:08:16.550 Number neurons, we talked about that. 01:08:16.550 --> 01:08:18.859 Yes, so we'll give the last word to Stan Dehaene, who 01:08:18.859 --> 01:08:21.380 started off with this very extreme view 01:08:21.380 --> 01:08:25.340 and has evolved to a still interesting but slightly less 01:08:25.340 --> 01:08:26.510 extreme view. 01:08:26.510 --> 01:08:29.540 He says, the brain treats number like a specific category 01:08:29.540 --> 01:08:33.439 of knowledge requiring its own neurological apparatus 01:08:33.439 --> 01:08:35.390 in the parietal lobe. 01:08:35.390 --> 01:08:38.149 But when it comes to subtler distinctions, such as number 01:08:38.149 --> 01:08:42.740 versus length, space, or time, the specificity of hIPS 01:08:42.740 --> 01:08:43.819 vanishes. 01:08:43.819 --> 01:08:46.520 No part of hIPS appears to be involved 01:08:46.520 --> 01:08:50.160 in numerical computations alone. 01:08:50.160 --> 01:08:53.689 In fact, he goes further to say that the human brain, 01:08:53.689 --> 01:08:57.229 in general, is neither anisotropic white paper, 01:08:57.229 --> 01:09:00.859 like equipotential, where all the regions are equivalent, 01:09:00.859 --> 01:09:02.779 nor a neat arrangement of tightly 01:09:02.779 --> 01:09:05.750 specialized and well-separated modules. 01:09:05.750 --> 01:09:06.265 All right? 01:09:06.265 --> 01:09:07.640 Anyway, OK, there was a question. 01:09:07.640 --> 01:09:08.555 Sorry. 01:09:08.555 --> 01:09:18.979 AUDIENCE: [INAUDIBLE] just then having this, I guess, 01:09:18.979 --> 01:09:21.830 easier time with approximate numbers, 01:09:21.830 --> 01:09:24.740 given more of an interest in that. 01:09:24.740 --> 01:09:27.500 NANCY KANWISHER: That's a really, really good question. 01:09:27.500 --> 01:09:29.607 And I am sure there are data on that. 01:09:29.607 --> 01:09:30.899 And I don't know what they are. 01:09:30.899 --> 01:09:32.300 But I will go look. 01:09:32.300 --> 01:09:33.170 I always say that. 01:09:33.170 --> 01:09:36.470 But, Dana, will you send me an email right now to go 01:09:36.470 --> 01:09:39.649 look up whether the prediction from childhood 01:09:39.649 --> 01:09:43.760 ANS to adult arithmetic abilities 01:09:43.760 --> 01:09:45.500 has to do with an interest or you might 01:09:45.500 --> 01:09:48.050 say just an emotional response. 01:09:48.050 --> 01:09:50.600 Like if you suck at it, it feels bad. 01:09:50.600 --> 01:09:52.550 And you become avoidant, and you get all 01:09:52.550 --> 01:09:54.860 dysfunctional about it, right? 01:09:54.860 --> 01:09:59.000 We all have-- I mean, most of us have domains where we do that. 01:09:59.000 --> 01:10:01.400 And math phobia is a real thing. 01:10:01.400 --> 01:10:02.290 And who knows. 01:10:02.290 --> 01:10:03.290 It could start in there. 01:10:03.290 --> 01:10:04.458 So yeah, good question. 01:10:04.458 --> 01:10:05.000 I don't know. 01:10:05.000 --> 01:10:07.230 I will look that up. 01:10:07.230 --> 01:10:10.100 Other questions? 01:10:10.100 --> 01:10:13.870 OK, see you guys on Wednesday.