WEBVTT - From the Vault: Odds and Evens, Part 2 0:00:06.280 --> 0:00:09.119 Hey, welcome to Stuff to Blow your Mind. My name 0:00:09.200 --> 0:00:11.719 is Robert Lamb. Today is Saturday, so we have another 0:00:11.800 --> 0:00:13.920 VAULD episode for you. This is going to be Odds 0:00:13.960 --> 0:00:18.160 and Evens Part two. This one originally published nine ten, 0:00:18.360 --> 0:00:20.840 twenty twenty four. Let's jump right in. 0:00:24.239 --> 0:00:28.000 Welcome to Stuff to Blow Your Mind, the production of iHeartRadio. 0:00:33.800 --> 0:00:35.480 Hey, welcome to Stuff to Blow your Mind. 0:00:35.560 --> 0:00:38.680 My name is Robert Lamb, and I am Joe McCormick, 0:00:38.760 --> 0:00:41.560 and we are back with Part two in our series 0:00:41.600 --> 0:00:46.839 on the psychology and cultural significance of number parody p 0:00:47.040 --> 0:00:50.600 A r it y parody, meaning whether a number is 0:00:50.760 --> 0:00:54.960 odd or even. In Part one, we described the principle 0:00:55.000 --> 0:00:57.800 of number parody, and we talked about evidence that in 0:00:57.840 --> 0:01:02.760 some cases people seem to have surprising feelings about associations 0:01:02.760 --> 0:01:07.160 with and even preferences for odd and even quantities. And 0:01:07.240 --> 0:01:10.240 so one of the big examples we discussed in that 0:01:10.280 --> 0:01:13.640 first episode was the concept in various branches of visual 0:01:13.800 --> 0:01:17.680 art theory, that people have a preference for, say, three 0:01:17.760 --> 0:01:20.920 part divisions of imagery over two part divisions, or that 0:01:20.959 --> 0:01:24.440 people prefer an image composed with an odd number of 0:01:24.480 --> 0:01:27.520 subjects over an even number, even to the extent that 0:01:27.720 --> 0:01:31.760 even numbers of subjects will sometimes be subdivided into groups 0:01:31.760 --> 0:01:34.440 of odd numbers, so you know, instead of four subjects, 0:01:34.840 --> 0:01:37.480 you would get a painting with three and one. But 0:01:37.600 --> 0:01:40.440 we also got into a bit of empirical research interrogating 0:01:40.480 --> 0:01:43.560 these ideas and questioning to what extent they're truly natural 0:01:43.600 --> 0:01:46.840 aesthetic preferences. Maybe they're just sort of random conventions that 0:01:47.360 --> 0:01:50.360 people latched onto, including you know, one thing that came 0:01:50.440 --> 0:01:53.920 up in part one was the domain of food plating 0:01:53.960 --> 0:01:57.120 and food styling, with us just you know, shooting from 0:01:57.160 --> 0:02:00.600 the hips saying I think three little slide are better 0:02:00.640 --> 0:02:03.240 than four. We're going to come back to that later today. 0:02:03.520 --> 0:02:04.800 You might be surprised. 0:02:05.960 --> 0:02:08.600 I mean it is still you still see this idea 0:02:08.639 --> 0:02:11.720 out there, But how does it hold up to any 0:02:11.760 --> 0:02:13.920 manner of study. Well, we'll take a look at that. 0:02:14.600 --> 0:02:17.680 So one thing I wanted to talk about today was 0:02:17.840 --> 0:02:24.200 the cognitive psychology of number parity, how we process the 0:02:24.400 --> 0:02:27.919 idea of numbers being odd and even in the brain. 0:02:28.919 --> 0:02:31.639 So I came across a very interesting paper about this 0:02:31.800 --> 0:02:34.639 that was published in the journal Frontiers in Psychology in 0:02:34.680 --> 0:02:38.000 the year twenty eighteen by Hubner at All and it's 0:02:38.040 --> 0:02:42.640 called a mental odd even continuum account some numbers may 0:02:42.720 --> 0:02:46.480 be more odd than others, and some numbers may be 0:02:46.760 --> 0:02:50.400 more even than others. And so if you're not initially 0:02:50.520 --> 0:02:53.120 thrilled about the idea of that the cognitive psychology of 0:02:53.200 --> 0:02:57.320 numbers how we represent number properties internally, stick around. I 0:02:57.360 --> 0:03:00.359 think this might be more interesting than you would at 0:03:00.360 --> 0:03:03.360 first suspect, because it's kind of it kind of reveals 0:03:03.480 --> 0:03:06.320 deeper ways that our brains work in general, at least 0:03:06.360 --> 0:03:08.400 I think. So we can come back to that after 0:03:08.400 --> 0:03:10.840 we look at the findings of the study. But anyway 0:03:10.880 --> 0:03:15.840 to start with the mathematical fact is that number parity 0:03:16.080 --> 0:03:21.480 is binary. In math, natural numbers are either odd or even. 0:03:21.840 --> 0:03:25.320 Any positive integer is even if it can be represented 0:03:25.360 --> 0:03:28.920 as two times in wherein is also a positive integer, 0:03:29.360 --> 0:03:31.799 and it's odd if it can be represented as two 0:03:31.880 --> 0:03:36.000 times in plus one. All positive whole numbers are either 0:03:36.160 --> 0:03:39.480 odd or even. But this paper is focused not on 0:03:39.560 --> 0:03:43.120 the question of the mathematics of parity, but on the 0:03:43.200 --> 0:03:47.120 question of how number parity is represented in the brain, 0:03:47.320 --> 0:03:50.640 how we think about quantities that are odd and even. 0:03:51.480 --> 0:03:55.680 And the authors propose an interesting hypothesis that people do 0:03:55.760 --> 0:04:00.119 not think about odd and even as a mathematical binary, 0:04:00.160 --> 0:04:04.560 but rather as a spectrum of odd ness and even ness, 0:04:04.680 --> 0:04:07.960 where some numbers can be relatively more odd or even 0:04:08.000 --> 0:04:11.720 than others, and in a kind of amusing aside. The 0:04:11.760 --> 0:04:14.400 authors acknowledge that if this is true, it may prove 0:04:14.600 --> 0:04:18.280 irritating to some researchers. But you know, this is the 0:04:18.360 --> 0:04:20.320 kind of thing I like reading about, because I think 0:04:20.680 --> 0:04:24.839 it's when you observe the mismatch between how a concept 0:04:24.920 --> 0:04:28.600 is technically defined and how we actually think about it. 0:04:28.640 --> 0:04:31.520 When we know when we consider it in practice, it's 0:04:31.560 --> 0:04:33.480 a great way to get insights into our brains. 0:04:34.120 --> 0:04:36.760 Yeah. Yeah, And I'm already thinking about thinking about ways 0:04:36.760 --> 0:04:39.880 that I might qualify certain numbers as more even or 0:04:39.880 --> 0:04:42.120 more odd than others. But I want to see where 0:04:42.120 --> 0:04:44.080 you're taking us here and see if any of these 0:04:44.080 --> 0:04:47.520 are are the examples that are coming to my mind. 0:04:47.839 --> 0:04:49.680 So to provide a model for how this would be 0:04:49.760 --> 0:04:53.160 happening in the brain, the authors refer to a psychology 0:04:53.200 --> 0:04:57.720 concept called prototype theory, which has been established going at 0:04:57.800 --> 0:05:01.839 least as far back as the nineteen sixties. As they explain, quote, 0:05:02.080 --> 0:05:06.719 prototype theory has long suggested that certain members of distinct 0:05:06.800 --> 0:05:12.400 categories are more typical examples of that category than others, 0:05:12.680 --> 0:05:17.200 and that membership to such a category may be graded. Now, 0:05:17.320 --> 0:05:19.520 they don't use the following example, And in fact, I 0:05:19.520 --> 0:05:22.200 don't know if this is strictly a perfect example of 0:05:22.240 --> 0:05:25.120 prototype theory, because the category I'm going to use is 0:05:25.160 --> 0:05:28.280 not strictly defined, But I think this will still illustrate it. 0:05:28.839 --> 0:05:35.120 Both Pumpkinhead and Grover from Sesame Street are examples of 0:05:35.160 --> 0:05:41.159 the category monster. And yet while they are undoubtedly both monsters, 0:05:41.200 --> 0:05:43.599 and if you doubt Grover is a monster, go read 0:05:43.680 --> 0:05:47.000 up about them. Grover's a monster, one of them just 0:05:47.040 --> 0:05:51.080 seems like a better example of the category monster than 0:05:51.120 --> 0:05:55.039 the other. Now, there are no real objective criteria for 0:05:55.120 --> 0:05:57.680 what is and is not a monster, but you could 0:05:57.800 --> 0:06:01.640 learn a lot about how people mentally construct the idea 0:06:01.680 --> 0:06:05.680 of a monster by studying how easy it is to 0:06:05.839 --> 0:06:11.720 associate particular examples of creatures with the category monster. And 0:06:11.960 --> 0:06:15.440 one way of studying this would be time latency. So 0:06:16.040 --> 0:06:19.680 imagine you're in a psychological study and you're given a task. 0:06:20.600 --> 0:06:23.560 Somebody's going to show you a series of images of creatures, 0:06:24.240 --> 0:06:26.560 and it's your job to say as quickly as you 0:06:26.640 --> 0:06:29.680 can whether the creature in the image is a monster 0:06:29.880 --> 0:06:33.200 or not. In this kind of test, the speed with 0:06:33.320 --> 0:06:36.839 which you make the categorization could be one piece of 0:06:36.880 --> 0:06:41.880 evidence for how easily you associate the example with the category. 0:06:42.080 --> 0:06:44.719 So even if everybody who takes this kind of test 0:06:44.839 --> 0:06:48.920 correctly recognizes that Grover is a monster, I would still 0:06:49.000 --> 0:06:52.080 bet that on average people would say Pumpkinhead is a 0:06:52.120 --> 0:06:55.159 monster a good bit faster. It would just it takes 0:06:55.240 --> 0:06:57.800 less thinking to get there, so you can click the 0:06:57.839 --> 0:06:59.000 monster button faster. 0:06:59.720 --> 0:07:02.680 Yeah, you don't have to catch yourself and go, oh, well, yes, 0:07:02.720 --> 0:07:04.279 of course he is the monster at the end of 0:07:04.320 --> 0:07:04.640 the book. 0:07:04.920 --> 0:07:07.640 Yeah, exactly. And so with this kind of study you 0:07:07.640 --> 0:07:09.960 could maybe get some insights. For example, you could look 0:07:10.000 --> 0:07:13.760 at these specific attributes that make an individual picture of 0:07:13.760 --> 0:07:17.440 a creature a better prototype example of the monster category 0:07:18.360 --> 0:07:21.240 as measured by people selecting it as a monster faster. 0:07:21.960 --> 0:07:26.360 Maybe creatures that have sharp teeth or claws or threatening 0:07:26.440 --> 0:07:28.640 posture or something like that. It just clicks in the 0:07:28.680 --> 0:07:30.840 brain faster that it's a monster. You got to think 0:07:30.840 --> 0:07:33.720 about it less. And so in this paper, the authors 0:07:34.080 --> 0:07:37.880 do the same thing with odd and even numbers. They're 0:07:37.880 --> 0:07:40.840 going to study the degree to which different numbers are 0:07:41.040 --> 0:07:44.640 prototypes of their parity class, and then they're going to 0:07:44.720 --> 0:07:47.000 try to look for the different factors that make a 0:07:47.080 --> 0:07:51.200 number more easily identifiable as odd or even. And this is, 0:07:51.320 --> 0:07:53.240 by the way, not the first study ever to do this. 0:07:53.360 --> 0:07:55.360 There have been studies in the past that have used 0:07:55.400 --> 0:07:58.760 processing time as a measure of prototypicality for odd and 0:07:58.800 --> 0:08:03.400 even numbers, like they mentioned one study that showed six 0:08:03.800 --> 0:08:07.600 took people longer to classify as even than two four 0:08:07.760 --> 0:08:08.440 or eight did. 0:08:08.920 --> 0:08:09.160 Why. 0:08:09.320 --> 0:08:12.600 I don't know. That's kind of interesting. I mean, two, four, six, 0:08:12.640 --> 0:08:16.080 and eight are all equally even in real mathematics, but 0:08:16.200 --> 0:08:19.600 apparently two four and eight are just easier to identify 0:08:19.640 --> 0:08:22.000 as even. Something's a little different about six. 0:08:23.000 --> 0:08:24.040 Huh. Interesting. 0:08:24.680 --> 0:08:26.960 So in their introduction, the authors lay out a bunch 0:08:27.000 --> 0:08:31.000 of different numerical reasons that they think a number might 0:08:31.080 --> 0:08:34.920 be more easily recognizable as even or odd, and the 0:08:35.360 --> 0:08:41.199 hypothetical explanations they include are first of all, ease of divisibility. 0:08:41.640 --> 0:08:44.640 So the easier a number is to divide, the more 0:08:44.880 --> 0:08:48.840 even and less odd it should feel. And this principle 0:08:48.880 --> 0:08:52.440 could subconsciously be applied within the categories and not just 0:08:52.480 --> 0:08:56.680 between them. So twenty five and twenty seven are both odd. 0:08:57.080 --> 0:08:59.839 But the author's idea here is that twenty five may 0:08:59.880 --> 0:09:03.559 feel less odd and take longer to classify as odd 0:09:03.640 --> 0:09:05.200 because it's easy to divide it. 0:09:05.640 --> 0:09:08.000 Now, this is where my mind was headed that. Yeah, 0:09:08.160 --> 0:09:11.000 just thinking about the way I divide numbers is if 0:09:11.040 --> 0:09:13.840 it's easier to divide, then yes, on some level, it 0:09:13.960 --> 0:09:17.000 is more even than an even number that I have 0:09:17.040 --> 0:09:19.560 to sort of like pause a second with then do 0:09:19.640 --> 0:09:21.520 a little extra math in my head. 0:09:21.840 --> 0:09:25.000 Yeah, I think that's a strong instinct that they had 0:09:25.040 --> 0:09:27.880 the same idea to begin with. Here. Another thing they 0:09:28.000 --> 0:09:31.120 hypothesize would make a number feel more even is powers 0:09:31.160 --> 0:09:34.560 of two, so that would be two, four, eight, sixteen, 0:09:34.760 --> 0:09:39.320 thirty two. They think these are cognitively more even. Another 0:09:39.360 --> 0:09:43.800 factor is whether a number is prime. The authors argue 0:09:43.840 --> 0:09:49.200 that prime numbers may feel more odd than non prime odds, 0:09:49.920 --> 0:09:52.240 and one piece of evidence for this is that a 0:09:52.280 --> 0:09:55.280 couple of different previous studies have found that people are 0:09:55.440 --> 0:10:00.439 quicker to flag three, five, and seven as odd than 0:10:00.440 --> 0:10:03.800 they are to flag nine. That's interesting, Now, this is 0:10:03.840 --> 0:10:06.960 kind of like the inverse of the six not feeling 0:10:07.040 --> 0:10:09.960 as even as the other even numbers under ten. In 0:10:10.000 --> 0:10:13.320 this case, apparently, maybe nine does not feel as odd 0:10:13.400 --> 0:10:17.040 as the other odd numbers under ten, and the authors 0:10:17.160 --> 0:10:19.560 argue that this may be because the other three odd 0:10:19.679 --> 0:10:22.160 numbers under ten, three, five, and seven are all prime. 0:10:22.400 --> 0:10:25.320 Nine is not prime three times three is nine, so 0:10:25.800 --> 0:10:28.600 the divisibility of it maybe makes it feel less odd. 0:10:29.880 --> 0:10:34.679 The authors also hypothesize maybe being part of a standard 0:10:34.760 --> 0:10:39.040 multiplication table that children memorize in school that might make 0:10:39.160 --> 0:10:43.400 numbers feel more even and less odd, but we'll have 0:10:43.440 --> 0:10:46.400 to look at the results and see if that bears out. However, 0:10:46.800 --> 0:10:49.640 the authors point out that previous studies have shown that 0:10:49.760 --> 0:10:54.520 it is probably not only the mathematical properties of a number. 0:10:54.559 --> 0:10:58.360 The number properties of a number that influence how long 0:10:58.520 --> 0:11:02.080 we take to make judgments about other factors, such as 0:11:02.360 --> 0:11:06.360 linguistic factors, appear to play a role as well. And 0:11:06.559 --> 0:11:10.040 illustrate this, the authors bring up a really interesting concept 0:11:10.400 --> 0:11:12.920 that I don't think I'd ever read about before, but 0:11:13.040 --> 0:11:16.640 this really stuck with me. So they refer to previous 0:11:16.679 --> 0:11:20.640 research by Hines in the journal Memory and Cognition in 0:11:20.760 --> 0:11:24.600 nineteen ninety and this paper found that if you give 0:11:24.720 --> 0:11:29.200 people random numbers, especially in pairs or in triples, and 0:11:29.320 --> 0:11:32.000 ask them to judge whether the numbers are odd or even. 0:11:32.520 --> 0:11:36.920 People simply take longer to recognize oddness than they do 0:11:37.040 --> 0:11:41.760 to recognize evenness. So odd numbers were just harder to 0:11:41.920 --> 0:11:46.800 judge overall, So people more quickly recognize that fifty two 0:11:46.880 --> 0:11:50.520 and fifty four are even than that fifty three and 0:11:50.559 --> 0:11:53.800 fifty five are odd. Now that that's kind of weird, 0:11:53.960 --> 0:11:58.120 Like why would oddness itself take longer to process? Pretty 0:11:58.200 --> 0:12:02.120 much across the board? In older paper, the author argued 0:12:02.240 --> 0:12:05.480 that part of the explanation may lie in the idea 0:12:05.480 --> 0:12:10.079 of what are called marked and unmarked terms in language. 0:12:10.160 --> 0:12:13.960 Marked and unmarked This is a concept in linguistics, and 0:12:14.120 --> 0:12:18.640 it goes like this, So there exist in languages pairs 0:12:18.760 --> 0:12:24.000 of adjectives that have opposite meanings, so long and short, 0:12:24.640 --> 0:12:30.120 old and young, even an odd, alive and dead, things 0:12:30.160 --> 0:12:34.320 like that. Linguistic markedness theory says that usually when you 0:12:34.440 --> 0:12:38.720 have pairs of adjectives like this, one of the terms 0:12:38.840 --> 0:12:42.560 in the pair is treated as the more basic and 0:12:42.760 --> 0:12:45.960 natural of the two in the brain. So we think 0:12:46.000 --> 0:12:48.800 about one of these two terms in a way that 0:12:48.840 --> 0:12:52.160 what they call they call it unmarked. It is the 0:12:52.320 --> 0:12:56.640 natural state of this measure, and then the other term 0:12:57.200 --> 0:13:02.800 is treated as mentally more complex, complicated, and unnatural. This 0:13:02.880 --> 0:13:05.959 is the marked word in the pair, and there are 0:13:06.040 --> 0:13:09.080 experiments that will show this. But the unmarked word in 0:13:09.160 --> 0:13:13.840 the pair, for example, is used more frequently than the 0:13:13.840 --> 0:13:17.920 marked word. It's learned earlier in language acquisition, when you're 0:13:17.960 --> 0:13:22.280 a child, and it is considered usually the default measure. So, 0:13:22.400 --> 0:13:26.679 for example, you say how old are you, not how 0:13:26.800 --> 0:13:30.000 young are you? Because in old and young, old is 0:13:30.040 --> 0:13:35.400 treated as the unmarked word and young is the marked concept. Similarly, 0:13:35.480 --> 0:13:38.600 you will say how long will it take? Not how 0:13:38.679 --> 0:13:41.400 short will it take? I thought this was interesting. They 0:13:41.400 --> 0:13:44.240 say also that in some cases you can create the 0:13:44.280 --> 0:13:47.840 same meaning as the marked word simply by adding a 0:13:47.920 --> 0:13:51.560 negative prefix to the unmarked word. So you can say 0:13:51.840 --> 0:13:55.280 uneven to mean the same thing as odd, But nobody 0:13:55.280 --> 0:13:57.400 says un odd to mean even. 0:13:58.080 --> 0:13:59.559 Oh that's true. There's a great point. 0:13:59.800 --> 0:14:03.160 Now, whatever this division between marked and unmarked comes from, 0:14:03.520 --> 0:14:07.719 it seems that it results in different processing times in 0:14:07.760 --> 0:14:12.800 the brain, that we just deal with unmarked concepts faster 0:14:13.000 --> 0:14:16.000 and more easily, and it takes us you know, maybe 0:14:16.040 --> 0:14:19.480 a split second longer to think about, or deliver or 0:14:19.560 --> 0:14:23.640 deal with a marked concept. And so if even is 0:14:23.800 --> 0:14:27.120 unmarked and odd is marked, it may in fact be 0:14:27.400 --> 0:14:30.440 that we just deal with the concept of evenness a 0:14:30.520 --> 0:14:33.160 little bit more easily in the brain than oddness. It's 0:14:33.280 --> 0:14:36.800 oddness is linguistically marked, and so it takes us a 0:14:36.840 --> 0:14:40.760 split second longer to kind of process this concept whenever 0:14:40.760 --> 0:14:43.160 we're dealing with it or producing a judgment about it, 0:14:43.560 --> 0:14:45.560 and this may play a role in explaining the so 0:14:45.640 --> 0:14:49.320 called odd effect that was discovered in this paper in 0:14:49.360 --> 0:15:01.960 the nineties. Moving on from that, there's another linguistic effect 0:15:02.040 --> 0:15:06.040 that actually shows up when you compare judgments about parody 0:15:06.080 --> 0:15:10.960 across different languages, and this is the inversion property of 0:15:11.120 --> 0:15:15.080 multiple digit numbers. So in English, when we want to 0:15:15.280 --> 0:15:17.840 say or write out in words the number that is 0:15:17.920 --> 0:15:21.200 one quarter of one hundred, we say twenty five, we 0:15:21.280 --> 0:15:24.640 write the twenty first and then the five, or we 0:15:24.680 --> 0:15:27.160 say the twenty first and then the five. So for 0:15:27.200 --> 0:15:31.160 two digit numbers, it's always the decade digit first in language, 0:15:31.240 --> 0:15:34.120 and then the unit digit. But not all languages work 0:15:34.160 --> 0:15:38.000 this way. For example, in German, twenty five is and 0:15:38.040 --> 0:15:40.400 I'm sorry, I'm sure I'm pronouncing this wrong. It is 0:15:40.440 --> 0:15:46.880 something like fun Fundzwanzig, meaning five and twenty, And this 0:15:46.960 --> 0:15:49.600 has been found to have all sorts of interesting effects 0:15:49.680 --> 0:15:54.560 on number cognition. For example, German speakers our studies have 0:15:54.600 --> 0:15:58.680 shown more likely to make trans coding errors when writing 0:15:58.800 --> 0:16:03.080 numbers out, so likely to write fifty two when they 0:16:03.160 --> 0:16:07.440 mean twenty five. In terms of digits. Also, compared to 0:16:07.640 --> 0:16:12.400 non inverted languages, German speakers pay relatively more attention to 0:16:12.560 --> 0:16:16.320 the unit digit in a multi digit number, and so 0:16:16.400 --> 0:16:20.080 the authors write quote. This prioritizing of either the unit 0:16:20.200 --> 0:16:24.359 or decade digit might influence participants' performance in number processing 0:16:24.400 --> 0:16:28.320 tasks in which units play a decisive role. Parity judgment 0:16:28.400 --> 0:16:31.520 is clearly one of those tasks, because only the unit 0:16:31.760 --> 0:16:35.640 parity is relevant for answering correctly, which is true when 0:16:35.640 --> 0:16:37.560 you look at you can judge whether it's odd or 0:16:37.600 --> 0:16:40.800 even without knowing any of the numbers before the last one. 0:16:41.280 --> 0:16:43.840 And just a couple of other factors the authors mention 0:16:44.120 --> 0:16:49.120 that have been possibly shown to influence parity judgments. Larger 0:16:49.240 --> 0:16:53.680 numbers may cause longer processing times, regardless of the parity 0:16:53.840 --> 0:16:56.080 or any other facts about them, just like the bigger 0:16:56.120 --> 0:16:58.960 the number is, the longer you have to think about it. Also, 0:16:59.200 --> 0:17:02.760 word frequently. Numbers that appear more often in language get 0:17:02.800 --> 0:17:05.760 faster responses, and this is not just true of numbers 0:17:05.840 --> 0:17:08.520 in any words. In general that are used more often 0:17:08.600 --> 0:17:13.280 are processed more efficiently. So this study tried to test 0:17:13.320 --> 0:17:18.080 the relative influence of number prototypicality and the linguistic factors 0:17:18.080 --> 0:17:20.520 we were just talking about. And the way they did 0:17:20.560 --> 0:17:24.800 this was by getting a group of subjects and giving 0:17:24.800 --> 0:17:29.120 them auditory prompts of numbers between twenty and ninety nine, 0:17:29.240 --> 0:17:31.560 and then they would try to analyze how long it 0:17:31.600 --> 0:17:34.280 took people to classify these numbers as odd or even 0:17:35.040 --> 0:17:39.000 to test the linguistic factors. The author's recruited subjects from 0:17:39.080 --> 0:17:42.600 three different language groups. They had English speakers, German speakers, 0:17:42.600 --> 0:17:47.000 and Polish speakers. In Polish, two digit numbers are expressed 0:17:47.000 --> 0:17:50.600 with the decade digit first, like in English. And I'm 0:17:50.640 --> 0:17:52.720 not going to discuss all of their findings, but just 0:17:52.760 --> 0:17:55.879 to summarize and pick a few highlights, they do say 0:17:55.960 --> 0:18:00.560 that quote. Overall, the results suggests that perceived parody is 0:18:00.600 --> 0:18:03.720 not the same as objective parity, and some numbers are 0:18:03.800 --> 0:18:09.800 more prototypical exemplars of their categories, and specifically with regards 0:18:09.800 --> 0:18:14.840 to these mathematical or numerical factors influencing things, they found 0:18:14.960 --> 0:18:19.400 that some but not all, of the characteristics they hypothesized 0:18:19.920 --> 0:18:24.159 actually did play a role in perceived parity. So, for evens, 0:18:24.960 --> 0:18:29.199 the numbers that people identified as even the fastest tended 0:18:29.240 --> 0:18:33.200 to be even squares, so a square being the product 0:18:33.200 --> 0:18:36.479 of a number multiplied by itself. Sixteen is a square 0:18:36.520 --> 0:18:39.320 because it's four times four, sixty four is a square 0:18:39.400 --> 0:18:42.720 because it's eight times eight. Thirty six is a square 0:18:42.760 --> 0:18:46.000 because it's six times six. So in the results, you 0:18:46.040 --> 0:18:50.080 would find that sixty four was significantly easier to identify 0:18:50.160 --> 0:18:54.159 as even than sixty two, So squares tended to be 0:18:54.359 --> 0:18:58.840 very fast. Multiples of four also did really good. For 0:18:58.880 --> 0:19:02.560 some reason, our brains love noticing that multiples of four 0:19:02.680 --> 0:19:06.600 are even. Now, when it came to recognizing odd numbers, 0:19:06.680 --> 0:19:09.480 things got a little more complicated, and the authors say 0:19:09.560 --> 0:19:12.560 that there's a good reason for this. It may have 0:19:12.640 --> 0:19:17.200 to do with multiple hypothesized effects working against one another, 0:19:17.280 --> 0:19:20.639 and these would be number prototypicality on one hand, but 0:19:20.880 --> 0:19:25.639 linguistic markedness on the other. So to refresh. The explanation 0:19:25.760 --> 0:19:29.639 based on linguistic markedness says that because even is an 0:19:29.760 --> 0:19:34.240 unmarked concept and odd is marked, we will usually recognize 0:19:34.320 --> 0:19:38.160 evens faster than odds across the board, and it may 0:19:38.280 --> 0:19:42.240 also possibly mean that numbers that seem odder to us 0:19:42.720 --> 0:19:47.440 will take longer to recognize. So this effect, if present, 0:19:47.520 --> 0:19:51.840 would work in opposite directions depending on parity. For example, 0:19:52.040 --> 0:19:56.080 the super even numerical properties like say being a multiple 0:19:56.119 --> 0:19:59.879 of four, will make a number feel more even, but 0:20:00.040 --> 0:20:03.520 they will also make it easier to process the evenness 0:20:03.520 --> 0:20:06.920 of the number quickly from a linguistic standpoint, because now 0:20:06.960 --> 0:20:10.720 the number is especially unmarked. On the other hand, as 0:20:10.760 --> 0:20:14.359 a number becomes more subjectively odd by say being a 0:20:14.400 --> 0:20:19.000 prime number, the prototypicality explanation would predict that we can 0:20:19.400 --> 0:20:24.679 notice that it's odd faster, but because it's especially numerically odd. 0:20:25.160 --> 0:20:28.720 Working against this would be the linguistic markedness, which might 0:20:28.840 --> 0:20:32.960 predict that the more odd number seems, the more linguistically 0:20:33.040 --> 0:20:36.160 complicated it will feel, and thus the longer our reaction 0:20:36.320 --> 0:20:39.960 time before we can say anything about it. So with evens, 0:20:40.320 --> 0:20:44.040 these two explanations stack, but with odds they work against 0:20:44.119 --> 0:20:47.760 each other and so they said that the results with 0:20:47.880 --> 0:20:51.440 odd numbers were more muddled, but they did find basically 0:20:51.480 --> 0:20:56.200 that primes and numbers divisible by five took the longest 0:20:56.320 --> 0:21:01.040 to classify as odds. Odd squares were the fastest, kind 0:21:01.040 --> 0:21:05.840 of counterintuitively a couple of other results. They also found 0:21:06.080 --> 0:21:10.280 effects from what's called paroity congruity. That's whether the two 0:21:10.359 --> 0:21:13.280 digits in the number are the same parody, so whether 0:21:13.480 --> 0:21:17.960 you know, like sixty eight, they're both even, sixty seven 0:21:18.119 --> 0:21:20.360 one is even and one is odd. That had an effect, 0:21:20.920 --> 0:21:25.480 and also decade magnitude, so how high the first number 0:21:25.480 --> 0:21:28.280 in the pair was had an effect on how long 0:21:28.320 --> 0:21:30.879 it took to process. As it gets bigger, it takes 0:21:30.920 --> 0:21:34.560 longer to think about. They also did find some major 0:21:34.600 --> 0:21:38.399 differences in reaction times by language group. In general, German 0:21:38.480 --> 0:21:42.600 speakers identified two digit numbers as odd or even faster 0:21:42.720 --> 0:21:45.840 than English or Polish speakers, and this could be due 0:21:45.880 --> 0:21:49.679 again to this linguistic inversion principle that you say the 0:21:49.840 --> 0:21:53.240 unit number first when you're speaking German, and the unit 0:21:53.320 --> 0:21:55.439 number is actually all you need to know whether a 0:21:55.520 --> 0:21:58.520 number is odd or even. But anyway, I found this 0:21:58.600 --> 0:22:02.600 whole thing so interesting because it sort of reveals to 0:22:02.640 --> 0:22:08.160 me that while the actual, you know, the mathematical algorithm 0:22:08.720 --> 0:22:12.080 for determining whether a number is even or odd is 0:22:12.440 --> 0:22:18.080 extremely simple, and it's and it's totally binary, and yet 0:22:18.359 --> 0:22:21.480 when we think about it, apparently we must be using 0:22:21.640 --> 0:22:27.240 all these different kind of heuristics and influences and different 0:22:27.320 --> 0:22:30.680 kinds of little rules to make these judgments about numbers 0:22:30.720 --> 0:22:33.320 as fast as we can. And the study did find 0:22:33.359 --> 0:22:35.200 that people get the right answer most of the time, 0:22:35.240 --> 0:22:37.520 and people rarely get it wrong when asked to judge 0:22:37.520 --> 0:22:40.400 whether a number is even or odd, but they're they're 0:22:40.400 --> 0:22:44.680 clearly using like different, little, different little principles are at 0:22:44.720 --> 0:22:47.480 work in helping them get to that answer as fast 0:22:47.520 --> 0:22:51.560 as they can. And some numbers are just easier to 0:22:51.680 --> 0:22:54.640 judge faster than other ones, meaning that they're just more 0:22:54.800 --> 0:22:58.720 represented as as a correct answer within this category than 0:22:58.800 --> 0:23:02.600 others are. And no number in reality is any more 0:23:02.640 --> 0:23:04.439 even or any more odd than another. 0:23:05.080 --> 0:23:07.199 Yeah, I mean, I can't help but think about the 0:23:07.359 --> 0:23:12.000 basic reality of when I'm using real world math, particularly 0:23:12.000 --> 0:23:14.320 say with money, Uh, you know, any amount of money 0:23:14.400 --> 0:23:16.920 is divisible by two, you just get into change. And 0:23:17.040 --> 0:23:19.479 that holds true elsewhere as well. I mean, it's not 0:23:19.600 --> 0:23:23.080 like something that an odd number cannot be split into 0:23:23.359 --> 0:23:27.200 two equal portions. It's it's just it's just you're gonna 0:23:27.200 --> 0:23:28.760 have to go into the decimal points to do so. 0:23:29.119 --> 0:23:31.560 But when you do have to divide an even number 0:23:32.200 --> 0:23:34.960 into in the real world, it does feel like a 0:23:35.000 --> 0:23:38.720 more wholesome act. Yeah, maybe I just hate doing math, 0:23:38.760 --> 0:23:39.800 but that's the way I feel. 0:23:40.400 --> 0:23:42.760 Well, no, no, I see, yeah, what you're saying. I mean, 0:23:42.880 --> 0:23:48.000 so when you're talking about whole number division, obviously dividing 0:23:48.040 --> 0:23:50.359 an even number is you know, you can get to 0:23:50.440 --> 0:23:52.800 an unproblematic answer to that, And if you have an 0:23:52.800 --> 0:23:54.720 odd number, you're going to have a problem. You're gonna 0:23:54.760 --> 0:23:57.120 have to figure out what to do about the fact 0:23:57.160 --> 0:23:59.600 that it doesn't split down the middle correctly. If you're 0:23:59.800 --> 0:24:01.720 dealing with some kind of like whole I don't know, 0:24:01.760 --> 0:24:03.439 if you're trying to figure out how to split the 0:24:03.480 --> 0:24:04.800 three scallops on your plate. 0:24:05.080 --> 0:24:06.480 M yeah, yeah. 0:24:06.520 --> 0:24:08.479 But this also it just makes me think about all 0:24:08.520 --> 0:24:12.119 the ways that you know, you might have categories in 0:24:12.160 --> 0:24:15.800 the real world, whether it's mathematical or whatever, that you 0:24:15.840 --> 0:24:18.600 know are are technically distinct in the way that they 0:24:18.600 --> 0:24:21.719