WEBVTT - Odds and Evens, Part 2 0:00:03.080 --> 0:00:07.080 Welcome to Stuff to Blow Your Mind production of iHeartRadio. 0:00:12.640 --> 0:00:14.760 Hey, welcome to Stuff to Blow Your Mind. My name 0:00:14.800 --> 0:00:15.400 is Robert. 0:00:15.200 --> 0:00:18.560 Lamb, and I am Joe McCormick, and we are back 0:00:18.640 --> 0:00:21.920 with Part two in our series on the psychology and 0:00:21.960 --> 0:00:26.680 cultural significance of number parody p A R I T 0:00:26.960 --> 0:00:30.800 Y parody meaning whether a number is odd or even. 0:00:31.560 --> 0:00:34.640 In Part one, we described the principle of number parody, 0:00:34.760 --> 0:00:38.240 and we talked about evidence that in some cases people 0:00:38.280 --> 0:00:42.360 seem to have surprising feelings about associations with and even 0:00:42.400 --> 0:00:46.519 preferences for odd and even quantities. And so one of 0:00:46.520 --> 0:00:50.280 the big examples we discussed in that first episode was 0:00:50.360 --> 0:00:53.920 the concept in various branches of visual art theory that 0:00:54.000 --> 0:00:57.560 people have a preference for, say, three part divisions of 0:00:57.600 --> 0:01:00.720 imagery over two part divisions, or that people prefer an 0:01:00.720 --> 0:01:04.240 image composed with an odd number of subjects over an 0:01:04.280 --> 0:01:07.440 even number, even to the extent that even numbers of 0:01:07.720 --> 0:01:11.360 subjects will sometimes be subdivided into groups of odd numbers, 0:01:11.560 --> 0:01:14.160 so you know, instead of four subjects, you would get 0:01:14.160 --> 0:01:17.040 a painting with three and one. But we also got 0:01:17.080 --> 0:01:20.000 into a bit of empirical research interrogating these ideas and 0:01:20.080 --> 0:01:23.720 questioning to what extent they're truly natural esthetic preferences. Maybe 0:01:23.720 --> 0:01:27.280 they're just sort of random conventions that people latched onto. 0:01:27.720 --> 0:01:30.119 Including you know, one thing that came up in Part 0:01:30.160 --> 0:01:33.840 one was the domain of food plating and food styling, 0:01:33.880 --> 0:01:36.360 with us just you know, shoot, shooting from the hips 0:01:36.360 --> 0:01:39.920 saying I think three little sliders are better than four. 0:01:40.240 --> 0:01:42.440 We're going to come back to that later today. You 0:01:42.520 --> 0:01:43.640 might be surprised. 0:01:44.840 --> 0:01:47.440 I mean it is still you still see this idea 0:01:47.480 --> 0:01:50.559 out there, but how does it hold up to any 0:01:50.600 --> 0:01:52.760 manner of study. Well, we'll take a look at that. 0:01:53.440 --> 0:01:56.520 So one thing I wanted to talk about today was 0:01:56.680 --> 0:02:03.120 the cognitive psychology of number parity, how we process the 0:02:03.240 --> 0:02:06.760 idea of numbers being odd and even in the brain. 0:02:07.760 --> 0:02:10.480 So I came across a very interesting paper about this 0:02:10.680 --> 0:02:13.480 that was published in the journal Frontiers and Psychology in 0:02:13.520 --> 0:02:16.839 the year twenty eighteen by Hubner at All and it's 0:02:16.880 --> 0:02:21.520 called a mental odd even continuum account some numbers may 0:02:21.560 --> 0:02:25.160 be more odd than others, and some numbers may be 0:02:25.600 --> 0:02:29.279 more even than others. And so if you're not initially 0:02:29.360 --> 0:02:32.440 thrilled about the idea of that, the cognitive psychology of numbers, 0:02:32.480 --> 0:02:36.359 how we represent number properties internally. Stick around. I think 0:02:36.840 --> 0:02:39.919 this might be more interesting than you would at first suspect, 0:02:39.960 --> 0:02:43.200 because it's kind of it kind of reveals deeper ways 0:02:43.240 --> 0:02:45.560 that our brains work in general, at least I think. 0:02:45.600 --> 0:02:47.520 So we can come back to that after we look 0:02:47.560 --> 0:02:50.080 at the findings of the study, But anyway to start 0:02:50.120 --> 0:02:55.840 with the mathematical fact is that number parity is binary. 0:02:56.480 --> 0:03:00.920 In math, natural numbers are either odd or even. Any 0:03:01.000 --> 0:03:04.359 positive integer is even if it can be represented as 0:03:04.440 --> 0:03:08.320 two times in, wherein is also a positive integer, and 0:03:08.360 --> 0:03:11.040 it's odd if it can be represented as two times 0:03:11.040 --> 0:03:15.280 in plus one. All positive whole numbers are either odd 0:03:15.360 --> 0:03:18.480 or even. But this paper is focused not on the 0:03:18.639 --> 0:03:22.359 question of the mathematics of parity, but on the question 0:03:22.480 --> 0:03:26.359 of how number parity is represented in the brain, how 0:03:26.360 --> 0:03:30.440 we think about quantities that are odd and even, And 0:03:30.639 --> 0:03:34.720 the authors propose an interesting hypothesis that people do not 0:03:35.080 --> 0:03:39.080 think about odd and even as a mathematical binary, but 0:03:39.200 --> 0:03:43.440 rather as a spectrum of odd ness and even ness, 0:03:43.560 --> 0:03:46.800 where some numbers can be relatively more odd or even 0:03:46.840 --> 0:03:50.560 than others. And in a kind of amusing aside, the 0:03:50.600 --> 0:03:52.960 author is acknowledge that if this is true, it may 0:03:53.000 --> 0:03:57.040 prove irritating to some researchers, but you know, this is 0:03:57.080 --> 0:03:58.920 the kind of thing I like reading about, because I 0:03:58.920 --> 0:04:03.040 think it's when you observe the mismatch between how a 0:04:03.160 --> 0:04:07.280 concept is technically defined and how we actually think about 0:04:07.320 --> 0:04:10.640 it when we consider it in practice, it's a great 0:04:10.680 --> 0:04:12.320 way to get insights into our brains. 0:04:12.960 --> 0:04:15.600 Yeah. Yeah, And I'm already thinking about thinking about ways 0:04:15.600 --> 0:04:18.719 that I might qualify certain numbers as more even or 0:04:18.760 --> 0:04:20.960 more odd than others. But I want to see where 0:04:20.960 --> 0:04:22.920 you're taking us here and see if any of these 0:04:22.920 --> 0:04:26.360 are are the examples that are coming to my mind. 0:04:26.680 --> 0:04:28.560 So to provide a model for how this would be 0:04:28.600 --> 0:04:32.000 happening in the brain, the authors refer to a psychology 0:04:32.040 --> 0:04:36.560 concept called prototype theory, which has been established going at 0:04:36.640 --> 0:04:40.680 least as far back as the nineteen sixties. As they explain, quote, 0:04:40.920 --> 0:04:45.560 prototype theory has long suggested that certain members of distinct 0:04:45.640 --> 0:04:51.240 categories are more typical examples of that category than others, 0:04:51.520 --> 0:04:56.000 and that membership to such a category may be graded. Now, 0:04:56.160 --> 0:04:58.359 they don't use the following example, and in fact, I 0:04:58.360 --> 0:05:01.080 don't know if this is strictly a perfect example of 0:05:01.080 --> 0:05:03.960 prototype theory, because the category I'm going to use is 0:05:04.000 --> 0:05:07.120 not strictly defined, but I think this will still illustrate it. 0:05:07.680 --> 0:05:13.960 Both Pumpkinhead and Grover from Sesame Street are examples of 0:05:14.000 --> 0:05:20.000 the category monster. And yet while they are undoubtedly both monsters, 0:05:20.040 --> 0:05:22.440 and if you doubt Grover is a monster, go read 0:05:22.520 --> 0:05:25.839 up about them, Grover's a monster, one of them just 0:05:25.880 --> 0:05:29.920 seems like a better example of the category monster than 0:05:29.960 --> 0:05:33.880 the other. Now, there are no real objective criteria for 0:05:33.960 --> 0:05:36.520 what is and is not a monster, but you could 0:05:36.680 --> 0:05:40.480 learn a lot about how people mentally construct the idea 0:05:40.520 --> 0:05:44.520 of a monster by studying how easy it is to 0:05:44.680 --> 0:05:50.560 associate particular examples of creatures with the category monster. And 0:05:50.800 --> 0:05:54.279 one way of studying this would be time latency. So 0:05:54.880 --> 0:05:58.520 imagine you're in a psychological study and you're given a task. 0:05:59.440 --> 0:06:02.400 Somebody's going to show you a series of images of creatures, 0:06:03.080 --> 0:06:05.440 and it's your job to say as quickly as you 0:06:05.480 --> 0:06:08.520 can whether the creature in the image is a monster 0:06:08.720 --> 0:06:12.040 or not. In this kind of test, the speed with 0:06:12.160 --> 0:06:15.680 which you make the categorization could be one piece of 0:06:15.720 --> 0:06:20.760 evidence for how easily you associate the example with the category. 0:06:20.960 --> 0:06:23.559 So even if everybody who takes this kind of test 0:06:23.680 --> 0:06:27.760 correctly recognizes that Grover is a monster. I would still 0:06:27.839 --> 0:06:30.920 bet that on average people would say Pumpkinhead is a 0:06:30.960 --> 0:06:34.359 monster a good bit faster. It just it takes less 0:06:34.440 --> 0:06:37.040 thinking to get there, so you can click the monster 0:06:37.080 --> 0:06:37.880 button faster. 0:06:38.600 --> 0:06:41.520 Yeah, yeah, you don't have to catch yourself and go, oh, well, yes, 0:06:41.560 --> 0:06:43.120 of course he is the monster at the end of 0:06:43.160 --> 0:06:43.520 the book. 0:06:43.760 --> 0:06:46.479 Yeah, exactly. And so with this kind of study you 0:06:46.480 --> 0:06:48.840 could maybe get some insights. For example, you could look 0:06:48.839 --> 0:06:52.599 at these specific attributes that make an individual picture of 0:06:52.600 --> 0:06:56.320 a creature a better prototype example of the monster category 0:06:57.200 --> 0:07:00.000 as measured by people selecting it as a monster faster. 0:07:00.800 --> 0:07:04.479 Maybe maybe creatures that have sharp teeth or claws or 0:07:04.839 --> 0:07:07.360 threatening posture or something like that. It just clicks in 0:07:07.400 --> 0:07:09.480 the brain faster that it's a monster. You got to 0:07:09.520 --> 0:07:12.040 think about it less. And so in this paper, the 0:07:12.080 --> 0:07:16.480 authors do the same thing with odd and even numbers. 0:07:16.520 --> 0:07:19.480 They're going to study the degree to which different numbers 0:07:19.520 --> 0:07:23.400 are prototypes of their parity class, and then they're going 0:07:23.440 --> 0:07:25.720 to try to look for the different factors that make 0:07:25.760 --> 0:07:29.720 a number more easily identifiable as odd or even. And 0:07:29.760 --> 0:07:31.720 this is, by the way, not the first study ever 0:07:31.760 --> 0:07:33.480 to do this. There have been studies in the past 0:07:33.560 --> 0:07:36.920 that have used processing time as a measure of prototypicality 0:07:37.000 --> 0:07:40.160 for odd and even numbers, like they mentioned one study 0:07:40.200 --> 0:07:45.119 that showed six took people longer to classify as even 0:07:45.600 --> 0:07:47.280 than two four or eight did. 0:07:47.800 --> 0:07:48.040 Why. 0:07:48.160 --> 0:07:51.480 I don't know. That's kind of interesting. I mean, two, four, six, 0:07:51.480 --> 0:07:54.920 and eight are all equally even in real mathematics, but 0:07:55.040 --> 0:07:58.440 apparently two four and eight are just easier to identify 0:07:58.480 --> 0:08:00.840 as even something something's a little for about six. 0:08:01.840 --> 0:08:02.880 Huh. Interesting. 0:08:03.520 --> 0:08:05.800 So in their introduction, the authors lay out a bunch 0:08:05.840 --> 0:08:09.840 of different numerical reasons that they think a number might 0:08:09.920 --> 0:08:13.800 be more easily recognizable as even or odd, and the 0:08:14.200 --> 0:08:20.040 hypothetical explanations they include are first of all, ease of divisibility. 0:08:20.480 --> 0:08:23.520 So the easier a number is to divide, the more 0:08:23.720 --> 0:08:27.680 even and less odd it should feel. And this principle 0:08:27.720 --> 0:08:31.280 could subconsciously be applied within the categories and not just 0:08:31.320 --> 0:08:35.520 between them. So twenty five and twenty seven are both odd, 0:08:35.920 --> 0:08:38.760 but the author's idea here is that twenty five may 0:08:38.960 --> 0:08:42.400 feel less odd and take longer to classify as odd 0:08:42.480 --> 0:08:44.040 because it's easy to divide it. 0:08:44.480 --> 0:08:46.840 Now, this is where my mind was headed that. Yeah, 0:08:47.040 --> 0:08:49.840 just thinking about the way I divide numbers is if 0:08:49.880 --> 0:08:52.680 it's easier to divide, then yes, on some level, it 0:08:52.800 --> 0:08:55.840 is more even than an even number that I have 0:08:55.880 --> 0:08:58.400 to sort of like pause a second with then do 0:08:58.480 --> 0:09:00.000 a little extra math in my life. 0:09:00.679 --> 0:09:00.920 Yeah. 0:09:01.960 --> 0:09:03.959 I think that's a strong instinct that they had the 0:09:04.000 --> 0:09:07.480 same idea to begin with. Here. Another thing they hypothesize 0:09:07.480 --> 0:09:10.760 would make a number feel more even is powers of two, 0:09:10.800 --> 0:09:14.079 so that would be two for eight, sixteen, thirty two. 0:09:14.520 --> 0:09:18.920 They think these are cognitively more even. Another factor is 0:09:19.080 --> 0:09:23.280 whether a number is prime. The authors argue that prime 0:09:23.360 --> 0:09:28.920 numbers may feel more odd than non prime odds, and 0:09:29.200 --> 0:09:31.400 one piece of evidence for this is that a couple 0:09:31.440 --> 0:09:34.720 of different previous studies have found that people are quicker 0:09:35.200 --> 0:09:39.439 to flag three, five, and seven as odd than they 0:09:39.480 --> 0:09:42.839 are to flag nine. That's interesting, now, this is kind 0:09:42.840 --> 0:09:46.000 of like the inverse of the six not feeling as 0:09:46.120 --> 0:09:49.800 even as the other even numbers under ten. In this case, apparently, 0:09:49.920 --> 0:09:52.800 maybe nine does not feel as odd as the other 0:09:52.920 --> 0:09:56.640 odd numbers under ten, and the authors argue that this 0:09:56.760 --> 0:09:59.679 may be because the other three odd numbers under ten, three, five, 0:09:59.720 --> 0:10:02.960 and seve are all prime. Nine is not prime. Three 0:10:03.000 --> 0:10:06.160 times three is nine, so the divisibility of it maybe 0:10:06.160 --> 0:10:11.880 makes it feel less odd. The authors also hypothesize maybe 0:10:11.920 --> 0:10:16.439 being part of a standard multiplication table that children memorize 0:10:16.440 --> 0:10:19.680 in school that might make numbers feel more even and 0:10:19.760 --> 0:10:23.160 less odd, But we'll have to look at the results 0:10:23.160 --> 0:10:26.360 and see if that bears out. However, the authors point 0:10:26.400 --> 0:10:29.839 out that previous studies have shown that it is probably 0:10:29.880 --> 0:10:33.880 not only the mathematical properties of a number the number 0:10:33.960 --> 0:10:37.840 properties of a number that influence how long we take 0:10:37.920 --> 0:10:42.679 to make judgments about it. Other factors, such as linguistic factors, 0:10:42.800 --> 0:10:46.080 appear to play a role as well. And illustrate this, 0:10:46.200 --> 0:10:49.760 the authors bring up a really interesting concept that I 0:10:49.800 --> 0:10:52.280 don't think I'd ever read about before, but this really 0:10:52.280 --> 0:10:56.280 stuck with me. So they refer to previous research by 0:10:56.559 --> 0:11:01.120 Hines in the journal Memory and Cognition in nineteen and 0:11:01.200 --> 0:11:05.000 this paper found that if you give people random numbers, 0:11:05.120 --> 0:11:08.679 especially in pairs or in triples, and ask them to 0:11:08.800 --> 0:11:12.280 judge whether the numbers are odd or even, people simply 0:11:12.480 --> 0:11:17.480 take longer to recognize oddness than they do to recognize evenness. 0:11:17.760 --> 0:11:23.120 So odd numbers were just harder to judge overall, so 0:11:23.200 --> 0:11:26.640 people more quickly recognize that fifty two and fifty four 0:11:26.880 --> 0:11:30.680 are even than that fifty three and fifty five are odd. 0:11:31.160 --> 0:11:35.199 Now that's kind of weird, like why would oddness itself 0:11:35.360 --> 0:11:38.720 take longer to process? Pretty much across the board. In 0:11:38.800 --> 0:11:42.040 this older paper, the author argued that part of the 0:11:42.120 --> 0:11:45.040 explanation may lie in the idea of what are called 0:11:45.400 --> 0:11:50.440 marked and unmarked terms in language. Marked and unmarked This 0:11:50.520 --> 0:11:53.720 is a concept in linguistics, and it goes like this, 0:11:54.280 --> 0:11:59.480 So there exist in languages pairs of adjectives that have 0:11:59.600 --> 0:12:05.680 opposite meanings, so long and short, old and young, even 0:12:05.920 --> 0:12:11.160 an odd, alive and dead, things like that. Linguistic markedness 0:12:11.200 --> 0:12:15.199 theory says that usually when you have pairs of adjectives 0:12:15.280 --> 0:12:18.760 like this, one of the terms in the pair is 0:12:18.840 --> 0:12:23.040 treated as the more basic and natural of the two 0:12:23.240 --> 0:12:25.800 in the brain. So we think about one of these 0:12:25.840 --> 0:12:29.079 two terms in a way that what they call they 0:12:29.120 --> 0:12:33.560 call it unmarked. It is the natural state of this measure, 0:12:34.080 --> 0:12:39.880 and then the other term is treated as mentally more complex, complicated, 0:12:39.960 --> 0:12:43.440 and unnatural. This is the marked word in the pair, 0:12:44.240 --> 0:12:46.840 and there are experiments that will show this. But the 0:12:46.960 --> 0:12:51.559 unmarked word in the pair, for example, is used more 0:12:51.600 --> 0:12:56.439 frequently than the marked word. It's learned earlier in language acquisition, 0:12:56.480 --> 0:12:59.320 when you're a child, and it is considered usually the 0:12:59.480 --> 0:13:03.880 default to measure. So, for example, you say how old 0:13:04.040 --> 0:13:07.520 are you, not how young are you? Because in old 0:13:07.559 --> 0:13:11.120 and young, old is treated as the unmarked word and 0:13:11.200 --> 0:13:15.200 young is the marked concept. Similarly, you will say how 0:13:15.320 --> 0:13:18.520 long will it take? Not how short will it take? 0:13:19.000 --> 0:13:21.320 I thought this was interesting. They say also that in 0:13:21.320 --> 0:13:24.440 some cases you can create the same meaning as the 0:13:24.480 --> 0:13:28.120 marked word simply by adding a negative prefix to the 0:13:28.280 --> 0:13:32.000 unmarked word. So you can say uneven to mean the 0:13:32.040 --> 0:13:35.560 same thing as odd, but nobody says un odd to 0:13:35.640 --> 0:13:36.240 mean even. 0:13:36.920 --> 0:13:38.400 Oh, that's true. That's a great point. 0:13:38.720 --> 0:13:42.000 Now, whatever this division between marked and unmarked comes from, 0:13:42.400 --> 0:13:46.560 it seems that it results in different processing times in 0:13:46.600 --> 0:13:51.640 the brain. That we just deal with unmarked concepts faster 0:13:51.840 --> 0:13:54.840 and more easily, and it takes us, you know, maybe 0:13:54.880 --> 0:13:58.320 a split second longer to think about, or deliver or 0:13:58.400 --> 0:14:02.480 deal with a marked concept. And so if even is 0:14:02.640 --> 0:14:05.960 unmarked and odd is marked, it may in fact be 0:14:06.240 --> 0:14:09.360 that we just deal with the concept of evenness a 0:14:09.360 --> 0:14:11.960 little bit more easily in the brain than oddness. It's 0:14:12.120 --> 0:14:15.640 oddness is linguistically marked, and so it takes us a 0:14:15.679 --> 0:14:19.600 split second longer to kind of process this concept whenever 0:14:19.640 --> 0:14:22.000 we're dealing with it or producing a judgment about it, 0:14:22.400 --> 0:14:24.400 And this may play a role in explaining the so 0:14:24.480 --> 0:14:28.200 called odd effect that was discovered in this paper in 0:14:28.200 --> 0:14:40.840 the nineties. Moving on from that, there's another linguistic effect 0:14:40.880 --> 0:14:44.920 that actually shows up when you compare judgments about parody 0:14:44.920 --> 0:14:49.800 across different languages, and this is the inversion property of 0:14:49.960 --> 0:14:53.920 multiple digit numbers. So in English, when we want to 0:14:54.120 --> 0:14:56.680 say or write out in words the number that is 0:14:56.760 --> 0:15:00.120 one quarter of one hundred, we say twenty five, we 0:15:00.160 --> 0:15:03.480 write the twenty first and then the five, or we 0:15:03.520 --> 0:15:06.000 say the twenty first and then the five. So for 0:15:06.040 --> 0:15:10.000 two digit numbers, it's always the decade digit first in language, 0:15:10.080 --> 0:15:12.960 and then the unit digit. But not all languages work 0:15:13.000 --> 0:15:16.840 this way. For example, in German, twenty five is and 0:15:16.880 --> 0:15:19.240 I'm sorry, I'm sure i'm pronouncing this wrong. It is 0:15:19.280 --> 0:15:25.720 something like fun fundzwanzig, meaning five and twenty. And this 0:15:25.800 --> 0:15:28.520 has been found to have all sorts of interesting effects 0:15:28.520 --> 0:15:33.400 on number cognition. For example, German speakers are studies have 0:15:33.440 --> 0:15:37.520 shown more likely to make trans coding errors when writing 0:15:37.640 --> 0:15:41.760 numbers out, so more likely to write fifty two when 0:15:41.800 --> 0:15:46.200 they mean twenty five. In terms of digits, Also, compared 0:15:46.240 --> 0:15:50.960 to non inverted languages, German speakers pay relatively more attention 0:15:51.200 --> 0:15:55.040 to the unit digit in a multi digit number, and 0:15:55.080 --> 0:15:58.480 so the authors write quote. This prioritizing of either the 0:15:58.640 --> 0:16:02.600 unit or decade digit might influence participants' performance in number 0:16:02.640 --> 0:16:06.680 processing tasks in which units play a decisive role. Parity 0:16:06.760 --> 0:16:09.960 judgment is clearly one of those tasks, because only the 0:16:10.080 --> 0:16:14.320 unit parity is relevant for answering correctly, which is true 0:16:14.320 --> 0:16:16.280 when you look at you can judge whether it's odd 0:16:16.320 --> 0:16:19.000 or even without knowing any of the numbers before the 0:16:19.080 --> 0:16:21.920 last one. And just a couple of other factors the 0:16:21.960 --> 0:16:26.600 authors mention that have been possibly shown to influence parity judgments. 0:16:27.520 --> 0:16:31.640 Larger numbers may cause longer processing times, regardless of the 0:16:32.080 --> 0:16:34.480 parity or any other facts about them. Is just like 0:16:34.520 --> 0:16:36.400 the bigger the number is, the longer you have to 0:16:36.400 --> 0:16:40.560 think about it. Also, word frequency, numbers that appear more 0:16:40.600 --> 0:16:43.680 often in language get faster responses, and this is not 0:16:43.720 --> 0:16:46.520 just true of numbers any words in general that are 0:16:46.600 --> 0:16:51.400 used more often are processed more efficiently, So this study 0:16:51.440 --> 0:16:55.160 tried to test the relative influence of number prototypicality and 0:16:55.400 --> 0:16:58.840 the linguistic factors we were just talking about. And the 0:16:58.840 --> 0:17:02.440 way they did this was by getting a group of 0:17:02.480 --> 0:17:06.640 subjects and giving them auditory prompts of numbers between twenty 0:17:06.680 --> 0:17:09.760 and ninety nine, and then they would try to analyze 0:17:09.800 --> 0:17:12.399 how long it took people to classify these numbers as 0:17:12.480 --> 0:17:16.399 odd or even to test the linguistic factors. The author's 0:17:16.480 --> 0:17:20.640 recruited subjects from three different language groups. They had English speakers, 0:17:20.680 --> 0:17:25.200 German speakers, and Polish speakers. In Polish, two digit numbers 0:17:25.240 --> 0:17:29.040 are expressed with the decade digit first, like in English. 0:17:29.080 --> 0:17:31.280 And I'm not going to discuss all of their findings, 0:17:31.280 --> 0:17:34.120 but just to summarize and pick a few highlights, they 0:17:34.160 --> 0:17:38.240 do say that quote. Overall, the results suggest that perceived 0:17:38.359 --> 0:17:41.800 paroity is not the same as objective paroity, and some 0:17:42.000 --> 0:17:47.680 numbers are more prototypical exemplars of their categories. And specifically, 0:17:48.040 --> 0:17:52.960 with regards to these mathematical or numerical factors influencing things, 0:17:53.200 --> 0:17:57.119 they found that some but not all, of the characteristics 0:17:57.160 --> 0:18:02.080 they hypothesized actually did play a role imperceived paroity. So, 0:18:02.160 --> 0:18:06.479 for evens. The numbers that people identified as even the 0:18:06.560 --> 0:18:11.399 fastest tended to be even squares, so a square being 0:18:11.480 --> 0:18:14.840 the product of a number multiplied by itself. Sixteen is 0:18:14.880 --> 0:18:17.760 a square because it's four times four, sixty four is 0:18:17.800 --> 0:18:21.200 a square because it's eight times eight. Thirty six is 0:18:21.240 --> 0:18:23.919 a square because it's six times six. So in the 0:18:23.960 --> 0:18:28.240 results you would find that sixty four was significantly easier 0:18:28.280 --> 0:18:32.600 to identify as even than sixty two, so squares tended 0:18:32.640 --> 0:18:36.920 to be very fast. Multiples of four also did really good. 0:18:37.560 --> 0:18:41.080 For some reason, our brains love noticing that multiples of 0:18:41.160 --> 0:18:45.440 four are even. Now, when it came to recognizing odd numbers, 0:18:45.520 --> 0:18:48.320 things got a little more complicated, and the authors say 0:18:48.400 --> 0:18:51.400 that there's a good reason for this. It may have 0:18:51.480 --> 0:18:56.040 to do with multiple hypothesized effects working against one another, 0:18:56.119 --> 0:18:59.439 and these would be number prototypicality on one hand, but 0:18:59.640 --> 0:19:04.480 linguistic markedness on the other. So, to refresh the explanation 0:19:04.600 --> 0:19:08.479 based on linguistic markedness, says that because even is an 0:19:08.600 --> 0:19:13.080 unmarked concept and odd is marked, we will usually recognize 0:19:13.160 --> 0:19:17.000 evens faster than odds across the board. And it may 0:19:17.119 --> 0:19:21.040 also possibly mean that numbers that seem odder to us 0:19:21.560 --> 0:19:26.320 will take longer to recognize. So this effect, if present, 0:19:26.359 --> 0:19:30.680 would work in opposite directions depending on parity. For example, 0:19:30.880 --> 0:19:34.920 the super even numerical properties like say being a multiple 0:19:34.960 --> 0:19:38.760 of four, will make a number feel more even, but 0:19:38.800 --> 0:19:42.359 they will also make it easier to process the evenness 0:19:42.359 --> 0:19:45.760 of the number quickly from a linguistic standpoint, because now 0:19:45.800 --> 0:19:49.560 the number is especially unmarked. On the other hand, as 0:19:49.600 --> 0:19:53.159 a number becomes more subjectively odd by say being a 0:19:53.240 --> 0:19:57.840 prime number, the prototypicality explanation would predict that we can 0:19:58.240 --> 0:20:03.520 notice that it's odd faster, but because it's especially numerically odd. 0:20:04.000 --> 0:20:07.560 Working against this would be the linguistic markedness, which might 0:20:07.680 --> 0:20:11.840 predict that the more odd number seems, the more linguistically 0:20:11.880 --> 0:20:15.000 complicated it will feel, and thus the longer our reaction 0:20:15.160 --> 0:20:18.800 time before we can say anything about it. So with evens, 0:20:19.160 --> 0:20:22.960 these two explanations stack, but with odds they work against 0:20:22.960 --> 0:20:26.600 each other. And so they said that the results with 0:20:26.720 --> 0:20:30.280 odd numbers were more muddled. But they did find basically 0:20:30.320 --> 0:20:35.040 that primes and numbers divisible by five took the longest 0:20:35.160 --> 0:20:39.880 to classify as odds. Odd squares were the fastest. Kind 0:20:39.920 --> 0:20:44.679 of counterintuitively, a couple of other results They also found 0:20:44.920 --> 0:20:49.120 effects from what's called parody congruity. That's whether the two 0:20:49.200 --> 0:20:52.119 digits in the number are the same parody, so whether 0:20:52.320 --> 0:20:56.840 you know, like sixty eight, they're both even, sixty seven 0:20:56.960 --> 0:20:59.240 one is even and one is odd. That had an effect, 0:20:59.760 --> 0:21:03.879 and also decade magnitude, so the how high the first 0:21:04.000 --> 0:21:06.960 number in the pair was had an effect on how 0:21:06.960 --> 0:21:09.399 long it took to process. As it gets bigger, it 0:21:09.440 --> 0:21:12.720 takes longer to think about. They also did find some 0:21:13.119 --> 0:21:16.800 major differences in reaction times by language group. In general, 0:21:16.880 --> 0:21:20.920 German speakers identified two digit numbers as odd or even 0:21:21.080 --> 0:21:24.439 faster than English or Polish speakers, and this could be 0:21:24.520 --> 0:21:28.359 due again to this linguistic inversion principle that you say 0:21:28.440 --> 0:21:31.760 the unit number first when you're speaking German, and the 0:21:31.840 --> 0:21:34.200 unit number is actually all you need to know whether 0:21:34.200 --> 0:21:37.200 a number is odd or even. But anyway, I found 0:21:37.280 --> 0:21:41.320 this whole thing so interesting because it sort of reveals 0:21:41.359 --> 0:21:46.280 to me that while the actual, you know, the mathematical 0:21:46.359 --> 0:21:50.280 algorithm for determining whether a number is even or odd 0:21:50.880 --> 0:21:57.440 is extremely simple and it's totally binary, and yet when 0:21:57.480 --> 0:22:00.680 we think about it, apparently we must be using all 0:22:00.760 --> 0:22:06.440 these different kind of heuristics and influences and different kinds 0:22:06.440 --> 0:22:09.760 of little rules to make these judgments about numbers as 0:22:09.800 --> 0:22:12.280 fast as we can. And the study did find that 0:22:12.320 --> 0:22:14.159 people get the right answer most of the time, and 0:22:14.200 --> 0:22:16.560 people rarely get it wrong when asked to judge whether 0:22:16.600 --> 0:22:19.640 a number is even or odd. But they're they're clearly 0:22:19.800 --> 0:22:23.840 using like different little, different little principles are at work 0:22:23.960 --> 0:22:26.520 in helping them get to that answer as fast as 0:22:26.520 --> 0:22:30.920 they can. And some numbers are just easier to judge 0:22:30.960 --> 0:22:34.320 faster than other ones, meaning that they're just more represented 0:22:34.400 --> 0:22:38.200 as a correct answer within this category than others are. 0:22:38.680 --> 0:22:41.960 And no number in reality is any more even or 0:22:42.000 --> 0:22:43.280 any more odd than another. 0:22:43.920 --> 0:22:46.080 Yeah, I mean, I can't help but think about the 0:22:46.200 --> 0:22:50.840 basic reality of when I'm using real world math, particularly 0:22:50.880 --> 0:22:53.320 say with money. You know, any amount of money is 0:22:53.359 --> 0:22:56.000 divisible by two, you just get into change, And that 0:22:56.040 --> 0:22:58.520 holds true elsewhere as well. I mean, it's not like 0:22:58.960 --> 0:23:03.560 an odd number cannot be split into two equal portions. 0:23:03.880 --> 0:23:06.159 It's it's just it's just you're going to have to 0:23:06.200 --> 0:23:08.280 go into the decimal points to do so. But when 0:23:08.280 --> 0:23:11.800 you do have to divide an even number into in 0:23:11.840 --> 0:23:16.040 the real world, it does feel like a more wholesome act. Yeah, 0:23:16.280 --> 0:23:18.200 maybe I just hate doing math, but that's the way 0:23:18.280 --> 0:23:18.640 I feel. 0:23:19.240 --> 0:23:21.600 Well no, no, I see, yeah, what you're saying. I mean, 0:23:21.720 --> 0:23:26.840 so when you're talking about whole number division, obviously dividing 0:23:26.880 --> 0:23:29.199 an even number is you know, you can get to 0:23:29.280 --> 0:23:31.639 an unproblematic answer to that, and if you have an 0:23:31.640 --> 0:23:33.439 odd number, you're going to have a problem. You're going 0:23:33.480 --> 0:23:35.760 to have to figure out what to do about the 0:23:35.760 --> 0:23:38.280 fact that it doesn't split down the middle correctly. If 0:23:38.320 --> 0:23:40.320 you're you're dealing with some kind of like whole I 0:23:40.320 --> 0:23:41.879 don't know, if you're trying to figure out how to 0:23:41.880 --> 0:23:43.639 split the three scallops on your plate. 0:23:43.920 --> 0:23:44.360 Mm hmm. 0:23:44.480 --> 0:23:44.720 Yeah. 0:23:44.920 --> 0:23:45.320 Yeah. 0:23:45.359 --> 0:23:47.360 But this also it just makes me think about all 0:23:47.359 --> 0:23:50.960 the ways that you know, you might have categories in 0:23:51.000 --> 0:23:54.639 the real world, whether it's mathematical or whatever, that you 0:23:54.680 --> 0:23:57.439 know are are technically distinct in the way that they 0:23:57.440 --> 0:24:00.560 are defined, and yet our brains are just not going 0:24:00.600 --> 0:24:04.040