September 10, 2024 • 44 mins
In this episode of Stuff to Blow Your Mind, Robert and Joe explore the world of odd and even numbers. How does it factor into our psychology, our art and our culture? Find out…
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Speaker 1 (00:03):
Welcome to Stuff to Blow Your Mind production of iHeartRadio.
Speaker 2 (00:12):
Hey, welcome to Stuff to Blow Your Mind. My name
is Robert.
Speaker 3 (00:15):
Lamb, and I am Joe McCormick, and we are back
with Part two in our series on the psychology and
cultural significance of number parody p A R I T
Y parody meaning whether a number is odd or even.
In Part one, we described the principle of number parody,
and we talked about evidence that in some cases people
(00:38):
seem to have surprising feelings about associations with and even
preferences for odd and even quantities. And so one of
the big examples we discussed in that first episode was
the concept in various branches of visual art theory that
people have a preference for, say, three part divisions of
imagery over two part divisions, or that people prefer an
(01:00):
image composed with an odd number of subjects over an
even number, even to the extent that even numbers of
subjects will sometimes be subdivided into groups of odd numbers,
so you know, instead of four subjects, you would get
a painting with three and one. But we also got
into a bit of empirical research interrogating these ideas and
questioning to what extent they're truly natural esthetic preferences. Maybe
(01:23):
they're just sort of random conventions that people latched onto.
Including you know, one thing that came up in Part
one was the domain of food plating and food styling,
with us just you know, shoot, shooting from the hips
saying I think three little sliders are better than four.
We're going to come back to that later today. You
might be surprised.
Speaker 2 (01:44):
I mean it is still you still see this idea
out there, but how does it hold up to any
manner of study. Well, we'll take a look at that.
Speaker 3 (01:53):
So one thing I wanted to talk about today was
the cognitive psychology of number parity, how we process the
idea of numbers being odd and even in the brain.
So I came across a very interesting paper about this
that was published in the journal Frontiers and Psychology in
(02:13):
the year twenty eighteen by Hubner at All and it's
called a mental odd even continuum account some numbers may
be more odd than others, and some numbers may be
more even than others. And so if you're not initially
thrilled about the idea of that, the cognitive psychology of numbers,
how we represent number properties internally. Stick around. I think
(02:36):
this might be more interesting than you would at first suspect,
because it's kind of it kind of reveals deeper ways
that our brains work in general, at least I think.
So we can come back to that after we look
at the findings of the study, But anyway to start
with the mathematical fact is that number parity is binary.
In math, natural numbers are either odd or even. Any
(03:01):
positive integer is even if it can be represented as
two times in, wherein is also a positive integer, and
it's odd if it can be represented as two times
in plus one. All positive whole numbers are either odd
or even. But this paper is focused not on the
question of the mathematics of parity, but on the question
(03:22):
of how number parity is represented in the brain, how
we think about quantities that are odd and even, And
the authors propose an interesting hypothesis that people do not
think about odd and even as a mathematical binary, but
rather as a spectrum of odd ness and even ness,
(03:43):
where some numbers can be relatively more odd or even
than others. And in a kind of amusing aside, the
author is acknowledge that if this is true, it may
prove irritating to some researchers, but you know, this is
the kind of thing I like reading about, because I
think it's when you observe the mismatch between how a
concept is technically defined and how we actually think about
(04:07):
it when we consider it in practice, it's a great
way to get insights into our brains.
Speaker 2 (04:12):
Yeah. Yeah, And I'm already thinking about thinking about ways
that I might qualify certain numbers as more even or
more odd than others. But I want to see where
you're taking us here and see if any of these
are are the examples that are coming to my mind.
Speaker 3 (04:26):
So to provide a model for how this would be
happening in the brain, the authors refer to a psychology
concept called prototype theory, which has been established going at
least as far back as the nineteen sixties. As they explain, quote,
prototype theory has long suggested that certain members of distinct
categories are more typical examples of that category than others,
(04:51):
and that membership to such a category may be graded. Now,
they don't use the following example, and in fact, I
don't know if this is strictly a perfect example of
prototype theory, because the category I'm going to use is
not strictly defined, but I think this will still illustrate it.
Both Pumpkinhead and Grover from Sesame Street are examples of
(05:14):
the category monster. And yet while they are undoubtedly both monsters,
and if you doubt Grover is a monster, go read
up about them, Grover's a monster, one of them just
seems like a better example of the category monster than
the other. Now, there are no real objective criteria for
what is and is not a monster, but you could
(05:36):
learn a lot about how people mentally construct the idea
of a monster by studying how easy it is to
associate particular examples of creatures with the category monster. And
one way of studying this would be time latency. So
imagine you're in a psychological study and you're given a task.
(05:59):
Somebody's going to show you a series of images of creatures,
and it's your job to say as quickly as you
can whether the creature in the image is a monster
or not. In this kind of test, the speed with
which you make the categorization could be one piece of
evidence for how easily you associate the example with the category.
(06:20):
So even if everybody who takes this kind of test
correctly recognizes that Grover is a monster. I would still
bet that on average people would say Pumpkinhead is a
monster a good bit faster. It just it takes less
thinking to get there, so you can click the monster
button faster.
Speaker 2 (06:38):
Yeah, yeah, you don't have to catch yourself and go, oh, well, yes,
of course he is the monster at the end of
the book.
Speaker 3 (06:43):
Yeah, exactly. And so with this kind of study you
could maybe get some insights. For example, you could look
at these specific attributes that make an individual picture of
a creature a better prototype example of the monster category
as measured by people selecting it as a monster faster.
Maybe maybe creatures that have sharp teeth or claws or
(07:04):
threatening posture or something like that. It just clicks in
the brain faster that it's a monster. You got to
think about it less. And so in this paper, the
authors do the same thing with odd and even numbers.
They're going to study the degree to which different numbers
are prototypes of their parity class, and then they're going
to try to look for the different factors that make
(07:25):
a number more easily identifiable as odd or even. And
this is, by the way, not the first study ever
to do this. There have been studies in the past
that have used processing time as a measure of prototypicality
for odd and even numbers, like they mentioned one study
that showed six took people longer to classify as even
than two four or eight did.
Speaker 2 (07:47):
Why.
Speaker 3 (07:48):
I don't know. That's kind of interesting. I mean, two, four, six,
and eight are all equally even in real mathematics, but
apparently two four and eight are just easier to identify
as even something something's a little for about six.
Speaker 2 (08:01):
Huh. Interesting.
Speaker 3 (08:03):
So in their introduction, the authors lay out a bunch
of different numerical reasons that they think a number might
be more easily recognizable as even or odd, and the
hypothetical explanations they include are first of all, ease of divisibility.
So the easier a number is to divide, the more
(08:23):
even and less odd it should feel. And this principle
could subconsciously be applied within the categories and not just
between them. So twenty five and twenty seven are both odd,
but the author's idea here is that twenty five may
feel less odd and take longer to classify as odd
because it's easy to divide it.
Speaker 2 (08:44):
Now, this is where my mind was headed that. Yeah,
just thinking about the way I divide numbers is if
it's easier to divide, then yes, on some level, it
is more even than an even number that I have
to sort of like pause a second with then do
a little extra math in my life.
Speaker 4 (09:00):
Yeah.
Speaker 3 (09:01):
I think that's a strong instinct that they had the
same idea to begin with. Here. Another thing they hypothesize
would make a number feel more even is powers of two,
so that would be two for eight, sixteen, thirty two.
They think these are cognitively more even. Another factor is
whether a number is prime. The authors argue that prime
(09:23):
numbers may feel more odd than non prime odds, and
one piece of evidence for this is that a couple
of different previous studies have found that people are quicker
to flag three, five, and seven as odd than they
are to flag nine. That's interesting, now, this is kind
of like the inverse of the six not feeling as
(09:46):
even as the other even numbers under ten. In this case, apparently,
maybe nine does not feel as odd as the other
odd numbers under ten, and the authors argue that this
may be because the other three odd numbers under ten, three, five,
and seve are all prime. Nine is not prime. Three
times three is nine, so the divisibility of it maybe
(10:06):
makes it feel less odd. The authors also hypothesize maybe
being part of a standard multiplication table that children memorize
in school that might make numbers feel more even and
less odd, But we'll have to look at the results
and see if that bears out. However, the authors point
(10:26):
out that previous studies have shown that it is probably
not only the mathematical properties of a number the number
properties of a number that influence how long we take
to make judgments about it. Other factors, such as linguistic factors,
appear to play a role as well. And illustrate this,
the authors bring up a really interesting concept that I
(10:49):
don't think I'd ever read about before, but this really
stuck with me. So they refer to previous research by
Hines in the journal Memory and Cognition in nineteen and
this paper found that if you give people random numbers,
especially in pairs or in triples, and ask them to
judge whether the numbers are odd or even, people simply
(11:12):
take longer to recognize oddness than they do to recognize evenness.
So odd numbers were just harder to judge overall, so
people more quickly recognize that fifty two and fifty four
are even than that fifty three and fifty five are odd.
Now that's kind of weird, like why would oddness itself
(11:35):
take longer to process? Pretty much across the board. In
this older paper, the author argued that part of the
explanation may lie in the idea of what are called
marked and unmarked terms in language. Marked and unmarked This
is a concept in linguistics, and it goes like this,
So there exist in languages pairs of adjectives that have
(11:59):
opposite meanings, so long and short, old and young, even
an odd, alive and dead, things like that. Linguistic markedness
theory says that usually when you have pairs of adjectives
like this, one of the terms in the pair is
treated as the more basic and natural of the two
(12:23):
in the brain. So we think about one of these
two terms in a way that what they call they
call it unmarked. It is the natural state of this measure,
and then the other term is treated as mentally more complex, complicated,
and unnatural. This is the marked word in the pair,
(12:44):
and there are experiments that will show this. But the
unmarked word in the pair, for example, is used more
frequently than the marked word. It's learned earlier in language acquisition,
when you're a child, and it is considered usually the
default to measure. So, for example, you say how old
are you, not how young are you? Because in old
(13:07):
and young, old is treated as the unmarked word and
young is the marked concept. Similarly, you will say how
long will it take? Not how short will it take?
I thought this was interesting. They say also that in
some cases you can create the same meaning as the
marked word simply by adding a negative prefix to the
(13:28):
unmarked word. So you can say uneven to mean the
same thing as odd, but nobody says un odd to
mean even.
Speaker 2 (13:36):
Oh, that's true. That's a great point.
Speaker 3 (13:38):
Now, whatever this division between marked and unmarked comes from,
it seems that it results in different processing times in
the brain. That we just deal with unmarked concepts faster
and more easily, and it takes us, you know, maybe
a split second longer to think about, or deliver or
deal with a marked concept. And so if even is
(14:02):
unmarked and odd is marked, it may in fact be
that we just deal with the concept of evenness a
little bit more easily in the brain than oddness. It's
oddness is linguistically marked, and so it takes us a
split second longer to kind of process this concept whenever
we're dealing with it or producing a judgment about it,
And this may play a role in explaining the so
(14:24):
called odd effect that was discovered in this paper in
the nineties. Moving on from that, there's another linguistic effect
that actually shows up when you compare judgments about parody
(14:44):
across different languages, and this is the inversion property of
multiple digit numbers. So in English, when we want to
say or write out in words the number that is
one quarter of one hundred, we say twenty five, we
write the twenty first and then the five, or we
say the twenty first and then the five. So for
(15:06):
two digit numbers, it's always the decade digit first in language,
and then the unit digit. But not all languages work
this way. For example, in German, twenty five is and
I'm sorry, I'm sure i'm pronouncing this wrong. It is
something like fun fundzwanzig, meaning five and twenty. And this
has been found to have all sorts of interesting effects
(15:28):
on number cognition. For example, German speakers are studies have
shown more likely to make trans coding errors when writing
numbers out, so more likely to write fifty two when
they mean twenty five. In terms of digits, Also, compared
to non inverted languages, German speakers pay relatively more attention
(15:51):
to the unit digit in a multi digit number, and
so the authors write quote. This prioritizing of either the
unit or decade digit might influence participants' performance in number
processing tasks in which units play a decisive role. Parity
judgment is clearly one of those tasks, because only the
unit parity is relevant for answering correctly, which is true
(16:14):
when you look at you can judge whether it's odd
or even without knowing any of the numbers before the
last one. And just a couple of other factors the
authors mention that have been possibly shown to influence parity judgments.
Larger numbers may cause longer processing times, regardless of the
parity or any other facts about them. Is just like
(16:34):
the bigger the number is, the longer you have to
think about it. Also, word frequency, numbers that appear more
often in language get faster responses, and this is not
just true of numbers any words in general that are
used more often are processed more efficiently, So this study
tried to test the relative influence of number prototypicality and
(16:55):
the linguistic factors we were just talking about. And the
way they did this was by getting a group of
subjects and giving them auditory prompts of numbers between twenty
and ninety nine, and then they would try to analyze
how long it took people to classify these numbers as
odd or even to test the linguistic factors. The author's
(17:16):
recruited subjects from three different language groups. They had English speakers,
German speakers, and Polish speakers. In Polish, two digit numbers
are expressed with the decade digit first, like in English.
And I'm not going to discuss all of their findings,
but just to summarize and pick a few highlights, they
do say that quote. Overall, the results suggest that perceived
(17:38):
paroity is not the same as objective paroity, and some
numbers are more prototypical exemplars of their categories. And specifically,
with regards to these mathematical or numerical factors influencing things,
they found that some but not all, of the characteristics
they hypothesized actually did play a role imperceived paroity. So,
(18:02):
for evens. The numbers that people identified as even the
fastest tended to be even squares, so a square being
the product of a number multiplied by itself. Sixteen is
a square because it's four times four, sixty four is
a square because it's eight times eight. Thirty six is
a square because it's six times six. So in the
(18:23):
results you would find that sixty four was significantly easier
to identify as even than sixty two, so squares tended
to be very fast. Multiples of four also did really good.
For some reason, our brains love noticing that multiples of
four are even. Now, when it came to recognizing odd numbers,
(18:45):
things got a little more complicated, and the authors say
that there's a good reason for this. It may have
to do with multiple hypothesized effects working against one another,
and these would be number prototypicality on one hand, but
linguistic markedness on the other. So, to refresh the explanation
based on linguistic markedness, says that because even is an
(19:08):
unmarked concept and odd is marked, we will usually recognize
evens faster than odds across the board. And it may
also possibly mean that numbers that seem odder to us
will take longer to recognize. So this effect, if present,
would work in opposite directions depending on parity. For example,
(19:30):
the super even numerical properties like say being a multiple
of four, will make a number feel more even, but
they will also make it easier to process the evenness
of the number quickly from a linguistic standpoint, because now
the number is especially unmarked. On the other hand, as
a number becomes more subjectively odd by say being a
(19:53):
prime number, the prototypicality explanation would predict that we can
notice that it's odd faster, but because it's especially numerically odd.
Working against this would be the linguistic markedness, which might
predict that the more odd number seems, the more linguistically
complicated it will feel, and thus the longer our reaction
(20:15):
time before we can say anything about it. So with evens,
these two explanations stack, but with odds they work against
each other. And so they said that the results with
odd numbers were more muddled. But they did find basically
that primes and numbers divisible by five took the longest
to classify as odds. Odd squares were the fastest. Kind
(20:39):
of counterintuitively, a couple of other results They also found
effects from what's called parody congruity. That's whether the two
digits in the number are the same parody, so whether
you know, like sixty eight, they're both even, sixty seven
one is even and one is odd. That had an effect,
and also decade magnitude, so the how high the first
(21:04):
number in the pair was had an effect on how
long it took to process. As it gets bigger, it
takes longer to think about. They also did find some
major differences in reaction times by language group. In general,
German speakers identified two digit numbers as odd or even
faster than English or Polish speakers, and this could be
(21:24):
due again to this linguistic inversion principle that you say
the unit number first when you're speaking German, and the
unit number is actually all you need to know whether
a number is odd or even. But anyway, I found
this whole thing so interesting because it sort of reveals
to me that while the actual, you know, the mathematical
(21:46):
algorithm for determining whether a number is even or odd
is extremely simple and it's totally binary, and yet when
we think about it, apparently we must be using all
these different kind of heuristics and influences and different kinds
(22:06):
of little rules to make these judgments about numbers as
fast as we can. And the study did find that
people get the right answer most of the time, and
people rarely get it wrong when asked to judge whether
a number is even or odd. But they're they're clearly
using like different little, different little principles are at work
in helping them get to that answer as fast as
(22:26):
they can. And some numbers are just easier to judge
faster than other ones, meaning that they're just more represented
as a correct answer within this category than others are.
And no number in reality is any more even or
any more odd than another.
Speaker 2 (22:43):
Yeah, I mean, I can't help but think about the
basic reality of when I'm using real world math, particularly
say with money. You know, any amount of money is
divisible by two, you just get into change, And that
holds true elsewhere as well. I mean, it's not like
an odd number cannot be split into two equal portions.
It's it's just it's just you're going to have to
(23:06):
go into the decimal points to do so. But when
you do have to divide an even number into in
the real world, it does feel like a more wholesome act. Yeah,
maybe I just hate doing math, but that's the way
I feel.
Speaker 3 (23:19):
Well no, no, I see, yeah, what you're saying. I mean,
so when you're talking about whole number division, obviously dividing
an even number is you know, you can get to
an unproblematic answer to that, and if you have an
odd number, you're going to have a problem. You're going
to have to figure out what to do about the
fact that it doesn't split down the middle correctly. If
you're you're dealing with some kind of like whole I
(23:40):
don't know, if you're trying to figure out how to
split the three scallops on your plate.
Speaker 2 (23:43):
Mm hmm.
Speaker 3 (23:44):
Yeah.
Speaker 2 (23:44):
Yeah.
Speaker 3 (23:45):
But this also it just makes me think about all
the ways that you know, you might have categories in
the real world, whether it's mathematical or whatever, that you
know are are technically distinct in the way that they
are defined, and yet our brains are just not going
to be bound by that for having like strict inclusion criteria. Well,
like we'll get into these like ways of thinking about
(24:06):
it as some kind of gradient, and that's just kind
of interesting that we tend to work that way.
Speaker 2 (24:11):
Yeah. Yeah, Like now that I think about it. I'm
pretty sure that five and seven especially are just like
disgustingly odd, you know. Oh okay, I mean it gets
more disgusting the more sevens you have. I guess, like
like seventy seven, seven hundred and seventy seven. Just I
don't even want to think about those.
Speaker 3 (24:29):
Oh, that's starting to make me think about the stacking
of sevens in the Bible.
Speaker 1 (24:33):
You know.
Speaker 3 (24:34):
Sometimes they really like to get into the There will
be like seven seven seven of something that they're seventy
seven of on the seventh day.
Speaker 2 (24:41):
Yeah, I mean it kind of gets into the know,
the idea of something Okay, well, you know it's not
easily divisible. I guess it's you know, it's more solid,
it's more holy in that regard. It depends on how
you want to spend all right, now, it's time to
come back to the idea of three sliders on a plate,
(25:06):
the supposed rule of odds. So in part one I
mentioned the rule of odds and visual composition, and yeah,
I want to come back and discuss it a bit
more here, so refresh. This is the idea that if
you're going to present multiple objects or subjects in an image,
you should gravitate toward odd numbers rather than evens. The
basic concept here, as described by David Taylor in Understanding
(25:28):
Composition from twenty fifteen, is that a presentation of odd
numbers is always more esthetically pleasing. With an odd number,
there's always a central object or subject framed by the others. Meanwhile,
even numbered subjects or objects will read as symmetrical with
no central subject or object unless they are, as we discuss,
grouped in a manner that reads more as odd than even.
Speaker 3 (25:53):
Yeah, And we talked about examples of that last time,
with like paintings that will have four people in them
and it's like three standing together, one standing apart.
Speaker 2 (26:01):
Right, And I and I know, I've seen this pointed
out as something that factors into food photography as well.
And I kind of like ended on that point on
a Friday afternoon and then spent the whole weekend thinking
about it, and like went into a restaurant with my family,
and you know, at one point, appetizer just came out
in a pair of two and I was, I was,
you know, thinking about that a lot. I was like,
(26:22):
why is it too? It should be three? Right? That
is that the whole sense here and so then I
came back to it Monday morning and read a bit
more about it. So I'm going to come back to
the food spin on this in just a minute, but
just this idea of Okay, if you have odd images,
there's always a central and if you have even there's
no like centrality. It's it's symmetrical. It's like a group
(26:45):
of two and two, and that's just how our brains
end up taking it all in. Now. I started wondering,
what is this reminding me of. There's some sort of
image in my head, and I realized I was thinking
of a particular puppet on display in the museum at
the Center for Puppetry Art here in Atlanta. The puppet
is of the demon king Ravana from the Hindu epic
(27:06):
the Ramayana. This is the demon king, the villain of
that particular work. He rules over the island of Lanka
and famously abducts Lord Rama's wife Sita. So yeah, he's
the big bad and he's often depicted as having ten heads,
though for reasons I'll get into, he also sometimes is
(27:27):
depicted is having nine heads. These heads are generally presented
lined up ear to ear, with only a single head
connected by a neck to a single humanoid body. Now
the puppet that's on display in the Center for Public Arts,
this is a West Bengal puppet in the tradition of
and I'm maybe mispronouncing this, my apologies, don jier Puto knock.
This is a style of wooden rod puppetry. Literally it
(27:49):
means dance of the wooden dolls. This puppet has ten heads,
and you can guess what that means. It means that
a tin headed Ravena presented in this fashion does not
have an even number of heads on either side of
the bodied head. The Center for Puppetry Arts puppet Ravena
has a row of four heads to one side of
the main head and a row of five heads to
(28:10):
the other side of the main head. It's also hard
to portray that with nonlinear depictions of Ravena. So I
came across a likely AI generated depiction of Ravena on
Shutterstock with a different grouping that does read is more balanced,
you know, to the average observer. But I should note
that this is non through traditional means of depicting the character.
(28:32):
This one has like a group of four on one side,
group of four on the other and then one above
the central head. I also ran across a statue of
Ravena from Statue Park in Muraswar, India that seems to
have a circular representation, So I guess kind of like
radial alignment of the heads. But I believe this is
a more modern interpretation. It's not what you tend to
(28:54):
see in sculpture, puppetry, masks and so forth. And it
is a depiction of Ravena attempting to lift a mountain
in order to impress or intimidate Lord Shiva. Now, meanwhile,
like I said earlier, Ravena is sometimes depicted as having
nine heads, and when presented in the traditional fashion, this
does even things out and gives us a central bodied
(29:15):
head with four heads to either side. Why does Rabina
sometimes only have nine heads? Well, remember the tail of
him lifting the mountains to impress Lord Shiva. Well, according
to this telling, Lord Shiva was not impressed and merely
put one toe on the mountain to squash Ravena beneath
it like a bug. He howls out in pain, but
he realizes, Oh, the only way I'm going to escape
(29:36):
this is if I can play a sweet hymn, a
sweet song for Shiva about how great he is. But
I need an instrument to do that. So what does
he do? He plucks off one of his heads, He
plucks off one of his twenty arms, some of his
intestines and tendency plucks out as well, and he makes
himself a traditional stringed instrument known as a vina to play.
(29:56):
And there are some there are different depictions of this.
I think sometimes Ravena is seen to basically just be
holding a traditional stringed instrument here, but other times, for instance,
there's at least one temple example, saw an image of this.
This is a photograph from Sri Lanka. It is the
Konswaram Hindu temple, and we see this kind of I
(30:19):
guess mildly grizly musical instrument that Ravena has made out
of his body parts and he's playing it there. And
in this image he does have foreheads to either side
of the central head instead of again that kind of
visually reading lop sided arrangement that we see in a
tin headed rabna. Now you may wonder why does Ravena
(30:42):
have tin heads to begin with? Well, I was reading
different examples and different stories regarding this number, and one
in particular, there's an article titled the Untold Story of
Ravena on the Hindu American Foundation website by Maha Kashuk
from twenty twenty two. The author here recounts the story
(31:03):
of how Ravena came to have ten heads to begin
with in some tellings, and this one involves Ravena seeking
atonement from Shiva by annexing his head, which I'm to
assume means a form of self decapitation. And he does
this enough times that when the head grows back each time,
he ends up with ten. Now, symbolically, the author also
(31:23):
has that ten heads represent the six Shastras or say,
these are sacred scriptures of Hinduism, as well as the
four Vedas. Thus it's a manifestation of Ravena's scholarly mastery
over these subjects. So multiple heads can mean great knowledge.
Another take on the ten heads that the author points
out here, and I've seen this sighted elsewhere as well,
is that they stand in for the ten emotions lust, anger, delusion, greed,
(31:46):
pride in the mind, intellect, will, and ego. And the
idea here apparently is that you want intellect to overpower
all the rest. But Ravena is instead controlled by all
of them, which leads him to make the choices, the
result in his downfall now in him. I do iconography,
As with most religious iconography, we have to remember that
these images are meant to convey ideas. So multiple arms
(32:06):
on a deity are more about displaying their power and
via the objects in said hands, other particularities about the deity.
But power is definitely key, which is why you'll definitely
see multiple hands when various deities are depicted as being
in battle or overcoming an adversary. Again, multiple heads may
(32:26):
likewise speak to the intellect of a particular entity or
various other aspects of that deity and their differing nature. So,
for instance, Siva is sometimes depicted with a triple head
blissful and wrathful aspects to either side, and of course
this also lines up with the general tradition of the
great triad, you know, a triple face or triple headed
(32:47):
god that is depicted in religions around the world. Other times,
Shiva is depicted with five heads, each representing the five
divine activities creation, preservation, destruction, concealing grace and revealing grace,
and Brahma may be depicted with four faces and four arms.
Four arms is very common in Hindu symbolism for multiple gods. Now,
as to the particular fondness for odd numbers and Hindu traditions,
(33:10):
I haven't run across anything that draws a fine line
on the matter. In large part this is not surprising because,
as we've discussed in the show before, Hinduism is not
a monolith. It's a deep well of belief that's thousands
of years old and contains many di her schools. And
while one does see a tendency towards odd numbers a
law of odds to a certain extent, I guess in
(33:30):
Hindu traditions it's probably easier to loop all of that
in to what might seem like a global tendency towards
sacred odd numbers as opposed to anything that is particular
to Hinduism. And I was reading about this in a
book from nineteen eighty three titled The Mystery of Numbers
by Anne Maurice Shimmel, and the author here points to
various examples from the ancient Mediterranean, from Christian, Muslim, and
(33:53):
Jewish traditions as well that dwell on odd numbers, particularly
in ritual acts prayer and incantations. She writes, one performs
acts of magic three or seven times and repeats a
prayer or the concluding amen thrice. In earlier times, physicians
and medicine men used to give their patients pills in
(34:15):
odd numbers. Magic knots, too, had to be tied in
odd numbers. The Talmud offers numerous examples of the use
of odd numbers and the avoidance of even ones, and
the Muslim tradition states that the prophet Muhammad broke his
fast with an odd number of dates. When performing witchcraft
or black magic, an odd number of persons should be present,
and even today it is the custom in Europe at
(34:36):
least to send someone bouquets containing an odd number of flowers,
with the exception of a dozen hm hm Yeah.
Speaker 3 (34:43):
I think it's so interesting to consider why these kinds
of patterns emerge.
Speaker 2 (34:49):
Now.
Speaker 3 (34:50):
On one hand, I do think there can be a temptation,
probably to quickly jump to some kind of like universal
in you know, built in thing in our brain is like, oh,
we just everybody around the world. Something about being human
prefers odd numbers or thinks they're more sacred, And I
wouldn't rule that out. It could be possible, but I
(35:10):
wouldn't jump to that conclusion either, because you know, you
can think about all kinds of ways that that sort
of accidents of history can become ingrained in a culture
or literary tradition and then just get amplified from there
that maybe something about you know, initial bits of storytelling
that happened to include an odd number of something or
(35:30):
an even number of something can build up over time
and suddenly that starts to feel just like the fabric
of reality.
Speaker 2 (35:37):
Yeah. Yeah, I mean, we definitely don't want to overstate
it because from on one hand, any given faith that
we mentioned just now, there are going to be examples
in both odd and even. You know, you can come
up with plenty of examples of wholly even numbers or
the use of even numbers, and you know, some sort
of sacred tradition of one sort or another. And likewise, yeah,
(35:59):
there's information that is being related, ideas that are being
related that may just incidentally be even odd. It's not like,
you know, it's not like they were putting together the
Ten Commandments and they're like, well, this is a good
even number of commandments. We don't need to add or
subtract one or it's not like they were. Oh, we
have nine nine commandments, we better come up with one more.
We want an even ten.
Speaker 4 (36:18):
Well, who knows, maybe maybe, But I mean, at the
same time, with the example of the Bible, like I
was saying earlier, like it is hard not to if
you just read through the Old Testament, notice a huge
amount of odd numbers, especially a lot of sevens.
Speaker 3 (36:34):
I don't know that that that's meaning something.
Speaker 2 (36:37):
Yeah, I couldn't help but think about this one as well.
Over the weekend because I went with my family to
see the new Beetlejuice movie. Oh and of course one
uh summons the character in question by saying his name
three times in why not two or four? Yeah, And
we see the same with you know other you know
folk traditions, the old idea of bloody Mary, you know
(36:59):
some and her scaring yourself by seeing her in the
mirror by saying her name three times in a row,
that sort of thing.
Speaker 3 (37:07):
I got real freaked out about that. When I was
a kid, I had a phase where that was just
like super scary to me.
Speaker 2 (37:15):
I mean, I still am not going to do it.
I don't believe it, but I'm not gonna still not
going to say her name three times in front of
a mirror.
Speaker 3 (37:22):
And I mess around.
Speaker 2 (37:24):
Yeah, yeah, okay, So coming back to the law of
odds in general, Yeah, it's often touted as a deciding
factor in various various approaches to visual imagery, and I
have seen it mentioned as as lining up with food
imagery as well. You know, again, I think the example
used before was, if you're gonna have a appetizer of
(37:45):
sliders at a restaurant, you want as your menu photo
or your Instagram food photo, you want an image of
three sliders, not two. You want an image of three
sliders and not four, because three is going to be
an odd number. It's more attractive. And yeah, you can
throw in these other ideas of like, well there's a
central slider, I know which one is the lead slider.
(38:06):
But the thing is, when I started looking around for
studies about this, it seems like that experiments don't back
this up. So according to odd versus even a scientific
study of the rules of plating by woods at all
published in twenty sixteen in pere j Law and Environment. Yeah,
according to this paper, it just doesn't seem to work
(38:31):
quite as strongly as some might have you believe, they
actually conducted some experiments. I want to say it was
over a thousand folks involved in this, but you know,
they ended up contending that we have to take various
cultural factors into consideration here. You know, there's a lot
going on when we look at an image and if
(38:51):
we add but if we add that that image is
image of food, and it's food that we are on
some level considering eating, then it seems that overall portion
size is more important than odd or even numbers when
it comes to human perceptions of food.
Speaker 3 (39:07):
Okay, so we would rather have on average, would rather
have four sliders than three.
Speaker 2 (39:12):
Right, We'd rather have three than two, yes, but not
because three is odd, but because three is more sliders.
And of course this seems like a gross over statement
of the obvious, right, because it's like you go to
a restaurant. You're like, I'm paying you know, close to
twenty dollars for this plate of sliders. Of course I
want it to be four and not three, because I'm
getting more slider for my buck. Also, when you're hungry,
(39:34):
you're hungry, and your hunger is not always a great
gauge of how many sliders you need to satisfy yourself
and or those around you, you know, so you know,
on that level, of course four sliders sound better. Let
it be four and not three. Three is just maybe
a little less likely to satisfy everyone's cravings.
Speaker 3 (39:55):
But so on my understanding this right there, it's not
necessarily that they found that people prefer for evens to odds.
It's just that maybe, like if there is a preference
for odds, it doesn't play that big of a role
when compared to people just wanting more food.
Speaker 2 (40:10):
Right, right, And they provide some wiggle room there, because
again there's a lot going on when you're considering an
image or you're considering a presentation. I think there could
based on what I was reading here, I mean, there
could easily be a situation where ultimately having an odd
number is more important. Like maybe it's a very you know,
ritualistic presentation of food. Maybe it's a situation where the
(40:34):
present where the presentation is more about just having a
great photograph as opposed to, you know, making the potential
customer salivate. Again, there are a lot there's so much
going on when we look at an image, but you
cannot discount the importance of hunger when that image is
of food.
Speaker 3 (40:52):
It's it's about tricking people into believing that if you
get this sandwich the tomato on it will be red
and juicy.
Speaker 2 (40:59):
Yeah, in reality, it may not, may be very anemic
look at it. It may not have much flavor to it.
It may merely be wet and hopefully cold. In some cases,
that's fine, Maybe it's gonna work well within the context
of the slider the studying question. They also looked at like,
you know, they were looking at it like horizontal versus
vertical plating scenario. So I would be very interested to
(41:20):
hear from anyone out there who is involved in plating,
either professionally or you know, on an amateur chef level,
what your thoughts are on this.
Speaker 3 (41:28):
Oh yeah, I actually just got interested in how much
of say you're at, you know, sort of some kind
of elite level, you know, you're working at like a
very fancy, expensive restaurant or something plating choices. How much
of that is is an art and how much is
a science? Are you just sort of going off of
some kind of chef or stylists instinct there, or do
(41:50):
you actually do research on what people dining there prefer
in terms of plating in appearance.
Speaker 2 (41:56):
Yeah, I mean, and then there's also the whole the
economic value of there, right, because I mean, you have
to have to factor in like can we afford to
have a four slider platter? Shouldn't it just be a
three slider platter? Are we really going to lose business
because everyone thinks they need a fourth one? If they
need a fourth one, they can buy that out a
la cart perhaps, I don't know. There are a number
of factors involved.
Speaker 3 (42:17):
You know, I'm a big fan of chips and dips,
and for some reason, I really like it when there
are two dips. Oh okay, there were two different dips.
It seems like there should be three. Though there should
be three tips. I mean yeah, but then you start
once they're three, that's just like that's like a buffet
of dips. You get two dips, that's like really focused.
Do you get like one I don't know, one roasted
(42:38):
tomato salsa and one guacamole or something.
Speaker 2 (42:41):
Yeah, when there are three dips, I do find that
one dip is definitely going back in the fridge for dinner.
And then because you think, well, I'll use that later.
I'll definitely dip something in that later, and you don't
you just wash that up out and recycle it like
a week or two later. All right, I guess we're
out of time for this, but we didn't even get
into the whole idea of the seven layer burrito. So
(43:01):
just leave listeners to contemplate the seven layer burrito and
if that is an appropriate number of layers or should
it be less or more?
Speaker 3 (43:09):
I don't know the magic burrito.
Speaker 2 (43:12):
All right, Just a reminder for everyone that Stuff to
Blow Your Mind is primarily a science and culture podcast,
with core episodes on Tuesdays and Thursdays, short form episode
on Wednesday and on Fridays. We set aside most serious
concerns to just talk about a weird film on Weird
House Cinema and let's see what else to remind you of.
Oh yeah, if you were on Instagram, follow us on Instagram.
We are STBYM podcast. That's our handle, and you know
(43:36):
you can keep track of keep up a little bit
with what we're putting out in the podcast.
Speaker 3 (43:40):
Feed Huge thanks as always to our excellent audio producer
JJ Posway. If you would like to get in touch
with us with feedback, on this episode or any other.
To suggest a topic for the future, or just to
say hello, you can email us at contact at stuff
to Blow your Mind dot com.
Speaker 1 (44:02):
Stuff to Blow Your Mind is production of iHeartRadio. For
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or wherever you're listening to your favorite shows.
Welcome to Stuff to Blow Your Mind production of iHeartRadio.
Speaker 2 (00:12):
Hey, welcome to Stuff to Blow Your Mind. My name
is Robert.
Speaker 3 (00:15):
Lamb, and I am Joe McCormick, and we are back
with Part two in our series on the psychology and
cultural significance of number parody p A R I T
Y parody meaning whether a number is odd or even.
In Part one, we described the principle of number parody,
and we talked about evidence that in some cases people
(00:38):
seem to have surprising feelings about associations with and even
preferences for odd and even quantities. And so one of
the big examples we discussed in that first episode was
the concept in various branches of visual art theory that
people have a preference for, say, three part divisions of
imagery over two part divisions, or that people prefer an
(01:00):
image composed with an odd number of subjects over an
even number, even to the extent that even numbers of
subjects will sometimes be subdivided into groups of odd numbers,
so you know, instead of four subjects, you would get
a painting with three and one. But we also got
into a bit of empirical research interrogating these ideas and
questioning to what extent they're truly natural esthetic preferences. Maybe
(01:23):
they're just sort of random conventions that people latched onto.
Including you know, one thing that came up in Part
one was the domain of food plating and food styling,
with us just you know, shoot, shooting from the hips
saying I think three little sliders are better than four.
We're going to come back to that later today. You
might be surprised.
Speaker 2 (01:44):
I mean it is still you still see this idea
out there, but how does it hold up to any
manner of study. Well, we'll take a look at that.
Speaker 3 (01:53):
So one thing I wanted to talk about today was
the cognitive psychology of number parity, how we process the
idea of numbers being odd and even in the brain.
So I came across a very interesting paper about this
that was published in the journal Frontiers and Psychology in
(02:13):
the year twenty eighteen by Hubner at All and it's
called a mental odd even continuum account some numbers may
be more odd than others, and some numbers may be
more even than others. And so if you're not initially
thrilled about the idea of that, the cognitive psychology of numbers,
how we represent number properties internally. Stick around. I think
(02:36):
this might be more interesting than you would at first suspect,
because it's kind of it kind of reveals deeper ways
that our brains work in general, at least I think.
So we can come back to that after we look
at the findings of the study, But anyway to start
with the mathematical fact is that number parity is binary.
In math, natural numbers are either odd or even. Any
(03:01):
positive integer is even if it can be represented as
two times in, wherein is also a positive integer, and
it's odd if it can be represented as two times
in plus one. All positive whole numbers are either odd
or even. But this paper is focused not on the
question of the mathematics of parity, but on the question
(03:22):
of how number parity is represented in the brain, how
we think about quantities that are odd and even, And
the authors propose an interesting hypothesis that people do not
think about odd and even as a mathematical binary, but
rather as a spectrum of odd ness and even ness,
(03:43):
where some numbers can be relatively more odd or even
than others. And in a kind of amusing aside, the
author is acknowledge that if this is true, it may
prove irritating to some researchers, but you know, this is
the kind of thing I like reading about, because I
think it's when you observe the mismatch between how a
concept is technically defined and how we actually think about
(04:07):
it when we consider it in practice, it's a great
way to get insights into our brains.
Speaker 2 (04:12):
Yeah. Yeah, And I'm already thinking about thinking about ways
that I might qualify certain numbers as more even or
more odd than others. But I want to see where
you're taking us here and see if any of these
are are the examples that are coming to my mind.
Speaker 3 (04:26):
So to provide a model for how this would be
happening in the brain, the authors refer to a psychology
concept called prototype theory, which has been established going at
least as far back as the nineteen sixties. As they explain, quote,
prototype theory has long suggested that certain members of distinct
categories are more typical examples of that category than others,
(04:51):
and that membership to such a category may be graded. Now,
they don't use the following example, and in fact, I
don't know if this is strictly a perfect example of
prototype theory, because the category I'm going to use is
not strictly defined, but I think this will still illustrate it.
Both Pumpkinhead and Grover from Sesame Street are examples of
(05:14):
the category monster. And yet while they are undoubtedly both monsters,
and if you doubt Grover is a monster, go read
up about them, Grover's a monster, one of them just
seems like a better example of the category monster than
the other. Now, there are no real objective criteria for
what is and is not a monster, but you could
(05:36):
learn a lot about how people mentally construct the idea
of a monster by studying how easy it is to
associate particular examples of creatures with the category monster. And
one way of studying this would be time latency. So
imagine you're in a psychological study and you're given a task.
(05:59):
Somebody's going to show you a series of images of creatures,
and it's your job to say as quickly as you
can whether the creature in the image is a monster
or not. In this kind of test, the speed with
which you make the categorization could be one piece of
evidence for how easily you associate the example with the category.
(06:20):
So even if everybody who takes this kind of test
correctly recognizes that Grover is a monster. I would still
bet that on average people would say Pumpkinhead is a
monster a good bit faster. It just it takes less
thinking to get there, so you can click the monster
button faster.
Speaker 2 (06:38):
Yeah, yeah, you don't have to catch yourself and go, oh, well, yes,
of course he is the monster at the end of
the book.
Speaker 3 (06:43):
Yeah, exactly. And so with this kind of study you
could maybe get some insights. For example, you could look
at these specific attributes that make an individual picture of
a creature a better prototype example of the monster category
as measured by people selecting it as a monster faster.
Maybe maybe creatures that have sharp teeth or claws or
(07:04):
threatening posture or something like that. It just clicks in
the brain faster that it's a monster. You got to
think about it less. And so in this paper, the
authors do the same thing with odd and even numbers.
They're going to study the degree to which different numbers
are prototypes of their parity class, and then they're going
to try to look for the different factors that make
(07:25):
a number more easily identifiable as odd or even. And
this is, by the way, not the first study ever
to do this. There have been studies in the past
that have used processing time as a measure of prototypicality
for odd and even numbers, like they mentioned one study
that showed six took people longer to classify as even
than two four or eight did.
Speaker 2 (07:47):
Why.
Speaker 3 (07:48):
I don't know. That's kind of interesting. I mean, two, four, six,
and eight are all equally even in real mathematics, but
apparently two four and eight are just easier to identify
as even something something's a little for about six.
Speaker 2 (08:01):
Huh. Interesting.
Speaker 3 (08:03):
So in their introduction, the authors lay out a bunch
of different numerical reasons that they think a number might
be more easily recognizable as even or odd, and the
hypothetical explanations they include are first of all, ease of divisibility.
So the easier a number is to divide, the more
(08:23):
even and less odd it should feel. And this principle
could subconsciously be applied within the categories and not just
between them. So twenty five and twenty seven are both odd,
but the author's idea here is that twenty five may
feel less odd and take longer to classify as odd
because it's easy to divide it.
Speaker 2 (08:44):
Now, this is where my mind was headed that. Yeah,
just thinking about the way I divide numbers is if
it's easier to divide, then yes, on some level, it
is more even than an even number that I have
to sort of like pause a second with then do
a little extra math in my life.
Speaker 4 (09:00):
Yeah.
Speaker 3 (09:01):
I think that's a strong instinct that they had the
same idea to begin with. Here. Another thing they hypothesize
would make a number feel more even is powers of two,
so that would be two for eight, sixteen, thirty two.
They think these are cognitively more even. Another factor is
whether a number is prime. The authors argue that prime
(09:23):
numbers may feel more odd than non prime odds, and
one piece of evidence for this is that a couple
of different previous studies have found that people are quicker
to flag three, five, and seven as odd than they
are to flag nine. That's interesting, now, this is kind
of like the inverse of the six not feeling as
(09:46):
even as the other even numbers under ten. In this case, apparently,
maybe nine does not feel as odd as the other
odd numbers under ten, and the authors argue that this
may be because the other three odd numbers under ten, three, five,
and seve are all prime. Nine is not prime. Three
times three is nine, so the divisibility of it maybe
(10:06):
makes it feel less odd. The authors also hypothesize maybe
being part of a standard multiplication table that children memorize
in school that might make numbers feel more even and
less odd, But we'll have to look at the results
and see if that bears out. However, the authors point
(10:26):
out that previous studies have shown that it is probably
not only the mathematical properties of a number the number
properties of a number that influence how long we take
to make judgments about it. Other factors, such as linguistic factors,
appear to play a role as well. And illustrate this,
the authors bring up a really interesting concept that I
(10:49):
don't think I'd ever read about before, but this really
stuck with me. So they refer to previous research by
Hines in the journal Memory and Cognition in nineteen and
this paper found that if you give people random numbers,
especially in pairs or in triples, and ask them to
judge whether the numbers are odd or even, people simply
(11:12):
take longer to recognize oddness than they do to recognize evenness.
So odd numbers were just harder to judge overall, so
people more quickly recognize that fifty two and fifty four
are even than that fifty three and fifty five are odd.
Now that's kind of weird, like why would oddness itself
(11:35):
take longer to process? Pretty much across the board. In
this older paper, the author argued that part of the
explanation may lie in the idea of what are called
marked and unmarked terms in language. Marked and unmarked This
is a concept in linguistics, and it goes like this,
So there exist in languages pairs of adjectives that have
(11:59):
opposite meanings, so long and short, old and young, even
an odd, alive and dead, things like that. Linguistic markedness
theory says that usually when you have pairs of adjectives
like this, one of the terms in the pair is
treated as the more basic and natural of the two
(12:23):
in the brain. So we think about one of these
two terms in a way that what they call they
call it unmarked. It is the natural state of this measure,
and then the other term is treated as mentally more complex, complicated,
and unnatural. This is the marked word in the pair,
(12:44):
and there are experiments that will show this. But the
unmarked word in the pair, for example, is used more
frequently than the marked word. It's learned earlier in language acquisition,
when you're a child, and it is considered usually the
default to measure. So, for example, you say how old
are you, not how young are you? Because in old
(13:07):
and young, old is treated as the unmarked word and
young is the marked concept. Similarly, you will say how
long will it take? Not how short will it take?
I thought this was interesting. They say also that in
some cases you can create the same meaning as the
marked word simply by adding a negative prefix to the
(13:28):
unmarked word. So you can say uneven to mean the
same thing as odd, but nobody says un odd to
mean even.
Speaker 2 (13:36):
Oh, that's true. That's a great point.
Speaker 3 (13:38):
Now, whatever this division between marked and unmarked comes from,
it seems that it results in different processing times in
the brain. That we just deal with unmarked concepts faster
and more easily, and it takes us, you know, maybe
a split second longer to think about, or deliver or
deal with a marked concept. And so if even is
(14:02):
unmarked and odd is marked, it may in fact be
that we just deal with the concept of evenness a
little bit more easily in the brain than oddness. It's
oddness is linguistically marked, and so it takes us a
split second longer to kind of process this concept whenever
we're dealing with it or producing a judgment about it,
And this may play a role in explaining the so
(14:24):
called odd effect that was discovered in this paper in
the nineties. Moving on from that, there's another linguistic effect
that actually shows up when you compare judgments about parody
(14:44):
across different languages, and this is the inversion property of
multiple digit numbers. So in English, when we want to
say or write out in words the number that is
one quarter of one hundred, we say twenty five, we
write the twenty first and then the five, or we
say the twenty first and then the five. So for
(15:06):
two digit numbers, it's always the decade digit first in language,
and then the unit digit. But not all languages work
this way. For example, in German, twenty five is and
I'm sorry, I'm sure i'm pronouncing this wrong. It is
something like fun fundzwanzig, meaning five and twenty. And this
has been found to have all sorts of interesting effects
(15:28):
on number cognition. For example, German speakers are studies have
shown more likely to make trans coding errors when writing
numbers out, so more likely to write fifty two when
they mean twenty five. In terms of digits, Also, compared
to non inverted languages, German speakers pay relatively more attention
(15:51):
to the unit digit in a multi digit number, and
so the authors write quote. This prioritizing of either the
unit or decade digit might influence participants' performance in number
processing tasks in which units play a decisive role. Parity
judgment is clearly one of those tasks, because only the
unit parity is relevant for answering correctly, which is true
(16:14):
when you look at you can judge whether it's odd
or even without knowing any of the numbers before the
last one. And just a couple of other factors the
authors mention that have been possibly shown to influence parity judgments.
Larger numbers may cause longer processing times, regardless of the
parity or any other facts about them. Is just like
(16:34):
the bigger the number is, the longer you have to
think about it. Also, word frequency, numbers that appear more
often in language get faster responses, and this is not
just true of numbers any words in general that are
used more often are processed more efficiently, So this study
tried to test the relative influence of number prototypicality and
(16:55):
the linguistic factors we were just talking about. And the
way they did this was by getting a group of
subjects and giving them auditory prompts of numbers between twenty
and ninety nine, and then they would try to analyze
how long it took people to classify these numbers as
odd or even to test the linguistic factors. The author's
(17:16):
recruited subjects from three different language groups. They had English speakers,
German speakers, and Polish speakers. In Polish, two digit numbers
are expressed with the decade digit first, like in English.
And I'm not going to discuss all of their findings,
but just to summarize and pick a few highlights, they
do say that quote. Overall, the results suggest that perceived
(17:38):
paroity is not the same as objective paroity, and some
numbers are more prototypical exemplars of their categories. And specifically,
with regards to these mathematical or numerical factors influencing things,
they found that some but not all, of the characteristics
they hypothesized actually did play a role imperceived paroity. So,
(18:02):
for evens. The numbers that people identified as even the
fastest tended to be even squares, so a square being
the product of a number multiplied by itself. Sixteen is
a square because it's four times four, sixty four is
a square because it's eight times eight. Thirty six is
a square because it's six times six. So in the
(18:23):
results you would find that sixty four was significantly easier
to identify as even than sixty two, so squares tended
to be very fast. Multiples of four also did really good.
For some reason, our brains love noticing that multiples of
four are even. Now, when it came to recognizing odd numbers,
(18:45):
things got a little more complicated, and the authors say
that there's a good reason for this. It may have
to do with multiple hypothesized effects working against one another,
and these would be number prototypicality on one hand, but
linguistic markedness on the other. So, to refresh the explanation
based on linguistic markedness, says that because even is an
(19:08):
unmarked concept and odd is marked, we will usually recognize
evens faster than odds across the board. And it may
also possibly mean that numbers that seem odder to us
will take longer to recognize. So this effect, if present,
would work in opposite directions depending on parity. For example,
(19:30):
the super even numerical properties like say being a multiple
of four, will make a number feel more even, but
they will also make it easier to process the evenness
of the number quickly from a linguistic standpoint, because now
the number is especially unmarked. On the other hand, as
a number becomes more subjectively odd by say being a
(19:53):
prime number, the prototypicality explanation would predict that we can
notice that it's odd faster, but because it's especially numerically odd.
Working against this would be the linguistic markedness, which might
predict that the more odd number seems, the more linguistically
complicated it will feel, and thus the longer our reaction
(20:15):
time before we can say anything about it. So with evens,
these two explanations stack, but with odds they work against
each other. And so they said that the results with
odd numbers were more muddled. But they did find basically
that primes and numbers divisible by five took the longest
to classify as odds. Odd squares were the fastest. Kind
(20:39):
of counterintuitively, a couple of other results They also found
effects from what's called parody congruity. That's whether the two
digits in the number are the same parody, so whether
you know, like sixty eight, they're both even, sixty seven
one is even and one is odd. That had an effect,
and also decade magnitude, so the how high the first
(21:04):
number in the pair was had an effect on how
long it took to process. As it gets bigger, it
takes longer to think about. They also did find some
major differences in reaction times by language group. In general,
German speakers identified two digit numbers as odd or even
faster than English or Polish speakers, and this could be
(21:24):
due again to this linguistic inversion principle that you say
the unit number first when you're speaking German, and the
unit number is actually all you need to know whether
a number is odd or even. But anyway, I found
this whole thing so interesting because it sort of reveals
to me that while the actual, you know, the mathematical
(21:46):
algorithm for determining whether a number is even or odd
is extremely simple and it's totally binary, and yet when
we think about it, apparently we must be using all
these different kind of heuristics and influences and different kinds
(22:06):
of little rules to make these judgments about numbers as
fast as we can. And the study did find that
people get the right answer most of the time, and
people rarely get it wrong when asked to judge whether
a number is even or odd. But they're they're clearly
using like different little, different little principles are at work
in helping them get to that answer as fast as
(22:26):
they can. And some numbers are just easier to judge
faster than other ones, meaning that they're just more represented
as a correct answer within this category than others are.
And no number in reality is any more even or
any more odd than another.
Speaker 2 (22:43):
Yeah, I mean, I can't help but think about the
basic reality of when I'm using real world math, particularly
say with money. You know, any amount of money is
divisible by two, you just get into change, And that
holds true elsewhere as well. I mean, it's not like
an odd number cannot be split into two equal portions.
It's it's just it's just you're going to have to
(23:06):
go into the decimal points to do so. But when
you do have to divide an even number into in
the real world, it does feel like a more wholesome act. Yeah,
maybe I just hate doing math, but that's the way
I feel.
Speaker 3 (23:19):
Well no, no, I see, yeah, what you're saying. I mean,
so when you're talking about whole number division, obviously dividing
an even number is you know, you can get to
an unproblematic answer to that, and if you have an
odd number, you're going to have a problem. You're going
to have to figure out what to do about the
fact that it doesn't split down the middle correctly. If
you're you're dealing with some kind of like whole I
(23:40):
don't know, if you're trying to figure out how to
split the three scallops on your plate.
Speaker 2 (23:43):
Mm hmm.
Speaker 3 (23:44):
Yeah.
Speaker 2 (23:44):
Yeah.
Speaker 3 (23:45):
But this also it just makes me think about all
the ways that you know, you might have categories in
the real world, whether it's mathematical or whatever, that you
know are are technically distinct in the way that they
are defined, and yet our brains are just not going
to be bound by that for having like strict inclusion criteria. Well,
like we'll get into these like ways of thinking about
(24:06):
it as some kind of gradient, and that's just kind
of interesting that we tend to work that way.
Speaker 2 (24:11):
Yeah. Yeah, Like now that I think about it. I'm
pretty sure that five and seven especially are just like
disgustingly odd, you know. Oh okay, I mean it gets
more disgusting the more sevens you have. I guess, like
like seventy seven, seven hundred and seventy seven. Just I
don't even want to think about those.
Speaker 3 (24:29):
Oh, that's starting to make me think about the stacking
of sevens in the Bible.
Speaker 1 (24:33):
You know.
Speaker 3 (24:34):
Sometimes they really like to get into the There will
be like seven seven seven of something that they're seventy
seven of on the seventh day.
Speaker 2 (24:41):
Yeah, I mean it kind of gets into the know,
the idea of something Okay, well, you know it's not
easily divisible. I guess it's you know, it's more solid,
it's more holy in that regard. It depends on how
you want to spend all right, now, it's time to
come back to the idea of three sliders on a plate,
(25:06):
the supposed rule of odds. So in part one I
mentioned the rule of odds and visual composition, and yeah,
I want to come back and discuss it a bit
more here, so refresh. This is the idea that if
you're going to present multiple objects or subjects in an image,
you should gravitate toward odd numbers rather than evens. The
basic concept here, as described by David Taylor in Understanding
(25:28):
Composition from twenty fifteen, is that a presentation of odd
numbers is always more esthetically pleasing. With an odd number,
there's always a central object or subject framed by the others. Meanwhile,
even numbered subjects or objects will read as symmetrical with
no central subject or object unless they are, as we discuss,
grouped in a manner that reads more as odd than even.
Speaker 3 (25:53):
Yeah, And we talked about examples of that last time,
with like paintings that will have four people in them
and it's like three standing together, one standing apart.
Speaker 2 (26:01):
Right, And I and I know, I've seen this pointed
out as something that factors into food photography as well.
And I kind of like ended on that point on
a Friday afternoon and then spent the whole weekend thinking
about it, and like went into a restaurant with my family,
and you know, at one point, appetizer just came out
in a pair of two and I was, I was,
you know, thinking about that a lot. I was like,
(26:22):
why is it too? It should be three? Right? That
is that the whole sense here and so then I
came back to it Monday morning and read a bit
more about it. So I'm going to come back to
the food spin on this in just a minute, but
just this idea of Okay, if you have odd images,
there's always a central and if you have even there's
no like centrality. It's it's symmetrical. It's like a group
(26:45):
of two and two, and that's just how our brains
end up taking it all in. Now. I started wondering,
what is this reminding me of. There's some sort of
image in my head, and I realized I was thinking
of a particular puppet on display in the museum at
the Center for Puppetry Art here in Atlanta. The puppet
is of the demon king Ravana from the Hindu epic
(27:06):
the Ramayana. This is the demon king, the villain of
that particular work. He rules over the island of Lanka
and famously abducts Lord Rama's wife Sita. So yeah, he's
the big bad and he's often depicted as having ten heads,
though for reasons I'll get into, he also sometimes is
(27:27):
depicted is having nine heads. These heads are generally presented
lined up ear to ear, with only a single head
connected by a neck to a single humanoid body. Now
the puppet that's on display in the Center for Public Arts,
this is a West Bengal puppet in the tradition of
and I'm maybe mispronouncing this, my apologies, don jier Puto knock.
This is a style of wooden rod puppetry. Literally it
(27:49):
means dance of the wooden dolls. This puppet has ten heads,
and you can guess what that means. It means that
a tin headed Ravena presented in this fashion does not
have an even number of heads on either side of
the bodied head. The Center for Puppetry Arts puppet Ravena
has a row of four heads to one side of
the main head and a row of five heads to
(28:10):
the other side of the main head. It's also hard
to portray that with nonlinear depictions of Ravena. So I
came across a likely AI generated depiction of Ravena on
Shutterstock with a different grouping that does read is more balanced,
you know, to the average observer. But I should note
that this is non through traditional means of depicting the character.
(28:32):
This one has like a group of four on one side,
group of four on the other and then one above
the central head. I also ran across a statue of
Ravena from Statue Park in Muraswar, India that seems to
have a circular representation, So I guess kind of like
radial alignment of the heads. But I believe this is
a more modern interpretation. It's not what you tend to
(28:54):
see in sculpture, puppetry, masks and so forth. And it
is a depiction of Ravena attempting to lift a mountain
in order to impress or intimidate Lord Shiva. Now, meanwhile,
like I said earlier, Ravena is sometimes depicted as having
nine heads, and when presented in the traditional fashion, this
does even things out and gives us a central bodied
(29:15):
head with four heads to either side. Why does Rabina
sometimes only have nine heads? Well, remember the tail of
him lifting the mountains to impress Lord Shiva. Well, according
to this telling, Lord Shiva was not impressed and merely
put one toe on the mountain to squash Ravena beneath
it like a bug. He howls out in pain, but
he realizes, Oh, the only way I'm going to escape
(29:36):
this is if I can play a sweet hymn, a
sweet song for Shiva about how great he is. But
I need an instrument to do that. So what does
he do? He plucks off one of his heads, He
plucks off one of his twenty arms, some of his
intestines and tendency plucks out as well, and he makes
himself a traditional stringed instrument known as a vina to play.
(29:56):
And there are some there are different depictions of this.
I think sometimes Ravena is seen to basically just be
holding a traditional stringed instrument here, but other times, for instance,
there's at least one temple example, saw an image of this.
This is a photograph from Sri Lanka. It is the
Konswaram Hindu temple, and we see this kind of I
(30:19):
guess mildly grizly musical instrument that Ravena has made out
of his body parts and he's playing it there. And
in this image he does have foreheads to either side
of the central head instead of again that kind of
visually reading lop sided arrangement that we see in a
tin headed rabna. Now you may wonder why does Ravena
(30:42):
have tin heads to begin with? Well, I was reading
different examples and different stories regarding this number, and one
in particular, there's an article titled the Untold Story of
Ravena on the Hindu American Foundation website by Maha Kashuk
from twenty twenty two. The author here recounts the story
(31:03):
of how Ravena came to have ten heads to begin
with in some tellings, and this one involves Ravena seeking
atonement from Shiva by annexing his head, which I'm to
assume means a form of self decapitation. And he does
this enough times that when the head grows back each time,
he ends up with ten. Now, symbolically, the author also
(31:23):
has that ten heads represent the six Shastras or say,
these are sacred scriptures of Hinduism, as well as the
four Vedas. Thus it's a manifestation of Ravena's scholarly mastery
over these subjects. So multiple heads can mean great knowledge.
Another take on the ten heads that the author points
out here, and I've seen this sighted elsewhere as well,
is that they stand in for the ten emotions lust, anger, delusion, greed,
(31:46):
pride in the mind, intellect, will, and ego. And the
idea here apparently is that you want intellect to overpower
all the rest. But Ravena is instead controlled by all
of them, which leads him to make the choices, the
result in his downfall now in him. I do iconography,
As with most religious iconography, we have to remember that
these images are meant to convey ideas. So multiple arms
(32:06):
on a deity are more about displaying their power and
via the objects in said hands, other particularities about the deity.
But power is definitely key, which is why you'll definitely
see multiple hands when various deities are depicted as being
in battle or overcoming an adversary. Again, multiple heads may
(32:26):
likewise speak to the intellect of a particular entity or
various other aspects of that deity and their differing nature. So,
for instance, Siva is sometimes depicted with a triple head
blissful and wrathful aspects to either side, and of course
this also lines up with the general tradition of the
great triad, you know, a triple face or triple headed
(32:47):
god that is depicted in religions around the world. Other times,
Shiva is depicted with five heads, each representing the five
divine activities creation, preservation, destruction, concealing grace and revealing grace,
and Brahma may be depicted with four faces and four arms.
Four arms is very common in Hindu symbolism for multiple gods. Now,
as to the particular fondness for odd numbers and Hindu traditions,
(33:10):
I haven't run across anything that draws a fine line
on the matter. In large part this is not surprising because,
as we've discussed in the show before, Hinduism is not
a monolith. It's a deep well of belief that's thousands
of years old and contains many di her schools. And
while one does see a tendency towards odd numbers a
law of odds to a certain extent, I guess in
(33:30):
Hindu traditions it's probably easier to loop all of that
in to what might seem like a global tendency towards
sacred odd numbers as opposed to anything that is particular
to Hinduism. And I was reading about this in a
book from nineteen eighty three titled The Mystery of Numbers
by Anne Maurice Shimmel, and the author here points to
various examples from the ancient Mediterranean, from Christian, Muslim, and
(33:53):
Jewish traditions as well that dwell on odd numbers, particularly
in ritual acts prayer and incantations. She writes, one performs
acts of magic three or seven times and repeats a
prayer or the concluding amen thrice. In earlier times, physicians
and medicine men used to give their patients pills in
(34:15):
odd numbers. Magic knots, too, had to be tied in
odd numbers. The Talmud offers numerous examples of the use
of odd numbers and the avoidance of even ones, and
the Muslim tradition states that the prophet Muhammad broke his
fast with an odd number of dates. When performing witchcraft
or black magic, an odd number of persons should be present,
and even today it is the custom in Europe at
(34:36):
least to send someone bouquets containing an odd number of flowers,
with the exception of a dozen hm hm Yeah.
Speaker 3 (34:43):
I think it's so interesting to consider why these kinds
of patterns emerge.
Speaker 2 (34:49):
Now.
Speaker 3 (34:50):
On one hand, I do think there can be a temptation,
probably to quickly jump to some kind of like universal
in you know, built in thing in our brain is like, oh,
we just everybody around the world. Something about being human
prefers odd numbers or thinks they're more sacred, And I
wouldn't rule that out. It could be possible, but I
(35:10):
wouldn't jump to that conclusion either, because you know, you
can think about all kinds of ways that that sort
of accidents of history can become ingrained in a culture
or literary tradition and then just get amplified from there
that maybe something about you know, initial bits of storytelling
that happened to include an odd number of something or
(35:30):
an even number of something can build up over time
and suddenly that starts to feel just like the fabric
of reality.
Speaker 2 (35:37):
Yeah. Yeah, I mean, we definitely don't want to overstate
it because from on one hand, any given faith that
we mentioned just now, there are going to be examples
in both odd and even. You know, you can come
up with plenty of examples of wholly even numbers or
the use of even numbers, and you know, some sort
of sacred tradition of one sort or another. And likewise, yeah,
(35:59):
there's information that is being related, ideas that are being
related that may just incidentally be even odd. It's not like,
you know, it's not like they were putting together the
Ten Commandments and they're like, well, this is a good
even number of commandments. We don't need to add or
subtract one or it's not like they were. Oh, we
have nine nine commandments, we better come up with one more.
We want an even ten.
Speaker 4 (36:18):
Well, who knows, maybe maybe, But I mean, at the
same time, with the example of the Bible, like I
was saying earlier, like it is hard not to if
you just read through the Old Testament, notice a huge
amount of odd numbers, especially a lot of sevens.
Speaker 3 (36:34):
I don't know that that that's meaning something.
Speaker 2 (36:37):
Yeah, I couldn't help but think about this one as well.
Over the weekend because I went with my family to
see the new Beetlejuice movie. Oh and of course one
uh summons the character in question by saying his name
three times in why not two or four? Yeah, And
we see the same with you know other you know
folk traditions, the old idea of bloody Mary, you know
(36:59):
some and her scaring yourself by seeing her in the
mirror by saying her name three times in a row,
that sort of thing.
Speaker 3 (37:07):
I got real freaked out about that. When I was
a kid, I had a phase where that was just
like super scary to me.
Speaker 2 (37:15):
I mean, I still am not going to do it.
I don't believe it, but I'm not gonna still not
going to say her name three times in front of
a mirror.
Speaker 3 (37:22):
And I mess around.
Speaker 2 (37:24):
Yeah, yeah, okay, So coming back to the law of
odds in general, Yeah, it's often touted as a deciding
factor in various various approaches to visual imagery, and I
have seen it mentioned as as lining up with food
imagery as well. You know, again, I think the example
used before was, if you're gonna have a appetizer of
(37:45):
sliders at a restaurant, you want as your menu photo
or your Instagram food photo, you want an image of
three sliders, not two. You want an image of three
sliders and not four, because three is going to be
an odd number. It's more attractive. And yeah, you can
throw in these other ideas of like, well there's a
central slider, I know which one is the lead slider.
(38:06):
But the thing is, when I started looking around for
studies about this, it seems like that experiments don't back
this up. So according to odd versus even a scientific
study of the rules of plating by woods at all
published in twenty sixteen in pere j Law and Environment. Yeah,
according to this paper, it just doesn't seem to work
(38:31):
quite as strongly as some might have you believe, they
actually conducted some experiments. I want to say it was
over a thousand folks involved in this, but you know,
they ended up contending that we have to take various
cultural factors into consideration here. You know, there's a lot
going on when we look at an image and if
(38:51):
we add but if we add that that image is
image of food, and it's food that we are on
some level considering eating, then it seems that overall portion
size is more important than odd or even numbers when
it comes to human perceptions of food.
Speaker 3 (39:07):
Okay, so we would rather have on average, would rather
have four sliders than three.
Speaker 2 (39:12):
Right, We'd rather have three than two, yes, but not
because three is odd, but because three is more sliders.
And of course this seems like a gross over statement
of the obvious, right, because it's like you go to
a restaurant. You're like, I'm paying you know, close to
twenty dollars for this plate of sliders. Of course I
want it to be four and not three, because I'm
getting more slider for my buck. Also, when you're hungry,
(39:34):
you're hungry, and your hunger is not always a great
gauge of how many sliders you need to satisfy yourself
and or those around you, you know, so you know,
on that level, of course four sliders sound better. Let
it be four and not three. Three is just maybe
a little less likely to satisfy everyone's cravings.
Speaker 3 (39:55):
But so on my understanding this right there, it's not
necessarily that they found that people prefer for evens to odds.
It's just that maybe, like if there is a preference
for odds, it doesn't play that big of a role
when compared to people just wanting more food.
Speaker 2 (40:10):
Right, right, And they provide some wiggle room there, because
again there's a lot going on when you're considering an
image or you're considering a presentation. I think there could
based on what I was reading here, I mean, there
could easily be a situation where ultimately having an odd
number is more important. Like maybe it's a very you know,
ritualistic presentation of food. Maybe it's a situation where the
(40:34):
present where the presentation is more about just having a
great photograph as opposed to, you know, making the potential
customer salivate. Again, there are a lot there's so much
going on when we look at an image, but you
cannot discount the importance of hunger when that image is
of food.
Speaker 3 (40:52):
It's it's about tricking people into believing that if you
get this sandwich the tomato on it will be red
and juicy.
Speaker 2 (40:59):
Yeah, in reality, it may not, may be very anemic
look at it. It may not have much flavor to it.
It may merely be wet and hopefully cold. In some cases,
that's fine, Maybe it's gonna work well within the context
of the slider the studying question. They also looked at like,
you know, they were looking at it like horizontal versus
vertical plating scenario. So I would be very interested to
(41:20):
hear from anyone out there who is involved in plating,
either professionally or you know, on an amateur chef level,
what your thoughts are on this.
Speaker 3 (41:28):
Oh yeah, I actually just got interested in how much
of say you're at, you know, sort of some kind
of elite level, you know, you're working at like a
very fancy, expensive restaurant or something plating choices. How much
of that is is an art and how much is
a science? Are you just sort of going off of
some kind of chef or stylists instinct there, or do
(41:50):
you actually do research on what people dining there prefer
in terms of plating in appearance.
Speaker 2 (41:56):
Yeah, I mean, and then there's also the whole the
economic value of there, right, because I mean, you have
to have to factor in like can we afford to
have a four slider platter? Shouldn't it just be a
three slider platter? Are we really going to lose business
because everyone thinks they need a fourth one? If they
need a fourth one, they can buy that out a
la cart perhaps, I don't know. There are a number
of factors involved.
Speaker 3 (42:17):
You know, I'm a big fan of chips and dips,
and for some reason, I really like it when there
are two dips. Oh okay, there were two different dips.
It seems like there should be three. Though there should
be three tips. I mean yeah, but then you start
once they're three, that's just like that's like a buffet
of dips. You get two dips, that's like really focused.
Do you get like one I don't know, one roasted
(42:38):
tomato salsa and one guacamole or something.
Speaker 2 (42:41):
Yeah, when there are three dips, I do find that
one dip is definitely going back in the fridge for dinner.
And then because you think, well, I'll use that later.
I'll definitely dip something in that later, and you don't
you just wash that up out and recycle it like
a week or two later. All right, I guess we're
out of time for this, but we didn't even get
into the whole idea of the seven layer burrito. So
(43:01):
just leave listeners to contemplate the seven layer burrito and
if that is an appropriate number of layers or should
it be less or more?
Speaker 3 (43:09):
I don't know the magic burrito.
Speaker 2 (43:12):
All right, Just a reminder for everyone that Stuff to
Blow Your Mind is primarily a science and culture podcast,
with core episodes on Tuesdays and Thursdays, short form episode
on Wednesday and on Fridays. We set aside most serious
concerns to just talk about a weird film on Weird
House Cinema and let's see what else to remind you of.
Oh yeah, if you were on Instagram, follow us on Instagram.
We are STBYM podcast. That's our handle, and you know
(43:36):
you can keep track of keep up a little bit
with what we're putting out in the podcast.
Speaker 3 (43:40):
Feed Huge thanks as always to our excellent audio producer
JJ Posway. If you would like to get in touch
with us with feedback, on this episode or any other.
To suggest a topic for the future, or just to
say hello, you can email us at contact at stuff
to Blow your Mind dot com.
Speaker 1 (44:02):
Stuff to Blow Your Mind is production of iHeartRadio. For
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