The Origins of Numeracy

What are numbers and where does our number sense come from? Is numeracy an innate part of our biology, or is it a cultural invention? In this episode of Stuff to Blow Your Mind, Robert and Joe explore the origins of numbers.

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Episode Transcript

Speaker 1 (00:03):
Welcome to stot to Blow Your Mind, the production of
My Heart Radio. Hey you welcome to Stuff to Blow
your Mind. My name is Robert Lamb and I'm Joe McCormick.
And today we're going to be looking at an interesting
question about the human mind and some of it's uh,

(00:23):
possibly innate or possibly learned capacities, and those capacities have
to do with numbers were Today we're gonna be looking
at the question of the uh, the origins in the
brain of numerous e Yeah, this is this is one
of those topics that I found rather interesting for for
a while, and it's it's a great one to dive

(00:44):
into because on one hand, there's the there's this the
stuff about it that just seems to be true that
you and you that we take for granted, and then
when you dive into it, you find all sorts of,
you know, arguments on two different sides of the equation.
And sometimes a lot of the argumentation is about like
where you're drawing the line between in this case, like
what is what is like preloaded hardware and software and

(01:08):
what is is learned what is transmitted educationally, culturally, etcetera.
Like where what is what is our innate uh number
since and how does that then allow us to build
upon it numerous e mathematics, etcetera. Uh, you know when
did the numbers come in? You know, is there is
there something that is five, that is that is already

(01:31):
in the brain or is easily um more easily acquired
by the human brain for as compared to you know,
animal brains. Uh. You know you can ultimately can kind
of chase your tail through all of this and it's uh,
it's it's a wonderful Uh. It's a wonderful experience, wonderful
topic we're gonna get into here now. Like so many
of the most interesting topics, one thing about it is

(01:52):
that you quickly realize, like the central now and that
you're discussing is much harder to define than you might
guess if you know, like everybody knows what a number is, right,
you just know that intuitively. But could you give the
definition of a number please? I'll wait, Yeah, I would
love for everyone out there to think about that for

(02:12):
just a second. So what comes to your mind for me?
Like if you just ask me that question and don't
give me a chance to sort of like back it up.
The first thing I think about our our shorts cartoon
shorts on Sesame Street, you know, because ultimately, like that's
the case we're sort of hit very early on with
with numericle and counting essentially propaganda, you know, like like

(02:34):
let us show you the way of the numbers. But yeah,
as we'll get into here, like what what is what
is actually already there, what is built upon, etcetera. Um, So, yeah,
I thought it would be good to get into a
you know, basically a brief discussion of just what numbers are. Now,
this might seem a bit elementary to many of you,
but first of all, I'd invite you to sit in

(02:55):
on an elementary math class or or just take a
look through an elementary student math textbook and see how
that comparison stacks up with what you think. You know,
I find it as often a uh learning experience about
oneself in one's own mathematical skills by by looking at
how kids today are learning math. Well, one of the

(03:15):
interesting things is that I don't think you can begin
to teach mathematics or even arithmetic sense by starting with
the most basic questions like what is a number? You
actually have to start at a higher level and work
your way down to that. Yeah, I think ultimately numbers
and math are are things that we we so easily
take for granted and we forget at a basic level

(03:36):
what they actually are. Uh So I think it's helpful
to sort of take a back step before moving forward
from in order to have that maximum wonder with the topic. So, UM,
I remember looking at a couple of sources on this,
uh several years back when I wrote an article on
math for how Stuff Works. So one of those was

(03:58):
uh Stanis Laws de Hainey's what are Numbers Really? A
Cerebrial basis for number since and that was published on Edge.
But I also had a wonderful book that sadly is um.
I think it's at the office, so I haven't. I
don't have access to it right now, but it was
by by two authors, Richard Kurant and Herbert Robbins, titled
What is Mathematics? That was from University Oxford University Press,

(04:20):
published back in But um, that's a great book if
you just want to, like, all right, I'm going to
start from the basics, let me learn what maths uh
and and and you know it been builds up from there.
So as as for what a number is Again you
probably don't need more than that moment of contemplation to
to state that it's a word, and it's a symbol,

(04:41):
and it represents account, not a vampire count, because that's
where myn my mind goes instantly as well. Again the
sesame street. But but account as in UM an understanding
of how many things, an analysis of how many things
there are a quantity. I came across a definition of
numbers that I thought was very useful, and this was

(05:02):
one that was derived from a study that was authored
by Raphael Nunez at All, who is a figure I'll
come back to later in this episode. But this definition
was that numbers are discrete entities with exact values that
are represented by symbols in the form of words and signs.
So like each part of that definition I think contributes

(05:24):
something important about what a number is. So, first of all,
discrete meaning each one is different from the others. Uh,
exact values is very important to the concept of a
number because because a number is different than a quantity,
quantities can be fuzzy, right Like you can look at
a quantity of something and say, I don't know, this

(05:45):
seems approximately more than this other thing. But seven is
not just more than five, It is exactly two more
than five, and that never changes. So yeah, so the
numbers are exact. And then finally, and you alluded to
this in your definition you just cited it's represented by
symbols such as words or signs. And this also seems

(06:08):
like a very important thing, and that a number can
exist independently of a concrete object being counted. So in
a world without human mathematics, obviously there could still be
a pile of five rocks if there were no humans
and no math. But could there still be the number
five without any rocks? That's an open question. I think

(06:29):
with numbers you can store, manipulate, and interpret the symbols
themselves independent of any observable material reality. To be counting,
you can just say, what's five plus two? Not like
five apples plus two apples? Yeah, Like, I guess it's
irresistible to compare it to words in this respect, even
though it's not a one to one here. But we

(06:50):
we think about how you have, like a word, the
word cat that stands in for that thing you're looking at,
that that furry creature may is distinct from other furry creatures.
And once you have that label for it, that enables
communication and various other more advanced uses of set information
as opposed to just having to describe the beast every

(07:10):
time you need to tell someone about it and not
having like an easy peg for what a cat is.
If you you want to, you know, engage in metaphors, etcetera.
You don't have to say the four legged boss of
my house every time? Right, Um? And and so I
mean this is where we get into like the idea
of Okay, what if I what if I didn't have
the word for cat, what could I still do? Uh?

(07:33):
You know? And likewise, if I did not have the numbers,
what could I still do? And this is ultimately going
to be a question that we're going to go back
and forth on throughout this episode. And and uh, you know,
their their arguments on both sides essentially, But um, it
comes down to this idea of of of of of
numeracy and also numbers. Since so I've read that the

(07:56):
numeracy seems to entail first of all, approximate representation of
numerical magnitude and then two precise representation of the quantity
of individual items. And there's there's an argument to be
made that this is an innate ability of human beings.
Will discuss more what this means in a bit, and
will also discuss animals um and the idea here is

(08:17):
that it that whatever is innate there does not depend
on individual or cultural acquisition of mathematical knowledge. That mathematical
knowledge is then built upon what is already innate. Yeah,
and I guess one of the big questions we're looking
at here is when humans manipulate numbers with their brains,
when they count, when they do arithmetic, additions, attraction and

(08:39):
all that, what part of what they're doing is innate?
What part is just they're already in the brain without
any education whatsoever. And what part, if any, is a
product of culture is something that was invented at some
point in history and has to be learned. Yeah. Now,
it's interesting to to realize that that numerous e appears

(08:59):
to predate literacy in human culture by several thousand years.
Neolithic societies used clay and stone counters to keep track
of quantities of stored goods, for example, and uh and
counting was a primary function of written records in the
earliest state societies of the late fourth millennium b c.
This was pointed out by anthropologist Brian and Fagan and

(09:21):
Eleanor Robson uh in the you know the Great Inventions
of the Ancient World. But Robinson is the author of
Mesopotamian Math and the Literature of Ancient Sumer so um
I found their thoughts on this rather interesting that they
point that the first large scale evidence of mathematics as
an intellectual activity probably dates to the Middle Bronze agent

(09:43):
Egypt around fifteen sixty b C. He but that would
be mathematics as an intellectual activities more in the realm
of what you might see people doing with you know,
pure math today. Obviously the more functional things like counting
go even further back much for the Yeah, this is
getting down to like how do we keep track of
these goods? How do we trade with these goods? We

(10:03):
need things to stand in for certain quantities. And this
definitely came up in the past when we've talked about
some of the earliest written records that exist. A lot
of that you might think, well, what are the earliest
written records? Is it you know, is it mythology? Is
it telling like a great poem about the creation of
the world. I mean, we do have very ancient examples
of that, but actually older than that are written records

(10:25):
that seem to usually denote uh quantities of property. Who
has how much of this? Or how much of this
do you owe me? And so forth. Yeah, and I
think this, like a number of topics we've discussed in
the show, I think a lot of it comes back to,
you know, some some key aspects of human cognition, that
there are limits to what we've evolved to deal with,

(10:46):
you know, and then we have to to build upon
upon that natural ability. So, for example, it's one thing
to know how many bags of coffee you need to
buy on each grocery store visit in order for you
or your immediate family to get through to the next week.
You know, Like, if you're like me, you may not
even need to get into numbers at all. You know.
You just realize, well, I have less than one bag,

(11:07):
I'll need more than that to get through the week.
So I guess, get one bag, and then once I
have that patterned down, I can just keep doing that
for the rest of my life. Wait, what what if
your need for coffee explodes? What if it just increases exponentially? Well,
That's a great question, because you could, I guess hand
you could. You can handle that at least for a while.

(11:27):
Um uh. That's the thing when complications enter the picture,
be it um, you know, fluctuations or or just increase change.
Or how about this. What if you were buying coffee
for two different houses, so your house and I don't know,
maybe you have a vacation home, or maybe you agreed
to buy all the coffee for your parents house and

(11:48):
your siblings house. Oh and then how about this, You
also have a business and it sells coffee and you
need to provide it with coffee. Oh, now you have
two locations with two different streams of client tell. So
I mean the details of this, I guess is important.
It's just the idea that, like whatever is in your
immediate sphere, regarding some level of number since uh and

(12:11):
even numerousy like, you're gonna have to You're gonna have
to build upon that if you're going to deal with
some sort of larger experience that emerges out of human invention.
This just goes to highlight something that I think will
be increasingly apparent as as we talked throughout the episode,
that what kind of sense of numbers you need has
very much to do with how you're making a living,

(12:32):
with what you have to do to get by, And
so some people may have ways of making a living
that are essentially almost totally devoid of need for for
numbers of more than a handful, whereas other people have
ways of making a living that are heavily exact number dependent.
And I like how you mentioned a handful of numbers, um, so,

(12:53):
so I think we're gonna do another episode in the
future that is going to deal more specifically with like
the creation of now is the invention of numbers and
different number of systems. But it is interesting to think
of our fingers and ultimately our toes as well, because
one of the initial steps here is that humans had
to come up with ways to augment their number, since

(13:14):
we already mentioned using little tokens to stand in you know,
clay tablets and what not to stand in for things.
But another method, of course, is just that's immediately available,
is turning to fingers and or toes. You have tin fingers,
you have tin toes. And for this reason, various numerical
systems depend on groups of five, ten, or twenty based
ten or decimal systems stem from the use of both hands,

(13:35):
while based twenty or vegicimal systems are based on the
use of fingers and toes. So the argument here is
that you know this is ultimately an externalization of number,
since that this is the root, the roots of mathematics,
or the bedrock upon which mathematics may be built. Um.
And I guess that's the way I keep coming back
to thinking about it, or the way that i've I've
thought about it for a while, the idea of of

(13:57):
of of number since and numerousy and mathematics. It's like
building this tower. You know that we we keep building. Uh,
that you have these different types of numbers that are utilized. Uh,
you have different um types of mathematics. And the higher
the tower a sense, the greater height, the greater power,

(14:17):
the greater your vantage point from which to understand the cosmos,
a cosmos that that some for instance Max teg Mark
goes as far as to describe as a single vast
mathematical object. So I actually got the idea to talk
about this today when I was reading a couple of
recent articles that I found very interesting. One was a
news feature in Nature from June of one by Colin

(14:40):
Barris called how did Neanderthals and other ancient humans learned
to count? Uh? And so? So that got my brain
going on this, But also I was reading an article
by Philip Ball and Eon Magazine called how natural is
numerous e? Uh? Now as to the specific archaeological evidence,
linguistic evidence, and other stuff about how how humans in

(15:00):
fact first started displaying number. Since we might come back
to that more in a future episode, I wanted to
focus more today on this question of how natural is numerousy?
To what degree is our number? Since in eight and so?
Of course, you know, like we were saying at the beginning,
it can feel very natural to be able to count

(15:22):
to a hundred and thirty seven, but basic numerical literacy
that includes counting up to arbitrary numbers and the ability
to do basic math. It might seem so natural that
you just assume it is an evolved biological capacity, right,
something that any human brain could just automatically do naturally.
But actually there's some question about this. The question would be,

(15:45):
how do we know that numerousy is not to some
degree and invented cultural capacity, more like the ability to
read sheet music or the ability to play football, something
that generally people can do if they're how to do it,
but it's not something that's like in our biology is
a part of our ancestral evolved capabilities. One of the

(16:08):
things I love about this discussion or or even argument,
if you want to frame it that way, and it's
you know, it's been going on for a while, is
it Also it lines up rather well with the the
the argument slash discussion of whether mathematics is a human
invention or a human discovery, you know, And and it's
one of those two where I don't know, being one

(16:29):
that's not like professionally engaged with either side, I tended
to I tend to sort of fall in the middle
and think, well, it seems like it's it's it's both, right.
I mean, it's it's both this thing that we uh
that is the universe and is a description of the universe.
It is both this thing that we have some level
of innate capability for, and yet it is also there's

(16:51):
also undeniably um, you know, plenty of it that is
acquired that is uh, that is written down in a
textbook and then or or put into a set semi
street short and then related in you know, into the mind.
So uh again, that's that's that's something that I just
find um fascinating about the topic. Well yeah, I mean
I think you could argue that it is to some degree,

(17:14):
Like you could say, it's like the rules of chess.
So like chess is not something that exists outside of
human invention. Humans had to invent it. But once you
have invented the rules, it's not up to human chess
players to say, like what is the most advantageous move
or something like that. That's just objectively true, right you know,

(17:35):
So like you you have created a system of rules
and symbols, but it turns out within that system of
rules and symbols you can discover objectively true facts about
the universe. Right, So I guess it would be like
if chess gave you objective understanding of actual warfare. I
don't know, maybe it does to a certain extent um

(17:55):
except that horses. Horses are not limited by going what
to space one to the side or or one space
up and then one diagonal, depending on how you which
is it, which is it considered? Is the horse going
two up and then one over, or is it going
one up and then one diagonal? The night. Rather, it's
not the horse. I think it's two up and one over.
It's two up and one over. Yeah, it is a

(18:17):
fact of physics that bishops can only move diagonally. Have
you ever seen a bishop streithe? I have not, It's true.
But anyway, to come back to the main question we're
talking about here again, it's not so much the bigger
question about like is math a a pre existing sort
of fact of the universe or is it a human
invention that merely describes the universe? The question here would

(18:40):
be uh, is numerous e a baseline evolved capability in
the human brain, meaning like, do you have number meat
in your head? Or is it a cultural invention that
makes use of some meat in your head? It makes
use of the brain's natural capacities, but is not itself
and evolved in a copa pacity, not something that would

(19:01):
be arrived at unless you were taught it. Thank thank now,
there is plenty of evidence that researchers point to as
as supporting the idea of a biologically endowed number sense,
And one thing that often gets pointed to here is

(19:22):
the is the capacity of other animals for certain kinds
of number consciousness and so the question would be to
what extent are non human animals capable of numerous e
what kinds of number consciousness, if any, can they demonstrate,
and where do they differ from us? So studies with animals,
including monkeys, some apes, marine mammals like dolphins, and dogs,

(19:45):
have shown that these creatures do have some basic innate
sense of quantity. For example, they can look at two
groups of food items and they can usually tell which
one has more items in it, provided that the numbers
of items are small enough. So if it's like, you know,
fewer than ten items, more often than not a dog

(20:06):
can look at that and tell which which pile has
more food items in it and go to that pile,
And in fact it sometimes even to surprising extents. There.
There was one study that Ball linked to that I
thought was interesting called Quantity Based Judgments in the Domestic Dog,
published in the journal Animal Cognition in two thousand seven

(20:26):
by camill Ward and Barbara be Smuts. And these authors
tested dogs on the mental management of different food quantities,
and one of the things they found was that in
in a second experiment, they did so well. Their first
experiment was that they would simultaneously visually present two options
to a dog and they would see, you know, does

(20:47):
the dog reliably choose the larger quantity of food instead
of the smaller quantity? And they found that yes, dogs
on average do tend to go more for the larger
quantity when they can see both. But they found that
numerically close comparisons were more difficult, So like, you know,
if if it's like five versus six, the dog's going
to have a harder time going to the six than

(21:08):
if it's you know, eight versus three. But they also
found interestingly that in a second experiment, they had some
additional conditions where the food was not visually available to
the dog at the time they made their choice, So
the food would be shown to the dog and then
hidden from the dog, and then the dog would have
to make a choice. So the question is does the

(21:30):
dog remember the differences in quantities when it can't see
it right in the moment, And they found yeah, even
in this case, quote subjects still chose the larger quantity
more often than the smaller quantity when the food was
not simultaneously visible at the time of choice. And they
also said that they worked to exclude other cues like

(21:50):
olfactory cues. You know, maybe the dog can smell more
food in one case than the other and uh and
queuing by the experiment ers they work to eliminate those influences.
Uh So, it seems if there's really no chance that
dogs have culturally learned or invented number systems that they're
working from. Uh So, if they can more often than

(22:11):
not visually assess numbers of food items, tell which grouping
has more even when they can no longer see them,
maybe you could interpret that as a sign that there's
some kind of innate capacity, not just in human brains,
but in the broader mammalian brain structure for understanding numbers,
at least in a rudimentary way. Right. Well, maybe not

(22:31):
so fast. We we will come back to that. Um.
But there are plenty of other examples that have been
cited of of animals showing some kind of sense of
what could be called numerous e or maybe would be
called appreciation of quantities, if not numbers. Uh. For example,
I was I was looking at one study published in
Philosophical Transactions of the Royal Society b Biological Sciences by

(22:53):
Rosa Rugani that was studying, uh the appreciation of differences
in quantities in day old chicks. So these are chickens
they've hatched for one day, and they can tell some
differences in quantities of food looking like pellet items. So
you know when the dogs the chicks this, this is

(23:13):
not something they learned in school. It's not a cultural
invention here that this is playing on some innate capacity
they have. Now, I found that really interesting what you
said earlier about about them accounting for the idea that
the dog might smell more meat in one direction as
opposed to the other. Uh and and and instead focusing
more on visual stimuli. And then perhaps this is something

(23:34):
to get into in the study for sure, But is
that fair? I wonder was something like a dog who's
who's um olfactory senses are far superior to that of humans,
like ult like if they are thinking based on olfactory
data as opposed to visual data, Like, aren't we ultimately
talking about the same thing? Well, I mean, so that

(23:57):
gets into something interesting about numbers, right, So a dog
dog could maybe smell that one pile of meat has
more meat in it than another pile, But would that
involve numbers? Because so you could take like one ounce
of meat and cut it in half, and that's now
like two pieces of meat, but it would be the
same mass of it, right, so you'd imagine it would

(24:17):
admit probably about the same amount of smell. But it
seems like the visual sense is especially useful for distinguishing
numbers of objects, you know. I don't know how exactly
that would change our understanding of the study, but it
is a good point, and it raises this interesting question
about gross quantity versus discrete numbers of quantity. You know,

(24:38):
like doesn't make make a difference to a dog to
have like, uh, if it's the exact same weight of meat,
but it's cut into more pieces, clearly, And this may
be a product of our numerical education. We're very primed
to think about numbers of pieces of something, you know,
which is why it seems like it's better to get
like more smaller pieces of candy than one big piece

(24:58):
of candy, right, mean candy being key, I guess, because
you know, also candy plays into some of these experiments
with humans, which we'll get into in a bit here.
But like you think about the like the chocolate bar
that you get and it already has the divisions, you know,
mapped out, uh, so that you can be completely fair
about how how the pieces are broken up. This is
a no cheat candy bar. Yeah, but anyway, I also

(25:21):
wanted to mention that there is research also in human
babies for evidence of a biologically endowed number, since that
has not learned through culture. And so, for example, in
his article, Philip Ball sites a cognitive neuroscientists named Daniel
and Sorry of the University of Western Ontario in London, Canada,
and am Sorry says quote, studies with newborns and infants

(25:44):
show that if you show them eight dots repeatedly and
then change it to sixteen dots, areas in the right
parietal cortex of the brain respond to a change in numerosity.
This response is very similar in adults. And this this
might not be exactly what it seems. We can come
back to this later, but that's at least sort of
a baseline finding. And there are plenty of people who

(26:07):
adhere to this view and interpret this uh this evidence favorably.
For example, Ball sites a researcher named Andreas Nieder who
is a neuroscientist at the University of Tubingen in Germany.
And actually Needer recited in both of the articles I
mentioned earlier, but this researcher argues that the neuroscience helps
support the idea that human number since is innate and

(26:28):
a product of biological evolution rather than just culture. And
one big clue here is the is the brain imaging
showing similarities between what happens when non human animals and
babies process quantities in the brain and when adult humans
process quantities and the numbers in the brain. And so
Nater would argue that the similarity in the biological substrate

(26:51):
here points to an innate biological capacity that you've got
some number meat. And this debate has actually been going
on for quite some time. I think there's sort of
a revival and interest in it with some recent research
that's made coming out, But people have been talking about
this question for a while. Yeah, I mean to the
point where I mean, I feel like it would be difficult.
Like sometimes when we're discussing two sides of a given debate,

(27:14):
we can sort of easily point to like the key papers,
the key studies um or just or or just you know,
pick a handful, And I feel like this is one
of those situations where you, yeah, you just have kind
of like ebbs and flows of of of as far
as the whole research goes. But uh, yeah, like as
far back as there was an article that was looking

(27:34):
at it I found rather helpful by Robert Schwartz Um
discussing the debate, and this was in the Philosophy of
Science journal, and the article was titled is mathematical competence
in eight? And schwartz points out that you know, some
of the earlier principles before models, Uh, the theseum, the
work here relates to uh, some published and findings in

(27:58):
the nineteen seventies by Gellman and Galastel, and that they
quote argue that innate number specific principles underlie children's ability
to count. Okay, so maybe we invent the words for numbers,
but that the the number since is already something in
the brain that they're innately harnessing. Right. But then a
counter argument that shwarts mentions uh, And this was one

(28:20):
that that he sides Karen c uh Fusan from who
argued a principles after model, and the idea here would
be that children begin by mechanically repeating sequences of counting
words and it it builds up from there. Um. And
so Schwartz goes from there, you know, in relatively short time, uh,

(28:43):
to discuss the divide as follows. Want to read this
quote because I thought it was pretty helpful quote. In
many discussions of mathematical cognition, the principles before model is
identified as a Nativist thesis and the principles after model
as non nativist. I believe this is a miss steak.
For suppose the principles before model is correct, that children

(29:03):
understand the basis for counting before they are able to count,
and that this understanding guide skill development nothing specifically mathematical.
Is thereby innate A principles before model is a Nativist
thesis only if the how to count principles are themselves
not learned. Showing this, however, is not easy. So I
don't know. I like that breakdown. I like the way

(29:25):
that he approaches, uh, the divide, if you will, Oh,
I see. Okay. So even if there is some underlying
capacity that's being harnessed, um when when you are learning
to count, you would still have to show that the
underlying capacity was not itself something that had been learned
before the counting education took place. Yeah, Like, I mean, well,

(29:49):
one way that I was thinking about some of this earlier,
was too, was thinking about like, Okay, what happens when
I pick up a paintbrush, you know, because on one level,
I have more or lessoning, you know, I have this
in a ability where I can pick up something and
it and my body schema updates to incorporate it. I mean,
that's just basic tool use. That is, that is something
that that our our species has going for it, right,

(30:11):
But that doesn't mean you can pick up a paint
brush and then uh, you know, you know, reproduce the
works of Michelangelo or what have you, or create just
anything of of you know, of of of usefulness. There's
a place where you cross the threshold of innate ability
and you get into um education and skill acquisition. Right. Well,

(30:31):
and I guess it's a good question in the case
of the paint brush also like what part of it
is the innate part? Like what what's the part that
is just what kind of animal you are? And what
is the part where your seat you're watching things that
other people have done and you've learned from them. Um,
there's a good distinction. In that Philip Ball article where
he talks about the example of tennis you know, so like, uh,

(30:56):
nobody would seriously argue that we are biological evolved to
play tennis. And yet, of course tennis does make use
of tons of things that are biologically evolved. Good tennis
players make use of a range of capacities that probably
evolved based on their ancestral pressures, involving things like hunting

(31:16):
or searching the environment for movement or escape behaviors, perhaps throwing.
You know, so like there were things that originally shaped
what our bodies are nervous systems and our muscles could do.
And that's not exactly what we're doing, but it's somehow
close enough that we can use those uh skills and
capacities for this highly artificial thing like tennis. Yeah, like

(31:40):
what is tennis? But hit thing with stick? Right, but
hit thing with stick that is taken to um a
very specific and advanced degree um you know, with you know,
hitting things with stick is something innate ever since the
monolith came. But but you know, you could like take
like all the Olympics, like a lot of it kind
of well not all of it, but there's a lot
of stuff there that kind of comes down to hit

(32:02):
thing with stick or do things with rock and or
do things with rock and stick. Yeah, there are very
few competitions there that you could say are just like
purely biologically adapted to though maybe things that are just
purely like running or jumping or wrestling or something like that.
But yeah, once you're involving uh, once you're involving rules
and a ball and all that, you're getting increasingly abstract

(32:22):
away from the ancestral environment. But but the environment is
also key here because, like you you pointed out earlier,
like it's not just a situation of you know, you
pick up the paintbrush and then you figure out in
a vacuum what to do with it. You're immersed in
in in a culture, in an environment in which it
is used in a particular way. You're seeing it used
in a particular way. Even if if and then you know,

(32:45):
there's probably going to be some level of of of
direction and education uh there as well. Uh so I
love For instance, there was another treatment on this by
Baruti at all Um in a titled The Development of
Young Children's Early number in Operations since and its Implications
for Early Childhood Education from two thousand and six, and

(33:05):
they pointed out that you know that that a young
child's whatever their spontaneous number attention is, that's how they
defined its spontaneous number attention. It's then going to be
also affected by their age, by their language, by their
collective makeup. I mean, they're they're all these other factors
that come in. Yeah, of course that's true. And so
one thing that might be helpful for sorting this out

(33:28):
is looking at the question of what is it that
that typically non human animals and and babies have not
been documented to do in any known case that that
can be done once you have a numerical education. And uh,
I think one of the important things here is the
ability to make fine distinctions between differences in quantities of

(33:51):
again more than a handful. So animals of all sorts
with non symbolic quantitative senses might be able to tell
a difference between one and two, or between two and three,
or between three and five, and they might be able
to tell a difference between one hundred and two hundred.
But what seems really unique of humans with a number

(34:12):
of education is the ability to tell the difference between
something like twenty one and twenty two, or the difference
between a hundred and fifty and a hundred and fifty three,
and I do think the researchers who favor the biological
adaptation argument would would acknowledge this point that this seems
to be a unique and different kind of thing. So

(34:33):
I guess this brings us to the other camp of researchers,
the ones who are more sympathetic to the idea that
that numerous e is in some sense a cultural invention
and it needs to be learned. Now, there's one major
figure in this area of research who is mentioned in
both of the articles I talked about up top that
that got me interested in the subject, and it is
the cognitive scientist Raphael Nuniez of the University of California

(34:57):
at San Diego. Now, I knew his name seemed familiar
when I read it, so I was like, I bet
we've talked about him in the past. I looked it
up in our previous notes. And actually he was involved
in some of the cultural variance research on pointing that
we talked about in the pointing episodes. You remember this, Rob, Yeah,
it was the study that found that while pointing with
the index finger is very common around the world, there

(35:20):
are some cultural and language groups that have a preference
for facial pointing, pointing often with the nose instead of
with the the index finger, And the specific example they
looked at was the you know, people of Papua New Guinea.
I remember that study was really interesting because it was
trying to find, well, what could be some possible explanations
why this, uh, why this one group of people, this

(35:41):
language group tends to prefer pointing with the nose instead
of with the hand, or at least a higher relative
frequency of pointing with the nose than you find in
other cultures and language groups. And there were a number
of possible explanations there. I didn't want to get into
all of them, but one that seemed interesting and perhaps
very relevant to today's topic was the possibility that it

(36:01):
could have something to do with the yep No language
having more geographically specific demonstratives than many other languages do. So,
for example, not just like this and that, which we
have in English, but versions of this and that that
encode more specific location information with quote uphill downhill distinctions

(36:24):
and a three way distance contrast. So imagine if you
had words for like this and that that sort of
included something akin to uh north southeast west nearer further
and that kind of thing. Would you need to do
as much pointing in your life? Oh, that's a good point,
or at least would you need to do the kind
of precise pointing that's achievable with the finger, or would

(36:47):
a more general kind of facial point or not in
the direction be more suitable anyway, I remember. So that
was really interesting. But we're talking about the same researcher here, Nunez,
and he has an idea to make sense of some
of the research with animals and babies showing some limited
numerical distinctions they can make. And his idea here is

(37:08):
that of making a distinction between numerical cognition and what
he calls quantical cognition. So here to read from Philip
Ball's article and summarizing this quote, the perceptual rough discrimination
of stimuli differing in numerousness or quantities seen in babies
and other animals is what he calls quantical cognition. The

(37:29):
ability to compare a hundred and fifty two and a
hundred and fifty three items in contrast, is numerical cognition.
Quantical cognition cannot scale up to numerical cognition via biological
evolution alone, Nuniaz said, And this seems to correspond with
the possibility that without education. To the contrary, humans naturally

(37:51):
tend to process quantities in terms of a long arrhythmic
scale rather than an arithmetic scale. Yeah, there was one
study that I was looking at. There are a lot
of studies that deal with logarithmic thinking in in infants,
but I was just looking at Duke Institute for Brain
Science and study that found that babies that were good

(38:14):
at discerning between large and small groups of items before
learning to count, we're more likely to do better with
numbers in the future. And so the idea here again
is that there there's you know, some sort of primitive number,
since that the acquisition of the numerocy and mathematics is
built a top off. But even that, the researchers, they
were you know, quick to point out that this doesn't
mean you can totally predict an infants mathematical future off

(38:37):
of this data, but rather that there's some sort of
cognitive overlap between the two. Yeah, and I think there
there probably is. But to explain the difference more, we
should say, so the arithmetic scale is the one that
you learn in school, the arithmetic scale is the one
with a number line where each number increases by one
and is evenly spaced, So one fifty one is more

(38:59):
than one fifty, So when you picture the number line,
it's like that each plus one is is evenly spaced.
But under log arrhythmic cognition, in contrast, the difference between
numbers is about ratios rather than about absolute magnitude, increasing
one unit at a time. So under log rhythmic cognition,

(39:19):
the difference between something like you know, like one thousand
and two thousand can be viewed as similar to the
difference between one and two, even though on the arithmetic
scale it's a thousand times more of a difference, and
you can you can again see this kind of comes
back to the coffee description from earlier, like you know,
on on on a on a log rhythmic level, there's

(39:41):
really what's the difference between a thousand and a thousand
and one? Right? And if and if you're just dealing
on an individual level, I mean, there's ultimately no experiential
difference between the two. But if you're dealing in units
of like a thousand things you know, like and then
then there is a real difference between a thousand and
a thousand in one, you know, or between a thousand

(40:02):
and you know. Well, this actually brings me back to
do you remember when we did the episode on Fermi estimation. Uh,
it's a slightly different thing, but so, Uh, you're physicist
like Enrico Fermi, you obviously need arithmetic sense of numbers
in order to do precise calculations to do your science.
But also, Fermi was famous for being good at estimating

(40:25):
quantities when other whereas other people, you know, other colleagues
of his, who had very good arithmetic number sense, might
not be. He was really good at just looking at
something and guessing some huge number that would turn out
to be actually quite accurate. And apparently a lot of
his reasoning was based on not getting bogged down in particulars,
but thinking about things in terms of orders of magnitude,

(40:47):
which actually seems closer to the long rhythmic type consciousness.
Maybe that kind of thing is really good for fast
and dirty estimation of meaningful quantities. Yeah, But another way
of putting it is that on the log arrhythmic scale,
differences between numbers become smaller or less important as the
numbers increase so you know, the difference between one and

(41:09):
two is huge. The difference between two and three is
still pretty big. Once you're getting into the differences between
twenty eight and twenty nine, these are not very meaningful
distinctions anymore. And there is some research indicating that some
people who live in hunter gatherer societies today tend to
conceive of quantities in a log arrhythmic sense rather than

(41:30):
an arithmetic sense. And I think you could make a
very strong case that this way of looking at numbers,
the logarhythmic version, is biologically adaptive in a way that
that arithmetic numerousy is not necessarily so. Again, to read
from balls article, quote, attributing more weight to the difference
between small than between large numbers makes good sense in

(41:51):
the real world and fits with what a researcher named
vim Fis says about judging by difference ratios. A difference
between families of two and three people is of comparable
significance in a household as the difference between two hundred
and three hundred people in a tribe, while the distinction
between tribes of a hundred and fifty two and a

(42:12):
hundred and fifty three is negligible. And so I think
this could be a very insightful way of breaking through
this issue. It seems to be quite conceivable that logarithmic
cognition is the baseline for reasoning about quantities, is just
sort of what our brains naturally do, and that we
have to harness that innate capacity for for logarithmic thinking

(42:36):
and retrain it to use the equally spaced arithmetic number
line through education in school, since that arithmetic number line
is useful for certain types of work, work that we
often end up getting trained for, like if you need
to be an engineer or an architect or something. Yeah,
And I think it's also worth worth stressing that that
the idea that say a human infant can engage in

(42:59):
logarithm thinking, Like that's incredible, Like that's that's I find
that really amazing. I like, I don't think the read
on it should be if if that is indeed the
cut off, if that is the base upon which the further,
you know, tower of numbers and mathematics is built, Like
that's still really incredible. I think that's that's that's amazing
to think about the way that the you know, that

(43:21):
this developing mind is able to to view the world
and look at, you know, one pile of marshmallows versus another.
I know I'm taking the difficult um stance of babies
are are are good as opposed to babies are bad
and dumb? Well, yeah, I think one very important takeaway
from this is that the more log arrhythmic style of
conceiving of numbers is in no way an indication of

(43:44):
like a lack of sophistication or anything like that. Instead,
it has to do with what kinds of quantity concepts
are useful for your way of life, like what do
you need to do to get through a day? And
for some ways of making a living, arithmetic cognition maybe
more useful, but for other ways of living, a more
approximate log arrhythmic cognition might be more useful. So it's

(44:06):
really just a question of what do you need in
order to do what you do to survive. Than There's
another thing Philip Ball sites in this article that I
had read before but I had forgotten about until now,
which I thought was pretty interesting. He talks about some
of the research of Jean Piage in the nineteen sixties

(44:30):
that was about how young children often instinctively use visual
features of quantities rather than explicit counting, in order to
judge the magnitude of a quantity. For example, if you're
to ask a child you know which group of marbles
has more in it, you take the same number of
marbles and you line them up widely spaced versus densely spaced.

(44:51):
Young children will tend to think that the group with
wider spacing has more marbles in it, even if they're
the same number. And I could be wrong about this,
but I think I recall when I was reading this
that the studies showed that the kids thought even if
you just moved the same number of marbles around right
in front of their eyes, they still thought when you
space them out there was more in them. Interesting. Yeah, this, Uh,

(45:14):
this reminds me of of something that I've accused my
child and my cat off on many occasions. Not so
much might keeping my child anymore more when he was younger,
but uh, the idea of crumb blindness, where once the
larger portions of the particular meal have been consumed, there
is an inability to realize that there is still a
substantial amount of food on the plate, albeit in smaller spread.
Out form. Um. I think the boys figured it out,

(45:37):
but the cat still seems clueless to this. Well. I
think it could also highlight something, which is that, Um,
there's some indication I mentioned earlier that I was going
to come back to that that research indicating that babies
are maybe using what is sometimes believed to be a
number module in the brain when they're judging different quantities
of objects in their visual field. Uh. There was that

(45:59):
researcher Dan Elend sorry who I mentioned earlier, um who
who also is cited in this article talking about how
the neuroscience research into human infants actually became more complicated
as it went on, where he said that more recent
research has revealed some dissimilarities between the way that brains
process non symbolic numbers. So that would mean like something

(46:20):
that you can look at, you know a number of
dots in your visual field, versus symbolic numbers, you know,
numbers that you're manipulating based on their symbols nine plus
five or something, and this more recent research has found
that they're not always correlated. Uh. To read part of
a quote from n sorry here that challenges the notion
that the brain mechanisms for processing culturally invented number symbols

(46:42):
maps onto the non symbolic number system. I think these
systems are not as closely related as we thought. So
maybe the brain is actually doing something importantly kind of
different when it's judging quantities based on you know, visual cues,
like looking at a number of physical objects, versus when
it is manipulating abstract numbers through learned symbols. But this

(47:06):
makes you wonder about like, is it possible that a
crucial element in number since is actually language or some
form of language. Does having a naming system for numbers
unlock types of numeracy that aren't really there if you
don't have that naming system. Yeah, I mean it kind
of comes back to what I said earlier about the cat.
Right if without the word for cat, you can't engage

(47:27):
in more complex uses of its basic catinus. You know,
I can't. I can't make comparisons and analogies and metaphors
regarding the cat. Uh if I don't have some sort
of word for it, right, Well, this here you go.
Maybe this will tie way back to our first Monster episode.
Could you imagine a cat human hybrid if you didn't

(47:49):
have a word for cat, does having a word for
these animals allow you to start mixing and matching them
in a way that you wouldn't if you didn't have
the word. Yeah, doing theeomorphic arithmetic. Yeah, I don't know,
just a weird idea now that I think about it,
Maybe I'm doubting that. Actually surely you could picture that
with it. Well, I don't know who knows what Well,

(48:12):
I'm doubting myself every which way. Now, Well, but I
guess that I guess on one level, Yes, if you're
you're a human being, you're standing next to a human
being and you see the lion, I'm I'm gonna, you know,
be generous enough to imagine that that, even the ancient
and imaginative force was enough to put the two together.
But then for that combination then itself to have some

(48:33):
sort of value that can be easily transmitted. Then it
helps to have those two words, right, because then you
can say, like, imagine the wolf man or you know,
imagine the lion man, as opposed to saying, hey, you
ever seen that credit out there that you know, that big,
the big scary one that that that uh you know,
I don't know, eight dog the other day, Well, imagine

(48:54):
that but with Tharg's head no instead of its own.
You know, it just becomes more complicated. Try and put
the two together. But if you have the words, then
you have the like then the combination of the two words,
the hybrid is also so much more easily um conveyed
to other people and discussed and used and has kind
of weight all its own. Yeah, we may have to

(49:15):
come back to this, but also back to numbers, because
I think there's another way of looking at this question
that that we'll have to revisit in a future episode,
which is, do we have any evidence that could answer
the question of how in fact our prehistoric ancestors first
started using the concept of numbers and displaying signs of
a number since as opposed to just a quantity sense. Uh,

(49:37):
And if there are signs of that, can that shed
any light on this question? I think we will come
back to that in a future episode, maybe a very
near future episode. Yeah, yeah, I mean it might. We
were talking about this earlier. Should we do Apart one
in a part two? Maybe it is ultimately a part
one in part two instead of like two related standalone episodes.
I don't know, but but yeah, we'll think it will

(49:58):
definitely be back to this, guss this more like ultimately
like it's kind of the invention of numbers. Uh what
what what does it mean? And then where is the
invention taking place? And what cultures, what different systems, etcetera.
And what can we what can we learn from those? Yeah,
I'm chassed jazz for numbers. Man, I'm gonna have one
more small child story. And I don't know how this

(50:20):
relates to anything we've discussed. Maybe it doesn't. Um, but
when my son was either less than two or more
than two, I'm not sure how old he was, he
was approximately a two year old, I think, Um. I
remember there would be these situations where he would have
a snack or a meal or something generally a snack
situation or some sort of like shared um uh you

(50:43):
know plate situation, and he would be excited about eating something.
And then if if if I, as an adult, were
to come up and have a piece of it, like
to take a cheerio for myself or what have you
or a chip. Uh. There were on more than one occasion,
he would look dejected. He would say why you eat
it all? And uh? And I keep coming back to that,

(51:06):
like it like a you know, some level of like
food and security or just like or this, you know,
how does it relate to our discussing of of uh
you know of of you know, thinking about quantities and numbers,
because like, clearly I didn't eat at all, and never
did I come around and just eat all of the
food that was allocated to him. Uh oh, this seems
very related to the general psychological phenomenon of um of

(51:30):
totalizing the single experience. You ever notice, how like if
you if you do something one time that somebody doesn't like,
sometimes they'll be like, why are you always doing this?
But in fact, you did it once, but if like
they really didn't like it, it feels like it's always happening,
right right, So anyway, like I said, that may have

(51:51):
may have no connection to anything we discussed here, but
I've been thinking about it in the background the whole time,
so I had to share it. Why you eat them all?
All Right, We're gonna go and close it out for today,
but we'll be back with more discussion of numbers and
more discussion of various other topics in the immediate future.
In the meantime, if you would like to listen to
other episodes of Stuff to Blow your Mind. You can

(52:11):
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(52:31):
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and we just discuss some manner of b movie or
weird film or whatever it happens to be huge things
as always to our excellent audio producer Seth Nicholas Johnson.
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(52:53):
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(53:13):
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