**Robert** and **Jad** about whether you could live without numbers. Jad introduces his newborn son, **Amil**, and insists that he has no concept of numbers whatsoever. Like father, like son? Producer **Lulu Miller** talks to Stanislas Dehaene, whose work in neuroimaging suggests that Amil probably does have a number sense. You and I might not even know what logarithmic counting is, but apparently we used it as babies. Susan Carey explains why counting pennies is no small feat. Using an experiment designed by Karen Wynn, Susan breaks down the trick that separates us from the animal world: the counting song. Producer **Amanda Aronczyk's** daughter **Mina** demonstrates how complicated this whole penny business really is.

## Comments [47]

In the research involving "the middle number between 1 and 9" that was asked to the Amazon people, how exactly was the question phrased to elicit their response?

It seems there's one area of life where we all retain child-like logarithmic counting all the way into adulthood: money. If somebody offers us a $5 discount off a $100 mobile phone, we get a lot more excited than if offered a $5 discount off a $1000 large-screen TV. In theory, money is completely fungible: $5 is $5, right? What I can do with that extra $5 doesn't change. Yet it seems we react emotionally to the $5 as a proportion rather than an absolute number.

The research of Alison Gopnik, you may have had her on a previous program, observed that a keen sense of mathematics is at the core of the best machine learning programs that we have now. And some 10 years ago, she suggested that babies might be doing the same thing; " babies are actually making complicated calculations with conditional probabilities that they're revising to figure out how the world works."

Could it be that children are doing statistical analysis?

See Gopnik's TEDTalk

Engrossing. But: Astounded how the hosts just cut off the female guest mid-sentence and then go on and on, enjoying hearing themselves talk. I kept waiting for them to let her finished her thought, but no...

Jad & Robert, I was thinking this over and suddenly realized we do still think logarithmically with numbers! so we learn 1+1=2+1=3+1=4 and so on. but this doesn't hold true if you visualize objects in your mind. imagine 10 quarters on a table, add one. (no problem, you can imagine it differently) but now imagine 10,000 quarters, now add one (I don't think you can "feel" the difference) its too small! Logarithmically! (you probably cant even really image 10,000 quarters in your head) I tapped out around 100 (10x10) past that and its fuzz.

I think we are born with natural number sense. If we stimulate babies brain for math early we will help children to harness this natural number sense. To get babies started we can introduce them to a number song that wires their brain to become more connected to mathematics. For a suggested number song, please visit www.numbersong.com

Great piece, thanks for creating it.

But at the end, there seems to be a conflation of "mathematics" and "integers" or "arithmetic" by the hosts. You don't need an intuitive understanding of a linear number system (as opposed to a logarithmic one) to understand the broader world of mathematics. (Thinking so seems a bit ethnocentric to me.) Linear thought supports thinking about absolute quantities, but absolute quantities are still expressible and fathomable on a logarithmic scale.

Anyway, number scale is quite separate from many mathematical concepts beyond arithmetic. Some areas of mathematics are much easier to understand intuitively with a logarithmic scale, particularly those that pertain to biology, sociology (group power as a function of the *magnitude* of group size), psychophysics (decibels, richter scale, brightness/luminous flx), and all sorts of practical problems (length of a line for coffee, size of a crowd). *So even as adults, we do continue to think and perceive in logarithmic terms naturally.*

It's not uncommon to say "the line was three times longer than usual", but that's somewhat imprecise. I think the "sad" thing about the logarithmic scale is that it's essentially a language we can intuitively understand, but have unlearned the ability to speak in precise terms.

I completely disagree with the results of Stanislas Dehaene's research, discussed here.

In his experiment on young babies (2-3 months old), he was measuring the neural response from different regions of the brain. And he found that when the repetitive series of images was changed from that of, let's say ducks, to trucks, a response was measured in region "A" of the brain. And then he repeated the same experiment (not clear if he repeated it on the same babies -- who's brains have in some way been "primed" for this experiment).

On the second run, he didn't change the object displayed (duck vs. truck) but rather the quantity. He didn't explain how he changed the quantity. Meaning, he was using a completely visual stimuli to "engage" and "test" the babies.

And with visual presentation of pictures, when you simply add more (in his case, much more -- presumably a "logarithmic" increase -- the resulting image will take up a logarithmically larger AREA (spacial area). For the visual regions of our brain, they are very efficient at discerning logarithmic changes in spacial area.

So how does he know the different response (signals in a different area) that he found wasn't due to the increased spacial area, and not this notion of "numbers"? If he's equating area-increase with numerical-increase, he may be correct.

But a neuroscientist/researcher would not say the two (area-increase and numerical-increase) are the same.

It appears to me he confused what the babies were observing. They had a different response (in a different area) because they correct "noticed" that the video frame was much more filled-in (since 20 ducks would take-up much more space than 10 ducks).

He needs to normalize this "spacial" issue out of his experiment, by using "smaller ducks" when showing 20. Thus, the overall size (area of "duckyness") of the image -- to the babies -- would remain similar. And in this case, I believe he will find no difference.

I don't think his research proves that babies are born with a logarithmic sense of numbers. It believe these experiments he discusses prove instead that babies' sense of visual/spacial area is logarithmic.

This is one of several common errors still made by scientific researchers today..... their inability to correctly assess the exact variable they actually change (as opposed to the one they intend to change) in an experiment. In his experiment, the error was between spacial-area versus number.

Hi Beattie. Being a descendant, I am always looking for possible Lesnick relatives

Andrew at andrew.berman@sbcglobal.net

I liked the Pi song. I would really like to make it into a ringtone. Anyway, I read that in Fundamentalist Christian states, there is a law that makes Pi exactly 3.:(π

The funniest thing about the episode was the radio host's absolute dislike of numbers, or should I say, arithmetic. I'm currently a math major and I do less arithmetic than anyone. I don't ever use numbers, but greek letters! If you want numbers to be totally out of your life, go into math! HA!

Anyway, as math major, this episode is very dear to me, on many levels. I have explored the different infinties deeper than the average person for many hours, but that part was still hauntingly beautiful. A+ overall

Relistening to podcast: had to pull over car to type this in.

I'm convinced our perception of time is logarithmic. My second and third summer vacations from school stretched out like an endless road. And now in my forties the seasons hurtle by, and I can actually make plans to catch up with old friends a year in advance.

Behavioral economists have identified emotional, irrational behavior which is logarithmic. Dan Ariely observes logarithmic sense of number when he talks of how we will go out of our way to save $100 on a $200 purchase, but not to save $100 on a $10,000 purchase.

And it's been said before here, but I'll reiterate: Jad's interest in logarithmic number sense is entirely predictable. Jad is foremost a musician. Wavelength change from octave to octave is logarithmic. Db is a logarithmic scale. Jad swims in logarithmic seas.

Thanks. JW

That was so cool. I've never really enjoyed numbers until this. I always thought numbers were kinda boring and never realized they could be such a huge, and important part of us especially from the time we are born.

Thanks for the interesting piece, but I'd like to share some gentle criticism. From 31:30 to 32:00, Krulwich leaves the topic of Benford's law as something "too numeric", "very complicated and deeply mathematical" and claims he doesn't really understand it.

1. It's disappointing that Krulwich propagates the attitude that math is inherently difficult. Studies show that people succeed far better at things when told that they are easy. We need to just start acting like math isn't that hard.

2. A lot of times, math seems difficult because the way it's explained is confusing. People's minds fall asleep when they see equations with confusing symbols. I think it's much more effective to explain math in plain English, or with pictures. Let me try making Benford's law intuitive:

Do you think more bank accounts have balances between $100 and $200 or balances between $1,000 and $1,100? It's the same range, right? How about $100,000 and $100,100? Intuitively, the higher a $100 range, the fewer bank accounts fall within the range. Now, think of bank accounts from $900 to $1,000. You'd expect to see about the same number as from $1,000 to $1,100, maybe slightly more because the range is smaller. But far fewer than from $100 to $200. Benford's Law is based on the fact that there are more accounts from $100 to $200 than from $900 to $1000, more accounts from $1,000 to $2,000 than from $9,000 to $10,000, more from $10,000 to $20,000 than $90,000 to $100,000, and so on. There's a sawtooth-like graph on Wikipedia's Benford's Law article that illustrates that idea as a picture.

It would have also been neat if the program had tied back Benford's law to the fact that 1, 3, and 9 are equally spaced. Because our number system has more digits between 3 and 9, we see those digits less often than those from 1 to 3. In fact, 3 is the median digit from Benford's distribution. Isn't that interesting?

Not through listening yet, but having heard the first segment with "25 minutes to go", I just wanted to point out that even if it is Johnny Cash singing, the song is written and originally performed by madman Shel Silverstein.

Sadie (age: 22 mos.) has become fascinated with her two ceramic piggie banks; she loves emptying them and listening to the coins hit the kitchen table and running her fingers through the resulting piles. Today I said "Sadie can I have ONE coin?" and she quickly selected a penny and gravely handed to me; then I said "Sadie can I have TWO coins" and she screwed up her face in furious concentration and--after a moment--elatedly scooped up two fistfulls, and dumping them out said TWOOOOO!

There are a few Radiolab episodes that don't leave my mp3 player and "Numbers" is one of them. In fact, I recently noticed I had two copies of the file on my player. :-)

Learning to count in base 10 may be the way we start to learn to quantify, but if you've ever met an engineer you know that applied mathematics quickly moved back to logarithmic thinking when it came to engineering and science. Engineers do much of their work with quantifying signals in dB - decibels, a system where ratios are the method of counting -

1 is 0dB

10 is 10dB

100 is 20dB

1000 is 30dB

Besides being mathematically useful (multiplication of signals in normal numbers becomes addition in decibels), looking at engineering signals and systems in dB just "seems" natural (at least to me!)

I am surprised that there was no connection made between the first and last segments of this piece. The first segment implies that our natural inclination is to view numbers as ratios; the last, Benford's Law, suggests that in fact the collections of numbers we encounter are distributed in a logarithmic fashion. Human perceptions of sight, sound and I suspect pressure (or touch) are also logarithmic, most likely because only that way can we survive in a world of stimuli of large dynamic range. I see a logarithmic number sense as the one which deals best with our experiential world.

The NY Times had a recent article about how young children are more capable of learning mathematical concepts than previously thought: http://www.nytimes.com/2009/12/21/health/research/21brain.html?em

Hi folks - the Pi Song is a group called Hard N Phirm, and there's a great education kids program video of it on Youtube: http://www.youtube.com/watch?v=Mfr7xG6smhU

Enjoy!

I want to find that song too!! I love it. Who found that song and what is it called?

Thanks!

I've been trying to track down the music played at the end of the segment. Can anyone identify the music that plays at about 18:09 and continues until the credits? I think it sounds great and I'd love to purchase it.

This is up to your typical quality. Thinking in Log space may also explain why zero has been a typically difficult concept for the majority of humanity

I wish the show would have explored how civilizations developed this non-log way of thinking about values/given some examples of stimuli that might force people to organize values as integers. You described how infants make the switch over a matter of months because they are raised by people who think in integers, but don't address the larger issue of how so many groups developed numerical systems based on integers. While listening, my big question was: "Well, a great many civilizations have made this switch independently, so how? And why have some not?"

and our ears!

The fact that we naturally think in terms of logs doesn't surprise me. There are many processes in our bodies that rely on them, our retinas for example.

Hey Adam-

you're right about us not entirely losing our logarithmic thinking. In fact, Stan cited almost the exact example you bring up: you'll haggle over $1 for the price of coffee, but when buying a house, say, $10,0000 may not seem like such a big deal. Though we didn't include this in the piece, it's a great example of moments where we slip back... to our intuitive thinking.

thanks for posting!

Lulu

Obviously I didn't read the first words of the first sentence. ..

What is the name of the Johnny Cash song at the beginning of the episode?

I just wanted to point out that generalizations of logarithms play a very important role in the study of algebraic and analytic number theory (which is an area of very active research in mathematics). They are called l-functions. So after all the logarithmic view of numbers might really be more natural.

"Susan breaks down the trick that separates us from the animal world: the counting song."

Humans are not the only animal that can count by integers. Alex the Parrot learned to count and add numbers (up to six).

http://www.alexfoundation.org/index2.html

http://en.wikipedia.org/wiki/Alex_(parrot)

@Curt from California "ZAP from WHNP in Boston" is a parody of the kids TV show "Zoom" from WGBH in Boston.

(I grew up with the original show in the 70s)

Hey Chuck,

I think you meant

e ^ (j * pi) + 1 = 0

Euler's identity also contains one instance of each simple mathematical operation -- exponentiation, multiplication, addition, and assignment.

I don't think we lose log numbers altogether. How about spending money? Spending $10 on something that costs $1 would seem ridiculous; spending $1,000,010 on something that costs $1,000,001 isn't as big a deal. You're spending $9 more than you should be in both cases, but $9 seems like a lot more in the first case. Or think of obtaining possessions: getting a second tv when you only have one vs. getting your 21st tv when you already own 20. One more tv in both cases; but one more tv seems like a lot more when you only have one.

I came here to ask the same question about the "Pi digits" song and am glad to see it's already been answered.

Looks like the mp3 can be found at http://zorin.org/share/Hard-n-Phirm-Pi.mp3

Your segment on numbers called to mind a section of Carl Jung's Memories, Dreams and Reflections in which he describes how as a child he simply couldn't get it about numbers; that in essence he had to fake it to get by. I thought of this when the program spoke of both the innate sense of number by children and some primitive cultures that sees numbers in an altogether way than we are raised up to see them. Does the indoctrination of numbers that we undergo also have something with blocking the perceptions that came to Jung and seem so odd to most of the rest of us.

Curt from Minneapolis and other friends,

The music you heard was excerpted from "Pi" by Hard N Phirm, which forms part of the sound track to a WHNP Boston production for children, "Zap." Here is a link to the video on You Tube: http://www.youtube.com/watch?v=Mfr7xG6smhU

If you want a shorter version (with "lyrics"!), see: http://www.youtube.com/watch?v=KgeKx6O2cLQ&feature=related

"Radiolab dedicates this hour to an exploration of numbers..." great topic, mildly interesting stories. However, you missed some much more interesting stories.

Numbers are far more bizarre and interesting than you would imagine. Check out the "Road to Reality" by Roger Penrose.

Some ideas for your next numbers show:

1) the history of zero - historically, people found zero to be pretty disturbing

2) the concept of "real numbers": the pythagaereans thought everything was a ratio of integers. Until someone proved that the length of the diagonal of a 1x1 square could not be the ratio of two integers!

3) this brings in the concept of infinity. Did u know that there are different infinities of different sizes? The person who developed these ideas (Cantor) went insane.

4) which brings us to the unfortunately named "imaginary" numbers. The richness of complex numbers is mind boggling and somewhat disturbing. The Mandelbrot set is an example of this richness.

Also the equation:

e^(2*pi*i) -1 =0

is amazing. It relates the 5 most important number in math (e,i,pi,0,1)

5) There is a debate among mathematicians about whether or not they are discoveres or inventors.

6) why does pi show up on so many equations that seem to have nothing to do with circles?

Keep up the good work,

Chuck

Like "Curt from Minneapolis" I was intrigued by the fragment of song whose lyric is a string of integers. I also felt it likely the song was about pi. I also have looked fruitlessly for some indication of the title or credit, but it seems totally forgotten by the producers. Any info would be appreciated!

PS @ Alan W - I believe you are misinterpreting the statement you've quoted. Basically, I inferred from it that humans are born with a natural "ratio-nal" way of interpreting amounts, and it takes training to allow us to see things integral-ly. He's speaking here of the bulk of humanity, not the outliers like Newton, Einstein, or the middle eastern genius that came up with "arabic numerals". Think about it! Numbers as we use them are recent enough an invention that we can attribute them, at least to a culture. Much more recent than fire, for example, or agriculture, or even the wheel! That boggles me - I had never really considered numerals as a created concept, since because they seem to describe how the universe works they seem as basic as, well, light or gravity, not an invented way of describing them and such...

Alan, I think what Dehaene was saying is that if an infant were raised in a society in which discrete number relations were not a necessity, then that infant wouldn't grow to know non-log number relations. You said that all civilizations have switched from log to non-log, but that clearly isn't true as you've cited the example of the Amazon people who think logarithmically. Those folks are part of a civilization that has no use for non-log number relations.

Also, you're making the assumption that the jump from log to non-log thinking occurred a.) in one individual and b.) that this switch occurred spontaneously. Rather than one person fiddling with numbers in their head until non-log thinking just happened to occur to them and then catch on like wild fire, wouldn't it be a more reasonable to assume that non-log number relations arose when a whole society was faced with the need to deal with discrete number increments? Perhaps when trading with another civilization?

Also also, I too would like to know the name of the song which appears briefly at the end of this segment.

"logharism" of the numbers!!! And again radiolab ceases to fail to entertain me

Continue...

Even under the assumption that non-log switch is inevitable, Stan's observations can be explained. An action exhibited by an infant does not imply the action will stuck with him for his whole life, case in point: drooling. The Amazon tribe's lack of non-log thinking can be explained by many alternatives, like the small size of the tribe (so the first "enlightened non-logic" person has not been born yet), that their civilization is not advanced enough for non-logic thinking to take place (like man-made tools, from stone bronze to iron), or their environment does not provide the necessary stimulus for the switch (an area without iron ore will not breed a civilization with iron tools). Until all the feasible alternatives can be explained away, claiming that "the switch will NEVER happen" is not scientifically acceptable.

We can not take a "pinhole perspective" and generalizing the incomplete observation to untested territory. The tale of "Blind men and an elephant" should have illustrated this point clearly.

I'm not a neuroscientist. But I know logic. Stan's conclusion can not emerge from his observations, even if the conclusion were to be right. Jumping to the conclusion with the premises mentioned in the conversation, is like looking at a 4-week old human fetus and say "Gee, it looks at a fish, so a human will look like fish forever".

BTW, I have been a fan of your show for years. Keep up the good work.

I have a commonsensical question regarding the conversation between Lulu Miller with Stanislas Dehaene.

At 09:29, the conversation goes "...we just naturally switch from logarithmic thinking to the numbers we all know now, but this is not true... According to Stan, you'd never switch...stay in this logarithmic world forever."

We, the so-called civilized people, use non-logarithmic thinking. It means either "we'd never switch" is not true, or "we" are different from the indigenous people in the Amazon. These are the ONLY logical explanations. Since the second is highly unlikely to be true, Stan's assertion must be false.(If the second were to be true, then the conclusion drawn from the Amazon people may not apply to all humans.Either way, Stan's assertion is false.)

The fact is, all civilizations had, at some point, switched from log to non-log thinking, which means non-log thinking may be an inevitable stage of a human society.

Also, the switch happened because someone had switched his/her thinking from log to non-log, SPONTANEOUSLY. (Divine interventions, either from God or from the black monoliths in "2001 a space odyssey", are not considered.) This implies the switch can happen to SOME individuals instinctively, even if not everyone.

Are there any logical holes in my deduction so far?

To be continued .....

The "pennies" segment went out on a song that had to be digits of Pi -- I can't find a credit for it anywhere. Anybody know it?

As someone who has never really internalized the concept of 'e' I feel cheated to think I once understood logarithms and lost it. Intergers do have their value but it is shame we have to destroy ratios to gain it.

"Innate Numbers?" shows that perception of quantity, like perception of sound pitch, light intensity, or most other things, is a matter of ratios. The ancient Pythagoreans would have loved it: "All is ratio!"

We perceive equal ratios as equal linear steps, the effects being proportional to the logarithm of the cause. ("Logarithm" comes from "Logos," Greek for "word," "idea," and "ratio."

Benfords's Law suggest that logarithms apply to things much more fundamental than just human perception.

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