Number: The Language of Science by Tobias Dantzig
What Counts: How Every Brain is Hardwired for Math by Brian Butterworth
Bertrand Russell, a famous philosopher, mathematician, historian, logician, and all around super smart and sexy guy, once wrote:
“It must have required many ages to discover that a brace of pheasants and a couple of days were both instances of the number 2: the degree of abstraction involved is far from easy.” — Introduction to Mathematical Philosophy by Bertrand Russell, page 3
Really? Cause it sounds pretty easy to me. So how does all this squishy stuff inside our skulls enable us to do math, anyway? Well, according to phrenology, a pseudoscience of the 1800s, the math organ, the part of the brain that does calculations, is right about… here. (Pointing to just behind my.)
The phrenologists thought you could predict a person’s abilities by feeling the bumps on their skull. And they found a bunch of people who were good at math, who also had really big bump in the same place, so they figured: ah, that must be the math organ! Seems legit to me.
There is some truth to it, though. Modern neuroscience has found that there is a specific part of the brain that seems to control our intuitive ability to do things like estimating how many marbles are in a jar, converting Farenheit to Celsius, and other fun stuff like that. That intuition is usually thanks to the approximate number system, usually just called our number sense. It seems to be located in the parietal lobe — more specifically, in a structure called the intraparietal sulcus. I hope I’m saying that right. I’m not exactly a neuroscientist… Ooh! But I can quote one!
“We now know that the human brain has evolved specialized circuits to exploit the fact that much of the perceptible world is countable. This is why neurological damage can affect numbers and nothing else, and why people are born dyscalculic.” — The Number Sense: How the Mind Creates Mathematics by Stanislas Dehaene, page 351
Just like some people are dyslexic, meaning their brain has trouble with letters, a similar percentage of people are dyscalculic — their brains have trouble with numbers. One study found that children who are dyscalculic actually have less grey matter in the intraparietal sulcus. And a handful of people even have what’s called arithmetic epilepsy. Their epileptic seizures are triggered whenever they do math. (I know I shouldn’t say this, but a really small part of me wishes I had that, just because it would’ve been the perfect excuse not to do my math homework!)
Anyway, it seems we have a calculator built into our brains. But unlike a real calculator, our brain calculators only handle estimates. The more objects there are, the less accurate your estimation becomes. This has been called the magnitude effect. And there’s also what’s called the distance effect: the closer together two numbers are, the harder it is to tell them apart.
All that stuff might help to explain why we can only perceive up to about four objects at a time without counting them. There’s even a name for that ability:
Subitizing: the “ability to take in the numerosity of a visual array of objects at a glance, and without counting” — What Counts: How Every Brain is Hardwired for Math by Brian Butterworth, page 113
Now once there are more than about four objects, we have to either count or estimate. And it turns out that everyone’s estimating ability, or number sense, is different. Professor Justin Halberda created a test that measures your number sense. You see a bunch of blue and yellow dots for only an instant, and you have to guess which color has more dots. Since you don’t have time to count them, you’re forced to rely on your number sense.
(Taking the number sense test at Panamath.org.) OK, let’s see how I do… I feel like I’m just guessing randomly, but I seem to be getting most of them correct! Correct, correct, correct… aww, darn.
(Looking at the results.) Dude… I’m awesome at this! I wasn’t expecting this. I’m terrible at arithmetic! Oh, OK, hang on. This explains it. I’m really good, but I’m also really, really slow. That actually explains a lot.
The very latest research, just published in January 2012, created an artificial brain that taught itself how to estimate the number of objects in an image without being preprogrammed to do so. Wow. As the lead researcher says,
“It answers the question of how numerosity emerges without teaching anything about numbers in the first place.” — Marco Zorzi, quote from Neural network gets an idea of number without counting by Celeste Biever, New Scientist
So if an artificial neural network can teach itself to estimate the number of objects, regardless of size or color or stuff like that, then clearly abstraction thinking isn’t that difficult after all, and it doesn’t take ages to discover that a brace of pheasants and a couple of days are both instances of the number two. Sorry, Bertrand Russell. Despite your sexy mustache, I guess you were wrong. That’s OK, though. I forgive you.
- Introduction to Mathematical Philosophy by Bertrand Russell
- Number: The Language of Science by Tobias Dantzig
- What Counts: How Every Brain is Hardwired for Math by Brian Butterworth
- Gut Instinct’s Surprising Role in Math by Natalie Angier. The New York Times.
- Kucian, K., Loenneker, T., Dietrich, T., Dosch, M., Martin, E., & von Aster, M. (2006). Impaired neural networks for approximate calculation in dyscalculic children: a functional MRI study. Behavioral and Brain Functions, 2, 31. http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1574332/
- The Number Sense: How the Mind Creates Mathematics by Stanislas Dehaene
- Halberda, J., Mazzocco, M., & Feigenson, L. (2008). Individual differences in nonverbal number acuity predict maths achievement. Nature, 455, 665-668. http://panamath.org/papers/HalberdaEtAl2008Nature.pdf (PDF)
- Neural network gets an idea of number without counting by Celeste Biever. New Scientist. January 20, 2012.
- Zorzi, M. & Stoianov, I. (2012). Emergence of a ‘visual number sense’ in hierarchical generative models. Nature Neuroscience, 15, 194-196. http://www.nature.com/neuro/journal/v15/n2/full/nn.2996.html
- “Gallon of marbles” by Rex Hammock
- Photo of thermometer by OliBac on Flickr
- Photo of rock pile by Pascal Blachier
- “Shiro Kuro (2008)” by Sk’p
- What Makes a Genius? from BBC Horizon, hosted by Marcus du Sautoy. (Watch on YouTube)
- Number sense test footage and results recorded from Panamath.org
- Video game sound effect by HardPCM on FreeSound.org
- “James Mireau – Gary’s Netbook” by chipsounds
And public domain media:
- Flute Concerto in G Major, K. 212 by Wolfgang Amadeus Mozart, from Musopen.org
- Portrait of Bertrand Russell (1907)
- Illustration of the common pheasant from The Birds of Europe by John Gould (1837)
- Phrenology diagram from People’s Cyclopedia of Universal Knowledge (1883)
- Engraving of craniometer from “Elements of phrenology” (1835) by George Combe
- Scan of the American Phrenological Journal, Volume 10, Issue 3 (1848)
- Principal fissures and lobes of the cerebrum viewed laterally. Figure 728 from Anatomy of the Human Body by Henry Grey (1918)
- Lateral surface of left cerebral hemisphere. Figure 726 from from Anatomy of the Human Body by Henry Grey (1918)