Quick: What's halfway between 1 and 9?
If you're like me, you answered "5," which is (approximately) the right answer. But if you've got a 6-year-old kid on hand, try asking him or her. According to a recent study, most little kids will answer "3," as will many people in non-literate cultures. And as it turns out, there's a solid evolutionary reason why that guess makes sense.
The reason, strange as it may seem, is that our neural circuits are evolved to count logarithmically rather than linearly. This strikes me as ironic, because logarithms always drove me nuts with their anti-intuitiveness back in high school math class. The basic idea, if you remember, is that a logarithm is the inverse of an exponential, so if 102 = 100, then the log10(100) = 2. Unlike me, many scientists and artists are in love with logarithms, because these functions elegantly describe all sorts of ever-tightening curves and spirals, such as fractal patterns, ratios between musical octaves and, most famously, the golden ratio itself.
It's likely that we love logarithmic shapes so much because logarithms are hardwired into our sensory pathways. Stevens' power law explains that we perceive logarithmic increases in light, sound and heat (among lots of other stimuli) as if they were linear increases. For example, as heat rises along a tightening logarithmic curve -- say, from 80º to 84º to 86º to 87º -- our sense of touch is growing correspondingly more sensitive to each increase, causing us to perceive smaller and smaller changes as if they were equal. In short, we might describe that logarithmic temperature change as a linear rise from 80º to 84º to 88º to 92º.
From a survival perspective, this relativistic estimation is actually really practical, because it provides reasonable guesses while saving on energy. Imagine that you're a stone-age warrior stalking through the forest with your tribe, when suddenly a pack of wolves leap out of the shadows. If you can instantly guess whether it's more like a three-wolf pack or a nine-wolf pack, you can make a snap decision about whether to stand and fight or turn and run. On the other hand, distinguishing between a nine-wolf pack and, say, a 10- or 11-wolf pack isn't nearly as helpful. In other words, your brain is evolved to estimate in a way that minimizes relative errors rather than absolute ones; the larger the sample size, the less individual variations matter. (As you might've noticed, this logarithmic curve is the reverse of the temperature curve above, where smaller and smaller changes became more and more significant as they progressed.)
In fact, the differences between logarithmic and linear counting only leap into sharp contrast when we compare our brains' estimates with the precision of mathematics, which brings us back to my original point about numbers between 1 and 9. Because 30 = 1, 31 = 3, and 32 = 9, the number 3 is logarithmically halfway between 1 and 9, and young children somehow intuitively know that. This gives new meaning to the term "baby genius," doesn't it? The folks at Radiolab certainly thought so; they devoted a whole podcast to infant intuition about numbers.
This all points back to one of the stranger ideas in the philosophy of science: Numbers themselves don't actually exist in the same way forests and wolves do; we made them up to count and classify what we observe in the world around us. Math may be the "language of the universe," but it's still a language -- a map, but not the territory itself. So what's "really" halfway between 1 and 9? In the end, the answer depends on how you're counting.
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Steven Strogatz: The 3 Most Confusing Things Your Math Teacher Ever Told You
But your article is amazing. Makes me want to go back to that class and start all over again. Your visualizations at least explain WHAT logarithms are and I love the concept of "1, 2, ... a lot". Need to go find me a math class for old folks now. :) You just stoked my long-dead curiosity again!
Just a little "visual aid" for those still coming up with 3. Or 4.5 lol.
"Stevens' power law explains that we perceive logarithmic increases in light, sound and heat (among lots of other stimuli) as if they were linear increases"
is incorrect. Stevens' power law is, as the name suggests, a power law: it describes a polynomial, not a logarithmic, relationship between stimulus and sensation. Your reference to Stevens' power law therefore tends to undermine rather than support the rest of your post. You might want to try reading your links before posting.
Take this example: y=x^2 (y equals the square of x). If we write down some (x,y) pairs, then we have (0,0),(1,1), (2,4), (3,9), (4,16), (5,25), ..., (9,81), (10,100). This is an example of a power law.
Now look at this example: y=2^x (y equals two to the power of x). Here are some (x,y) pairs for this equation: (0,1),(1,2),(2,4),(3,8),(4,16),(5,32),...(9,512),(10,1024). This is an example of an exponential relationship. You can see that, as x goes higher, y increases much more rapidly than for the power law example. The two types of relationships are qualitatively very different.
I do not claim to know anything at all about Stevens' power law apart from what I read in your link, and I also know nothing at all about the science of the perception of stimuli. I only wanted to call attention to the fact that the description of Stevens' power law in your link does not appear to provide support for the rest of the material in your post. I don't know the relationship between Stevens' power law and other work that does suggest a logarithmic relationship between stimulus and response. Perhaps the explanation is as simple as a log having already been implicitly taken in the definition of "sensation magnitude" in your link.
We invented numbers. Before we invented them it makes sense we would just say "One, two, a lot".
We don't say "Look, 349 trees", we say "Look, a forest".
Less civilized societies often do not have words for numbers beyond three. The Yanomamo have words for one, two and many. That is more typical of primitive societies. Children learn to count by memorizing the numbers and learning to use their fingers and later their minds to tally up objects, a process on top of the language instinct.
Knowing how to count tens of objects is useful for societies with money or farms with dozens of sheep or bushels of crops. Not so important when you're hunting - if there are "a lot" of buffalo that's is all you need to know and if you bag three it was a blockbuster day.
I bet if you showed the kids a picture of nine blocks they would pick the fifth block most of the time as the middle. (Which would be the middle, 4 on each side, not "approximately" 5 as this article wrongly states.)
We notate and think of musical pitches as steps, but they are actually ratios. If we transpose any melody up one octave we add 12 semi-tones on paper, but the actual change in frequency is multiplication by 2.
Another interesting point is that simple ratios give us the more common consonant harmonies. In "just intonation":
1:2 = octave
2:3 = perfect fifth
3:4 = perfect fourth
4:5 = major third
5:6 = minor third
3:5 = major sixth
5:8 = minor sixth
Generally speaking, the more complex the ratio, the more dissonant it will seem to the ear. The topic gets much more complex when you consider equal temperament. For the curious: http://en.wikipedia.org/wiki/Equal_temperament
children see if easier because they are not as influenced by peoples views of the world
like neon trees says everybody's talking
well if everybody is talking no one is listening
kids see things very simply for what they are because they do not have the capacity to manipulate people yet
look at kids at a park they will play with any kid of any color any financial background and religious background
because they understand if they don't look past the boundaries in our minds they would be sitting on the bench all a lone having no fun
SIMPLE LOGIC
take out the preconceived beliefs and just have fun building an amazing home planet and start populating others
lets have fun focusing on creation not destroying it thats simply logic
our environment changes all the time so our scientific answers theories and answers should
science evolves as well it is not a permanent form it changes so must equations
you can see logical events and follow mans patterns
you can follow a path and predict how due to a persons evolution taking in just small aspects of their beliefs and thoughts you can predict how they will react to different situations
our past and our knowledge are our greatest gifts, combine that with our curiosity thats an energy source and supercomputer made up of the best organic brains processing and evolving
thats the funny thing about computers is that we have to give them the sequence order and equations otherwise the sequence is wrong and so will the numbers
the fact is logic is just a series of steps in a sequence that follows certain laws of nature at its smallest form to grow in to a large form micro to macro
if we can do it on a small scale you can replicate on a larger scale and that is how we learn and apply
Archimedes took logical events like the the crown and the bath story to prove fake gold used
science is logic math is logic problem is everyone has different strengths
i like logic and words but the math,chemistry.physics,biology etc i am not good at but their are people who excel at it so they are the ones who should do the calculations and so on
we need to work as a team
companies are so scared of people stealing ideas that they shield them when in fact if you put the ideas to gether they may actually be the answer you need
.... 5 is exactly the halfway between 1 and 9. I even counted it out for you.
Another way to look at it is like this -- how far apart are 1 and 9? 8 (9-1). So half of that distance is 4. What's 4 away from 1 and 4 away from 9? It's 5.