
What number is halfway between 1 and 9? - aurelianito
http://web.mit.edu/newsoffice/2012/thinking-logarithmically-1005.html
======
cs702
According to the research discussed in this article, when the intensity of
some sensory stimuli increases exponentially, our minds somehow
_unconsciously_ perceive the increase as linear. Our perception, in other
words, is often logarithmic.

What I find most fascinating about this is that our _conscious_ minds are not
very good at thinking about exponential growth. My favorite example is that of
the twin brothers who invest with annual gains of 12% each: one twin saves
$2,000 from age 19 to 26 (a total of $16,000) and then stops saving; the other
twin doesn't start saving until age 27 but then puts away $2,000 every year
until age 65 (a total of $78,000). Do the math, and you will see that the
savings of the second twin, growing at 12% every year, _never_ catch up with
those of the first one.

Our perception might often be logarithmic, and we might understand the math of
exponentiation, but our brains are not very good at 'seeing' exponential
growth!

~~~
citricsquid
> My favorite example is...

This exact thing caught me out a few weeks ago, it still boggles my mind. I
started a pension recently and I was given examples of how much I would be
worth if I started as soon as possible (my 21st birthday) or if I waited until
I was 22, 23, 24... if my total monthly contributions are $2,000 and I start
on my 21st birthday I will be worth hundreds of thousands of dollars more than
if I started at 23, or 25, but my brain thinks that is insane and even though
I _know_ it's true, I still can't get over it. The difference between 21 and
23 is $160,000!

going even more on a tangent, it's really a great shame this sort of thing
isn't taught in schools. It seems obvious _now_ after being taught that
savings can vary so much this way with so little time, but until I was
explained it I had no idea. I figured waiting until 25 or 30 to start a
pension made most sense!

~~~
jerf
Those estimates are often wrong though, because the idea of annual 12% returns
is complete fiction. In fact the idea of any particular percentage annually is
fiction. For instance, I only recently started a pension, however if you look
at the stock market history (for simplicity), had I started in 2000, the net
gain I could have expected was almost precisely zero, if not negative. If I
could have found something that paid 12% from 2000 to 2012, sure, I would have
missed out on a lot, but there in fact not only was no such beast (in the
broad sense), there was also a lot of ways to experience negative gain.

This is one of my favorite visualizations ever:
[http://www.nytimes.com/interactive/2011/01/02/business/20110...](http://www.nytimes.com/interactive/2011/01/02/business/20110102-metrics-
graphic.html) Be sure to _carefully read_ what it is actually visualizing;
I've posted this a couple times and people often knee-jerk a reaction to it
based on a complete misreading of the chart.

~~~
itry
Interesting chart.

Strange, at one point it says "High inlfation led to negative returns". When
money is worth less, then you get more money when you sell something. In this
case your share of a company. So why would the returns of investing in the S&P
be affected by inflation?

~~~
mikeash
Inflation affects the number that you use to value the worth of a company.

Say you buy one share at $10 of a company with a billion outstanding shares.
That company is worth, on the market, $10 billion.

Let's say that next year, the company's share price is still $10. Let's also
say that inflation in that period was 4%. The company is still worth $10
billion, but each dollar is worth 4% less. The company's total value has
dropped.

Inflation didn't cause the total value to drop. The company's value dropped
for other reasons (bad sales, company president went crazy, market jitters,
whatever causes these things) and inflation simply moved the numbers a bit in
the opposite direction.

In short: inflation doesn't affect your returns, but it affects the numbers
you use to measure them.

~~~
itry
You say "In short: inflation doesn't affect your returns". And thats what I
said. But the linked article says the opposite: "High inlfation led to
negative returns".

------
Robin_Message
1.8 (harmonic mean)

Seriously though, I thought we already knew that human perception is mostly
logarithmic (for example, the eye performs well across 9 magnitudes of
brightness – a linear system doesn't make sense). On the other hand, human
perception of future value has generally found to be discounted
hyperbolically, which is illogical
(<http://en.wikipedia.org/wiki/Hyperbolic_discounting>).

I wonder is there is a correspondence with the harmonic mean but I can't get
my head around it at the moment.

~~~
pessimizer
Not illogical:
[http://en.wikipedia.org/wiki/Hyperbolic_discounting#Uncertai...](http://en.wikipedia.org/wiki/Hyperbolic_discounting#Uncertain_risks)

~~~
davetannenbaum
The reason why it is usually considered irrational/illogical is because
hyperbolic discounting leads to dynamically inconsistent preferences. For
example, under hyperbolic discounting I may prefer $100 today to $150 in one
month, but prefer $150 in 61 months to $100 in 60 months. It's the same rate
of discounting in both cases ($50 for an additional month of delay) but my
preferences are not consistent.

Of course, there are other criteria for rationality besides
coherence/consistency (as the wikipedia article alludes to).

~~~
pessimizer
But if you view the payoff as not certain, $100 now is worth a lot more than
$150 in a month - because you are here now with $100. The $150 is worth its
normal discount rate multiplied by the strength of the probability that you
will be here in a month with $150. The discount rate here is entirely due
differences in risk of nonpayment - which are marginal when comparing 60
months to 61 months, but massive when comparing now to an hour from now.

According to criteria that would label hyperbolic discounting irrational or
illogical, debt collectors are equally confused for not accepting the promise
of a $100,000 payoff 50 years from now rather than a settling of the debt
today.

------
woodchuck64
"But pose the same question to small children, or people living in some
traditional societies, and they're likely to answer 3."

Citation needed! I mean, if "half" is not understood as mathematically half,
how can traditional societies even have an economy?

~~~
inemnitable
It's not "half," though, it's "halfway between." This implies a mean, but
which mean to use is not explicitly expressed. Choosing the arithmetic mean
gives 5, choosing the geometric mean gives 3.

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kevindication
Radiolab's Numbers episode covered this fascinating topic back in 2009:
<http://www.radiolab.org/2009/nov/30/>

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gilgoomesh
Where is the study that showed "But pose the same question to small children,
or people living in some traditional societies, and they're likely to answer
3."

Without a source on this, it's a difficult article to take the article
seriously. It offers no support for its most prominent claim.

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jarel
"Hi, quick question: Which number is halfway between 100 and 10,000?"

"1000. No, wait..."

(I might actually say that)

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chinpokomon
Interesting... this has been something I've postulated for several years now.
My reasoning had most to do with our vision system. As we perceive distances,
the distance between something 10 feet, 100 feet, 1000 feet away, etc, becomes
less about knowing the absolute difference and more about relative distances;
someone is more likely to determine that something is 110 feet away than they
are to realize something is 1010... but realizing something is 1100 rather
than 1000 feet away will feel about as off as suggesting something 110 feet
away is really 100. Our vision measurements are further reinforced as we
observe parallax scrolling as we are driven around the country in our youth.

I've also felt that teaching a number system that is linear means that it is
harder to understand the difference between 100 and 1000000000. Even many
adults find it difficult to understand the magnitude of numbers when you
approach a billion or a trillion. I think this is largely because we are
taught to think in a relatively small range of the number line and perceiving
numbers outside that range, we try to relate it back to what we were taught in
schools.

I'd love to see schools teach a logarithmical number line instead of the
linear system kids are taught today.

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mratzloff
Perhaps the fact that human brains seem hardwired to perceive relative
differences by orders of magnitude is an argument for graphing logarithmically
by default, instead of linearly.

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cr4zy
Musical scales are exponential as well. (octaves are frequency doublings)
There's also this experimental evidence about sensing loudness, weight,
vibration, lightness, saltiness, thickness and several dozen other things.
From: <http://en.wikipedia.org/wiki/Stevens%27_power_law> ...

The general form of the law is

ψ(I) = kI^a

where I is the magnitude of the physical stimulus, ψ(I) is the psychophysical
function relating to the subjective magnitude of the sensation evoked by the
stimulus, a is an exponent that depends on the type of stimulation and k is a
proportionality constant that depends on the type of stimulation and the units
used.

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artursapek
Radiolab did a good piece about this.
<http://www.radiolab.org/2009/nov/30/innate-numbers/>

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jamesbressi
What does it mean when you can't answer the question properly and said "4.5"?

~~~
sukuriant
You're either a math major or an engineer. We can't do simple math anymore.
Please give it to us in the form of an integral.

Also, there's the ambiguity of "inclusive or exclusive? And, which side?"

[1,9], (1,9], [1,9) or (1,9)?

~~~
kylebrown
Those ambiguities make no difference to the answer. Answering 4.5 just means
you forgot that starting a count from 0 gives a different total than starting
from 1.

~~~
sukuriant
If you're using the set of integers, then [1,9) is equivalent to [1,8]; and
that would have a different answer from 5 for "what the center is"; so, it
does affect the answer.

~~~
kylebrown
[1,9) is not equivalent to [1,8]. The center of [1,9) is infinitesimally close
to the center of [1,9]. Its exactly equal when you take the limit.

~~~
sukuriant
... I was using that nomenclature for the short hand of "inclusive/exclusive".
1 to 9 exclusive is 1 to 8 in the integer set. Sorry for the lack of clarity.
Perhaps now we can get to my original point?

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hcarvalhoalves
For anyone who hikes or goes outdoor, this might be obvious.

If you notice how we perceive distances, it's logarithmic too. It's really
hard to judge accurately the time to arrive over long distances. Like the
child's example on the OP, people often perceive 1/4 of the distance as being
"halfway there". It's some kind of psychological phenomenon.

------
smoyer
I'm wondering if somehow our predisposition to think logarithmically is
related to Benford's law - <http://en.wikipedia.org/wiki/Benfords_law>.

~~~
BenoitEssiambre
I think it is related to Benford's law in that they are both a manifestation
of the fact that a log scale is the most natural (most uninformative, highest
entropy) distribution (or prior) for things that have magnitude (having a
parameter that can't go into the negatives such as height or length or volume
as opposed to a positional parameter which can be negative).

I do believe this is a property of the universe and not just a psychological
phenomenon. Our mind here is simply following the natural distribution and
Benford's law is just the observed manifestation of this pattern.

This type of pattern is not odd if you are believer in the Bayesian
interpretation of probability theory. The shape of Benford's distribution
follows the shape of a maximum entropy, most uninformative distributions for a
value of magnitude (as I mentioned, positions do not follow benford's law).

Take street lengths for example. Assuming that they follow a log prior simply
means that for a length of street L, if you pick another random street, you
are as likely to pick a street within the length range L/2 to L than L to 2*L.
From the original street length, to get a street twice as short, you need to
subtract much less than you would have to add to get a street twice as long.
That is why this distribution is not linear or rather it is linear on the
multiplication and division operation, not on additions or subtractions.

If instead you'd assume that a street x meter longer is as likely as a street
x meter shorter you would end up with impossible probabilities. For example,
for a street of 1 km, a 3 kilometer street would be as likely as a -1km
street? Even if you'd assume probabilities were equal for all lengths between
0 and infinity that would mean you think there are as many streets measuring a
tredecillion billion km long as there are street 5 km long. This is simply not
how things are sized in the universe. Smaller things are in greater numbers.
Log priors are one of these areas where the math predicts the universe
logically and the universe is mirrored by the math beautifully iff you do your
calculations properly (using Bayesianity). It seems evolution has made our
psychology reflect this reality.

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andrewcooke
i wish it had described _how_ people were asked. i guess it's through some
graphical means. anyone know (original paper is paywalled)?

~~~
takluyver
That question only seems to appear in the press release - it may be a
reference to an earlier study, but press releases don't tend to cite sources.
The paper in question is theoretical, although it gives an example with the
intensity of various human and animal sounds. (I have access through my
university)

~~~
derekp7
One thing not mentioned in the paper is also music. Each octave increases
exponentially (each is twice the frequency of the previous). However our ears
perceive the difference between notes as linear (it is actually linear if you
take the log of the frequency). One theory is due to the shape of the inner
ear, where the individual hairs that hear sounds each octave apart are spaced
lineally.

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maratd
Did anyone permit for the possibility that _some_ people may be wired to have
an affinity for one and some for the other?

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malingo
Could this be related to the recent post about software estimates?

~~~
saraid216
Software estimates are just another unit of distance. That's a valid
application of this general idea: Halfway between a day and a month is a week.

