What number is halfway between 1 and 9?

http://web.mit.edu/newsoffice/2012/thinking-logarithmically-1005.html

by aurelianito • 14 years ago

189 74 14 years ago

web.mit.edu

mratzloff• 14 years ago

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.

Robin_Message• 14 years ago

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.

pessimizer2• 14 years ago

Not illogical: http://en.wikipedia.org/wiki/Hyperbolic_discounting#Uncertai...

davetannenbaum• 14 years ago

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).

smoyer• 14 years ago

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

atamyrat• 14 years ago

I don't think Benford's law has anything to do with human psychology, it's more of mathematical property that applies to random sample from certain distributions.

baddox• 14 years ago

I suspect only in the sense that both are related to http://en.wikipedia.org/wiki/Power_law.

BenoitEssiambre• 14 years ago

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.

maratd• 14 years ago

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

andrewcooke• 14 years ago

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

takluyver• 14 years ago

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• 14 years ago

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.

cs702• 14 years ago

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!

tomrod• 14 years ago

This is why the second twin should put in $5000 every year :D

sk5t2• 14 years ago

Modern humans no more need to appreciate the economics of consistent 12% gains than they need to know how to fend off a polar bear, or how to date three supermodels on a modest budget--it just ain't gonna happen.

rcfox• 14 years ago

When I first learned about this investment problem (although, my version had 18-30 and 30-65), I worked out the math, and I think it held for as low as 2% interest. I could be remembering wrong though... Surely, someone's done a formal write up on this?

taejo• 14 years ago

OTOH, sometimes our perception is multiplicative when it should be relative: charge me $100 extra on a house, I probably won't even notice; charge me $100 extra on a bar of chocolate, there's no way I'm going to buy it. But either way I lose $100!

StavrosK• 14 years ago

You only buy a house once, you buy chocolate bars all the time. $100 on a chocolate bar every time compounds.

scott_s• 14 years ago

Dan Arielly's "Predictably Irrational" (http://www.amazon.com/Predictably-Irrational-Revised-Expande...) is a great book that covers such phenomenon.

citricsquid• 14 years ago

> 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!

keithpeter• 14 years ago

The contribution from your first year is earning interest for N years, so contributes (1 + R/100)^N (assuming fixed rate R, which is a fiction. That could be a lot of dosh...

jerf• 14 years ago

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... 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.

Dove2• 14 years ago

Starting early is only half the battle. The other half is the rate of return. If you start up a retirement account at age 17 with $2000 in it, but you just put it in a generic savings account, by age 37 you'll have . . . $2209.

People encouraging you to save often give unrealistically high rates of return. 12% is unrealistic for any investment vehicle I know of (though people sometimes say it for the stock market -- but they're crazy optimistic).

Here's a spreadsheet to show you the impact of both rate of return and time:

https://docs.google.com/spreadsheet/ccc?key=0AqTfjQUnJdLkdFl...

eCa• 14 years ago

You are not alone: www.youtube.com/watch?v=F-QA2rkpBSY

ed209• 14 years ago

on this topic - this is a video worth watching http://www.youtube.com/watch?v=F-QA2rkpBSY

csomar• 14 years ago

What kind of investment brings 12% every year, for 40 years without the risk of losing all my money? With the investment options I have now (today), I'm better of consuming my money.

petercooper• 14 years ago

I think your example illustrates the surprises of compound growth well, but with a realistic growth of 4.1% per year, twin A ends up with a pot of $88k and twin B ends up with $183k. So your example provides food for thought but probably shouldn't form the basis for a retirement strategy.

sliverstorm• 14 years ago

When the intensity of some sensory stimuli increases exponentially, our minds somehow unconsciously perceive the increase as linear

By necessity, really. If you've ever spent much time with electronic sensors, you may discover as I did that seemingly half of the work is making nonlinear input linear, because nonlinear input is a mess to compute with.

001sky• 14 years ago

SW and HW for digital imaging had/have to deal with this quite a bit.

Xion• 14 years ago

> What I find most fascinating about this is that our conscious minds are not very good at thinking about exponential growth.

I thought about it too, and come up with idea that some equivalent of Weber-Fechner law [1] might be working at it this level. So I have made a small write-up about this idea that you might find interesting [2].

[1] http://en.wikipedia.org/wiki/Weber%E2%80%93Fechner_law [2] http://xion.org.pl/2012/04/11/self-reinforcement-and-exponen...

artursapek• 14 years ago

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

kevindication• 14 years ago

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

woodchuck64• 14 years ago

"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?

evincarofautumn• 14 years ago

Probably because there is a difference in perception between half of an amount and half of a number.

inemnitable• 14 years ago

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.

chinpokomon• 14 years ago

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.

hcarvalhoalves• 14 years ago

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.

malingo• 14 years ago

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

saraid216• 14 years ago

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.

jamesbressi• 14 years ago

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

sukuriant3• 14 years ago

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)?

kylebrown3• 14 years ago

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.

nostromo• 14 years ago

4.5 is halfway between 0 and 9. Halfway between 1 and 9 is 1+(9-1)/2 = 5

JoeAltmaier• 14 years ago

Nu-uh. By that math, halfway between 1 and 2 is.... 1?!

jarel• 14 years ago

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

"1000. No, wait..."

(I might actually say that)

gilgoomesh• 14 years ago

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.

cr4zy• 14 years ago

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.

What number is halfway between 1 and 9?

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