The problem often starts like this: “If Jane pays £5 for 10 grapefruits, how many grapefruits will she get for £50?”
The answer to this question implies that the idealized world of mathematics is the only place where you can buy 100 grapefruits without anyone batting an eyelid.
To arrive at the answer to this question, many of us are conditioned to use straight-line reasoning and assume that for 10 times the money, Jane will get 10 times the grapefruit.
The word “straight-line” describes a special relationship between two variables – input and output.
If the relationship is linear, a change in one quantity by a fixed amount will always produce a fixed change in the other quantity.
It is a good model for all kinds of relationships in the real world.
With a fixed exchange rate, for example, one pound sterling might be worth two NZ dollars, 10 would be 20 NZ dollars, and 100 would be 200 NZ dollars.
It is a special type of straight line relationship.
As you increase the amount of pounds you want to change, the number of dollars you get in return changes in direct proportion – if I double the input, I double the output.
If I can buy three bars for £2, then surely I can buy six bars for £4.
The number of chocolates I can buy increases linearly with the money I’m willing to spend on them.
Linearity does not assume that there could be a three-for-two offer in play.
(And, of course, in reality the exchange rate varies drastically with the change in the financial market.)
However, not all linear relationships are directly proportional.
To convert from Celsius to Fahrenheit you need to multiply the temperature in Celsius by 1.8 and add 32 to that.
In this relationship, doubling the input does not double the output, but because it is linear, a fixed change in the input always corresponds to a fixed change in the output.
A rise of 5 degrees Celsius is always a rise of 9 Fahrenheit, no matter what temperature you start from.
These relationships can be represented in straight lines, which is why we call them rectilinear.
I may have overemphasized the point of rectilinear relationships, especially since linearity is a very familiar idea.
But therein lies the problem: we are so familiar with the concept of linearity that we impose our rectilinear reference point on all the data we encounter in the real world.
This is linearity bias in its simplest form.
As I discuss in my new book How to expect the unexpectedmany systems do not obey these simple linear relationships.
For example, if I leave money in my bank account or forget to pay a debt, then that amount of money will grow non-linearly (specifically, it will grow exponentially) – the interest will stick to the interest.
The more money I have (or owe), the faster it will grow.
Because many of us are subject to linearity bias, we underestimate how quickly these sums of money grow, which makes saving for the future less attractive, but making borrowing more attractive.
It has been shown that people with with a higher degree of linearity bias, they have a higher debt-to-income ratio (how much they borrow in relation to the income they have).
Pseudo-linearnost
It seems that the most important explanation for our overdependence on linearity comes from math classes.
Investigations into the origin of this bias show that it is our tendency to assume linearity is present long before we finish school.
These studies ask students questions where linearity is not the right tool to gauge how they respond.
These so-called pseudo-linear problems they can look like this:
“Laura is a sprinter. Her best time for 100 meters is 13 seconds, how long will she run the kilometer?”
It is not possible to arrive at the right answer based on the information from the task.
However, most students still reach for the straight-line solution, regardless of the unrealistic nature of its inherent assumptions.
They increase the time it takes to run 100 meters by a factor of 10 to cover 10 times the distance, giving the time it takes to run a kilometer at 130 seconds.
This can obviously only be the lower bound of the right answer as it does not take into account the fact that no athlete can maintain their best 100m speed for the entire kilometer.
And indeed, according to the linear answer, Laura would greatly exceed the world record for running a kilometer – two minutes and 11 seconds.
A complicating factor is the lack of appreciation in math classes that the real world is usually not as simple as a math problem.
Even artificial intelligence manages to make these mistakes: ChatGPT, a chatbot built to mimic human interactions, learned these same biases.
When I asked him, “Three towels dry on a rack in three hours, how long will it take nine towels to dry?” he replied, “nine hours,” reasoning that if you triple the number of towels, you’ve tripled the amount of time it takes to dry them. they dry out.
And really, if your shirt is long enough, nine towels shouldn’t be drying in parallel for more than three hours.
Nonlinear world
I make cupcakes with the kids and if we want to make twice as many cupcakes as defined in the recipe, we have to use twice as much of each ingredient.
The ingredients are mixed linearly to make a double mixture.
That sounds about right.
But to assume that this is true of every phenomenon in our world would be to deny the existence and magic of unforeseen phenomena—for example, that no molecule of H2O is wet, or that the unique fractals that make up snowflakes do not add individual crystals to the pile, but are one complex structure.
Even our own lives are much more than the simple sum of atoms and molecules that make up our physical embodiments.
Although we are often unaware of them, many of the most important relationships we experience every day are non-linear.
But the idea of linearity is drilled into our heads so early and so often that we sometimes forget that other types of relationships might even exist.
Our over-familiarity with linear relationships means that when something non-linear happens, it can catch us off guard and defy our expectations.
By making the implicit assumption that inputs are in proper relationship with outputs, it is easy for our predictions to be far from the actual result and for our own plans to fall on our heads.
We live in a non-linear world, but we are so used to linear thinking that we often don’t even notice it.
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Source: www.vijesti.me
