How are mathematical ideas discovered

[Fascheue]

If I need to solve the equation x^2 + x = 5, I can recognize that it’s a quadratic equation, change it to x^2 + x - 5 = 0, and then plug into the quadratic formula. In general the way that math is taught, I recognize what type of problem it is, then use the techniques that I was told to use to solve that sort of problem.

How though did people figure out this sort of stuff? If I was given that same equation, but didn’t yet know how to solve quadratic equations, I’m not even sure where one would begin to try to solve it or to find some way to solve it.

 

[fresh_42]

If I need to solve the equation x^2 + x = 5, I can recognize that it’s a quadratic equation, change it to x^2 + x - 5 = 0, and then plug into the quadratic formula. In general the way that math is taught, I recognize what type of problem it is, then use the techniques that I was told to use to solve that sort of problem.

How though did people figure out this sort of stuff? If I was given that same equation, but didn’t yet know how to solve quadratic equations, I’m not even sure where one would begin to try to solve it or to find some way to solve it.

They rely on what is already known.

Bernard of Chartres used to say that we [the Moderns] are like dwarves perched on the shoulders of giants [the Ancients], and thus we are able to see more and farther than the latter.

https://en.wikiquote.org/wiki/Isaac_Newton

It is often along this important pattern:

1. Pose the question as precise as possible.
2. List what you already have in the context.
3. Try to figure out a path from 2. to 1.

In the case you've mentioned, it is: We want to know the roots of $x^2+x-5$. We already know, that $(a+b)^2=a^2+2ab+b^2$. Now we simply try to get from one to the other:
$$
x^2+x-5=x^2+2\cdot x \cdot \dfrac{1}{2} + \left( \dfrac{1}{2} \right)^2 - \left( \dfrac{1}{2} \right)^2 -5 = \left( x + \dfrac{1}{2} \right)^2 - \dfrac{11}{2}
$$

 

[mfb]

Discovering how to approach a new type of problem is typically much more difficult than applying an approach that has been found before. Sometimes it is easy to reduce the new problem to a known one (e.g. the first step you made in your post), sometimes it is much more difficult.

How to approach it: try different steps that change the problem to different forms until you find one that is easier to solve. With the quadratic equations this is typically done in class to see where the formula comes from. With cubic functions it gets much more complicated. Wikipedia has a description. Hundreds of things were tried, a few of them worked.

 

[LCKurtz]

Sometimes such problems take a flash of insight or the perspective of a genius. As an old saying goes, if you had been born into a civilization where the wheel had not been invented, you probably wouldn't have thought of it either.

 

[Ssnow]

First of all are mathematical ideas discovered or invented ?

Ssnow

 

[fresh_42]

First of all are mathematical ideas discovered or invented ?

Ssnow

This is a purely philosophical question and has been discussed on PF several times before, most recently here:
https://www.physicsforums.com/threa...onsidered-a-discovery-or-an-invention.924120/

In addition it leads to a different topic, because how to develop mathematical ideas - the OP's question - does not mean, how to label mathematical ideas. And this is it: invention or discovery is only a label and a question of taste and philosophical schools.

Thread closed.