Stephen Hawking and Math


(Josh Cohen) #1

I saw the movie The Theory of Everything a few days ago and highly recommend it. Here’s the trailer:

I was wondering how he does math in his head. Wikipedia says: “As he slowly lost the ability to write, he developed compensatory visual methods, including seeing equations in terms of geometry.

The citations for that sentence are:

  • Ferguson 2011, pp. 76–77 - http://books.google.com/books?id=nZLMWpujVUcC
  • White & Gribbin 2002, pp. 124–25 - http://books.google.com/books?id=NoE6FAnDtRYC

Does anyone have the books or know more information about how he does math?


#2

I remember seeing that “stephen hawking sees geometry and math in terms of shapes” before, on these forums somewhere.

I’ve also read his books, the universe in a nutshell and a brief history of time(two books combined into one), would highly recommend to anyone.

Will also definitely watch this movie soon.

Don’t recall him ever mentioning his visual abilities.

Bateman


#3

Also, doesn’t it come out New Years?


(Josh Cohen) #4

The Stephen Hawking movie? I’m not sure – it’s playing here in California.


#5

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#6

In some situations, using geometry makes math easier to understand. I don’t know if this is the case in the field that Stephen works in though. But here is an example.

Famous example is that of imaginary numbers or the square root of -1.
It is difficult to imagine a number that, when squared, results in -1.

I can see the number ‘3’ as a process that takes a line segment from 0 to 1 and makes it 3 times longer.
Then 3^2 is just this process twice.

Then 1/3 is a process that takes a line segment and makes it smaller.
-1 is a process that takes a line segment from the right side and moves it to the left side.
Or takes a negative number on the left and moves it to the right.

This way I can think of all numbers as processes that do something on this line.

If I imagine all line segments from 0 to infinity I am working on the right side of zero.
Next to the right side of zero I can imagine the left side of zero.
All numbers on this line (left and right of zero) when squared result in a number on the right side.
So therefore it is difficult to imagine a process, when done 2 times results in numbers on the left side.

A line has only 1 dimension.
However; if I extend the line from 1 dimension to 2 dimensions I am working in a plane.
Now it is easy to imagine a process that when done 2 times results in numbers on the left.
It is a rotation by 90 degrees!
Think about it. Take a line segment, let’s say from zero to 3 and rotate it in the plane.
This is a process, when done 2 times, that takes any number on the right side of zero to the left side.
How easy is that.

What we have done is some simple geometry to understand imaginary numbers.
Sometimes things are easier using geometry.


(Josh Cohen) #7

That’s something I would like to learn more about. There’s a visual representation of the idea in this series:

(It’s one of the best films I’ve ever seen. I’ve watched some of the episodes 3 or 4 times already.)


#8

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#9

This was my fear also; that explaining imaginary numbers and relating it to geometry would become a difficult to read story and/or understand.

For me this story is about beauty. It turns a difficult to imagine subject into one that is very easy to imagine. By imagining it a profoundly deep understanding ensues.

Another related example is Eulers identity. When I first saw it, my jaw dropped to the floor.
If you have never seen it, here is it without explanation:

e i π = -1

This is one of those things that are easy to write down and yet have a world of meaning behind it.

It is impossible to understand this formula without geometry.
It is also impossible to not see the profound beauty in this!

Read this for the background:

Back on topic. I don’t know exactly what mathematics Stephen Hawking uses for his research and in what way geometry helps him in reaching his goals. But it is fantastic to see examples of geometry helping math for deep understanding and beauty!


#10

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#11

Thanks!

I wholeheartedly agree. Beauty married to practical application.


(selmo'i cu se nintadni) #12

I don’t remember it off the top of my head, but there is a really cool proof/derivation of that from trigonometric identities using infinite series. When my calculus teacher showed us that, that was the first time I really wholeheartedly believed (not just knew) that Euler’s identity was true. Geometry may be a very powerful tool in understanding it, but I don’t think it’s strictly required.


#13

I don’t have any information on Stephen Hawking’s approaches in particular, but I can recommend a great source on making any type of math intuitive and visual: https://betterexplained.com/

A few articles on that site which relate in particular are:

Colorized Math Equations: https://betterexplained.com/articles/colorized-math-equations/

Learn Difficult Concepts with the ADEPT Method: https://betterexplained.com/articles/adept-method/