Is geometry the heart of mathematical insight?

[pergradus]

Richard Feynman once remarked during a lecture that the invention of analytical methods such as calculus allowed people to "be more stupid", in reference to solving problems. I think he was on to something with that remark.

Often when faced with a problem which requires me to set up some kind of picture, I have to resort to "brute force" methods of calculus or lots of algebraic manipulation, and I find myself unable to offer a geometric argument for what I'm trying to show - even though I suspect the geometric argument is far more simple and elegant.

I was thinking this summer maybe I should really focus on learning some geometry to help me develop some real mathematical skills, instead of what I feel like is the systematic and uninspired way you learn in school. (I am a physics student btw, not a math student). Starting with the master, Euclid and seeing where that leads me. Just wondering what some opinions are about this and if you agree or disagree with this sentiment.

 

[micromass]

Well, in my opinion, there are two different questions you ask: "does geometry offer more insight?" and "are geometric arguments more simple and elegant?"

As for the first question, my answer would be yes. Working out the surface area of a parabola by calculus shows not much insight. It simply appears to be symbolic manipulation. If you work it out geometrically, then you can see why it is true.

However, the geometric approach is by far the most complicated approach. Lots of things are simplified by the use of algebra and calculus. And calculus is by far more elegant than the most geometric arguments. As an example: if you want to prove that the ratio of the circumference of a circle to the diameter is a constant, then this is a very easy exercise in calculus. However, it is much more complicated in geometry.

Historically, geometry was seen to be superior to algebra. And in my opinion, this prevented mathematics to develop significantly. It was only with the invention of calculus by Newton and Leibniz, and with the invention of analytic geometry by Descartes, that mathematics could develop. The same is true with physics. Without calculus and algebra, physics would be very hard. Calculus simplifies a lot of arguments!

As for learning geometry: I do think it's a very good idea. But I don't think that this would enhance your physics and mathematics by a lot. But if you're curious about it, then by all means: study it! However, I don't know if learning Euclid is the best thing to do. Euclid contains some errors and is very outdated. A nice alternative is Hilbert's "The Foundations of Geometry". See http://www.gutenberg.org/ebooks/17384 But it's less intuitive and comprehenisve then Euclid...

 

[mathwonk]

for other opinions of Euclid, see:

https://www.amazon.com/dp/1888009195/?tag=pfamazon01-20and:

https://www.amazon.com/dp/1441931457/?tag=pfamazon01-20

 

[AlephZero]

Learning some geometry certainly won't do you any harm. If you get "bored" with Euclid after a while, try some other flavors. e.g. Appollonius on conic sections if you want to stick to the Greeks, or something more modern like projective geometry.

The bottom line is, you have to get insight (as opposed to the ability to plug-and-chug) from anywhere you can. There aren't any "rules" about what is the best place to find it. If geometry floats your boat, then go with that!

 

[morrobay]

I would appreciate some geometric insight as to how the area below the inverse curve
between 1 and x is equal to ln x.
ln x = integral 1 to x 1/t dt

 

[AlephZero]

I would appreciate some geometric insight as to how the area below the inverse curve
between 1 and x is equal to ln x.
ln x = integral 1 to x 1/t dt

That is how ln x is defined.

The "insight" is figuring out that ln x actually behaves like a logarithm function, in other words proving that ln (xy) = ln x + ln y from the definition of ln.

The area between 1 and xy is ln xy. Divide this into two parts, from 1 to x and from x to xy.

The area between 1 and x is ln x.

If you compare the areas from 1 to y and form x to xy, the second one is "x" times as wide and "1/x" times as high as the first one, so the areas are the same.

In other words, the area from x to xy is ln y.

So ln xy = ln x + ln y.

You can prove that the "ln" function has other properties of logarithms with similar arguments.

 

[pergradus]

Learning some geometry certainly won't do you any harm. If you get "bored" with Euclid after a while, try some other flavors. e.g. Appollonius on conic sections if you want to stick to the Greeks, or something more modern like projective geometry.

The bottom line is, you have to get insight (as opposed to the ability to plug-and-chug) from anywhere you can. There aren't any "rules" about what is the best place to find it. If geometry floats your boat, then go with that!

Sounds really interesting. I've always wondered what a treatment of conic sections would look like without modern analytic geometry.

Also thanks for the links Mathwonk.