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How do you understand/appreciate math on a more philosophical level?

atreides

Graduate Student
I have been wondering how someone can really understand math on a more philosophical level... I would assume most people are exposed to 'plug and chugg' or 'formulaic' math/calculus in their college years. When you hear that the second derivative is 1 or the integral is 0, how does that register on a philosophical note as opposed to learning a bunch of rules to tell you what's going on?

Is this level of enlightenment / understanding only achievable with exposure to some higher level mathematics?
 
I've always tended towards the "enlightened understanding is an innate ability" opinion. In the same way that most people can probably produce a half decent painting after lessons, they'll likely never produce truly inspired works. Art that transcends time and space is only going to be produced by someone with the right combination of crazy/genius/flair etc.

In the same way, a lot of people will be exposed to and learn mathematics, but only a handful of people will ever really be able to push the limits of understanding and/or really discover those "beautiful" mathematical truths. When I think about this myself, I've accepted the idea that I'm never going to be able to imagine folding coffee cups into doughnuts in 11 dimensions etc, but I think I'll always be envious of the minds that are able to do so ;)
 
By taking an advanced Algebra and Analysis class or a good class in Discrete Math.

After taking discrete math with a very good prof. I've always wondered how colleges get about teaching Linear Algebra and Calculus while skipping such a huge chunk of what's needed to really understand these topics.

It also made me realize that a lot of the questions asked can never be solved from frist principles alone and require either the intuition of Einstein and newton or you having seen the problem and solution before. Students therefore solve more questions (read: memorize more), read about math less and the creative side is stifled.
 
Agree. Analysis class in grad school was the first time I realized that all the Math I learned in college during my engineering degree was very superficial.
So I promptly went ahead and took 3 more analysis courses.
 
Math, like programming, and anything else, is a mere tool for solving problems. Anything for its own sake that doesn't solve problems IMO is superficial. One thing I will never understand is how theoreticians can simply write papers about theoretical and abstract concepts so disconnected from the rest of the world.
 

koupparis

Carpe noctum
Math, like programming, and anything else, is a mere tool for solving problems. Anything for its own sake that doesn't solve problems IMO is superficial. One thing I will never understand is how theoreticians can simply write papers about theoretical and abstract concepts so disconnected from the rest of the world.
It is often those theoretical papers, which will 50 years down the line solve some of the most interesting and deep problems we'll be encountering.
 

Dave Haan

aka nnyhav
The more exposure to different disciplines, the better. And not just the higher or more theoretical versions. An interdisciplinary view can contribute insights across the joints (which are to some extent artificial anyway).
 

Sanket Patel

i do stuff
By not acting as if you know everything
 
I found Jacques Hadamard's "The Psychology of Invention in the Mathematical Field" to be useful. It's a slim book and inexpensive (since it was published by Dover). It may be going by another name today. Just checked Amazon -- the new title is "The Mathematician's Mind."

Postscript: I also found Lakatos' "Proofs and Refutations" useful.
 

CGiuliano

Lowly Undergrad
I'm about half way into my first term of analysis. As a resistant-to-change student, I've come up with a fairly "formulaic" method to understand proofs, and more abstract mathematics.

First off, choose a topic, -- or better yet: a single problem. Time is finite; you can learn anything, but not everything. Then, beat the hell out of the problem in the formulaic manner you have been taught, i.e., understand all the operations and concepts used by practicing them yourself, repeat. Briefly revisit your problem whenever you get a minute, or an hour. The "how" is always our first question, and generally comes in a fashion we are accustomed. The "why" follows promptly with repetition.
 
I think Galileo Galilei said it best:

"Philosophy is written in this grand book - I mean the Universe - which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles and other geometrical figures, without which it is humanly impossible to understand a single word of it."
 
I think 'enlightenment' comes from discovering something on your own, not necessarily in higher math. I still vividly remember the happiness I got from finding a formula for number of diagonals of a polygon, or the area of a quarter sphere, many many years ago. And just recently I discovered that convergent sequences form a sub-space in real numbers and I just went 'wow'.

The joy of discovery that math provides is almost unmatched in academics, because whenever you discover something, you can also 'almost' every time also prove it. I don't know if there is a deeper philosophical level to math, but does it have a meaning in my life; Absolutely!
 
everyone in this thread is talking about the appreciation of maths such as analysis, discrete maths, etc.

I think it was G.H hardy a pure mathematician in the early 20th century(major contributor to number theory) who wrote a book called a A mathematician's Apology. He spoke about the joys of solving a mathematical problem, not for the uses of mathemtics but the fact that you can actually find a solution to a problem is a beauty itself. Though he actually thought number theory could never be applied until we implemented into computer science... Aristotle(or Pythagoras) was also against applying maths to solve real world problems, it was newton who changed all that...

Anyways Im just wondering how many people have the same appreciation towards financial maths as opposed to maths in physics, cs, engineering, etc. On one hand were applying maths to solve problems but whether or not it really works is still questionable considering the GFC. I think the maths in financial maths/quant finance is really overused and has lost its meaning in the sense its able to accurately describe the world we live in.

One things for sure, I'm not in awe to mathematics like I used to be when I studied physics/maths in undergrad, now studying quant finance

Also one thing to also be aware of is the difference between first 50 yrs of last century in physics and the next 50, is it philosophical foundations, perhaps maths finance needs a bit more intuition/philosophy applied to better its practicality.

String theory faces a similar problem......
 
Aristotle(or Pythagoras) was also against applying maths to solve real world problems, it was newton who changed all that...

Galileo, and before Galileo, Bacon. You will find more details of the program to use math as a master key to the cosmos in one of Frances Amelia Yates' books (Giordano Bruno and the Hermetic Tradition? The Rosicrucian Enlightenment? The Occult Philosophy in the Elizabethan Age?)

I think the maths in financial maths/quant finance is really overused and has lost its meaning in the sense its able to accurately describe the world we live in.

It never had it (with regard to applications in finance). Most of quant finance will probably go the same way as the Ptolemaic theory of planetary motions. The math will of course remain (and predates the applications to finance), but its applications to finance will increasingly be questioned, are already being questioned.

String theory faces a similar problem......

String theory as a physical theory is rubbish. In Woit's words, it's "not even wrong."
 
It never had it (with regard to applications in finance). Most of quant finance will probably go the same way as the Ptolemaic theory of planetary motions. The math will of course remain (and predates the applications to finance), but its applications to finance will increasingly be questioned, are already being questioned.

I've noticed that you feel strongly about this. What in your opinion is the best approach to understand/study capital markets? Or do you consider such an attempt futile?
 
I've noticed that you feel strongly about this. What in your opinion is the best approach to understand/study capital markets? Or do you consider such an attempt futile?

No, it's not futile. What is futile is the attempt to mathematise the field -- as if it's subject to general laws like celestial mechanics, rather than being subject to ad hoc and expedient political decisions. If it's to be studied, it should be empirically, without theoretical preconceptions, and using rules of thumb one is prepared to jettison the moment they look as if they are not working. Finance is an artificial man-made system, where the ground rules keep getting changed by the people at the pinnacle. The math is there to dress it up as something subject to laws and regularities. And to provide gainful employment to dishonest nincompoop professors who drone on about sigma-algebras.
 
And yet, BBW, the people who have done the best in the markets--the top quants--are those who have taken that exact advice to heart.

(It's also why I decided to master in statistics instead of mathematical finance at Rutgers)

I'm just wondering what types of quantitative methods are best amenable to quick gear-shifting.
 
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