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This is more "How to study pure mathematics", when the aim is to understand how the theory is build, so that you learn to contruibute to the theory by discoveri

by sampo 9y ago

This is more "How to study pure mathematics", when the aim is to understand how the theory is build, so that you learn to contruibute to the theory by discovering and rigorously proving your own theorems. I don't think I've ever seen a "How to study applied mathematics". How do applied mathematicians, physicicst and engineers (who apply mathematics to real world problems) study mathematics, when they use it as a tool? How much or little emphasis do they give to proofs and theorems?

dboreham 9y ago
All mathematics gets applied sooner or later, no?
- aaachilless 9y ago
  Maybe, but “applied mathematics” is conventionally more similar to “how to apply mathematics” than it is to “mathematics that is applied”. Studying how to apply mathematics is qualitatively different in many ways than studying pure mathematics.
  - dboreham 9y ago
    Ah. I genuinely did not know that. In Scotland we did not have Applied Mathematics (as a school exam subject) so I guess I just assumed it was about physics problems (there was an A-level with that name in England). It never came up again in my Mathematics career..
yiransheng 9y ago
Some personal thoughts: the key for me at least, is a bit like the hammer-nail mentality. An software engineer for example has a (small) mathematical toolkit (a hammer), sometimes forcing your problem at hand into the shape of nail is sufficient to solve it. A dump example, consider the problem of detect if two date ranges overlap (for a calendar app perhaps). By recognizing a date range, DateRange(a, b) as an interval on the real line: [a, b] or a set {x | a <= x <= b}, the problem becomes find if two sets' intersection is empty. Set {x | a <= x <=b } and {x | c <= x <= d}'s intersection is {x | (x >= a and x >= c) and (x <= b and x<=d)} which simplifies to {x | max(a, c) <= x <= min(b, d) }. Therefore, the set is empty iff max(a, c) > min(b, d). Translating the problem into mathematical structures here (intervals), and manipulate them with math tools (inequalities) would count as applied math I guess. By having a repository of math knowledge and an eager attitude to look for opportunities to apply them, one can find success in solving practical problems. In this context, rigor and formal proofs are less important and math intuition is paramount. An real world example I like to motivate myself with is Richard Feynman and connection machine[1], a story about Feynman using partial differential equations to model on boolean circuits. I do believe even with a limited set of mathematical tools, by trying them on every problem you encounter, you can still get good results. Something like, if one squint hard enough, you can find a monad anywhere :) [1] http://longnow.org/essays/richard-feynman-connection-machine/ http://longnow.org/essays/richard-feynman-connection-machine...

by sampo 9y ago

dboreham 9y ago

All mathematics gets applied sooner or later, no?

aaachilless 9y ago
Maybe, but “applied mathematics” is conventionally more similar to “how to apply mathematics” than it is to “mathematics that is applied”. Studying how to apply mathematics is qualitatively different in many ways than studying pure mathematics.
- dboreham 9y ago
  Ah. I genuinely did not know that. In Scotland we did not have Applied Mathematics (as a school exam subject) so I guess I just assumed it was about physics problems (there was an A-level with that name in England). It never came up again in my Mathematics career..

yiransheng 9y ago

Some personal thoughts: the key for me at least, is a bit like the hammer-nail mentality. An software engineer for example has a (small) mathematical toolkit (a hammer), sometimes forcing your problem at hand into the shape of nail is sufficient to solve it. A dump example, consider the problem of detect if two date ranges overlap (for a calendar app perhaps). By recognizing a date range, DateRange(a, b) as an interval on the real line: [a, b] or a set {x | a <= x <= b}, the problem becomes find if two sets' intersection is empty. Set {x | a <= x <=b } and {x | c <= x <= d}'s intersection is {x | (x >= a and x >= c) and (x <= b and x<=d)} which simplifies to {x | max(a, c) <= x <= min(b, d) }. Therefore, the set is empty iff max(a, c) > min(b, d). Translating the problem into mathematical structures here (intervals), and manipulate them with math tools (inequalities) would count as applied math I guess. By having a repository of math knowledge and an eager attitude to look for opportunities to apply them, one can find success in solving practical problems. In this context, rigor and formal proofs are less important and math intuition is paramount. An real world example I like to motivate myself with is Richard Feynman and connection machine[1], a story about Feynman using partial differential equations to model on boolean circuits. I do believe even with a limited set of mathematical tools, by trying them on every problem you encounter, you can still get good results. Something like, if one squint hard enough, you can find a monad anywhere :) [1] http://longnow.org/essays/richard-feynman-connection-machine/ http://longnow.org/essays/richard-feynman-connection-machine...