Is applied mathematics proof-based?

[Bipolarity]

I have noticed that I love proving things in math. People have told me that proof-based work is more the specialty of the pure mathematician whereas math used for practical purposes is the specialty of the applied mathematician but I cannot imagine this to be so.

What I've realized is that I like to prove ideas in math that are indeed applied to other fields. In other words, I dislike pure math such as topology but I love to prove things in other types of math such as statistics/calculus. But I've heard that the applied math don't care much for proofs and that they use intuition rather than formalizing everything as a pure mathematician does. This view of applied mathematics scares me, and it cannot be right because I have never imagined mathematics to be devoid of proof! But for example, I've heard that engineers for example, are often oblivious to "proof-based work" by mathematicians, even if it pertains to engineering. So is there any place for proof-based work in applied mathematics? Surely there must be! But the attitude of the engineers I have met is discouraging me so much.

As an example, I am hugely fascinated by things such as Arrow's Impossibility Theorem, the Minimax theorem, things like that... which are indeed based on mathematical proof, but which could never be considered as "pure mathematics". But the economists I met seem largely oblivious to the "proof-based mindset of a mathematician". So there is this disconnect I am observing which I cannot account for!

So do applied mathematicians actually do proof-based work like the pure mathematicians or do they eventually rely on simulations/experiments like the engineers and natural scientists? Because what I want is proof-based work which can be applied in real life.

BiP

 

[MathWarrior]

Check the curriculum at the university you intend to go to, they usually differ.

Being someone who is doing Applied Mathematics, I am expected to take a introduction to proof writing, and a real analysis class, both of which are heavily proof based. Most of my upper division classes require the introduction to proof class before I can take them, because they require the ability to read and write proofs.

The remainder depends on what your university expects of you, but their is a requirement to understand and write proofs for even applied mathematics in most cases.

Colleges are usually pretty flexible mine even allows me to take upper division curriculum from the pure mathematics department, applied mathematics department, and even he computer science department. This however depends on the college.

 

[AlephZero]

You might have got a misleading idea about how mathematics is actually DONE (i.e. how new math is invented). What you see in textbooks and even in journals is the end product, arranged in a nice tidy logical format. The "doing" part is often a mess of bad ideas, mistakes, stuff that leads nowhere, etc. I've no idea how Arrow's Impossibility Theorem was actually thoght up, but I would bet a few dollars of my own money there were a few months or even years of messing around with ideas that didn't work, before getting to the final version.

You don't need "proofs" to USE mathematics that has already been invented, which is what many engineers and applied mathematicans do most of the time. On the other hand inventing new applied math is no different from inventing new pure math. If you dream up with a new computer algorithm for solving differential equations or whatever, you had better PROVE how accurate it is and in what situations it fails, if you want anybody to take the results seriously.

Look at some of the SIAM journals like Applied Math, Numerical Analysis, Scientific Computing, and see whether there are enough "proofs" to keep you happy.

 

[Integral]

An applied mathematician takes very rigorous course work in Differential equations (ODE and PDE) as well as numerical methods. While an engineer learns how to use DE's and numerical methods an Applied Mathematician learns how they are derived and must understand when and how they are applied. This means understanding the underlying proofs. That is why, when an engineer can't get the results he wants, he can call on a Applied Mathematician to find why the methods being used are not working.

So while an AM needs to go through and understand proofs, writing proofs will not be part of the day to day job.

 

[Matrix0]

:

I can assure you that applied mathematics is indeed proof-based. While there may be a perception that pure mathematics is solely focused on proofs and applied mathematics is more focused on practical applications, this is not entirely accurate.

In applied mathematics, the focus is on using mathematical theories and techniques to solve real-world problems. This often involves developing and proving new mathematical models and algorithms that can be applied to various fields such as engineering, economics, and natural sciences.

Just like in pure mathematics, proofs play a crucial role in applied mathematics. They serve as a way to validate the accuracy and effectiveness of the mathematical models and algorithms being developed. In fact, many applied mathematicians are highly skilled in both developing practical applications and providing rigorous proofs to support their work.

It is true that in some fields, such as engineering, simulations and experiments may be used in addition to mathematical proofs. This is because some problems may be too complex to solve purely through mathematical analysis, and simulations can provide valuable insights and predictions. However, this does not diminish the importance of proofs in applied mathematics.

Furthermore, the examples you mentioned, such as Arrow's Impossibility Theorem and the Minimax theorem, are prime examples of how proof-based work in applied mathematics can have significant real-world applications. These theories have been used to solve problems in economics, game theory, and decision-making processes.

In conclusion, applied mathematics is a field that values both practical applications and rigorous mathematical proofs. I can assure you that there is a place for proof-based work in applied mathematics, and it is highly valued and respected in the scientific community. So do not be discouraged by the attitudes of some individuals, as proof-based work in applied mathematics is essential in solving real-world problems.