Is a formal, rigorous math education useful in problem solving?

[fatpotato]

I'm very curious as to know how a person who pursued a formal, strict and rigorous education in mathematics would fair in comparison with a person who learned applied math by "intuition" (that is without doing any proofs and relying more on the computational part), when confronted with problem solving.

For example, suppose a person follows a rigorous education in pure mathematics (let's say at a Graduate level), takes part in Math Olympiads, proves every theorem she encounters in her homework and is generally pretty math savvy. I assume such a person would have a good grasp on abstract reasoning and generalization.

Now, how would such a person, who learned to think in an abstract manner would fair when confronted with "hands-on" problems, like designing an electronic amplifier circuit? What is her thought-process like? Since the calculations required to design an amplifier are just a special case of a broader set, would it be de facto an easy task to perform?

I never had the opportunity to take formal math courses and I can't help but wonder if taking more abstract math courses gives an edge on solving "simpler" applied math problems.

EDIT:
Question is directed towards applied maths disciplines problem solving, specifically how a theoretical point of view helps when solving more practical problems (for example, engineering related problems).

 

[S.G. Janssens]

I very strongly feel that "applied math" without proofs is not mathematics. Mathematics of any kind requires proofs, not just intuition. The proofs are there to make sure that your intuition was, in fact, correct. The intuition is there to guide the proof, but it is no substitute.

I also very strongly feel that it is perfectly fine to skip complete and rigorous proofs and rely more on physical intuition, but then the result should not be advertised as mathematics, but rather as the use of mathematics in science or engineering. This use is very honorable and not of any less value.

In fact, it seems to me that it is perfectly well possible (and, of course, common practice) to - for example - design your amplifier circuit without first proving well-posedness of the underlying system of (possibly nonlinear) differential equations. Mathematicians may or may not be horrible at this, depending on how much they know about electronics. In any case, the engineer is likely to beat them, not only because of his specialized knowledge, but also because of his orientation towards problem solution rather than rigor and abstraction.

However, mathematics (pure and applied alike) is about discovering common structures in - for example, again - electronic circuits and proving the properties of these structures. In order to do that, a rigorous and abstract approach can be very fruitful.

 

[nuuskur]

Are specific electronics problems regarded as day-to-day problems in this topic? How does one measure 'usefulness' of education? I regard my highly abstract education 'useful' in my everyday life.

As a seemingly very unlikely example, learning mathematics has helped me teach myself to cook and elevate my past diet which consisted of at most five colours to diverse and healthy meals.

I can also use specific knowledge from linear programming to optimise my finances or at the very least decide what sort of spending is 'feasible'.

By becoming a more logical creature, I am also able to (easily) resist wasting money (street corner hamburger joints or impulse buying to name some).

One could argue all of the above are some kind of "common sense", but I surely didn't possesses that common sense before studying mathematics for years.

 

[gleem]

Now, how would such a person, who learned to think in an abstract manner would fair when confronted with "hands-on" problems, like designing an electronic amplifier circuit? What is her thought-process like? Since the calculations required to design an amplifier are just a special case of a broader set, would it be de facto an easy task to perform?

I would say no. As a physicist I learned all my math in strictly math courses some relatively abstract like Linear Vector Spaces and Analysis and other like PDEs and Vector Calculus more applied. For problem solving I depended on the results of the proofs and relationships and my experience in solving problems. I fact it was my experience with the type of problem that I would be routinely meeting that helped the most.

You do not need to know how a tool is made to use it well. But you do need to use it often to use it well.

Alan Schoenfeld (Stanford University) has studied problem solving for decades and I believe he has not found as a rule that mathematicians have an advantage over physicists in problem solving.

One could argue all of the above are some kind of "common sense", but I surely didn't possesses that common sense before studying mathematics for years.

Perhaps it wasn't the subject of math per se that was responsible for your improved ability to effectively/efficiently deal with life's problems and issues, but the discipline that is required to be a successful mathematician. Add to that the natural maturation process by which we tend to become wiser.

I can also use specific knowledge from linear programming to optimise my finances or at the very least decide what sort of spending is 'feasible'.

Linear programming as such is not needed by the average person to be financially responsible. For you this may be only a crutch on which you depend.

As a seemingly very unlikely example, learning mathematics has helped me teach myself to cook and elevate my past diet which consisted of at most five colours to diverse and healthy meals.

Here again it is the critical thinking that math develops that is probably the factor. But one does not need math to develop critical thinking.

 

[fatpotato]

I also very strongly feel that it is perfectly fine to skip complete and rigorous proofs and rely more on physical intuition, but then the result should not be advertised as mathematics, but rather as the use of mathematics in science or engineering. This use is very honorable and not of any less value.

I find this very true. We should clearly separate mathematics per se and its uses, without denigrating the lack of proof and seeing it as less pure, as I have heard so many times.

However, mathematics (pure and applied alike) is about discovering common structures in - for example, again - electronic circuits and proving the properties of these structures. In order to do that, a rigorous and abstract approach can be very fruitful.

Now that I think about it, I believe the abstract approach would be very useful when confronted with unusual problems, where applying common formulas and solving axiomatically do not suffice.

For problem solving I depended on the results of the proofs and relationships and my experience in solving problems. I fact it was my experience with the type of problem that I would be routinely meeting that helped the most.

Thank you for your contribution. It is interesting to hear about someone who received formal education for comparison purpose. It seems that generally speaking, experience in the subject might count more than proofs.

 

[CrysPhys]

EDIT:
Question is directed towards applied maths disciplines problem solving, specifically how a theoretical point of view helps when solving more practical problems (for example, engineering related problems).

My career has spanned a wide range of fields, from basic physics research to applied device development to quality process engineering to systems engineering to patent law. The mathematical formalism I've found useful throughout in approaching a problem is to frame it in terms of boundary conditions, constraints, governing equations (or rules), knowns, and unknowns. I don't think in the theorem-proof mode.

 

[rkr]

For example, suppose a person follows a rigorous education in pure mathematics (let's say at a Graduate level), takes part in Math Olympiads, proves every theorem she encounters in her homework and is generally pretty math savvy. I assume such a person would have a good grasp on abstract reasoning and generalization.

Now, how would such a person, who learned to think in an abstract manner would fair when confronted with "hands-on" problems, like designing an electronic amplifier circuit? What is her thought-process like? Since the calculations required to design an amplifier are just a special case of a broader set, would it be de facto an easy task to perform?

For context, I competed in the Olympiads and the Putnam. My coursework was on a theoretical track, e.g. my first introduction to statistics was Casella, which I got to after measure theory, and I skipped more practical coursework in probability and statistics. I now run a startup.

Sampling from people I've worked with, and doing some self-assessment, I think there's 2 transferrable skills that I had picked up from theoretical work.

The first is that people with theoretical backgrounds seem to be more dogged and gritty in chasing down every detail. I think that's because we're used to having slow payoff. This is great for a certain class of problems where there's a lot of plumbing that tends to deter the younger, CS crowd that like using shiny new frameworks and abstractions, moving on after 1 year of mastery of any technique, and skipping the lower level, foundational work.

The second is that theoretical folks, surprisingly, tend to be a lot more willing to use simple, parsimonious solutions to problems that also in turn tend to be easier to communicate and explain. More applied folks seem to feel that the more sophisticated the machinery you throw at a problem, the better of a solution you get. I see this practice in ML literature and applications all the time. I don't have a good explanation for this observation. Putnam and Olympiad problems tend to be characterized by simple, elegant solutions that are difficult to find (see "Jewish Problems" - https://arxiv.org/pdf/1110.1556.pdf). Like a clever substitution that cracks a tight bound. Perhaps it's something we're conditioned to.

Even if you don't intend to do theoretical work later in your career, I highly recommend taking 1 course that requires rigorous proof writing. Group theory was incredibly rewarding and opened up many epiphanic moments for me, and I think I'd have been very satisfied even if I had walked away from further theory at that point.

 

[clope023]

I'm very curious as to know how a person who pursued a formal, strict and rigorous education in mathematics would fair in comparison with a person who learned applied math by "intuition" (that is without doing any proofs and relying more on the computational part), when confronted with problem solving.

For example, suppose a person follows a rigorous education in pure mathematics (let's say at a Graduate level), takes part in Math Olympiads, proves every theorem she encounters in her homework and is generally pretty math savvy. I assume such a person would have a good grasp on abstract reasoning and generalization.

Now, how would such a person, who learned to think in an abstract manner would fair when confronted with "hands-on" problems, like designing an electronic amplifier circuit? What is her thought-process like? Since the calculations required to design an amplifier are just a special case of a broader set, would it be de facto an easy task to perform?

I never had the opportunity to take formal math courses and I can't help but wonder if taking more abstract math courses gives an edge on solving "simpler" applied math problems.

EDIT:
Question is directed towards applied maths disciplines problem solving, specifically how a theoretical point of view helps when solving more practical problems (for example, engineering related problems).

Depending on how simple your circuit is, all you need to know how to do is effectively add, divide, and/or multiply a couple of numbers together to calculate an amplification gain factor; if you're philosophically waxing about whether you have sufficient proof that your algebraic operations are valid, you're overthinking it...

 

[fatpotato]

The mathematical formalism I've found useful throughout in approaching a problem is to frame it in terms of boundary conditions, constraints, governing equations (or rules), knowns, and unknowns. I don't think in the theorem-proof mode.

That is exactly what I wanted to know : "Do people think in the theorem-proof mode". Thank you for your answer. Also, thinking in terms of boundary conditions is a habit I should strive to form.

This is great for a certain class of problems where there's a lot of plumbing that tends to deter the younger, CS crowd that like using shiny new frameworks and abstractions, moving on after 1 year of mastery of any technique, and skipping the lower level, foundational work.

The second is that theoretical folks, surprisingly, tend to be a lot more willing to use simple, parsimonious solutions to problems that also in turn tend to be easier to communicate and explain. More applied folks seem to feel that the more sophisticated the machinery you throw at a problem, the better of a solution you get.

I do prefer elegant solutions to more complex and involved ones. Alas, when one is rushed to solve a problem as fast as possible, I fear one is tempted to prefer the latter which is, generally speaking, more obvious although less satisfying.

Even if you don't intend to do theoretical work later in your career, I highly recommend taking 1 course that requires rigorous proof writing. Group theory was incredibly rewarding and opened up many epiphanic moments for me, and I think I'd have been very satisfied even if I had walked away from further theory at that point.

Thank you for your contribution. I don't know if I'm allowed to ask another question on top of this thread, but I'll try anyway ; why are you especially suggesting group theory? Would this field be "better" to learn than, say, real analysis? I have read a bit to get a grasp of what group theory is and I found it pretty interesting, could you tell me what felt rewarding about it please?

 

[gleem]

It seems you are more interested in math than EE.

 

[marcusl]

Since you ask about hands-on problems such as designing electronic circuits, I will reply that the few mathematicians I've come across in industry who try it seem miserable at it. They are at their best when analyzing complex phenomena and when designing simulation codes for something where a commercial code falls down, and here they are without peer.

 

[FactChecker]

Pure math concentrates on proofs and probably does not help much with problem solving. I would say it is the long way around to improve applied problem solving.