in the US, most engineers take the standard two-year lower division sequence (calculus, linear algebra, a bit of diffeqs). for the most part, you learn technique rather than proving things. upper division engineering math courses teach more technique (e.g., more diffeqs).

but as madhadron says, you can't read/write proofs of upper division or graduate level math without the "foundations" material, which includes naive set theory.

do you need any of that to do engineering math? well, there are a couple of standard quotes, relating to the fact that the technique taught is brittle, in weird and subtle ways. the claim is that understanding the proofs tells you what the limits of applicability are.

"[F]or more than 40 years I have claimed that if whether an airplane would fly or not depended on whether some function that arose in its design was Lebesgue but not Riemann integrable, then I would not fly in it." - richard hamming, "mathematics on a distant planet"

"It is customary to begin courses in mathematical engineering by explaining that the lecturer would never trust his life to an aeroplane whose behaviour depended on properties of the Lebesgue integral. It might, perhaps, be just as foolhardy to fly in an aeroplane designed by an engineer who believed that cookbook application of the Laplace transform revealed all that was to be known about its stability." - tom korner, fourier analysis

I am a scientist/engineer/mathematician who studied the hell out of abstract mathematics at an extremely rigorous undergrad program. In the 20 years since, not once has that knowledge been useful in my academic or industrial work, not even remotely. I am all for studying theory for its own sake, for the career theoretician and the interested hobbyist, but as an investment I regret those four years of my life as a colossal waste of time.

Math is not just a tool; it is an area of intellectual exploration. By the same logic studying history or philosophy is a colossal waste of time, too.

so, in real world application, what's the most useful/valuable topic in mathematics?

Although I wonder for 90% people of this world that math is just a tool to pass the exam at school, not any real application (or they just can't sense it).

> in real world application, what's the most useful/valuable topic in mathematics?

Basic arithmetic: addition, subtraction, multiplication, division.

It depends on what your relative judgements are for what is deemed 'valuable' or 'useful'. Are the number theory foundations that allow digital transactions to occur more valuable than the subtraction done to figure out how many minutes until the next hour?

From what I've observed, most engineers are glad to be done with math when they finish college. Most engineering is qualitative: Organizing and arranging things, making things fit together, and troubleshooting. Maybe 10% of engineering is quantitative, and that work often goes to the handful of people in the department who have maintained an interest in it.

Some of the engineers who attract quantitative work are people who came from outside of the mainstream engineering training, such as scientists and math people.

When you refer to engineers, do you mean actual engineers (BSc in Engineering)? or your web designer with jquery skills who calls himself engineer?

In my experience (as a civil then software engineer with a MS in CS in ML/Big Data) "real" engineering is much more prescriptive than software.

When you're designing a real-world engineering project, the entire specifications are defined legally (through national, state and local laws) and technically in manuals/books. Many engineering specifications will describe the work done to a T before you even need to think about it i.e. "water main shall be constructed of 12'' coated DIP at depth no less than 2 ft". A lot of the challenges are managerial and logistical.

All of this is on purpose. "Traditional" engineering disciplines are more mature and have the constraint of being safe for the general public. There isn't much room at all to creatively deviate from what's already specified.

I've found software design to be a lot more technically demanding in regards to designing and building things. There's a lot less precedent, more moving parts and many different ways to do one thing.

I'm referring to people with degrees in mechanical, electrical, etc., or programmers with computer science or related degrees. These are all people who got through some level of college math requirement such as calculus.

And I'm not blaming anybody -- for one thing college math is often badly taught, and there's a pervasive message that you won't use any of your math or theory after you finish your degree. And then we get them so busy with CAD and bureaucracy, that they forget a lot of their school stuff.

But anything requiring calculus or above, goes to a handful of "math people" in the department, who accept those tasks in return for avoiding the CAD and organization stuff. (I'm one of those people at my workplace, my degree is in physics).

To make it a bit harder, virtually all math these days is done with computation, which means a person has to be good at both math and programming at some level.

Depends on what you're doing. My father did a huge amount of mathematical analysis over the years (roofs, bridges, trusses, ship hulls), but, again, he had the mathematical ability to do it. One of my best friends gets called into big institutional HVAC systems when they're misbehaving, and pretty much makes his living from qualitative understanding of dynamical systems that requires this kind of foundation.

Come to think of it, most of the math majors know (who didn't get PhDs and now work as programmers or data scientists) have forgotten the greater part of what they learned about groups/rings/lattices, and they seem to be doing pretty well too! ;)

Most people I know have not bothered to learn any math beyond algebra, and try to forget what algebra they did learn, they still seem to be doing pretty well. You can always limit yourself and pretend that's "enough" but it can very well be self-limiting. You simply don't know what you don't know, and that includes not knowing all of the things you might miss out on by not studying. Set theory and abstract algebra has a huge range of applications, simply being able to earn a living without studying those things is an insufficient reason to avoid studying them.

Depends what you work on. My suggestions were based on what has served me well professionally. Another way of looking at it is that if you don't have this, you're not going to get to work on things that require them.

Another thing most engineering courses skimp upon is variational methods/calculus of variations, and that has effect in how they do classical mechanics. Physics majors and engineers do classical mechanics in very different manner.

Neither did most historians, or most apple farmers. What's your point?