sakky

I would add that the loss of mathematics academic leadership of the United States would almost certainly result in the concomitant loss of economics, finance, and (probably) accounting academic leadership of the United States: 3 academic fields where Americans clearly dominate. Heck, it's hard for me to think of more than a handful of econ/finance/accounting departments that are not in US universities (and of the rare few that do exist, most seem to be in the UK). Now, whether you may think such dominance is good or bad for the nation depends on whether you think economics/finance/accounting is harmful for the nation (you could arguably make the case that they're actually harmful for the nation), but undoubtedly it would be bad for all of the Americans who comprise the shockingly well-paid faculties of those disciplines. {A newly minted assistant professor of finance or accounting will probably make $125-250k a year to start including summer support, without need for a postdoc. Not bad for somebody in their mid-to-late 20's}. Like I said, academic economics, finance, and even to some extent accounting nowadays are arguably just sub-branches of math. Frankly, most top mathematicians could walk in and quickly dominate most economics and finance departments.

Projectile

Lol, proofs can't be used to solve computational problems. I should know, I've tried solving computational problems in the pure computational calculus class at my school trying to use the techniques I learned in analysis I. What you're saying is akin to an engineering student learning the mathematical proofs for string theory to help solve their engineering problems


Really? Did you ever have to use mathematical induction? Proof by contradiction? Because those proof techniques would never be useful in taking a derivative or an integral. Also, taking a derivative by first principle isn't a proof nor are delta-epsilon "proofs". Those are computations and at best algebraic excercises with notation and require no creativity or deep thinking to solve.

aegrisomnia

At first I was going to say that this was the most wrong statement ever. Then I realized that was hyperbole; your statement is less clearly false than, for instance, "false iff true" or "This is not a statement." Still this statement is completely and utterly false, as false as those statements. It is very often the case that a constructive proof will imply a computational technique in computer science. Computer programs are equivalent to mathematical proofs via an isomorphism. Curry?Howard correspondence - Wikipedia, the free encyclopedia . The idea that proofs can't be used to compute something is an idea you need to abandon right now.

Where do I even begin? First, delta-epsilon proofs are proofs, even if you don't find them particularly complicated. In mathematics, the proof of a statement x is a sequence of statements, such that the nth statement is either an axiom or implied by statements 1, 2, ..., n-1 and some rules of inference, where the last statement is x. That certainly applies to delta-epsilon proofs.

Besides, who cares whether engineers need to learn how to do proofs? We rely on mathematicians to prove things, as it should be {if you consider computer science to be "engineering", you should know that computer science majors do real proofs using the same kinds of techniques and reasoning as math majors, by the way; so if your argument is that nobody needs to know how to do proofs because proofs are useless, you're lumping CS in with math, despite the seemingly fantastic opportunities for CS majors, even compared to engineering}.

Projectile

I meant for the types of proving techniques available to a high school student or freshmen/sophomore math major, not the types of proofs and skills a math graduate student would know.

You really think a high school student would try to learn complex analysis and number theory just so they could do computations in a different manner in high school calculus?



Yes, by definition they are proofs. But qualitatively, no real math student believes delta-epsilon to be on the same rigor as other proving techniques learned in a proof-y math class such as analysis or set theory.

Engineers learn delta-epsilon proofs in their undergraduate calculus classes. So, can we then conclude that engineers "know how to do proofs" despite the fact you admitted the contrary in a previous post?

The simple fact that CS majors need to do proofs does not do much to support your argument. In industry when they are programming or in software development, are they constantly proving theorems on a day-to-day basis? Nope. Even Sakky mentioned in another thread that engineers do not use most of the theoretical concepts they learned in undergraduate from physics and math, and that a senior engineer couldn't even remember how to take a derivative.

sakky

Projectile, I agree with your general point that most engineers do not need to know how to construct proofs or otherwise engage in theoretical complexities. But I would temper that by saying that the vast majority of people in the world do not really need to know whatever they had studied in college. Let's face it - the vast majority of English majors will never actually analyze literature as a profession.

Theoretical mathematics knowledge is therefore probably useful only in academia. But again, that's why I encourage you (and all other math majors) to seriously encourage using your math knowledge within social science academia, especially within in a business school. Like I said, business school professors are shockingly well-paid and enjoy arguably the best job prospects of any academic discipline. And believe me, they are deeply intimidated by anybody who knows theoretical math, as most of them don't know it. The business academic journals become ever more mathematically oriented every year - to the point that qualitatively oriented business researchers bemoan the lack of publication (and hence career-advancement) opportunities.

In my previous posts, I failed to mention perhaps the clearest pathway for a math undergrad to enter business academia: enter operations research, which truly is just a mathematics subdiscipline. Many articles from A-leading operations journals such as the eponymously named Operations Research (the journal) and Management Science are truly indistinguishable from a math journal, consisting of little more than theorems and proofs. Basically, you can devote your entire career to what is effectively theoretical mathematics research while enjoying the luxurious salary and facilities of a business school professor of operations. Frankly, I'm shocked that more math professors and PhD students do not pursue this route (along with the aforementioned economics/finance/accounting routes).

sakky

To clarify my previous post and the others I have written on this thread, I am not necessarily endorsing the growing mathematicization of business academia or of the social sciences in general. In fact, I actually consider one of the greatest philosophical questions of the world to be whether the social sciences can be adequately analyzed mathematically, particularly in light of the fact that social phenomenon often changes without warning. {As an analogy, how useful would mathematical physics be if physical constants such as the speed of light not only changed repeatedly, but did so without warning?}

But be that as it may, whether we endorse or it not, the fact remains that social science academia has become ever-more-mathematical which therefore provides a no-brainer opportunity for the mathematically-trained to dominate the space. It's a perfect 'rent-seeking' opportunity: conduct highly theoretical mathematical research not dissimilar to what the mathematics faculty does, all while cashing a paycheck of double or triple the amount as a supposed "business" or "economics" professor (while actually just being a mathematics professor in disguise). Granted, you do have to 'teach' business or economics courses, but the (sad) reality is that most research universities place little weight upon your teaching abilities anyway. And if you publish prolifically - which may well be possible if you're mathematically skilled - you'll probably be able to negotiate a light or perhaps even a zero teaching load anyway. Either that, or jump to a business-school or economics department at another university that won't require that you teach.

sakky

Projectile, after checking around, I can now verify that the avenue I delineated above is also available to the physics students. It's not that hard for even a half-decent physics student to be admitted to an economics/finance/business/accounting PhD program , or perhaps a MFin/MFE. After the Phd, you should be able to obtain an academic placement at a business school or economics department, or after the MFin/MFE, you should be able to obtain a lucrative placement in the finance industry.