Other Should I Become a Mathematician?

[Bourbaki1123]

A) is not all mathematicians are turkeys. ii) says that all mathematicians solve puzzles, its not necessarily true from that, that all puzzle solvers are mathematicians. My conclusion would be that thanksgiving is for the birds.

In earnest, I would say that the conclusion would be that all of your recent guests are faithful crossword puzzle solvers and turkeys. Since it isn't really implicated that all crossword puzzle solvers are mathematicians, what happens is that your guests and the set of all mathematicians are subsets of the set of all crossword puzzle solvers, and these subsets can still have a null intersection.

 

[uman]

By ii), all mathematicians are faithful crossword puzzle solvers. By iii), all faithful crossword puzzle solvers Dr. Smith knows are his recent Thanksgiving guests. By i), all these are turkeys.

I guess the conclusion should read: All mathematicians that Dr. Smith knows are turkeys.

 

[Tickitata]

I believe these are the answers (fun puzzles by the way! I wish my intro to proofs class used this idea)

1. Some mathematicians may be turkeys
2. Some American missionaries ended up doing quite well
3. Dr. Smith spends all his money on comic books
4. Dr. Smith should bury his head in the sand

 

[mathwonk]

ok please forgive me if these answers do not make sense. After all, I made up the answers before I made up the questions.

\my personal answers are:
1) all the mathematicians I personally know are turkeys.

2) Some Americam missionaries who went to hawaii to do good, ended up doing quite well.

3) If I am intelligent I will spend money only on comic books.

4) Dr. Smith should bury his head in the sand at his wonderful beach.

 

[mathwonk]

the other famous quote i refer to involved dan quayle, and ran roughly as follows:

"I knew Jack kennedy,.. Jack kennedy was a friend of mine,.. and you senator are no Jack kennedy!"

 

[mathwonk]

I am encouraged to post some more of my challenging exam questions: (provided you can handle them.)

In “A few good men”, after a marine named Santiago was killed by two soldiers of the colonel’s command, Tom Cruise cross - examined the colonel (Jack Nicholson) as follows: “Colonel, you told us you ordered Santiago transferred off the base because he was in grave danger, and that your men always do exactly as you tell them.” “That’s right”. “I just have one question: If you told them Santiago wasn’t to be touched, and if your men always do exactly what you say, then why would Santiago be in danger?”

A) Clarify Cruise’s implication, by giving the contrapositive of the statement “If you told your men he was not to be touched, then Santiago was not in danger.”
...

Later, Cruise elicited from the Colonel a list of items he had packed for a weekend trip, plus several phone calls he had made in preparation. Then he observed, “Colonel, you were leaving for two days, and you packed two bags and made three phone calls. I’m just puzzled, since according to you, at 5am the next morning, Santiago was leaving for the rest of his life, but he hadn’t called anybody, and he hadn’t packed a thing.”

B) Clarify this implication by giving the contrapositive of the statement:
“If Santiago knew he was being transferred off the base first thing in the morning, he would have made some phone calls or at least packed some clothes.”
...

C) Based on the contrapositives of these statements, what would you say Tom Cruise is implying the colonel did (or did not do)?
...

D) Do you think Cruise’s arguments raise sufficient reasonable doubt, to counter the prosecution’s charge that the two soldiers acted without the colonel’s approval, or do you think he needed to go after a full confession by the colonel? Why or why not?

 

[uman]

A) If Santiago was in danger, then you did not tell your men he was not to be touched.
B) Since Santiago neither made phone calls nor packed some clothes, he could not have known he was being transferred off the base first thing in the morning.
C) Cruise is implying that the colonel did nothing to prevent his men from killing Santiago.
D) Cruise does raise sufficient reasonable doubt: Although he does not obtain a direct confession from the colonel, assuming his premises (those listed in A and B above, as well as the implicit assumption that the fact that the Colonel did nothing to stop Santiago's murder is equivalent to his tacit approval) are true, he uses valid logic to lead from what the colonel did say to the conclusion that the Colonel's men acted with his approval. Assuming his premises are true, this is just as good as a full confession. His premises may or may not be true, but they are certainly plausible enough to meet the standard of "reasonable doubt" required for an acquittal.

 

[mathwonk]

so why in the movie did tom cruise go for the confession, at the risk of blowing the whole case?

 

[uman]

I don't know. I haven't seen it. What did you think of my answers? I'm fairly sure they're correct, but obviously your eyes are better than mine.

 

[mathwonk]

i like your answers. that's why i ask another question. my own opinion is that the forced confession is a dramatic device, which as you argue was not necessary for the judicial purpose of the trial.

 

[uman]

Or maybe the lay jurors (or whoever is judging the case -- I'm not familiar with how military trials work), having not been trained in mathematics and logic, are at risk of not grasping an indirect and therefore more subtle argument.

The idea that it's just a device to move the plot along is plausible too. Again, I haven't seen the movie.

Did you really give these questions on a test?

 

[mathwonk]

what, aren't they standard?just kidding... I get the "are you serious?" question a lot.

 

[atyy]

A) is not all mathematicians are turkeys. ii) says that all mathematicians solve puzzles, its not necessarily true from that, that all puzzle solvers are mathematicians.

1) all the mathematicians I personally know are turkeys.

What about Bourbaki1123's analysis? I don't understand how (ii) is linked to (i) and (iii).

4) Dr. Smith should bury his head in the sand at his wonderful beach.

Beautiful!

 

[atyy]

mathwonk and uman, thanks! I have just learned what a contrapositive is

so why in the movie did tom cruise go for the confession, at the risk of blowing the whole case?

I saw the movie and loved it. Unfortunately, I don't remember details except what you are supplying, so I am only going on them. I think Cruise needed a confession, or at least something beyond reasoning from the axioms ("that your men always do exactly as you tell them"), since he had not established that the axioms were (experimentally?) true.

 

[uman]

A defense attorney doesn't have to prove innocence, just that there is a reasonably believable possibility if innocence. Thus, unlike in mathematics, an argument may rely on hypotheses that aren't certainly true but that may be.

These were more interesting than the UGa application essays. You should talk to the admissions people... ;-)

 

[mathwonk]

Thank you, I live for the feedback from intelligent readers.

 

[arshavin]

hi mathwonk or anyone,

Are gauss's books, in particular "General Investigations of Curved Surfaces" accessible to someone with only calculus and linear algebra knowledge?

 

[mathwonk]

I have not read this book, but I recall that Michael Spivak had his differential geometry class read this work as part of the course he taught that led to (especially the second volume of) his own book on differential geometry.

I believe he said his class enjoyed, or perhaps was impressed by, Gauss's book. So just plunge in and see for yourself. Or get hold of Spivak's second volume of his opus on diferential geometry.

 

[Bourbaki1123]

Mathwonk,

Have you read Hausdorff's Set Theory? If so, do you feel it gives one a good sense of set theory?

Also, what are the usual prerequisites for learning Category Theory? It seems that the actual material is not too far removed from just understanding homomorphisms, but it also speaks of metric spaces, isometries ect. Since the goal is to generalize mathematical structures, it seems that the need for broad knowledge is substantial.

 

[mathwonk]

i have not read all of hausdorff's set theory but have read some and was quite impressed by its high quality. i recommend reading as much as one finds appealing. in general i recommend reading also einstein, euclid, archimedes, riemann, and other brilliant geniuses.

as to category theory, it is a useful language for most people, and a specialty for a few people. i myself enjoyed reading the book abelian categories by peter freyd, and never read further. as to prerequisites, there are not too many, and abstract algebra should suffice.

 

[Wretchosoft]

Would you recommend taking graduate level classes in mathematics even if you don't plan to go to graduate school? I will be in a position to do so by my senior or possibly junior year, if everything goes well.

 

[mathwonk]

if you do not plan on grad school in math, but love math, then grad courses are still fine if you have completed all relevant undergrad courses.

in general, take what you enjoy. courses are just courses. there is no firewall between grad and undergrad.

 

[Bourbaki1123]

Mathwonk,

Would you say that it would be a good idea for an undergraduate to join the AMS and attend the meetings/talks?

 

[mathwonk]

I don't know what to advise on this. I have not myself been very active in the AMS. They do have a lot of meetings, and there can be some very good talks at them on up to the minute research work. These general talks might be useful for undergrads.

maybe someone knows more abut this than I do. have any undergrads here been to an AMS meeting and enjoyed it, or have any had other positive experiences with the AMS?

 

[qntty]

Mathwonk, after taking Calculus with one and many variables, which university class(es) will typically give you a taste of what Mathematicians do? (I am in High school so I can only take two or three classes per year and have one year to taken them)

the classes that I can take are
Linear Algebra, Diff Eq, More Calculus, Geometry, Probability, Proofs/Logic, possibly Abstract Algebra

 

[mathwonk]

that is hard to answer. college classes are typically presented in a way that does not give the feel for what mathematicians do. basically what they do is discover the stuff you are learning there.

so in a way, what we do is solve problems like the ones in your courses, only harder. or prove theorems like the ones in your courses. to get a feel for what we do, you could take any good course and try to prove the theorems yourself.

there are special seminars and courses some places that are meant to introduce students to research, but not everywhere.

as to specific courses, abstract algebra might give the best idea. or you could try reading a good creative geometry book like thurston's.

Three-Dimensional Geometry and Topology: Volume 1 (Hardcover)
by William P. Thurston (Author), Silvio Levy (Editor)

 

[abelian jeff]

I don't know what to advise on this. I have not myself been very active in the AMS. They do have a lot of meetings, and there can be some very good talks at them on up to the minute research work. These general talks might be useful for undergrads.

maybe someone knows more abut this than I do. have any undergrads here been to an AMS meeting and enjoyed it, or have any had other positive experiences with the AMS?

I went to MathFest two years ago, and it was amazing. I'll be attending the JMM next week, and if I remember, I'll let you know how it was, but my guess is that it, too, will be amazing. (I'm currently a senior undergraduate.)

 

[samspotting]

In my university the pure math degree does not require any applied math courses like ode's or pde's. Are understandings of these fields that you get in applied math courses good for a pure mathematician?

 

[Bourbaki1123]

I don't think so, unless you want to do analysis, then maybe a rigorous treatment of them, but ode's are pretty much strictly application. I have found that the only reason to take these courses if you are not going into applied math is that many grad programs want you to have taken them.

 

[mathwonk]

i would say that ode and pde are not at all applied courses, just courses with important applications. some of those applications are in pure math subjects like topology and differential and algebraic geometry.

wouldn't you say differential equations were the key to perelmans recent solution of the poincare conjecture?

i know the complex heat equation is just crucial in much beautiful work on moduli of abelian varieties.

 

[samspotting]

The problem is that I don't have enough time to take everything, with the pure math major I can either do a load of combinatorics classes (graph theory and enumeration) which look interesting, or a few courses in the applied math department:

Calculus 4
Vector integral calculus-line integrals, surface integrals and vector fields, Green's theorem, the Divergence theorem, and Stokes' theorem. Applications include conservation laws, fluid flow and electromagnetic fields. An introduction to Fourier analysis. Fourier series and the Fourier transform. Parseval's formula. Frequency analysis of signals. Discrete and continuous spectra. [Offered: F,W,S]

Introduction to Differential Equations
Physical systems which lead to differential equations (examples include mechanical vibrations, population dynamics, and mixing processes). Dimensional analysis and dimensionless variables. Solving linear differential equations: first- and second-order scalar equations and first -order vector equations. Laplace transform methods of solving differential equations. [Offered: F,W,S]

Ordinary Differential Equations 2
Second order linear differential equations with non-constant coefficients, Sturm comparison, oscillation and separation theorems, series solutions and special functions. Linear vector differential equations in Rn, an introduction to dynamical systems. Laplace transforms applied to linear vector differential equations, transfer functions, the convolution theorem. Perturbation methods for differential equations. Numerical methods for differential equations. Applications are discussed throughout. [Offered: F,S]

Partial Differential Equations 1
Second order linear partial differential equations - the diffusion equation, wave equation, and Laplace's equation. Methods of solution - separation of variables and eigenfunction expansions, the Fourier transform. Physical interpretation of solutions in terms of diffusion, waves and steady states. First order non-linear partial differential equations and the method of characteristics. Applications are emphasized throughout. [Offered: W,S]

Partial Differential Equations 2
A thorough discussion of the class of second-order linear partial differential equations with constant coefficients, in two independent variables. Laplace's equation, the wave equation and the heat equation in higher dimensions. Theoretical/qualitative aspects: well-posed problems, maximum principles for elliptic and parabolic equations, continuous dependence results, uniqueness results (including consideration of unbounded domains), domain of dependence for hyperbolic equations. Solution procedures: elliptic equations -- Green functions, conformal mapping; hyperbolic equations -- generalized d'Alembert solution, spherical means, method of descent; transform methods -- Fourier, multiple Fourier, Laplace, Hankel (for all three types of partial differential equations); Duhamel's method for inhomogeneous hyperbolic and parabolic equations.

 

[mathwonk]

calc 4 maybe?

 

[samspotting]

I'll talk to the profs about it, I don't really know much about differential equations except the part in calculus 2 where you solve these really simple first order linear equations that you had to build from confusing and complex word problems.

Differential equations deal with change, I have heard from one professor that pure mathematics does not care so much about solutions to differential equations as to whether solutions exist.

 

[mathwonk]

here is a statement: a compact manifold on which there exists a smooth function with exactly two critical points, which may be assumed non degenerate, is a sphere.

how would you go about proving this? consider the flow given by the gradient of the function, and use the solutions structure theorem for odes.

 

[samspotting]

Ill check it out once I take calculus 3 (this semester). Thanks.

Oh btw, I'm entering a version of calculus 3 where multivariable calculus is taught from a more rigorous perspective. I never took the more rigorous version of the single variable calculus, is there a quick primer somewhere?

I'm only done up to sequences in rudin's book, I feel like I am entering this course really unprepared.

 

[altcmdesc]

samspotting, I'm currently taking a class similar to that. If yours will be any thing like mine, learn some linear algebra while you're at it (be sure you're comfortable working with matrices and linear mappings, even at the basic levels).

Also, if you haven't covered basic point-set topology (interior, boundary, limit points, open/closed balls) or vector geometry cover that too.

 

[Linear Space]

Mathwonk,

Would you say that Spivak's Calculus fully prepares you for his Calculus on Manifolds?...or What else do you need?

Also, after both of these books I plan on tackling Spivak's Differential Geometry series. I noticed that you said in the prereq to diff geometry thread that any amount of algebra and topology will only broaden your knowledge. Where within this sequence of books would you recommend picking this stuff up?

Thank you.

 

[arshavin]

What is the typical session length(in terms of weeks) at a standard university in the US or UK?


In Australia we have 2 semesters (12 weeks each) and we have a summer break of almost 4 months (during which no maths courses are offered).

This just seems completely ridiculous to me.

Is it the same everywhere else?

 

[snits]

I imagine most universities in the US offer math courses during the summer session.

 

[mathwonk]

yes spivak's pwn calculus is adequate preparationm for his calc on manifolds.

then volume one of his diff geom is chock full of great topology and basic manifolds.

a little heavy going in the basic theory of manifolds.

the problems are also wonderful, and the extra chapters on lie groups and de rham cohomology are terrific.

just volume one is very valuable information on manifolds and cohomology.

then volume 2 is the world's best historically oriented but modern version of an analysis of the most important gadget in diff geom, the curvature tensor.

those are the only two volumes i own. i am tempted by volume 5 i think it is, on the chern version of the gauss bonnet theorem?

but i myself have little time for learning now. maybe soon.

 

[uman]

Mathwonk or others,

Long ago I was reading both an Algebra book (Artin) and a Mathematical Analysis one (Apostol). I had to take the Algebra book back to the library, so I've since been doing only Apostol's book. I'd like to get back into Algebra to give me a change of pace, but I don't have access to Artin's book; my local library now only has Dummit and Foote (long story). Is it worth buying the Artin book (which I really liked) or will D&F suffice? Is it accessible to someone who doesn't know much about Algebra?

 

[desti]

Dummit and Foote is very easy to follow. It has loads of examples and the exercises vary from trivial to moderately hard, so it's easy to find toy problems to test understanding of definitions. I think its biggest downside is that it doesn't have any harder exercises. There is some stuff in Artin that you won't find in Dummit and Foote though. There's a very brief intro to algebraic number theory in Artin and then there is the section on wallpaper symmetries. It's hard to say which is better though. Artin also focuses a lot more on connections with linear algebra by using matrices as examples for almost anything.

If you consider your "mathematical maturity" as ok (i.e. you're comfortable writing proofs), I would recommend reading Herstein's, Topics in Algebra, for a quick intro to group theory and then go straight into e.g. Lang's Algebra and use Dummit and Foote mainly for examples and easy exercises if you get stuck. This gets you quicker to advanced material. If you read Lang, I would recommend that you teach yourself LaTeX and write your own account of any major theorems after you've read about them in Lang. It's slow and takes time, but when you go back to the stuff later, you'll notice that having your own notes makes you recall the stuff a lot faster.

 

[abelian jeff]

I went to MathFest two years ago, and it was amazing. I'll be attending the JMM next week, and if I remember, I'll let you know how it was, but my guess is that it, too, will be amazing. (I'm currently a senior undergraduate.)

So, the Joint Math Meetings were awesome. I highly advise any undergraduates who can get funding (or pay their own way) to attend MathFest and JMM. Both are great experiences.

 

[rodigee]

I (undergrad) went to the JMM this year as well to present a poster. I'll echo what Jeff said. Overall, it was quite fun. No matter your interests, you should be able to find a session of talks about something intriguing. However, understanding the talks may be another matter all together.

I really enjoyed seeing D.C. as well.

 

[uman]

desti: thanks. I'm not sure how "mathematically mature" you would consider me: I got through five chapters or so of Artin without too much trouble before I give it back. I guess I'll look into Herstein if you think D&F is too trivial... In any case having multiple resources can't hurt.

Anyone else have an opinion on this?

 

[jacksonpeeble]

Hell no. Maths and economics majors know jack about maths either pure or applied. Economists struggle to add up, never mind do maths properly (including applied maths).

I find this entertaining (because it's absolutely true); my economics teacher seems clueless about math in general, much less the complicated things.

Besides, economists mess up all the time. Mathematicians don't seem particularly apt or happy to make errors.

 

[PhysicalAnomaly]

Does that mean that if I study some stochastic and financial maths units, they'll hire me to do their sums? =P

 

[matqkks]

Mathematics is such a powerful tool that we all need some. Here are a few examples.
Number theory had no serious application for 2 500 years but since the advent of the digital computer number theory has had a major impact on everyday life - online transactions would be impossible without number theory.
Most physical systems involve rate of change so we need calculus to describe these physical processes.
Partial differential equations in Fluid Dynamics explains how water passes by submarines, air flows past aircraft and air flows over formula 1 racing cars.
The entire field of physics - heat, light, sound, fluid flow, gravitation, electricity and magnetism - can all be described by differential equations.
More advanced technologies such as radio, tv, jet commercial aircraft rely on the mathematics of differential equations.
There are millions of other examples.

 

[Orphen89]

I have a question regarding a math major:

Basically, for my degree I'd like to double major in Applied and Computational Mathematics. However I am unsure what I should add to my degree from here - at the moment, I have a few extra units to add to it, but I don't know whether I should add some Computer Science units to it so that I can get a Double Major in Applied and Computational Maths and have a Computer Science Minor, or if I should add more Applied Mathematics units so that I can have a better mathematics major.

Initially, I wanted to have a CS minor, however there are a few Applied Mathematics unit I will be missing out on if I do get one e.g. Fluid Dynamics, Mathematics in Finance (in case I want to go into banking/economics later), and Real Analysis (which I heard is an important unit in any math major).

What would be better, and what would employers prefer in a degree? A double major in maths and a minor in CS, or only a double major in maths but with extra units? (I don't know if this is important, but I do intend on doing a graduate degree once I've finished my bachelors)

 

[timur]

People will give better answers than mine, but it should depend on what you want to do with your degree after graduation. For instance if you want to do actual computation and so a lot of programming, a CS minor would be good. But if you want to do research in applied math, like designing or improving algorithms, proving theorems about numerical algorithms etc, I would suggest the more math the better.

I have my own question from people here: Does it make sense for somebody who is trying to get into grad school to get a recommendation letter from his brother who is a postdoc?