Should I Become a Mathematician?

[mathwonk]

help? why is a cube not called a hexahedron?

 

[morphism]

mathwonk, the text used for the classical geometry course at my school is Survey of Classical and Modern Geometries by Arthur Baragar. I can't comment on its quality because I have never taken the course or read the book, but I thought I might as well suggest it. You can find the table of contents in the following link: http://isbn.nu/toc/9780130143181.

 

[mathwonk]

looks interesting, but ouch! $114.

 

[Werg22]

A sheet of paper is big enough for all of high school geometry to be printed on.

 

[honestrosewater]

Hey, everybody, I just decided that I want to be a mathematician! Yay! Actually, I just realized that I'm not happy doing anything else. So, yeah, I'm going to go read this thread now.

Has anyone yet touched on mathematical logic or areas relating to language, e.g., model theory, proof theory?

 

[mathwonk]

yes the book modern geometries by james R smart (is that a joke?) has it all on both sides of one page in the appendix.

but the "one page every 8 weeks" density seems high for presentation in the first course.

 

[quasar987]

Hey, everybody, I just decided that I want to be a mathematician! Yay! Actually, I just realized that I'm not happy doing anything else. So, yeah, I'm going to go read this thread now.

Has anyone yet touched on mathematical logic or areas relating to language, e.g., model theory, proof theory?

GOnna be taking 'Logic' next semester. Why do you ask?

 

[mathwonk]

Logic reminds me again of eucldean geometry, where the logic is complicated by our over familiarity with the subject matter.

and the irony is that as a lifelong profesional algebraic geometer i know hardly anything about plane euclidean geometry.

What I have learned is roughly this (about the logic). Consider a set of statements ("axioms").

they are "consistent" iff one cannot deduce a statement of form P and notP from them, iff there exists a "model" universe in whiuch all the statements are true of the model.

Even this is probably wrong, but I am a beginner in logic.

Questions one asks about axiom sets include:

are they consistent?i.e. does at least one model exist?

does more than one model exist? i.e. do they fully characterize some one model geometry?

e.g. if you look at the postulates given in the list of postulates for geometry in Harold Jacobs book 3rd edition, you will see they all hold not only in the euclidean plane, but also in euclidean 3 space.

hence it is imposible to prove from them the theorem of pasch, that a line which meets one side of a triangle away from a vertex, must also meet another side. but this property is cruciaL to all plane geometry of triangles. or that two circles which meet at a point which is not collinear with their centers must meet a second time.

i also recall SAS congruence being a theorem from high school, but it is properly an axiom, since there exist geometry models in which all other axioms of protractor geometry hold, including pasch, but in which SAS is false.

another question is whether axioms are independent, i.e. given one of them, can it be proved from assuming only the others? if not then apparently there is a model in which all the others hold but this one does not, and vice versa.

this makes it really cool and fun to look at various different models, and see what is true of each one.

e.g. if you assume all euclidean postulates except the euclidean parallel postulate, then it seems there can by triangles whose angles do not add up to 180 degrees.

and although mathematicians searched unfruitfully for thousands for evid3nce as to whether the parallel postulate was indeed independent of the others (it is), the almost trivial example of "table top geometry" i.e. lines on a table top that reach from one edge to another, almost give an example.

I.e. they immediately satisfy ll other postulates except the ability to lay off infinitely many copies of a line segment on any line, but this can be rescued if you just realize that you can change the meaning of length as you get closer to the edge, so that you never fall off.

I.e. think of walking along a line, and that you walk slower if it gets colder. Then just drop the temperature near the edge of the table. then you can take as many steps as you want along a line without going off the table if you keep walking slower and slower, i.e. if it gets colder and colder.

 

[mathwonk]

honest rosewater, please read the advice on fields medalist terry tao's webpage. that is much better than anything I wrote here.

 

[mathwonk]

this baragar's preface: so far so good:

From the Inside Flap
Preface for the Instructor and Reader
I never intended to write a textbook and certainly not one in geometry. It was not until I taught a course to future high school teachers that I discovered that I have a view of the subject which is not very well represented by the current textbooks. The dominant trend in American college geometry courses is to use geometry as a medium to teach the logic of axiomatic systems. Though geometry lends itself very well to such an endeavor, I feel that treating it that way takes a lot of excitement out of the subject. In this text, I try to capture the joy that I have for the topic. Geometry is a fun and exciting subject that should be studied for its own sake.

 

[honestrosewater]

yes the book modern geometries by james R smart (is that a joke?) has it all on both sides of one page in the appendix.

Haha. Oh, I see.

GOnna be taking 'Logic' next semester. Why do you ask?

Because that's what I'm most interested in at the moment. I meant to ask whether it has been talked about yet (as some other subjects have) in this thread.

Logic reminds me again of eucldean geometry, where the logic is complicated by our over familiarity with the subject matter.

Maybe it helps to step back and consider other logics (as you might other geometries).

What I have learned is roughly this (about the logic). Consider a set of statements ("axioms").

they are "consistent" iff one cannot deduce a statement of form P and notP from them, iff there exists a "model" universe in whiuch all the statements are true of the model.

Even this is probably wrong, but I am a beginner in logic.

Right, that is a theorem of model theory: a theory's consistency and its having a model are equivalent. Although, come to think of it, that might be due to completeness (or just a restatement of it), so I should say it's specifically a theorem of first-order model theory (which is usually what is meant, I think).

Questions one asks about axiom sets include:

are they consistent?i.e. does at least one model exist?

does more than one model exist? i.e. do they fully characterize some one model geometry?

Yes, I think consistency, completeness (syntactic and semantic variations), and independence (of the axioms) are three big, basic properties that you want to know about a theory. Whether it is categorical (i.e., has exactly one model up to isomorphism) might be another.

e.g. if you look at the postulates given in the list of postulates for geometry in Harold Jacobs book 3rd edition, you will see they all hold not only in the euclidean plane, but also in euclidean 3 space.
[snip]
I.e. think of walking along a line, and that you walk slower if it gets colder. Then just drop the temperature near the edge of the table. then you can take as mnay steps as you want along a line without going off the table if you keep walking slower and slower, i.e. if it gets colder and colder.

Ah, you got independence. Thanks for the ideas. I guess I am really hungry for some (useful) problems to solve, or I'm ready to start accumulating solutions. I imagine you've heard of George Carr's http://books.google.com/books?id=FTgAAAAAQAAJ". This is the book of theorems, definitions, and such that Ramanujan got (and kept) his hands on. I was looking at it the other day, and I find it quite handy, as just a source of lots of problems to solve (theorems to prove), laid out in somewhat logical progressions. Does anyone know of another, perhaps more recent, book like this? I'm not looking for a full treatment of any subject or a "how to solve problems" book. I'd like just a list of theorems with whatever additional notes are necessary.

I suppose I already have my guy for model theory, if anyone else is looking: http://www.maths.qmul.ac.uk/~wilfrid/" . He's super. He's good for logic too.

 

[honestrosewater]

honest rosewater, please read the advice on fields medalist terry tao's webpage. that is much better than anything I wrote here.

Yeah, I've seen it. I assume you mean his http://www.math.ucla.edu/~tao/advice.html" (a very memorable phrase). I guess I didn't mention that I've loved math and been around it for a while. I'm just now deciding to give up and dive in.

 

[mathwonk]

i recommend the book by gelbaum and olmsted, counter examples in analysis. it was really fascinating to me as a freshman to see all the exotic things that are true about the reals.

Counterexamples in Analysis
Bernard R. Gelbaum|John M.H. Olmsted
Bookseller: WebBookStore
(Pittsburgh, PA, U.S.A.) Price: US$ 7.17
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US$ 3.95
[Rates & Speeds]
Book Description: Dover Publications, 2003. Paperback. Book Condition: Brand New. Brand new as book, not a remainder, no marks. As published by Dover Publications. Paperback edition. Book Size: Length: 8.27 inches, Width 5.43 Height inches 0.55 Inches. Book weight is 0.57 pounds. This book will require no additional postage. Orders processed on AbeBooks Monday - Friday and ships 6 days a week. Synopsis: These counterexamples deal mostly with the part of analysis known as "real variables." The 1st half of the book discusses the real number system, functions and limits, differentiation, Riemann integration, sequences, infinite series, more. The 2nd half examines functions of 2 variables, plane sets, area, metric and topological spaces, and function spaces. 1962 edition. Includes 12 figures. Barcode/UPC of the book/13 digit ISBN # 9780486428758. 10 digit ISBN # 0486428753. Brand New. Bookseller Inventory # 9780486428758_N

 

[Werg22]

I have a simple question: what kind of people are mathematicians?

 

[J77]

I have a simple question: what kind of people are mathematicians?

You have to have a third nipple, but don't tell anyone...

 

[pivoxa15]

I have a simple question: what kind of people are mathematicians?

From my experience and hearing from what they say, all types. But one thing that unites them is that they are perfectionists. Are there any mathematicians who are not perfectionists?

However perfectionists interested in analytical objects tend to produce personalities that are introverted and so not too socially oriented. However there are exceptions and some are more extroverted. Perfectionists interested in other things might be very different.

 

[fopc]

Thank you for the suggestions. I have already ruled out Hartshorne, Euclid and Beyond; Millman and Parker; Moise, Elementary Geometry from an Advanced standpoint; Modern Geometries by James Smart;... all as excellent but too difficult.

I see you gave a review of the second edition of Millman's book at Amazon.
I'm trying to find out more about the first edition.
Can you make a comaprison between the two editions?

 

[mathwonk]

i have not seen but one edition of millman parker, but all they say that added to the second edition was a collection of "expository exercises" to implement the program "writing across the curriculum". so it makes almost no difference to the presentation and the way I teach the course, if that's all they did.

by the way i am reconsidering hartshorne, and some other books recommended here like Baragar and Bass et al, if I can find them. Thanks very much for the sugestions!, and looking for review copies in libraries, since the publishers make it so hard to get review copies.

after all one can teach out of anything if you handle it well in class. actually most sudents don't read the book anyway in calculus at least, so better to give them a good book and hope they read it than a bad book they claim they can read.

 

[mathwonk]

as to what kind of people mathematicians are, it seems fair to call them perfectionists, but others exist as well.

when i grade papers, i always write a lot of detailed comments on them, and yet 99% of the students never come to look at them, so all the hundreds? of hours spent doing that over the last 30 years are wasted. but i do it anyway, for the one in a hundred who might want to see them.

we don't do things to a standard that will pass muster from others, but to a standard of our own choosing. no one may ever see or read what we do, but we do it to our own standard of perfection anyway. when i write a paper it usually goes through dozens of iterates, some just changing a few words, some just removing superfluous spaces between words.

not everyone is like this. i think this perfectionism is an enemy in many cases to creativity, and some of the most creative people just try to forge ahead, not nit picking their own work at every stage. indeed this is essential. that's why math is so hard, it ideally requires both sides of the brain, real hard creative work, followed by very precise critical review.

just the subject of plane geometry we are talking about here has gone through what, 2,000 years? of critical review by mathematicians and still geting fresh looks, like Hartshorne's book from 2005. they want to get it right.

now I am finally beginning to think my students at least should not be held to this standard and I allow a lot of leeway.

 

[mathwonk]

mathematicians come in all stripes. there is no personality litmus test for one. If you enjoy doing math, and preferably have some success at it, or can learn to, you can be one too.

to see the variety just look at a photo of g. perelman and compare to a photo of eli cartan, and compare their biographies.

http://www.englishrussia.com/?p=250

http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cartan.html

 

[mathwonk]

werg, the best high school geometry teacher i know, Steve Sigur, of the Paideia school in atlanta, taught me the basic principle of geometry prep for the SAT's "angles that look equal are equal".

 

[Werg22]

Thanks for the advice, but I took my SAT's a long time ago already! They're not much to prepare for nowadays anyway; the problems are more about intuition than analysis. This said, even if what you said is true, I'd still be inclined to verify the value of the angles at every problem; you never know!

 

[mathwonk]

i think, rather than advice, it was agreement with post 600.

 

[Werg22]

Oh I see. I have a poor short memory, I'm afraid.

 

[mathwonk]

well i had trouble remembering myself, but that seemed plausible. its like a password you make up. it seems so clever and memorable at the time and later, huhh?

 

[Werg22]

I only keep one password for everything. Should someone discover it, I'd be in deep trouble.

 

[honestrosewater]

I only keep one password for everything. Should someone discover it, I'd be in deep trouble.

I made up a simple algorithm for making up passwords that are acronyms formed from a sentence based on something persistent about the specific site (or whatever). Sentences are easier to remember, I find, and you don't actually have to remember it anyway since you can just rerun the program that generated it. Although, I suppose you could always forget the program or how to execute it, since it's just some instructions in your head. But if you write it in your native language, you'll have bigger problems if you ever forget how to execute it.

Also, with acronyms, you avoid the common spelling patterns of words (based on a language's http://en.wikipedia.org/wiki/Phonotactics" ), which can rule out a lot of combinations and make others more or less likely.

I'm not actually worried about anyone guessing my passwords, by the bye. I just like language and solving problems.

 

[mathwonk]

i also have one universal password,

qerii23849504434528888nmartw@!&@@

but i still keep forgetting it.

 

[Werg22]

I find it easier to remember a sequence of numbers or a word by associating them to either an idea or an image. For example, I remember one of my friend's phone # by remembering "the inverse of my regional code, two similar numbers (69) and the day of st-valentine (14)". If I remember these steps, I remember the number. Same thing goes with formulas: I like to remember formulas conceptually rather than just as expressions.

 

[morphism]

I have two base passwords, and I use different arrangements of each. Like uppercasing, adding numbers, switching o's with 0's and i's with 1's, etc. It usually takes me a couple of tries to finally figure out what the password should be!

 

[fopc]

i have not seen but one edition of millman parker, but all they say the added to the second edition was a collection of "expository exercises" to implement the program "writing across the curriculum". so it makes almost no difference to the presentation and the way I teach the course, if that's all they did.

by the way i am reconsidering hartshorne, and some other books recommended here like Baragar and Bass et al, if I can find them Thanks very much for the sugestions!, and looking for review copies in libraries, since the publishers make it so hard to get review copies.

after all one can teach out of anything if you handle it well in class. actually most sudents don't read the book anyway in calculus at least, so better to give them a good book and hope they read it than a bad book they claim they can read.

Thanks. First edition is about $50 cheaper.

By the way, I wouldn't give up on Smart.
Yes, the axioms from high school geometry (32 of them) are relegated to an appendix along with other axiom sets (Hilbert, Birkhoff), but you could supplement this material with you own notes. Also, how much time can you devote
to it in a quarter (or semester) anyway?
If I understand your comments, your class will be made
up primarily of college students intending to be *high school math teachers*. I'd say if that's their goal, then it should be expected that they'll come to class with
prerequisites satisfied (which at the very least should include good understanding of the
axioms from *high school geometry*).

 

[mathwonk]

well that is true, they should, but in fact they don't. this is the problem facing the teacher today.

by the way i found a copy of the first edition of millman parker in a library yesterday and compared the two editions for you. the first edition is only 15 pages shorter than the second, and has the same chapter headings, and every single chapter section has the same title.

oh yes and the quality of the paper was superior in the first edition and the print was larger and more readable. so the first edition seems to be a better book, as is usual.

 

[fopc]

well that is true, they should, but in fact they don't. this is the probl;em facing the teacher today.

by the way i found a copy of the first edition of millman parker in a library yesterday and compared the two editions for you. the first edition is only 15 pages shorter than the second, and has the same chapter headings, and every single chapter section has the same title.

oh yes and the quality of the paper was superior in the first edition and the print was larger and more readable. so the first edition seems to be a better book, as is usual.

Ordered a copy for $29.25. Thanks for information.
Price not too bad compared to the $79.50 price tag on the current ed.

 

[mathwonk]

i also like Hartshorne's recent book, geometry: euclid and beyond.

edit: I now recommend everyone to learn plane geometry from Euclid with Hartshorne as a guide.

 

[quasar987]

Did you ever try your luck at the Hodge conjecture mw?