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I have a request of the mathematics-minded.


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Showing 1-25 of 156 posts in this discussion
Initial post: Feb 19, 2013 5:29:13 PM PST
jpl says:
Please tell me the order in which you studied mathematics from high school through college. I'd really appreciate it, because this was a precious time that I wasted in my life.

All I want is a list from the beginning to where you're at now.

I want to learn as much math as I can before I die. Thanks ahead of time for helping me out.

jpl

In reply to an earlier post on Feb 19, 2013 5:38:21 PM PST
Ninth grade Algebra. Tenth grade Algebraic Geometry. Eleventh Grade Trignonometry. Twelfth Grade Calculus I (calculus of one variable). College only one math course, retook Calculus I.

Starting years later, when I began to learn it on my own: Calculus I (calculus of one variable), Calculus II (multivariable calculus), Calculus III (vector calculus), Linear Algebra, Ordinary Differential Equations, Partial Differential Equations, Complex Analysis, Group Theory, Number Theory, Galois Theory, Differential Geometry, Calculus of Variations, Topology, Algebraic Topology (very preliminary on the last two). For most of these I bought introductory texts on amazon.

Math is a truly enormous subject. If you want a nice overview before plunging into study of one particular area I would recommend Mathematics 1001: Absolutely Everything That Matters About Mathematics in 1001 Bite-Sized Explanations by Dr. Richard Elwes. It's inexpensive, very clearly written, and absolutely fascinating.

In reply to an earlier post on Feb 19, 2013 5:50:35 PM PST
Last edited by the author on Feb 19, 2013 5:52:48 PM PST
jpl says:
Thanks a million, arpard! There seems to be much confusion on the order in which different maths should be taught.

I also appreciate your telling me what you studied after your schooling.

I've done much hunting on the internet and have found that mathematics is virtually an endless subject, but it sure is nice of you to tell me how to get started with what you learned in school.

Thanks also for the recommendation of the book you cite by Dr. Elwes. I'll check it out immediately.

Many thanks,
jpl

In reply to an earlier post on Feb 19, 2013 5:52:24 PM PST
You are welcome. I really love math. Unfortunately I tend to forget what I've studied fairly quickly. Sometimes it seems like a walk on the beach: you think you're moving toward a goal, but when you look back all your footsteps have been erased by the waves.

Posted on Feb 19, 2013 5:55:35 PM PST
Last edited by the author on Feb 19, 2013 6:16:22 PM PST
jpl says:
I imagine you have as many questions as you do answers because of your knowedge. You probably feel crazy with questions sometimes.

You're one of the most polite geniuses I've ever known. I don't mean that as a slight. I really mean it.

Posted on Feb 19, 2013 9:49:05 PM PST
I majored in math in the years 1966-1970, with classes similar to Arpard's. In addition to his recommendation of Elwes, which I want to read, I can mention James Newman's 4-volume "The World of Mathematics" and any/all of Martin Gardner's "Mathematical Recreations" books (from his Scientific American columns in the 1950s, 60s, and 70s). I haven't yet seen any of The Teaching Company's math courses, but my wife and I have appreciated their series in science and philosophy. Whether on your own or in formal class, best of luck and enjoyment in your studies.

Posted on Feb 19, 2013 11:41:38 PM PST
jpl says:
arpard fazakas says . . ..If you want a nice overview before plunging into study of one particular area I would recommend Mathematics 1001: Absolutely Everything That Matters About Mathematics in 1001 Bite-Sized Explanations by Dr. Richard Elwes. It's inexpensive, very clearly written, and absolutely fascinating.

jpl: Thanks, arpard. I can walk to where that very book is in five minutes. I've browsed through it a few times. Because of your recommendation, I may buy it.

In reply to an earlier post on Feb 19, 2013 11:59:23 PM PST
jpl says:
Eugene R. Walker says: I majored in math in the years 1966-1970, with classes similar to Arpard's. In addition to his recommendation of Elwes, which I want to read, I can mention James Newman's 4-volume "The World of Mathematics" and any/all of Martin Gardner's "Mathematical Recreations" books (from his Scientific American columns in the 1950s, 60s, and 70s). I haven't yet seen any of The Teaching Company's math courses, but my wife and I have appreciated their series in science and philosophy. Whether on your own or in formal class, best of luck and enjoyment in your studies.

jpl: Thanks, Eugene. I'll definitely check out your suggestions.

Posted on Feb 20, 2013 4:54:18 AM PST
Brian Curtis says:
The sequence at my high school was

Algebra
Geometry
Trig
Calculus

In reply to an earlier post on Feb 20, 2013 7:40:18 AM PST
It seems to me that:
A) It is not possible to learn all the math that might be relevant.
B) After the basic H.S. curriculum, it is necessary to "choose your battles", so to speak, and focus on the stuff which is of particular interest or use to you.
C) This is not to say that many of af's categories don't have very general applicability, but most of these fields are broad and deep enough that taking a course in them is unlikely to be anything like exhaustive.

Thus, to actually use them, it is probable that you need to have a problem to solve, a "project". For instance, when Einstein started to get into the problem of general relativity, he saw that he was going to have to learn about Riemann's analytic geometry in much more detail. His "project" propelled him into assimilating this material. I suspect that much of our learning only sticks when we take this approach, because only then does it pass from the abstract to the concrete.

In reply to an earlier post on Feb 20, 2013 9:17:00 AM PST
Last edited by the author on Feb 20, 2013 9:46:27 AM PST
Jack Shandy says:
In college:
Calculus/
Vector Calculus (Marsden)/
Linear Algebra(Fraleigh)/
Differential Equations(Boyce)

Privately:
Basic concepts (Zakon)/
Analysis(Baby Rudin)/
Linear Algebra(Halmos)/
Tensors (Lovelock)/
Set theory (Hrbacek)/
Combinatorics(Chen)/
Groups and Rings(Dummit)/
Differential Geometry(Do Carmo)/
Real+Complex Analysis(Big Rudin)/

Currently doing Mathematical Physics(Hassani) , which comprises Linear Algebra, Complex Analysis, Operator theory, Differential Equations, Green's Functions, Group (Representation) Theory, Differential Geometry, Lie Groups, and Calculus of Variations in a really coherent and rigorous manner.

Next on the list:
Calculus of Variations(Gelfand)/
Probability(Feller)/
Topology(Munkres)/
Mathematical Logic(Shoenfield)/
Differential Forms(Darling)/
Manifolds, Tensor Analysis(Marsden)/
Semi-Riemannian Geometry(O'neil)/
Functional Analysis(Rudin)/
Group Representations(Barut)/
Geometry of Minkowski Spacetime(Naber)/
Probability(Feller2)/
Set Theory(Jech's Milennium ed.)

Also quite some physics texts in between these. How long will it take me? Two lifetimes?

Edit: reading the list, it may seem that math is a collection of an enormous amount of disconnected subjects, but surprisingly, the deeper you delve into particular subjects, the more other disciplines get interwoven with it. Linear algebra-operator theory-functional analysis-real/complex analysis-topology-differential geometry-group theory-differential equations-set theory-combinatorics-logic are ALL inter related, and boundaries between the subjects fade away.

Posted on Feb 20, 2013 11:05:06 AM PST
Mike W says:
GCSE mathematics; AO-level mathematics; A-level pure mathematics and A-Level statistics; an undergraduate degree in mathematics with emphasis in stats/probability; a masters degree in mathematics with emphasis in pure; PhD in mathematics with a dissertation in convex analysis.
If you wish to learn mathematics in a purely logical format, you could try the works of Bourbaki but I would not recommend it; building from the axiomatic foundations of mathematics tends not to work very well from an educational standpoint. (It is easier to learn the multiplication table than it is to understand the axiomatic construction of the natural numbers; even though logically you should construct the natural numbers before you do any operations with them.)

What I recommend to anyone who is interested in learning mathematics starting at the American college level is as follows:
Calculus (review College Algebra and Trig as necessary, but do not study College Algebra for its own sake because it's an incoherent mishmash of disconnected topics).
Vector Calculus, Elementary Linear Algebra and ODEs [these should be studied concurrently]
Foundations [an intro to proof writing, usually with an emphasis on the basics of modern algebra]

After that, a decision has to be made about what emphasis you would like to go in; pure, applied, stats or math ed are the most common threads. With regard to pure, I'd recommend Real Analysis, Complex Analysis, Modern Algebra, Linear Algebra, Topology, non-Euclidian Geometry, Set Theory and PDEs. The order probably doesn't matter much. My experience has been that the connections tend to happen most when you try to teach the subject. In any event, there is a certain expectation that a mathematics professor should be capable of teaching the entire undergraduate curriculum in mathematics.

In reply to an earlier post on Feb 20, 2013 12:02:29 PM PST
jpl says:
I appreciate that, Brian. I see this particular order more than others.

In reply to an earlier post on Feb 20, 2013 12:05:50 PM PST
jpl says:
Thanks for your thoughts, Randall. What you say makes sense. It seems to me that math is endless and that it would be up to the individual to move in the direction that he or she finds most fascinating.

In reply to an earlier post on Feb 20, 2013 12:09:09 PM PST
jpl says:
Hi, J. Shandy. Thanks for the info. I didn't know whether these different disciplines complemented one another. I can tell that you are math nut. I imagine you go to sleep thinking of whatever you've been studying.

I appreciate the information.

In reply to an earlier post on Feb 20, 2013 12:17:06 PM PST
jpl says:
Mike W., I appreciate your taking the time to explicitly explain the options you present. I feel better informed now about how to approach mathematics.

In reply to an earlier post on Feb 20, 2013 12:31:04 PM PST
Jack Shandy says:
Sometimes I even wake up in the night, suddenly grasping that which eluded me by daylight. Not too often though. I mostly just bang my head against the wall to reshuffle my neural pathways, until understanding follows.

In reply to an earlier post on Feb 20, 2013 12:36:12 PM PST
Mike W says:
The same here- except then when I do write down what I figured out in the middle of the night, I realize that my nighttime reasoning was seriously flawed :)

In reply to an earlier post on Feb 20, 2013 1:00:17 PM PST
It seems to me that these categories are more a byproduct of the practical necessity that curricula need to have course titles attached, so that they may be packaged for "sale". The math itself comes to us with a much more amorphous set of attachments, more like a seething mass of squids, with suckers and tentacles reaching out and grabbing onto everything that gets anywhere nearby.

In reply to an earlier post on Feb 20, 2013 3:01:44 PM PST
Last edited by the author on Feb 20, 2013 6:51:03 PM PST
Physicist ... 40 years in leading edge kind of technology here ... same academic math curriculum as previous commenters (except slide rules, before calculators). An affinity for mathematics is something brains are wired or not wired for. Yours is an ambitious objective for the older unschooled brain!

In practice ... measurement, statistics & stochastic processes (DOE's, Monte Carlo's, simulations, etc) comprise most of the applied mathematics in industry. At least 95% of the academic stuff will never cross your path again. If I need a mathematician, I hire one on contract ... if I've paid 160 hours in 40 years, I'd be surprised. If you have a lick of advanced training, the grunt work of applied mathematics is automated in software like Mathematica, Maple, MATLAB, LabVIEW, et. al.

However ... The Philosophy of Mathematics is either not well taught or I had a hangover on the day it was. The Philosophy of Mathematics has been a hobby for some years. It's a curiosity in quirky or supposed unrelated relationships that appear by coincidence (or not) and these fascinate. The Philosophy of Mathematics should be introduced in high schools and continue to be advanced as it parallels the curriculum (these are thought experiments that deserve more than footnotes). The rote nature of academic mathematics seems to be a failure in the general poulation. The `why?' of number and set theory is a more interesting topic at this stage. I have enough ex-engineering science/math teacher friends to believe that it's not introduced in engineering or secondary educators. The Philosophy of Mathematics provides an expanded plain of understanding of the quantitative nature of everything physical and parametric.

Before jumping into Algebra 1, I'd suggest a couple appetite wetting reads that are not all that technical:

Tractatus Logico-Philosophicus

The Irrationals: A Story of the Numbers You Can't Count On

In Pursuit of the Unknown: 17 Equations That Changed the World

Gamma: Exploring Euler's Constant (Princeton Science Library)

The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles Of Our Time

... I've offered the P=NP? problem significance several times on this forum. It is not a computer science problem, rather a basic hurdle if one is to imagine a unification theory in some distant future. Perhaps the significance is not understood until one has an understanding of the Philosophy of Mathematics.

In reply to an earlier post on Feb 20, 2013 5:35:09 PM PST
If you're talking to me, I assure you I'm no genius. My IQ is moderately above average. I do worse on math than verbal questions. I do enjoy reading about what the geniuses in math and science have discovered, though.

In reply to an earlier post on Feb 20, 2013 5:36:15 PM PST
Excellent points.

In reply to an earlier post on Feb 20, 2013 5:39:07 PM PST
Speaking of everything being related and all the boundaries fading away, I recently bought an introductory book on Category Theory, or "abstract nonsense" as it's sometimes dubbed. Interesting to see how everything can be joined up.

In reply to an earlier post on Feb 20, 2013 5:43:35 PM PST
Sounds similar to the various math major pathways at MIT, which has all its course materials available online.

Years ago I got the idea of "majoring in physics" at MIT by simply doing self-study of all the courses they require for a physics major. I managed to make it through the sophomore year. I even e-mailed one of the profs with a question and he answered back.

In reply to an earlier post on Feb 20, 2013 5:44:38 PM PST
One of the best feelings one can have: getting the answer to a difficult question through persistence and insight.
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Discussion in:  Science forum
Participants:  23
Total posts:  156
Initial post:  Feb 19, 2013
Latest post:  Mar 5, 2013

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