Where To Start?

[SaxonViolence]

Friends,

 

I've long been fascinated with Mathematics.

 

A long time ago, when I was young, I had all hopes of ever being a Scientist Dashed, when I failed out at Purdue three times.

 

The biggest cause of my downfall was Calculus. It was a five credit hour course--and I spent so many hours failing Calculus, that I failed everything else too.

 

#1} I am a very "Left Brain" thinker.

 

I really excelled at Euclidean Geometry in High School

 

I really grooved on the method of Proving Theorems by justifying each step by citing an Axiom, Definition; or Prior Proven Theorem.

 

I believe that's how all Mathematics was once done.....

 

But somewhere along the way, those Devious and Degenerate Right-Brainers mounted a silent coup; and substituted the curious and quaint method of simply solving equations instead of using Formal Proofs.

 

#2} Part of my problem was being very uptight about learning--particularly after my first failure.

 

 

#3} The non-professional Teachers Assistants had an invincible tendency to Demonstrate rather than Explain.

 

I don't understand that equation that you just did. Where did you get the number you brought in at step #3?

 

TA works another problem on the board.....

 

I Still don't understand. This time you brought in a completely different number at step #3--But now I have no idea where Either Number came from.

 

TA erases, and silently works another problem on the Blackboard.

 

Eventually I'd get "We have to move on. Everyone else understands this."

 

I didn't realize at the time, to know that they were giving me Demonstrations in lieu of an Comprehendible Explanation.

 

 

Anyway, I've always had a Fascination with Higher Math. I really groove on reading the "Popular and Non-Mathematical" books on math.

 

I also liked to pick up old math texts at used bookstores or yard sales. I loved the look of page after page of undecipherable Hieroglyphics.

 

I used to dream that some day, I'd really get into some sort of Ambitious Home Study Program on my own.....

 

I wanted to finally conquer the Calculus that had so wrecked my life so long ago.....

 

And get into the far more interesting Fields of Topology, Non-Euclidean Geometry, Number Theory, Formal Logic, etc.

 

A large cooler that I'd filled with Math books was stolen out of a storage trailer. Don't know why whoever took it, didn't dump the books. It was a nice cooler, but how many folks desire fifty pounds of old Textbooks on higher math?

 

So I don't get around to university book sales anymore. Hardly leave the house.....

 

So now I have a much more modest collection of math texts from Dover, that I add to occasionally (Much more expensive than paying a buck per book)

 

I'm 55 years old--not as sharp as I once was--memory not so good.....

 

And I figure that if I ever want to learn higher math, I'd better get at it.

 

I'd really like to be able to fully understand: The Riemann Conjecture; String Theory; Hyperbolic Geometry and Fractals--along with Topology, Matrices and Vectors and Fluidics. Oh yes, and Chaos Theory along with Catastrophe Theory.

 

Does anyone have any good idea how to get started?

 

What sort of things to study first?

 

Any hints how to get away from wanting to use "Brute Force" methods at every breakdown in understanding?

 

Thanks.

 

Saxon Violence.

[7DSUSYstrings]

When I first began studying astrophysics, it had spawned from a journey through Wheeler and Thorne's "Gravitation." This sometimes referred to "The Phone Book," because it is a bulky book. It may be a public domain PDF by now, you'd have to check. It then proceeded to Thorne's translation of Zel'dovich and Novikov "Relativistic Astrophysics. Vols. 1 and 2" where the former really started making sense.

 

That would depend, though, on having taken a class in at least equilibrium thermodynamics.

Edited April 7, 2012 by 7DSUSYstrings

[maddog]

I've long been fascinated with Mathematics.

A long time ago, when I was young, I had all hopes of ever being a Scientist Dashed, when I failed out at Purdue three times.

The biggest cause of my downfall was Calculus. It was a five credit hour course--and I spent so many hours failing Calculus, that I failed everything else too.

#1} I am a very "Left Brain" thinker.

...

But somewhere along the way, those Devious and Degenerate Right-Brainers mounted a silent coup; and substituted the curious and quaint method of simply solving equations instead of using Formal Proofs.

... Ambitious Home Study Program on my own.....

I wanted to finally conquer the Calculus that had so wrecked my life so long ago.....

And get into the far more interesting Fields of Topology, Non-Euclidean Geometry, Number Theory, Formal Logic, etc.

... I'm 55 years old--not as sharp as I once was--memory not so good.....

And I figure that if I ever want to learn higher math, I'd better get at it.

I'd really like to be able to fully understand: The Riemann Conjecture; String Theory; Hyperbolic Geometry and Fractals--along with Topology, Matrices and Vectors and Fluidics. Oh yes, and Chaos Theory along with Catastrophe Theory.

Does anyone have any good idea how to get started?

What sort of things to study first?

I am sorry to condense the quote of your post (just so I could comment on pertinent parts). I somehow feel affinity for you in that I also went to Purdue (though I graduated - in Physics). I am sadly Right-Brained (being left-handed) though I am told by some test that I am somewhat oddball in that I tend to use both sides of my brain in some integrated fashion. Psychiatrists/Psychologists ?!?

 

I find it odd that you liked High School Geometry yet had problem with Calculus. The principles of Euclid study in High School were of a discrete axiomatic nature whereas what is at the heart of Calculus is the notion of the approximation of continuity. This is the resolution of the paradox by Zeno. You could never get to a wall by halving the distance without an approximation of near zero (not in discrete terms).

 

Calculus owes its similarity to Algebra in solving problems whereas Geometry if about proofs. In Algebra there are equations (polynomials) that you solve for the variable (be it x or otherwise) or factor in simplest terms.

 

The added value in Calculus is you can get the Derivative (also thought as Slope or Velocity) of a function use Integration to compute the Area under a curve (or function).

 

I would think you need to ground yourself in the Algebra before you tackle the "phone-book" of Gravitation by Misner, Kip and Thorne. This book (I have a copy) is aggressive tour of General Relativity using Vector and Tensor Analysis, Linear Algebra which would likely presume a second or third year undergraduate knowledge of university mathematics.

 

Though for a good book on Calculus try one by Taylor (its been through a lot of printings). Know that Analysis is just a rigorous version of Calculus after studying Linear Algebra.

 

There is some good books on Topology (I remember finding one in my 8th grade library by Chinn). It was from this that I learned a doughnut and a coffee cup are actually the same object or that I could take my vest off that was under a coat without taking off the coat. With this I wowed and amazed my friends. Like you I am also getting on in age (I am 56).

 

As for understanding String Theory (or M-Theory for that matter) - you will need to understand Complex Analysis, Linear Algebra, Non-Euclidian Algebra, some Group Theory, and the notion of the Harmonic Oscillator from Physics.

 

It does all start to fit together once you have the basics down (Algebra/Calculus). Good Luck!

 

maddog

Edited April 10, 2012 by maddog

[phillip1882]

i afraid i don't personally know of any good complete studies of the topics you're interested in. it's too sparced and varied. i might suggest www.kahnacademy.com for some good refresher courses, they have everything from basic addition to calculus.

beyond that, i might suggest doing some standard google searches for fractal math textbooks and see what kind of hits you get.

[Turtle]

Friends,

 

I've long been fascinated with Mathematics.

 

...

I'm 55 years old--not as sharp as I once was--memory not so good.....

 

And I figure that if I ever want to learn higher math, I'd better get at it.

 

I'd really like to be able to fully understand: The Riemann Conjecture; String Theory; Hyperbolic Geometry and Fractals--along with Topology, Matrices and Vectors and Fluidics. Oh yes, and Chaos Theory along with Catastrophe Theory.

 

Does anyone have any good idea how to get started?

 

What sort of things to study first?

 

Any hints how to get away from wanting to use "Brute Force" methods at every breakdown in understanding?

 

Thanks.

 

Saxon Violence.

 

never memorize something that you can look up. ~albert einstein ;)

 

for fractals i recommend going to the horse's mouth and getting ahold of a copy of The Fractal Geometry of Nature by Benoit Mandelbrot. to just check it out, go to your library. (:doh:) if you want to own it, new copies go for ~$35.00 but you can probably do better at e-bay. my copy, which i have since passed on to someone else in my pursuit of dematerialization, i got for $2.00 by joining one of those science book club's introductory offers.

 

in any case, don't get hung up on the calculus. if it's not your bag, it's not your bag and there's plenty else out there to keep you interested and sharp.

Edited April 11, 2012 by Turtle

[SaxonViolence]

Maddog

 

I never studied, nor had any interest in anything Academic in High School--yet somehow I got by.....

 

Natural Intelligence; Near Total Recall; Shameless Cheating and the reluctance of the Instructors to fail someone who was always in class and never a Discipline Problem got me by.

 

{Euclidean Geometry was a Pleasant exception to the tremendous distaste that I felt for almost everything taught in School.}

 

When I got to Collage, I resolved to turn over a new leaf--but I hadn't a clue how to study.

 

I thought that it was extremely important to grasp everything 101% before moving on to the next sentence or paragraph. I thought that trying to understand line nine of text, without having understood lines one through eight--would be like trying to climb a ladder that had a two or three foot gap in the middle--even if you could somehow get past the gap--what would support the levitating portion of the ladder?

 

I was in my forties when I learned that if you'll simply skip over a part that isn't clear--momentarily--but keep the question in the back of one's mind--often something further ahead will clarify.

 

I would read my Calculus Text. Come to a sentence that I didn't understand. Reread the sentence--10; 20; 50; 100 times. Go back to the beginning of the Paragraph--maybe I missed something earlier. Reread five to a dozen more times. Go back to the start of the chapter--reread three to five times.

 

As the semester wore on, and the prospect of failing out loomed--something that horrified me beyond words.....

 

Throw tantrum--Beat self over head with heavy Calculus text; throw text across room; break pencil; strike wall with fist....

 

After fifteen minutes to a half hour, calm down; feel foolish; retrieve text; get out another pencil, start over.

 

At some point, even opening the book would bring on an irresistible urge to say, "To Hell with it!" and take a nap instead.

 

About mid-semester I'd realize that I had dropped too many points to pass the course anyway--stop going to class--start sleeping eighteen to twenty hours a day, and dread my folks finding out.

 

Nonetheless, over the years I've come to have a pretty fair grasp of what Calculus is and does--largely find the slope of a single point or the area under an irregular curve.

 

I can do a Differentiation--albeit very slowly and haltingly.

 

I'm fuzzier about Integrals. I know that they're the converse of Differentials; Necessarily more General; And related to the summations under a Sigma Sign.

 

If I'm reading an Engineering Text--I'm a part-time inventor--or a text on Cosmology--and I see the long "S" Integration Sign, I understand the principle, and can carry on my reading--

 

But I'd be hard pressed to work any equations with the Formula.

 

I found talking to TAs or Professors unproductive.

 

Private tutoring might very well have saved me--but it seemed like a form of "Cheating" to me back then.

 

I've had folks absolutely amazed and infuriated by that last remark--and some who just can't comprehend.

 

The Old Aristocratic British Amateur Athletes felt that it was very poor form, even if it wasn't outright cheating, to hire a Professional Coach or Trainer.

 

If you are a true "Amateur" there is a limit to how much Sweat and Effort one puts into Training.

 

If one can't win gracefully with a modest amount of effort, then one should lose like a Good Sport--anything else smacked of an Ungentlemanly Professionalism.

 

I had never quite formulated my beliefs back then. I felt that the only real measure of a man was his Physical Prowess--but whatever minor acclaim was due Academic Achievement, it was only due achievements made gracefully and without undue strain.

 

If you had to be a Grind to get your grades, then they were nothing to be particularly proud of.

 

If I'd have realized the horrors of having to work for a living, I'd have gotten a Degree by hook or crook--but I was young and naive......

 

And I'd never worked fourteen hour days under the Blazing-Hot Sun, or experienced Heat Exhaustion.....

 

Saxon Violence

[maddog]

Natural Intelligence; Near Total Recall; Shameless Cheating ...

 

I would read my Calculus Text. Come to a sentence that I didn't understand. Reread the sentence--10; 20; 50; 100 times. Go back to the beginning of the Paragraph--maybe I missed something earlier. Reread five to a dozen more times. Go back to the start of the chapter--reread three to five times.

I have sympathy with on reading/rereading text. There are some books that I continually loose my place and have to reread again and again. I know in my case that this is because of my ADHD (I was diagnosed in the last five years). Was Algebra in High School or Jr High difficult for you ? The brute force methods of solving equation starts there.

 

I can do a Differentiation--albeit very slowly and haltingly.

I'm fuzzier about Integrals. I know that they're the converse of Differentials; Necessarily more General; And related to the summations under a Sigma Sign.

Yes, these are the opposite of each other, in all cases.

 

There is one method I learned that when I got interested in something it could do violence to any boredom I might be suffering. So here is something, as you did mention Fractals in your first post.

 

Fractals are in general based upon a method known as Iterating Function Systems. Just as Turtle said, Mandelbrot does use the common example he had discovered that generates the Mandlebrot Set.

 

Say we have a simple polynomial in the Complex plane (oops, Complex numbers are of the form a + bi where i = sqrt(-1)).

 

Let z = a + bi where a, b are Real variables.

 

So to iterate say we square the complex variable z and add a complex constant value c.

 

z^2 + c

 

Now use this as the next generation of Z (this is one iteration).

 

From this we plot a the value in the complex plane say around zero (0). You can do this on a calculator or write a simple program to plot.

 

Z <= Z^2 + c where c is a constant value for the whole plane.

 

The picture you get near the surface of the big black space is just amazing!

 

Do a google search on Fractals or Mandlebrot and learn from what you find.

 

maddog

Edited April 12, 2012 by maddog

[SaxonViolence]

I probably have ADHD.

 

I may also be a High-Functioning Aspergers.

 

I am also a Verbal Literalist.

 

Neither Diagnosis was common when I was younger.

 

I'm on Disability for a grab-bag of Symptoms, both Physical and Mental--probably few of them sufficient, in and of themselves--to qualify me.

 

I have Depression--perhaps Bi-Polar with an ever-shrinking Manic Phase.

 

One Psych summed it up:

 

"When he's down, he couldn't work fast enough to meet Production Standards.

 

When he's up, his incessant talking would annoy everyone around him."

 

Thanks for the Explanation of Fractals--that will take a bit of cogitating to absorb.

 

In my current incarnation, I frequently get all excited about an Idea one day; and the next day, I'm asking myself why I should bother.

 

I'm a reasonably popular On-Line Author--under another Handle.

 

Being an On-Line Author is rather like being a Champion Softball Player--Even playing on the team that won the Softball World Series (if there were such a thing)--would be a very mild accomplishment. I mean, if you had skills, why not play Baseball?

 

At any rate, I conceived the plan for a Mainstream, fairly realistic story intended for Publication called "Gunstore Knights", centered around a small group of dedicated Martial Artists who hang around a Gunstore, and think of themselves as Modern Day Ronin--stuck in a World that has very little use for their skills or attitudes.

 

You also encounter a Good many "Tin-Foil" Nut-Cases in Gunstores along the way, some of them hysterically funny--and while I have scores of good first-hand accounts, I sent out a call for all my friends to send me any good Anecdotes they might have.....

 

And by the time that they started coming in the next day, I was in a "Why do I trouble myself to write? I don't enjoy it. When the Urge comes on me, it is more a Compulsion than anything enjoyable.....and if I could just permanently purge myself of this Writing Compulsion, I'd have much more time to Draw and Paint" (Which I do enjoy).

 

I've been on and off the "Gunstore Knights" project a Dozen times since.....

 

You may guess--I work off much of my Writing Compulsion on the Forums.....

 

Which I enjoy modestly.

 

Shame that I have no such Compulsion to do Art.

 

Day after day, I promise myself that I'll get off the computer fairly early today, and work on my Art.....

 

But minute by minute, the time slips away.

 

{I have to stop about 3:30 PM, to fix Supper. Rarely feel creative after Supper--plus the lighting isn't the best.....)

 

Saxon Violence

Edited April 12, 2012 by SaxonViolence

[maddog]

I probably have ADHD. I may also be a High-Functioning Aspergers.

I am also a Verbal Literalist. Neither Diagnosis was common when I was younger.

I'm on Disability for a grab-bag of Symptoms, both Physical and Mental--probably few of them sufficient, in and of themselves--to qualify me.

I have Depression--perhaps Bi-Polar with an ever-shrinking Manic Phase.

The more I listen to your story, the more similarities I see with mine. Like you (during a time of job loss), I was recently diagnose with a "mild" Aspergers condition. I am not sure what a "touch of Aspergers" does to ADHD. Yet it's gotta' be quite a concoction if you ask me!

 

I knew that early on in my childhood that I was "different" than other kids in many respects. Different than you though that I excelled at math. Though admittedly, I did flunk my High School Algebra class and had to take it again that summer. I found out in High School (or rather my Chemistry teacher found before me) that if I became interested in a subject there was no stopping me (you couldn't shut me "off"). Likewise if I was not interested - you couldn't "turn me on" either.

 

Shame that I have no such Compulsion to do Art.

Day after day, I promise myself that I'll get off the computer fairly early today, and work on my Art.....

My drawing abilities are quite lacking, so I let my wife pursue that venue.

For myself, I like puzzles (sudoku, and some math ideas -- like to solve the generalized Stokes Theorem). Just call me ambitious. Then there is this conundrum between QM and GR (where gravity does not well behave in a Quantum realm). One form of literature I took to early on (about 6th grade or so) was Science Fiction. I'm an avid reader. When an author writes well, the book will disappear and I am amazed at how quickly I read. When the opposite is true, I will reread a page 20 times to understand and still do not. I guess that is why I am afraid to read Ulysses, by James Joyce as I have been told it is hard to read.

 

maddog

Edited April 13, 2012 by maddog

[SaxonViolence]

This has kinda turned into a Private Conversation between us--but that's okay.

 

I was in 3rd grade when the "New Math" came out. I was so good at it, that I used to have to explain some of it to my Teachers.

 

5th and 6th grade, we had different Teachers for different classes by then. The Math teacher was a big Muscular, Swag-Bellied Brute of a man, who got off on totally dominating a bunch of Children and often handed out five or six paddlings in a single hour.

 

I think he taught me to approach Math with Fear and Loathing.

 

Then my 7th and 8th grade Math Teacher was a very unimaginative accountant sort of man.

 

I often would solve very complex problems with a flash of Intuition...

 

He continually told me that A.} Such Flashes were unreliable; and B.} That they were a form of "Cheating" and C.} That I should mercilessly suppress such Flashes; because the soul of Mathematics was to keep one's head down, and slog along dutifully, from point to point.

 

In High School, I got bored and sat and daydreamed myself as the Hero in some sort of Robert E Howard Phantasy the first few weeks. I got behind, and there was no way to catch up.

 

I was too intimidated by personal interactions to ask a teacher for any after-class help. I had no friends--and if I had, and they'd have been good at Math, they'd undoubtedly been limited to "Demonstrating" rather than "Explaining"--like the TAs at Purdue.

 

My father often offered a to hire a Tutor--But I was Absolutely Horrified by the Idea--as previously discussed, it felt like cheating--and it seemed unmanly to me, to be that concerned about School Subjects anyway. Algebra wasn't going to add a single pound to my Bench Press.

 

I took Geometry I & II in my Junior Year, instead of taking it between Algebra I & II; and Algebra III & IV like most students did--and one of the grand things, was that it required no prior knowledge. Also, the Instructor was also the Baseball Coach--and while I didn't play Baseball, being a Coach earned him my respect.

 

At Purdue, I took the two semester Algebra and Trig Refresher--had a Grand Instructor, everything Clicked, Got A's.....

 

And when I switched my major to Science, and had to take Calculus--When I started having trouble--I reverted to hating the Class--like in 5 & 6; I finally internalized the Nutty "Kill your Intuition" urgings of my 7 & 8 instructor; and like in the four semesters of High School Algebra, I constantly felt that I was in way over my head--only this time I cared.

 

On top of all that--I do very poor intellectual work under pressure--That is, anxiety severely impedes my ability at understand and absorb new material.

 

A character in one of the "Dune" books says (approximately), "You Force the Body to Work. You have to learn to do the opposite, and Relax and Let the Mind Work."

 

Actually, while that's true enough so far as the Mind goes, you aren't going to learn much Martial Arts, Gymnastics, Batting or anything else--if you approach it like you'd just been given a huge injection of Crystal Meth, and then hit with a Taser--and expected to be hit with the Taser again and again, at random intervals, all through your training session.....

 

Saxon Violence

[maddog]

This has kinda turned into a Private Conversation between us--but that's okay.

Yeah, I notice that & I'm ok with that too.

 

I was in 3rd grade when the "New Math" came out. I was so good at it, that I used to have to explain some of it to my Teachers.

I'm not sure for me where I was whether it was 3rd, 4th or later. By 6th it definitely was.

 

5th and 6th grade, we had different Teachers for different classes by then. The Math teacher was a big Muscular, Swag-Bellied Brute of a man, who got off on totally dominating a bunch of Children and often handed out five or six paddlings in a single hour.

Yes, in my 6th I was given an assignment (with 2 others I think) to help other students as I took this like bread to butter.

 

I think he taught me to approach Math with Fear and Loathing.

I have to admit a similarity here as well. My sixth grade teacher was also the principal and an ex?-preacher (so fire & brimstone was his rhetoric). ;-)

 

Then my 7th and 8th grade Math Teacher was a very unimaginative accountant sort of man.

I often would solve very complex problems with a flash of Intuition...

He continually told me that A.} Such Flashes were unreliable; and B.} That they were a form of "Cheating" and C.} That I should mercilessly suppress such Flashes; because the soul of Mathematics was to keep one's head down, and slog along dutifully, from point to point.

Here we differ, my 7th grade (I don't even remember 8th) teacher had a wildly different approach. We were allowed to move at our own pace through

the textbook. When I finished my text halfway through the course, I was allowed to go to the library to pick out other books. This where I found the book on Topology.

 

In High School, I got bored and sat and daydreamed myself as the Hero in some sort of Robert E Howard Phantasy the first few weeks. I got behind, and there was no way to catch up.

I was too intimidated by personal interactions to ask a teacher for any after-class help. I had no friends--and if I had, and they'd have been good at Math, they'd undoubtedly been limited to "Demonstrating" rather than "Explaining"--like the TAs at Purdue.

In my freshman year, I flunked Algebra and had to take it again in summer school where I got a C. I was so scare of math then, I was afraid to take any more math classes ever!

 

I took Geometry I & II in my Junior Year, instead of taking it between Algebra I & II; and Algebra III & IV like most students did--and one of the grand things, was that it required no prior knowledge. Also, the Instructor was also the Baseball Coach--and while I didn't play Baseball, being a Coach earned him my respect.

I think I got a D half-way through the first but pulled out a C in the end. Yeah, in this course I had an ex-drill Sargent for a teacher. Kinda' made math un-fun if you know what I mean. Yechh!

 

At Purdue, I took the two semester Algebra and Trig Refresher--had a Grand Instructor, everything Clicked, Got A's.....

By my Senior year of HS I had worked up enough courage to take Algebra II - got an A as well. I realized I needed Trigonometry & Analytic Geometry

to be able to take Calculus in College. So I signed up for summer session before the first semester (I started at IU, in Bloomington). I got an A in this

too. As I had an aggressive schedule, I had to take Calculus I in the second session to comply with having Calculus before to take Physics I&II for

majors (it required working knowledge of Calculus).

 

On top of all that--I do very poor intellectual work under pressure--That is, anxiety severely impedes my ability at understand and absorb new material.

Yes I have the same problem. While taking a Graduate math class, on the final (which was oral), I was so nervous, I forgot L'Hospital's Rule

(which is something from Caclulus II for testing if a series is divergent or not). In my Senior level undergraduate Abstract Algebra final, I was given

credit for putting my name on the test and still only got a 20 out of 100. The instructor said I continually proved the contrapositive (the opposite).

 

A character in one of the "Dune" books says (approximately), "You Force the Body to Work. You have to learn to do the opposite, and Relax and Let the Mind Work."

I think that is from the first book when Paul is given the test of the Gom Jobar. "Fear is the Mindkiller!"

 

I must say there is a lot that is similar in our life paths... :D

 

maddog

 

ps: If ever you need any input/or answer to a math question, I might be able to help...

[belovelife]

why not khan academy

 

 

kinda like playing a video game

[SaxonViolence]

why not khan academy

 

 

kinda like playing a video game

 

WOW!

 

Thanks!

 

Saxon Violence