> It would be like comparing wiles proof for FLT to an entire book about Modular Forms, Elliptic Curves and Galois Representations.

I have (mostly) read that book. It's nothing like Mochizuki's work.

The issue isn't that the theory has a bunch of definitions, most theories do. It's that the entire theory seems to be a bunch of definitions with no nontrivial work or minor applications, except one incredibly grand claim. It this is actually true, there would be nothing even remotely like that in the history of mathematics.

Edit:

> He has made an extremely terrible job communicating with the westerners yeah, but it does not seem to be the case for other japaneses as one of his colleagues wrote a 300 page summary on it

He's convinced a few people, maybe like 10-20, but that's not the same thing as convincing the entire japanese community. And the people he's convinced have also been incapable of explaining it to their colleagues (and not for lack of trying, it seems), which kind of makes it questionable whether they actually understand it, or just think they do. My understanding is that the 300 page summary didn't really help much.

What do you mean by the "Whole Proof"? How much do you want to assume? If you assume everything that is learned in a standard phd program that is loosely related to number theory (so including things like Class Field Theory, basic theory of Elliptic curves and Modular forms), you would need the Langlands-Tunnell Theorem, which is a whole book on its own, Ribet's Theorem and the analysis of Frey Curves, Deformation Theory, Hecke Algebras, a boatload of advanced Commutative Algebra and many computational results on particular elliptic/modular curves. Then you can begin to talk about Wiles' contributions. It wouldn't be just one book.

If you want something that contains the general knowledge of the proof, but is brief when it needs to be, then Modular Forms and Fermat's Last Theorem is pretty solid.

You are in a very special position right now where many interesing fields of mathematics are suddenly accessible to you. There are many directions you could head. If your experience is limited to calculus, some of these may look very strange indeed, and perhaps that is enticing. That was certainly the case for me.

Here are a few subject areas in which you may be interested. I'll link you to Dover books on the topics, which are always cheap and generally good.

• The Nature and Power of Mathematics, Donald M. Davis. This book seems to be a survey of some history of mathematics and various modern topics. Check out the table of contents to get an idea. You'll notice a few of the subjects in the list below. It seems like this would be a good buy if you want to taste a few different subjects to see what pleases your palate.
  
  
• Introduction to Graph Theory, Richard J. Trudeau. Check out the Wikipedia entry on graph theory and the one defining graphs to get an idea what the field is about and some history. The reviews on Amazon for this book lead me to believe it would be a perfect match for an interested high school student.
  
  
• Game Theory: A Nontechnical Introduction, Morton D. Davis. Game theory is a very interesting field with broad applications--check out the wiki. This book seems to be written at a level where you would find it very accessible. The actual field uses some heavy math but this seems to give a good introduction.
  
  
• An Introduction to Information Theory, John R. Pierce. This is a light-on-the-maths introduction to a relatively young field of mathematics/computer science which concerns itself with the problems of storing and communicating data. Check out the wiki for some background.
  
  
• Lady Luck: The Theory of Probability, Warren Weaver. This book seems to be a good introduction to probability and covers a lot of important ideas, especially in the later chapters. Seems to be a good match to a high school level.
  
  
• Elementary Number Theory, Underwood Dudley. Number theory is a rich field concerned with properties of numbers. Check out its Wikipedia entry. I own this book and am reading through it like a novel--I love it! The exposition is so clear and thorough you'd think you were sitting in a lecture with a great professor, and the exercises are incredible. The author asks questions in such a way that, after answering them, you can't help but generalize your answers to larger problems. This book really teaches you to think mathematically.
  
  
• A Book of Abstract Algebra, Charles C. Pinter. Abstract algebra formalizes and generalizes the basic rules you know about algebra: commutativity, associativity, inverses of numbers, the distributive law, etc. It turns out that considering these concepts from an abstract standpoint leads to complex structures with very interesting properties. The field is HUGE and seems to bleed into every other field of mathematics in one way or another, revealing its power. I also own this book and it is similarly awesome. The exposition sets you up to expect the definitions before they are given, so the material really does proceed naturally.
  
  
• Introduction to Analysis, Maxwell Rosenlicht. Analysis is essentially the foundations and expansion of calculus. It is an amazing subject which no math student should ignore. Its study generally requires a great deal of time and effort; some students would benefit more from a guided class than from self-study.
  
  
• Principles of Statistics, M. G. Bulmer. In a few words, statistics is the marriage between probability and analysis (calculus). The wiki article explains the context and interpretation of the subject but doesn't seem to give much information on what the math involved is like. This book seems like it would be best read after you are familiar with probability, say from Weaver's book linked above.
  
  
• I have to second sellphone's recommendation of Naive Set Theory by Paul Halmos. It's one of my favorite math books and gives an amazing introduction to the field. It's short and to the point--almost a haiku on the subject.
  
  
• Continued Fractions, A. Ya. Khinchin. Take a look at the wiki for continued fractions. The book is definitely terse at times but it is rewarding; Khinchin is a master of the subject. One review states that, "although the book is rich with insight and information, Khinchin stays one nautical mile ahead of the reader at all times." Another review recommends Carl D. Olds' book on the subject as a better introduction.
  
  Basically, don't limit yourself to the track you see before you. Explore and enjoy.

So you believe correlation is causation. Did you know that:

• The lower your consumption of butter the more likely you are to get divorced.
  
  
• The more movies Nicalas Cage makes the more people drown in swimming pools.
  
  
• The more our country spends on science the more people commit suicide by hanging.
  
  
• The more lemons we import from Mexico the lower our highway accident fatality rate is.
  
  
• And don't even get me started on how eating chocolate is the sure way to win nobel prizes.
  
  Here is a book recommendation for you:
  
  https://www.amazon.com/Counterfactuals-Causal-Inference-Principles-Analytical/dp/1107694167
  
  I'm sure someone with as well informed opinions as you must have an interest in reading basic textbooks and learning new things. Right?

This book supposedly covers the proof and much of the background material: Modular Forms and Fermat's Last Theorem (Springer). Of course, you'll probably have to consult many, many other books along the way, but this looks good as a point of reference.

By far, the best resource is Andrew Gelman's book Data Analysis Using Regression and Multilevel/Hierarchical Models.

Also, the book Extending the Linear Model with R by Julian Faraway touches on it.

Other than that, here are some blog posts that I found helpful:

https://ourcodingclub.github.io/2017/03/15/mixed-models.html

https://www.jaredknowles.com/journal/2013/11/25/getting-started-with-mixed-effect-models-in-r

http://www.bodowinter.com/tutorial/bw_LME_tutorial2.pdf

And here's the paper written by Doug Bates, the author of the r mixed effects package lme4.

Spivak is probably the best intro to real analysis book out there, so I would say start there. Also, go dig out your old pre-calc textbook and be sure you've really mastered the basics. For diff eq, just keep your calc tools extra sharp - think deeply about the fundamental theorem and get right with taylor series. Maybe go back and work through the material on continuously compounding interest and radioactive decay, that sort of thing. For probability, start with khan and then try to find introduction to probability by Bertsekas at your library. Most importantly: have fun and try to solve something nifty with your new math-toys :-)

>Analysis If you want to keep learning analysis, check out Introductory Real Analysis by Rudin, Principles of Mathematical Analysis by Kolmogorov & Fomin

Permuted the authors :P. That being said, Rudin's first book I don't think is the best introduction to real analysis (and Kolmogorov's I think is better as a bridge towards functional analysis). My recommendation for a first real analysis book is Real Analysis by N.L. Carothers.

Nate Silver is a perfect reason why you need to read "Lies, Damn Lies, and Statistics: The Manipulation of Public Opinion in America" https://www.amazon.com/Lies-Damn-Statistics-Manipulation-Opinion/dp/0393331490

Was required to read it in my college statistics class.

It does not, and the set of confounding variables in fitness is legion. The particular study you linked is a pure observational exercise that makes no attempt to deal with selection issues and the idea that one might find causality in the results is fantasy.

Consider checking out some causality literature, either the Ruben-Imbens thread or the Pearl devotees. Join us in believing nothing very few extant studies.

Or just read people like Andrew Gelman and dispense with learning about causality, there are ten dozen other reasons most published research findings are false.

Related.

For theoretical foundations and readability, the Theory of Probability: Explorations and Applications by Santosh S. Venkates is my favorite. Venkates' passion for the subject is contagious, and his historical references give a you additional perspective on the foundations of the subject https://www.amazon.com/Theory-Probability-Explorations-Applications/dp/1107024471/

For practical, no-nonsense approach with real-world examples John Bertsekas & Tsitsiklis' Intro to Probability is excellent (although in my opinion more of a manual and reference) https://www.amazon.com/Introduction-Probability-Dimitri-P-Bertsekas/dp/188652940X/

If you would like to become an expert in probability theory, you need to have a solid ground in measure theory. I would suggest to study analysis out of Carothers. This covers most of what Rudin covers but I find it easier to read, and it goes into more detail about measure and Lebesgue integration on the real line. If you work through this, you'll have a solid background for heavier measure theory books and for upper level probability theory.

I thought these were decent:

https://www.amazon.com/Statistical-Models-Practice-David-Freedman/dp/0521743850
good for no nonsense straight forward derivations of a lot of the basic results (and which assumptions are needed for each of the basic results)

Applied Regression Analysis and Generalized
Linear Models, J. Fox

Plane Answers to Complex Questions - more theory, geometry heavy emphasis

*There is no way to determine causation from a single correlation without further assumptions. There is a large body of literature devoted to estimating causal relationships without experimental data. Here are a couple of standard textbooks in this literature.

National Wrestling Alliance: The Untold Story of the Monopoly That Strangled Pro Wrestling

It's a great, very detailed book that goes back to even before the NWA was initially started.

Pain and Passion: The History of Stampede Wrestling

Another great book all about the Harts promotion.

These were the most enlightening for me on their subjects:

• Differential and Integral Calculus, Vol. 1&2 by Richard Courant
  
• Linear Algebra by Georgi Shilov
  
• Differential Equations and Their Applications by Martin Braun
  
• Introduction to Geometry by H.S.M. Coxeter
  
• A Course in Mathematical Analysis, Vol. 1&2 by Edouard Goursat
  
• Introduction to Topology and Modern Analysis by George Simmons
  
• The Theory of Functions of a Complex Variable by A.I. Markushevich
  
• Basic Topology by M.A. Armstrong
  
• Topology by James Dugundji
  
• Ordinary Differential Equations by V.I. Arnold
  
• Partial Differential Equations by Fritz John
  
• Introduction to Probability Theory by Paul Hoel, Sidney Port and Charles Stone
  
• Linear Programming by Katta Murty
  
• Nonlinear Programming: Theory and Algorithms by M. Bazaraa, H. Sherali and C. Shetty
  
• A First Course in Numerical Analysis by Anthony Ralston and Philip Rabinowitz
  
• A First Course in Stochastic Processes by Samuel Karlin and Howard Taylor
  
• Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields by John Guckenheimer and Philip Holmes
  
• Homology Theory by James Vick

I really believe that Michael Kelly's "Humongous Book of" series are the best resources for getting through all math classes up to Calculus II. These books contain every single type of problem you will ever encounter in Algebra I & II, Geometry, Trig, and Calc I & II, all solved in great detail. They are like Schaums Outlines but much more reliable.

https://www.amazon.com/Humongous-Basic-Pre-Algebra-Problems-Books/dp/1615640835

https://www.amazon.com/Humongous-Book-Algebra-Problems-Books/dp/1592577229

https://www.amazon.com/Humongous-Book-Geometry-Problems-Books/dp/1592578640

https://www.amazon.com/Humongous-Book-Trigonometry-Problems-Comprehensive/dp/1615641823

https://www.amazon.com/Humongous-Book-Calculus-Problems-Books/dp/1592575129

I think that is where statistics are very wrong. He said he's done a bunch of HITs that only take a few seconds. And then when you couple that with HITs that take a few minutes and pay big, then you get the average that you're talking about. I do the same thing sometimes. I'll HIT a batch that pays OK that takes a short amount of time, and then I'll get a batch that pays $2.00 that takes me 5 minutes to do.


If you want someone to tell you who those people are and what requirements you need, you won't get very far. You have to find them. It's a turk-eat-turk world. We're all trying to make good money. HWTF and those type of forums are mostly for surveys that you can only complete one HIT for. If someone knows a good batch..... I'm sorry to say that the less people that know about it, the better.

TL:DR Lies, Damn Lies, and Statistics OR Damned Lies and Statistics

1. Ross - First Course in Probability (Calculus based probability, undergraduate level, good introduction to probability)
   
   http://www.amazon.com/First-Course-Probability-9th-Edition/dp/032179477X
   
   
2. Rice - Mathematical Statistics (introduction to statistics, focuses on applications with data, great book, includes good probability review)
   
   http://www.amazon.com/Mathematical-Statistics-Analysis-Available-Enhanced/dp/0534399428
   
   
3. Billingsley - Probability and Measure (graduate, measure-theoretic probability)
   
   http://www.amazon.com/Probability-Measure-Patrick-Billingsley/dp/1118122372
   
   
4. Bickel & Doksum - Mathematical Statistics (graduate level statistical theory, much more theoretical than Rice, can be a difficult book to learn from but it is a great reference)
   
   http://www.amazon.com/Mathematical-Statistics-Selected-Topics-Edition/dp/0132306379
   
   EDIT:
   
   Most likely Rice will be the best book for a comprehensive look at prob/stat, and it is sufficiently technical.
   

I mean it in a kind of casual sense. It seems like after reading and writng so much mathematics, a lot of people converge to a writing style that's characteristic to math writing. They'll default to writing in first person plural whenever possible ("we have that...", "we define"), they'll use the word "therefore" almost every other sentence, and they'll generally write in a rather dry but logically clear manner.

I'd contrast this with the more "didactic" writing style of the usual high school texts (e.g. something like this.)

Take a 'probability theory' class that covers or read something like this Dover book:

http://www.amazon.com/Probability-Theory-Concise-Course-Mathematics/dp/0486635449/

The statistics survey courses are generally horrible.

Lady Luck: The Theory of Probability, by Warren Weaver

http://www.amazon.com/Lady-Luck-Theory-Probability-Science/dp/0486243427

This is a great introduction to probability theory, an entertaining read, and written precisely for the motivated high school student. It also represents a nice little slice of history. Warren Weaver was an adviser of and co-author with Claude Shannon, the founder of modern information theory.

These are listed in the order I'd recommend reading them. Also, I've purposely recommended older editions since they're much cheaper and still as good as newer ones. If you want the latest edition of some book, you can search for that and get it.

The Humongous Book of Basic Math and Pre-Algebra Problems https://www.amazon.com/dp/1615640835/ref=cm_sw_r_cp_api_pHZdzbHARBT0A


Intermediate Algebra https://www.amazon.com/dp/0072934735/ref=cm_sw_r_cp_api_UIZdzbVD73KC9


College Algebra https://www.amazon.com/dp/0618643109/ref=cm_sw_r_cp_api_hKZdzb3TPRPH9


Trigonometry (2nd Edition) https://www.amazon.com/dp/032135690X/ref=cm_sw_r_cp_api_eLZdzbXGVGY6P


Reading this whole book from beginning to end will cover calculus 1, 2, and 3.
Calculus: Early Transcendental Functions https://www.amazon.com/dp/0073229733/ref=cm_sw_r_cp_api_PLZdzbW28XVBW

You can do LinAlg concurrently with calculus.
Linear Algebra: A Modern Introduction (Available 2011 Titles Enhanced Web Assign) https://www.amazon.com/dp/0538735457/ref=cm_sw_r_cp_api_dNZdzb7TPVBJJ

You can do this after calculus. Or you can also get a book that's specific to statistics (be sure to get the one requiring calc, as some are made for non-science/eng students and are pretty basic) and then another book specific to probability. This one combines the two.
Probability and Statistics for Engineering and the Sciences https://www.amazon.com/dp/1305251806/ref=cm_sw_r_cp_api_QXZdzb1J095Y1


Differential Equations with Boundary-Value Problems, 8th Edition https://www.amazon.com/dp/1111827060/ref=cm_sw_r_cp_api_sSZdzbDKD0TQ9



After doing all of the above, you'd have the equivalent most engineering majors have to take. You can go further by exploring partial diff EQs, real analysis (which is usually required by math majors for more advanced topics), and an intro to higher math which usually includes logic, set theory, and abstract algebra.

If you want to get into higher math topics you can use this fantastic book on the topic:

This book is also available for free online, but since you won't have internet here's the hard copy.
Book of Proof https://www.amazon.com/dp/0989472108/ref=cm_sw_r_cp_api_MUZdzbP64AWEW

From there you can go on to number theory, combinatorics, graph theory, numerical analysis, higher geometries, algorithms, more in depth in modern algebra, topology and so on. Good luck!

I'm a math tutor and I use these books with almost all my students. They go into pretty good detail about the why's and have quite challenging problems. They include chapter tests, chapter reviews, tons of word problems, challenging multiple SAT-type questions every other chapter, and cumulative reviews. They also thoroughly prepare you for every calculus topic, as well as probability and statistics. They cover Algebra through Pre-calc:

Algebra

Geometry

Algebra 2

Precalculus

To supplement those, you could also use this British math series. It should fill in any possible gaps or clarify certain topics:

9th grade

10th/11th grade

12th grade

Disclaimer: I'm an engineer, not a mathematician, so take my advice with a grain of salt.

Early in my grad degree I wanted to master probability and improve my understanding of statistics. The books I used, and loved, are

DeGroot, Probability and Statistics

Rozanov, Probability Theory: A Concise Course

The first is organized very well, with ever increasing difficulty and a good number of solved problems. I also appreciate that as things start to get complicated, he also always bridges everything back to earlier concepts. The books also basically does everything Bayesian and Frequentist side by side, so you get a really good idea of the comparison and arbitraryness.

The second is a good cheap short book basically full of examples. It has just enough math flavor to be mathier, without proofing me to death.

Also, if you're really just jumping into the subject, I would recommend some pop culture math books too, e.g.,

Paulos, Innumeracy

Mlodinow, The Drunkards Walk

Have fun!

Measure, Integral and Probability Pretty good book. As an undergrad I used that for the third in a sequence for analysis. The first was real in one variable with the first 9 chapters of Bartle, and the second was multi-dim with Spivak's little white book.

Introduction to Probability and Mathematical Statistics This is another senior level math book, but can be used as a first year intro grad level book. It seems people either love it or hate it. I really enjoyed it.

A first Course in Probability This is the standard book. Personally, I hate it. But, you can't love all the books!

Hope that helps.

I'm from a social science background and, like you, I often find myself hopelessly lost when it comes to what feels like very basic concepts in statistics. I think that's partly due to how statistics is taught in all non-mathematics disciplines - in theory we're taught how to use and evaluate quite complex statistical procedures, but with only 1-2 hours per week teaching, it's impossible for our lecturers to cover the fundamental building blocks that help us to understand what's actually going on.

Because of this, I've recently started a few MOOCs on Coursera, and I've found these massively helpful for covering research methods and statistics in far more depth than my undergraduate and postgraduate lecturers ever had time to delve into. In particular, a couple of courses I'd recommend are:

• Methods and Statistics in Social Sciences - This is particularly focused on quantitative methods in the social sciences (including quite a bit on behavioural and self-report research) so I'm not sure if it will be directly relevant with respect to neuroimaging and cognitive neuroscience, but this gives a great introduction to research methods in general. I've actually only done the first course in this series (Quantitative Research Methods), but they're very comprehensive and well made, so I'm confident that the whole series will be useful for any researcher.
  
• Probability and statistics: To p or not to p? - This one is a little bit more maths-heavy so might be a bit intimidating if you don't find that sort of material easy, but it's a good introduction to some of the core concepts in quantitative research, including some you mentioned (e.g. probability distributions). You don't really have to fully engage with or grasp the maths for it to be useful either.
  
  In terms of textbooks, I personally use Andy Field's Discovering Statistics Using R, and find that very helpful. Field is a psychologist who is very open about his difficulties with learning statistics, and I've found it quite useful and re-assuring to learn from someone with that mindset. He's also tried writing a statistics textbook in the form of a graphic novel, An Adventure in Statistics: The Reality Enigma, so if that sounds like something that might help you, check it out.
  
  I think a few people from a 'purer' statistics background are a bit more critical about Field's books because they're not as comprehensive as a book written by, for example, a statistics professor - and there might be some advice in there that's a little bit out-of-date or not quite correct. He also has a very hit-and-miss cheesy sense of humour, which you'll either love or find very annoying. But I think he takes the right sort of approach for helping people who aren't necessarily mathematically-inclined to dip their toes into the world of statistics.

Here is Occupational Health Psychology: https://www.amazon.com/Handbook-Occupational-Health-Psychology-Second/dp/1433807769/ref=sr_1_2?crid=23K4PM6UI8F10&keywords=handbook+of+occupational+health+psychology&qid=1574832541&sprefix=handbook+of+occupation%2Caps%2C198&sr=8-2

​

Here is also a great stats textbook: https://www.amazon.com/Discovering-Statistics-Using-IBM-SPSS/dp/1526436566/ref=sr_1_1?keywords=andy+field+statistics&qid=1574833320&sr=8-1

The same author also has a interesting version of a stats book: https://www.amazon.com/Adventure-Statistics-Reality-Enigma/dp/1446210456/ref=sr_1_3?keywords=andy+field+statistics&qid=1574833347&sr=8-3

> Or that communism creates starvation (joke)

I don't think this is a joke. While causal designs would be difficult to apply, the spatio-temporal correlation is hard to ignore.


>Regarding causality- as you know that’s nearly impossible to prove in the social sciences.

Actually, these days the application of designs and approaches that provide strong support for causal claims have become quite prevalent. Some standard references-



1

2

3

4

good framework reference or a slightly heavier read

and the old classic


In fact, the Nobel prize in economics this year went to some people who have built their careers doing exactly that

It's actually become quite hard to publish in ranking journals in some fields without a convincing (causal) identification strategy.


But we digress.


>We will never be able to do an apples to apples study between heterosexual and homosexual child rearing for some of the reasons you mentioned above. (Diversity of relationship styles, not both biological parents within gay/lesbian couples)

In this case it isn't far fetched at all. The data collection for the survey data used in the study you linked could just as easily have disagregated the parents involved in same sex romantic relationships instead of pooling them. If I understood correctly, the researcher had obtained the data as a secondary source, so they didn't have control over this.

Outcomes for children in the foster care system are well studied, so one could in principal easily replicate the study comparing outcomes between children in the foster care system and those adopted into homes shared by stable same sex couples (you couldn't likely restrict it to married same sex couples, though, because laws permitting same sex civil marriage are too recent to observe outcomes).

>My bottom line-that I don’t see many disagree with if they are being intellectually honest, is a stable monogamous heterosexual family structure is the best model for immediate families. Or would you disagree?

But that's not the question at hand, is it? What we are interested in here is comparing kids bouncing around the state care system to those adopted into homes with two same-sex parents in a stable relationship.

That is exactly my point. The comparison you propose is uninformative relative to the question of permitting same sex couples to "foster to adopt". Because the counterfactual for those children is not likely to be a "stable monogamous heterosexual family". It is bouncing around the foster care system.

I used Carothers in my Real Analysis class, but if you're looking for a readable, introductory book which doesn't skip any steps, Bruckner is a great, free choice. I used this textbook in an Advanced Calculus class and really enjoyed reading it.

That's pretty cool. Unfortunately finding geodesics is a pain because you end up trying to solve evil nonlinear systems of differential equations. This is a great book if you're interested in learning some more about calculus of variations. If you have any questions I can try to answer them.

1. Don't be disappointed in yourself. Give yourself time - you'll get there. If I can do it - and I'm a math teacher now - you can.
   
   
2. I know you asked for web sites, but I want to mention this. It's less than 13 bucks.
   The Humongous Book of Basic Math and Pre-Algebra Problems: Translated for People Who Don't Speak Math

Depending on how strong your math/stats background is you might consider Statistical Inference by Casella and Berger. It's what we use for our first year PhD Mathematical Statistics course.

That might be a little too difficult if you're not very comfortable with probability theory and basic statistics. If you look at the first few chapters on Amazon and it seems like too much I recommend Mathematical Statistics and Data Analysis by Rice which I guess I would consider a "prequel" to the Casella text. I worked through this in an advanced statistics undergrad course (along with Mostly Harmless Econometrics and the Goldberger's course in Econometrics).

Let's see, if you're interested in Stochastic Models (Random Walks, Markov Chains, Poisson Processes etc), I recommend Introduction to Stochastic Modeling by Taylor and Karlin. Also something I worked through as an undergrad.

Usual hierarchy of what comes after what is simply artificial. They like to teach Linear Algebra before Abstract Algebra, but it doesn't mean that it is all there's to Linear Algebra especially because Linear Algebra is a part of Abstract Algebra.

Example,

Linear Algebra for freshmen: some books that talk about manipulating matrices at length.

Linear Algebra for 2nd/3rd year undergrads: Linear Algebra Done Right by Axler

Linear Algebra for grad students(aka overkill): Advanced Linear Algebra by Roman

Basically, math is all interconnected and it doesn't matter where exactly you enter it.

Coming in cold might be a bit of a shocker, so studying up on foundational stuff before plunging into modern math is probably great.

Books you might like:

Discrete Mathematics with Applications by Susanna Epp

Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers

Building Proofs: A Practical Guide by Oliveira/Stewart

Book Of Proof by Hammack

Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand et al

How to Prove It: A Structured Approach by Velleman

The Nuts and Bolts of Proofs by Antonella Cupillary

How To Think About Analysis by Alcock

Principles and Techniques in Combinatorics by Khee-Meng Koh , Chuan Chong Chen

The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!) by Carol Ash

Problems and Proofs in Numbers and Algebra by Millman et al

Theorems, Corollaries, Lemmas, and Methods of Proof by Rossi

Mathematical Concepts by Jost - can't wait to start reading this

Proof Patterns by Joshi

...and about a billion other books like that I can't remember right now.

Good Luck.

What is your background?

http://www.amazon.com/Statistical-Inference-George-Casella/dp/0534243126
Is a fairly standard first year grad textbook with I quite enjoy. Gives you a mathematical statistics foundation.

http://www.amazon.com/All-Statistics-Concise-Statistical-Inference/dp/1441923225/ref=sr_1_1?ie=UTF8&s=books&qid=1278495200&sr=1-1
I've heard recommended as an approachable overview.

http://www.amazon.com/Modern-Applied-Statistics-W-Venables/dp/1441930086/ref=sr_1_1?ie=UTF8&s=books&qid=1278495315&sr=1-1
Is a standard 'advanced' applied statistics textbook.

http://www.amazon.com/Weighing-Odds-Course-Probability-Statistics/dp/052100618X
Is non-standard but as a mathematician turned probabilist turned statistician I really enjoyed it.

http://www.amazon.com/Statistical-Models-Practice-David-Freedman/dp/0521743850/ref=pd_sim_b_1
Is a book which covers classical statistical models. There's an emphasis on checking model assumptions and seeing what happens when they fail.

Book is An Adventure in Statistics: The Reality Enigma: https://www.amazon.com/Adventure-Statistics-Reality-Enigma/dp/1446210456

For a book that's less likely to defeat the reader (because even Baby Rudin is very tough), I'd like to recommend N. (Neal) L. Carothers - Real Analysis.

Metric spaces and Lebesgue measure on the real line only, but quite well written. Honestly, most of what's in there (on metric spaces) should probably be covered in your next analysis course, but owning more good books never hurt, right?

hey nerdinthearena,

i too find this area to be fascinating and wish i knew more on the upper end myself. i'm just going to list off a few resources. in my opinion, graduate school will concentrate a lot on progressing your technical knowledge, but will likely not give you a lot of time to hone your intuition (at least in the first few years). so, the more time you spend in undergraduate school doing so, the better.

helpful for intuition and basic understanding

• geometrical vectors by gabriel weinreich - why there's more to vectors and how to draw them. this basically shows visual examples of differential forms and the like.
  
• advanced calculus: a differential forms approach by harold edwards - a fantastic approach to differential forms. it's a treasure, so just read it. :) it is actually well built for skimming and skipping around.
  
• vector calculus, linear algebra, and differential forms: a unified approach - this is a big book, but you could just jump ahead to the later sections covering differential forms for an interesting approach and technical skill building.
  
• div, grad, curl are dead by william burke - pure intuition and similar to geometrical vectors. don't look for anything else here, but definitely read it.
  
• applied differential geometry by william burke - a more complete work than the above and showcases a lot of the connections between geometry and physics.
  
  

  more advanced but still intuitive
  

• an introduction to manifolds - this is at the same level and covers essentially the same material as lee, but in my opinion, it is much better. it is much more concise and to the point with more focused exercises.
  
• differential forms in algebraic topology by raoul bott and loring tu - the sequel to the above, but i haven't read beyond the first chapter. it is well known as a must read classic, but having some topology background will help with this book.
  
• the laplacian on a riemannian manifold: an introduction to analysis on manifolds - an interesting introduction to geometric analysis.
  
• differentiable manifolds: forms, currents, harmonic forms by georges de rham - a classic by one of the greats and also one of the only sources i've found that covers de rham currents (besides research papers that just use them).
  
• riemannian manifolds: an introduction to curvature by john lee - the sequel to lee's smooth manifolds book.
  
  hopefully this helps. if i were to revisit geometric analysis, i would basically use the above books to help bone up my understanding, intuition, and technical skill before moving on. these are also mainly geometry books, so learning analysis (like functional analysis) would be good as well. i mainly have three suggestions there.
  
  

  three general analysis favorites
  

• introductory functional analysis with applications by erwin kreyszig - used everywhere.
  
• calculus of variations by gelfand and fomin
  
• linear differential operators by cornelius lanczos - a geometrical look at the subject. also take a look at the variational principles of mechanics by him as well.

For Variational Calculus, the best references are Landau and Lifchitz and Gelfand and Fomin. The former is really a mechanics book that incorporates variational calculus in a very rigorous manner that one would expect from a theoretical physicist. The latter is a straight-up variational calculus book. Both are relatively cheap (you can find landau for cheaper than the amazon price).

For non-commutative geometry, there is this classic paper. /u/hopffiber gave the classic references for the rest of the topics, although you should think about learning quantum field theory since all the applications of Lie algebras come from QFT and String Theory. There are some excellent notes by David Tong that you can find with google-fu.

> mi posti qualche link come si deve?

Basi: http://www.amazon.com/Introduction-Probability-Theory-Paul-Hoel/dp/039504636X

http://www.amazon.co.uk/Introduction-Statistical-Theory-Houghton-Mifflin-Statistics/dp/0395046378/ref=sr_1_1?ie=UTF8&qid=1407165301&sr=8-1&keywords=hoel+port+stone

(pigliali usati che nuovi costano troppo)

Poi, mi dicono che questo e' un corso di base, ma pratico e accessibile

https://www.coursera.org/specialization/jhudatascience/1

• topology: Introduction to Topology and Modern Analysis by G. Simmons
  
• real analysis: A First Course in Real Analysis by Protter and Morrey
  
• complex analysis: Theory of Functions of a Complex Variable by A.I. Markushevich
  
• differential equations: Ordinary Differential Equations by V.I. Arnold
  
• linear algebra: Linear Algebra by G. Shilov
  
• calculus: Calculus with Applications and Computing (vol.1) by P. Lax
  
• probability: Introduction to Probability Theory by Hoel, Port and Stone

Imbens and Rubin have a relatively new book:

Causal Inference for Statistics, Social, and Biomedical Sciences: An Introduction

http://www.amazon.com/Causal-Inference-Statistics-Biomedical-Sciences/dp/0521885884

Razanov's Probability Theory was nice. It certainly won't cover everything you need to learn, but it requires the reader to put a great deal of effort into processing each proof, which helped me a lot with retention and application.

edit: Introductory Econometrics for Finance is great. I'd accidentally given you the wrong book earlier. Sorry!

edit #2: this is entirely unrelated but I'm reading it now and felt like sharing.

Start with this question, "What is PDF of the second largest of 5 iid normal RVs?" If you can answer that, then wiki, below, will make sense. (Sorry, on Mobile)

https://en.m.wikipedia.org/wiki/Order_statistic

If not, then the following book is the starting text for most mathematically trained statisticians. They derive the distributions there.

https://www.amazon.com/Introduction-Probability-Mathematical-Statistics-Duxbury/dp/0534380204

If you want to get to the bottom of the comparison, look up the PDF of trimmed/truncated means (means without outliers) and compare them to the usual sample mean. One should appear closer to the true mean. Keep in mind that everything changes as you change assumptions.

You could easily do this exercise with normal RVs. Assuming a second moment in this analysis would be a hard homework problem. Using distributions with many outliers (those without a second moment) should be looked up since it might not be that easy to answer.

If you take a look at the cover of this book you will see an ellipse (you can imagine this as a point cloud) and two lines running through the ellipse- a solid line, and a dotted line.

By eye, you may think the dotted line seems to cut through the ellipse the best, but the solid line is actually the regression line.

Imagine that you have an x-value, and you want to predict the corresponding y-value. The solid line is the best for this prediction. If you draw a vertical line anywhere on the graph, (fixing x), you will see that if you consider the intersection of the x-line with the ellipse, half of the intersection is above the solid line, and half below. The dotted line here does not fit as well. At the x-extrema of the ellipse, drawing a vertical line will place most of the intersection above or below the dotted line.

The assumptions here is that your x value has no error, and the whole shape of the ellipse, or the variation in y, comes from noise.

If you repeat the whole thing, but instead fix y, and draw horizontal lines, and consider the intersection with the ellipse, you are now attempting to predict an x from a fixed y. Now, the solid line is abysmally bad, but the dotted line is ok, but not the best possible line.

The dotted line is the major axis regression, and it is the line that both predicts x best from y, and y best from x.

Mathematical Statistics and Data Analysis.

I learned from this textbook and have found it quite good. It's pretty expensive, but may be what you're looking for. I really don't know how much statistics your classes covered, but the table of contents should give you a good idea on what to expect.

I also had success with cheap supplemental books from Dover, which can cover quite a lot of undergraduate statistics at an affordable price. I found good use in Statistical Inference by Rohatgi.

I like Faraway as an introduction, it's very approachable.

Highly recommend this one on the history of Stampede Wrestling.

https://www.amazon.com/Pain-Passion-History-Stampede-Wrestling/dp/1550227874

There’s also one on the history of the NWA I’ve heard is a bit dry but informative.

http://www.amazon.com/Applied-Regression-Analysis-Generalized-Linear/dp/0761930426

http://www.amazon.com/Regression-Categorical-Dependent-Quantitative-Techniques/dp/0803973748

http://www-stat.stanford.edu/~tibs/ElemStatLearn/

http://www-bcf.usc.edu/~gareth/ISL/

http://www.amazon.com/Extending-Linear-Model-Generalized-Nonparametric/dp/158488424X/ref=sr_1_2?s=books&ie=UTF8&qid=1380716057&sr=1-2

http://www.amazon.com/Generalized-Edition-Monographs-Statistics-Probability/dp/0412317605

hope it helps man. good luck. im learning this stuff for my project too

I actually just read the proof of the Cramer-Rao lower bound the other night in Bain and Englehardt which made a lot of sense to me. It's on page 305.

Edit: I'd be happy to talk about that proof with you if you like. You might also read about the score function. (Fisher Information is the variance of the score function.)

Have you had a rigorous course on Analytical Mechanics? You will learn all about Noether's theorem there. How does Noether's theorem relate to charge conservation? For ANY continuous symmetry of the Lagrangian, we observe an associate Noether current made up of Noether charge. A continuous symmetry of the lagrangian is a symmetry that is generated by that lagrangian's Lie group. For example, the Lie group associated with electricity and magnetism is U(1). U(1) is the unitary group in 1 dimension and represents complex rotations in 1D. This is equivalent to SO(2), the 2 dimensional rotations in real space. If you apply this symmetry to the electromagnetic lagrangian using the proper covariant derivatives, you will obtain an associated four-current density that contains terms relating standard electrical current density. As for your question about special relativity and local gauge invariance. Strictly speaking, special relativity only has a continuous global symmetry, the poincare group, which is made up of the lorentz group (spacetime boosts, spacetime rotations) with the addition of spacetime translations. Jumping back to electricity and magnetism, enforcing local gauge invariance requires that the photon is massless. This is a definitely important for special relativity because the photon is assumed to be as such; only massless particles can keep pace with light. Neat info: because of the link to this gauge symmetry, you can actually experimentally verify charge conservation by measuring a zero mass photon. If the photon were massive, then the gauge symmetry is destroyed and you lose your conserved current. This is why you must have local charge conservation. No local charge conservation => massive photon => speed of light is not an invariant quantity.

Edit: Here is a link that has some information about the lorentz group. I wanted to mention the the four connected subgroups in my original post but didn't want to drone on. From them, you can derive the CPT symmetries and so forth.

http://en.wikipedia.org/wiki/Lorentz_group


Edit 2: Here is my favorite book on the topic of calculus of variations. This theoretical machinery is the foundation for mechanics, and really, your most important tool in theoretical physics. With it, you derive all of the fun contained in Noether's theorem. It is my opinion that no physics student should be without a copy of Weinstock's book.

http://www.amazon.com/Calculus-Variations-Applications-Physics-Engineering/dp/0486630692

Edit 3: Last one, I promise haha. Here is my other favorite, if you are interested in cutting your teeth in a more mathematically rigorous way. Also an excellent book on the topic, it contains a lot the the other book is missing. I want to say that Weinstock doesn't cover the calculation of the second variation(and beyond), which you use to prove that your extremized functional is a minimum or maximum.


http://www.amazon.com/Calculus-Variations-Dover-Books-Mathematics/dp/0486414485/ref=pd_bxgy_b_img_y

Ah yes, if you ever get stuck there's a lovely book to consult along the way!

I would try Mathematical Statistics and Data Analysis by Rice. The standard intro text for Mathematical Statistics (this is where you get the proofs) is Wackerly, Mendenhall, and Schaeffer but I find this book to be a bit too dry and theoretical (and I'm in math). Calculus is less important than a thorough understanding of how random variables work. Rice has a couple of pretty good chapters on this, but it will require some mathematical maturity to read this book. Good luck!

Hello! I'm interested in trying to cultivate a better understanding/interest/mastery of mathematics for myself. For some context:

 




To be frank, Math has always been my least favorite subject. I do love learning, and my primary interests are Animation, Literature, History, Philosophy, Politics, Ecology & Biology. (I'm a Digital Media Major with an Evolutionary Biology minor) Throughout highschool I started off in the "honors" section with Algebra I, Geometry, and Algebra II. (Although, it was a small school, most of the really "excelling" students either doubled up with Geometry early on or qualified to skip Algebra I, meaning that most of the students I was around - as per Honors English, Bio, etc - were taking Math courses a grade ahead of me, taking Algebra II while I took Geometry, Pre-Calc while I took Algebra II, and AP/BC Calc/Calc I while I took Pre-Calc)

By my senior year though, I took a level down, and took Pre-Calculus in the "advanced" level. Not the lowest, that would be "College Prep," (man, Honors, Advanced, and College Prep - those are some really condescending names lol - of course in Junior & Senior year the APs open up, so all the kids who were in Honors went on to APs, and Honors became a bit lower in standard from that point on) but since I had never been doing great in Math I decided to take it a bit easier as I focused on other things.

So my point is, throughout High School I never really grappled with Math outside of necessity for completing courses, I never did all that well (I mean, grade-wise I was fine, Cs, Bs and occasional As) and pretty much forgot much of it after I needed to.

Currently I'm a sophmore in University. For my first year I kinda skirted around taking Math, since I had never done that well & hadn't enjoyed it much, so I wound up taking Statistics second semester of freshman year. I did okay, I got a C+ which is one of my worse grades, but considering my skills in the subject was acceptable. My professor was well-meaning and helpful outside of classes, but she had a very thick accent & I was very distracted for much of that semester.

Now this semester I'm taking Applied Finite Mathematics, and am doing alright. Much of the content so far has been a retread, but that's fine for me since I forgot most of the stuff & the presentation is far better this time, it's sinking in quite a bit easier. So far we've been going over the basics of Set Theory, Probability, Permutations, and some other stuff - kinda slowly tbh.

 




Well that was quite a bit of a preamble, tl;dr I was never all that good at or interested in math. However, I want to foster a healthier engagement with mathematics and so far have found entrance points of interest in discussions on the history and philosophy of mathematics. I think I could come to a better understanding and maybe even appreciation for math if I studied it on my own in some fashion.

So I've been looking into it, and I see that Dover publishes quite a range of affordable, slightly old math textbooks. Now, considering my background, (I am probably quite rusty but somewhat secure in Elementary Algebra, and to be honest I would not trust anything I could vaguely remember from 2 years ago in "Advanced" Pre-Calculus) what would be a good book to try and read/practice with/work through to make math 1) more approachable to me, 2) get a better and more rewarding understanding by attacking the stuff on my own, and/or 3) broaden my knowledge and ability in various math subjects?

Here are some interesting ones I've found via cursory search, I've so far just been looking at Dover's selections but feel free to recommend other stuff, just keep in mind I'd have to keep a rather small budget, especially since this is really on the side (considering my course of study, I really won't have to take any more math courses):
Prelude to Mathematics
A Book of Set Theory - More relevant to my current course & have heard good things about it
Linear Algebra
Number Theory
A Book of Abstract Algebra
Basic Algebra I
Calculus: An Intuitive and Physical Approach
Probability Theory: A Concise Course
A Course on Group Theory
Elementary Functional Analysis

http://www.amazon.com/Introduction-Probability-Dimitri-P-Bertsekas/dp/188652940X

Chapter 4 and beyond should be a good point to start from where you leave off. There's important stuff like convergence and then stochastic processes. You shouldn't need an analysis or measure theory class to understand the material but it would be helpful.

How would you compare this with Freedman's Statistical Models? (https://www.amazon.com/Statistical-Models-Practice-David-Freedman/dp/0521743850)

I enjoyed

Introduction to Probability Theory, Hoel et. al

Also,

Probability Theory, Jaynes

is essential. For probabilistic programming I would also look into

Bayesian Methods for Hackers

> Is your claim that, as you put it, "causal" algebraic equations are in fact not algebraic, and do not conform to the rules of algebra? Can you point me to your source of this new math?

I'm sorry, but I cannot stress this enough. It wasn't ME who developed the use of math to describe causal relationships. After reading the linked wikipedia pages, you're going to have to take that up with the relevant mathematicians/engineers/scientists, etc.

> Can you point me to your source of this new math?

SURE THING. Here's another wikipedia page. But just to ask, I'm curious why you want to use the word "new" for this, given that the wikipedia page cites the standard textbook written in the year 2000, (16 years ago). In fact, by now the concept is so old and established that there are textbooks of it

> The lunacy in your statements is highlighted by the fact that algebraic equations are bidirectional "causal", this is the point of the equal sign. Please read about the equal sign.

See the wikipedia page for causality and then get back to me. Here is a relevant piece of text:

• Nobel Prize laureate Herbert A. Simon and philosopher Nicholas Rescher claim that the asymmetry of the causal relation is unrelated to the asymmetry of any mode of implication that contraposes. Rather, a causal relation is not a relation between values of variables, but a function of one variable (the cause) on to another (the effect). So, given a system of equations, and a set of variables appearing in these equations, we can introduce an asymmetric relation among individual equations and variables that corresponds perfectly to our commonsense notion of a causal ordering.
  
  I highlighted the author's name. That way there should be no confusion about WHO published WHAT (hint: I wasn't me, I'm just a guy who learned how to use causal modeling in college, and who occasionally uses it on the job.).
  
  If using math to model causality is something you personally do not believe has any merit (for whatever personal reasons), that is your own prerogative. I get that you want to pretend this specific use of math doesn't real, in the same way that you said earlier that traders using models for trading doesn't real (I guess it means that in our opinion, the use of proprietary trading models is ALSO not a thing. right?). But yeah, modeling causal relationship is a real thing. And you can rant about it on the internet all day long I suppose, but if you ever go to the doctor, you might find out that causal modeling is used not just in finance, but also in medical sciences.
  
  
  > The lunacy in your statements
  
  DO you mean the part where I cite the research and publication of others? Did you write to any of those authors yet to tell to share any sort of evidence which can challenge anything they've published? I encourage you to read up. So far this conversation is all ME sharing sources and you shouting "doesn't real" (while not being able to produce any sources whatsoever to reinforce that view).
  
  

1. Lies, damned lies, and statistics
   
   https://en.m.wikipedia.org/wiki/Lies,_damned_lies,_and_statistics
   
   
2. https://www.amazon.com/Lies-Damn-Statistics-Manipulation-Opinion/dp/0393331490

A bunch of fairy land assumptions counts as science these days? How's lowering tax rates working out for Kansas?


https://www.amazon.com/Lies-Damn-Statistics-Manipulation-Opinion/dp/0393331490

Here's a good book to start with:

https://www.amazon.com/Introduction-Probability-Statistics-William-Mendenhall/dp/1133103758/ref=sr_1_2?ie=UTF8&qid=1487730146&sr=8-2&keywords=introduction+to+statistics+textbook

If you do better with physical materials than online reading, it sounds like renting a textbook would be the way to go for you? Shop around for yourself, but something like this might be good since it covers Algebra 2 and Trig, and renting instead of buying usually only costs around $20. Just remember that if you get stuck or need help there's a lot of resources online too.

Would online lecture videos potentially be helpful for you? Since they're audio visual rather than written, I'm curious if that might be better than learning through reading. Professor Leonard is a personal favorite of mine, although I have to adjust the speed to x1.5 because I feel like he talks really slowly. He has lectures for Algebra, Stats, Precalc, all the way up to Multi Variable Calculus.

As for Kahn Academy, it's definitely more than a lesson supplement, if you wanted to you could learn of all of math from the basics up to college level without having to step foot in a classroom. I personally enjoy it because the online practice problems are much more fun to me than written homework, since you get instant feedback if you get an answer wrong, and help to see how to correct it.

I'll acknowledge that some of the practice problems on Kahn Academy don't get as rigorously difficult as what you might come across in a textbook, as the goal is more to make sure you understand the basic concepts than to test the limit of your problem solving skills.

If you did want to give Kahn a try, then I'd recommend to start from the beginning in Algebra. It should be easy initially since you already completed Algebra 1, but it will also serve as a good review. On the KA home page click on "Subjects" in the upper left corner and select "Algebra." Near the top it should say "Explore" "Classes" "Practice" and "Mission".

Select "Practice," and then start with the first problems and work your way through. If it's too easy initially, then scroll down to the section titled "Functions," as this will be closer to where you left off in Algebra 1. And like I said, you can probably skip the video lessons, and just use the "hint" button, but the relevant videos are also linked with every practice problem if you do get stuck or don't understand something.

Three very good introductory textbooks for statistics are:

Mendenhall, Statistics https://www.amazon.com/Introduction-Probability-Statistics-William-Mendenhall/dp/1133103758

Larson and Farber https://www.amazon.com/Elementary-Statistics-Picturing-World-Books/dp/0321901118

Peck and Devore https://www.amazon.com/Statistics-Exploration-Analysis-Available-Titles/dp/0840058012

Buy used, older editions of each, and you can get them very cheap. These are all well explained, beginner texts.

First year MS student, currently am completely lost in my course. Been hanging on my HW problems by the edge of my seat and its just isn't enough.

I'm not sure if it is because I am from life sciences background (currently have an MS in Biochemistry) but I have not been able to follow along in my lectures and the textbook we are using has been using examples that aren't very explicit or explaining what is going on.

We are using Introduction to Probability and Mathematical Statistics 2nd Edition and I'm not sure whether this textbook is the norm for an beginning level graduate course in Statistics or what.

I am willing and struggling to learn statistics, but I think there is a gap in my math knowledge and this book. Is there anything someone could recommend to me as a supplement, with ample and detailed examples that I can practice?

Hey, a probability and geometry book is on sale as well.

Then there's "Mathematics for Non-mathematicians"

I just finished a newly released book about Brian Pillman. Highly recommend it - one of the best I've ever read. To go with that, I'd add Bret's book as well as Pain & Passion (book about Stampede Wrestling).

Thank you! This is exactly what I was looking for!!!! I didn't think anyone was going to give me a sufficient reply because there are a lot of books (sorry), but this is what I wanted. Where would you place the two books I linked, Principles and Techniques in Combinatorics and Introduction to Combinatorial Mathematics, Liu, in that list or would you consider studying them a redundant exercise? I also did not include this book in the list, but where would you place Problems from the Book and its accompanying Straight from the Book?

I will likely end up replacing the Graph Theory book I have in the list, by Berge, with Modern Graph Theory by Bellobas, since Berge doesn't have exercises, but I will assume it stays in the same order of the sequence.


I apologize for not initially including them. I did not realize that I did not. Also, are there any other topics you would recommend I cover for establishing a solid foundation. I didn't buy Rudin's Complex Analysis because I didn't know if that kind of thing was necessary. I don't even know what other branches of mathematics Complex Analysis relates to. There could be other topics I'm not aware of as well. Please don't hesitate to make more recommendations. I appreciate it.

http://www.amazon.com/Lady-Luck-Theory-Probability-Mathematics/dp/0486243427 This is the one.

ISBN: 9780395977255
Title: Algebra and Trigonometry, Grades 10-12 Structure and Method Book 2: Mcdougal Littell Structure & Method
Author: Houghton Mifflin Company
Copyright: 1/26/99
Publisher: HBC
Amazon link: https://www.amazon.com/dp/0395977258/

Okay, I see where you're coming from.


This is a common mistake. An idea that makes you uncomfortable is not necessarily wrong.


Here is a book that might help.