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Mathematical Logic books:
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Don't tell me I don't support a thing that I support so that you can criticize me for not supporting it, lmao.
You lot aren't too bright, eh? I guess it's easier to argue against us when you just make shit up on the spot. Here's a book for you, since you like reading so much.
you should definitely give harder problems multiple tries, you might not be able to solve them right away. Go back and re-learn the concepts the problem needed. Sometimes you might need to use a concept which you are not familiar with at the moment. I recommend reading "How to Solve it " by G. Polya https://www.amazon.com/How-Solve-Mathematical-Princeton-Science/dp/069116407X
It covers different problem solving approaches in a agnostic fashion.
I highly recommend reading "Mathematical Proofs: A Transition to Advanced Mathematics" by Gary Chartrand et. al. It helped me get a better understanding of how to write a proof as well as organize my own thoughts.
Here's the Amazon link: Mathematical Proofs: https://www.amazon.com/dp/0321797094/ref=cm_sw_r_cp_apa_i_V1UDDb4JBGWFX
A lot of times with a course I was studying there was just something I didn't get and it took a while for it to click, then once it did the rest of the course was cake. I had this problem with exterior derivatives. The text we were using didn't introduce or motivate them well so I ended up in the profs office for an hour once and he helped motivate it and after that the course was no problem.
With proofs there's a big jump to get over and often times the profs don't motivate what you're doing really well. I found a good book on proof technique was helpful because it helped me understand what I was reading. It turns out for most undergrad math you can break a proof down into one of a few patterns and once you know the patterns when you're reading a text you know what to look for and recognize the flow of the proof.
This is a good text on the subject
https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995
Check out the book Descartes Error. He goes into the split-brain studies, and yes, two distinct consciousnesses do seem to form.
I'm 33 and I did not learn coding in school either.
<em>How to Prove It: A Structured Approach, 2nd Edition</em> by Daniel Velleman, is very good. He does reference computer coding, but no background in coding or proofing is required. Literally anyone can pick it up and get started.
A lot of my thinking has very much become similar to an SAT question. "If X is to Y as Y is to Z..." or however those went. There's a book called Bad Arguments worth checking out. People use these all the time, especially in political debates, and it's nice to know them and be able to call them out. Plus, fun illustrations!
I would recommend the book "How To Prove It".
https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995
It helped me in my transition into proof based mathematics. It will teach common techniques used in proofs and provides a bunch of practice problems as well.
This is a great book. Godel Escher Bach... mind expanding is what I would call it.
https://www.amazon.com/G%C3%B6del-Escher-Bach-Eternal-Golden/dp/0465026567
This book looks good:
An Illustrated Book of Bad Arguments: The Lost Art of Making Sense.
> I don't believe anyone has a satisfying answer though.
The question may well be unanswerable. This leads to one of my favorite unanswerable questions: why is there something instead of nothing? (Favorite because anyone who says they've got an answer to this question is almost certainly either delusional or lying... so it's a good litmus test for 'woo'.)
I highly recommend Hofstadter's Godel, Escher, Bach: An Eternal Golden Braid for a very readable exploration on why this is unanswerable.
- Yes, it's possible, although it varies by person.
- Yes - almost any process that can be done by hypnosis can, at least in theory, be done through self-hypnosis.
- The mind is organic, complex, and unpredictable. This might even be structurally built into the study of cognition.
So... nothing is really engraved in a subconscious, because it's constantly changing and it's highly complex. There isn't a function. There's no deactivation switch, because there's no switch in the first place. The mind is not a machine and hypnotists aren't programmers.
Hypnotists are guides. They specialize in navigating some of this very poorly mapped territory. They're often quite good at it. In some cases - like smoking cessation or phobia reduction - they're reliably good at particular functions - so much so that it's published and statistically significant.
Don't let the reliability of some operations fool you, though. The mind isn't a series of mapped switches and mechanical functions, and IMO it never will be. As a result, the reliable answers in spaces like this will generally be frustratingly vague, just because no one can say "yup, I just slap that tear switch and call tech support if it doesn't work."
There is a popular book devoted to this topic entitled "Godel Escher Bach - The Eternal Golden Braid". It is not a particularly Christian book, it was written by a Buddhist mathematician in the 1970's, as he contemplated computers, math, artificial intelligence, and the like.
One of the central ideas of the book involves "Godel's Incompleteness Theorem", which says that you can never prove a "set" from only the data/information inside that set. For example, you cannot prove 2+2=4 unless there is something else, outside of the equation, that proves what a "2" or a "4" even are. Taken to the extreme -- we cannot even prove there is a universe (from inside the universe, as we are), unless there is something outside of the universe to give it some sort of context/meaning. When people say "you can't prove there's a god!", I always say "according to Godel's Theorem, you can't even prove there's a YOU".
In short - my argument would be... if there's a universe, then there "must" be "something" outside of the universe that gives the universe a context... is this "God"? Is it "Ultimate Truth"? This gives new insight into the meaning of the Hebrew name for God -- "I Am".
https://www.amazon.com/Gödel-Escher-Bach-Eternal-Golden/dp/0465026567
Yeah, I've just never been shown a problem where this stuff gives deep insight, and until I see one and understand it these are just gonna be arbitrary definitions that slide right out of my brain when I'm done reading them. I'll definitely give the book a look - is it motivated with examples?
The only book I have on category theory is Conceptual Mathematics: A First Introduction to Categories, and I must say, I'm not a fan of it - too intuitive, not detailed enough, not well organized, not formal enough - should have gone for MacLane instead.
The first required one is Sets and Logic. Usually they use: How to Prove It: A Structured Approach, 2nd Edition https://www.amazon.com/dp/0521675995/ref=cm_sw_r_cp_apa_rE.eybZMYQ8NV If you want to take a peek before you decide to do math.
Check out this book: http://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995/
You can definitely handle it and it'll give you a feel for exactly what being a math major is like. It's a lot of fun too. This material is the basis for all higher math. The things you do in the future are just more specialized versions of this.
If you enjoy math because you like solving puzzles and figuring out how things work, you'll love real math. If you like the idea of constructing arguments, you'll love real math. If you enjoy math because you like applying things that you've memorized and having immediate answers to problems, you probably won't like real math.
The way real math is written and taught is really fantastic. It's nothing to be afraid of once you get the hang of it. Proofs have a fairly standard lingo that's written in an inclusive and relatively unintimidating way. "We'll do this", "Suppose that is true." The math itself can certainly be intimidating but the prose is very welcoming. That might not be as true for professional publications but it's definitely true for almost all learning materials. For learning new things, almost every piece of information comes with a label. Definition, Theorem, Proposition, Corollary, Lemma, Proof, Remark. It looks very robotic at first but it's done that way to help you and it makes it much easier to piece together knowledge. I'm at the point now where, if my professor doesn't give something a label, I assign it one myself.
The stuff you're learning now certainly comes in handy but it's more of a toolbox for doing math rather than math itself.
edit: This is the style guide I used when I was first learning to write proofs, in case anyone is curious where all of that nonsense came from: https://www.math.hmc.edu/~su/math131/good-math-writing.pdf
Two of my favorites:
- Gödel, Escher, Bach: An Eternal Golden Braid. Not precisely about math but it's very logic-driven and most of the arguments D. Hofstadter makes are math related. This book has given me a lot of insight.
- Flatland. Basically a kids book regarding geometry and how do we look at "space" and dimensions. It's simple, yet intriguing.
220 - do all your projects early and don't cheat. this class will teach you how to really "program". start the programs early and you will be fine.
250 - i recommend this book: http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995
this book is pretty good when combined with barrington's "teaching" style. this book doesn't have all the topics covered, but if you find yourself struggling to understand concepts i recommend ordering this book (or find a PDF online of it) and just work through every practice problem you can. answers for a bunch are in the back. i think this book does a better job explaining conceptually what everything means in the class.
good luck!
How to Solve it and The Art and Craft of Problem Solving are good books.
If you want to become better at writing proofs, a book that helped me immensely was the book I used for my introduction to adv. math class.
The whole book is written with extreme clarity (in my opinion) and the proof chapters end with a TON of exercise proofs (for direct, and indirect) that are all easily doable by a novice. Moreover, each odd exercise (even the proofs) have a pretty detailed answer in the back if you get stuck.
I've been working through Mathematical Proofs, started chapter 3 last night. I've started both Dudley's Elementary Number Theory and Spivak's Calculus but have them on hold until I'm through the Proofs book.
I haven’t gotten a chance to look at it but I've heard good things about Mathematical Proofs: A Transition to Advanced Mathematics
The common metaphor is like a sequence of dominoes.
If
(1) You line up all the dominoes so that each one would knock over the next one
and
(2) You knock over the first one
Then they will all fall.
The first one will fall because you knocked it over (2).
The second one will fall because you knocked the first one over (2), and because the first one will knock over the second one (1).
The third one will fall because you knocked over the first one (2), and because the first one will knock over the second one (1), and because the second one will knock over the third one (1).
Mathematically, if you want to prove some statement about every number, say greater than or equal to 0, then you have to prove it about 0 and about 1 and about 2, and so on.
So what you do is you prove that if a statement about each number is true then the statement about the next number is true (1)
And that the statement about zero is true (2).
Then you've proved the statement about all numbers.
You've proved it about 0 from (2)
And you've proved it about 1 because you've proved it about 0 from (2) and because you've proved that if you've proved it about 0 then you've proved it about 1 from (1).
And you've proved it about 2 because you've proved it about 0 from (2) and because you've proved that if you proved it about 0 then you've proved it about 1 from (1), and you've proved that if you proved it about 1 then you've proved it about 2 from (1),
and so on.
For a text, I recommend Velleman
Just pick up an intro to proof book. Velleman's Book is well-regarded, or if you want something free you could read The Book of Proof.
I saw the word entity and started to scroll past the post, but I think this is actually a really interesting theory that makes sense. If you haven't, you should read Descartes' Error. He explores the studies done on blind-sight and split brains, and it does appear that separate, distinct consciousnesses can emerge in one brain under the right conditions (this is entirely different from Dissociative Identity Disorder, which isn't really multiple distinct consciousnesses the way it's popularly portrayed in media).
I've been fascinated by consciousness and the mind/body problem since I was a teen. Never quite found the "answer" (spoiler: no one has), but some of my reading on how human consciousness works inadvertently left me with coping strategies for our mortality.
It's a long, dense read, but I would always recommend Gödel, Escher, Bach as my favorite among these types of books. The follow-up, I Am A Strange Loop is great as well.
You should read Gödel, Escher, Bach if you are looking for the links between math, music, and language.
This book is a little outdated now, but when I was an undergrad philosophy major, this was THE book for the materialist viewpoint: https://mitpress.mit.edu/books/neurophilosophy
I'd also recommend reading https://www.amazon.com/Descartes-Error-Emotion-Reason-Human/dp/014303622X . Again, it's a little dated but should help you understand where most materialists are coming from.
> There is no point in debating true believers of any variety.
That seems tautologically true. Having observed and interacted with John W. Loftus while he was an atheist, I'm guessing you would have characterized his Christian self as a "true believer". Yes? No? If yes, then we could see whether atheists debating with him were any part of his deconversion. If yes, then perhaps it is impossible to tell who is actually a "true believer", making your observation pragmatically iffy.
> Beliefs are not points of debate, they are emotional premises upon which all else is built.
Is this true only of theists, or atheists as well? If only theists, can you point to peer-reviewed science which establishes your point?
> Emotions and reason (the mechanism of debate) have little intersection.
I suggest checking out Antonio Damasio 1994 Descartes' Error. Damasio found that people with brain lesions which disconnected them from their emotions made it very difficult for them to pursue long-term goals in life, while leaving their ability to solve logic puzzles and the like unaffected. Here's how Damasio summarizes his findings:
> When emotion is entirely left out of the reasoning picture, as happens in certain neurological conditions, reason turns out to be even more flawed than when emotion plays bad tricks on our decisions. (xii)
That book stands at 35,000 'citations'. While it might not be 100% right, it also isn't 100% wrong.
Book of Proof by Richard Hammackis available for free as a PDF; it is a great resource.
How to Prove It by Daniel Velleman is not free, but is another great resource.