I would like to learn higher or abstract math for the sake of fun and beauty of it, where could I start?

[oneweekleft]

I don't have a solid background in math. I read, that the mathematician G.H. Hardy pointed out the beauty and asthetics of math and the joy it brings. I want an intellectual challenge and something I can dive into.

I also heard that Bertrand Russell said, that dealing with math helped him to overcome his suicidality and depression as a young man, which is something, that is also bothering me.

So I would really like to get into higher maths, especially a field that involves more abstract concepts, I find that fascinating. I really don't care for usability or applications of math, it could be something, that is not relevant for my life at all.

I'm certainly willing to really dedicate myself to learning math, but I don't want to repeat all the school stuff and feel like a child again.

Could you please tell me what field of maths would meet my expectations: abstraction, beauty (not in the sense of visualisations like geometry, rather elegant formulas) and not too much prior knowledge. And with which book could I start? It really could be a rigorous book, not something like "math for dummies". I would prefer learning from a book rather than videos or apps.

 

[eternalmelancholy]

The most exciting fields of mathematics has to be experimental ones like game theory and nonlinear dynamics. These fields are an attempt to explain the strings that control the system. The so called invisible hand that guides human behavior and the seemingly chaotic nature of the physical world.

You have to be a genius level intellectual to even begin grasping these high level concepts. The equivalent would be like a plumber trying to figure out how to split an atom.

 

[oneweekleft]

The most exciting fields of mathematics has to be experimental ones like game theory and nonlinear dynamics. These fields are an attempt to explain the strings that control the system. The so called invisible hand that guides human behavior and shapes society.

You have to be a genius level intellectual to even begin grasping these high level concepts. The equivalent would be like a plumber trying to figure out how to split an atom.

I just want to do something rather abstract, that is a managable challenge for a person, that is not a genius. It really doesn't have to be useful or give deeper insights or help to understand the world.

 

[eternalmelancholy]

I just want to do something rather abstract, that is a managable challenge for a person, that is not a genius. It really doesn't have to be useful or give deeper insights or help to understand the world.

That is the problem. All the exciting fields are reserved for the top elites. Because you need to wade through decades of education, training and experience to understand abstract concepts.

Experimental fields are the cutting edge of human knowledge and technology. Only brightest minds on the planet are able to pursue these endeavors. I'm sorry to say if you don't have a strong background in mathematics it will be impossible to enter these fields.

To you start you need about 8 years of university and post graduate level education in mathematics. You can't just skip the fundamentals.

 

[oneweekleft]

That is the problem. All the exciting fields are reserved for the top elites. Because you need to wade through decades of education, training and experience to understand abstract concepts.

Experimental fields are the cutting edge of human knowledge and technology. Only brightest minds on the planet are able to pursue these endeavors. I'm sorry to say if you don't have a strong background in mathematics it will be impossible to enter these fields.

To you start you need about 8 years of university and post graduate level education in mathematics. You can't just skip the fundamentals.

Okay, I accept that. Are a lot of fundementals needed for self-studying mathematical analysis or "a pure course of mathematics" by G.H. Hardy, who I mentioned in my first post?

 

[eternalmelancholy]

Okay, I accept that. Are a lot of fundementals needed for self-studying mathematical analysis or "a pure course of mathematics" by G.H. Hardy, who I mentioned in my first post?

I would start by looking up a local university website for their undergrad mathematics program. Look for course descriptions that pique your interest. These will most likely be advanced senior level classes. Then work backwards and see what prerequisite courses are required. Set a plan in motion from where to start and where you want to end up.

This will take some time and research to plan out. It will take years of hard work to follow through but if this is your life's calling you might as well try. Maybe schedule an appointment with an academic advisor for advice.

//edit: I left academia along time ago when I realized that I was simply not smart enough to compete with my peers or fully understand complex concepts. So I might be giving bad advice unintentionally. I haven't used my brain in any serious capacity in a long time and as a result I have regressed into a moron. Not that I was particularly bright to begin with.

 

[Meretlein]

You can take some classes at Arizona State University online. You only have to pay for the class if you want credit for it.

 

[Tempest]

You don't need to be a genius level to study mathematics. To do research, maybe, but that's not what you're after. I'd recommend starting with set theory. Irving Kaplansky's book was what my professor used and I enjoyed it; it's moderately rigorous but still accessible. You might also like abstract algebra. To understand that you might want to study linear algebra first, but honestly you can jump in and see where your gaps in knowledge are and what you want to study further. If you don't already know calculus, I'd recommend studying that. There's a lot that deals with infinity and the infinitesimal, and the process through which people "discovered" calculus is inspiring. They burst through into a whole new echelon of conceptualizing mathematics and the phenomena that mathematics is used to model. Topology is another field that has the potential to evoke awe when you study it. You said you don't prefer videos, but there are a lot of good ones out there about math and related subjects. Vihart is a YouTuber who combines math with aesthetics. Watch her channel and you might find specific topic to pique your interest.

 

[eternalmelancholy]

You don't need to be a genius level to study mathematics. To do research, maybe, but that's not what you're after.

You are right. I only assumed so since I am too incompetent to recognize my own shortcomings. I did not mean to be discouraging.

 

[irememberinnocence]

You can just look up a short course on coursera, edx, or future learn. I actually did a type of maths paper at university (won't get too specific) and loved it, despite not feeling competent in maths beforehand. The patterns and the formulas are indeed beautiful.

 

[Shadowplay]

The most exciting fields of mathematics has to be experimental ones like game theory and nonlinear dynamics. These fields are an attempt to explain the strings that control the system. The so called invisible hand that guides human behavior and the seemingly chaotic nature of the physical world.

You have to be a genius level intellectual to even begin grasping these high level concepts. The equivalent would be like a plumber trying to figure out how to split an atom.

As someone who enjoys study for the sake of it, though not in maths, I do a disagree. Geniuses, in any fielf are rare; that is part of being a genius. But you can still enjoy studying maths without going for the Fields Medal.

@oneweekleft maybe you learn differently to me, but I like learning from podcasts and YouTube. I find they're a good way to dip your toe in the water; they can give you some guidance before you start reading books, or doing online courses; point you in a direction (obviously people are different and hence they learn differently). And as someone studying philosophy good to hear you're inspired by Bertrand Russell. Tell me how you go.

 

[SentientCreature]

Okay, I accept that. Are a lot of fundementals needed for self-studying mathematical analysis or "a pure course of mathematics" by G.H. Hardy, who I mentioned in my first post?

I just took a quick glance at it and it seems accessible, although in my opinion the topics that are covered require more attention than they're given. I'd use this as a reference perhaps after getting a broader understanding from another resource. Michael Spivak's book is often recommended as an introductory textbook. It's proof-based which you'll like as it will allow you to fully understand each theorem.

That's as far as calculus is concerned. Now linear algebra is something that also satisfies your criteria. I'd recommend Stephen Friedberg's book. You could then delve into abstract algebra.

I think you'd also find discrete mathematics interesting. Kenneth Rosen's textbook is a good resource in that case.

It's been a while since I've studied some of these things so I could actually get back and review some of the material myself.

 

[oneweekleft]

Thank you for your answers. I just want to point out, that I really admire the hard work, that mathematicians do. I don't think, that I could just start somewhere and understand everything with ease. Basics are important and I am certainly willing to (re)learn some of them. I just don't want to start from high school math and feel like a stupid child and get the negative assosiciation to my time in school.

If someone cares, why I want to learn math: I don't consider myself as an intelligent person, but I am able to understand some rather complex ideas and read some complicated books, so I hope I'm at least able to grasp some of the stuff. I don't have a lot of feelings and I'm overwhelmed by this world and my very limitited intellectual abilities help myself to deal with this for me hardly bearable world. I'm trying to intellectually defend myself against others, that question my existence. My intellect (and I don't claim that I've got a lot of it) is the only thing I have and that keeps me together. So I want to do something, that occupies me intellectually.

But this rather philosophical thinking disturbs me sometimes, I feel lost, I'm tired of this normative thinking. Talking to people , hear people in the media making judgements, having unfounded opinions, share their beliefs with the whole world and imposing their worldview on others. I want an intellectual field, that is neutral, understandable and reliable. I don't expect any understandings of the world, I just want something to dive into.

I can't take this world and want a place to go, basically escapism. I feel math could do that for me. Maybe that sounds ridiculous to you, but sometimes certain hobbies or occupations can distract you and give you a feeling of self esteem or a reason to live for a while.

 

[SentientCreature]

Basics are important and I am certainly willing to (re)learn some of them. I just don't want to start from high school math and feel like a stupid child and get the negative assosiciation to my time in school.

I totally relate to this feeling as someone who wasted 4 years of their life in high school. It was an overall negative experience for me and I wouldn't lose anything of value if I completely erased that period from my memory. I didn't even gain any knowledge there.

None of the books I recommended require an extensive and systematic review of high school mathematics, you'll be more than fine picking things up on the go, so there's nothing to worry about.

The entire post sums up my experience quite well. I am also easily overwhelmed by the complexity of human social systems and "worldly" phenomena. This is why the simplicity of abstract mathematics appealed to me.

I wish I could stop being chronically fatigued now so I could return to that form of escapism, at least until I obtain all the required supplies for my exit strategy.

 

[irememberinnocence]

Thank you for your answers. I just want to point out, that I really admire the hard work, that mathematicians do. I don't think, that I could just start somewhere and understand everything with ease. Basics are important and I am certainly willing to (re)learn some of them. I just don't want to start from high school math and feel like a stupid child and get the negative assosiciation to my time in school.

If someone cares, why I want to learn math: I don't consider myself as an intelligent person, but I am able to understand some rather complex ideas and read some complicated books, so I hope I'm at least able to grasp some of the stuff. I don't have a lot of feelings and I'm overwhelmed by this world and my very limitited intellectual abilities help myself to deal with this for me hardly bearable world. I'm trying to intellectually defend myself against others, that question my existence. My intellect (and I don't claim that I've got a lot of it) is the only thing I have and that keeps me together. So I want to do something, that occupies me intellectually.

But this rather philosophical thinking disturbs me sometimes, I feel lost, I'm tired of this normative thinking. Talking to people , hear people in the media making judgements, having unfounded opinions, share their beliefs with the whole world and imposing their worldview on others. I want an intellectual field, that is neutral, understandable and reliable. I don't expect any understandings of the world, I just want something to dive into.

I can't take this world and want a place to go, basically escapism. I feel math could do that for me. Maybe that sounds ridiculous to you, but sometimes certain hobbies or occupations can distract you and give you a feeling of self esteem or a reason to live for a while.

I really understand your need for a neutral unbiased hobby. To me the media is completely toxic. Sometimes I like to just immerse myself in philosophy and poetry. And I find your desire to delve into maths refreshing.

 

[oneweekleft]

I just took a quick glance at it and it seems accessible, although in my opinion the topics that are covered require more attention than they're given. I'd use this as a reference perhaps after getting a broader understanding from another resource. Michael Spivak's book is often recommended as an introductory textbook. It's proof-based which you'll like as it will allow you to fully understand each theorem.

That's as far as calculus is concerned. Now linear algebra is something that also satisfies your criteria. I'd recommend Stephen Friedberg's book. You could then delve into abstract algebra.

I think you'd also find discrete mathematics interesting. Kenneth Rosen's textbook is a good resource in that case.

It's been a while since I've studied some of these things so I could actually get back and review some of the material myself.

Just to make sure, you mean this books:

Michael Spivak - Calculus
Amazon product ASIN 0521867444
Stephen H. Friedberg: Linear Algebra
Amazon product ASIN 9332549648
Kenneth Rosen:
Amazon product ASIN 9332549648

 

[SentientCreature]

Just to make sure, you mean this books:

Michael Spivak - Calculus
Amazon product ASIN 0521867444
Stephen H. Friedberg: Linear Algebra
Amazon product ASIN 9332549648
Kenneth Rosen:
Amazon product ASIN 9332549648

Yep, except you linked linear algebra twice :)


Amazon product ASIN 0073383090


That's the one on discrete math, but holy cow it's pricy! I'd just get a free pdf version online.

 

[oneweekleft]

Yep, except you linked linear algebra twice :)


Amazon product ASIN 0073383090


That's the one on discrete math, but holy cow it's pricy! I'd just get a free pdf version online.

I think, there's a cheaper version, I hope that's the same. I'll probably buy one of them, but they should be available at libgen.

Just one question, I prefer serious books and don't mind acadamic approaches to things, as long they are possible to understand without being exceptional smart. I'm skeptical about stuff like "Linear Algebra for Dummies" or that things. I don't need encouraging comments, funny stories or unnecessary informations.

But I stumbled upon a book called "No Bullshit guide to Linear Algebra". The title put me off at first, but it seems to include some basic knowledge, that I'm missing and is probably not written too jovial, rather written pretty concisely. What do you think about it?

Here's a preview, if you would be so kind to take a look:

https://www.google.de/url?sa=t&source=web&rct=j&url=https://minireference.com/static/excerpts/noBSLA_v2_preview.pdf&ved=2ahUKEwicwY-n-eXzAhUmRPEDHdpuB5YQFnoECDoQAQ&usg=AOvVaw2Blzydq1S8uOXMhekaHr8Q

Amazon product ASIN 0992001021

 

[GenesAndEnvironment]

Not sure why people say you need university. Just but whatever books they have you read in uni. Unless you're one of those that ask the teacher for help all of the time, I don't see how that's any worse.

 

[callme]

Unless you're one of those that ask the teacher for help all of the time

Yeah, you know, one of the uh... stupids?! Oh no, they are trying to learn something much above what they can understand!

 

[SentientCreature]

I think, there's a cheaper version, I hope that's the same. I'll probably buy one of them, but they should be available at libgen.

Just one question, I prefer serious books and don't mind acadamic approaches to things, as long they are possible to understand without being exceptional smart. I'm skeptical about stuff like "Linear Algebra for Dummies" or that things. I don't need encouraging comments, funny stories or unnecessary informations.

But I stumbled upon a book called "No Bullshit guide to Linear Algebra". The title put me off at first, but it seems to include some basic knowledge, that I'm missing and is probably not written too jovial, rather written pretty concisely. What do you think about it?

Here's a preview, if you would be so kind to take a look:

https://www.google.de/url?sa=t&source=web&rct=j&url=https://minireference.com/static/excerpts/noBSLA_v2_preview.pdf&ved=2ahUKEwicwY-n-eXzAhUmRPEDHdpuB5YQFnoECDoQAQ&usg=AOvVaw2Blzydq1S8uOXMhekaHr8Q

Amazon product ASIN 0992001021

The first chapter of that book which gives a brief overview of math fundamentals is something that you should take a look at regardless of what book you choose in the end, as everything else builds on top of those concepts. So basically you should understand what a function is, what it means for a function to be injective or surjective, what an inverse function is, and function composition. Those are all relatively simple concepts that you could look up anywhere on the internet.

As for the rest of the book, this is just my biased opinion, but I'm not a fan of literature which states results as matters of fact without providing a thorough explanation of where the result is coming from, and that's what I see in this book. The content is quite scattered leading to a discussion of eigenvalues after as little as 100 pages. My point is that it seems quite sparse and that it might leave you wondering how the hell anyone came up with those ideas and why.

The book I suggested is extremely rigorous, but I wouldn't say that it requires the reader to be exceptionally smart(I studied from it after all). In fact, its rigorous nature is what makes it simpler and easier to follow, as every step of the way is showed to follow logically from previous results and simple definitions and all that's required to follow it is logic. It's much less computational and much more abstract than the one you linked which, judging by what you've said so far, seems to be preferable. You'd only need some familiarity with constructing proofs, but you'll find everything you need to know about that in the first chapter of the discrete math book I mentioned(the third one). Actually, the first two chapters of that book cover a lot of the essentials and I'd recommend reading them both.

All that said, you could be more of an intuitive thinker than I am, which would allow you to grasp certain ideas without seeing the whole system built from the foundations, and in that case you might find this book more valuable than the one I used.

 

[motel rooms]

Why do people only use that gif with the lady?

 

[UseItOrLoseIt]

I don't have a solid background in math.

Try set theory. Bertrand Russel's "Introduction to Mathematical Philosophy" was one of the most interesting reads in my life.

Why do people only use that gif with the lady?

 

[motel rooms]

Try set theory. Bertrand Russel's "Introduction to Mathematical Philosophy" was one of the most interesting reads in my life.

View attachment 77273

Meet my bf

 

[Shadowplay]

Thank you for your answers. I just want to point out, that I really admire the hard work, that mathematicians do. I don't think, that I could just start somewhere and understand everything with ease. Basics are important and I am certainly willing to (re)learn some of them. I just don't want to start from high school math and feel like a stupid child and get the negative assosiciation to my time in school.

If someone cares, why I want to learn math: I don't consider myself as an intelligent person, but I am able to understand some rather complex ideas and read some complicated books, so I hope I'm at least able to grasp some of the stuff. I don't have a lot of feelings and I'm overwhelmed by this world and my very limitited intellectual abilities help myself to deal with this for me hardly bearable world. I'm trying to intellectually defend myself against others, that question my existence. My intellect (and I don't claim that I've got a lot of it) is the only thing I have and that keeps me together. So I want to do something, that occupies me intellectually.

But this rather philosophical thinking disturbs me sometimes, I feel lost, I'm tired of this normative thinking. Talking to people , hear people in the media making judgements, having unfounded opinions, share their beliefs with the whole world and imposing their worldview on others. I want an intellectual field, that is neutral, understandable and reliable. I don't expect any understandings of the world, I just want something to dive into.

I can't take this world and want a place to go, basically escapism. I feel math could do that for me. Maybe that sounds ridiculous to you, but sometimes certain hobbies or occupations can distract you and give you a feeling of self esteem or a reason to live for a while.

I can understand. If I won Tattslotto tomorrow, and I was set for life; I would quit my job, finish my PhD in philosophy, and then do a PhD in economics. It is nice to lose yourself in a world of ideas if you can. But since I don't have any great hope for winning the lottery I'm hoping to buy a bus ticket...

 

[netrezven]

I would start with paradoxes. There are some in quantum dynamics too.

 

[Anathema]

If you want, I can direct you to some books where you can start learning mathematics in some organised fashion rigorously.
There's many subjects to dip your toes in. You don't have to be 'smart'.

I'd even be willing to help you with questions if I can, I love math.
My interests are formal logic and foundations. I'm familiar with most undergrad topics.

You should probably start with the more fundamental parts of mathematics. Learning calculus as a first exposure is just for entertainment purposes. You will not be able to appreciate what is going on. And if you're not planning on becoming an engineer, you might as well do mathematics right.

As a first exposure I would HIGHLY recommend you read Joel David Hamkin's 'Lectures on the Phil. of Math'
It's a recent book (2020) and is suited for people who don't know mathematics that much. READ IT, it's so good.
It will give you exposure on buzzwords surrounding set theory, logic, etc. Fundamental stuff.

Then, the subjects you study should roughly follow this order :

1. Basic, naive set theory. Only finite sets. Don't bother with infinite sets until you understand what's going on with finite set theory. Book : Halmos, Naive Set Theory. This book is one of the easier reads on ZFC, a first order set theory seen as 'standard' within the mathematical community. It is not an easy book for beginners though. You should do some googling as you read it. The book does cover infinite sets fairly early. No good book on set theory doesn't eventually cover infinite sets... The reason you want to know some set theory is that modern mathematics assumes some reasonable set universe as a metasystem. I.e. 99% of the things you will be facing in the literature end up being interpreted as sets. Graphs? Sets. Functions? Sets. This? Set. That? Set. Set of sets. Sets of sets of sets. Everything is a set.
2. First order logic. Although in everyday parlance it is 'trivial'/familiar, and you might already understand it intuitively, make the effort to go through a formal introduction to logic a la model theory. If the book doesn't mention models, it's trash. There's many horrible books on logic. There's many horrible logic teachers. Don't ever read a book that only covers propositional logic - because authors are lazy and will optimise a lot of detail in favour of simplicity. Then, you go to first order logic knowing nothing useful. Don't read 'easy' books on logic, waste of time. I recommend you read Rautenberg's A Concise Introduction To Mathematical Logic. The title is accurate - it is concise. And dense. But it will guide you with the keywords it uses to look up information on google. This book will teach you things far beyond what an undergraduate understands about the general situation of formal mathematics. A lot of Unis skip logic altogether, and the end result is students who don't know what they're doing by the time they graduate, being carried only via parroting back the proofs they learnt by heart.
3. Look up Fitch style deduction. It's immensely, immensely helpful if you learn how to properly organise logical arguments within mathematics. It is a skill you transfer in every field of math. Try to learn Fitch style deduction of first order logic.
4. If you learn the above 3 to a decent level, congratulations. You're more mathematically mature than most uni students learning math. And you haven't even touched algebra, or geometry, or anything traditionally mathematical. Now you can begin learning mathematics properly, and appreciate what you're actually doing. Go nuts. Now's the time to explore.

If you want a very formal, by the book approach to mathematical logic, George Tourlakis' book is very good. It goes all the way to the incompleteness theorems in the most step by step manner I've seen to date.

 

[Gustav Hartmann]

Start with knot theory.