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April 27

Grasping maths from scratch

If somebody was looking for a really good (standard) work explaining mathematics and its principles to a layperson from scratch to its finest, most complex ramifications, with a rather wholesome, philosophical approach, and ideally in a way as if one had never had any maths at school, what would you recommend?--Hildeoc (talk) 21:22, 27 April 2023 (UTC)[reply]

The first thing that springs to mind for me is the work of Morris Kline. He wrote about 20 books, about half of which I think fit your description. Mathematics is a huge subject though, bigger than most people realize. I'd recommend the video The Map of Mathematics by Domain of Science to get you started. The first comment there says "The funny thing is every single item on the map could probably have a full-scale map of its own." So "finest, most complex ramifications" seems overly ambitions unless you have access to an immortality pill. --RDBury (talk) 00:17, 28 April 2023 (UTC)[reply]

from scratch to its finest, most complex ramifications – If you truly want to go from scratch to the "most complex” you’ll need to spend years of intense effort, and the most appropriate advice is probably to sign up for a full-time undergraduate degree program, as the support, structure, and feedback will save a lot of bother compared to trying to self-study. If you are looking for a less substantial commitment, you’ll need to moderate your expectations. –jacobolus (t) 02:05, 28 April 2023 (UTC)[reply]

You'd be surprised how good some of the mathematics content on YouTube is. I always recommend the works of 3Blue1Brown (Grant Sanderson), for their clarity and easy of understanding for the uninitiated. He has done series on calculus, linear algebra, differential equations, and he's currently working through a series on statistics. He does exactly as you describe in each series, starting from first principles and helping the viewer "invent" the concepts themselves, building to more and more complex issues. I recommend starting with the calculus series, playlist here. --Jayron 32 12:15, 28 April 2023 (UTC)[reply]

H. Steinhaus wrote Mathematical Snapshots to answer a friend's question, "just what do mathematicians do??" It won't make anyone a mathematician, but it gives a flavor of the field, and may help the reader choose a subfield of interest. —Tamfang (talk) 14:05, 28 April 2023 (UTC)[reply]

@Tamfang, Jayron32, Jacobolus, RDBury: Thank you all so much for your tips! To clarify: I would, of course, certainly step down from my initial demand in terms of getting to the "finest, most complex ramifications"! This would indeed not be my priority, and I realize this is a rather impossible undertaking, particularly in a single book. I would actually like to have some nice book on my bedside table that explains the essence of maths as if I were a completely blank slate in this field (I had maths in an advanced course at Gymnasium). I guess I will then resort to Mathematical Snapshots to begin with. Hildeoc (talk) 00:21, 29 April 2023 (UTC)[reply]

PS: What a bummer: There seems to exist no current print edition of Mathematical Snapshots, just an ebook edition. I'd really prefer something on paper though. Hildeoc (talk) 00:44, 29 April 2023 (UTC)[reply]

There is a 1959 print edition of the German translation, Kaleidoskop der Mathematik, which may be available from antiquarian sources.^[1][2] --Lambiam 08:11, 29 April 2023 (UTC)[reply]

Thanks for pointing this out! Though I guess, in general, English would be fine as well. Hildeoc (talk) 15:15, 29 April 2023 (UTC)[reply]

@RDBury: What would probably be the most appropriate work by Kline to get started with in my case? Hildeoc (talk) 15:12, 29 April 2023 (UTC)[reply]

I can't speak for RDB, but I think Mathematics for the Nonmathematician is a good start. You can also use it to see which topics you'd like to dive into more deeply. --Lambiam 16:39, 29 April 2023 (UTC)[reply]

I haven't read them all, and I'm not exactly a layperson, but I found Mathematics: The Loss of Certainty very interesting. It gets into the history of the philosophical development of mathematics from ancient times to the 20th century. (Turns out there's a lot of history there and the 20th century was perhaps the most active.) Lambiam is pretty knowledgeable so you can probably trust his choice. I was thinking, if you want to get close to your original goal of "finest, most complex ramifications" you could try Bourbaki's Éléments de mathématique; it's only 20 or so volumes and most of it has even been translated into English. It does cover everything from foundations (logic and set theory) to advanced topics like algebraic topology. Seriously though, I'd recommend going to your local college library to peruse a copy, just to get an idea of what you'd be getting into. It's definitely not meant for laypeople though, more like people who already have a post-graduate degree. To get an idea of how detailed you can get, there's Russel and Whiteheads's Principia Mathematica. Famously, it gets around to proving 1+1=2 after nearly 400 pages. RDBury (talk) 04:03, 30 April 2023 (UTC)[reply]

If you want a used copy of Steinhaus's book in English it's about $6 for the cheapest paperback or $9 for the cheapest hardback at abebooks (or if those two sell there are a bunch more copies for <$20), https://www.abebooks.com/servlet/SearchResults?kn=steinhaus%20snapshots –jacobolus (t) 16:56, 29 April 2023 (UTC)[reply]