What is Reddit's opinion of How to Read and Do Proofs: An Introduction to Mathematical Thought Processes?

From 3.5 billion Reddit comments

The book How to Read and Do Proofs by Daniel Solow is a gem. It may be just what you're looking for. Here's a link

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https://www.amazon.com/How-Read-Proofs-Introduction-Mathematical/dp/1118164024/ref=sr\_1\_1?crid=1M2436RJP1Z1X&keywords=how+to+read+and+write+proofs&qid=1642397334&sprefix=how+to+read+and+write+proofs%2Caps%2C176&sr=8-1

Doing a proof is not doing a computation. You are making an argument by drawing conclusions from facts and relationships between facts. So it's not so surprising if it is challenging intially (and even then challenging in the future when you move to a new domain of discourse.)

See if a book like this helps:

https://www.amazon.com/How-Read-Proofs-Introduction-Mathematical/dp/1118164024

It's not an ELI5 but How to Read and Do Proofs is a really good book on the subject (at least the 2nd edition was). You can probably pick up an older edition in paperback really cheap.

I know this isn't what you're looking for, but when I was at your stage the following book was super helpful:

https://www.amazon.com/How-Read-Proofs-Introduction-Mathematical/dp/1118164024

The other thing I can say is working in groups makes this stuff go a lot better. Just make sure you're doing the work and understanding everything as you go along. Good luck!

What are the exact things that are giving you problems?

There are walls that I hit in math as well, but they involve things that are really hard to turn into a visualization since I am very much a visual math person. For example, in systems of differential equations I have to pursue the visualizations outside of class using this course and undergrad books by the same author. Most of real analysis is really easy to visualize, though sometimes it takes some help. For real analysis some like Mathematical Analysis I by Claudio Canuto and Anita Tabacco, and/or James Callahan's Advanced Calculus: A Geometric View.

The other thing that gives difficulty for some is the idea of writing proofs. If you haven't had explicit instruction in writing proofs, you should invest a lot more time into that. A book that helped me is How to Read and Do Proofs by D. Solow. It doesn't go through proofs for Real Analysis-- it uses much more simple math so that you can focus on the goal and methods that proofs can follow toward the goal.

Some suggestions: you're asking a favour, here – for someone to put in the effort to read and understand your argument. So you need to make it easy for them, and show that you value their time: write it out properly as a <em>math</em> proof, even if you think it's an exercise in basic meta-logic. In English. With brief background (What is the Collatz conjecture?, for instance. It's unclear why readers of /r/askphilosophy would know this), and links to proofs or results you're citing (e.g. Löb's theorem).

Don't use formalisms when those formalisms don't actually add any value to the proof (e.g. phrasing "Assume there's a proof that some number's Collatz sequence does not tend to infinity" as "Suppose T ⊢ ☐~I 2"). Or alternatively, if you disagree with me on this, perhaps chat to /u//StrangeGlaringEye, since they seem to hold a view more like yours: that formal notation is very important even for proofs intended for humans to read.

Explain briefly what notation you're using - there is no one universally accepted mathematical notation even for basic logic operators. (I take it you're using '~' to represent negation, but you need to make that clear.)

Show that you understand the current state of the art in solving the problem, and how your approach avoids difficulties they've run into.

Probably also best if you don't blame others for your mistakes ("due to an error on one of my professors’ part ... [I] was banned from r/logic") - it's not a good look, and makes you look crank-y.

Once you've done that - as r/mathematics/ mods have said, it's probably easiest to link to your nicely presented proof, rather than trying to cram it into a Reddit post).

Best of luck – I hope these ideas help.

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You might want to start with:

How to Read and Do Proofs: An Introduction to Mathematical Thought Processes by Daniel Solow.

and/or:

How to Prove It: A Structured Approach by Daniel J. Velleman.

Solow is quote a bit shorter than Velleman and there are some nice videos of him lecturing the material to high-school students on the book publisher's website if that's the sort of thing that floats your boat. I think other people also have some suggestions for other resources for getting into proofs but the two above are the ones I have some familiarity with.

IIRC Spivak admits in the foreword/introduction that with the benefit of hindsight he might have named the book something like "introduction to real analysis". I'd suggest getting a second introductory real analysis textbook so that you can compare and contrast between the two. I find having other sources is often useful when I get stuck with something. I only have 4th Ed. Spivak so I'm not sure what else to suggest. The analysis part of my OU degree (in M208 - Pure Mathematics) that I'm working on is based on/related to:

A First Course in Mathematical Analysis by David Brannan.

I've really liked the bits of Spivak I've read so far. I wouldn't say it was perfect but it's definitely good if the style suits you. It's helped me resolve several issues with understanding I had from my A-level Calculus 30 years ago!

EDIT: Don't forget there's also the solutions manual for Spivak 3rd/4th Ed which would probably be useful!

Daniel Sulow's How to Read and Do Proofs it may be a bit more advanced than you might be used to but it is an invaluable book. I know this is expensive but you can find a used copy, and just as good as this one, from Amazon.

If it's too much go back to the beginning and read Euclid's Elements and it's cheaper.