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My partner is a PhD student in philosophy and has recently developed a keen interest in learning pure mathematics. I am doing my best to teach her (I'm a pure maths PhD student myself) and it is turning out to be a rewarding experience for the both of us, but having virtually no teaching experience myself I have run into a few difficulties.

Some context:

She hated maths at school and did quite badly, mainly due to very poor teaching and undiagnosed autism. She wants to gain confidence by proving to herself that she can succeed in maths.

She is very intelligent and teaches formal logic to philosophy undergraduates. I think given her logical reasoning abilities she might have a lot of potential in mathematics, but is struggling with the formal language and style of proofs.

Currently she is exploring topics such as elementary set theory, complex numbers and an axiomatic approach to understanding the real numbers.

I am a little unsure what texts might be best to suggest given her mathematical background, since most people have a very thorough grounding in school-level mathematics (elementary algebra, calculus, graphs etc) before studying these things and she is rather rusty on these topics. Her interest is in pure mathematics, so ideally she would be able to take the fastest route to diving into typical first-year undergraduate pure maths.

How might I help her best? Any advice much appreciated :)

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    $\begingroup$ Essay by someone who learned math in his 60s: nytimes.com/2022/09/18/opinion/math-adolescence-mystery.html -- His circumstances seem different, though. $\endgroup$
    – Raciquel
    Jun 19 at 22:49
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    $\begingroup$ Somewhat old, but from what you've said maybe look carefully at Mathematics for the General Reader (1948; reprinted by Doubleday in 1959; reprinted by Dover in 1981 & 2017) -- Chapter I and Chapter XV. $\endgroup$
    – Dave L Renfro
    Jun 20 at 0:29
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    $\begingroup$ Also Foundations and Fundamental Concepts of Mathematics by Howard Eves (1997, 3rd revised edition) and The Nature and Growth of Modern Mathematics by Edna E. Kramer (1970 & 1982). Of course, there are hundreds of such books (I have well over 100 such on my bookshelves), but I've tried to pick those that would likely be relatable to someone in philosophy, and don't require much background while still pitched at a relatively high cognitive level, and teach/reinforce lower level math concepts. $\endgroup$
    – Dave L Renfro
    Jun 20 at 0:43
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    $\begingroup$ Halmos' Naive Set Theory is a very readable and mathematical introduction to the topic. from the person who wrote How to Write Mathematics. $\endgroup$
    – Cong Chen
    Jun 20 at 8:12
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    $\begingroup$ @Michael: Mathematicians turn a simple for loop with nice meaningful names into a mess of Σ and ∫ and the full Greek alphabet. -- I would be interested in a nontrivial example of mathematical exposition using notation that would be considered appropriate by mathematicians but which you think could be improved by using "meaningful names". $\endgroup$
    – Dave L Renfro
    Jun 21 at 2:14

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As a person self-studying pure maths right now, I recommend the book "How to think about Analysis" by Lara Alcock since its very readable and the way Lara explains things, the concepts covered are quite digestible + the book has many advice on how to learn and study pure maths. It also does go through set theory notation, reading and understanding proofs, etc. I think she has one on Abstract algebra as well.

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  • $\begingroup$ +1 for abstract algebra $\endgroup$
    – Nick Barnes
    Jun 20 at 15:59
  • $\begingroup$ She also has such a book on analysis. $\endgroup$
    – KCd
    Jun 27 at 20:09
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If she is more of a visual learner, there are excellent Youtube channels about mathematics.

3Blue1Brown is excellent and visually introduces many concepts that are quite scary when looking at the equations.

Numberphile is another one. This one is more oriented on interesting or surprising things in mathematics, and it is a door to areas of mathematics that otherwise may not have been known by the viewer.

There are many more (including actual courses).

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  • $\begingroup$ There's also Khan Academy, which goes over lower-level topics than the above two. And if you're looking for additional higher-level channels, check out PBS Infinite Series, Mathemaniac, Zetamath, Eigenchris, and Aleph 0 $\endgroup$
    – BlueRaja - Danny Pflughoeft
    Jun 21 at 3:17
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    $\begingroup$ Though I would add that if the goal is to learn math, then relying on 3Blue1Brown and Khan Academy is counter productive, or at least my experience tells me so. Sure, you understand the topics much faster, and in an easier manner, but this prevents you from learning how to learn math. I would say that 3Blue1Brown and Khan Academy isn't meant to help math students, rather help students study some math. $\endgroup$
    – AyamGorengPedes
    Jun 21 at 12:58
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    $\begingroup$ @AyamGorengPedes I thought you were going to say something like they give you a superficial understanding, and so they're not good in that sense. But you're really saying they give you an understanding of math in an easy way, and that's somehow a bad thing? Because that sure sounds good to me! $\endgroup$
    – Thierry
    Jun 22 at 20:26
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    $\begingroup$ @Thierry Oh they do explain things very well. However that still leaves you with a blackbox that you thing you know about. I strongly think that you can't steal understanding, it is hard earned. $\endgroup$
    – AyamGorengPedes
    Jun 24 at 18:29
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    $\begingroup$ Numberphile is the worst place to learn mathematics. Please don't recommend it. You end up stuffing your head with stupid conceptual errors that it feeds to you, and it's way harder to get nonsense out of your head than it is to get it in. $\endgroup$
    – user21820
    Sep 16 at 5:09
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You might consider that she is experiencing difficulties not because the formal proofs per see are difficult, but because she has not developed intuitions about mathematics. Starting a mathematical education with a focus on set theory leads to a situation where the student is spending most of the time making a colossal effort to prove in a very abstract way something that seems trivial while not knowing why. The New Math movement in the US failed drastically in this approach.

Supposing that understanding maths is synonymous with being able to talk to mathematicians and understand why they are interested in a given topic, I think it is important to try not only to grasp formal proofs, but to understand that they are necessary because our regular intuition fails to consider interesting structures that are logically possible.

Given all I said, I would suggest going back to geometry. It is a topic where everybody has some intuition about the objects involved and at the same time has plenty of interesting constructions that help motivate the need for proofs. One suggestion would be to use A. P. Kiselev's two volumes "Geometry". This is a soviet textbook for advanced high school, and has a very professional "proposition-proof" style for euclidean geometry, while trying to introduce more advanced topics, for instance the first book, on plane geometry, will introduce the concept of homothety and similarity transformations (which should help her have a first contact with the notion of a group and a higher level of abstraction) and the second book, on solid geometry, finishes by introducing vectors and having a first discussion on the foundations of geometry.

A follow-up could be Coxeter's "Introduction to Geometry", a book aimed at undergrads. It is divided in four parts, euclidean geometry, coordinate geometry, foundations and differential geometry. It is not meant to be thorough but to give an introduction to interesting problems in each of these parts of geometry. It has a lot of discussion on symmetry groups and non-euclidean geometry.

I must say I'm biased in liking a lot of group theory, but maybe she could profit from this kind of focus. Studying geometry from the ground up with a focus on symmetry is a way of using an intuitive topic, with plenty of room to pick up basic competencies (such as elementary algebra) and introducing a topic that motivates more abstract reasoning (from the symmetries themselves to the abstract concept of group grounded in set theory and algebraic structures). It also has the benefit of leading straight to a discussion of the Erlangen program and a discussion of the foundations of geometry (and maths), something a philosopher might find interesting.

Since you mentioned complex numbers, in the same vein I would recommend I. M. Yaglom's "Complex Numbers in Geometry", another soviet book for advanced high school. It requires being comfortable with elementary algebra, but introduces complex numbers with a focus on transformations on the plane. It also introduces dual numbers, which I find neat at this level to look at the myriad of interesting objects that are logically plausible, and motivate more abstract reasoning about structures.

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    $\begingroup$ I was a victim of New Math in Sweden. So I would say that it is better to start somewhere else than with set theory. To me, today, it seems like a necessary tool but until I understood why it is needed it stayed totally meaningless to me There are so many other branches of pure mathematics to immerse yourself in and getting the intuitive feel for things.. $\endgroup$
    – ghellquist
    Jun 20 at 20:16
  • $\begingroup$ The fastest route to diving into typical first-year undergraduate pure maths? Be a sophomore in highschool where they teach geometry. Then trig junior year. Then brush up on algebra II before you take pre-calc senior year. Ain't no shortcuts, +1. $\endgroup$
    – Mazura
    Jun 22 at 0:27
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    $\begingroup$ @Mazura Could you possibly rephrase that in a way does not rely on so many American educational terms? $\endgroup$
    – user829347
    Jun 22 at 16:07
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My experience is that getting the student to believe that they are capable of understanding the material is 80% of the work. Most 'not understanding' comes from a belief that they 'will never be able to understand'.

Make them feel safe, link your first few examples to stuff your student knows about and is already utilising the concepts in.

eg1: teaching an elementary school teacher about statistics. Ask: How do you know your class is doing well in a subject? (mean grade) Does this mean grade tell you things about each individual student? (how representative is this mean value? - variance, distribution etc).

eg2: teaching a aged care worker about mathematical formulas: If you had to say how many beds you'll have free next month, how would you do it? (they start giving you variables) Have them write down abbreviations for these variables and the calculations they use. Then it's a short step to show that mathematical formulae are just language.

Show them that they already know, you'll notice that they will start being playful and inquisitive about the material. Then give them the next level. Sometimes it even helps to just repeat: 'This is something that you can understand'

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Since she teaches logic, this should be accessible:

Set Theory and Logic by Robert R. Stoll. Corrected (1979) reprint.

This is a Dover book so quite inexpensive. Lots of examples and exercises. From a review: "Its clarity makes this book excellent for self-study."

Cover

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I don't think the books of Joel David Hamkins have been proposed yet. Beside posting beautiful answers on Math.SE, he has also published "Lectures on the Philosophy of Mathematics" and "Proof and the Art of Mathematics", and I don't think one could find better with clarifying the connection between formal philosophy and math.

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Have you considered Eugenia Cheng' work? I'm currently reading The Joy of Abstraction: an Exploration of Math, Category Theory, and Life, which might interest your partner. Eugenia Cheng has a Youtube channel dedicated the Category Theory.

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It might not be what you want but I'd recommend starting with the book "Engineering Mathematics" by KA Stroud, it is a standard text in the UK for Engineering undergrads, so is a little bit more slowly and clearly explained for someone who might be a bit rusty/ not actually taking a mathematics degree.

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I recently finished reading, and enjoying, Alec Wilkinson's 2022 book:

"A Divine Language: Learning Algebra, Geometry, and Calculus at the Edge of Old Age"

I found the book to be quite enlightening and highly readable; it often rambles off into wonderful byroads of the philosophy and history of mathematics. The author acts in the role of student, rather than teacher, which enhanced it value for me.

I highly recommend this book.

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"How to Prove It" is a good book for beginners in proofs in mathematics. But there is also "Nets, Puzzles, and Postmen" that is an excellent introduction to graph theory for lay people that even has interesting bits for advanced readers while keeping it simple and clear for beginners.

Regarding logic, I don't believe that "She [...] teaches formal logic to philosophy undergraduates" and "[she] is struggling with the formal language and style of proofs" are consistent. If she truly understands FOL (first-order logic) well enough to be able to teach it, then she certainly can learn all mathematics (regardless of area) given sufficient work and patience. But I actually have some doubts as to that. For a concrete assessment, look at this Fitch-style natural deduction system for FOL, which anyone (with just high-school knowledge) can learn to use and thereby gain a 100% crystal-clear understanding of formal proofs. I have had success with every student who has learnt this system.

Also, I always encourage teachers to use more interesting applications of mathematics, including teaching more concrete interesting mathematics, and engaging in solving logic puzzles. You will notice that these things are rarely taught in schools, which in my opinion is to the detriment of students. Mathematics should not be about learning to do what the teacher told you to do, but about learning truth via guided exploration with a curious mindset.

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What level of logic does she teach/ is she comfortable with?

If

Propositional logic -> she should learn to be comfortable with predicate logic and basic discrete math.

Predicate logic -> most intros to predicate logic are at the equivalent of an intro to proof math course, so she can safely being with any gentle intro pure math course, say Pinters abstract algebra or Axlers LA. Alternatively, work up to metalogic with Sider's logic for philosophy.

Metalogic -> this course is roughly equivalent to a first course in mathematical logic, say some natural deduction + completeness and applications. If shes here, she probably has enough mathematical maturity to choose topics at will.

Where in particular are her interests? There are several important places where math and philosophy intersect.

modal logic -> perhaps she is interested in modal logic? the semantics of modality formed a large part of phil lang discourse in the later part of the century. Van Benthems modal logic would be quite accessible as long as she is comfortable with induction + predicate logic.

proof theory -> structural proof theory features in say, Dummett's phil of lang. Also important in phil logic. Very few prereqs, could be done possibly before a metalogic course. Negri/von Plato have a good text.

phil math -> she'll want to go with a nice set theory text, maybe Jechs beginner text. Don't bother with naive set theory if you can help it, as knowledge of axiomatic set theory is assumed at higher levels. Some familiarity with analysis might be helpful.

formal epistemology -> bayesianism and interpretations of probability were (still are?) quite popular for sometime in epistemology. There are a lot of texts for probability geared for beginners, and many good texts for measure theory.

Also an intro computation course (say along the lines of Cutland) could be very helpful in discussions about computational theory of mind. This would lead straight to discussions of Godels incompleteness,an important topic in phil math.

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  • $\begingroup$ Zalta (of SEP fame) uses coq to investigate theories of abstract objects. Software foundations could be a fun intro to logic as well. $\endgroup$
    – emesupap
    Jun 20 at 23:24
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It's best to expose her to different topics at different levels of rigor. You can do set theory and basic algebra rigorously and at the same time also calculus in a more intuitive way. You can then later, when she has made progress with the more basic math, polish up het calculus by doing that rigorously. And when you arrive at that point, her non-rigorous calculus skills can be sufficient to study applied math with.

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I have tutored a few adults on this very topic and I have used: Mastering Algebra, An Axiomatic Approach (2nd Edition), by Roger W. Oster

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Since your partner is a philosophy student, why not use Russel or Frege as starting point to tackle pure mathematics? The book 'Introduction to mathematical Philosophy' by Russell is excellent at tackling foundational questions of mathematics.

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