I ran into a colleague the other day who’d never read Proofs and Refutations (full title: Proofs and Refutations: The Logic of Mathematical Discovery). He’d never even heard of it!

Posted by Andrew on 23 November 2018, 9:25 am

I ran into a colleague the other day who’d never read Proofs and Refutations (full title: Proofs and Refutations: The Logic of Mathematical Discovery). He’d never even heard of it!

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I’m certainly not a proof connoisseur, but is this kind of book still relevant now that we have automatic computer-based proving languages?

Yes. Automatic theorem proving is not yet at a point where it has any practical effect on the work of the large majority of mathematicians.

I found that book really boring. I couldn’t finish it.

By the time I picked it up, I’d already done abstract algebra and several graduate level math classes in logic and set theory. So maybe I was already too familiar with what it meant to present a proof. Or maybe I’d have enough dialectic after reading

Gödel, Escher, Bachin high school (as much as that book influenced me to specialize in logic and computation, I wouldn’t recommend rereading favorite childhood pop books after obtaining a graduate-level understanding of the material).@Joshua Pritikin: Computers are helpful for some proofs (finite simple groups, four color problem, etc.), but they’ve hardly replaced mathematicians.

Maybe your colleague is not so interested in the history of math. The books gives some examples of sloppy definitions being given by 19th century mathematicians. It is not necessarily relevant to later math.

Roger:

I disagree. I don’t think the book is about “sloppy definitions” at all. Mathematical definitions are, by their nature, as precise as they need to be. As the theory is expanded, the definitions become more specific. And I think the general point of the interplay between proofs and refutations is very relevant to modern mathematics.

I’m not sure if I’ve read the whole thing, but I do recall looking at it once — but it didn’t make a positive impression on me.

I don’t think most mathematicians hold philosophers’ takes on mathematics in high regard.

Hate to keep writing the same reply, but Lakatos was explicitly building on the work of a very well-known mathematician.

I have no idea if that’s true, but maybe they should!

Very much enjoyed your piece in the NYTimes (https://www.nytimes.com/2018/11/19/science/science-research-fraud-reproducibility.html?action=click&module=Discovery&pgtype=Homepage). Nowadays I only come here via your tweets as otherwise I get time warped and see nothing new. Sorry to be a complainer but … where’s the complaint dept?

Thanatos:

Yes, this is annoying me too. I’ll talk with the sysadmin on Monday.

Is it everyone now? I thought it was people using browsers other than the latest google chrome (always released last month) or something.

Andrew,

Congratulations on your NYTimes piece

I remember reading it in the 1980s, during peak post-modernism. It was not as dramatic to an 18-year old as the social construction of race, sex, and identity. But I agree with Andrew that it succeeded in emphasizing the importance of definition and the evolution of scientific knowledge. Today it makes a good antidote to reflexive anti-anti-science.

It was highly influential to me… It helped convince me to leave mathematics! So maybe mathematicians haven’t heard of it because if they had, they wouldn’t be mathematicians.

I’ve certainly heard of “Proofs and Refutations”, but somehow never got around to reading it. However, if you’ve never read “How to Solve it” by George Polya, I highly recommend it. I think that a lot of non-mathematicians have a distorted view of how mathematicians work. Even though writing a proof (and a good proof — cf. “Proofs from the Book” by Erdos et. al.) might be the end result of one’s work, finding such a proof isn’t the immediate motivator. There are a lot of guesses, blind alleys, and, most important, development of intuition. One of my objections to the “Satz-Beweis” style of mathematics, is that it seems to squeeze alll of that out.

Victor Miller

As mentioned below, HTSI and the other plausible reasoning books were a primary and acknowledged source for P and R.

I have two (possibly the same) issues:

1. I can get to new articles from the blog list of (for example)

http://observationalepidemiology.blogspot.com/ long before they appear on your home page

2. Comments in the Recent Comments list cannot be accessed for posts that do appear for a day or so.

I felt I learned a lot from Proofs and Refutations. For me, the core sentence went something like “Columbus didn’t find a new route to China, but what he found was very interesting.” In other words, we advance by learning from our mistakes. Lakatos, of course, goes somewhat deeper into this to show the process by which refutation leads to more precise framing etc. (Or so it seemed/seems to me as an amateur.) I take that lesson to my own work and especially to teaching, where I try to encourage students to make productive mistakes and take advantage of them. (School is a special institution in society where you can mess up and nothing out there in the world falls apart.)

Are there books on more empirical branches of science that do something similar by focusing on the productivity of error if it’s incorporated in a larger process of reflection?

As mentioned below, check out Mathematics and Plausible Reasoning

This might be obvious, but mistakes, serendipitous discoveries and the winding road of scientific progress are key ingredients in “The Structure of Scientific Revolutions” by Kuhn. A worthwhile well written book. Also “Against method” by Feyerabend, but here the whole process is hailed as even more chaotic. That one left me a bit confused.

Any thoughts on Lakatos’ more science-general “The Methodology of Scientific Research Programmes”? It’s on my reading list.

I never read it either. Eeekkkk~

Lakatos was explicitly building on the plausible reasoning work of George Pólya, particularly How to Solve It, and I’m still finding people who haven’t read that most readable of books.

Proofs and Refutations hews especially close to the principles of How To Solve It. Not coincidentally, Lakatos did the Hungarian translation of HTSI.

Mark:

I’m sure you’re right, but I suffer from Polya-phobia because way back when I did math team, the coaches were always pushing Polya and How to Solve It, and I associated this with other irritating attitudes they had, such as not wanting to use calculus or analytic geometry, or being opposed to applications.

For several years, I taught a course on Problem Solving for prospective secondary math teachers. I drew heavily on ideas from Polya’s How to Solve it, and also from the book Mathematical Problem Solving, by Alan Schoenfeld. But I did a lot of tailoring for the intended audience, many of whom had experience only with “routine” (i.e., learn the algorithm and apply it) problems. For example, one of Polya’s maxims is “draw a picture”. For this audience, I gave some examples, but also some “exercises” that stated a problem and asked the students to draw a suitable picture. (For example, one such problem that I recall is on figuring out how to design a window overhang that will shade from the sun certain times and let sun in at other times.)Then students presented their picture in class, and the class discussed them. This helped them understand how a suitable picture needed to be tailored to the problem being asked.

I’ve got some of the materials posted at http://www.ma.utexas.edu/users/mks/360M05/360M05home.html.

Very interesting.

thanks for the book suggestion – does anyone have any ideas on self teaching concepts higher than university level calculus? would like to actually understand the mathematical principles underlying ML…

Question is too general, I think you need to be more specific about the topics, but you may be in a situation where you don’t know what the topics are. in which case, I think a really useful thing to learn is Linear Algebra, both from an abstract and from a very concrete matrix basis. Can’t help you with suggestions for material offhand though.

Another thing to consider is perhaps combinatorics, which is very useful for understanding the underlying ideas behind probability (in a discrete setting, but it’s a good intuition pump), as well as ideas surrounding graphs, and decision trees, and soforth.

Question was definitely too general – my apologies! I am definitely in the situation you described – the many ‘paths’ of math are quite confusing from the basic center. I’ve found some openlib Linear Algebra courses, and so I’ll tackle those over the next couple months, spending extra times in gaps. Then perhaps some combinatorics. Thanks so much!

I’ve heard of the book although I didn’t recognize the title, and I looked at it once long enough to see that it was going to bore me to tears. As a mathematician one generally has one’s own ideas about proofs.

If I want to read an interesting reflection on proofs and progress in mathematics, I read Thurston or Gromov or Langlands, or someone similar.