The idea of a “category”–a sort of mathematical universe–has brought about a Written by two of the best-known names in categorical logic, Conceptual Mathematics is the first book to apply F. William Lawvere,Stephen H. Schanuel. Conceptual Mathematics: A First Introduction to Categories. Front Cover · F. William Lawvere, Stephen H. Schanuel. Cambridge University. I find Conceptual Mathematics creative, illuminating, and thought-provoking. Subobject classifiers for high school students! However, I’ve never.
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Has anyone here used it as such and what were the pros and cons you experienced in doing so?
It is also a gentle introduction to Category Theory, but with an emphasis on modeling scientific ideas. I find Conceptual Mathematics creative, illuminating, and thought-provoking. Subobject classifiers for high school students!
However, I’ve never taught from it and I mmathematics think it’s well-suited to the goals of a typical bridging course. The students in Lawvere and Schanuel’s dialogues remind me of the students in Proofs And Refutations, by Imre Lakatos — nominally naive, actually not likely to be tripped up by any of the above questions — and therefore more mathematically sophisticated than most students that would be taking a bridging course.
I’d stick with the suggestions from the other question. My instinctive reaction is that a “category error” is being made here in the philosophical sense, not the mathematical sense of category.
Conceptual Mathematics: A First Introduction To Categories by F. William Lawvere
Namely, category theory is an abstraction of standard, undergraduate level abstract algebra, which is itself an abstraction of the sort of very concrete mathematical manipulations most students have seen up to that point. In most undergraduate curricula I am familiar with, the sort of transition course you describe comes just before abstract algebra and gives students needed familiarity with i reading and writing proofs, ii very basic mathematical logic, and iii experience with the next level of abstraction in mathematics i.
I have taught the above bridge course twice at the University of Georgia. Admittedly there is a class of undergraduates who do not take this course, so it is somehow the opposite of an honors course.
Nevertheless I think the students there are representative of the sort of math majors one meets in many American universities. In this course I spend more lawveree two weeks on mathematical induction, and the abstraction of induction as a statement about subsets of the natural numbers is very challenging for the students. Each time I taught the course I ended up doing very little with cardinalities of infinite sets: I should admit that I do not own the book vonceptual Lawvere and Schanuel.
I looked at some of it on amazon just now, and it does look to be quite carefully written and unusually friendly. At a preliminary glance it looks plausible and even intriguing to use this text for some other undergraduate course. However, to use it for a transitions course would involve increasing the level of abstraction in such a course and therefore seems to be less appropriate for at least the standard versions of that course than for other courses.
Using this text would involve abandoning most of the traditional content of a transitions course and, for the clientele to which the traditional content matjematics pitched, that would be a loss. I don’t want to be too discouraging though: In fact my first undergraduate introduction to abstract algebra began with five mathejatics of category theory. Before we learned about groups, we learned about monoids and the free monoid functor called the “James construction”: Mathemtaics have not gone back to try to track down its provenance.
Before we studied monoids we studied sets and mapping from the perspective of universal mapping properties, e. The latter at least turned out to be extremely useful. Overall the course at the time looked eccentric, and doing something more traditional would probably have worked even better, but it did work, because the instructor — the still-present, great Arunas Liulevicius — had so much insight, enthusiasm and charm. It also worked because the students were very talented and enthusiastic: So you can make things work that sound like they shouldn’t, sometimes.
In case you have not yet seen it, I thought I would draw your attention to what is currently the most recent issue of the American Mathematical Monthly, and, in particular, concetual article:.
The American Mathematical Monthly, 5pp. An arXiv version can be found here. Mathematicians manipulate sets with confidence almost every day, rarely making mistakes. Few of us, however, could accurately quote what are often referred to as ‘the’ axioms of set theory. This suggests that we all carry around with us, perhaps subconsciously, a reliable body of operating principles for manipulating sets. What if we were to take some of those principles and adopt them as our axioms instead?
The message of this article is that this can be done, in a simple, practical way due to Lawvere. The resulting axioms are ten thoroughly mundane statements about sets. Leinster makes it a point to dispel a few misconceptions about Lawvere’s presentation, namely, 1 that an underlying goal is to replace set theory with category theory; 2 that the axiomatization requires greater mathematical maturity than other systems e. The primary motivation for this paper is that most working mathematicians use ZFC set theory without really paying attention to the axioms.
As the author somewhat humorously remarks:. The article does mention some pros and cons of using the text to teach “axiomatic set theory,” but perhaps they could transfer to a bridge course:. Rosebrugh, Sets for Mathematics. Cambridge University Press, Cambridge, Retrieved online from http: If you are interested conceptuwl trying this text for a bridging course, then maybe using Leinster’s presentation would be of help.
Or reading it over, making it available for students, etc. As a final, offhand comment about bridging courses: You can find more on the history of such courses at my MO response here.
However, the opportunity cost of not focusing mathemxtics proof coneptual might be too high. Clark 1, 6 Now, I wonder if there are benefits to introducing it earlier, counterintuitive as that may seem. Perhaps this should not be a replacement for a more conventional bridge course, but, as you mention, some other undergraduate course. I actually think it makes a great deal of sense to talk about quotients in the category of sets before introducing them in group theory.
In case you have not yet seen it, I thought I would draw your attention to what is currently the most recent issue of the American Mathematical Monthly, and, in particular, the article: In particular, the ten axioms stated informally are: As the author somewhat humorously remarks: The article does mention some pros and cons of using the text to teach “axiomatic set theory,” but perhaps they could transfer to a bridge course: The citations above are: Benjamin Dickman 16k 2 28 JW I have taught with neither, so am only pointing to possibly helpful supplementary materials if you decide to give it a shot.
The last time I was teaching for a ,awvere proof-course we used the first half of Wilder’s classic text: I do want to point out though that one is certainly not learning “ZFC set theory” in any transitions course I have ever seen. One is barely learning set theory at all but rather learning how to do some manipulations with sets.
Conceptual Mathematics: A First Introduction To Categories
Clark May 19 ’14 at When I taught transitions, I pointed out as an aside that one should in theory probably define “ordered pair” and mentioned one possible way to do so. This entire consideration turned out to be too “formalized” for most students, almost to the point that I regret mentioning it.
But maybe other transitions courses are different. I have also not seen ZFC covered in a transition course, and I don’t think it would be wise to do so outside of certain exceptional circumstances.
The nice idea about the approach outlined here is that the axioms look very digestable – at least in their informal presentation. You might need to make additional comments, e. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name.