What do you think?
Rate this book
Mathematical Rigour and Informal Proof
- GenresMathematicsLogic
90 pages, Hardcover
Published March 28, 2024
1 person is currently reading
12 people want to read
Ratings & Reviews
Friends & Following
Create a free account to discover what your friends think of this book!
Community Reviews
Displaying 1 of 1 review
March 10, 2024
Thank you, Tanswell, for giving me something I could focus on for a several hour period, after days of distractedly trying to memorize definitions and sketches of proofs for my upcoming final exams.
Now, this review might come off as critical, but I assure you, this is a well-written, well-researched book, and it is relatively short, less than seventy pages.
I loved this enough to give several hours of my life to it. Dearest Dr. Francis Su should take notes for the second edition of “Mathematics for Human Flourishing” (no shade, and in fact, respect for giving Chris Jackson a platform to tell his prison math story).
Now, at times, Tanswell makes retro-active remarks that save this text.
For example, lists unless they are perhaps chronological or some such are almost always terrible. Thankfully Tanswell at least once remarks, “The list of purposes in the previous section is an example of this messiness: some of them are normative ideals, some of them are descriptive, and others can be read either way” (p. 7).
One of my favorite parts of the “standard view” section was the reference to De Toffoli’s 2023 article “Who’s afraid of Diagrams?” where the section “Diagrams that are not Abbreviations” gave a brief account of the Venn II system even more compelling than Tanswell’s later gesture (pp. 13, 23). I now want to read some other Toffoli references, like “Objectivity and rigor in classical Italian algebraic geometry” and “An inquiry into the practice of proving in low-dimensional topology.”
I will admit, my eyes glazed over for the talk of potential errors in the view of rigor as formality—or the so-called standard view. Many of the points were boring, from incompleteness, to redundancy, to explanatory power, informal proofs which can be formalized (gaps filled) and still incorrect, infinite regress, et cetera.
Even though Lowe and Muller (L&M) later seem to have insightful remarks on the role of skill in “mathematical knowledge”, the proposal of “conceptual modeling” is too close to the internet infatuation with Yudkowsky for me (p. 29).
Similarly, Tanswell dedicates the first section after his introduction to the formalist, even non-pluralist perspective. Thankfully, he concludes it with a gesture toward Godel and even Graham Priest. “In my own work, I have argued that the place where this argument breaks is in adopting the standard view in the first place. A proponent of the standard view is therefore challenged to offer a different solution, or to accept Priest’s conclusion that mathematics is inconsistent” (pp. 22-23).
Now, plural views can overwhelm a reader. This especially holds for someone who is reading ninety pages in one or two sittings because they should be doing something else, like analyzing some least squares or learning Gram-Schmidt.
Tanswell’s reference that algebraic concepts maintain a fixedness because they do not assert the existence of any objects but merely refer to any structure satisfying the definition (p. 38). We can see then why algebraic diagrams appeal to me, as “informal” but potentially rigorous and “fixed” communications.
Funnily enough, I kept reading this from the start only for the dialogical section, and it may have disappointed me more than the “standard view” formality section. Toulmin is ick, and reminded me of community college writing classes (p. 31). I was disappointed by the account on Dutilh Novaes’ “The Dialogical Roots of Deduction”, which seemed promising to me (p. 39). It was just kind of a similar line of reasoning to Toulmin’s writing 121 notions of clarity, concision, generosity, et cetera all draped with phrases like “truth preservation” and “belief bracketing” (p. 40).
The discussion of “Proofs and Refutations” by Lakatos is one of the clearest and most concise expositions of the book I have read. Corfield and Zalamea better watch out. Still, it is wild that a footnote mentions the connection between Lakatos and Hegel, with a bibliographical reference to “The Phenomenology of Spirit” (p. 36). Why not e.g., “The Science of Logic”? But, further, there is no mention of Lautman, Lawvere, or Schreiber, et cetera. Please, at least gesture toward one person working with dialectical or explicitly Hegelian views of math (other than the reference to a 2001 journal article by Larvor).
I can understand, though, that Hegel is not a serious topic for most mathematicians. Further, I would have loved to read more on Tanswell’s earlier work of his own that he mentions: “Conceptual engineering for mathematical concepts” (p. 36). Tanswell does reference this article of his one other time to say there are no “safe havens” for mathematicians to fully control their meanings (p. 39). Did you have a Deleuzean streak in your younger years, tanswell?
It is also wild that Novaes has to say math dialogue is “semi-cooperative” where she splits the dialogue into a prover and a skeptic, and she has to argue that the “skeptic” is not mostly implicit. Why can’t math dialogue just be fully “cooperative”? This feels like that hermeneutic strain of suspicion, like the theology of math has to now move through history’s great critics or something.
Of course, for me, if it is not dialogical, then the diagrammatic must save me. Gilles Chatalet be my guide. Tanswell’s account of low-dimensional topology requiring communication of proofs through words and diagrams, neither without the other, is wonderful. Of course he refers to Tenswell’s account of the topic (p. 48).
Tanswell brings together (1) Rav’s claim that “mathematical skill” determines what one comes to know from a “proof” with (2) L&M’s claim that proofs require the “whole arsenal” of drawing connections and performing techniques and so “proof” bears the knowledge of math (p. 50).
The claim from L&M that theorems are just headlines is wild. Supposedly, this slogan comes from L&M’s advocacy for mathematical knowledge as mathematical skill. Tanswell challenges this claim as it allows for too many breaking cases. I would also argue the term “skill-based” echoes too clearly the jargon of “test scores, achievement gaps, calculation abilities” as Orlin put it in his review of Phillips’ “The New Math”.
But, Tanswell only addresses the issues of L&M’s skill-based knowledge via some “knowledge-how” and “knowledge-that” distinction which reeks of Heidegger’s at-hand mumbo jumbo (p. 51). It is the misguided but well-intentioned analytic philosopher’s habit of verging on new age neologisms to save the argument’s tractability.
Admittedly, by this point my mind began to fog again, right as Tanswell names J.L. Austin and starts discussing speech acts. Though, I guess Tanswell refers back to De Toffoli’s discussion of proofs in low-dimensional topology, where the words and diagrams together form “inferential actions” that are not speech acts (p. 53). Then, there’s business about “epistemic” actions which sort of lost me in a similar vein as the “epistemology of proofs section”, only to be saved where, again, diagrams give issues to the epistemics (p. 54).
I guess the “recipe” view was the culmination here, but, meh. Tanswell also says that the recipe view might not work well with the formalist view of metamathematics or computer checkability, but this seems misinformed. Almost all proof languages have a “tactics” layer that is precisely the intersection of “recipes” and “metamathematics” (p. 57).
Now, Tanswell, why did you have to go talking about virtues? The return to virtue ethics is a tragic misstep. I will never forgive Elizabeth Anscombe or Alasdair MacIntyre. And, I certainly will never forgive Francis Su. No way, I wrote this sentence-long tirade about Anscombe and MacIntyre before beginning the chapter, and their names appear in the first paragraph (p. 58). Who would have thought?
This chapter is all the more intolerable in its use of surveys. I just cannot. Tanswell discusses epistemic injustice and mentions the infamous 1993 Jaffe and Quinn article as well as its 1994 responses in an article from Atiyah and, of course, the one from Thurston. Who doesn’t love Thurston’s article on proofs and progress in mathematics? It almost single-handedly inspired me to finish undergraduate calculus, to continually search for materials for Stokes’ Theorem on manifold and chains, to continually re-read the infamous articles by Grisha Perelman and Richard Hamilton.
But, Tanswell concludes with “rigorous pluralism” as he calls it, as the formal, dialogical, and virtuous views are all flawed one way or another. The book could be one long ode to statistician George Box, famous for coining and re-stating the slogan that “all models are wrong, but some are useful.”
https://coconifer.substack.com/p/brie...
Now, this review might come off as critical, but I assure you, this is a well-written, well-researched book, and it is relatively short, less than seventy pages.
I loved this enough to give several hours of my life to it. Dearest Dr. Francis Su should take notes for the second edition of “Mathematics for Human Flourishing” (no shade, and in fact, respect for giving Chris Jackson a platform to tell his prison math story).
Now, at times, Tanswell makes retro-active remarks that save this text.
For example, lists unless they are perhaps chronological or some such are almost always terrible. Thankfully Tanswell at least once remarks, “The list of purposes in the previous section is an example of this messiness: some of them are normative ideals, some of them are descriptive, and others can be read either way” (p. 7).
One of my favorite parts of the “standard view” section was the reference to De Toffoli’s 2023 article “Who’s afraid of Diagrams?” where the section “Diagrams that are not Abbreviations” gave a brief account of the Venn II system even more compelling than Tanswell’s later gesture (pp. 13, 23). I now want to read some other Toffoli references, like “Objectivity and rigor in classical Italian algebraic geometry” and “An inquiry into the practice of proving in low-dimensional topology.”
I will admit, my eyes glazed over for the talk of potential errors in the view of rigor as formality—or the so-called standard view. Many of the points were boring, from incompleteness, to redundancy, to explanatory power, informal proofs which can be formalized (gaps filled) and still incorrect, infinite regress, et cetera.
Even though Lowe and Muller (L&M) later seem to have insightful remarks on the role of skill in “mathematical knowledge”, the proposal of “conceptual modeling” is too close to the internet infatuation with Yudkowsky for me (p. 29).
Similarly, Tanswell dedicates the first section after his introduction to the formalist, even non-pluralist perspective. Thankfully, he concludes it with a gesture toward Godel and even Graham Priest. “In my own work, I have argued that the place where this argument breaks is in adopting the standard view in the first place. A proponent of the standard view is therefore challenged to offer a different solution, or to accept Priest’s conclusion that mathematics is inconsistent” (pp. 22-23).
Now, plural views can overwhelm a reader. This especially holds for someone who is reading ninety pages in one or two sittings because they should be doing something else, like analyzing some least squares or learning Gram-Schmidt.
Tanswell’s reference that algebraic concepts maintain a fixedness because they do not assert the existence of any objects but merely refer to any structure satisfying the definition (p. 38). We can see then why algebraic diagrams appeal to me, as “informal” but potentially rigorous and “fixed” communications.
Funnily enough, I kept reading this from the start only for the dialogical section, and it may have disappointed me more than the “standard view” formality section. Toulmin is ick, and reminded me of community college writing classes (p. 31). I was disappointed by the account on Dutilh Novaes’ “The Dialogical Roots of Deduction”, which seemed promising to me (p. 39). It was just kind of a similar line of reasoning to Toulmin’s writing 121 notions of clarity, concision, generosity, et cetera all draped with phrases like “truth preservation” and “belief bracketing” (p. 40).
The discussion of “Proofs and Refutations” by Lakatos is one of the clearest and most concise expositions of the book I have read. Corfield and Zalamea better watch out. Still, it is wild that a footnote mentions the connection between Lakatos and Hegel, with a bibliographical reference to “The Phenomenology of Spirit” (p. 36). Why not e.g., “The Science of Logic”? But, further, there is no mention of Lautman, Lawvere, or Schreiber, et cetera. Please, at least gesture toward one person working with dialectical or explicitly Hegelian views of math (other than the reference to a 2001 journal article by Larvor).
I can understand, though, that Hegel is not a serious topic for most mathematicians. Further, I would have loved to read more on Tanswell’s earlier work of his own that he mentions: “Conceptual engineering for mathematical concepts” (p. 36). Tanswell does reference this article of his one other time to say there are no “safe havens” for mathematicians to fully control their meanings (p. 39). Did you have a Deleuzean streak in your younger years, tanswell?
It is also wild that Novaes has to say math dialogue is “semi-cooperative” where she splits the dialogue into a prover and a skeptic, and she has to argue that the “skeptic” is not mostly implicit. Why can’t math dialogue just be fully “cooperative”? This feels like that hermeneutic strain of suspicion, like the theology of math has to now move through history’s great critics or something.
Of course, for me, if it is not dialogical, then the diagrammatic must save me. Gilles Chatalet be my guide. Tanswell’s account of low-dimensional topology requiring communication of proofs through words and diagrams, neither without the other, is wonderful. Of course he refers to Tenswell’s account of the topic (p. 48).
Tanswell brings together (1) Rav’s claim that “mathematical skill” determines what one comes to know from a “proof” with (2) L&M’s claim that proofs require the “whole arsenal” of drawing connections and performing techniques and so “proof” bears the knowledge of math (p. 50).
The claim from L&M that theorems are just headlines is wild. Supposedly, this slogan comes from L&M’s advocacy for mathematical knowledge as mathematical skill. Tanswell challenges this claim as it allows for too many breaking cases. I would also argue the term “skill-based” echoes too clearly the jargon of “test scores, achievement gaps, calculation abilities” as Orlin put it in his review of Phillips’ “The New Math”.
But, Tanswell only addresses the issues of L&M’s skill-based knowledge via some “knowledge-how” and “knowledge-that” distinction which reeks of Heidegger’s at-hand mumbo jumbo (p. 51). It is the misguided but well-intentioned analytic philosopher’s habit of verging on new age neologisms to save the argument’s tractability.
Admittedly, by this point my mind began to fog again, right as Tanswell names J.L. Austin and starts discussing speech acts. Though, I guess Tanswell refers back to De Toffoli’s discussion of proofs in low-dimensional topology, where the words and diagrams together form “inferential actions” that are not speech acts (p. 53). Then, there’s business about “epistemic” actions which sort of lost me in a similar vein as the “epistemology of proofs section”, only to be saved where, again, diagrams give issues to the epistemics (p. 54).
I guess the “recipe” view was the culmination here, but, meh. Tanswell also says that the recipe view might not work well with the formalist view of metamathematics or computer checkability, but this seems misinformed. Almost all proof languages have a “tactics” layer that is precisely the intersection of “recipes” and “metamathematics” (p. 57).
Now, Tanswell, why did you have to go talking about virtues? The return to virtue ethics is a tragic misstep. I will never forgive Elizabeth Anscombe or Alasdair MacIntyre. And, I certainly will never forgive Francis Su. No way, I wrote this sentence-long tirade about Anscombe and MacIntyre before beginning the chapter, and their names appear in the first paragraph (p. 58). Who would have thought?
This chapter is all the more intolerable in its use of surveys. I just cannot. Tanswell discusses epistemic injustice and mentions the infamous 1993 Jaffe and Quinn article as well as its 1994 responses in an article from Atiyah and, of course, the one from Thurston. Who doesn’t love Thurston’s article on proofs and progress in mathematics? It almost single-handedly inspired me to finish undergraduate calculus, to continually search for materials for Stokes’ Theorem on manifold and chains, to continually re-read the infamous articles by Grisha Perelman and Richard Hamilton.
But, Tanswell concludes with “rigorous pluralism” as he calls it, as the formal, dialogical, and virtuous views are all flawed one way or another. The book could be one long ode to statistician George Box, famous for coining and re-stating the slogan that “all models are wrong, but some are useful.”
https://coconifer.substack.com/p/brie...
Displaying 1 of 1 review