Timeline for How can I improve my formal definitions?
Current License: CC BY-SA 4.0
30 events
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Sep 1, 2019 at 2:41 | comment | added | Daniel R. Collins | This is incredibly lovely and I'm so glad you took the time to write it. One of the biggest barriers I find in my undergraduate math majors with their first proof-based course is the apparent belief that "writing math" must be all symbols and not mostly sentences. | |
| Aug 31, 2019 at 4:15 | comment | added | anomaly | I don't think adding an example contributes anything to the definition. It's merely an unwinding of the definition (as opposed to, e.g., adding to the definition of a group examples "in the wild" like $\mathbb{Z}_n, GL_n(k)$, etc.), and the reader should be able to do that for him- or herself. | |
| Aug 30, 2019 at 20:19 | comment | added | Izar Urdin | @NoahSchweber and company, all of you are very kind and I am very satisfied with your comments and advicing. I know the comments are not for this proposal, but I don't know how to thank you all this awsome response. Sincerely, thankyou very much. | |
| Aug 30, 2019 at 17:59 | comment | added | user21820 | @NoahSchweber: Sure; I agree with your remarks. My initial comment was just in reaction to Dave's comment that implicitly suggested that the best was no symbolic notation at all even for students, and that would be a severe pedagogical mistake, as my teaching experience attests. Thanks for putting up with the pinging! =) | |
| Aug 30, 2019 at 17:55 | comment | added | Noah Schweber | Anyways, I'm going to tentatively suggest a moratorium on further comments (if only because I keep getting pinged!). | |
| Aug 30, 2019 at 17:55 | comment | added | Noah Schweber | @user21820 Since $[S]^{<\omega}$ isn't very known outside of set theory, I'd say Mario's point is meaningfully correct. Regardless, none of this is in tension the point: when one should use symbols is determined in large part by readability, which in turn is highly context-dependent. And while there are certainly situations where the symbolism is crucial, either as a language (like computer-verified proofs) or an object of study in and of itself (in logic), readability is indeed almost always the dominating concern. | |
| Aug 30, 2019 at 17:52 | comment | added | user21820 | @MarioCarneiro: No. It is well-established in mathematical logic (proof theory, reverse mathematics, set theory) that $[S]^{<κ}$ denotes the set of subsets of $S$ of cardinality less than $κ$, and $[S]^κ$ denotes the set of subsets of size $κ$. And you will almost surely see this notation in infinitary combinatorics. At different levels in your mathematical learning, different notations will be more suitable for your mental processing. There is no single notational convention that is best for all at the same time. | |
| Aug 30, 2019 at 7:06 | comment | added | Mario Carneiro | ... For me, this is usually what determines which parts of a statement to render in english and which parts to render in symbols. | |
| Aug 30, 2019 at 7:06 | comment | added | Mario Carneiro | Another consideration is that some definitions have no symbolic analogue, and so you will want to use them in "text mode" when possible. For example, while "subset" has the associated notation $\subseteq$, "finite" has no analogue, so it should be used in an english phrase when possible (i.e. "finite subsets of $S$"), or in embedded text as a last resort (i.e. $\{T:T\subseteq S\wedge T\mbox{ is finite}\}$). Conversely, some symbols like $+$ or ${}^2$ are best rendered as symbols because the english locutions are more cumbersome. ... | |
| Aug 30, 2019 at 5:56 | comment | added | user21820 | @MarioCarneiro: Yes, commonly used notions should be packaged into definitions. But also note that it's not just definitions that help; the natural language grammar is convenient for this particular phrase. More generally, if we have "[adj] [adj] ... [noun]" where each [adj] is a single word, then it is almost always better to use that than to use a formal logical conjunction. Anyway, your example can be phrased cleanly as "subsets of $S$ with less than $5$ elements different from $3$". | |
| Aug 30, 2019 at 5:46 | comment | added | Mario Carneiro | @user21820 The reason "finite subsets of S" is clearer is primarily because of the increased clarity associated with the use of the definition "finite". It would be more difficult to correctly render $\{T:T\subseteq S\land \#(T\setminus\{3\})<5\}$; if you attempt to use english locutions to avoid introducing the variable $T$ you end up with something like "the set of subsets of $S$ such that removing $\{3\}$ results in a set with cardinality less than $5$" so clearly the word "finite" was pulling a lot of weight. | |
| Aug 30, 2019 at 5:23 | comment | added | user21820 | That said, we should employ the cleanest presentation possible, and sometimes this does mean less symbols. For example, "finite subsets of $S$" is much better than "$\{ T : T⊆S ∧ \#(T) < ω \}$", and expanding the definition of cardinality would make it even worse. | |
| Aug 30, 2019 at 5:19 | comment | added | user21820 | Mathematical symbols are part of the mathematical language that develops over time. In the past, we had less symbols and hence much less ability to handle complicated mathematics. Before the rudiments of first-order logic were introduced, many mathematicians employed dubious or even incorrect reasoning due to unclear quantifier order. Now we have the ability to handle arbitrarily complicated mathematics, thanks to formalization over FOL. Just try doing measure theory or higher set theory while avoiding symbols! | |
| Aug 30, 2019 at 5:13 | comment | added | user21820 | @DaveLRenfro: I don't agree with that at all. In my opinion and experience, only students who can completely grasp the formal (first-order) definition of limits and the induction schema can truly understand them. I do think that it is convenient to use natural language when there is low logical complexity, but I find it unwise to suggest avoiding symbols in situations of high logical complexity. After all, symbols were designed to attain precision, whereas natural language cannot always do it. People today happily use symbols like $+,−,×,÷,^2$. Imagine eschewing them for words! | |
| Aug 30, 2019 at 0:27 | comment | added | Martin M. W. | This is truly fantastic advice, and should be a standard reference for anyone writing mathematics. | |
| Aug 29, 2019 at 23:10 | comment | added | Dave L Renfro | @Izar Urdin: When I was a student of Computer Since (25 years ago) teachers forced us to use symbols instead of words. --- Nearly 50 years ago (1970), Paul Halmos wrote "The best notation is no notation; whenever it is possible to avoid the use of a complicated alphabetic apparatus, avoid it.". | |
| Aug 29, 2019 at 22:01 | history | edited | Noah Schweber | CC BY-SA 4.0 |
added 78 characters in body
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| Aug 29, 2019 at 21:54 | vote | accept | Izar Urdin | ||
| Aug 29, 2019 at 21:53 | comment | added | Izar Urdin | That´s incredible and a good news ! When I was a student of Computer Since (25 years ago) teachers forced us to use symbols instead of words. Things are going better ;) | |
| Aug 29, 2019 at 21:49 | comment | added | Noah Schweber | @IzarUrdin I've modified my answer to better capture your definition. Re: "it seems that each definition is going to be very extensive including motivation," that may be unavoidable - but the advice I gave still applies. | |
| Aug 29, 2019 at 21:49 | comment | added | Noah Schweber | @IzarUrdin Yes - symbols don't inherently add clarity. Don't use them just because you can; a clear, precise natural language definition is (in almost every case) optimal. Basically, which is generally more readable: $$\{x: A\wedge B\}$$ or $$\{x: A\mbox{ and }B\}?$$ When the conditions $A$ and $B$ are at all complicated, it's usually the latter. | |
| Aug 29, 2019 at 21:48 | comment | added | Izar Urdin | One question ... you use "and" insteed of "$\land$" ?? | |
| Aug 29, 2019 at 21:48 | history | edited | Noah Schweber | CC BY-SA 4.0 |
added 52 characters in body
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| Aug 29, 2019 at 21:48 | comment | added | Mikhail Borovoi | @IzarUrdin: Ok, Noah understood you better than me. Note that if you like Noah's answer (I do!), you can accept it. | |
| Aug 29, 2019 at 21:43 | comment | added | Izar Urdin | @MikhailBorovoi that was I would to say in the previous comment. | |
| Aug 29, 2019 at 21:41 | comment | added | Izar Urdin | @NoahSchweber It sounds very reasonable, I'm glad to read you. However, I feel that it is not exactly the same definition, it seems not to be an indexed family. Furthermore, it seems that each definition is going to be very extensive including motivation. I usually present some definitions to "build" another bigger showing the motivation of them togother. I don't know it can be weak syllogism. | |
| Aug 29, 2019 at 21:35 | history | edited | Mikhail Borovoi | CC BY-SA 4.0 |
Typos corrected
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| Aug 29, 2019 at 21:28 | comment | added | Mikhail Borovoi | OP writes "family" rather than "set". Therefore, I would write: We then let the $m$-crown of $S$ be the family $$(X_i: i\in S).$$ For example, if $S=\{1,2,3,4\}$ and $m=2$ then e.g. $$X_2=\{\{1,3\}, \{1,4\}, \{3,4\}\},$$ and the whole $m$-crown of $S$ is $$($$ $$(1)\ \{\{2,3\}, \{2,4\}, \{3,4\}\},$$ $$(2)\ \{\{1,3\}, \{1,4\}, \{3,4\}\},$$ $$(3)\ \{\{1,2\}, \{1,4\}, \{2,4\}\},$$ $$(4)\ \{\{1,2\}, \{1,3\}, \{2,3\}\}$$ $$).$$ | |
| Aug 29, 2019 at 21:16 | history | answered | Noah Schweber | CC BY-SA 4.0 |