# How can I improve my formal definitions?

I am a Software Architect and not very familiarized with standard notation in mathematics. Nonetheless, I would like to write a paper explaining a normalization of a computing model for expert systems. It has a very deep background on geometry, logic and group theory, so I have to define some [new] unusual mathematical objects, and in order to get it accepted by the reader (some of them scientists of different disciplines) I would like to be as clear and correct as possible. How can I improve these definitions? Does anyone or any company offer this "help" as a service?

I have about 20 definitions like the following (only for instance) to be improved.

The $$m$$-crown of a set $$S$$, denoted by $$S^m$$, is the family of sets of every subset of its index set of cardinality $$m$$ not containing its index, such that

$$\forall(X_i \in S^m : i \in S) \rightarrow X_i = \{ x \subset S :(i\notin x \land|x|=m) \}$$

(Note: I am sure that it is a correct definition but may be not easy to understand with a not very standarized notation)

• @RodrigodeAzevedo it is possible to send private messages to avoid this conversations in comments? – Izar Urdin Aug 31 at 7:01
• That will be interesting ... how can we do it? – Izar Urdin Aug 31 at 7:26
• Jajajaj ... it is not cool. But the answer has a lot yet. It will be desirable to have an "space" like teams for this kind of open-team-reviews. – Izar Urdin Aug 31 at 7:29
• I have created the chat room "Pioneers" ( chat.stackexchange.com/rooms/98162/pioneers ) to discuss this type of "innovations"; everyone is welcome;) – Izar Urdin Sep 1 at 11:11

I don't know about a definition-checking service, but I can give some general advice which I think will help.

Let me begin by rewriting your definition (hopefully correctly!):

Suppose I have a set $$S$$ and a natural number $$m$$. For $$i\in S$$, let $$X_i=\{a\subseteq S: \vert a\vert=m\mbox{ and }i\not\in a\}.$$ We then let the $$m$$-crown of $$S$$ be the indexed set (with indexing set $$S$$ itself) $$(X_i)_{i\in S}.$$ 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 $$($$ $$X_1=\{\{2,3\}, \{2,4\}, \{3,4\}\},$$ $$X_2=\{\{1,3\}, \{1,4\}, \{3,4\}\},$$ $$X_3=\{\{1,2\}, \{1,4\}, \{2,4\}\},$$ $$X_4=\{\{1,2\}, \{1,3\}, \{2,3\}\}$$ $$).$$

Now, what's behind this?

First, AND MOST IMPORTANTLY, I've broken the definition into separate pieces and worked from the inside out. My version of the definition ends by introducing the $$m$$-crown, and starts by introducing the much simpler $$X_i$$s. Moreover, my version breaks into multiple separate sentences. Think of this all as a kind of cognitive piton: it lets the reader digest the definition in discrete steps, without ever having to be kept guessing what things mean. It also helps prevent errors on your end, by making you think carefully about what exactly is going on at each step along the way - when you write an "outside-in" definition, it's easy to experience a kind of "precision fatigue" and wind up finishing with something unclear or garbled. Finally, it winds up helping you use natural language in a precise way, for the same reason.

Second, I've provided nontrivial explicit examples of everything. Moreover, I haven't tried to mix notations to be more efficient - the only thing I've done is throw in some spacing for readability (which is actually quite useful in some situations, and sets-of-sets is one of those since the curly braces can blur together). This helps the reader both understand your definition and repair it if you have made any errors.

Besides all that, examples also provide a more high-level kind of cognitive piton: in the course of a slew of definitions, they help me keep straight what each thing is and how the various things differ from each other. Really, you should always give examples of everything.

Now let me point out two things I haven't done.

First, I avoided using too many symbols. In particular I avoided quantifiers. There's a positive and a negative aspect to this. The positive aspect is that precise natural language is easier to read than symbolic expressions; it's almost always better to focus on making your explanation precise than using symbols to shorten everything (and the "piton" stuff above is a big help here).

The negative aspect is that unless one has some experience with the relevant formalism, it's easy to misuse - and this is exceptionally true of quantifiers for whatever reason. In the case of your definition, when blindly rendered into English your expression $$\forall(X_i \in S^m : i \in S) \rightarrow X_i = \{ x \subset S :(i\notin x \land|x|=m) \}$$ translates to

• "If for all [missing bound variable], $$X_i$$ is in $$S^m$$ [grammatically incorrect symbol - maybe "such that"?] $$i$$ in $$S$$, then $$X_i$$ is the set of subsets of $$S$$ of size $$m$$ not containing $$i$$."

The second half of that is what you want, but the first half is thoroughly garbled. Indeed, my first guess at rewriting this would be

• "If $$X_i$$ is in $$S^m$$ for every $$i\in S$$, then [rest],"

which is very much not what you want. And reading not being much easier than writing, even if you'd gotten the symbols right your readers might still have trouble following the definition.

Second, I didn't put motivation inside the definition proper. This isn't something you did either, but it is a common issue (and like everything else makes it much easier to mess up natural-language definitions) so it's worth mentioning here. You should definitely include motivation, but put it before or after the definition (or both); keep the definition itself nice and clean.

Here's an example of the sort of thing I'm railing against:

Let FOO be the tensor product of BLAH, which by Theorem 11.36 characterizes the FLEEN completely, with (in order to de-VORP the resulting algebra) BLEELG.

It would be much better to write this as:

Let FOO = BLAH $$\otimes$$ BLEELG.

Remember that BLAH completely characterizes the FLEEN (Thm. 11.36); meanwhile, bringing BLEELG prevents VORPiness.

Note that I've also broken the motivation itself into separate pieces: why the BLAH?, and why the BLEELG?, are separate issues and I've treated them as such.

Now it's important to note that these rules don't always have to be followed. But I think one should follow them very closely if one isn't already rather experienced with this sort of writing, especially if the intended readership isn't necessarily either.

• @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. – Noah Schweber Aug 29 at 21:49
• @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.". – Dave L Renfro Aug 29 at 23:10
• This is truly fantastic advice, and should be a standard reference for anyone writing mathematics. – Martin M. W. Aug 30 at 0:27
• @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! – user21820 Aug 30 at 5:13
• @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. – Noah Schweber Aug 30 at 17:55