Also, if your examples are made only of the thing you're trying to explain, they're not examples.
A cake is a baked good containing one or more edible substances, created by a baker. Cakes are made in a kitchen, using appliances, implements and utensils. An appliance is a powered mechanical device used to augment the baker's abilities, different from an implement and a utensil. Implements and utensils are unpowered tools used by hand to produce batter from ingredients. A batter is a mixture produced in stages, consisting of a base starch, proteins and flavoring agents.
For a concrete example of a cake, consider a cake containing one or more fruits. The fruit may be uniformly distributed among the cake's contents or clustered in toppings. This produces the cake's characteristic flavor and color, though color may also result from artificial coloring agents.
Through prolongued exposure to heat, we can transform batter into a cake. Molds may be used for this. Any bowls, spoons or whisks used in the process may contain residual batter suitable for direct consumption. [1]
In some ways I've stopped just when things were about to get good, but now you should have a solid understanding of the fundamentals.
[1] Licking is actually quite natural in this context.
No, seriously. Just look what this example does - it recursively provides a short description for each of the important concepts in the process of baking cakes. After reading this, I now feel I have a decent map of the topic - both what the things / processes involve and what they don't involve. I've also been introduced to domain terminology that will help me understand other sources.
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As for original question, why definitions - I believe it's good to have precise definitions next to casual explanations of the same things, just so that the reader can quickly connect the casually discussed aspect with the precise definition, and answer some of their own questions (which authors can't predict) through thinking.
Note e.g. that I never once cited an actual cake, I merely provided an example of a category of cakes. This too is a problem with mathematics, so in love with their abstractions, they no longer notice the lack of the concrete.
>Sorts and function symbols are just symbols. Something is a sort if it is in the collection of sorts. Nothing else is required.
If you gave up before reading this, then you didn't even read the 4th paragraph.
She even appeared on the late show (https://m.youtube.com/watch?v=mA402F5K47o).
Basically, after you've been exposed to a lot of mathematics you develop a skill where you can just accept abstract definitions and go with it.
The syntactic approach is really useful as a programmer. A Logical Approach to Discrete Math[0] teaches programmers how to develop proofs in this style using logic and syntactic substitution. Once shown how easy it is to achieve rigour it is hoped that work-a-day programmers can formulate their own proofs and develop stronger specifications using mathematical tools.
I'm not a formally trained mathematician and am completely self-taught as a programmer but with some background in set and graph theory I found it approachable. It's certainly a good companion text to learning more practical systems like TLA+ or Lean.
[0] https://www.amazon.ca/Logical-Approach-Discrete-Math/dp/0387...
As a writer or a teacher, if you want to reach a broad audience, give some simple, concrete examples, then explain the abstraction that covers them all. Assuming mathematical maturity won't help you reach your audience.
One thing that is required by a formal notation is that before you write a statement in your formalised notation, you need to define the terms and symbols and how they interact with one another or the statement makes no sense to the reader.
I would say this is true even when you're trying to use words rather than some type of syntax, because what you are trying to do is paint a very specific picture (or concept) in peoples minds, and you need these definitions there or people wouldn't understand them.
If you're trying to be concise and to constrain the space of possible interpretations of the words you have written, to only contain the concept that you are trying to explain, and not contain other valid interpretations of the words, that may disagree with the concept you are trying to get the reader to explain, the easiest way to do so is probably with up-front definitions of the terms you are about to use.
TL;DR I posit that an up-front definition of terms simplifies explanation of formal notation (or words) which is necessary to constrain the semantic idea-space to only the valid definition trying to be conveyed.
You probably already "developed it" if you're looking back on even just a few years of full-time coding, typically "multi-paradigm" anyway (imperative, "functional" when you dig out SQL, denotative/descriptive for the common web DSLs html/css).
You never formalized or looked for existing formalizations of your organically acquired & accumulated intuitions and "instincts" (covering numerous little assimilated-by-osmosis laws and relations and "ittt" implications) because no real need ever arose, nothing in the real world prompted you to, and if the passing thought to do so ever occurred to you briefly, you saw no particular promise in going down that rabbit hole. I think that's the bulk of C.T. (just covering logical/mathy sciences/fields from a higher-vantage-point in a blurrier meta way). Once you would do so, you'd find it also shockingly hard to really interrelate and out-formulate and staking-off-boundaries those, figuring out "the laws of the laws" etc.
Admittedly, this my current assessment of "category theory" may be quite incomplete, pending further future insights & compelling persuasion to reassess it from scratch.
((( NB. imho we current crop of developers (and indeed most other "brainy" disciplines I'd wager) in many ways (besides of our syntaxes, schemas, algos, type systems that form the core "laws of nature") are still quite a bit like our stone/ice-age ancestors: in a way they enjoyed quite a grasp of the various facets (related to survival & thrival) of physics, biology, botanics, some chemistry, some "meteorology", geography and much more --- BUT strictly local, not formalized, formulated, no formal core/base language such as maths, and seeing no need for any of it to put fresh mammoth on the table. Quite an enjoyable state of affairs, too --- until you run out of mammoths and need to invent agriculture, thus cities, thus money, thus maths, thus science, not necessarily in that order =) so even though coding seems rigid and formal, a large part in reality is somewhat primal/instinctive/intuition-from-experience and stuff like C.T. is the "meta" attack on this organically-grown primitive (though again, tremendously enjoyable for the coder himself) state of affairs.. )))
Uh, no? Are you really arguing that (e.g.) a book teaching a programming language has no examples unless some of them are in another language? How on earth do you learn your first one?
That's just recursive definition. Some concepts demonstrably can't be exemplified otherwise.