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I don't understand why anyone thinks step 1 in conveying any concept is definitions. This is not how brains work. Definitions are how you formalize your underst
by stillkicking 9y ago
I don't understand why anyone thinks step 1 in conveying any concept is definitions. This is not how brains work. Definitions are how you formalize your understanding once you've developed it. If this is meant to be an introduction, it's entirely backwards.
Also, if your examples are made only of the thing you're trying to explain, they're not examples.
- stillkicking 9y agoIntroduction to Cake Making 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.
- TuringTest 9y agoThis would actually be a very good introduction for a race of alien visitors who'd never heard of cakes, baking, or kitchens.
- bbcbasic 9y agoIf they are Vulcans or androids.
- shmageggy 9y agoNo it wouldn't for the same reason the article isn't a good introduction -- it relies on concepts that a naive reader doesn't understand. In your example the aliens are unfamiliar with kitchens, but the text mentions kitchens without any explanation. Similarly the article mentions and uses "sort" without any explanation. It seemed to be a fundamental concept but without any idea what a sort was I gave up. Every article on category theory does this.
- jjaredsimpson 9y agoThis isn't true. >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.
- MichaelBurge 9y agoThere are 2 sentences between the first use of 'sort' and its explanation, which is that they are symbols with no interpretation. It looks like he defines function symbol before sort, presumably because function symbols are more important than sorts.
- Xophmeister 9y agoMore recipe books should be written by mathematicians ;)
- 1001101 9y agoEuginia Cheng, a category theorist, has written Cakes, Custard and Category Theory: Easy Recipes for Understanding Complex Maths, and How to Bake Pi: An Edible Exploration of the Mathematics of Mathematics.
- seanmcdirmid 9y agoFascinating! https://en.wikipedia.org/wiki/Eugenia_Cheng https://en.wikipedia.org/wiki/Eugenia_Cheng She even appeared on the late show (https://m.youtube.com/watch?v=mA402F5K47o https://m.youtube.com/watch?v=mA402F5K47o).
- andrzejsz 9y agoThis first title disappeared in favor of the second that is "How to bake Pi" I guess pure marketing move
- gumby 9y agoAfter the proof that one may substitute slivered almonds for chopped crystalized fruit, I have a lemma that says one may substitute slivered almonds for a subset of the chopped crystalized fruit.
- danidiaz 9y agoIndeed, like "Burritos for the Hungry Mathematician" http://emorehouse.web.wesleyan.edu/silliness/burrito_monads.pdf http://emorehouse.web.wesleyan.edu/silliness/burrito_monads....
- vanderZwan 9y agoSo basically, the Guy Steele approach to cooking https://www.youtube.com/watch?v=_ahvzDzKdB0 https://www.youtube.com/watch?v=_ahvzDzKdB0
- diyseguy 9y agohar! please do keep going
- TeMPOraL 9y agoThis is Actually Good, and I'd love more stuff written that way! 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. -- 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.
- justifier 9y agoYou may enjoy: http://www.cookingforengineers.com http://www.cookingforengineers.com
- stillkicking 9y agoNo it's terrible, and the fact that you don't notice is exactly the problem. See also: monads. If you understand the basic process of what's going on (preparing food) and what it's meant to do (produce a delicious treat), formalizing your understanding in this way feels like concise and clear insight. If you don't yet know anything, it will be meaningless word salad where each 'answer' produces a multitude more questions. 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.
- wfisher 9y agoAmazing. I couldn't agree more this is obviously a useless way to learn baking at first.
- lmm 9y agoThis is explicitly a syntactic approach. It's treating this as a game you play with symbols. I found it very helpful, so maybe some brains work differently from others.
- dualogy 9y ago> Definitions are how you formalize your understanding once you've developed it 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.. )))
- johncolanduoni 9y ago> Also, if your examples are made only of the thing you're trying to explain, they're not examples. 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?
- klibertp 9y agoStrictly speaking all programming language books have the examples shown in two languages: the programming one and the natural language.
- mej10 9y agoBrains can work that way. It is a skill, and this skill even has a name: mathematical maturity. 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.
- agentultra 9y agoI would expect no less even in an introductory text. Mathematics is meant to be precise and formal mathematics even more so. Prosaic pedagogy might suit a liberal arts introduction but it won't help the reader to convince themselves that the statements and theorems proposed are true and valid. 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/0387941150 https://www.amazon.ca/Logical-Approach-Discrete-Math/dp/0387...
- skybrian 9y agoGreat that it works for you, but it won't convince anyone of anything if they don't understand what it means or why they should care. 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.
- mijoharas 9y agoThis makes me think about formalised notation. If you have a mathematical formalised notation, you can express things either more expressively, or with less ambiguity (or both) than without it. 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.
- gone35 9y agoIf your examples are made only of the thing you're trying to explain, they're not examples. That's just recursive definition. Some concepts demonstrably can't be exemplified otherwise.