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People who hate math aren't going to be inspired by a collection of axioms and equations. Nor are they likely to be wowed by the examples of matrices and vector
by alex-g 15y ago
People who hate math aren't going to be inspired by a collection of axioms and equations. Nor are they likely to be wowed by the examples of matrices and vectors. Some examples from geometry, or permutations, might be more likely to teach the concepts.
People who know something about math are going to spot several errors in the presentation. For example, the reals with multiplication don't form a group, since 0 has no multiplicative inverse.
Not an error, but a subtlety: the collection of axioms here says that the group has a right identity and right inverses. It's later assumed that the identity is also a left identity (in the proof of uniqueness). This is true, as it happens, but it's not entirely obvious. The usual presentation of groups has two-sided identities and inverses up-front.
Actually, the alternative axiom set of a right identity, but left inverses, has models that are not groups. An example is a left zero semigroup, where ab = a for all a and b. We have right identity (ae = a; no problem) and left inverses (for all a, there exists b such that ba = e; yes, just take b = e). This isn't a group! (OK, unless it has only one element.)
Maybe this illustrates an interesting point for developers: the axiomatic approach is like having a common interface for some class of objects, and it's helpful to be able to reason about them without knowing the details of how they're implemented. But reasoning about the expressive power of different interfaces or specifications can be horribly difficult. The Robbins conjecture (about the equivalence of two axiom sets for Boolean algebra) was open from 1933 to 1996 - and proved in an automated theorem prover, no less.
- dhaivatpandya 15y agoUpvoted. Very true, and thanks for the mistakes and stuff. I'll fix the ones I can, and I'm adding some more stuff that actually relates to CS.
- alex-g 15y agoAre you also going to fix the assertion that matrices form a group under multiplication? (They don't: not all matrices have inverses. The general linear group is what you are probably looking for.) It's a bit embarrassing that of your three examples of groups, only one is actually a group...
- dhaivatpandya 15y agoSorry, again. I'm really trying my best to keep it all correct, but, I do make mistakes.
- wging 15y agoAnother mistake you might want to fix: >(like, matrix multplication isn't commutative, that means: A∗B≠B∗A when B and A are matrices) You want to say something like "A∗B isn't necessarily equal to B∗A when B and A are matrices." Neither A∗B≠B∗A nor A∗B=B∗A are true in general when A and B are matrices.
- dhaivatpandya 15y agoFixed. Thanks for telling me these things, it really helps.
- akkartik 15y agoThe historical context was useful.
- deleted 15y ago[deleted]
- sirclueless 15y agoI think that your last point is a worthy tie-in to the developer's world. For example, for those who use C# or Java regularly: groups, Abelian groups, and integers under addition are roughly analogous to the Enumerable interface, the List interface, and the ArrayList class. Obviously they serve different purposes, but their role in their respective professions is approximately the same. Groups appear all over the place in math, and the Enumerable interface is baked right into the for(each) loop. Abelian groups add commutativity to general groups, and the List interface adds indexing and insertion to Enumerables. Meanwhile integers/ArrayLists are basically the go-to concrete implementation that mathematicians/programmers default to when it comes to Abelian groups/lists.
- dhaivatpandya 15y agoyeah, that's very true. That analogy works excellently until you consider the implementation of these things, and then its a complete mess because there are side effects all over the place, except in languages like haskell.
- ArbitraryLimits 15y ago> People who hate math aren't going to be inspired by a collection of axioms and equations. So true. My favorite quotation on this subject (admittedly, there aren't that many): "Many a graduate student has come to grief when they discover, after a decade of being told they were "good at math," that in fact they have no real mathematical talent and are just very good at following directions. Math is not about following directions, it's about making new directions...It would be bad enough if the culture were merely ignorant of mathematics, but what is far worse is that people actually think they do know what math is about - and are apparently under the gross misconception that mathematics is somehow useful to society!...In any case, do you really think kids even want something that is relevant to their daily lives? You think something practical like compound interest is going to get them excited? People enjoy fantasy, and that is just what mathematics can provide - a relief from daily life, an anodyne to the practical workaday world." - Paul Lockhart Also relevenat, by the same author: "You don't start with definitions, you start with problems. Nobody ever had an idea of a number being "irrational" until Pythagoras attempted to measure the diagnonal of a square and discovered that it could not be represented as a fraction. Definitions make sense when a point is reached in your argument which makes the distinction necessary. To make definitions without motivation is more likely to cause confusion."
- dhaivatpandya 15y agoI love Paul Lockhart's writings...