Timeline for Why are polynomials so useful in mathematics?
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Jun 16, 2014 at 1:59 | comment | added | Todd Trimble | One origin from which addition is abstracted is the idea of taking a union of two disjoint collections, which with hindsight we recognize as a coproduct. Similarly the idea of forming rectangular arrays to represent repeated addition is abstracted as multiplication; with hindsight we recognize rectangular arrays as categorical products. So yes, addition and multiplication and their "laws" (e.g., commutativity) evolved by abstracting more concrete instantiations, but with hindsight many such concrete processes can be seen (and their "laws" explained) as instances of categorical constructions. | |
| Jun 15, 2014 at 17:00 | history | made wiki | Post Made Community Wiki by Todd Trimble | ||
| Jun 15, 2014 at 16:04 | comment | added | Qiaochu Yuan | @Tom: but you can run it forward; the yoga of Lawvere theories implies that you can define commutative algebras as precisely those things whose elements you can apply polynomials to. So everything comes full circle. | |
| Jun 14, 2014 at 13:16 | comment | added | Joseph O'Rourke | Indeed, my question was too broad to admit a sharp answer. But I learned the most from Qiaochu's viewpoint. | |
| Jun 14, 2014 at 6:12 | comment | added | Qiaochu Yuan | Yeah, honestly I'm not sure this really addresses the question either, but then again the question is extremely broad. | |
| Jun 14, 2014 at 4:14 | comment | added | Todd Trimble | Joseph, while I agree Qiaochu gave a good answer, it doesn't address your question with your particular choices of coefficient rings ($\mathbb{Q}, \mathbb{R}$, etc.). In other words, your question seems a bit more specific than what this answer answers. | |
| Jun 13, 2014 at 21:57 | vote | accept | Joseph O'Rourke | ||
| Jun 13, 2014 at 21:57 | comment | added | Joseph O'Rourke | "...polynomials are ... the only natural operations on ...": This is, to me, the compelling point. | |
| Jun 13, 2014 at 1:52 | comment | added | Todd Trimble | Of course, this is basically the same as the comment that polynomials constitute free commutative algebras over a given coefficient ring: the Lawvere theory is dual to the category of finitely generated free objects. | |
| Jun 13, 2014 at 1:49 | history | answered | Qiaochu Yuan | CC BY-SA 3.0 |