A side note - these types of problems (finding patterns) tend to be fun and very approachable math problems. You can almost guarantee that the exact same thing has been solved before, but like solving cross-words, sudoku, or whatever - it's easy to get drawn to them.
Formally, if one expands <math>(x-r_1) (x-r_2) \cdots (x-r_n),</math> the terms are precisely <math>(-1)^{n-k}r_1^{b_1}\cdots r_n^{b_n} x^k,</math> where <math>b_i</math> is either 0 or 1, accordingly as whether <math>r_i</math> is included in the product or not, and ''k'' is the number of <math>r_i</math> that are excluded, so the total number of factors in the product is ''n'' (counting <math>x^k</math> with multiplicity ''k'') – as there are ''n'' binary choices (include <math>r_i</math> or ''x''), there are <math>2^n</math> terms – geometrically, these can be understood as the vertices of a hypercube. Grouping these terms by degree yields the elementary symmetric polynomials in <math>r_i</math> – for ''x<sup>k</sup>,'' all distinct ''k''-fold products of <math>r_i.</math>
Math is the subset of fiction that, for some reason inherent in the human brain and\or the universe, is always invented/re-invented with the exact same essential details across several non-colluding minds.
Mathematical mental artifacts are special stories with incredible mutation-resistent properties, memes which compel their host with astonishing power and insistence to interpret and extract from them the same conclusions and patterns and behaviours that other hosts infected with the same meme did\do, or at least prevent one host from interpreting or concluding the opposite of what most other hosts interpreted\concluded.
That is, Mathematical Reasoning is a distributed mind-consensus 0-communication algorithm, a way for minds (instantiated in brains, silicon chips,spacetime foams,...) to agree on something while never actually explicitly arguing or debating or handshaking in any way. The set of all objects and states Mathematical Reasoning is capable of agreeing on is by definition Mathematics.
Discovering sentient aliens would shed much light on this.
There's a distinction between whether the objects of mathematics are "invented" vs. whether they are subjective. These arent equivalent notions.
It's possible to maintain mathematics is discovered and its objects are subjective. This would be akin to the kantian position, in which mathematics constitues something like the formal structure of subjective experience itself.
'
Either way, the options arent "invented subjectivity, vs. discovered platonism
TrackerFF|3 years ago
graderjs|3 years ago
eneas|3 years ago
Banana699|3 years ago
Mathematical mental artifacts are special stories with incredible mutation-resistent properties, memes which compel their host with astonishing power and insistence to interpret and extract from them the same conclusions and patterns and behaviours that other hosts infected with the same meme did\do, or at least prevent one host from interpreting or concluding the opposite of what most other hosts interpreted\concluded.
That is, Mathematical Reasoning is a distributed mind-consensus 0-communication algorithm, a way for minds (instantiated in brains, silicon chips,spacetime foams,...) to agree on something while never actually explicitly arguing or debating or handshaking in any way. The set of all objects and states Mathematical Reasoning is capable of agreeing on is by definition Mathematics.
Discovering sentient aliens would shed much light on this.
mjburgess|3 years ago
It's possible to maintain mathematics is discovered and its objects are subjective. This would be akin to the kantian position, in which mathematics constitues something like the formal structure of subjective experience itself. ' Either way, the options arent "invented subjectivity, vs. discovered platonism
ux|3 years ago
Joke aside, yeah various people pointed that out to me so I quickly added an edit at the end of the post. Thanks :)
unknown|3 years ago
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