One of the ways to answer 0!=1 in math is by "completing the pattern" (4!=5!/5,…

1F1ZT6VXT4LCQcg13Tdp7x99zzoSQLS6ix (2936) · May 13, 2020

One of the ways to answer 0!=1 in math is by "completing the pattern" (4!=5!/5, 3!=4!/4... 0!=1!/1) but if you complete the pattern in the opposite direction going from negative numbers(-4!= -5!/5, -3!=-4!/4)... 0! = -1?
Is this incompleteness?
https://twetch.app/t/e576a2f5dcbcdf03a2dbf94f02ec227881f417bc123fdfaf81d9d1b42f2ea032

Replies

16SgJCRH7fB5ny6wCHabmiGaa6wcbve6WS (21877) · May 13, 2020

Yeah, not so sure about that. The factorial is defined on positive integers by induction. What value to assign the base case of 0 is in a sense a free choice, but you know you'll always want 0!=1 for binomial theorem, etc...

16SgJCRH7fB5ny6wCHabmiGaa6wcbve6WS (21877) · May 13, 2020

As it turns out, of the countably many continuous extensions of factorial to C, there is a unique meromorphic log convex one called the gamma function, but even she has poles at all negative integers. There really is a problem with thinking -1!=0!/0.

1F1ZT6VXT4LCQcg13Tdp7x99zzoSQLS6ix (2936) · May 13, 2020

Thanks for the clarification, I need to read up on my definitions! I'm no mathematician by any means but i do find philosophy of mathematics very interesting.

16SgJCRH7fB5ny6wCHabmiGaa6wcbve6WS (21877) · May 13, 2020

I would certainly agree. I strongly recommend the book Proofs and Refutations by Imre Lakatos if you haven't read it. It's rather accessible and a gem!