Peon wrote:You seem to be conveniently ignoring cases where n is 51 or greater, despite claiming in your "proof" that it works for all n > 1.
AmagicalFishy wrote:The problem isn't the induction, nor is it needing to have a base-case begin at 1 (which isn't a necessity, I just did it because your mother wears combat boots). The problem is trying to apply rigorous mathematical methods to vague and totally arbitrary concepts.
that's your mother, specifically
When
n = 51, it is implied that
-1 people is a party.
We know it's true for 1. We also know that if it's true for
n, it's true for
n+1. Thus, let
n = 1. This means we know it's true for 2. Now we also know it's true for 3—and 4, and 5, and 6, and 7, and 8, and 9, and 10, and 11, and 12, and 13, and 14, and 15, and 16, and 17, and 18, and 19, and 20, and 21, and 22, and 23, and 24 and ... and 51. This is the power of induction. If you successfully prove something
that is well defined by induction, you've successfully proven it. There are no caveats or anything—it's a solid, water-tight proof.
So, I'm not ignoring the case (and am thus doing nothing convenient!!)—it is implied by
n = 1 being true. My proof is solid (except for the fact that it's applied to vague and totally arbitrary concepts). Vague and totally arbitrary concepts ≠ well-defined. The induction method isn't the problem, it's what it's being applied to.
(Don't get me wrong: This doesn't mean we can't apply induction to
non-math things. They just have to be well defined. For example, say I have a pitri dish with a single bacterium which, over time, multiplied into a whole colony. This bacteriam had
Gene A. Let's index all bacteria in my pitri dish with the counting numbers (1 to
∞) in the order of their spawning; if
n spawned at the same time, choose the
n next numbers arbitrarily for each—but the first bacterium is always
1, and all other numbers came from that bacterium.
Now, say we have some biological law that if a bacterium has
Gene A, then all bacteria which spawn directly from it have
Gene A. We can use induction to show that
all bacteria in my pitri dish must have
Gene A (even though they all didn't spawn
directly from the first one). I'm not going to write out the proof, but I imagine the idea seems pretty intuitive.
Of course all bacteria have
Gene A.
But that's all induction is—it's that same deductive (lol) logic, but rigorously math.)