The Alphabet Story

[anoni]

Juxtaposing my earliness and your speed, we can see that you are indeed faster (maybe, I'm not sure, lets just use a proof by assumption. Assume True, therefore True, done!)

[Trixsie Vixen]

Karl Popper would suggest we assume false and rely on falsification rather than verification when testing hypotheses

[anoni]

Lousy method if used in experimental science I would think xD

"Look! The substance did this! This is proof the substance does this!"
"Nope, doesn't matter, now if the substance didn't do that, what would happen?"

[Trixsie Vixen]

Mathematics is well suited to verification.  Experimental methodology requires failed attempts at falsification to create inductive evidence.


"look the substance appears to have done this, did it really do something else?  No.  Then we can have confidence it really did what we thought"


*does a little dance*

[anoni]

Not REALLY, because then that requires both validation and falsification xD Also some claims don't require any falsification, such as in a mathematical proofs like "There is at least one prime that is odd", then you could say "3 is prime" and that would be a sufficient proof that at least one odd is a prime. Some existential proofs require no falsification.

[Trixsie Vixen]

Only induction requires falsification.  Deduction requires verification. Abduction uses both to provide an inference to the best explanation.


*cartwheels*

[anoni]

Perhaps, but mathematical induction doesn't necessarily require falsification either :O I'm sure it could be proven with falsification, but some most mathematical inductions don't require it and are based on verification not falsification.

And the abductions is a whole different story, especially if they're alien abductions xD

[Trixsie Vixen]

Quite possibly.  I can't think of a mathematical induction... can you give an example?

[anoni]

Really! With pleasure!

Here's a simple one.

Hypothesis:
1 + 2 + 3 + 4 + ... + n = n(n+1)/2; for n ∈ Z⁺

First of all, prove it exists for one case

Case 1: Let n = 1
1 = 1(2)/2
1 = 1

Therefore holds true for n = 1.

Step 2:
ASSUME it works for the kth test where k ∈ Z⁺

Let n = k
1 + 2 + 3 + ... + k = k(k + 1)/2

Step 3:
Now all we need to do is show it works for the kth + 1 case, seeing as case 1 is true, if we prove it for case k + 1, we can say that if case 1 is true, then case 2 is true. If case 2 is true, then case 3 is true, etc to infinity.

Therefore, let n = k + 1

1 + 2 + 3 + ... + k + (k + 1) = (k + 1)(k + 2) / 2
From step 2 we can reduce LHS with the RHS result in step 2

(k(k + 1) / 2) + (k + 1) = (k + 1)(k + 2)/2 
(k(k+1) + 2(k + 1)) /2 = (k + 1)(k + 2) / 2                  // Make a common denominator
(k² + 3k + 2) / 2 = (k + 1)(k + 2) / 2         // Expand
(k + 1)(k + 2) / 2 = (k + 1)(k + 2) / 2                         // factorize
Therefore LHS = RHS and true

Step 4:
As we concluded that as long as the kth test is true, then the (k + 1)th test will hold true. Seeing as the test held through when k was equal to 1, then by induction, all possible tests for k being a positive integer hold true. Proven by mathematical induction!

[Trixsie Vixen]

Sorry, I was unclear.  When I said induction I meant induction as synonymous with inductive reasoning, whereas mathematical induction is an example of deductive reasoning. It was a silly equivocation on my part which the wikipedia entry on mathematical induction warns me against in the 3rd paragraph. http://en.wikipedia.org/wiki/Mathematical_induction


*rolls eyes*


"silly Trixsie!"

[anoni]

That is pretty silly, I hate when names do that. >.<

[Trixsie Vixen]

Unless I am mistaken, we have scared everyfur away?

[anoni]

Very interesting, deductive proofs scare furs away, can we come up with an inductive proof to prove this?

[Trixsie Vixen]

Well this give a good example of my point.  Any finite number of examples of deductive proofs scaring away furs does not prove it, but one counter example disproves it, therefore we attempt counter examples ie falsification and either disprove or fail to disprove, therefore showing it is probably (but not definitely) true.

[anoni]

X + 1, I dunno. ANYWAY, I'm not saying there aren't proofs that should be based on falsification, counter example or contradiction, just that there are proofs that don't need that xD