Anti-Infinity

Great Debates
161

With all due respect, that’s not a very good characterization of mathematical induction. Induction does not generalize in any haphazard manner. Mathematical induction states:

If 1 has property P, and, whenever n has property P, n+1 must also have property P, then every natural number has property P.

To characterize it as reviewing a few numbers, seeing that they have property P, and concluding that all the numbers have P is not what’s happening at all–it’s more like a “domino effect” that forces all the numbers to have the property. By the way, it’s also equivalent to the well-ordering property of the natural numbers, which is that every nonempty set of natural numbers has a smallest element.

In particular, it’s not even slightly similar to the argument “I personally know of some black people who are bad or lazy, therefore black people as a whole must be bad or lazy.” I know this will sound weird, but induction requires a well ordering to be put on the set, and black people are not well ordered.

162

And, in the interest of equality, neither are white people.

163

Cabbage is right.

Induction, done properly, can lead to valuable insights, and in fact is the method by which we draw our axioms that serve as the basis of our deductions. We all use induction, for example to solve integrals.

But the caveat stands. There is a lot of abuse out there by people who induce carelessly.

“It is lucky for rulers that men do not think.” — Adolf Hitler

164

Libertarian

I’ll drink to that! …oops, my glass is empty.

.

165

I’ll send you a new bottle, Father. What’s yer pleasure?

“It is lucky for rulers that men do not think.” — Adolf Hitler

166

This is what I thought Lib. was referring to when he said that proof 2 was originally an inductive proof.

Now, is there any reason to prefer an inductive proof of this type to a deductive one? I know from experience that the first couple times you do 'em, they’re a bit hard to grasp, but are they weaker in any sense?

BTW: even if people were well ordered, it’d be hard to argue that if person n is bad and lazy, that person n+1 must be.

167

This is what I thought Lib. was referring to when he said that proof 2 was originally an inductive proof.

Now, is there any reason to prefer a deductive proof to an inductive proof of this type? I know from experience that the first couple times you do 'em, they’re a bit hard to grasp, but are they weaker in any sense?

BTW: even if people were well ordered, it’d be hard to argue that if person n is bad and lazy, that person n+1 must be.

168

Hunsecker:

The essential “weakness” in any induction is its potential for refutation by a single counter-example. There is no such weakness inherent in deduction, because a set of premises follow from universally accepted axioms.

A physicist friend, for example, pointed out that induction failed for the derivation of 0.0[n-1]1 because the process of extrapolation in Premises 2 through 6 did not necessarily carry through to Premise 7. I solved the problem by presenting the string of numbers to the right of the decimal as a sequence of natural numbers in their own right (i.e., digits) in Axioms 4 and 5.

Deduction is generally more satisfactory because parties are agreeing to certain postulates a priori. The conclusion is therefore inescapable so long as all the premises follow from precedent premises.

“It is lucky for rulers that men do not think.” — Adolf Hitler

169

It’s easy to well-order people; there’s only a finite :wink: number of us.

All you have to do is dig a nice deep well, dump everybody in the world in there, one by one (last person in has to jump rather than be dumped), and we’ve got a well-ordering, without having to invoke the Axiom of Choice.

Speaking of which, what’s yellow and equivalent to the Axiom of Choice?

170

It’s not a banana, RTE. I’m just happy to see you.

:wink:

171

Would that be Zorn’s Lemon?

172

Got it in one, Cabbage! And if you can take more of this punishment (here’s where you say, “I’ll take bad math jokes for $200, Alex”), what’s purple and commutes?

Spiritus - ooh, that was baaaad.

Maybe they could use your banana in Athena and Byzantine’s naked pillowfight on MPSIMS… :wink:

173

How about an abelian grape?

Here’s one for you: Why can’t a computer tell the diffence between Christmas and Halloween?

174

Sorry to interrupt the math hijinks (What do you do with a broken limit?), but I think Lib. and I are talking about 2 different kinds of proofs by induction. (Lib., let me know if I’m putting words in your mouth you never intended)

If you’re trying to prove property P(n) holds, for all integers n…

Lib is saying that if a proof goes:
*
P(1) is true
P(2) is true
…
P(500) is true
so P(n) must be true for all integers
*
Which is obviously a flawed proof, because you haven’t checked all the integers.

I’m talking about induction as in proofs of the form:
Step 1) P(1) is true
Step 2) If P(n) is true, then P(n+1) has to be true.

So for any integer, you could show a deductive proof that started at 1, and repeated step 2 until you got to n.

Proofs of this type can’t be refuted by a counter example, because if one existed, you would never have been able to show step 2.

OK, so I think we’ve been talking about different things when we say induction, and my original question thus goes “poof”.

175

Libertarian

As Hunsecker points out, there is a difference between ordinary run-of-the-mill induction and formal induction. Formal deduction is as susceptible to refutation by a single counter-example–since that would call into doubt some of your premises.

Let’s hear it! Anybody want any more beer? Lib’s buying.

.

176

OK, why can’t a computer tell the difference between Christmas and Halloween?

Because 31[sub]OCT[/sub] = 25[sub]DEC[/sub]!

Hey, stop groaning!

177

This thread is the most difficult, frustrating, incomprehensible thread I’ve ever come across in all my years on the Straight Dope MB. Even “Moron on a Moving Truck” doesn’t come close. You geniuses who understand this stuff have my salute! Because it’s a total WHOOOOOOOSH!! for me. :wink:

“God would never let me be happy. He’d kill me first.”
“I thought you didn’t believe in God.”
“I do for the bad things.” - George Costanza

178

Cabbage:

Yes, you’re right to that extent. We have been talking past each other a bit.

But it is important to keep in mind that you may induce P(n+1) to be true when P(n) is true because the induction hypothesis is axiomatic. It is the fifth of the five Peano Axioms: “any property that belongs to zero, and also to the immediate succesor of any natural number to which it belongs, belongs to all natural numbers.” Along with that, and the other four axioms, Peano deduced that 1 + 1 = 2.

Induction was first formalized by Augustus de Morgan in Formal Logic, based on the principle he called quantification of the predicate. He said that for an inductive proof to be valid, that is, to be universally true, it must be complete, which means that what is known to apply for a particular case must be shown to apply for any and every particular case of the given kind.

Deductively, de Morgan (famous for De Morgan’s law, Not(A and B) <=> Not A or Not B) proved that, by following these three steps, you may draw reasonable conclusions concerning any natural number n: (1) establish that your theorem is true for some starting value N; (2) assume that it is valid for a certain value n = p >= N (the “induction hypothesis”); and (3) prove that it then also holds true for the next higher value of n.

(sources: Mathematics from the Birth of Numbers, Jan Gullberg, W. W. Norton & Company, 1997; The Nature and Growth of Modern Mathematics, Edna E. Kramer, Princeton University Press, 1982)

RMMentock:

[quote]
Formal deduction is as susceptible to refutation by a single counter-example–since that would call into doubt some of your premises.

[quote]

With all due respect, Father, counter-examples do not apply to deduction because you are reasoning from the general to the particular. In other words, what you would propose as a counter-example is itself an element of the general set. The only way to knock down a deductive proof is either to show that an axiom is false or else that a premise is wrongfully derived (as you did for Proof 1).

Forgive me.

“It is lucky for rulers that men do not think.” — Adolf Hitler

179

One more question for socialists that has me intrigued: why is you fetish directed exclusively toward material wealth and wealth-equity?

Why don’t you force people who are smart to tutor people who are dumb? Why don’t you make people who have two perfectly good kidneys “donate” one to people who don’t? Why don’t you make Michael Jordan play basketball in leg braces so the playing field will be more level? Why don’t you make people with extra bedrooms house people who are homeless?

In other words, why aren’t you consistent?

“It is lucky for rulers that men do not think.” — Adolf Hitler

180

One more question for socialists that has me intrigued: why is your fetish directed exclusively toward material wealth and wealth-equity?

Why don’t you force people who are smart to tutor people who are dumb? Why don’t you make people who have two perfectly good kidneys “donate” one to people who don’t? Why don’t you make Michael Jordan play basketball in leg braces so the playing field will be more level? Why don’t you make people with extra bedrooms house people who are homeless?

In other words, why aren’t you consistent?

“It is lucky for rulers that men do not think.” — Adolf Hitler