## Search found 327 matches

### Re: Free Will

Well, yeah. That's how tests work. Generally, tests provide evidence, not proof. See: math tests, blood tests, smog tests, citizenship tests, etc. A student who passes a math test is reasonably likely to understand the material, but they may have merely cheated, or crammed, or otherwise scraped by w...

### Re: Free Will

The Turing Test is just a sort of rule of thumb. I don't think a program that can convince you it is a person through a single casual conversation necessarily is. You might want a more rigorous test than that. A simple example of the point: a program with a fixed output that just happens to make th...

### Re: Free Will

The other problem with the Chinese Room is that lookup tables are so incredibly bad for the task that you'd need something closer to a world-sized supercomputer rather than a guy in a room with a book, and "it's just a world-sized supercomputer feeding you responses based on the contents of its...

- Tue May 16, 2017 8:47 pm UTC
- Forum: Mathematics
- Topic: Alien Mathematics: A Math System Beyond our Comprehension?
- Replies:
**24** - Views:
**3657**

### Re: Alien Mathematics: A Math System Beyond our Comprehension?

Soupspoon wrote:Meteoric wrote:And if he isn't, I am, because that would be fun.

Would it make your life complete?

Could it?

Sure, just not consistent.

- Thu May 11, 2017 8:30 pm UTC
- Forum: Mathematics
- Topic: Alien Mathematics: A Math System Beyond our Comprehension?
- Replies:
**24** - Views:
**3657**

### Re: Alien Mathematics: A Math System Beyond our Comprehension?

I think Yakk is suggesting a fictional "third incompleteness theorem", which is even worse than the first two and essentially fatal to the field of logic as we know it.

And if he isn't, I am, because that would be fun.

And if he isn't, I am, because that would be fun.

- Wed May 03, 2017 4:45 pm UTC
- Forum: Mathematics
- Topic: Mathematical Induction - Introductory Question
- Replies:
**252** - Views:
**11914**

### Re: Mathematical Induction - Introductory Question

Right, exactly. You're correct that if we take "odd" to mean "not even", then there's nothing to prove. But under the standard definition about 2m and 2m+1, it takes a little bit of justification. Clearly, if I take 2m and add 1, then I've got 2m+1 which is odd. But what if I tak...

- Wed May 03, 2017 4:27 pm UTC
- Forum: Mathematics
- Topic: Mathematical Induction - Introductory Question
- Replies:
**252** - Views:
**11914**

### Re: Mathematical Induction - Introductory Question

Yes! That is a valid argument, and a perfect example of proof by induction. You have the base step, the induction step, and you correctly justify your conclusion using known properties of even and odd numbers.

- Wed May 03, 2017 7:46 am UTC
- Forum: Mathematics
- Topic: Mathematical Induction - Introductory Question
- Replies:
**252** - Views:
**11914**

### Re: Mathematical Induction - Introductory Question

Demki, your example is a good one, and may sidestep issues about "or" statements in proofs.

- Wed May 03, 2017 6:24 am UTC
- Forum: Mathematics
- Topic: An Eigenbasis Problem
- Replies:
**5** - Views:
**1496**

### Re: An Eigenbasis Problem

On a conceptual level, I understand what needs to be done here. I need some change of basis matrix B and its inverse B -1 , and I need to compute B -1 A n B. I don't think that is what you want to do. Fundamentally, the trick you're trying to use is that exponents of diagonal matrices are easy, but...

- Wed May 03, 2017 5:00 am UTC
- Forum: Mathematics
- Topic: Mathematical Induction - Introductory Question
- Replies:
**252** - Views:
**11914**

### Re: Mathematical Induction - Introductory Question

It seems that you have questions about the even/odd proof, but the questions you're asking map exactly onto things you're struggling with in the original problem. Let me give some explicit examples of this. Here, you are asking why the even/odd proof is valid, even though we haven't established spe...

- Wed May 03, 2017 3:25 am UTC
- Forum: Mathematics
- Topic: Mathematical Induction - Introductory Question
- Replies:
**252** - Views:
**11914**

### Re: Mathematical Induction - Introductory Question

If you want to continue down the even-or-odd induction proof path, then please (for my sanity) explain, using the contents and steps of that even-or-odd proof, how each piece of content and each step from that proof relates to if/then statements and the steps of the original induction proof that I'...

- Tue May 02, 2017 10:55 pm UTC
- Forum: Mathematics
- Topic: Mathematical Induction - Introductory Question
- Replies:
**252** - Views:
**11914**

### Re: Mathematical Induction - Introductory Question

There is not.

You can write the proof a different way, in which you use substitution INSTEAD of adding to both sides. But since you're talking about step 2, where you have already added to both sides, you are definitely not using that way.

You can write the proof a different way, in which you use substitution INSTEAD of adding to both sides. But since you're talking about step 2, where you have already added to both sides, you are definitely not using that way.

- Tue May 02, 2017 10:50 pm UTC
- Forum: Mathematics
- Topic: Mathematical Induction - Introductory Question
- Replies:
**252** - Views:
**11914**

### Re: Mathematical Induction - Introductory Question

There is no substitution step. We add stuff to both sides, rearrange, and that's it.

- Tue May 02, 2017 10:45 pm UTC
- Forum: Mathematics
- Topic: Mathematical Induction - Introductory Question
- Replies:
**252** - Views:
**11914**

### Re: Mathematical Induction - Introductory Question

I will be happy to give such a step-by-step when I am back at a real keyboard, but that will have to wait a few hours. The reason I am avoiding your question about step 2 is that, last time, we spent a full page on the same question, concluding with you saying "I give up". We explained and...

- Tue May 02, 2017 10:26 pm UTC
- Forum: Mathematics
- Topic: Mathematical Induction - Introductory Question
- Replies:
**252** - Views:
**11914**

### Re: Mathematical Induction - Introductory Question

INDUCTION STEP: Assume that k is either even or odd. If k is even, then k+1 is odd. If k is odd, then k+1 is even. So either way, k+1 is either even or odd. I assumed k is either even or odd, then used it to prove that k+1 is. That's how you say I should prove "If k is either even or odd, then...

- Tue May 02, 2017 9:57 pm UTC
- Forum: Mathematics
- Topic: Mathematical Induction - Introductory Question
- Replies:
**252** - Views:
**11914**

### Re: Mathematical Induction - Introductory Question

Isn't that what I did above?

- Tue May 02, 2017 9:49 pm UTC
- Forum: Mathematics
- Topic: Mathematical Induction - Introductory Question
- Replies:
**252** - Views:
**11914**

### Re: Mathematical Induction - Introductory Question

And how do you prove if-then statements? Like, can you prove "If n is even then n^2 is a multiple of 4"?

- Tue May 02, 2017 9:32 pm UTC
- Forum: Mathematics
- Topic: Mathematical Induction - Introductory Question
- Replies:
**252** - Views:
**11914**

### Re: Mathematical Induction - Introductory Question

But you don't seem to know how to prove them, which is what we're doing here. We keep circling around this same problem. The induction step is just a conditional proof.

- Tue May 02, 2017 9:18 pm UTC
- Forum: Mathematics
- Topic: Mathematical Induction - Introductory Question
- Replies:
**252** - Views:
**11914**

### Re: Mathematical Induction - Introductory Question

You're struggling here for the same reason I've noted before: you don't seem to understand the difference between a conditional and its parts.

You CANNOT understand induction without first understanding conditionals. They're what induction is made of.

You CANNOT understand induction without first understanding conditionals. They're what induction is made of.

- Tue May 02, 2017 8:54 pm UTC
- Forum: Mathematics
- Topic: Mathematical Induction - Introductory Question
- Replies:
**252** - Views:
**11914**

### Re: Mathematical Induction - Introductory Question

That IS the proof.

Suppose I claimed that, somewhere out past six billion, there is a number that is neither even (a multiple of 2) nor odd (a multiple of 2, plus 1). This proof by induction shows that isn't true: the number before it is either even or odd, so my number is either odd or even.

Suppose I claimed that, somewhere out past six billion, there is a number that is neither even (a multiple of 2) nor odd (a multiple of 2, plus 1). This proof by induction shows that isn't true: the number before it is either even or odd, so my number is either odd or even.

- Tue May 02, 2017 8:48 pm UTC
- Forum: Mathematics
- Topic: Mathematical Induction - Introductory Question
- Replies:
**252** - Views:
**11914**

### Re: Mathematical Induction - Introductory Question

I don't know how that counts as an induction proof. I've established only two alternatives, and not a single answer. In other words, there are two rungs and I haven't told anyone which rung we're actually on. The rungs in induction are always the natural numbers. Even/odd are not rungs. You've prov...

- Tue May 02, 2017 8:41 pm UTC
- Forum: Mathematics
- Topic: Mathematical Induction - Introductory Question
- Replies:
**252** - Views:
**11914**

### Re: Mathematical Induction - Introductory Question

Ah. Could we skip Step 1 (the k case) and proceed directly to the case in which we connect up the k case and the (k + 1) case? No, because step 1 is a part of the process of connecting those. Also, I don't understand how Step 2 and (whatever step involves substituting (k + 1) for every instance of ...

- Tue May 02, 2017 8:32 pm UTC
- Forum: Mathematics
- Topic: Mathematical Induction - Introductory Question
- Replies:
**252** - Views:
**11914**

### Re: Mathematical Induction - Introductory Question

Great! Then since k is either even or odd, we know that k+1 is odd (in the first case) or even (in the second case). That is, k+1 is either even or odd. That's the induction step. We took a fact about k, and used it to demonstrate a fact about k+1. Yes, but I don't understand how that's an example ...

- Tue May 02, 2017 8:26 pm UTC
- Forum: Mathematics
- Topic: Mathematical Induction - Introductory Question
- Replies:
**252** - Views:
**11914**

### Re: Mathematical Induction - Introductory Question

Great! Then since k is either even or odd, we know that k+1 is odd (in the first case) or even (in the second case). That is, k+1 is either even or odd.

That's the induction step. We took a fact about k, and used it to demonstrate a fact about k+1.

That's the induction step. We took a fact about k, and used it to demonstrate a fact about k+1.

- Tue May 02, 2017 8:20 pm UTC
- Forum: Mathematics
- Topic: Mathematical Induction - Introductory Question
- Replies:
**252** - Views:
**11914**

### Re: Mathematical Induction - Introductory Question

Okay, so according to that definition, is 1 even, or is it odd? And if I wanted to be pedantic, would it be valid to just answer "yes"? We could answer "yes" and be correct but that wouldn't yield any meaningful information. Right, but it would show that the base step holds. In ...

- Tue May 02, 2017 8:18 pm UTC
- Forum: Mathematics
- Topic: Mathematical Induction - Introductory Question
- Replies:
**252** - Views:
**11914**

### Re: Mathematical Induction - Introductory Question

Okay, so according to that definition, is 1 even, or is it odd?

And if I wanted to be pedantic, would it be valid to just answer "yes"?

And if I wanted to be pedantic, would it be valid to just answer "yes"?

- Tue May 02, 2017 8:14 pm UTC
- Forum: Mathematics
- Topic: Mathematical Induction - Introductory Question
- Replies:
**252** - Views:
**11914**

### Re: Mathematical Induction - Introductory Question

Look at the base step. Is 1 even or odd?

- Tue May 02, 2017 8:10 pm UTC
- Forum: Mathematics
- Topic: Mathematical Induction - Introductory Question
- Replies:
**252** - Views:
**11914**

### Re: Mathematical Induction - Introductory Question

The outline of an induction proof always looks like this:

BASE STEP: 1 is either even or odd, because [???]

INDUCTION STEP: Assume that k is either even or odd. Then [???], so k+1 is either even or odd.

What should go in the [???] boxes?

BASE STEP: 1 is either even or odd, because [???]

INDUCTION STEP: Assume that k is either even or odd. Then [???], so k+1 is either even or odd.

What should go in the [???] boxes?

- Tue May 02, 2017 8:04 pm UTC
- Forum: Mathematics
- Topic: Mathematical Induction - Introductory Question
- Replies:
**252** - Views:
**11914**

### Re: Mathematical Induction - Introductory Question

I am so invested in this one that I can't reasonably stop working on it now. You can, and I really recommend you do. I'm not telling you to give up, I am telling you to back up and get a running start. And the even-odd problem strikes me as even harder than this one (in fact, the even-odd problem l...

- Tue May 02, 2017 7:50 pm UTC
- Forum: Mathematics
- Topic: Mathematical Induction - Introductory Question
- Replies:
**252** - Views:
**11914**

### Re: Mathematical Induction - Introductory Question

I don't understand that question, sorry. Are you suggesting that the left-hand sides are always supposed to be the same? No. We have lost sight of the goal again. This is not your fault: it is impossible to keep your eye on the ball when we constantly have page-long detours onto other topics. So se...

- Tue May 02, 2017 7:37 pm UTC
- Forum: Mathematics
- Topic: Mathematical Induction - Introductory Question
- Replies:
**252** - Views:
**11914**

### Re: Mathematical Induction - Introductory Question

So do I simply add (k + 1) to both sides or do I also substitute (k + 1) in place of every single reference to k? That's what I'm currently pondering. The thing you are trying to prove is that these both amount to the same thing, is one way to look at it. You have to show that they're equivalent. I...

- Tue May 02, 2017 7:35 pm UTC
- Forum: Mathematics
- Topic: Mathematical Induction - Introductory Question
- Replies:
**252** - Views:
**11914**

### Re: Mathematical Induction - Introductory Question

1 + ... + k = k(k+1)/2 [???] 1 + ... + k + (k+1) = (k+1)(k+2)/2 What steps could go in the [???] box to make this a valid proof? Well, the left-hand sides sure look similar, yeah? We just need an extra (k+1) term. So let's add that in. 1 + ... + k = k(k+1)/2 1 + ... + k + (k+1) = k(k+1)/2 + (k+1) [b...

- Tue May 02, 2017 7:25 pm UTC
- Forum: Mathematics
- Topic: Mathematical Induction - Introductory Question
- Replies:
**252** - Views:
**11914**

### Re: Mathematical Induction - Introductory Question

But, when I did that before, someone said it was wrong. When you did that before, we said it was not the final goal . It was absolutely, incontrovertibly correct - but you were trying to figure out where the proof should end , and that's not the last step. So which is it, and why? Do I add (k + 1) ...

- Tue May 02, 2017 7:15 pm UTC
- Forum: Mathematics
- Topic: Mathematical Induction - Introductory Question
- Replies:
**252** - Views:
**11914**

### Re: Mathematical Induction - Introductory Question

MathDoofus wrote:But, when I did that before, someone said it was wrong.

When you did that before, we said it was not the final goal. It was absolutely, incontrovertibly correct - but you were trying to figure out where the proof should end, and that's not the last step.

- Tue May 02, 2017 6:58 pm UTC
- Forum: Mathematics
- Topic: Mathematical Induction - Introductory Question
- Replies:
**252** - Views:
**11914**

### Re: Mathematical Induction - Introductory Question

Then I'm lost again. That's OK. That's how math goes: you are constantly lost, all the time. Before, you wrote that the k+1 case was 1 + ... + (k+1) = k(k+1)/2 This wasn't structurally wrong, it was the right kind of statement. However, you had an error: 1 + ... + (k+1) = (k+1)(k+2) /2 There is no ...

- Tue May 02, 2017 6:46 pm UTC
- Forum: Mathematics
- Topic: Mathematical Induction - Introductory Question
- Replies:
**252** - Views:
**11914**

### Re: Mathematical Induction - Introductory Question

No, you're only trying to show the last part: 1 + 2 + 3 + ... (k + 1) = (k +1)((k + 1) + 1)/2 The preceding equations are not what you're trying to show. Moreover, they are false: for example, the sum of 1 through k is certainly not equal to the sum of 1 through k+1. It seems like you might be using...

- Tue May 02, 2017 6:05 pm UTC
- Forum: Mathematics
- Topic: Mathematical Induction - Introductory Question
- Replies:
**252** - Views:
**11914**

### Re: Mathematical Induction - Introductory Question

You're talking about how to get to the goal. Before you can do that, you have to know what the goal is.

The k+1 case is not inherently connected to the k case. You can state the k+1 case, on its own, without reference to anything else. When you do that, what is the equation?

The k+1 case is not inherently connected to the k case. You can state the k+1 case, on its own, without reference to anything else. When you do that, what is the equation?

- Tue May 02, 2017 6:00 pm UTC
- Forum: Mathematics
- Topic: Mathematical Induction - Introductory Question
- Replies:
**252** - Views:
**11914**

### Re: Mathematical Induction - Introductory Question

Now you've got the left-hand side from the k+1 case, but the right-hand side from the k case.

- Tue May 02, 2017 5:50 pm UTC
- Forum: Mathematics
- Topic: Mathematical Induction - Introductory Question
- Replies:
**252** - Views:
**11914**

### Re: Mathematical Induction - Introductory Question

The (k + 1) case would be replacing all references to k with (k + 1), yes? But I'm not sure whether I do that on both the left and right sides of the equation above! What exactly does "replacing all references" mean? You're not wrong, you just need to be more precise. The (k+1) case is a ...

- Tue May 02, 2017 5:41 pm UTC
- Forum: Mathematics
- Topic: Mathematical Induction - Introductory Question
- Replies:
**252** - Views:
**11914**

### Re: Mathematical Induction - Introductory Question

That makes sense - the testing occurs at the base-case level, right? The base case, plus the k case (i.e., any arbitrary number), and the (k + 1) case (because you can always get to the next-greatest natural number when one starts at any k) do the work together, yes? Right, that's a much better way...