## Tuesday, July 10, 2012

### Platonic Calculations and the Business of Giving Assumptions

Given certain assumptions, there are certain logically inevitable conclusions that you can calculate. I realized I should ask myself, where do the axioms of Peano arithmetic come from in the first place? Who's business is it to dole them out?

Page 80. (Via.)
"I also emphasize that, when I insert the Platonic output of a computation as a latent node in a causal diagram, I am not making a philosophical claim about computations having Platonic existence. I am just trying to produce a good approximation of reality that is faithful in its predictions and useful in its advice."
Technically speaking to solve the problem you not only need the givens, but need to assume or be given the rules of logic themselves.

Isn't 678 * 987 equal to 669,186 regardless of how well the calculators are constructed, regardless of electronic glitches or damage in transit?

The normal thing to say is that it is equal to 669,186 regardless of any facts about the physical world. My answer is that logic and arithmetic are facts about the physical world. Arithmetic works because it ultimately describes physics, which means physics described it long before we did.

If the calculator doesn't display 669,186 I can conclude it is broken. There's the calculation it is doing and then the calculation it is (platonically?) supposed to be doing, and they're different.
Imagine I dump 987 piles of sand into one sandbox, each exactly 678 grains. If my subsequent fully vetted count isn't nearly 670,000, then either I'm not creating an analogue of multiplication, or multiplication doesn't work the way I think it does. For the former, my experiment design is broken, because physics doesn't work the way I think it does. For the latter, multiplication doesn't work the way I think it does, which means my understanding of physics is broken. Either way, not the sand's fault.

Thing is, in such a world, would it have ever occurred to me that the answer might be nearly 670,000? I submit: no. Even if it occurs to you that your world might be fundamentally different, it is impossible to work out how.

Verification: try describe a world where A=A is false. A!=A. What does a thing look like when it doesn't look like itself? If that works, maybe you can tell me what multiplication might have looked like if it wasn't what we've got. Or, what an alternate world would have instead of multiplication. If you try to tell me these things are platonically impossible, then I'll ask you how it knows it is supposed to be impossible. What stops it?

Things like this are probably impossible to prove. It questions questioning itself; it tries to verify verification. You need some framework to hang arguments off of, and here I am wondering how to get a grasp of the framework. A hand, trying to grab itself.

Which itself interests me. If I'm trying to make a hand grasp itself, how do I get the answer 'no' at all? Shouldn't my brain just crash?

Aretae said...

Inductive probabilistic bootstrapping solves this cleanly. Failure only occurs near certainty.

spandrell said...

There's this thing we call God to stop playing with logic and go on thinking about useful things.

Alrenous said...

Aretae,

So, given induction, you don't have a problem.
What gives you induction?

Spandrell,

It's impossible to know in advance what is and isn't useful to think about. Being able to tell if a line of thought is useful means knowing something about where it ends. Which means having already travelled it to some extent.

Also, same thing. Given God, problem solved. What gives you God?

spandrell said...

Come on you can't have just discovered that axioms aren't provable by themselves.

You have to take some things for granted. Less is better, and when you reduce it to one, you call that God.

That's just how it works. You can't exclusively rely on logic, otherwise you get trapped on Zeno's paradoxes and chickens and eggs.

Alrenous said...

It's not good if everyone thinks a thing and nobody says it. I like to fish around in the obvious for candidates.

Or I could just take logic for granted. That's a single thing. God is the Word is Logos is Logic. This isn't even a particularly radical position.

nydwracu said...

My answer is that logic and arithmetic are facts about the physical world. Arithmetic works because it ultimately describes physics, which means physics described it long before we did.

"Works" is ambiguous. Arithmetic is useful because it ultimately describes physics, but that 678 * 987 = 669,186 is true by the definitions of the terms of the equation, where the definitions take Zermelo-Fraenkel (I forget whether Peano requires the axiom of choice) as axioms... but those axioms are taken because they lead to things that are useful... and they're useful because they describe physics. Hmm.

But is that even important? What's important is that 678, *, 987, =, and 669,186 are consensus-defined in such a way that the phrase "678 * 987 = 669,186" is necessarily true. It's not Platonic; it's just deductive. If the calculator spits something else out, it's talking nonsense.

Alrenous said...

It's hard to consistently define 'nonsense' at this level of analysis.

At the level of math logic, the calculation is identical whether you do it with transistors or do it with a couple handfuls of sand. But if the sand gives the 'wrong' answer, it isn't nonsense. The consensus is wrong.

Which implies there's an interpretation where the calculator isn't broken either. It's not implementing the consensus rules...but is it following physics, it is calculating something.

Calculators calculate using machine representations of arithmetic, and the output is consistent because arithmetic is consistent. If the calculator's 'defect' was such that it wasn't doing arithmetic, but the system was still consistent...it isn't nonsense. (Though I supposed the buttons would be mislabelled.)

However, I submit that because the calculator is made of physics, no such defect exists. It implements arithmetic, or else it will be inconsistent. For example, perhaps all calculator defects are re-arrangements of arithmetic, such as adding when it is supposed to divide. (Not including catastrophic flaws.)

In turn this implies the same thing about people. The consensus implements arithmetic, or it is inconsistent. Riemann no doubt thought his spaces were whimsical, but in making them consistent, they ended up describing physics.

Which makes me wonder. Does that mean dimensions are more like a parameter than a foundation? I think so. What makes a space Riemannian? Its relations, not the number of numbers that are related.