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Sewing-Machine comments on About addition and truth - Less Wrong

-1 Post author: RolfAndreassen 02 June 2011 08:34PM

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Comment author: [deleted] 03 June 2011 09:28:59PM 1 point [-]

When someone says "2+2=4", it feels as though they are asserting a necessary truth, something that cannot possibly be otherwise.

This is an illusion. If I say "37460225182244100253734521345623457115604427833 + 52328763514530238412154321543225430143254061105 = 8978898869677433866588884288884888725858488938" it should not immediately strike you as though I'm asserting a necessary truth that cannot possibly be otherwise.

My question is, ought the thought experiment of a universe whose galaxies and stars are counted by arithmetic mod 3^^^^3 cause us to abandon this intuition?

Counting is an algorithm, or really a sketch of an algorithm. In order to make this a coherent question, i.e. to imagine running an algorithm on that many galaxies and stars and coming up with a certain answer, and then thinking about the consequences, we would need at least

  1. An airtight definition of "galaxies and stars"
  2. A ledger big enough to fit 3^^^3 tickmarks
  3. A reliable enough method of writing down tick marks when we see stars, that when we did it twice and got two different answers, it was not overwhelmingly likely that we had made a mistake someplace.

Each of these is preposterous!

Comment author: AlephNeil 03 June 2011 09:44:31PM 2 points [-]

If I say "37460225182244100253734521345623457115604427833 + 52328763514530238412154321543225430143254061105 = 8978898869677433866588884288884888725858488938" it should not immediately strike you as though I'm asserting a necessary truth that cannot possibly be otherwise.

It immediately strikes me that what you're asserting is either necessarily true or necessarily false, and whichever it is it could not be otherwise.

Comment author: [deleted] 03 June 2011 09:46:53PM *  1 point [-]

That's fine, but it's not at all the same thing.

Comment author: AlephNeil 03 June 2011 09:59:17PM 0 points [-]

Why is the difference relevant? I honestly can't imagine how someone could be in the position of 'feeling as though 2+2=4 is either necessarily true or necessarily false' but not 'feeling as though it's necessarily true'.

(FWIW I didn't downvote you.)

Comment author: [deleted] 03 June 2011 10:20:32PM 0 points [-]

I honestly can't imagine how someone could be in the position of 'feeling as though 2+2=4 is either necessarily true or necessarily false' but not 'feeling as though it's necessarily true'.

That seems to imply you think it would feel different than how you felt at first looking at my sum. Why, besides the fact that it's much simpler?

I sort of agree, in the sense that "2+2 = 4" is a huge cliche and I have a hard time imagining how someone could not have memorized it in grade school, but that's part of the reason why I regard the "self-evidence" of this kind of claim as an illusion. We take shortcuts on simple questions.

Comment author: ec429 14 September 2011 06:46:08AM 1 point [-]

I believe that "2+2=4 is either necessarily true or necessarily false". I believe 2+2=4 is necessarily true (modulo definitions). I don't believe it's necessarily true that "2+2=4 is necessarily true".

There's some pretty strong evidence that the proof that 2+2=4 doesn't have a mistake in it (heckuva lot of eyeballs). I have good reasons (well, reasons anyway) to believe that mathematical truths are necessary. Thus most of my mass is on "2+2=4 is necessarily true". Yet, even if it's necessarily true that "2+2=4 is either necessarily true or necessarily false", and 2+2=4 is true, it still needn't be necessarily true that "2+2=4 is necessarily true", even though 2+2=4 is necessarily true.

If your eyes have glazed over at this point, I'll just say that Provable(X) doesn't imply Provable(Provable(X)), and if you think it does, it's because your ontology of mathematics is wrong and Gödel will eat you.

Comment author: lessdazed 14 September 2011 09:19:48AM 3 points [-]

Gödel will eat you.

That's exceptionally unlikely for more reasons than one might think.

Comment author: [deleted] 14 September 2011 07:21:53AM 2 points [-]

Not sure what work "necessarily" is doing, but mostly I'm with you. Still, I think this is mistaken:

I'll just say that Provable(X) doesn't imply Provable(Provable(X)), and if you think it does, it's because your ontology of mathematics is wrong and Gödel will eat you.

Though it is true and important that Unprovable(X) does not imply Provable(Unprovable(X)).