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Eliezer_Yudkowsky comments on How to Convince Me That 2 + 2 = 3 - Less Wrong

53 Post author: Eliezer_Yudkowsky 27 September 2007 11:00PM

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Comment author: Eliezer_Yudkowsky 28 September 2007 04:16:45PM 8 points [-]

Wikipedia on a priori: Relations of ideas, according to Hume, are "discoverable by the mere operation of thought, without dependence on what is anywhere existent in the universe".

This points out clearly the problem that I have with "a priori". It is a fundamentally Cartesian-dualist notion. The "mere operation of thought" takes place INSIDE THE UNIVERSE, as opposed to anywhere else.

To observe your own thoughts is a kind of evidence, if the spikings of your neurons be entangled with the object of your inquiry (relative to your current state of uncertainty about both). If, for example, I do not know what will happen with two earplugs and two earplugs on the nightstand, I can visualize two apples plus two apples to find out. All of this takes place in the same, unified, physical universe, with no ontological border between the atoms in my skull and the atoms outside my skull. That's why the trick works. It would work just as well if I used a pocket calculator. Is the output of a pocket calculator an a priori truth? Why not call the earplugs themselves a priori truths, then? But if neither of these are a priori, why should I treat the outputs of my neurons as "a priori"? It's all the same universe.

It appears to me that "a priori" is a semantic stopsign; its only visible meaning is "Don't ask!"

Vassar: It sure seems to me that our evolution and culture constructed ethical attitudes are entangled with the world.

They're causal products of the world, and yes, if I was ignorant about some evolution-related factual question, I might be able to use my ethical attitudes as evidence about conditions obtaining in my ancestral environment. That's not the same as my stating an external truth-condition for it being wrong to slaughter the first-born male children of the subjects of an unelected Pharaoh. It is perfectly acceptable for me to say, "I can think of no encounterable situation that would transform the terminal value of this event from negative to positive."

Spear: The test of any religion is whether cultures believing it tend to thrive and improve the quality of their lives or not.

Ah, yes, the old theory that there are reasons to believe2 in an assertion-of-fact besides its being true.

Lee: If he proclaims "two and two makes three," then he must be talking about something other than the integers. You cannot be mistaken about the integers, you can only misunderstand them.

Just to be clear, when I say "be convinced that 2 + 2 = 3", I mean being convinced that the system of Peano axioms with standard deductive logic and:

\a.(a + 0 = a) \ab.(a + Sb = S(a + b))

does not have as a theorem

SS0 + SS0 = SSSS0

but does have as a theorem

SS0 + SS0 = SSS0

and is consistent. Just as I currently believe that PA is consistent and has a theorem SS0+SS0=SSSS0 but not SS0+SS0=SSS0. So yes, this blog post is about what it would take to convince me that 2 + 2 actually equalled 3. I am not supposed to be convinced of this, if I am sane, and if it is not true. But at the same time, my belief in it should not be unconditional or nonevidential, because there are particular evidences which convinced me that 2 + 2 = 4 in the first place.

I also note that if you do not believe that there is a finite positive integer which encodes a proof of Godel's Statement, then you clearly are not using Peano Arithmetic to define what you mean by the word "integer".

Comment author: shrimp 29 March 2011 07:08:47PM 0 points [-]

What would convince me that 2 + 2 = 3, in other words, is exactly the same kind of evidence that currently convinces me that 2 + 2 = 4: The evidential crossfire of physical observation, mental visualization, and social agreement.

What has this to do with Peano Arithmetic and a mathematical proof "PA proofs 2+2=4" which is merely a string of symbols? On the other hand, what has PA to do with reality of earplugs except the evidence that PA is a good model for them?

Please explain the miraculous correspondence to apples and earplugs, then.

There is no miraculous correspondence, there is in fact a lot of evidence that FALSIFIES 2 + 2 = 4, like if it is 11 o'clock and 3 hours pass, it is 2 o'clock, and you can pour one glass of water and one glass of water into one glass of water, not to mention the already mentioned photons.

So 2 + 2 = 4 seems acutally to be true only when we "know what we are doing", when we are applying it "correctly". (and I am sure that in the world where 2+2 earplugs lead 3 earplugs, you may still find instances where 2+2=4 (like photons or whatever).) But applying "correctly" bears a lot of information about how and where you should be entangled with reality in order to claim 2+2=4.

That information is the difference between pure and applied mathematics. Also that is why there are two meanings of 2+2=4 which seem to have been mixed up in some of the discussion above. And that is what is meant by "2+2=4 is true in (pure) mathematics independently on whether or not it is true in the reality [when applied]". Using "a priory" is misleading, here I agree.

Of course it is also concievable that you wake up one morning and PA proofs 1+1=3 BUT 2 ear plugs + 2 ear plugs is still 4 earplugs! Isn't it?

...now I'm getting confused... if 2 ear plugs placed next to 2 earplugs lead 3, then how can you reliably write more than 3 sybmols next to each other to give a proof of anything from PA? spooky

Comment author: AmagicalFishy 22 June 2011 03:33:46PM *  2 points [-]

I think you've pointed out an issue of semantics, not falsified 2 + 2 = 4. If you pour one glass of water into another glass of water, you have one glass of water—but " one glass", in that case, is qualitative and not quantitative; it's not math.

Comment author: mytyde 20 November 2012 11:12:48PM *  0 points [-]

In regards to Hume's interesting contributions to the question, I stumbled across this video a while back which I think will be interesting: http://www.youtube.com/watch?v=BVZG0G-jnAM (don't let the title throw you off; there is content within it).