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
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?
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