> On 17 Dec 2018, at 20:34, Philip Thrift <[email protected]> wrote: > > > > On Monday, December 17, 2018 at 1:12:32 PM UTC-6, Bruno Marchal wrote: > >> >> What I am curious to know is how how many of these statements you agree with: >> >> "2+2 = 4" was true: >> 1. Before I was born >> 2. Before humans formalized axioms and found a proof of it >> 3. Before there were humans >> 4. Before there was any conscious life in this universe >> 5. As soon as there were 4 physical things to count >> 6. Before the big bang / before there were 4 physical things > > > > > Let + be "concatenation" (perhaps a more primitive notion than addition). > > So > > ||+|| = |||| > > > Now I could not write or show that without matter. The | is made and stored > with electrons in computer memory and then transferred via internet to a > computer screen. The concatenation operation producing the right side is made > via a computer or brain putting "sticks" together. > > Before there were "sticks" there was no concatenation of "sticks". (See > question 6.) > > What your brain does with the pixels you see above is up to your brain.
This is like saying that group theory assumes the existence of CaCO3 because without a shlock we can’t explain group theory. That is simply a confusion of level. Group theory remains correct in a reality without CaC03. Similarly 2+2=4, or if you prefer, what you intended to describe with II + II = IIII, is an elementary consequence of the axioms: 1) 0 ≠ s(x) 2) x ≠ y -> s(x) ≠ s(y) 3) x ≠ 0 -> Ey(x = s(y)) 4) x+0 = x 5) x+s(y) = s(x+y) 6) x*0=0 7) x*s(y)=(x*y)+x Proof: s(0) + s(0) = s(s(0)+0) by axiom 7, = s(s(0)) by axiom 6. This is not argument for 2+2=4, for we knew that already. It is shows that 2+2=4 can be proven in a theory which does not assume anything in physics, nor even that there is a physical world. Obviously, to communicate such proof in between physical being like us, we have to use matter, but we don’t have to assume it in the derivation, or in the theory of numbers that we want to use. I can also prove that 2+2=4 using the combinators (as I did explicitly in the thread on combinators). There too, we have use only the following assumptions: 1) If x = y and x = z, then y = z 2) If x = y then xz = yz 3) If x = y then zx = zy 4) Kxy = x 5) Sxyz = xz(yz) And nothing more, besides the definition of the natural numbers and + and * in that context. Bruno > > - pt > > > -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected] > <mailto:[email protected]>. > To post to this group, send email to [email protected] > <mailto:[email protected]>. > Visit this group at https://groups.google.com/group/everything-list > <https://groups.google.com/group/everything-list>. > For more options, visit https://groups.google.com/d/optout > <https://groups.google.com/d/optout>. -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
