How is reductio ad absurdum a valid proof method?

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  • #26
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Sorry I Did Not Notice That You Want Me To Do 2x2=4 In Peano Arithmetic
 
  • #27
Hurkyl
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OK STEPS 1 AND 2 ,BY WHAT LAW YOU GET 3
ALSO EXPLAIN <identity of indiscemibles>
Stop shouting. :grumpy:

O.K With such a small proof WHAT ABOUT THE FOLLOWING PROOF??
for all x and for all y(x>=0 and y>=0 -------> ( sqroot(xy)=sqroot(x).sqroot(y) ))
CAN YOU DO IT STEP WISE??????
If you're challenging others to write explicit and complete formal proofs from scratch (i.e. without invoking known theory) -- the least you could do is to fully and accurately present what it is you want others to prove.

e.g. what is the range of the variables x and y? What theorems are we allowed to invoke about the relation >=, the constant 0, the unary function sqroot, and the binary function .?

Also, note that if x ranges over all real numbers and sqroot is meant to denote the ordinary real square root function, then you've written an ill-formed expression: x isn't restricted to the domain of sqroot, and thus sqroot(x) is a syntax error.
 
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  • #28
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O.K IWAITED to long and it is getting late .Here is aproof that 1+1=2
1)for all x and for all y(x +Sy =S( x+y)) A peano axiom
2) 1 +S(0) = S( 1+ 0) from 1 and using Univ.Elim. where we put x=1 and y=0
3) for all x (x + 0 =x ) A peano axiom
4) 1 + 0 = 1 from 3 and using Univ.Elim where we put x=1
5) 1+S(0) = S(1) BY substituting 4 into 2
6) S(0)=1 BY definition
7) S(1)=2 By definition
8) 1+1=2 By substituting 6 and 7 into 5
SO HERE WE USED TWO LAWS OF LOGIC namely Universal Elimination and the substitution law and two peano axioms together with two definitions
 
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  • #29
HallsofIvy
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If a theory is not consistent then any statement can be proved so NO proof tells you anything. Why bother to talk about non-consistent theories in regard to proof?
 
  • #30
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If for example the real Nos system is not consistent then nature itself is not consistent
 
  • #31
CRGreathouse
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If for example the real Nos system is not consistent then nature itself is not consistent

I'm not even sure what it would mean for nature to be inconsistent. In a formal system it means that there is a proposition P such that both P and not-P can be proved. Any inconsistent system containing classical first-order logic can, in fact, prove any statement. But what would the analogue for 'nature' be?
 
  • #32
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Suppose you let a mas m fall from the top of the building of height, h, then you can prove the following proposition P.
FOR all m and for all h [ if air resistance is 0 then the time taken for m to reach the ground will be, t=sqroot(2h/g)].
Now the negation of this P WOULD be that there exist an m and an h such that t=/=sqroot(2h/g),where g =10m/sec^2,provided of course that again air resistance is 0.
Here is your analogue
 
  • #33
CRGreathouse
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So nature's inconsistency would mean that everything can, and does, happen?
 
  • #34
Hurkyl
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then you can prove the following proposition P
How? This is, for sure, a theorem of Newtonian mechanics, but how can you prove it for 'nature itself'?
 
  • #35
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How? This is, for sure, a theorem of Newtonian mechanics, but how can you prove it for 'nature itself'?

I am not sure i get you point please be a bit more specific
 
  • #36
CRGreathouse
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I am not sure i get you point please be a bit more specific

I think Hurkyl is asking 'what does it mean for Nature to "prove" something?'.
 
  • #37
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you are inside your apartment and you ask your friend.How long do you thing it will take if i jump from the top of the Empire Building?your friend takes out his pencil and he does a few calculations and he tells you the time.Then you go to the top of the building,you set your watch,you jump and when you land on the ground you check your watch.If the time is the same with the time calculated by your friend then nature has |"proved" the Newtonian theorem in mechanics,which your friend used to find out the time.
 
  • #38
D H
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If the time is the same with the time calculated by your friend then nature has |"proved" the Newtonian theorem in mechanics,which your friend used to find out the time.
No, it hasn't. Newton's laws are not mathematical theorems. They are scientific theories. Mathematic theorems and scientific theories are quite different things.

Gathering evidence does not prove a scientific theory to be true. The evidence instead shows that the theory is consistent with reality to within experimental error, and only in the case of the evidence at hand. Experimental evidence provides confirmation. It does not provide proof. On the other hand, one crummy piece of well-validated conflicting evidence makes a scientific theory fall apart. In the case of Newton's theory of gravitation, that one crummy piece of conflicting evidence is the precession of Mercury. Newton's laws predict a different value for the precession of Mercury than observed. Those observations falsify Newton's laws.
 
  • #39
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No, it hasn't. Newton's laws are not mathematical theorems. They are scientific theories. Mathematic theorems and scientific theories are quite different things.

Gathering evidence does not prove a scientific theory to be true. The evidence instead shows that the theory is consistent with reality to within experimental error, and only in the case of the evidence at hand. Experimental evidence provides confirmation. It does not provide proof. On the other hand, one crummy piece of well-validated conflicting evidence makes a scientific theory fall apart. In the case of Newton's theory of gravitation, that one crummy piece of conflicting evidence is the precession of Mercury. Newton's laws predict a different value for the precession of Mercury than observed. Those observations falsify Newton's laws.

Lets not forget Einsteins relativity either :)
 
  • #40
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Lets put that way.
Suppose you kick MATHEMATICS to oblivion ,Can you have science
 
  • #41
Hurkyl
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Then it wouldn't make much sense to have the discussion in a math forum. :tongue:
 
  • #42
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true indeed
 

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