
When is proof by contradiction necessary? - luu
https://gowers.wordpress.com/2010/03/28/when-is-proof-by-contradiction-necessary/
======
vilhelm_s
See also Andrej Bauer's "reply article", Proof of Negation and Proof by
Contradiction: [http://math.andrej.com/2010/03/29/proof-of-negation-and-
proo...](http://math.andrej.com/2010/03/29/proof-of-negation-and-proof-by-
contradiction/)

In short, there are two different logical principles, and ordinary
mathematicians call both of them proof by contradiction. However,
constructivist (aka intuitionist) mathematicians make a distinction between
them: they "believe in" proof of negation, but not in proof by contradiction.

A lot of work has been done about formalizing mathematics in constructive
logic. One big component of this is figuring out which uses of proof by
contradiction are avoidable, and which one are essential (in the latter case
the theorem statement has to be weakened for the constructive version). But
this is all about proof by contradiction in the narrow sense. The usual proof
about the irrationality of √2 is valid constructively.

~~~
monochromatic
The "problem" is basically that constructivists don't believe that this is a
valid inference:

¬¬ϕ ⇒ ϕ

~~~
sireat
What is the justification for not believing that from a double negation of A
follows A is a valid inference?

I took formal systems in university some years ago and remember something
about different axiom systems and how one school of thought rejected one
particular rule of inference but was this the one?

I am trying to think of a world where this rule is not valid and not
succeeding.

~~~
tel
The simplest intuition is to exchange "truth" as the airy concept of "is
valid/reflects the world" with "I possess tangible evidence to the case".

So, "2 + 2" is true because I can generate tangible evidence of it, but "the
Collatz conjecture" is not true because I cannot. It's also not false. To be
"false" I'd have to have tangible evidence of "the Collatz conjecture is
false", a counterexample. This "I lack evidence" as neither truth nor
falsehood is the characteristic of intuitionistic/constructive logic that
makes it so interesting.

To be clear, there is now an element of time involved. Or, as others have
stated, constructive logic reflects the "communicative nature" of proof. What
I should have said a second ago is "The Collatz conjecture is not true (for
me) (right now) (because I personally don't happen to have evidence for its
validity yet)". If you had a (valid) proof it'd be true for you and once you
communicated that proof to me it'd be true for me.

So in the case of rejecting "not not P => P", it's merely a statement of the
fact that "not (not P)" failing to actually generate any evidence for "P", it
merely shows that there isn't "not evidence". If you live in a world where
things must be true or false (e.g., excluded middle holds) then this is
sufficient, but if you demand I show you _why_ "P" holds, then "not (not P)"
is unsatisfactory.

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deckar01
I'm not sure "try not to use [proof by contradiction] unless you really need
it" is the optimal conclusion. In "Meta Math"[1], Gregory Chaitin explores the
question: "What makes a mathematical concept desirable?"

Simplicity was the solution. The best theorems seem obvious in retrospect,
because they are reduced into relationships between existing concepts.

I would recommend trying all angles of attack when searching for a proof. If
it starts to take too long or you hit a dead end, try a different angle. Even
if you discover a proof, try your other opening moves, look for a more elegant
solution.

I believe Wolfram Alpha converges on this solution by performing some sort of
weighted, breadth-first search on the permutations of an equation.

I highly recommend "Meta Math"[1] if you are interested in theories about
entropy and algorithms that generate proofs.

[1] ("Meta Math", Gregory Chaitin,
[http://arxiv.org/pdf/math/0404335.pdf](http://arxiv.org/pdf/math/0404335.pdf))

~~~
eru
> I'm not sure "try not to use [proof by contradiction] unless you really need
> it" is the optimal conclusion.

Oh, that's only a guideline for presenting your proves. Not for coming up with
them in the first place. Mathematicians do exactly what you are suggesting
already.

The first prove one finds is usually quite ugly, and then comes the
simplification. Writers call it editing, programmers refactoring.

~~~
EGreg
Many proofs by contradiction are simply results of wrapping a constructive
proof:

Suppose premises/axioms (B, C, ...) which don't depend on the value of A.
Then, B=true AND C=true ... implies A=true by some logical argument, such as

A Or Not (B And C And ...) = True

In this case, the proof by contradiction is

Assume A=False But B=True, C=True, ... (Since they dont depend on A) Apply
above constructive argument So A is true, _A CONTRADICTION_

Proofs by contradiction can often be of this type so the first and last lines
can be removed.

The question is, what other types can there be?

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leephillips
All difficult conjectures should be proved by reductio ad absurdum arguments.
For if the proof is long and complicated enough you are bound to make a
mistake somewhere and hence a contradiction will inevitably appear, and so the
truth of the original conjecture is established QED.

\-- John Barrow

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evitisopartnoc
I don't like the conclusion that students should avoid proving by
contradiction unless they really need it.

I almost always start with contradiction as my initial method of choice when
taking exams because you don't always have time to completely rewrite your
proofs. Contradiction helps with this since it naturally encompasses proving
by contrapositive also. Because if you're trying to prove that p->q, you can
assume p and ~q and getting to ~p is a contradiction since p^~p (and this is
the same structure as a proof by contrapositive if you got to this line
without assuming p). And of course if you find any other absurdities along the
way sooner, you're done quicker.

This is nice because a proof by contrapositive is more or less equivalent to a
direct proof. So simply by always starting with proof by contradiction, you
cover the cases where the theorem:

\- is simple enough for you to find out exactly why it's true (direct proof
hidden in the contrapositive).

\- is complex and wide reaching enough for you to hit an absurdity quickly

~~~
eru
> I don't like the conclusion that students should avoid proving by
> contradiction unless they really need it.

Where is that conclusion drawn? This is an blog post for research (and
armchair) mathematicians, not students trying to sit an exam.

~~~
evitisopartnoc
Last paragraph:

>In which case, perhaps the advice that I give to students — proof by
contradiction is a very useful tool, but try not to use it unless you really
need it — is, though not completely precise, about the best one can do.

~~~
eru
Oh, ok. I interpreted that as advice to students writing up their proves, not
trying to find them.

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raymondh
First, let's assume that proof by contraction isn't necessary, ...

:-)

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mattxxx
Pretty unrelated, but proof by contradiction has been really useful for me in
real life. It gives you a way to look at logical facts, and remove many
axioms, because it relies on empirical evidence and a set of assumptions that
you are trying to _invalidate_.

In that way, it avoids issues with deducing things from a set of pre-
prescribed beliefs. In fact, it fights against them.

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madamepsychosis
My intuition – proof by contradiction means we have to use one step that
inverts the set of values for which the statement is true – so whether we need
a contradiction step depends on the cardinality of this truth set. Perhaps it
depends on the relationship between the ‘truth set’ of our premises and our
conclusions (e.g. the square root of 2 is one number, but all of the rationals
is infinite) Supporting this intuition, getting rid of proof by contradiction
leaves us unable to construct uncountable sets. Maybe some proofs need several
contradiction steps (i.e., when their truth set cardinality is aleph-2,3,
etc.)

~~~
mattxxx
Yea, but, I mean, the isn't the diagonal argument just an extension of proof-
by-contradiction into countable sets?

The same way that induction is the extension of syllogism into countable sets?

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EGreg
When both X and Not X lead to contradictions, we have a paradox.

But what if Not X leads to a contradiction and we can't seem to find a
contradiction when we assume X, but we aren't sure there isn't one? Have we
proven X by contradiction, or is it really a paradox and there is some
nonsensical operation somewhere?

I remember wondering something likethat before.

~~~
ebola1717
In that case, you have proven X by contradiction. If you're rigorous enough,
you can rule out the 3rd option. As wondering if you're in a situation where
your axioms are inconsistent, well that's actually the world mathematics lives
in. Godel's Incompleteness Theorems show that, besides some weak axiom
systems, the only systems of axioms that can prove they are consistent are the
ones that are inconsistent. So we can only tell when systems are inconsistent,
never when they are consistent. You just kinda hope it's not all inconsistent.

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ikeboy
Why can't you prove irrationality as follows:

All rational numbers have the property that their square is not 2. (Should be
a simple proof by induction on the denominator.) The square root of two does
not have that property, and is therefore not rational.

Is the contradiction hiding in my implied definition, my implied induction
proof, somewhere else?

~~~
gamegoblin

        All rational numbers have the property that their square is not 2
    

==

    
    
       2 not in {r^2 for r in rationals}
    

==

    
    
       sqrt(2) not in {r for r in rationals}
    

==

    
    
       sqrt(2) not in rationals
    

It seems to me that you handwaved away the entire proof in your first
sentence, without really proving anything.

~~~
ikeboy
I did, although the statement I gave can be proven without contradiction. I
gave a proof in a different reply.

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brownbat
I especially liked the discussion under Example 1 about reformulating
"negative" sentences into "positive" sentences.

The briefest amount of work in mathematical deduction or symbolic logic will
cause you to raise an eyebrow whenever anyone insists "you can't prove a
negative."

Every negative statement has a positive formulation buried within it. Or, no
negative statement exists that lacks a positive formulation, and vice versa.

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nova
And then you have linear logic (which is to constructive logic what
constructive logic is to classical logic), with its two kinds of conjunctions
and two kinds of disjunctions, one of which is difficult to understand but
does have excluded middle again.

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amelius
Personally, I'd rather like to see a formal proof than a proof that is
"optimal" according to some vague metric.

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eterm
Minor mistake in stating the theorem about p^2 = 2 q^2.

As stated, p = q = 0 is a valid solution.

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lerpa
Empirical sciences are based upon this, you can only prove theories to be
false and you do that by demonstrating a contradiction regarding its
predictions and the observations.

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crimsonalucard
Outside of math and logic, in the world of science, not only is proof by
contradiction necessary, it is the ONLY way of proving anything.

~~~
eru
I have a strong suspicion you are mixing up different categories.

~~~
crimsonalucard
What categories? I'm differentiating between math/logic and science.

~~~
eru
Just to clarify: I assume your original comment was an allusion to Popper's
falsification?

~~~
crimsonalucard
yes.

