Are mathematical proofs incomparable with proofs in other disciplines?

Are mathematical proofs incomparable with proofs in other disciplines?

[John Baez]

Vaughan writes:

1.  Is mathematical proof so different from say legal proof that the two
> notions should be listed on a disambiguation page as being unrelated
> meanings of the same word, or should they be treated as essentially the same
> notion modulo provenance of evidence and strictness of sufficiency, both
> falling under the definition "sufficient evidence of the truth of a
> proposition."
>

Geoffrey Lloyd's book "Adversaries and Authorities: Investigations into
Ancient Greek and Chinese Science" has a discussion of how the Greek legal
system influenced Greek concepts of rationality.  While I haven't read it in
a long while, I think it makes a good case that the concepts of "proof" in
Western law, medicine, science, and mathematics are deeply intertwined.
It's very useful to compare Greece with China on these issues.  You could
get a taste of it here:

http://books.google.com/books?id=3820gVEQu1AC&printsec=frontcover&dq=adversaries+and+authorities&source=bl&ots=Z1qCx04ANb&sig=MROclDNl8-wjDKFi77o4z8Cs1_s&hl=en&ei=pz03TJLUHobEsAPglo1S&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBcQ6AEwAA#v=onepage&q=proof&f=false

but the whole book is fascinating.  It would also make a good citation - and
it contains many further references.

Best,
jb


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Re: Are mathematical proofs incomparable with proofs in other disciplines?

[Vaughan Pratt]

On 7/9/2010 12:55 PM, Joyal, André wrote:
> I agree with your definition:
>
> "A proof is sufficient evidence for the truth of a proposition,"

Meanwhile the existence of some who prefer adding "or argument" to 
"evidence" seems to have been established.  Maybe one day they'll 
surrrender, but this is not my highest priority.  Higher is to make the 
following title changes on Wikipedia:

Proof (truth)     --->     Proof
Proof             --->     Proof (disambiguation)

The first is the article on proof, the second is the Wikipedia 
disambiguation or dab page that takes you to other meanings such as 
"alcoholic proof" (an obsolete term even in the US, having been replaced 
by "alcohol by volume" or ABV) and proofreading.  The suggested change 
would make the article on proof the primary topic (a Wikipedia concept) 
having a so-called hatnote (note at the head of the article) pointing to 
a disambiguation page for the lesser meanings.  So far no but myself and 
one or two people have spoken up for this; until they do nothing will 
change.

> The article
>
> http://en.wikipedia.org/wiki/Proof
>
> does not discuss the idea (of Paul Lorenzen) that a mathematical proof
> is essentially a winning strategy in a formal game.
> I first learned the idea from Andreas Blass
> who introduced the game semantic of linear logic,
>
> http://arxiv.org/abs/math/9310211
>
> A proof can be viewed as an argumentation to convince others of the validity of a statement.
> In mathematics, the argumentation must be solid enough to resist
> any conter-argumentation by an ideal opponent.

I would divide proofs into two kinds, those where the intended audience 
can argue back, as in a courtroom or ordinary conversation, and those 
where they can't, for example the subscribers to a journal.  What you 
describe does cover both, but for the latter the game is very short.

In computation this distinction is that between alternating computation 
and nondeterministic computation.  Alternation computation is an 
on-going game, in non-deterministic computation one player makes one 
choice and the game ends.  Interestingly the notion of nondeterministic 
computation preceded that of alternating computation under that name by 
some 15 years or so.  But Fraïssé's alternation preceded both by a 
decade, although it took a decade for Ehrenfeucht to cast Fraïssé's 
approach as a game (1961).

Vaughan


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Re: Are mathematical proofs incomparable with proofs in other disciplines?

[Vaughan Pratt]

On 7/9/2010 7:10 AM, Michael Barr wrote:
> Nonetheless, the biggest difference between mathematical
> proof and, say, legal proof, is that the latter depends on real world
> evidence.

Yes, exactly.  Mathematical proof depends on evidence (or whatever you
feel it should be renamed as in that setting) found in mathematical worlds.

> Legal terms do not have definitions that can be understood
> without reference to the real world.

What about intellectual property?  Granted ideas can't be patented, but
is a method or algorithm (the case of software patents) a thing in the
real world or the conceptual world?

> we believe that, in principle,
> every proof we publish is a surrogate for a formal logical deduction,
> but I once tried that for a simple argument and gave up after I had
> filled a couple pages with indecipherable chicken scratchings (cf.
> Russell & Whitehead).

But R&W didn't give up, they published the full proof that 1 +_c 1 = 2
(the _c denotes cardinal addition), which they were able to complete as
*110.643 by the top of p. 83 of Volume II, with the modest remark, "The
above proposition is occasionally useful.  It is used at least three
times, in *113.66 and *120.123.472."  I always regretted that they
didn't use Roman instead of Arabic numerals and follow the convention of
omitting plus rather than times, so that *110.643 would then read

      II = II.

But I agree it's tedious.  In 1975 I published 23 lines of chicken
scratchings constituting a complete formal proof that 474,397,531 was
prime.  (Obviously this couldn't have been just an enumeration of its
possible factors up to its square root.)  This was not terribly tedious.
   Two years later, with Stephen Litvintchouk I published a complete
formal proof of correctness of the Knuth-Morris-Pratt pattern matching
algorithm, using a broadly construed notion of inference rule that
allowed the use of decision methods in checking some steps.  This was
somewhat more tedious, but it would have been worse without the decision
methods.

> I agree, however, that they [legal and mathematical proofs] are incomparable.

Along the dimension of provenance of evidence, yes.  And there is an
understandable reluctance on the part of some to refer to premises of
mathematical arguments as "evidence," which can be addressed by
broadening the notion to "evidence or argument."

Vaughan


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Re: Are mathematical proofs incomparable with proofs in other disciplines?

[Ronnie Brown]

My suggestion is that mathematical proofs still require a mathematical 
landscape in which to find our way. Michael Barr points out the problems 
of finding a completely `logical' proof. I propose the analogy of giving 
directions to the station, such as `go out of the house, turn right, go 
straight on until ..., etc., etc.' We don't need to specify all the cracks 
in the pavement, but we may need to warn of holes due to roadworks!

The aura of certainty in a mathematical proof is partly because the 
`conceptual landscape' has been worked up over centuries, in terms of 
convenience and usability, and tested by thousands. Hopefully, arguments 
come to be produced which seem inevitable, indeed aesthetic, rather than 
ad hoc.

Are there computer theorem provers which can work at a `landscape level'?

The assumption is also that there are no cracks in our mathematical universe.

On the more general point, I did hear of a University Vice Chancellor who 
asked his staff for a series of lectures on `The notion of validity in my 
subject'.

Ronnie Brown









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Are mathematical proofs incomparable with proofs in other disciplines?

[Joyal, André]

Dear Vaughan,

I agree with your definition:

"A proof is sufficient evidence for the truth of a proposition,"

The article

http://en.wikipedia.org/wiki/Proof

does not discuss the idea (of Paul Lorenzen) that a mathematical proof 
is essentially a winning strategy in a formal game.
I first learned the idea from Andreas Blass
who introduced the game semantic of linear logic,

http://arxiv.org/abs/math/9310211

A proof can be viewed as an argumentation to convince others of the validity of a statement.
In mathematics, the argumentation must be solid enough to resist
any conter-argumentation by an ideal opponent.
It can be compared to a winning strategy in a game with two players, 
one defending a statement and the other attacking it.
Lorenzen associates to every mathematical statement S a formal game with two players G(S),
the defender and the attacker. The defender has a winning strategy iff the statement has a formal proof.
The rules of the games for a proof in intuitinistic logic differ
from the rules for a proof in classical logic.
In other words, the rules of the games are determining the logic and vice versa.

I believe that game semantic is putting some light on the origin of logic.
I guess that logic was discovered by peoples debating in a democratic manner.
All communities need to choose between different courses of actions. 
There are many answers to the question: how should this choice made? 
One was given by Plato who favored a government 
by the "philosopher king" who "loves the sight of truth":

http://en.wikipedia.org/wiki/Plato#The_State

Plato does not like Athenian democracy because it is imperfect. 
He observes that its political debates are manipulated by sophists. 

I agree with Plato that democracy is imperfect.
But it should be improved, not condemned.

Logic is anti-authoritarian since it wishes to convince, not to coerce.


Best,
André



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Re: Are mathematical proofs incomparable with proofs in other disciplines?

[Michael Barr]

On Wed, 7 Jul 2010, Vaughan Pratt wrote:

> My questions are
>
> 1.  Is mathematical proof so different from say legal proof that the two
> notions should be listed on a disambiguation page as being unrelated
> meanings of the same word, or should they be treated as essentially the
> same notion modulo provenance of evidence and strictness of sufficiency,
> both falling under the definition "sufficient evidence of the truth of a
> proposition."

Let me begin my answer with an aphorism.  I don't know who said it first,
but I heard it from Charles Wells.

In principle, there is no difference between principle and practice, but
in practice...

If mathematical proof were simply logical deductions, there would never be
mistaken proofs published.  On the other hand, if proofs weren't logical
deductions, we could never find errors in proofs, only in their
consequences.  Nonetheless, the biggest difference between mathematical
proof and, say, legal proof, is that the latter depends on real world
evidence.  Legal terms do not have definitions that can be understood
without reference to the real world.  we believe that, in principle, every
proof we publish is a surrogate for a formal logical deduction, but I once
tried that for a simple argument and gave up after I had filled a couple
pages with indecipherable chicken scratchings (cf. Russell & Whitehead).


>
> 2.  Gandalf61 evidently feels his sources, Krantz and Bornat, prove the
> notions are incomparable.  Are there suitable sources for the opposite
> assertion, that they are comparable?
>

I agree, however, that they are incomparable.

Michael



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Are mathematical proofs incomparable with proofs in other disciplines?

[Vaughan Pratt]

There's an interesting dispute just started on Wikipedia concerning
whether it is reasonable to see some commonality of meaning between the
concept of proof in mathematics and in other areas such as rhetoric,
law, philosophy, religion, science, etc.  The dispute is at one or both of

http://en.wikipedia.org/wiki/Talk:Proof_(informal)#Disambig_page

(Editors keep changing the name of the article, which was Proof (truth)
when I wrote it and others have replaced "truth" first by "logic" and
then by "informal", neither of which are an improvement.)

The origin of the article in dispute is as follows. Some months ago I
went to Wikipedia to look up what it considered to be a proof and found
only a dab (disambiguation) page listing ten articles that seemed to
about proof as applied to propositions and about as many more to do with
testing and quality control as in galley proof, proof spirit, etc.

It seemed to me that the former kind were not so much different meanings
of the notion of proof as the same meaning arising in different areas
all depending on that meaning.  So, still some months ago, I wrote an
article on that common notion which began

    "A proof is sufficient evidence for the truth of a proposition,"

which as it happens is essentially the first entry in the definition at
dictionary.com.

The article enumerated the various notions of proof arising in different
disciplines (all of which have their own Wikipedia articles with much
more detail), and made a start on characterizing the scope of "evidence"
(need not be verbal, and need not contain the asserted proposition) and
"sufficient" (strict for formal proofs, less so elsewhere, to different
degrees).

The main dispute at the moment is Gandalf61's insistence that "Proof in
mathematics is not based on 'sufficient evidence' - it is based on
logical deductions from axioms. It is an entirely different concept from
proof in rhetoric, law and philospohy."  He backs this up with quotes
from Krantz---"The unique feature that sets mathematics apart from other
sciences, from philosophy, and indeed from all other forms of
intellectual discourse, is the use of rigorous proof" and
Bornat---"Mathematical truths, if they exist, aren't a matter of
experience. Our only access to them is through reasoned argument."

My position is that logical and mathematical proofs differ from proofs
in other disciplines in the provenance of their evidence and the rigor
of their arguments as parametrized by "sufficient."  Whereas evidence in
mathematics is drawn from the mathematical world, evidence in science is
drawn from our experience of nature.  And whereas formal logic sets the
sufficiency bar very high, mathematics sets it lower and other
disciplines lower still, at least according to the conventional wisdom.

Whereas I find my position in complete accord with the quotes of both
Krantz and Bornat when interpreted as in the preceding paragraph,
Gandalf61 does not.

My questions are

1.  Is mathematical proof so different from say legal proof that the two
notions should be listed on a disambiguation page as being unrelated
meanings of the same word, or should they be treated as essentially the
same notion modulo provenance of evidence and strictness of sufficiency,
both falling under the definition "sufficient evidence of the truth of a
proposition."

2.  Gandalf61 evidently feels his sources, Krantz and Bornat, prove the
notions are incomparable.  Are there suitable sources for the opposite
assertion, that they are comparable?

3.  Someone with a very heavy hand has tagged practically every sentence
with a "citation needed" tag.  For those that genuinely do need a
source, what would you recommend?

Vaughan Pratt

PS.  I hope this sort of argument doesn't put anyone off volunteering to
help out on Wikipedia.


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