
The Human Obsession With "Formal Proofs" is a Waste of Time - tyn
http://www.math.rutgers.edu/~zeilberg/Opinion94.html
======
fhars
Isn't this just a rehash of the lates 19th/early 20th century intuitionits vs.
formalists debate? Even though Gödel had shown that the ultimate goal of the
formalist program is unattainable, they still mostly won that debate by being
able to produce more meaningful results. The radical intuitionist, with whom
the author seems to sympathize, did not only reject the axiom of choice, but
even things like the law of the excluded middle (i.E. they assumed that "A or
not A" is not necessarily true). That makes many proofs considerably harder to
perform, if not totally impossible.

Not surprisingly, the formalist program was also more fruitful for endeavours
like computational logic and the theory of computation, which resonate with
the formalist tenet that mathematics is just the manipulation of strings of
abstract symbols according to specific rules.

This <http://en.wikipedia.org/wiki/Brouwer-Hilbert_controversy> gives some of
the context.

~~~
yummyfajitas
I think it's more a "Humans vs Computers" debate. Some background:

Tom Hales cleverly reduced the Kepler Conjecture to the solution thousands of
linear programming problems: if all the solutions are bigger than something,
the kepler conjecture is true. Then he had a computer solve the LP problems,
and the result was that Kepler was true.

However, his solution was informal in the following sense; he didn't have a
rigorous proof that the solution of the LP problems was accurate, nor were his
programs proven correct. Now he is attempting to redo the proof in HOL, a
formal theorem proving system.

More background: no math paper is completely formal. All involve some
handwaving, especially the 200 page monsters that appear in Annals of Math. An
example from a paper I'm currently writing: "It can be easily seen by
elementary calculus arguments that..."

Doron is arguing that the mathematical community is holding Hale's computer
proof to a higher standard than it would hold a comparable 200 page human-
written paper. Basically, he is claiming that we should accept an informal
computer-assisted proof just like we would accept an informal human-written
proof.

------
mgreenbe
One can prove, ab initio, that 1+1=2 in less than ten lines in Coq.

Algorithms are _meaningless_ without accompanying proofs. There should be ---
and is --- a continuum of formality. The 20th Century has been unique in
recognizing the fallacies and contradictions of completely informal reasoning;
why give up when we're finally ahead?

~~~
psyklic
>> Algorithms are _meaningless_ without accompanying proofs.

Most of what we write as coders (and in fact, most of what the world runs on)
are algorithms without proofs which "mostly" work.

In fact, you could call almost all of our strategies in social interactions
"heuristic algorithms"!

~~~
psyklic
To whomever rated me down -- why do you think that human-written software has
so many bugs? It's because we don't take the time (and don't have good
strategies yet) to effectively "prove" that our code/algorithms exactly meet
the program specs. To write code, we need to make certain idealizations (e.g.
thinking about "doubles" as actual real numbers) which actually have tricky
corner cases.

So many areas of life cannot easily be mathematically analyzed. Hence, our
brain resorts to algorithms which work "most" of the time, which is luckily
usually sufficient for survival.

~~~
wlievens
It's typically because we don't even write a spec :-)

------
RiderOfGiraffes
It's hard to see how a computer would produce a proof of the Banach-Tarski
"Paradox" ( <http://news.ycombinator.com/item?id=411043> ) without going
through effectively the same steps as a human. It doesn't seem likely that it
will be reduced to calculations, or simple formula manipulations such as can
be performed by Mathematica.

And in case you think that B-T is irrelevant, informal reasoning about
infinities produced all sorts of results that caused people problems. Formal
reasoning about infinities gave us calculus for real, that works wherever it's
known to work. And I use infinities on a daily basis designing and
implementing systems to assist with tasks similar to air traffic control and
automated target tracking.

Computers can be used to explore and experiment with numbers, geometry and
some structures, and will probably one day be able to assist with exploring
more abstract concepts, but as things stand today, there's a great deal of
mathematics that no one can see how to do with computers.

To say otherwise is to demonstrate ignorance of both fields.

------
sajidu
I'd agree with the author that only logicians are interested in formal proofs.

But what the author describes as the Hilbet-Bourbaki method is simply what
math _is_. Any mathematician will tell you that if you're not proving theorems
then you're not really doing math.

Anyway, what good is a computer proof if it doesn't provide any insight ?
(which is usually the case). A good proof usually illuminates _why_ some
mathematical statement is true, and that is why computer proofs are usually
looked upon with suspicion by many mathematicians.

After all, a computer simply churning out the answer (42 ?) doesn't advance
the state of human knowledge at all.

~~~
agentcoops
What the author is disagreeing with, the Hilbert-Bourbaki method, isn't
mathematics as the proving of theorems, but rather the proving of theorems as
a necessarily formal task; that is, that when one is doing a proof it ought to
be done in a (preferably intuitionistic) formal system with no steps omitted.
The alternative, and what almost all mathematicians do in practice, is to
informally lay out a proof that any professional mathematician (but, perhaps
not a computer) can follow, filling in details as necessary.

Further, formal computer proofs are rarely wholly generated by computer.
Rather the user specifies goals and tactics which the computer verifies or
refutes. If one is using a reasonably powerful logic (ie first-order and not
propositional) there are only relatively small tasks that can be automated.
Hence, the result is often equivalent, or, at least, close to a human
generated proof in terms of what is illuminated about the mathematical
statement, but every step is necessarily valid.

Further, there is a field of modern mathematical logic called proof mining
which concerns itself with the analysis of fully formal proofs and can often
obtain illuminating details from a formal proof that wouldn't be possible from
an informal equivalent.

This isn't to say that all mathematics should be formal in the sense
indicated, but just to point out that it remains an interesting endeavor that
is worth keeping an eye on. Also, note that the author /does/ think that
formal verification of this sort can be useful for software development which
is indeed what formal methods are mostly used for these days.

~~~
sajidu
Some quotes:

"and I believe that mathematicians who continue to do pure human, pencil-and-
paper, computer-less, research, are wasting their time."

"The axiomatic method is not even the most efficient way to prove theorems in
Euclidean Geometry."

The author _is_ disagreeing with math as practiced by most mathematicians:
proof using only pencil, paper and coffee.

BTW, formalism is only a fairy tale mathematicians tell their children when
putting them to bed at night. In our hearts, we're all closet platonists.

~~~
yummyfajitas
He isn't disagreeing with the _goals_ of math as practiced. All your quotes
illustrate is that he thinks the usual _methods_ are woefully inefficient.

More quotes:

"and I believe that mathematicians who continue to do pure human, pencil-and-
paper, computer-less, research, are wasting their time." _If they learned how
to program, in, say, Maple, and used the same mental energy and ingenuity
while trying to use the full potential of the computer, they would go much
further._

I.e., human + computer >> human.

"The axiomatic method is not even the most efficient way to prove theorems in
Euclidean Geometry." _Thanks to Rene Descartes, every theorem in Euclidean
Geometry is equivalent to a routine identity in high-school algebra,_

I.e., verifying algebraic identities is easier than euclidean proofs,
especially if done by maple.

His main argument is simply that computer-assisted proofs should not be held
to a higher standard than human-only proofs.

~~~
Tichy
"I.e., verifying algebraic identities is easier than euclidean proofs,
especially if done by maple."

If something like that has been discovered, fine, use it, and solve geometry
problems in that way in the future. But how would that have been discovered
without doing classical mathematical proofs? I don't think that some problems
can be solved algorithmically proves that the same holds for all problems.

~~~
yummyfajitas
He isn't arguing against classical mathematical proofs. He is arguing that
informal computer assisted proofs should not be held to a higher standard than
informal human created proofs.

------
homme
The maniacal obsession with unassailable formal proof gives mathematics its
credibility, above all else. There is a lesson here for all other endeavors
that seek truth: Rigor uber alles. Take nothing for granted.

------
gabriel
Here's the link to the Dec 2008 issue of the Notices of the American
Mathematical Society that this article is in response:
<http://www.ams.org/notices/200811/index.html>

------
Tichy
I think he forgot to make his point, and the whole article is just flamebait.

------
theoneweasel
The question I see here is whether you are interested in Mathematics as end,
or in the Application of Mathematics as an end. Certainly, if all you need is
an algorithm that works, don't write a formal proof. However, if your
mathematical work is foundational for some future algorithm, you need to know
that p(x) holds for every x, not just a bunch that you tested.

Most students do not need to be concerned with (and indeed, are not instructed
in) formal proofs. But, if we are going to push math forward, we need to lay a
solid foundation of axioms and proofs.

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lacker
It is not true that the best way to do geometry is by converting everything
into algebra. Proving simple geometric stuff like, the three altitudes of a
triangle all intersect at a single point, is a pain using coordinates. Just
because it's possible to convert problems in domain X into problems in domain
Y doesn't mean that domain X is pointless.

------
galo2099
What human obsession with "formal proofs"??? Have you heard of religions?

~~~
mjgoins
Funny you mention it, because when I hear "obsession with formal proofs" I
immediately think of Anselm and Plantinga.

