

This is what is wrong with contemporary mathematics - eriksank
http://erik-poupaert.blogspot.com/2012/12/this-is-what-is-wrong-with-contemporary.html

======
gizmo686
Is this article meant to be a joke?

He goes from the statement "For any formal effectively generated theory T
including basic arithmetical truths and also certain truths about formal
provability, if T includes a statement of its own consistency then T is
inconsistent." To "all basic claims are more or less false." If we could
actually prove that all claims were false, we would be in violation of the
Incompletness theorum.

Not to mention the absurtidy of his conclusion. Proving a statement for a set
of values you thought of is far more likely to be wrong than proving it with
formal mathamatics.

~~~
meta-coder
> Gödel's 1931 Second Incompleteness Theorem insists that it is strictly
> forbidden to ever claim the truth of any scientific or mathematical
> statement

That is grossly wrong interpretation of the theorem. I stopped reading the
article right there.

~~~
eriksank
Why don't you write a program to show that a particular statement in my
article would be wrong?

~~~
demetrius
Your article constains no assert statement, therefore, according to itself, it
is false.

~~~
eriksank
I like your attempt. At least, it is meaningful. Only the domain of statements
about facts can be verified with facts. The domain of statements about
statements cannot. You have indeed correctly underlined that the rules
governing scientific statements -- statements about statements -- cannot be
scientific themselves.

------
opminion
_A Popper-compliant claim must be phrased as an invitation to look for
counterexamples_

No, a _Popper-compliant_ claim must be falsifiable. It does not need to be
phrased in any particular way.

------
Zenst
I think the fact that I cant drop that article onto a computer and have it
tell me if its correct highlights the problem the author is trying to make.

I was also under the impression that programs like Mathimatica enabled you to
do this with mathematic equations. But I guess for many they would take a long
time to prove.

I suppose a easy way to sum up the article is we have a mathematic symbol for
infinity, yet we could never compute that on a compter as you could always add
a value greater than zero.

~~~
ReidZB
The reason we can't drop the article into a computer and have it magically
return "true" or "false" is because making computers that even have human
intelligence is _damn_ hard as it is, so until we can perfect language parsing
and the like and then input millennia of collective mathematical knowledge
(correctly, no less), we're not going to have any particular progress on that
front.

What does this have to do with "flaws" in contemporary mathematics though? If
you're trying to use this as an example of a flaw in anything, it would be the
computer industry for not figuring out artificial intelligence, not the
mathematics community. So, _if_ this was the author's intention, then I think
it's flawed right from the outset, but I don't think this was truly the
author's intent. (For one thing, Curry-Howard correspondence is talking about
lambda calculus, not what you think when you hear "computer" today.)

Really, the article's entire premise is predicated on _extreme_
misunderstandings of Godel. For instance:

> In fact, Gödel's 1931 Second Incompleteness Theorem insists that it is
> strictly forbidden to ever claim the truth of any scientific or mathematical
> statement:

What? Where does it say that? It says (quoting the article here):

> For any formal effectively generated theory T including basic arithmetical
> truths and also certain truths about formal provability, if T includes a
> statement of its own consistency then T is inconsistent.

If a theory includes a statement of its own consistency, then that theory is
inconsistent. How does that imply that it is "strictly forbidden to ever claim
the truth of any scientific or mathematical statement"? The honest answer is
that it doesn't, not in the slightest. When it comes to mathematics, we're not
saying the axioms are consistent, we're saying statements _within this set of
axioms_ are true. Are the axiomizations true? I don't know, but _a lot_ of
work has gone into these theories and so far they look pretty good. But at any
rate, the author uses this theorem in a completely incorrect manner. He is
right that we don't know if any of the various mathematical foundations are
consistent, true, but as I said, so far everything looks pretty good.

Then he throws this bombshell:

> Given Gödel's First Incompleteness we already know that for any claim making
> use of basic arithmetic, counterexamples must exist.

What? How? Here's Godel's First Incompleteness Theorem, per Wikipedia:

> Any effectively generated theory capable of expressing elementary arithmetic
> cannot be both consistent and complete. In particular, for any consistent,
> effectively generated formal theory that proves certain basic arithmetic
> truths, there is an arithmetical statement that is true, but not provable in
> the theory (Kleene 1967, p. 250)

How in the world does that say that "for any claim making use of basic
arithmetic, counterexamples must exist." (Counterexamples of what, anyway?
Basic arithmetic?) Actually, what the theorem says that you can't have a
theory capable of expressing elementary arithmetic that is _both_ consistent
and complete. The _both_ is the huge part. So far, for example, we think ZFC
is consistent. We _know_ it is not complete; see this:
[http://en.wikipedia.org/wiki/List_of_statements_undecidable_...](http://en.wikipedia.org/wiki/List_of_statements_undecidable_in_ZFC)
. So, the idea that there "must exist counterexamples" (of what, I don't know)
as per Godel's First Incompleteness Theorem is simply ridiculous. As a result,
we don't know that there are "simply bugs" in the "successor function." As a
result, the idea that "all basic claims are more or less false" is completely
shot down. Needs better proof.

> The claim above is not Popper-compliant.

Why not? It's falsifiable, so it is. (If it were false, you would need simply
to provide an extremely small epsilon and ask for the correspondingly large
delta.) It gives the "appearance" of truth because it is indeed true --
relative to the axioms under which it was proven. The author also seems to
conflate inductive and deductive disciplines. In mathematics, we create axioms
and follow them through to some meaning. In the vast majority of the sciences,
the process is very much inductive. So, this seeming focus on making
everything mathematical "scientific" is pretty flawed in and of itself. How
can you reason about mathematics if you don't have some way to check your
results? In the sciences, if I predict gravity will disappear in five minutes
for thirty seconds, that is a very testable claim (and a false one, most
likely, based on our past experimentation). How can I disprove or prove
something in mathematics? Well, we created all these wonderful axioms and they
seem consistent (although incomplete, of course), so I can use them to check
my results. A little PHP script is not going to cut it. For one thing, the
author's script's notion of "division" is predicated on the very idea of
division. Where did that come from? For another, the script is not a proof at
all. It's just a program to test some values. It doesn't _prove_ anything at
all. It's just further evidence towards the proof of something we already know
is true.

In sum: the article is bullshit. It misinterprets Godel in terrible ways. For
all the fancy French and Latin, it doesn't even provide a programmatic proof
of its own claims, the supposed holy grail, after all! The only program it
provides is a little test, not anything near a proof. I think if you actually
try to provide "Popper-compliant" statements and write programs to prove
everything, you end up reverting to all of the normal contemporary
mathematical language anyway. I mean, look at this:

> In order to reject this claim, find a number almostZero for which it will
> not be possible to find a large enough number x for which 1/x is smaller
> than almostZero.

That sentence is essentially a rewording of an old-fashioned delta-epsilon
proof. So, how does that indicate a flaw in contemporary mathematics at all?
Why should we stop teaching our kids these "symbols" since they are apparently
still so very convenient for reasoning about mathematics? The truth is that
there's no reason to. The "executable claims" are only found after all of the
"verbiage" is said and done.

~~~
eriksank
> That sentence is essentially a rewording of an old-fashioned delta-epsilon
> proof. So, how does that indicate a flaw in contemporary mathematics at all?

The first flaw is that it is not a proof but only a reduction. The second flaw
is that contemporary math does not make the slightest attempt at making its
claims easier to verify. What is wrong with an invitation/challenge in the
form of a program asking to show the claim wrong?

~~~
gizmo686
Contemporary math does not attempt to make its claims verifiable in the sense
you describe, because it proves them. There might be a flaw in the proof
itself (and there is active research in creating programs to check proofs).
Sometime proofs in math our of the form you describe. For example, the (most
well known) proof that sqrt(2) is irrational is a challenge to find the
simplist fraction of to intergers that equals sqrt(2). The proof then goes on
to show that any possible response you can provide to the challege is false.
Simmilarly, for any mathametical claim, if you can provide a single counter
example then you have dissproven the claim. Not being able to provide a
counter example does not prove the claim.

~~~
eriksank
Well, I have to insist that math does not prove a statement true. Math only
proves that the statement does not contain original untruth.

> Not being able to provide a counter example does not prove the claim.

It is never possible to prove a claim anyway. It is only possible to prove
that it does not contain original untruth.

~~~
lutusp
> Well, I have to insist that math does not prove a statement true.

Be careful to define your terms. Most people, including mathematicians, will
insist that Euclid's prime theorem proves that there are an infinity of primes
-- that the assertion is "true".

<http://en.wikipedia.org/wiki/Euclids_theorem>

But it will always be possible to redefine "true" arbitrarily to reject any
such claims, which partly explains philosophy's low standing among
intellectual disciplines.

> It is never possible to prove a claim anyway.

You've shifted ground in your wording, and contradicted the idea of a
mathematical proof. There really are mathematical proofs.

In general science, your claim is correct -- a theory cannot be proven true,
only false. But in mathematics, proofs exist.

~~~
eriksank
Math and science use exactly the same method of reducing their claims to their
basic claims and backing those with a failed search for counterexamples.
Consequently, all untruth is invariably always the result of the untruth
hidden in the basic claims. The untruth in the basic claims obviously exists
but it is sheer impossible to pinpoint precisely where exactly it is.

~~~
lutusp
> Math and science use exactly the same method of reducing their claims to
> their basic claims and backing those with a failed search for
> counterexamples.

This is false. You're overlooking a very important difference between math and
science. Science is empirical, which means any scientific theory can be
overthrown by new evidence arising in nature. This is not true for mathematics
-- once a conjecture becomes a theorem, it's irrevocable.

Your viewpoint is a philosophical one that fails to account for the key
difference between math and science.

~~~
eriksank
Probably worse than that. I simply refuse to make any such key difference
between math and science. In my opinion, both math and science are empirical.

~~~
gizmo686
In what way is math empirical? If one wants to prove that 1+1=2, one does not
take one apple, add another apple, and observe that one now has two apples.
One defines: x + 0 = x x + successor(y) = successor(x) + y 1=successor(0)
2=successor(1)

One then takes the true true statement 1+1=1+1 | Identity And finds:
1+successor(0)=1+1 | Definition of 1 successor(1) + 0 = 1+1 | Definition of +
2 + 0 = 1+1 | Definition of 2 2=1+1 | Definition of +

I made no observation to prove 1+1=2. What I do observe is that if we map the
counting numbers to the count of how many apples I have, and we map the
addition operation to combing to groups of apples, then the mathematical
result is consistent with reality. However, if we empirically discover that
the mathematical result is not constitent with reality, than that does not
mean that the math is wrong, it simply means that the reality does not behave
in a way that corresponds to the math.

~~~
eriksank
> it simply means that the reality does not behave in a way that corresponds
> to the math.

In that case, we should watch out to use that kind of math in engineering or
for example calculate expected airplane behaviour with it.

------
qwinter
sounds like a programmer took a philosophy of math & science class and didn't
fully grasp it

~~~
StevenXC
Agreed..

> Therefore, all basic claims are more or less false. We just do not know why
> they are false. The entire body of science and mathematics therefore does
> not rest on a set of true statements but on a set of difficult-to-prove-
> false statements. Disproving a claim may even require showing where we can
> find a bug in the successor function. This is seriously hard, but still
> possible, because we know that these bugs must exist.

We do NOT know that these bugs exist. Gödel has shown us that any useful
formalization of mathematics (such as the commonly utilized Zermelo–Fraenkel
"ZF" set theory) cannot prove its own consistency. However, that does not
imply that it must be inconsistant.

The fact that thousands of mathematicians have used ZF for about a hundred
years without finding a contradiction makes me confident that even if it's
inconsistant, we should be able to patch it up easily enough. But honestly I
don't expect that to happen.

~~~
eriksank
In my opinion, Kurt Gödel's claim points to something that seems to go wrong
in Alonzo Church's successor function. For the sake of the argument, let's
call that a bug. It would take a serious amount of work to point out how that
would affect Zermelo–Fraenkel.

~~~
gizmo686
Care to elaborate on how Godel's theroum implies a bug in the successor
function.

------
ReidZB
"A little learning is a dangerous thing" -- Alexander Pope

------
mshang
> Curry-Howard executable version of the claim

I don't see how the code provided has any relation to the Curry-Howard
correspondence, which is a statement relating _types_ and proofs.

~~~
eriksank
The original CH mapping, that is, "a proof is a program, the formula it proves
is a type for the program" is quite hard to achieve because it tries to
demonstrate that a particular statement is true. It is much easier to honestly
fail in demonstrating that it is false.

------
lsh123
IMHO, this article is a complete BS from someone who doesn't understand the
difference between Bourbaki's math and real modern math

(P.S. I used to be a mathematician in my previous life)

~~~
eriksank
The whole point was that we were going to write a program and invite each
other to find counterexamples for the statement proposed, so that we can
finally stop the bullshit of ad-hominem arguments. I know it must be hard for
you to switch but in the long term it will benefit you.

------
pshc
This had me grinning like a maniac the whole way through. Delightful. And the
PHP at the end is a masterstroke.

------
eriksank
Well, for any statement that you did not like in my article, write a program
to show me that the statement is wrong.

~~~
monochromatic
assert(0)

~~~
eriksank
If your claim is void (zero), you could put a bit more effort into elaborating
that ;-)

~~~
gizmo686
You state: "Gödel's 1931 Second Incompleteness Theorem insists that it is
strictly forbidden to ever claim the truth of any scientific or mathematical
statement"

That claim is false, therefore, the truth of that claim is equivilent to the
truth of 'false'. Becuase your article is true iff all of the statements
within it are true, and your article contains a statement which is false, then
your entire article is false. Using this, we can 'reduce' the 'proof' of your
into assert(false). I suspect that the parent is using 0 as an alias for
false.

More generally, for any contemporary mathamatical proof, what you call a
reduction, you can reduce the proof of the form you describe to either
assert(true) xor assert(false), depending on what the mathametical
proof/reduction shows.

~~~
monochromatic
_using 0 as an alias for false_

Oops, my python roots are showing.

