A little bit of knowledge is a dangerous thing.
There’s no shortage of stupidity in the world. And, alas, it comes in
many, many different kinds. Among the ones that bug me, pretty much
the worst is the stupidity that comes from believing that you know
something that you don’t.
This is particularly dangerous for people like me, who write blogs like
this one where we try to explain math and science to
non-mathemicians/non-scientists. Part of what we do, when we’re writing our
blogs, is try to take complicated ideas, and explain them in ways that make
them at least somewhat comprehensible to non-experts.
There are, arising from this, two dangers that face a math or science blogger.
- There is the danger of screwing up ourselves. I’ve demonstrated this plenty of
times. I’m not an expert in all of the things that I’ve tried to write about, and
I’ve made some pretty glaring errors. I do my best to acknowledge and correct those errors,
but it’s all too easy to deceive myself into thinking that I understand something better
than I actually do. I’m embarrassed every time that I do that.
- There is the danger of doing a good enough job that our readers believe that
they really understand something on the basis of our incomplete
explanation. When you’re writing for a popular audience, you don’t generally get into
every detail of the subject. You do your best to just find a way of explaining it in a way
that gives people some intuitive handle on the idea. It’s not perfect, but that’s life.
I’ve read a couple of books on relativity, and I don’t pretend to really fully understand it. I
can’t quite wrap my head around all of the math. That’s after reading several entire
books aimed at a popular audience. Even at that length, you can’t explain all
of the details if you’re writing for non-experts. And if you can’t do it in a three-hundred page
book, then you certainly can’t do it in a single blog post! But sometimes, a reader will
see a simplified popular explanation, and believe that because they understand that,
that they’ve gotten the whole thing. In my experience, relativity is one of the most common
examples of this phenomenon.
Todays post is
href="http://www.cosmicfingerprints.com/blog/incompleteness/">an example of
how terribly wrong you can go by taking an intuitive explanation of
something, believing that you understand the whole thing from that intuitive
explanation, and running with it, headfirst, right into a brick wall.
As any long-time reader of this blog knows, I’m absolutely fascinated by Kurt Gödel, and
his incompleteness theorem. Incompleteness is, without a doubt, one of the most important, most
profound, most surprising, and most world-changing discoveries in the history of mathematics.
It’s also one of the most misunderstood.
The problem is exactly what I described up above. It’s a really
complicated idea. You can’t fully grasp it without having a really good
understanding of logic and proof, and spending time going through the whole
proof, in all of its gory details. But you can get across the gist of it with
a simple explanation – and therein lies the problem. The gist that you can
grasp with a simple explanation isn’t the real meaning of the
incompleteness theorem. It’s an approximation – something close enough to what
the theorem says to help you understand it – but it’s not the real meaning of
the theorem. And if you don’t realize that you don’t understand all of the
details, you can wind up making some really serious errors.
One of the common ways that Gödel’s incompleteness theorem
is explained is by a metaphor. Incompleteness shows how, when you’re
working inside of a formal mathematical system, you can find statements
that can’t be proven true or false from within the system. So as an
approximation of that, people sometimes say something like “If you’ve
got all of the true statements you can prove inside of a circle, then
Gödel shows that there’s something outside of that circle.”
That’s a nice metaphor, which is certainly clearer, on an intuitive level
than the earlier, but more correct, statement.
People often try to make it even a bit clearer, by extending that
metaphor: If you’ve got a set of tools for drawing geometric
systems, and you use them to draw a circle, part of the field that you use to
draw on must be outside your circle. No matter how careful you are, you’ll
can’t draw a line around an area of the field without leaving part of the
field outside of it. Gödel’s theorem describes a mathematical form of the
same sort of problem: if you have a good enough set of mathematical tools for
showing what’s true and what’s false, there will be things that fall outside
of the range of those mathematical tools.
The problem is, that’s just an intuitive explanation. It misses the depth
of incompleteness. It both makes incompleteness seem like something
more than it really is, and also like something less than it
You can try to make the statement of the theorem closer to accurate.
That’s what I just did two paragraphs ago: I restated it in terms of a
mathematical toolkit. That’s closer. But it still stinks.
I can get even closer, by saying something like “In any valid, consistent,
formal mathematical system that’s capable of expressing Peano arithmetic,
there will be true statements that cannot be proved within the system.” That’s considerably
closer, but it still misses some of the essential points. After all, what does
“true” mean in a formal system? And it misses one of the big facets of incompleteness, which
is that no matter how careful you are to create a careful model that’s constrained
to prevent self-referential statements, you can always create an alternative and
equally valid model that does include problematic statement. Grasping that fact, that
there’s more than one model that can be fit to any consistent system, and what that
really means, is absolutely crucial to fully understanding incompleteness.
The point, however, is that just because you’ve understood some intuitive
explanation of something doesn’t mean that you really understand it. And using
your incomplete understanding as the basis for building a proof of something
else is, pretty much inevitably, going to be a total disaster.
Our target in this poist is the author of an argument that tries to use
Gödel’s incompleteness theorem as a proof of the existence of God. It’s a perfect
example of what I’ve just gone on at great length explaining. The author takes the
“no circle without something outside of it” explanation of Gödel, and abuses it
horribly. He really believes that he gets it, and that he’s doing valid reasoning on the
basis of incompleteness. But because he doesn’t know that he doesn’t really understand it,
he makes a mess.
Here’s his explanation of Gödel:
Gödel’s Incompleteness Theorem says:
“Anything you can draw a circle around cannot explain itself without referring
to something outside the circle – something you have to assume but cannot
You can draw a circle around all of the concepts in your high school geometry
book. But they’re all built on Euclid’s 5 postulates which we know are true
but cannot be proven. Those 5 postulates are outside the book, outside the
You can draw a circle around a bicycle. But the existence of that bicycle
relies on a factory that is outside that circle. The bicycle cannot explain
You can draw the circle around a bicycle factory. But that factory likewise
relies on other things outside the factory.
Gödel proved that there are ALWAYS more things that are true than you can
prove. Any system of logic or numbers that mathematicians ever came up with
will always rest on at least a few unprovable assumptions.
Gödel’s Incompleteness Theorem applies not just to math, but to everything
that is subject to the laws of logic. Everything that you can count or
calculate. Incompleteness is true in math; it’s equally true in science or
language and philosophy.
Anyone who knows math can tell you that most of that has nothing to do
with Gödel. It’s mostly confused babble. Gödel didn’t prove that you
need to start any proof with a set of unproven axioms. That was part of math
and logic long before Gödel ever came along. But our author believes
that that’s what Gödel actually talked about.
It isn’t. Gödel showed that given a formal mathematical system of
sufficient power, you can produce a statement in the system which is true, but
which is not provable within the system. Like I said before, that’s still a
wretched oversimplification, but it’s a whole lot closer to the real meaning.
What Gödel did was show how you can use simple arithmetic to encode
logical statements into numbers; and then that you could use that encoding to
create a number which encodes the statement “This statement cannot be proven
true within this system”. It’s true: you can’t prove it within the system. You can
use a different system to show that it’s true; but in that system, you can
do a similar construction, and show how that system includes statements that are true,
but not provable within it.
But he’s convinced that he understands it, and that what it really means is
that you need axioms that are outside of the system. He really believes that the
“something is outside of the circle” explanation really does express the full meaning
of Gödel, and that the things outside of the circle are the basic axioms.
Of course, he’s only just begun. Nothing demonstrates your command of a subject
better than your ability to take it and try to apply it to a domain where it makes
absolutely no sense at all.
A “theory of everything” – whether in math, or physics, or philosophy -
will never be found. Because it is mathematically impossible.
Does Gödel really say that you can’t describe the physics of reality mathematically?
No. We’re actually pretty close to nailing down something like a grand unification theory, which
would be a physical theory of everything. And Gödel’s theorem has nothing to do
with whether or not it’s possible.
OK, so what does this really mean? Why is this super-important, and not
just an interesting geek factoid?
Here’s what it means:
- Faith and Reason are not enemies. In fact, the exact opposite is
true! One is absolutely necessary for the other to exist. All reasoning
ultimately traces back to faith in something that you cannot prove.
Once again, completely wrong. Gödel’s theorem says nothing of the sort.
He’s still making the same basic mistake – that what Gödel did was show that
logic requires axioms. That’s not what it says, and even if it was, this kind of vague, fuzzy,
feel-good statement wouldn’t follow logically from it.
- All closed systems depend on something outside the system.
- You can always draw a bigger circle but there will still be something outside the circle.
You should be starting to see the pattern by now. He really doesn’t understand
what incompleteness means. But he’s got one silly metaphor about circles, which he’s
misinterpreted, and which he’s absolutely convinced is the whole truth. Gödel’s
theorem doesn’t say either of those things. It doesn’t come close to saying anything like those
things, and no one who even comes close to understanding what it says could possibly
make that mistake.
The problem with all of the statements above is (apart from his confusion about
axioms) the fact that Gödel’s incompleteness theorem is a statement about
formal logical systems, and statements within those systems. Incompleteness
doesn’t talk about religion, faith, god, circles, or open or closed systems. It talks about
formal logical inference systems.
Anyway, we’re just finally coming to the point of his argument. But first he needs
to take even more of logic and push it into his “circles” rubbish:
Reasoning inward from a larger circle to a smaller circle
(from “all things” to “some things”) is deductive reasoning.
Example of a deductive reasoning:
- All men are mortal
- Socrates is a man
- Therefore Socrates is mortal
Reasoning outward from a smaller circle to a larger circle (from “some things” to
“all things”) is inductive reasoning.
Examples of inductive reasoning:
- All the men I know are mortal
- Therefore all men are mortal
- When I let go of objects, they fall
- Therefore there is a law of gravity that governs all falling objects
Notice than when you move from the smaller circle to the larger circle,
you have to make assumptions that you cannot 100% prove.
For example you cannot PROVE gravity will always be consistent at all times.
You can only observe that it’s consistently true every time.
Nearly all scientific laws are based on inductive reasoning. All of
science rests on an assumption that the universe is orderly, logical and
mathematical based on fixed discoverable laws.
You cannot PROVE this. (You can’t prove that the sun will come up tomorrow
morning either.) You literally have to take it on faith. In fact most people
don’t know that outside the science circle is a philosophy circle. Science is
based on philosophical assumptions that you cannot scientifically prove.
Actually, the scientific method cannot prove, it can only infer.
(Science originally came from the idea that God made an orderly universe which
obeys fixed, discoverable laws – and because of those laws, He would not have
to constantly tinker with it in order for it to operate.)
Those are just about the worst definitions of “inductive” and “deductive”
that I’ve seen. But worse is that they’re part of a purportedly mathematical argument: in
math, “inductive” and “deductive” mean something specific, and it’s not this. In
math, inductive reasoning absolutely does produce proofs.
But the worst part of that is: “the scientific method cannot prove, it can only infer”.
What does “infer” mean? In mathematical terms – in the terms that we’re using because we’re talking
about the implications of a logical proof! – it means prove using mechanical inference rules within
a formal mathematical system”. So his statement, in mathematical terms, reduces to
a contradiction: “the scientific method cannot prove, it can only prove”.
And now, finally, we get to the point:
Now please consider what happens when we draw the biggest
circle possibly can – around the whole universe. (If there are multiple
universes, we’re drawing a circle around all of them too):
- There has to be something outside that circle. Something which we have to
assume but cannot prove
Nope. Doesn’t say that.
- The universe as we know it is finite – finite matter, finite energy, finite space and 13.8 billion years time
Nope. Doesn’t say anything like that. How can you possibly get from Gödel’s
theorem to a statement that the universe can’t be infinite? There’s a reason why he just
pretends to “prove” this, but doesn’t actually connect it to anything, even by the most
flimsy informal reasoning: because he can’t. The only reason that it’s here is because
he wants God to be the only infinite thing, so he just threw it in, even though it doesn’t come
close to making any sense.
- The universe (all matter, energy, space and time) cannot explain itself
Again, nope. Gödel’s theorem says nothing remotely like this. Gödel’s incompleteness
theorem doesn’t say anything about explanations. It only talks about proofs,
in the formal mathematical sense of proof. The whole concept of explanation is completely
outside the bounds of Gödel.
- Whatever is outside the biggest circle is boundless. So by definition it
is not possible to draw a circle around it.
Once again, this is a total non-sequitur. It simply does not follow from
- If we draw a circle around all matter, energy, space and time and apply
Gödel’s theorem, then we know what is outside that circle is not matter, is
not energy, is not space and is not time. Because all the matter and energy
are inside the circle. It’s immaterial.
And this is just word-games. “If we draw a circle around all of the gribble, then whatever isn’t
in the circle can’t be gribble”. It doesn’t mean anything.
- Whatever is outside the biggest circle is not a system – i.e. is not an
assemblage of parts. Otherwise we could draw a circle around them. The thing
outside the biggest circle is indivisible.
Isn’t this circle rubbish getting tiresome? Actually, it sort of makes sense: all of his
arguments are going in circles, so why not express circular arguments in term of a circular
- Whatever is outside the biggest circle is an uncaused cause, because you can
always draw a circle around an effect.
It’s right back to one of the classic crackpot arguments for god:
the uncaused cause. Every effect must have a cause; before the universe
was created, there was nothing to cause anything, so there must be something
outside of it, therefore god.
It’s a dreadful argument in general. But it’s worse here. He’s just spent all of this time
arguing that you can’t prove anything by what he calls inductive reasoning. Just because
every event that you’ve ever seen has a cause, by his own argument, that doesn’t mean
that you can conclude that every event must have a cause.
And even that isn’t the worst of it: he’s claiming that all of this is
“proven” by Gödel’s theorem. It’s not. The universe isn’t a formal mathematical
system. And even if it was, by this argument, God wouldn’t be what religious
folks think of as God; God wouldn’t be a sentient force that created the universe; God
would just be a self-referential statement encoded in arithmetic. Not exactly what most of
us religious folks believe.
And then, he needs to repeat that whole stupid argument again, this time using
“information” instead of “matter and energy”. It’s like he wants to make my argument for me. You
can substitute anything into that, and repeat the argument. Matter, space,
information, intelligence, consciousness, colors, shapes. Seriously – just try it. “If we
draw a circle around all of the colors in the universe, then anything outside of the circle can’t
have a color: it must be colorless! Therefore, by Gödel’s theorem, there must be a colorless thing outside
of the circle of color which is the creator of all color!”
A little bit of knowledge is a dangerous thing. It can convince you that an argument
this idiotic and this sloppy is actually profound. It can convince you to publicly make
a raging jackass out of yourself, by rambling on and on, based on a stupid misunderstanding of
a simplified, informal, intuitive description of something complex.