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:
GÃ¶del'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.
GÃ¶del 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.
GÃ¶del'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
GÃ¶del'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.
What if we draw a circle around god? There must be something outside!
(reminds me of that scene with the Djinni in G.E.B.)
Your point is well received. I file this strain of thinking under "practical epistemology." For numerical conclusions (say, of scientific studies),it's fairly straightforward to quantify your uncertainty using statistics, but what is the analog for non-numerical concepts that float around the brain? One helpful thing that I ask myself, as a check against over-certainty, is why I think I know something. If I think I understand something, it's usually the result of my having read about it somewhere; where did that information come from? Was it reputable? That kind of stuff. Once again, cheers on the post.
But maybe the universe is a formal mathematical system. Conservation of energy is just a different way of saying that the universe is symmetric with respect to translation in time, so all this "stuff" that we think is "real" is just a mathematical consequence of a symmetry.
Actually, there are equivalent of Godel's theorems for physical laws, see :
Physical limits of inference, Wolpert D, Physica D 237 (2008) 1257-1281
Philosophy of science: Theories of almost everything, P.-M. Binder, Nature 455 (2008), 884-885
The idea is very similar to Godel's proof : you define "inference device" which are basically physical theories, and then you show that for some families of inference devices, some laws can not be inferred (basically, as I understand it, they can not infer themselves, very classically). There are interesting results : just like there is a universal Turing machine, there is a unique universal inference device (i.e. one single "theory of everything"). Actually, it seems David Wolpert calls that a " monotheism theorem".
I summarized this in French here : http://tomroud.com/2008/11/04/le-demon-de-laplace-et-la-theorie-du-pres…
By the way, I find it quite fascinating to see how people remember the contribution of Godel but not Turing's, who was one of the major player of this field.
@Paul - I have to be honest, I do think the universe is computational in nature. Which is distinct from saying that it is deterministic, which I certainly don't buy. Just computational.
The incompleteness theorem is not about what you can prove with a mathematical language; it's about what you can represent with that language ('language' is a better word here, than 'system', because it automatically carries with it the idea that there is more than one approach to the 'same' set of knowledge). What the theorem itself proves is that (a) it is impossible to design a self-consistent mathematical language that is incapable of representing self-contradictory statements (regardless of their truth value), and that (b) it is, conversely, impossible to design a self-consistent mathematical language that can express every possible mathematical truth.
The importance of this is that (a) pointing out self-contradictory statements is not a valid criticism of any particular mathematical system; and (b) it's turtles all the way down.
How'd I do? XD
To pick a nit, you're omitting the important qualification of "first-order" in your descriptions of the incompleteness theorem. Second-order arithmetic is complete, in the sense that every true statement can be proved. Its proofs aren't recursively enumerable, so the computational force of incompleteness is still in force: you can't build a Turing machine that will prove or disprove arbitrary statements. But it is a formal system. Indeed, [i]the[/i] formal system that defines what arithmetic means.
I hope I'm not screwing up this nit, since it's late and I'm in my cups. Makes it risky to pick 'em. ;-)
If you think this guy's post on Incompleteness was incomplete, you should check out his post on 7 biological myths that no engineer would believe. It is an elegant proof of the Salem Hypothesis.
Lets consider that Goedel could mean our systems are inconsistent. Now lets go ahead and put a circle around that... lol
I read G.E.B. when you recommended it on this blog some years ago. I absolutely loved it. Thank you.
A Serious man:
Clive Park: I understand the physics. I understand the dead cat.
Larry Gopnik: You understand the dead cat? But... you... you can't really understand the physics without understanding the math. The math tells how it really works. That's the real thing; the stories I give you in class are just illustrative; they're like, fables, say, to help give you a picture. An imperfect model. I mean - even I don't understand the dead cat. The math is how it really works.
Clive Park: Very difficult... very difficult...
I need one of you guys' help to pin this guy down to the ground, while I draw a circle around him, "prove" to him that he's incomplete, and banish him to Timbuktoo.
After a user tried to correct Perry, in comment #31 the author writes:
"I believe the indivisibility of the trinity at the very least shows that God is not 'effectively generated' ie 'recursively enumerable.'"
Almost made reading the comments worth it.
Amusing "theory" Perry's got there. Let's see, if I draw a circle around all known and possible intelligence - man, animal, machine, extra-terrestrial, etc; then there is something outside the circle that is not intelligent but created all intelligence?
Hence, God is non-intelligent entity.
Somehow I feel Perry hasn't thought through all the ramifications of his "demonstration". Drawing circles around just about any positive attribute (power, goodness, honesty, etc.) leaves you with a "god" that can't have that attribute (or he'd be in the circle too).
A professor at my university wrote a textbook (quite a good one) on mathematical logic.
It has a nice statement: "All 'philosophical' comments based on Goedel theorem are false". With a footnote: "including this one".
Time and time again it shows to be true.
"The Incompleteness of the universe isnât proof that God exists. Butâ¦ it IS proof that in order to construct a consistent model of the universe, belief in God is not just 100% logicalâ¦ itâs necessary."
That's just nonsense, There isn't proof of god, but you MUST belive in him.
If the existence of god follows so immediately from Godel's Incompleteness Theorem, then why did Godel spend so much time working on his ontological proof of the existence of God? Quite an arrogant simpleton, to believe that he has found an immediate and important consequence of Godel's work that the great man himself missed...
Laroquod, I think you have it wrong. The incompleteness is about formal systems and provability. It says that a recursively axiomatized first-order logic as strong as arithmetic has statements that are neither provable nor disprovable. Which also means, via Lindenbaum, that the system isn't categorical, and can be extended. As I noted above, second-order arithmetic is categorical. But that just shifts the computational limit, since its set of proofs isn't recursively enumerable.
Roud, it may be that I come at logic as much from computer science as from mathematics, but I've never seen Turing's role downplayed vis-a-vis GÃ¶del. Church, on the other hand...
How about "Any system of mathematics that is complicated enough to include arithmetic is inherently either incomplete or inconsistent"?
Still not good.
First, it leaves "system", "arithmetic", "incomplete", and "inconsistent" undefined. Once you start providing enough information to get people to really understand what those mean, you've long-since left the realm of intuitive explanations.
But even then, it lacks some of the essential nuances that are part of incompleteness. For example, Gödel's theorem is fundamentally caught up in proofs - but not in models. Part of the reason that it works is because it relies on the structure of proofs, and the structure of statements.
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 metaphor?
That sentence is just full of win, right there. You came very close to owing me a new keyboard from some sprayed nose water. :-)
How about "Any system of mathematics that is complicated enough to include arithmetic is inherently either incomplete or inconsistent"?
Unlike Mark, it doesn't bother me as a summary, so long as you are aware that it's vague mush, and more importantly, change it to read: "Any system of mathematics that is complicated enough to include arithmetic is inherently either incomplete or inconsistent or noncomputable."
It doesn't bother me as a summary; my point is exactly that you need to be aware that it's vague mush. Intuitive explanations of complex ideas are extremely useful. Hell, most of what I do on this blog is write about intuitive explanations of complex ideas. The catch is that when what you're reading/learning is really just vague mush that gives you a tiny bit of intuition about something complicated, you need to be aware that it's just vague mush that gives you no more that a little bit of intuition.
People like the author of the post that I mocked take the vague mush intuitive explanation, and then because they understand the vague mush, believe that they fully grasp the actual complex idea. All that I'm really trying to say is *don't do what he did*. Recognize your limits; when you're using a vague explanation or an intuition, be aware that you're only using a vague explanation or intuition, and don't think that it makes you an expert.
Thanks for the post Mark. I've been using my vague mush understanding of Godel in some comments I've been posting against intelligent design at the Darwins-God blog. I'll have to watch myself.
I used to envy abstract thinkers: now I'm thankful that I'm pragmatic. From someone on the outside of the circle, it appears that what you are grappling with is the fact that you will never get outside of being human. The human brain is confined to its own languages: magical thinking being the primary and default way we perceive the environment. "Regular" people always have some innate number and quantity awareness. Then there are those whose familar language is mathematic, which curiously, can be revealed or invented. There's the mystery - and the problem. The universe does indeed seem to be a mathematical system - but it's a circle that we exist within. How could one get outside when we are a part, its product? Is there even an outside?
bomoore's comment reminded me of Russell's introduction to Wittgenstein's Tractatus
According to this view we could only say things about the world as a whole if we could get outside the world, if, that is to say, it ceased to be for us the whole world. Our world may be bounded for some superior being who can survey it from above, but for us, however finite it may be, it cannot have a boundary, since it has nothing outside it. Wittgenstein uses, as an analogy, the field of vision. Our field of vision does not, for us, have a visual boundary, just because there is nothing outside it, and in like manner our logical world has no logical boundary because our logic knows of nothing outside it. These considerations lead him to a somewhat curious discussion of Solipsism. Logic, he says, fills the world. The boundaries of the world are also its boundaries. In logic, therefore, we cannot say, there is this and this in the world, but not that, for to say so would apparently presuppose that we exclude certain possibilities, and this cannot be the case, since it would require that logic should go beyond the boundaries of the world as if it could contemplate these boundaries from the other side also. What we cannot think we cannot think, therefore we also cannot say what we cannot think.
"Gribble" is good; I'll have to remember that.
This post makes me a bit nostolgic for the days I spent on the talk.origins newsgroup. It was there that I learned that it isn't "a little knowledge" that's dangerous; it's the little intelligence that these jokers utilize.
Oh god, thank you for saying this. I've had stupid people try to rip my degree to shreds with bloody Goedel time and again. I hate how people missuse that damn theorem, when it really says something quite eloquent.
Brian: Gee whiz! I don't know what to think of "reminding" someone of Bertrand Russell or Wittgenstein, since they use a language that renders the simple opaque! Like every other species on earth we have evolved to survive in a specific environment, which for us is highly social - a competitive/cooperative hierarchy of cunning, but definately not rational, apes. The willingness to "screw up" your neighbor with lies and fraud, or direct violence may be a best bet strategy. Understanding the universe however, isn't a requirement for survival. I too thought that our brain was built to unravel the mysteries of the universe, but after years of thinking about it, I realized that this is silly. Our brain is poor at best at apprehending reality. A supernatural illusion caused by magical thought dictates our perceptions of, and interraction with, the environment. Our brain creates disasters we can't identify, let alone control. (Those circles again.) Mathematical ideas about the universe (however astounding) may not add one unit of adaptability to our repertoire when the environment changes. You can't eat Goedel's Theorem; you can eat your neighbor.
"The Danger When You Don't Know What You Don't Know" is the exact title I would have used to describe the complete nonsense penned by eminent PhD Dr. Carroll under the title "Shocking Fraud from financial scum".
Notwithstanding a complete ignorance of finance, to the extent of conflating finance with economics, morality, sociology and utilitarianism, you actually equate wealth with goods possessing physical, tangible value, which is laughable considering what your own employer does.
The day you figure out how to compute the price of a European put on an American call on the ATMF underlying using a Crank Nicholson finite difference algorithm, let us know if you think all that work represents something useful or just phooey.
In college, I took a math class for non-science majors on infinite set theory. The prof started with Cantor and then spent half the term walking us through Godel's proof. First he had to get us used to the symbolic language and the heavy levels of abstraction, and kept emphasizing how much he was simplifying and leaving out.
But that ah-ha moment when I got it, and saw how the proof worked in my English lit mind, was an unforgettable experience. I gasped, and the prof grinned. He was the best math teacher I ever had, and showed me how math could be plain fun.
He also left me with a great respect for Godel's proof. Like Heisenberg's incompleteness theorem, it gets used for all sorts of mushy things.
"Does GÃ¶del 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 GÃ¶del's theorem has nothing to do with whether or not it's possible."
Actually, there are many serious physicists that think Godel will prevent a final ToE. For example, read this from Stephen Hawking http://www.damtp.cam.ac.uk/strings02/dirac/hawking/. Freeman Dyson has also expressed this view. Godel blocking a finite theory of everything is still an open and valid question.
From my understanding on the issue, there are pretty good arguments on both sides, and neither can claim victory yet. Godel certainly prevents finite axiom systems capable of expressing multiplicative number theory, and if a ToE allows the same, it is limited by the same theorems.
you're aware that Godel thought the universe was something like abstract mathematics of some kind given that he was such a staunch platonist, so i'm curious if you think that the incompleteness theorems did have any influence on Godel's religious attitudes. I ask because thankfully, mathematical logic and theology are both areas where I know enough to know what I don't know. which is a wonderful thing really. but given that I tapped out of logic somewhere between axiomatic set theory and proving the completeness of predicate logic, Godel is a little outside my reach and while I have a vague suspicion that there's a connection between his thoughts on the foundations of mathematics and his theism, I am too ignorant to be able to draw the connections myself. Do you have any thoughts on that matter?
I don't know anything for sure. Gödel was a very private man, who didn't share a lot of his thoughts. And he was also a deeply disturbed man, who suffered from paranoia and delusions. He went through periods where he was convinced that people were trying to poison him.
To say that he was a complicated person whose private thoughts we know very little about is a pretty epic understatement. Most of what we know about what he thought about things is really just hearsay and speculation. So I don't think that we can really draw much, if anything, in the way of conclusions about his beliefs.
He was a very religious man. And because of things found in his notes after he died, we know that he at least dabbled with proofs of the existence of God. He famously translated one classical supposed proof into a form of modal logic. But he never published that, and it's not at all clear whether he thought it was really a convincing proof, or just an interesting logical exercise.
So the simple fact is, we really don't know.
Godel's mental illness does not invalidate his work. Brains do what they do, and much of what they do is nonsensical, contradictory or impossible. The brain has functions that have been added over evolution, and these are unaware of each other. Belief in god is an example. Someone who is a scientist accepts that reality is a system of physical processes independant of human belief or supernatural origin: ideas can be tested, proven, predictions made, and yet many scientists also believe that a supernatural being has preordained every event, enters one's mind, hears prayers, intervenes in some random fashion, and overthrows the laws of physics at will. These thought regimes are mutually impossible, and yet no one thinks this is odd. Godel's mental illness may have been on the "god" side of his mind; the rest of his thinking may have been sane.
Understanding the universe however, isn't a requirement for survival. I too thought that our brain was built to unravel the mysteries of the universe, but after years of thinking about it, I realized that this is silly
I don't think our brains are 'built' to unravel the mysteries of the universe either. But it doesn't follow that we can't try, and can't gain fulfillment from trying. Once our hunger is sated, and lust abates temporarily, we have to do something with the hodge-podge of neurons. And the scientific method allows us to overcome a lot of the 'defects' of our evolved brain viz biases, heuristics, irrationality to get a good handle on reality.
Sorry if my mentioning Russell (whom I think was pretty great, due to his willingness to stand up for his beliefs and go to jail for them as much as anything else) and Wittgenstein (don't know so much about him, seemed a pain to deal with by all reports) upset you. No insult intended.
Brian: I wasn't upset by your reference to Wittgenstein and Russell - I admire Russell also. I just think that "big boy brain stuff" - abstraction, can frustrate the quest for understanding. Philosophy may point to a direction of inquiry, but I've never found philosophy to be "true" in and of itself. Animal experience is the basis for who we are. "Our lust abates temporarily" Can't intellectual pursuits cooperate with, and complement other pleasures?
> Does GÃ¶del really say that you can't describe
> the physics of reality mathematically? No.
Agreed. GÃ¶del does say that the mathematical system itself will be limited, though. Whether or not you can contain the physics of reality inside the limited box is an open question.
> We're actually pretty close to nailing down
> something like a grand unification theory,
> which would be a physical theory of everything.
I'm not so sure we're close, Mark.
From a philosophical standpoint (as opposed to "mathematically correct" standpoint), my takeaway of the implications of GÃ¶del's work is that all formal systems have limitations in their boundary conditions. We might get a grand physical theory of everything, but that's still limited by its ability to formally express physical systems. We will probably someday get better formal structures that describe systems with independent agents in them (e.g. economics, sociology), but the ability of any formal system to describe what is inside of it is predicated on your understanding of the boundary conditions.
I didn't in any way mean to suggest that Gödel's mental illness invalidates anything about his work.
But the fact that he was an intensely private person, and at least sometimes paranoid, means that we don't have a whole lot of information about what *he* thought beyond the very formal things which he published.
It's always a problem to make guesses at what someone actually thought when they're not aronud to tell you. To do that when they're someone who went to great lengths to keep their private thoughts private - when they had personal problems that caused them to be incredibly protective of their privacy - is close to impossible.
C. Chu: Your assertion about "guesses" invalidates the entire fields of anthropolgy, archaelogy and the behavioral sciences, which make a virtue, and money, from a lack of information!
Can't intellectual pursuits cooperate with, and complement other pleasures? I'm not sure bloodlust fits under the category of pleasure. I meant lusts in a general sense. Anyway, I think we pretty much agree.
Brian: I never met a general lust.
I never met a general lust.
I think he's retired from the service some time now.
Surely you've heard of wonderlust, bloodlust, sexual lust (the more common usage), lust for life, etc, By lust in a general sense I mean whatever can grip us with a zeal that often stops us from doing anything rational until it's sated. Anyway, not much of a point, so doesn't matter.
A sadly misguided blog, Mark. You've been visiting too many crank sites: you are starting to write like them.
It seems to me that there are two ways one can read another person's opinions. You can read sympathetically and try to understand the point he/she is making. Or you can read antagonistically and willfully misinterpret whenever possible in order to attack it.
In the second case, your attacks do not disprove the other person's ideas, they only disprove your own misunderstandings.
Here you seem to be insisting that all Mr Cosmicfingerprints' statements are prefixed with "GÃ¶del's incompleteness theorem proves that ...". It is easy to demolish such a misinterpretation, but pointless, because that is not what he is arguing.
As I understand Mr Cosmicfingerprints, he is reasoning informally by analogy. He is putting forward a list of ideas related to or implied, in some way, by GÃ¶del's theorem. (Hofstader did the same thing all thru "G.E.B." after all.) Given that he is in no way attempting formal mathematics, his proposals seem generally reasonable. Much of what he says seems right to me, tho he does go off the rails at the end.
There are many profound implications of GÃ¶del's theorems, not all of them strictly mathematical. His theorems came at a time when people believed that mathematics could deliver absolute truth and that everything true was provable. He demolished that certainty. You could say that his work was about the limits of knowledge; or that mathematics is creative, not purely deductive; or that rationality can "see truths" which cannot be proved by logic. Of course, none of those meta-statements or non-mathematical opinions can be proved by maths. (Gosh! Is anyone surprised?) It does not mean that they are pointless, or wrong.
Now, if you had sympathetically followed Mr Cosmicfingerprints' argument and shown where his conclusions were actually wrong, then there might have been some purpose to your blog. Instead, you gave a long list of your obviously-wrong misinterpretations and said that he (not you and your misinterpretations) was wrong.
To my mind, what you have done does not teach good mathematics, nor correct thinking. What this teaches is how to nit-pick like a crank.
An example :
'Those are just about the worst definitions of "inductive" and "deductive" that I've seen.'
I disagree with you. I think those are jolly good ways of describing (not defining) deduction and induction in simple, everyday language. His deduction example is spot on. As for induction, I do not immediately see how to do better. If you start getting into "case(n) implies case(n+1) and case(0) is true", then there are all sorts of complexities like doing odds and evens separately, going backwards case(n-1), whatever.
Missed opportunity - you could have used this blog to say something interesting about mathematical induction.
Overall, you are both wrong. If forced to choose, I would say that Mr Cosmicfingerprints is less wrong than you are.
Still, thank you for your blog, Mark. Much good stuff. Just please stop writing like a crank about cranks.
(Suggestion for future topics : why do people keep inventing new programming languages that are almost indistinguishable from the N hundred already existing languages? Are there any new languages that are actually worth the effort of learning?
You started talking about Haskell for a while, without giving definitive examples of its benefits - but I may have missed the crucial blogs.)
You mention being practising Jew. How do you justify your belief in God, if not by GÃ¶del's incompleteness theorem?
Do you use Cantor's diagonal argument?
Ironic Post-Postscript :
I've just clicked the previous article at Cosmicfingerprints
The very first sentence makes me want to write a long rant, worse than your blog.
"In astronomy we have the Anthropic Principle, which is both an observation and a hypothesis that the universe appears to be fine tuned to support life."
No, it bloody well is not!
I was innocently surfing the web when I read about his new theory of evolution. I did the email subscription thing and when I got to read his explanation all I wanted to do is to puke, and I still do.
How can someone be so wrong about something and not know it?
PS: I know its' my fault, I should've never click that bloody subscribe button. Blame on me :(
I'll never get back about 100 wasted hours of my life. There was a group of eccentrics who "reply all" Yahoo emailed each other, mostly arguing crank theories of Electromagnetism, anti-Relativity, gravity -- but there were some smart and creative people in the group, so I hung in with them as all the PhDs and the 2 Nobel laureates dropped out. The one crank who I wasted the 100 hours on had a pet theory where he assumed that the aether is structured with what he absolutely insisted calling "double helix" of electrons and positrons. But, as I finally discovered, and could not budge him, he utterly misunderstood the half-century old meaning of "double helix" -- and refused to look at the single most famous scientific illustration of that half century. He persisted insanely in thinking that the double helix was a "twisted ladder" -- meaning, to him (2nd hand from pop science) twisted about the main axis, rather than "twisted around a cylinder of nonzero radius, and with minor groove and major groove." Eventually, the nominal host of the group backed the loon, on the basis that his model was "A double helix" even if not "THE double helix" and thus we were both right and should stop arguing.
I didn't use Yahoo at all for a couple of months, except to send myself backup stuff. Then Yahoo wiped out 30,000 cached emails of mine without notification, and refuses to restore them from backup.
My plan is to convince 1,000 paid Yahoo users (I paid for allegedly superior service) to drop Yahoo and switch to Google. Stupid me, for using Yahoo messages with attachments to myself to back up crucial documents.
I am attracted to the creativity of "crackpots" because a tiny percentage of them have clues to the next paradigm. But it can drive you crazy, trying to get through to someone who considers oversimplified hearsay or double hearsay as gospel.
Thank you very much for this article!
For a long time I have done exactly what you warn against. This article was a slap in the face - but a needed one.
That being said, what is the value of an intuitive explanation? Is it to give a lay person an "ah-ha" moment? Is it good to have SOME understanding, even if it is "vague and mush?"
Good point there ...
people do mixed the bounded time-space postulates to the unbounded time-space.
Siting putting a circle around GOD who is omni-present defying the boundaries of time-space.
A circle can be drawn in a paper which is part of time-space, there is a space where you postutes and there is a time (from the beginning of the circle to the time the end of the circle joined the beginning point)
The perimeter of the paper where the circle has been drawn can also be postulated but within the circle only ... meaning one can draw a rectangle inside the circle as a replica of the whole paper dimension.
This is what I understand of Godel's theorem ... in lay language that is ... LOLZ
A fatal flaw was that they failed to have any representative posts ready to go up when the blog went live.
Had they done so, and had the content been surprisingly acceptable, the reception might have been better.
Instead we get this "Hi! Welcome to ShillBlog!" (crickets) and everyone, quite reasonably, expects the worst.