### Reply to Kelly's post on Epistemology

[This post is actually a reply to an interesting proposal by Kelly on his blog. Supposedly, I am doing this because there is a bit of a technical problem interfering with the display on his page. Actually, I just spent so much time writing this, that I wanted to kill two birds with one stone by getting my comment and my post in at the same time.]

Very interesting post, Kelly! You may be onto something.

Here is my recent train of thought:

Euclid's book, the

*Elements*, introduced a system of geometry from which he deduced the properties of geometric objects from a set of only 5 postulates. He had invented a method of proving numerous truths about geometry and numbers that rocked the world. What was truly fascinating about his achievement was how he proved so many theorems (things proved) from such a small number of postulates or axioms.

His method was so nifty -- so frikkin' cool -- it inspired both geniuses and crackpots throughout the ages to use it as the model for almost any endeavor that required deductive reasoning, or the creation of proofs.

Today, modern math and logic make extensive use of the axiomatic system to prove whatever it is that they are concerned with proving. Informal logic is also heavily influenced by Euclid's great idea.

Euclid's idea was so impressive, that people tried to apply it to just about every area of human reasoning. Sometimes with absurd results. Several years ago, I encountered a book in a public library that was written in the 1800's (I think), and purported to teach chess as an axiomatic system. As far as I could tell, the book was complete nonsense.

The problem with incredibly brilliant ideas that rock the world, however, is that people don't often see their limitations. Sometimes for centuries -- or even millenia. They introduce notions that may become so pervasive, so much a part of our basic assumptions about reality, that we don't even notice them, let alone question them.

And that is what happened when people became so enamored with the axiomatic system. They thought it should be able to prove everything. All you had to do was to find the right basic set of axioms, and apply the right reasoning, and eventually you could prove all the truths in the system.

This idea was so pervasive, that it permeated virtually all of Western philosophy and science. [Okay, you should call me on this last statement.] And so did the fallacy that any truth should be provable if you just had the right axioms and the right method. Of course, there were always paradoxes and weirdities that occured in every tightly expressed logic or mathematical system. Like Bertrand Russell and a paradox involving the logic of classes. I just remember it had something to do with a class of all classes. I'm too tired think it through, so you can look it up yourself, if you want.

Anyway, as far as I remember learning it, Russell set out to try to resolve all the paradoxes. He hoped to come up with a complete little axiomatic system that resolved all paradoxes, and could prove all the truths of the system. He ended up producing the massive Principia Mathematica (with Alfred North Whitehead), which managed to avoid many paradoxes and problems of logic. It was a huge effort. But it was incomplete.

But he must have felt rather disappointed when Kurt Goedel came up with a proof that showed a that Russel never could have accomplished some of the goals that it aimed to do. He proved that it is impossible for any logic system (of a non-trivial size) to eliminate all paradoxes. He also showed that no logic system could be complete based on a limited number of axioms. There would always we truths that could not be proven from the axioms. If you wanted to prove them, you would have to increase the axioms. But then you will just get more paradoxes and more truths that are known to exist, but can not be proven.

And in one fell swoop, he knocked down some of the key assumptions driving western logic systems for the previous 2000+ years. In my opinion, it was a feat at least as important as Einstein's Theory of Relativity.

The problem is, that so many people didn't get the memo.

What does this have to do with anything? Well, it seems to me that this relates to the ID vs Evolution debate, epistimology, and whatever else you want to throw into the mix. (I am getting very, very sleeeepy).

It's the problem of there always being truths out there that can not be proven. We may know that there are such truths. We may suspect we know which truths they are. But we can not use deduction from any finite set of known axioms, or already proved things, to prove that they are true. So, if I understand Godel correctly (and I very well may not), it seems to me that there will always be these gaps in what the Scientific Method can and can't prove. (By the way, I should add that, because of Godel's proof, it does not seem to be a valid criticism of any theory that there are truths that it can not prove.)

But just as important, we have to ask the question, if it is proved that there will always be truths that the Scientific Method can not prove, then what options are open to us? Is religion the appropriate method for finding those truths?

I don't know. I guess that's a question to be answered in Kelly's epistemology class.

*Disclaimer -- I make no pretense of having doublechecked any of the above facts, or even having thought my conclusions and interpretations over in any detail. Please feel free to correct me if you have better knowledge than I.*

## 11 Comments:

I am no expert on Godel, but I believe the conclusion from his theorems is that the logical system COULD be contradictory... but probably isn't. I can't remember where I read that: probably in an Ian Stewart book (easily the best popular mathematics author).

But yes, there are always limits to what we know if only because we have to make assumptions. I mean, WHY does 1+1=2? It makes sense, sure, and we all agree that it does, but try to explain it without coming down to an argument that boils down to "it just does" or "we agree that it does"

Of course there was an interesting little factor, about Fermat's Last Theorem. Fermats last theorem says that pythagorus' theorem does not work for higher powers, essentially.

Well, theres a way to test theorems to see if they are unprovable- they can neither be proved incorrect or correct. However, if it turned out that that was true of fermats last theorem then fermats last theorem must have been true, because if was false a counter example would exist, so it would be easily possible to disprove fermat's last theorem.

Heh.

Mathematics is crazy and cooky, and you can actually, and this is just using simple logic, prove a lot of VERY counter intuitive results (for example, the interval [0,1] has the same amount of points as the interval [1,3]).

I don't have a point here other than mathematics can be very cool.

Good post! Mathematics is a little out of my depth (that's why I'm big on liberal arts and law!) but I'll respond by saying this: your observations are an excellent argument in favor of teaching epistemology to kids.

Never mind teaching those ignorant little brats -- teach it to me!

(I'm kidding)

I was just scanning the blogosphere for mention of Godel's result and found this site. I recently exchanged some comments with the owner of moderatesrus about the connection between Godel's incompleteness theorem and intelligent design, etc. I am not sure I agree with the statements here at copernicusnow about the connection between Godel's result and the scientific method. My sense is that Godel's result does not directly place limits on what we can deduce mathematically, but rather on what a computer with fixed programming can deduce. Sorry if this seems vague, but I have already said a lot about this at moderatesrus. My comments are at:

http://moderatesrus.blogspot.com/2005/08/cnn-and-evolution-vs-id-discussion.html

Check it out if you want, and I hope it doesn't make you too sleepy ;).

Another site with some explanations that I find helpful is:

http://plato.stanford.edu/entries/computability/

-Hal Knight

I checked out Hal's comments at http://moderatesrus.blogspot.com/2005/08/cnn-and-evolution-vs-id-discussion.html" and he makes an interesting point about the scope of what Godel was saying is not provable.

Nonetheless, it does seem to be intuitively true that there must always be some true statements in a system that can not be proved through reference to axioms (and are not the axioms themselves).

The most obvious case is the existence, or non-existence of God. Either God exists or God does not exist.

Logically, one of those statements must be true. But I think most people would admit that the Scientific Method can not provide the steps to deduce the true statement, whichever it should be.

At any rate, even if I am wrong about the scope of what Godel proved, it seems it must hold that there will always be truths that cannot be deduced using Scientific Method. We can know that such truths are out there, but we can not prove them.

So, back to epistemology.

Thanks, Hal, for wising me up about Godel.

David Chalmers is a philosopher who has studied questions related to consciousness. If I ever have time I will buy his book and try to figure out the philosophical side of the question (as opposed to the mathematical or scientific side).

My initial impression is that it is a mistake to say that something like consciousness is inherently non-quantifiable and therefore not susceptible to scientific inquiry. We can study consciousness scientifically and come up with theories that describe it to various degrees of approximation. The nature of consciousness is an interesting problem because it is on the boundary between the quantifiable and unquantifiable. (This is what I believe, at least.) I would guess that the explanation of evolution is another problem on the boundary. In some ways I have to question whether it is worthwhile investigating these subjects by any means other than the scientific method, although my statement about them being infinite mysteries cannot be established using the scientific method.

Anyway, thanks for reading my earlier comments. I will look over the earlier posts on this site and see how they relate to what I am saying.

-Hal

Sorry I didn't have a chance to get back to you, Hal. Unfortunately, I have a number of restrictions on my time.

When you mention consciousness in this way, are you referring to God (as a conscious entity)?

I am not referring to God. I am talking about our own minds. Sometimes Deepak Chopra mentions consciousness on intentblog (i.e. "Consciousness as the new civil rights issue") and I am not sure what he is talking about.

I said questions about consciousness were on the boundary between the quantifiable and the unquantifiable because we can study the brain scientifically and also test theories about the mind by attempting to simulate the workings of the brain (by computer) and using the Turing test.

Questions about God (if he/she/it even exists) are definitely unquantifiable, as far as I am concerned.

My views make me an agnostic, I guess. My only definite religious belief is that that we will never completely understand how our minds work. In other words, a computer will never be able to pass the Turing test.

I like writing about this, but honestly I don't take myself that seriously, so please don't give me the crackpot award :)

Just now I had a chance to review Deepak's post on "Intelligent Design without the Bible - Part I". His questions were interesting, as was his suggestion that ID should not be considered the sole property of fundamentalist Christians. The list of comments was pretty long, so I did't read it all, but the refutations by Cappie and Tom Haddon were enlightening for me.

I quickly checked out "Consciousness as the new civil rights issue" and I wasn't sure what he was trying to say. Perhaps because I am not sure what raising consciousness is, and I don't know who opposes it, or why. I don't think I am opposed to consciousness. I'm just not all that good at it.

Anyway, don't worry about getting that crackpot award, I have some theories of my own, and I was thinking I might get it myself -- my only problem is that the millions of viewers on awards night might suspect that the fix was in from the start.

Anyway, Hal, it seems to me that before we talk about quantifying anything -- consciousness, God, etc. -- there is the question of defining it.

Do you have any ideas about that?

Sorry, I hadn't checked this particular set of comments for a while and I didn't realize that you had posted a question.

As some of my other comments have indicated, I am not going to try to define God because this seems to be a purely theological matter. Instead of trying to define consciousness, let's just say that if a computer program can ever pass the Turing test then this will count as a definition and quantification of consciousness. Since I don't think this will ever happen, I am saying that consciousness cannot be quantified, but the Turing test is sort of a half-way quantification.

-Hal

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