[han]

mycroft, on Dec 17 2007, 06:37 PM, said:

The reason that the scientific method is useful is that it rejects the false; tests the potentially true; and continually searches to fill the holes in what it can't explain. So it's more likely to be right than most things. But then again, from the School of Hard Knocks, Old Wives' Tales are likelier to be right than one might think, because what was tried and failed didn't get passed down to the next generation of Old Wives.

Michael

Sounds like the old wives were quite scientific then.

It also sounds like you can appreciate the theory of evolution.

[mikeh]

Hannie, on Dec 18 2007, 02:11 PM, said:

mikeh, on Dec 17 2007, 04:22 PM, said:

It surely is not currently possible to state, with assurance, that 'science can never explain everything'

I already got lost in the first sentence of your rant. Before I get to the rest, could you explain why it is surely not possible? Does the work of Godel have any impact on this question?

I have only a layman's and therefore woefully inadequate understanding of Godel's work, and I have absolutely NO pretence to being able to follow any math beyond arithmetic: I did take some math in my engineering training, but that was a looong time ago, and while it seemed complex to me, I know that the toughest math I did was trivial to some of the people I knew doing doctoral or post-doc work.

So I confess I may (for neither the first nor last time) have stated something for which I have no convincing argument.

But it does seem to me that we cannot 'know' that 'science' (or, more accurately, knowledge gained through application of the scientific method) cannot eventually explain everything when we are at the very beginnings of our intellectual exploration of the universe. There has (it seems to me, from my readings on the subject) often been a feeling that 'modern' thought represents some form of pinnacle of knowledge, or at least a high elevation on the mountain (akin to the seemingly wide-spread belief that, even if Darwin is right, we form some form of pinnacle or end-result of evolution rather than merely being, as it were, part of a work in progress). This hasn't always been true: Europeans of 700 years ago could be forgiven for thinking that the ancient Greeks and Romans knew more than they themselves.

It seems to me to be more accurate to see ourselves as on the side of a mountain, of indefinite height. What we see as glimpses of the summit may only be local high points, with the slope continuing on beyond.

And maybe, as I posited before, no member of our species possesses or will ever possess the mental abilities to actually 'understand' some things, but that doesn't necessarily mean that other species or forms of intelligence, whether here or elsewhere, will always lack the ability, through scientific method (or through some approach that will be to scientific method as it is to alchemy) to 'explain' everything.

We are on a journey, and I think it is 'impossible' to state, with well-founded assurance' that no intelligence will ever complete the journey.

[helene_t]

I'm not sure Godel's theorem applies to Mike's statement. No scienticic theory has been "proven" in the mathematical sense, so for science to explain everything must mean something less ambitious. I think the concept of "explaining" is too vague for Mike's statement to be verified or refuted in absolute sense. It could be, for example, that we will one day have a stocastic model that could account for everything except for the error terms in the model, and then one could argue about if the error term is an essential shortcoming of the model.

But it could also be that one day we will run into some big problem which can be proved to be unsolvable. Suppose for example that it turns out to require an infinity of empirical information to decide whether the string theory is correct.

Then again, I find it difficult to imagine that we will ever have answers to everything, even if there is no particular question that can't be answered. I tend to think that scientists potential for making up new problems is literally infinite.

[mike777]

Some of these posts sound like they are talking about Justification.

Crypto-Inductivist: Someone who believes that the invalidity of inductive reasoning raises a serious philosophical problem, namely the problem of how to justify relying on scientific theories.

See Popper and John Worrall for more on this debate.

[mike777]

Other posts seem to be talking about the theory of solipsism.

[mike777]

If Mathematics is the study of absolutely necessary truths but Godel's incompleteness theorem says that Hilbert's tenth problem cannot be solved.

[han]

mike777, on Dec 18 2007, 07:14 PM, said:

If Mathematics is the study of absolutely necessary truths but Godel's incompleteness theorem says that Hilbert's tenth problem cannot be solved.

If this sentence but makes no sense.

[han]

helene_t, on Dec 18 2007, 06:39 PM, said:

I'm not sure Godel's theorem applies to Mike's statement.

Neither do I, I meant my question as a question. But it is clearly related, before Godel's work this:

Quote

We are on a journey, and I think it is 'impossible' to state, with well-founded assurance' that no intelligence will ever complete the journey.

might have been said about mathematics as well.

[cherdano]

mikeh, on Dec 18 2007, 02:20 PM, said:

Hannie, on Dec 18 2007, 02:11 PM, said:

mikeh, on Dec 17 2007, 04:22 PM, said:

It surely is not currently possible to state, with assurance, that 'science can never explain everything'

I already got lost in the first sentence of your rant. Before I get to the rest, could you explain why it is surely not possible? Does the work of Godel have any impact on this question?

(...)

But it does seem to me that we cannot 'know' that 'science' (or, more accurately, knowledge gained through application of the scientific method) cannot eventually explain everything when we are at the very beginnings of our intellectual exploration of the universe.

I don't understand your distinction between 'science' and 'knowledge gained through application of the scientific method'. But anyway, I think one can argue with a lot more assurance that science will never be able to explain 'everything', than for any of the many doubtful claims you made in your long rant, which in my view misrepresents both science and religion.

Anyway, as long as American intellectual think and talk about religion with as much disrespect and contempt as Mikeh does, there can hardly ever be a religious political movement that they will be able to talk to, or that will talk to them.

[mikeh]

cherdano, on Dec 18 2007, 07:42 PM, said:

Anyway, as long as American intellectual think and talk about religion with as much disrespect and contempt as Mikeh does, there can hardly ever be a religious political movement that they will be able to talk to, or that will talk to them.

I'm sorry you feel that way. But I am neither an intellectual nor an American, so I am not at all sure why you seem to correlate my rants with those categories of thinkers.

I know a number of devout religious believers, and I do not have contempt for any of them. I have seen how faith has enabled people to deal with tragedy. If and when tragedy strikes me and my family, as it already has in some ways, I will not have that comfort.. but I don't think that I would want it if it came at the cost of my abandoning the results of a lot of thinking over the years, sophomoric tho that thinking may appear to others. Faith works for others, and I would not take that away from anyone at a time of need, while I would regret that our society inculcated that perceived solution.

I do have contempt, tempered by a modest amount of fear, for all those whose beliefs are such that they feel that they have the right (in some cases, the duty) to try to make others live by their religious convictions, but none of the devout believers I have known as friends fall into that category. And I do not see that kind of preaching in posts here, even from those with who I disagree the strongest.

I truly believe that those who feel it necessary to prefer faith in a deity to acceptance of evolution (which is where this thread began, so many posts ago) are mistaken, but I have no problem with leaving them undisturbed in what seems to me to be their delusions IF they stop trying to impose their beliefs as 'science', which is, again, where this thread began.

I am sure that I come across as arrogant in many of my posts. I am sure that, on occasion, I come across this way in my personal life. I wish that were not true, but it is something I have struggled with for all my life.

But I can assure you that I am all too well aware of my capacity for error... well, maybe I underestimate it on occasion, but I am no pope, nor any evangelical christian leader of fundamentalist islamic iman, infallible because I have read the word of god. I am human, therefore I am (more than capable of) error

[mike777]

Hannie, on Dec 18 2007, 07:35 PM, said:

mike777, on Dec 18 2007, 07:14 PM, said:

If Mathematics is the study of absolutely necessary truths but Godel's incompleteness theorem says that Hilbert's tenth problem cannot be solved.

If this sentence but makes no sense.

I will attempt to rephrase.

I only present the following for discussion.

Some of the posters seem to be saying:
1) Mathematics is the study of absolutely necessary truths.
2) Hilbert's tenth problem is to establish once and for all the certitude of mathematica methods' by finding a set of rules of inference sufficient for all valid proofs, and then proving those rules consistent by their own standards
3) Godel's incompleteness theorem is a proof that Hilbert's tenth problem cannot be solved. For any set of rules of inference, there are valid proofs not designated as valid by those rules.
4) Godel tells us there will never be a fixed method of determining whether a mathematical proposition is true any more than there is a fixed way of determining whether a scientific theory is true.
5) Therefore progress in mathematics will always depend on the exercise of creativity.


Or to rephrase one would debate that proofs do not confer certainty upon their conclusions. The validity of a particular form or proof depends on the truth of our theories of the behaviour of the objects with which we perform the proof.

Do incomprehensible mathematical entitites, example Cantgotu environments, exist despite that they appear inextricable in our explanations of the comprehensible ones?

[kenberg]

This thread is generally about religion. Somehow Godel got into it.


I pose a question:

Has anyone, at any time, ever, known a person or heard of a person who has said either of the following two statements:

1. I used to believe in God but then I read Godel's theorem and I became an atheist
or
2. I used to be an atheist but then I read Godel's theorem and I came to believe in God.

I have never met such a person, I find it hard to imagine such a person.

Godel's theorem has not even affected the way I think about mathematics although I will concede that maybe it should. I do not concede that it has any significance for religion or philosophy in general. I suppose that it has some sort of significance to someone who maintains that life is reducible to logic. Who would think such a thing, with or without Godel?

[mike777]

" I do not concede that it has any significance for religion or philosophy in general. I suppose that it has some sort of significance to someone who maintains that life is reducible to logic. Who would think such a thing, with or without Godel?"


Ok we agree to disagree, I do think Godel does have a significance to philosophy in general. Do you you at least think Popper does?

[hrothgar]

mike777, on Dec 19 2007, 04:38 AM, said:

I only present the following for discussion.

Some of the posters seem to be saying:
1) Mathematics is the study of absolutely necessary truths.
2) Hilbert's tenth problem is to establish once and for all the certitude of mathematica methods' by finding a set of rules of inference sufficient for all valid proofs, and then proving those rules consistent by their own standards
3) Godel's incompleteness theorem is a  proof that Hilbert's tenth problem cannot be solved. For any set of rules of inference, there are valid proofs not designated as valid by those rules.
4) Godel tells us there will never be a fixed method of determining whether a mathematical proposition is true any more than there is a fixed way of determining whether a scientific theory is true.
5) Therefore progress in mathematics will always depend on the exercise of creativity.


Or to rephrase one would debate that proofs do not confer certainty upon their conclusions. The validity of a particular form or proof depends on the truth of our theories of the behavior of the objects with which we perform the proof.

Do incomprehensible mathematical entities, example Cantgotu environments, exist despite that they appear inextricable in our explanations of the comprehensible ones?

I don't believe that this is an accurate summation (In all seriousness, I don't think that you understand what you're talking about. For example, Item 2 seems to confuse a specific Hilbert problem with the entirety of the Hilbert Program)

This stuff is way beyond my math skills - I always steered very far away from symbolic logic - but I'm going to try to summarize Hilbert's 10th problem:

First of all, we need to define what's known as a Diophantine Equation:

A Diophantine equation is an equation that which has a set of integer's as roots. For example, the equation ax + by = 1 is a Diophantine equation.

Hilbert's 10th problem was to identify a universal algorithm that can be used to test whether or not any equation is a Diophantine equation.

Roughly 40 years ago a proof was completed that demonstrated that such an algorithm does not exist.

From what I can tell, this proof does not rely on Godel's Incompleteness theorem. The proof makes use of a halting problem - there is a bunch of discussion about recursively enumerated sets and the fact that all Diophantine equations are recursively enumerated sets - however, I don't believe that this is in any way equivalent to the Incompleteness Theorem.

Godel's Incompleteness Theorem does apply to what is known as Hilbert's program. (which is distinct from any one individual Hilbert problem). Here's how the Wikipedia summarizes Hilbert's program:

Quote

The main goal of Hilbert's program was to provide secure foundations for all mathematics. In particular this should include:

    * A formalization of all mathematics; in other words all mathematical statements should be written in a precise formal language, and manipulated according to well defined rules.
    * Completeness: a proof that all true mathematical statements can be proved in the formalism.
    * Consistency: a proof that no contradiction can be obtained in the formalism of mathematics. This consistency proof should preferably use only "finitistic" reasoning about finite mathematical objects.
    * Conservation: a proof that any result about "real objects" obtained using reasoning about "ideal objects" (such as uncountable sets) can be proved without using ideal objects.
    * Decidability: there should be an algorithm for deciding the truth or falsity of any mathematical statement.

Godel's second incompleteness theorem demonstrates that the Hilbert program can not be achieved. However, it doesn't necessarily mean that any one individual Hilbert problem can not be successfully solved. Indeed, several of Hilbert's problems have been successfully resolved.

In a similar vein, Turing proved "that that a general algorithm to solve the halting problem for all possible program-input pairs cannot exist." This doesn't mean that it is impossible to solve the halting problem for a specific program-input pair.

[kenberg]

mike777, on Dec 18 2007, 09:21 PM, said:

" I do not concede that it has any significance for religion or philosophy in general. I suppose that it has some sort of significance to someone who maintains that life is reducible to logic. Who would think such a thing, with or without Godel?"


Ok we agree to disagree, I do think Godel does have a significance to philosophy in general. Do you you at least think Popper does?

Probably.

I duck because it's been forty or so years since I thought much about such matters.

I got my Ph.D in Math from the University of Minnesota in 1967. At that time they had the odd idea that a PhD student had to have a minor. I have never heard of such an idea anywhere else. I decided to minor in philosophy, specializing (for my minor) in logic and the philosophy of science ( existentialism was big at the time and actually I do sort of believe that existence precedes essence but ...). So that's where I read some Popper. And a bunch of other guys. Interesting stuff, but I was there for the math.

My view of philosophy runs towards the pragmatic. It seems to me that to get from Newton to Einstein to Quantum Physics to strings really requires some philosophical reflection (of which I no doubt have done too little). Mathematics isn't enough. I doubt Godel has any effect, but I really don't know enough about Popper to say. Herbert Feigel was the big man in philosophy of science at Minnesota at that time and he was impressed with Popper.

Religion, though, is still a different bunch of problems. Most of us really can't give a logically impeccable accounting of our beliefs on that, or at least I cannot. For a whole host of reasons, not all of which I can defend with great logic, I have lived my adult life without any belief in any God. It seems, well, natural to me. My general experience, and it has been noted by many, is that people who tell you how often they pray are not to be trusted but this just is a reflection of how some people misuse religion, not a reflection on religion itself. I think it is more than that with me. I just don't believe that most religious claims are true. I don't claim I can prove they are false, but I do not believe that they are true. I strongly believe that neither science nor mathematics is the path to discovering religious truth. Except of course when religion makes historical claims such as the when and how of creation then science can pretty much say, often with a great deal of certainty, that it ain't so.

So, really, with Popper I plead ignorance. I'm willing to give it some thought. It won't bring me to Jesus. I'm sure.

Ken

[mike777]

hrothgar, on Dec 18 2007, 09:42 PM, said:

mike777, on Dec 19 2007, 04:38 AM, said:

I only present the following for discussion.

Some of the posters seem to be saying:
1) Mathematics is the study of absolutely necessary truths.
2) Hilbert's tenth problem is to establish once and for all the certitude of mathematica methods' by finding a set of rules of inference sufficient for all valid proofs, and then proving those rules consistent by their own standards
3) Godel's incompleteness theorem is a  proof that Hilbert's tenth problem cannot be solved. For any set of rules of inference, there are valid proofs not designated as valid by those rules.
4) Godel tells us there will never be a fixed method of determining whether a mathematical proposition is true any more than there is a fixed way of determining whether a scientific theory is true.
5) Therefore progress in mathematics will always depend on the exercise of creativity.


Or to rephrase one would debate that proofs do not confer certainty upon their conclusions. The validity of a particular form or proof depends on the truth of our theories of the behavior of the objects with which we perform the proof.

Do incomprehensible mathematical entities, example Cantgotu environments, exist despite that they appear inextricable in our explanations of the comprehensible ones?

I don't believe that this is an accurate summation (In all seriousness, I don't think that you understand what you're talking about. For example, Item 2 seems to confuse a specific Hilbert problem with the entirety of the Hilbert Program)

This stuff is way beyond my math skills - I always steered very far away from symbolic logic - but I'm going to try to summarize Hilbert's 10th problem:

First of all, we need to define what's known as a Diophantine Equation:

A Diophantine equation is an equation that which has a set of integer's as roots. For example, the equation ax + by = 1 is a Diophantine equation.

Hilbert's 10th problem was to identify a universal algorithm that can be used to test whether or not any equation is a Diophantine equation.

Roughly 40 years ago a proof was completed that demonstrated that such an algorithm does not exist.

From what I can tell, this proof does not rely on Godel's Incompleteness theorem. The proof makes use of a halting problem - there is a bunch of discussion about recursively enumerated sets and the fact that all Diophantine equations are recursively enumerated sets - however, I don't believe that this is in any way equivalent to the Incompleteness Theorem.

Godel's Incompleteness Theorem does apply to what is known as Hilbert's program. (which is distinct from any one individual Hilbert problem). Here's how the Wikipedia summarizes Hilbert's program:

Quote

The main goal of Hilbert's program was to provide secure foundations for all mathematics. In particular this should include:

    * A formalization of all mathematics; in other words all mathematical statements should be written in a precise formal language, and manipulated according to well defined rules.
    * Completeness: a proof that all true mathematical statements can be proved in the formalism.
    * Consistency: a proof that no contradiction can be obtained in the formalism of mathematics. This consistency proof should preferably use only "finitistic" reasoning about finite mathematical objects.
    * Conservation: a proof that any result about "real objects" obtained using reasoning about "ideal objects" (such as uncountable sets) can be proved without using ideal objects.
    * Decidability: there should be an algorithm for deciding the truth or falsity of any mathematical statement.

Godel's second incompleteness theorem demonstrates that the Hilbert program can not be achieved. However, it doesn't necessarily mean that any one individual Hilbert problem can not be successfully solved. Indeed, several of Hilbert's problems have been successfully resolved.

In a similar vein, Turing proved "that that a general algorithm to solve the halting problem for all possible program-input pairs cannot exist." This doesn't mean that it is impossible to solve the halting problem for a specific program-input pair.

1) Yes I am a referring to the tenth problem. I am paraphrasing so I make no claims of exact math. IT was to find a set of rules of inference with certain properties and by their own standards, to prove them consistent.
1b) Godel proved there is no hope that any set of rules of inference that is capable of validating even the proofs of ordinary arithmetic could never validate the proofs of its own consistency. There is a further second part to this that there must exist valid methods of proof that those rules fail to designate as valid. This all has to do with inference.
1c) IT will be necessary for smart math people to invent new types of proof to expand new mathematical knowldege ..in other words use creativity.
2) Yes the halting problem is germane, very, to the discussion as is Cantor's diagonal argument.
3) The math is way past me, I just approach this from a layman's viewpoint to further the discussion.

[mike777]

Another area of possible discussion is are the relativist and constructivist conceptions of truth and knowledge, which is basic orthodoxy in vast areas of the academic world, fundamentally flawed?

See such books as Understanding Scientific Reasoning by Ronald Giere or Theory and Evidence by Clark Glymour.

[Codo]

jdonn, on Dec 18 2007, 11:07 PM, said:

Codo, on Dec 18 2007, 08:55 AM, said:

2. What are christian ethics? From my point of view it is something like the Don´t do until others .., the ten requirements and that you should marry just one partner, not more at the same time. Surely there is more in it, but these are more or less the bottom lines.

Would you say that the western world is not more or less build on these basics? I would be quite surprised.

You don't seem to be understanding the difference between

"Treat others as you would like to be treated."

and

"Treat others as you would like to be treated because it is a part of Christian ethics."

As for the shock at your statement, the entire point is religion should have nothing to do with those decisions. As above, it is ok to say

"Killing stem cells should be illegal because I believe that is equivalent to killing a person."

if that's what you believe. But IMO it's ridiculous to say

"Killing stem cells should be illegal because I believe it violates Christian ethics."

BTW, in your post right before this you say

Quote

Of course we believe that "our" way is the right way. But we respect other ways too.

Which totally contradicts your belief that public policy decisions and political decisions should be based upon the ethics of 'your way'.

The discussion drifted into another direction, but I could not resist to return to this point:

I would nver claim (at least not intentionally) that etical descissions must be based on a christian view. I tried to say that they are based on this ethics, if we want to or not. And this is no matter of right or wrong ethics, it is a matter of historical development.

I tried to say that we all in the western world are very much influenced from christian ethics, no matter what we believe. In Western Europe the Christian Church had an unbelievable influence for more then 1000 years. They influenced the ethics and the thinking of your grandparents, their grandparents and so on. This is not a matter of right or wrong ethics, it is just a matter of influencing people thhrough 1000 years of indoctrination.
And when people with this background went to America some 300 years ago, they established the same way of thinking there.

I did not want to say in this context: Do this or that because in the sense of christian ethic it is right.
I wanted to say: We do this or that, but all our ethical descissions are based on our history. This history influences our descissions today. And this history is very much christian.

My apologisze for not being able to make this point clear at the beginning.

[Codo]

helene_t, on Dec 18 2007, 11:08 PM, said:

Also, you seem to impose on fellow Christians your own views on what true Christianity is. I would suppose GW is as Christian as you are, "Christian" simply being defined by self-identification. Now it is possible that GW knows about as much about theology as he knows about geography but then again, I suppose stupid people are entitled to their self-identification.

I find it hard to define what a real christian is. Maybe your idea of self-identification is correct. But I doubt that.
If I should try to find a definition it would be something like: Believe in God, try to follow the ten requirements, be honest, take people serious and try to do the right thing.

Without being able to define, I would still refuse to call GW a christian, but it is surely possible that I am in a very small minority in my views. And in my real world, there are many people who label themselves as christians but their life does not look like a christian life to me, so maybe my definition is wrong, but who knows...

[mike777]

It is interesting to see the debate that if knowledge is socially constructed either as truth or as justification or the common sense view is that there is a way the world is that is independent of human opinion and we are capable of arriving at beliefs about how it is that are objectively reasonable.

See Boghossian, Fear of Knowledge
Scientific Reasoning, Giese
Theory and Evidence, Glymour.