Difference between revisions of "Talk:Essay:Quantifying Openmindedness"

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::: Invariably math/physics types initially insist that it must be precisely 2 in order to make the integrals work nicely.  But physical observation is what counts, and there is no logical obstacle to mathematical inelegance.  Thanks for discussing this.--[[User:Aschlafly|Andy Schlafly]] 23:56, 1 October 2009 (EDT)
::: Invariably math/physics types initially insist that it must be precisely 2 in order to make the integrals work nicely.  But physical observation is what counts, and there is no logical obstacle to mathematical inelegance.  Thanks for discussing this.--[[User:Aschlafly|Andy Schlafly]] 23:56, 1 October 2009 (EDT)
:::: Mark and Jacob, I encourage you to ponder this question (#12) further.  I do think that an insistence on mathematical elegance as dictating physical phenomenon cannot withstand scrutiny.
:::: God and logic define nature, not mathematical elegance.  We have to observe first, and then come up with the best math to describe the observations.  There is no logical problem to 2.0000001 here.  Indeed, I find it implausible to expect that exponent to be precisely 2 for our calculating convenience.--[[User:Aschlafly|Andy Schlafly]] 00:22, 2 October 2009 (EDT)

Revision as of 23:22, 1 October 2009

Not a suitable criterion

I don't think that "a genuine willingness to accept something as possibly being true" is a suitable criterion. It doesn't take into account the difference between the following two scenarios:

  • A person who has never considered the evidence. For example, I would be close-minded to say that Noah could not have fitted all the animals on the ark if I didn't know how big the ark was and how many animals would have qualified.
  • A person who has considered the available evidence and has formed a considered opinion. For example, I would be not be close-minded to say that evolution is impossible if I had studied information theory and knew how it negated the possibility.

A better definition might be "a genuine willingness to consider the evidence before rejecting an idea".

Philip J. Rayment 04:17, 7 May 2007 (EDT)

Excellent! I've made the change. God bless you.--Aschlafly 12:31, 7 May 2007 (EDT)

Thanks! I'm glad you appreciated it.
However, I believe that it now renders your examples tests invalid. You can't test for a willingness to consider the evidence by asking someone if they accept the possibility of something. If they've already considered the possibility and have concluded that it is impossible, they have been open minded on it, but would still "fail" the test.
Philip J. Rayment 22:45, 8 May 2007 (EDT)
We can ask subjects a mixture of new ideas and ones they've already heard. It can be two questions:

1. Have you seriously considered the evidence for this idea? 2. If no, then would you? 3. If yes, then describe what evidence you looked at.

--Aschlafly 23:02, 8 May 2007 (EDT)

Would saying that something is absolutely the case be just as bad as impossible that something is the case? Are people who believe without equivocation that (say...) trickle down economics works just as close minded those who believe that without equivocation that it does not? I would assert that there are people who are just as close minded about the possibility the examples that you gave as there are people who are close minded about the impossibility of the examples. --Mtur 23:08, 8 May 2007 (EDT)
But look at the definition of openmindedness, "a genuine willingness to consider the evidence before rejecting an idea." That's the difference.--Aschlafly 23:47, 8 May 2007 (EDT)
You've got the Shroud of Turin as an example. What is your stance on that? What evidence have you considered? If presented with evidence to the contrary, are you willing to change your mind? The last two questions are potentially critical ones. This is not something that someone can eaisly say "your OMQ is such and such based on this test." What about situations where there is not 'evidence' to be considered? There are a multitude of social issues out there that people feel quite strongly upon with very little open mindedness on either side. One extreme difficulty with this test is that the questions can very easily be weighted to one political view or another to make it look like the other side are a bunch of close minded fools. --Mtur 23:59, 8 May 2007 (EDT)
No, I think the three questions are simple and straightforward. We can elaborate a bit as follows:

Have you seriously considered the evidence for this idea?

1a. If no, then is that because you have never heard of it?
1aa. If if you have never heard of it, then will you seriously consider the evidence?
1ab. If you have heard of it, but have never seriously considered the evidence, then on this question you lose a point for lack of openmindedness.
2b. If yes, then how much time have you spent reviewing the evidence? What evidence did you look at?
2ba. If less than 1 hour, then you lose a point for lack of openmindedness.
2bb. If more than 1 hour, then you gain a point for openmindedness.

(the time period may vary depending on the complexity of the topic).--Aschlafly 00:15, 9 May 2007 (EDT)

Now that's getting much closer to a suitable test. I'm glad you put that last point in; before I read it, my reaction was, "but one hour may not always be appropriate".
Another thought that needs developing: There are lots of things that I won't spend the time considering, simply because I don't have the time. But in those cases, I try not to criticise the idea because I haven't properly considered it. Philip J. Rayment 05:52, 9 May 2007 (EDT)
Good point. I'll incorporate all these concepts now.--Aschlafly 10:12, 9 May 2007 (EDT)
I have a question about your 1ab point. If for example, someone says no, they haven't seriously considered the evidence for the Shroud of Turin, but have heard of it, how does that make them less open minded? What if to that person the topic is just not found to be very interesting, and they prefer to spend time researching topics that are more interesting to them? --Colest 10:23, 9 May 2007 (EDT)
OK, in that case the follow-up question should be whether the person forms an opinion for himself about this topic without looking at the evidence. If the answer is yes, then that is quintessential closemindedness. I'll update the content page based on your excellent point.--Aschlafly 10:26, 9 May 2007 (EDT)

Something else I think you have to consider with this kind of test is that it is easy to misrepresent yourself on it. I believe "openmindedness" to be a trait which most people would deem to be a good trait. Therefore going into this exam, people will want the outcome to give them a more favorable score so it is heavily reliant on the test takers being 100% accurate in their responses. Whereas with an IQ test, you are asked questions with definitive answers where you either know the answer, or you don't. --Colest 10:33, 9 May 2007 (EDT)

Good point, but asking about specific amounts of time spent considering the evidence would seem to minimize the problem you describe. If further precautions are warranted, then follow-up questions about when, where, what, how could be asked concerning reviewing evidence. In fact, I'll add that now.--Aschlafly 10:52, 9 May 2007 (EDT)
Colest, you've done a better job then I could've explaining why I'm dubious (but not shut-off to the idea) about this test. If it is subjective enough that a person can misrepresent himself, are we really quantifying anything with this score? That political compass thing that some CPers advertise their scores from is neat and all, but I'd never try to use the number in any meaningful way.
If this is just a conversation starter then cool, but from the other endeavors Andy is involved in that sounds a little trivial to me. Aziraphale 20:32, 2 July 2007 (EDT) <- pursuing the trivial...

Example Questions and Topics

I see a couple of problems with the topics section of this essay. The first concern is a minor one. You have, as an example, “For example, did our subject think that President Ronald Reagan's exhortation, "Mr. Gorbachev, tear down this [Berlin] wall!" was an impossibility?” But, properly, speaking, it is not a statement that can be examined as a possibility (or impossibility) , since imperatives are neither true nor false. A better way to word this question would be, “When President Regan told Mr. Gorbachev to tear down the Berlin Wall, did you think that it was impossible for the Berlin Wall to be torn down?” Of course, this is not an especially good measure in the present, since, most people will report that they thought it was possible (even if they did not at the time, as people tend to misremember their past, to bring their past in line with there present reality).

The second problem I see is that the questions, as they now stand will clearly skew the results toward Conservatives being more open-minded—questions asking whether conservatives are open to liberal ideas should be included to make sure that the data is not inadvertently slanted. For example, “Did our subject think, or still think, that his/her understanding of the Bible may be wrong?” “Did our subject think, or still think, that evolution is possible?” “Did our subject think, or still think, that human behavior could warm the global climate?” and “Did our subject think, or still think, that homosexuality could be a natural behavior?”--Reginod 11:26, 9 May 2007 (EDT)

You rewording of the Reagan is longer, but fine. I'll change it per your comment.
I think we should add liberal-oriented questions. But your examples do not focus on a belief in impossibility, which is the essential element here. Perhaps you'd like to refine some liberal questions that ask about impossibility, for me to add?--Aschlafly 20:00, 9 May 2007 (EDT)
I’ll take a second stab at it:
“Does our subject think that it is impossible that evolution could have occurred?”
“Does our subject think that it is impossible that he/she could be mistaken about the truth of the Bible?” (I rather like this one as it cuts both ways, a believer is (implicitly) being asked if he/she thinks that atheism or some other religion could possibly be true, while a non-believer is asked if he/she thinks Christianity could be true.)
“Does our subject think that it is impossible that homosexuality is a natural behavior?”
“Does our subject think that it is impossible that the global climate is currently warming as a result of human behavior?”
How do those look to you? --Reginod 23:04, 9 May 2007 (EDT)
Those questions look fine to me. Some further refinements to limit them to facts might be helpful. But basically they look OK, and I'll add them.--Aschlafly 00:09, 10 May 2007 (EDT)
Couldn't this method of quantifying openmindedness be manipulated to receive any set of desired results? For example, if I was to ask "does our subject think that it is impossible for God to exist", then I could automatically chalk up one closed-mindedness point from anybody who is a Christian, Muslim, Hindu, Buddhist, etc. There are bound to be many questions like this, no? --Hojimachongtalk 17:43, 13 May 2007 (EDT)
For all queries for 'do you believe it is imposssible that x' there should be a matching 'do you believe that it is possible that not x'. X then merely has to cover all ideological issue. You could even have some a priori ones as control questions. —The preceding unsigned comment was added by Lookamoose (talk)

Curious About...

So, there's a conversation on the Main Page that directs to this essay; it's an interesting read. However, the poster is asked "By the way, what's your score on Essay:Quantifying Openmindedness?" I thought I'd be better able to follow the conversation if I found out my own score. Am I missing a link or something? I don't see any way that I can derive a score.

I have to admit, at the risk of sounding close-minded, I'm dubious about coming up with a useful measurement. That said, here I am, checking out the source. I'll even give it more than an hour. Aziraphale 10:41, 2 July 2007 (EDT) <-An open mind and a closed parentheses

I suspect your real objection is to trying at all. Six questions are posted. What's your score on those six?--Aschlafly 10:49, 2 July 2007 (EDT)
*blink* Andy, you and I have actually agreed on things on the front page. I know we're not of one mind on a lot of things (and I really do worry about isolating rather than engaging, but I know you'd disagree on that characterization), and I know you're too busy to remember who's who all the time (and I'm a pretty minor poster, I know), but an aggressive stance seems unwarranted.
Anyway, here are my answers:
1) No, but to be fair I wasn't particularly engaged in national and world affairs back then so I shouldn't get too much credit for that.
Evidence issue - maybe I'm favoring myself, but I choose not to deduct a point for failing to look at the evidence when it wasn't a conscious decision; I barely noticed the event taking place.
2) I never thought it was impossible. I have no advanced scientific knowledge so it would be ludicrous for me to assume that a technology is or is not possible. It actually sounds simple to me, but I've been told that it's harder than it looks.
Evidence issue - if I'm reading the essay correctly, I lose a point for having heard of the issue but not investigating the evidence, even though I agree it's possible. Is that correct?
3) Sure it can be authentic. There's a lot stranger things in the world than that.
Evidence issue - Same as #2. I accept the possibility but haven't looked into it. Do I lose a point?
4) I don't think the speed of light changing over time is impossible. I also don't know the arguments for or against it.
Evidence issue - same as #3.
5) I don't think it's impossible, but of course as I mentioned before I'm dubious. If you're interested in the conversation I'll tell you why, but you haven't asked and I won't assume.
Evidence issue - well, I'm working on it right now, so this would be incomplete.
6) Depends on how far you want to split hairs here. If you mean the evolution vs. creation debate, I absolutely accept the possibility that we were Created. If you mean ANY evolution whatsoever... well, I guess I live by "never say never" but we're talking about a very low level of accepting the possibility.
Evidence issue - Not to split hairs again, but on the one hand I've clocked in many hours going to church, reading books, and browsing sites like this one. On the other hand, I'm not particularly interested in the evolution debate, so I haven't tried to digest all of the arguments. So... point? No point? Again, I accept the premise, so it seems kind of moot.
So, my raw score for accepting the ideas presented is either 5 or 6, depending on how you score #6. Depending on how you adjust for the follow-up questions, my score ranges from 1 to 7 (and possibly 8 depending on how long this goes on).
Aziraphale 11:36, 2 July 2007 (EDT) <- my mind is open, but there's a bouncer


I see Andy "removed claims that no scientists or significant groups support". Perhaps "openmindness" should be redefined as a genuine willingness to consider the evidence for an idea that scientists and significant groups support before rejecting the idea. RSchlafly 21:12, 2 July 2007 (EDT)

Insert "some" before scientists and "or" rather than "and", which would be consistent with the quote above, and I wouldn't object to that modified definition: a genuine willingness to consider the evidence for an idea that some scientists or significant groups support--Aschlafly 21:22, 2 July 2007 (EDT)
Define it any way you want. But as it is, your questions don't really match your definition. RSchlafly 03:01, 3 July 2007 (EDT)

new question

We need a question in this essay that asks the user if they believe it is possible that God doesn't exist. This question drives to the heart of open-mindedness. Please do not remove it without legitimate discussion. --AntnyGonzo 19:34, 25 July 2007 (EDT)

No, Gonzo...YOU will ask the author of the essay first if you can alter it. As far as I'm concerned, you have no authorization to alter an essay of any kind. Karajou 19:36, 25 July 2007 (EDT)
And why do we need such a question?
FYI, the question you pose runs entirely counter to the thrust of the essay. An essay is not an ordinary article. It is an original work. As such, it is for the author of that original work to decide whether to include anything or not.
Publish your question here, by all means--and be prepared to take criticism as to the appropriateness of the question.
But you do not unilaterally change the essayist's work. If you are serious about whether being open-minded means deciding that God need not exist, write your own essay. But don't go ruining someone else's.--TerryHTalk 19:38, 25 July 2007 (EDT)
The user has been indef blocked, anyway. AManInBlack 19:39, 25 July 2007 (EDT)
So he has. But I'll let my comment stand anyway, as a warning to anyone who might feel the slightest temptation to repeat his bad example.--TerryHTalk 19:41, 25 July 2007 (EDT)
Gonzo's demand was we talk about his "alteration" on the talk page only, rather than him bring it up on the talk page, ar ask the author if he could change it first. Karajou 19:43, 25 July 2007 (EDT)
The essay started with the request comments or improvements welcome [1]. JohnMalin 19:50, 25 July 2007 (EDT)
Yes it did, but since it's an essay, those comments or improvements needed to be posted and disgussed on the talk page. Gonzo was more willing to shoot first and aks questions later. Karajou 19:55, 25 July 2007 (EDT)

Ok, well I'm interested in the discussion that was so poorly started. I don't really care if the question is about the existence of God, since that's probably a lightning rod of argument, but with the possible exception of the last question about the precision of Newtonian gravity, and I don't understand it so I say "possible", all of those questions clearly drive at "ok, lib, you really think you're open-minded?"

I believe that the Essay would benefit from example questions like "Is it possible that President Bush knowingly presented misleading evidence to the nation, rather than by mistake?" or "Do you think that it is impossible that, through mistranslation or any other reason, the age of the Earth is misunderstood by Young Earth Creationists?" Aziraphale 20:07, 25 July 2007 (EDT) <-ponderously...

I'm open to adding liberal questions. I deleted an earlier attempt to do this because the questions referred to crackpot theories for which there is neither scientific nor public support, like claiming that humans were populated by UFOs from outer space.
I don't have any problem with the questions posed by Aziraphale above as part of a test of open-mindedness.--Aschlafly 20:42, 25 July 2007 (EDT)

The gravity question

I just wanted to point out that this may not be a very good measure of open mindedness. 2 and 2.00001 are arbitrary numbers, yes, but that isn't why there is an inverse square relation ship. The square of the radius in this case comes from a derivation which is a direct result of geometry. That is to say, it isn't that the variable R was saddled with some arbitrarily accurate exponent, it specifically means that the equation holds that the value of R must be multiplied by itself. If I may use a thread of logic common at conservapedia, 1+1=2, not 2.0001. Likewise, r*r=r2 not r2.00001

Open mindedness is all well and good, but I feel this question overreaches a bit, considering the fact that this proportionality holds definitively that the exponent is exactly 2.

Thanks for your comment, but your remarks serve to illustrate how this question exposes close-mindedness. Gravity is a physical phenomenon. It's observed, not derived. Many people are trained (by atheists) to think that physics must be a certain way, rather than observing with an open mind what it actually is. Those who insist that physics must conform to mathematical expectations are ... unjustifiably closed-minded.
Open your mind and you'll never want to go back.--Andy Schlafly 14:13, 14 February 2009 (EST)
Just to clarify, as the above user mentioned, the law of gravity is an inverse square law for a specific geometric reason. If you envision a gravitational field as a series of concentric spheres radiating outward from a body, you will see that as the surface areas of the spheres become progressively larger, the gravitational force must be spread over a greater area, and will therefore be weaker. The fact that the exponent is 2 comes from the fact that the surface area of a sphere varies with the square of the radius, not with the 2.00001th power of the radius.
I agree with you that observations do not always agree perfectly with theoretical models, but this is because the universe does not behave in an ideal mathematical way. Other forces (such as the gravitational fields from other planets) require that correction factors be introduced to adjust the result given by a physical equation. This does not mean the equation is wrong, only that it isn't complex enough to account for everything that is at work. Hope this helps. --Economist 18:29, 14 February 2009 (EST)
It does help ... illustrate your closed-mindedness. The "law of gravity" is a physical observation, not a postulate of Euclid. Only the closed-minded would insist that gravity cannot be inversely proportional to r2.00001. Open your mind and you'll agree. Don't and you won't.--Andy Schlafly
The essay specifically says "Newtonian gravity", not "the law of gravity" (whatever that might be). In Newtonian gravity the force absolutely has to be inverse square; as was pointed out above it's a direct and simple consequence of the fact that the area of a sphere is proportional to radius squared. The same applies for other classical field theories, such as classical electromagnetism (eg, the Coulomb attraction between two point charges is inverse square). But "Newtonian gravity" is not the same thing as "the way gravity really works" - we have General Relativity after all, and doubtlessly other successful theories of gravity will be developed in future. BrianW 07:13, 15 February 2009 (EST)
Your comment misses the point. The phrase "Newtonian gravity" illustrates that we're talking about the law of gravity as being in rough proportion to one over distance squared. If you insist that it must be precisely r-squared, as you seem to, then your mind is inexplicably closed. Open it, and your only regret will be that you didn't open it sooner.--Andy Schlafly 08:48, 15 February 2009 (EST)
No Andy, you're the only one missing anything here. The phrase "Newtonian gravity" means something very specific, which is a theoretical framework which leads directly to the consequence that force is inversely proportional to distance squared. Newton didn't just randomly pick 2 as the power because he felt like it; it is an immediate consequence of developing a classical field theory in a 3D space. By definition, if you had a law of gravity that isn't inverse-square (and you will note that I never claimed such a thing was impossible) then it isn't Newtonian gravity. BrianW 09:00, 15 February 2009 (EST)
Brian, it's a fool's errand to argue with a closed-minded person, and I'm not interested in nitpicky semantics to justify it. Are you willing to admit that gravity, an observed phenomenon, could vary with the inverse of r2.00001? Euclidian geometry does not define gravity, and it's close-minded for anyone to imply that it does.--Andy Schlafly 09:11, 15 February 2009 (EST)
Yes, I agree it is entirely possible that the way gravity works in our universe is something other than an exact inverse-square law. I was only trying to point out that Newtonian gravity is a specific physical theory and also part of a specific theoretical framework. I don't see how using technical definitions in a precise and well-defined way is nit-picking: "gravity" and "Newtonian gravity" are not the same thing. BrianW 09:14, 15 February 2009 (EST)
Brian, you haven't admitted that "gravity, an observed phenomenon, could vary with the inverse of r2.00001."
I'm not going to waste any more time debating with a closed mind. Please contribute to the encyclopedia, but I'm not optimistic about your ability to add insights. Godspeed.--Andy Schlafly 09:55, 15 February 2009 (EST)
I never attempted to claim that mathematics "defines" physics, only that mathematical models are developed to describe physical behavior. Once again, I agree with you that real-world observations necessitate refinements to these models, but this does not invalidate the models themselves. I fail to see why this makes me "close-minded." We must understand gravity fairly well if we can land rovers on Mars! --Economist 23:56, 15 February 2009 (EST)

Unclear questions

Interesting idea, I answered the questions. However there are two questions that are not completely clear: 8, "Do you think that there must be a material explanation for remarkable homing and migration behavior of birds and butterflies?" Material is unclear. Of course there is some physical ability that they have that allows them to home, so do you mean "naturalistic" instead, as in shaped without God?

11, "Do you think that it is possible that evolution did not occur?" I take it you mean universal common descent, so perhaps it should read "...that the theory of evolution is false?" Evolution is a vague term and it encompasses speciation and descent with modification, which are accepted by virtually everyone. AddisonDM 13:23, 29 May 2009 (EDT)

Your point about question 8 is interesting, but I don't see how use of the term "naturalistic" improves it. Everyone understands what "material explanation" means, but "naturalistic" is far from clear. The question is whether the homing must be guided by something material, such as magnetism, or can it be guided by non-material phenomena, such as intelligent design that might allow for abstract programming features or divine guidance or another mechanism that transcends materialism.
As to question 11, I think the term "evolution" is widely understood to have the meaning that Darwin used for it. I realize that some have attempted to change its meaning to salvage the concept, such as redefining evolution to be any change, but I'm reluctant to use jargon like "universal common descent" rather than commonly understood terminology. Perhaps a footnote could explain further, which you're welcome to add!--Andy Schlafly 18:31, 29 May 2009 (EDT)

Unjustified Edit?

Regarding Question Two ("# Have you ever admitted that something you accepted for over a decade is, in fact, completely false? "), I reversed the question to "Have you ever realized that something you accepted for over a decade was, in fact, completely false?" and it was reverted as an unjustified edit. I did this because a yes means closed-minded, while accepting and admitting that you have changed your mind on one of your long-term beliefs is more openminded. It´s the same question, but phrased differently so it doesn´t corrupt the taker´s score. MaryC 17:43, 12 June 2009 (EDT)

You're right about the need to reverse the question to avoid corrupting the score. Thank you. I've made that change. But the reason for the reversion is that you conditioned the question on a realization. It is the closed mind that refuses to attain that realization, so that condition had to be reverted.
Thanks for your insightful edit and comment.--Andy Schlafly 19:23, 12 June 2009 (EDT)

Just my answers, for fun!

  1. No, of course not.
  2. A decade ago, I was 13...  so, no
  3. Who would?  I'VE never been to jail, unlike basically every celebrity I can name.
  4. What's that town in Montana with MANDATORY gun ownership, and ZERO crime?
  5. I was 11 months old.
  6. I thought SDI was demonstrated to work - haven't we shot down satellites?  Haven't the Chinese, as well?
  7. I think it is unlikely, not impossible.  I've researched it a bit.
  8. I think there must be an explanation, material or not.
  9. No, but I think if it was there would be evidence, possibly evidence we have yet to find.
 10. You can obviously measure it - take a man like Lee Strobel, compare to Richard Dawkins...  there ya go!
 11. It is possible, yes. 
 12. Actually, yes, I do think this is impossible.  Very simply, because the power of 2 wasn't arrived at by experimentation or observation, but by pure hard logic.  Additionally, in Newtonian gravity, it is 2 - that is part of what defines Newtonian gravity.  If it wasn't 2, then we wouldn't be talking about Newtonian gravity anymore.
 13. A lot of what I was taught was definitely false.
 14. There may be elements of Catholicism which are false, but on the whole, I'm pretty committed to it.  

I just did this for fun, don't go jumping down my throat for anything! JacobB 19:40, 1 October 2009 (EDT)

You're open-minded. Congratulations! (I altered two of the questions since you responded.)
I do think it is quite possible that the power of 2 in Newtonian gravity is not precisely exact, and that a slight deviation from 2 (such as 2.0000001) is to be expected. The physical world is not a perfect in a mathematical sense.--Andy Schlafly 23:05, 1 October 2009 (EDT)
Hmmm, that's an interesting question. It seems to me that the power of 2 would be forced to be exact if we take two other facts for granted, which seem plausible:
  • The universe is a 3-dimensional Euclidean space
  • Gravitational fields are divergence-free away from a source. This physically means that there is no contribution to the gravitational field of a mass from any point that's not part of the mass.
This is basically a consequence of the divergence theorem (if I'm thinking straight). These are both basic tenets of the Newtonian picture. It's really not possible to do Newtonian mechanics where gravity is an inverse-2.00001 law: it would look to bodies as if gravitational force was being exerted by a body outside the mass in question! Similarly the fact that Coulomb's law is precisely an inverse-square law is a consequence of Gauss's law, which also follows from very basic assumptions. On the other hand, Newtonian physics isn't exact in the first place. I think I have read that gravity seems to be weaker than an inverse square law over extremely large distances (on the astronomical scale). This is because of some non-Newtonian, and perhaps not fully understood, physics. But there's no way to describe gravity in GR by something as simple as an inverse-square law anyway! --MarkGall 23:31, 1 October 2009 (EDT)

That's a great analysis, Mark. Even before the invention of calculus, Newton figured this with basic geometry, on the assumption that the force of gravity caused by a point mass would be the same in any direction, that is, at any other point at some distance from the mass. This logically means the force would be spread out equally over a sphere at a given radius r. Then the strength of the force at a point at radius r would be some "inherent" strength, divided by the area of the sphere. The surface area of a sphere is 4*pi*r^2. That is where the exponent 2 comes from. That formula for the area is geometric; the exponent couldn't possibly be anything other than 2. ON THE OTHER HAND, this is all in Newtonian gravity. In GR, a derivation of the exponents works similarly, but the possibility of them being slightly off 2 is equally remote, that is, 0, but even then, GR is not an ultimate formulation of gravity. Who knows how the world really works. JacobB 23:53, 1 October 2009 (EDT)

Ah, that's a better way to picture it, no need for calculus. Thanks for sharing. If only I were as smart as Newton! --MarkGall 23:58, 1 October 2009 (EDT)
This question is a very interesting one, but I think you'll agree after pondering it further that there is no logical obstacle to observing a gravitational force that proportional to the inverse of, say, 2.0000001. Math is fine for what it does, but mathematical elegance does not dictate physical laws.
Invariably math/physics types initially insist that it must be precisely 2 in order to make the integrals work nicely. But physical observation is what counts, and there is no logical obstacle to mathematical inelegance. Thanks for discussing this.--Andy Schlafly 23:56, 1 October 2009 (EDT)
Mark and Jacob, I encourage you to ponder this question (#12) further. I do think that an insistence on mathematical elegance as dictating physical phenomenon cannot withstand scrutiny.
God and logic define nature, not mathematical elegance. We have to observe first, and then come up with the best math to describe the observations. There is no logical problem to 2.0000001 here. Indeed, I find it implausible to expect that exponent to be precisely 2 for our calculating convenience.--Andy Schlafly 00:22, 2 October 2009 (EDT)