I think Lonnie is right; if we completely discount dark matter/energy, then there’s about 70% of the universe that we can’t account for.
Granted this unknown 70% is probably not all of the same stuff (ie dark matter), but determining the exact properties of it is still beyond us or at the very forefront of what our tech can do. Give it a few decades.
Some might ask, Well, the alternate gravity theories fit observations such as rotational speed of galaxies and gravitational lensing, so doesn’t that mean that they have shown predictive ability?
No – predictive ability is shown by coming up with the prediction BEFORE the observation.
It’s not the same thing to for a theory to fit data already in existence – particularly when it is hand-fitted to that date – as it is for a theory to correctly predict something new and different that the old theory wrongly predicts.
It really is no surprise, for example, that the alternate gravity theories could do a better job of fitting speed of rotation of galaxies than does GR, as GR has no parameters in it that were adjusted to provide that fit, whereas these theories were hand-adjusted, indeed designed overall, precisely to meet that.
Very often many, many different equations will fairly closely fit some data. So when one of them is handfit to existing data, that alone doesn’t show a great deal.
I agree Bill, its so much easier to make a prediction when you know all the data isnt it?? Its like all the “psychics” who just knew something fishy was going to happen on 9/11 - Only they didnt know it on 9/10, they knew it on 9/12.
I cant be bothered to join in on internet Creationism debates any more, the level of ignorance (sometimes on both sides) is enough to make you go mad. For someone to REALLY understand evolution (Oh sorry, I mean “Darwinism”) and how we know it to be the case, you would have to spend hours reading/watching/listening to material on the subject. Trying to sum up ideas like genetic drift and molecular evolution (the basis of DNA testing) in an internet post and have ANY hope of someone reading it is exactly 0%
I just cant answer another “Then why are there still monkeys?” question or read another comment on how evolution is suppose to comment on how life began or the planets formed… I just cant do it anymore.
Everything is, in the final analysis, wrong.
Like mathematics? One day we’ll discover that in the ring of integers 2+2 doesn’t equal 4?
Besides, no one can ever disprove something like F=ma, it’s a damn definition! (Although, one can prove that the quantity we measure and call force doesn’t satisfy the definition we gave for force, which has already been done.)
I think that in this quote, you have successfully described a potential three pages of argument between us. There is a counter argument for every argument. They have been rehashed many times.
Just in case you’re wondering, I’m not some fool running the old “there’s no point in learning it all anyway, sure it changes every day” argument. I do have a bit of backround in this stuff. I’m saying that there will always be truths that lie outside the boundaries of our axioms, and we will have to change our axioms to accommodate them. But I’m sure you’ll agree that my original statement was a heck of a lot prettier!
I’m going to respond to the following few points all at once:
I do have a bit of backround in this stuff. I’m saying that there will always be truths that lie outside the boundaries of our axioms, and we will have to change our axioms to accommodate them. [/quote]
So, let me make a technical point, and then try to philosophical for a moment.
First, Godel’s incompleteness theorem does NOT say what you are implying it says. You are implying that Godel’s incompleteness theorem says that “there will always be truths (about the integers) that lie outside the boundaries of our (mathematical) axioms”. This statement, worded as ambiguously as it is, is 100% completely false. Godel’s theorem does not say this, and I cannot stress this point enough.
Now, you may ask, just what does Godel’s theorem say? Filling in the statement that you made, Godel’s theorem says the following: “there will always be truths (about the integers) that lie outside the boundaries of our (mathematical) axioms, if those axioms are formulated in a formal language in which the compactness theorem holds”. Since you have a bit of background, I assume you know what the compactness theorem says, if not, look it up.
So, you may ask, how does this subtle difference matter? Well, it makes all the difference! For example, if you formulate the Peano axioms in first-order predicate calculus–ie, in a language where one only quantifies over elements of the domain–or in, say, ZFC (which is also a first-order theory), then those axioms about the integers are incomplete–there are theorems that don’t follow from the axioms alone. BUT, if we formulate the Peano axioms in second-order predicate calculus–as is natural anyway–then they are complete–EVERY truth about the integers follows from these axioms (Or at least, follows in the sense that they are a model theoretic consequence, are entailed).
So that is the technical point, and it is an important one. Godel’s theorem isn’t that significant, because all it tells us is something very basic about certain types of formal systems. As a historical note, Godel’s result was unexpected and significant at the time he made it because there was no developed model theory at the time, and because the distinctions between first and higher-order logics weren’t well appreciated. Given these two advances though Godel’s result isn’t anything shocking. In fact, it’s such an elementary result that I can give a proof of incompleteness in less than 1 page (Simply construct nonisomorphic models for the axioms! This is were compactness comes in, as it’s needed to do this for the first-order Peano axioms).
So, now to the more philosophical stuff (briefly, at least). Someone may ask: How does this apply to duffy’s original point? Well, I believe duffy’s original point was that no matter how well we try to characterize, say, the integers, that that characterization is in some sense “wrong” because it is incomplete. There are things that are “true” about the integers that our characterization doesn’t capture. Well, since as I’ve pointed out that this point is false with respect to the integers–that is, we do as a matter of fact possess a characterization of the integers which is complete, ie, the second-order peano axioms–one may ask what’s left of Duffy’s claim.
Let me help duffy out–I hope you don’t mind. Your claim still has much force–and it is an important one–since not all of our mathematical theories are like the theory of integers. For example, there is no standard accepted formulation of set theory which goes beyond the limitations of compactness, and therefore for which a Godel-like incompleteness theorem doesn’t hold. So, it seems that here duffy is maybe right–here we have a mathematical theory–set theory–which we are in fact constantly revising.
In response to this, all I can say is that yes!, this is all true. If your claim is that there are mathematical objects–sets, for example–which our current well-accepted theories about them leave some statements indeterminate–the CH is a classic example–then yes, that’s true. In fact, I have had whole debates on this very forum with people who, besides lacking any technical experience in mathematics, could not believe that all the truths of mathematics weren’t somehow determined.
But, against this, I do not think this means that our theories are somehow “wrong”, or that in picking more powerful axioms we are “changing” what we know about sets. For example, if tomorrow the entire mathematical community decided to accept some axiom system for set theory in which compactness fails–ie, decided to accepted an axiom system for set theory on which there were NO undetermined facts about sets–that would not mean that somehow the original ZFC was wrong, or that those mathematicians who had earlier denied that axiom system were wrong either. All that would mean is that mathematicians have arbitrarily decided to study one certain type of mathematical object–whatever sort was picked out by that theory–and to call them “sets”. This arbitrary decision would not somehow invalidate all the other possible set theories that are consistent with ZFC.
If this sounds fantastic to you, it’s really not. It’s simply the same situation that we have in geometry, where there are many different geometries of interest that mathematicians want to study. That point, actually, is important. Ultimately mathematicians study what is of interest to them. There is no question of “what is right” at the heart of mathematics. For example, the second-order Peano axioms that pick out uniquely some objects aren’t “the right” characterization of “the integers”. They too are merely an arbitrary choice. That we privilege this set of axioms above other alternative “nonstandard models of arithmetic” is a mere social convention and has nothing to do with which model “really” is the integers.
So what is ultimately my point? My point is that in so far as one can choose one axiom set over another there is no question of “right” or “truth”. Mathematicians simply study those mathematical structures, models, that are of interest to them. Any attachment of special names like “integers” is arbitrary. Unless you hold to an extreme form of mathematical Platonism, whereby there actually are some “real” objects called numbers that mathematicians are trying to get at, there is no significance to what we call these things. My second point is that in so far as there is a question of truth regarding what’s true in a given model, there is no ambiguity here. It’s simply not true that no matter how hard we try we’ll never come up with a set of axioms that uniquely answers all our questions about some mathematical object (Again, at least in the sense of entailment). All we must do is use the right sort of language, one that is descriptively powerful enough. That is the lesson of Godel.
btw, I too have a bit of background in this stuff, it is my main research area.
When you’re talking about objects of astronomical size, there’s too many factors that could play into the deviations that are the basis of dark matter. That being said, the scale of the deviations is very large, and so it begs the question what exactly is causing it. Mot people pick a bone with dark matter because it’s non-intuitive to have matter that’s unobservable (which I agree), but there’s way more mind-fucks in physics than that. A few of my favorites are repulsive gravity and the universe having a uniform density within 1/100000 in all directions. For anyone who has an interest in Big Bang cosmology, these two things play a huge role in explain the cause of the Big Bang (inflation).
Physics is my love and my curse. Well, at least cosmology and quantum mechanics are.
I apologize for making that so long. Hopefully you find it engaging enough to wade through. [/quote]
This is in response to your post on peano axioms, I just didn’t want to quote the entire thing.
It is important to have formalized integer descriptions, but most of those axioms are common sense arithmetic. Their real value is in showing us the limitations of mathematics as it stands. In reality though, a lot of these issues are solved by complex numbers. A lot of people think that linear algebra is dumb because nth dimensions are meaningless, as are complex eigenvectors and such, but they’re actually essential in removing limitations of natural numbers that cause incongruities in not only math, but in physical systems that are described by mathematical expressions.