I take your point, but if we agreed that both of these examples are models representing an interaction on our part with mind independent reality, then yes we can get closer and closer to our observed reality (which is also an interaction on our part with mind independent reality) in terms of the predictions of GR over Newtonian mechanics, but that's the only comparison we can make. In terms of these models being able to represent mind independent reality (which I think of as the absolute reality of nature), no objective comparison can ever be made because we are fundamentally prohibited from examining the nature of that reality due to the ever presence of what I term "the notion of an observer".

If however we consider that, whilst mind independent reality cannot be reached by familiar notions, it may be able to be reached by notions from mathematics, then perhaps one would say that general relativity, rather than being a model in the manner I describe is actually accessing mind independent reality through mathematics. Is this how you see mathematics - as a means of accessing mind independent reality through mathematical notions that have no mandatory notion of an observer? Just for my interest, do you view GR as a model representing our interaction with mind independent reality, or would you see it as representing mind independent reality itself through its mathematical structure (rather than familiar notions)?

Thanks for the link to Karl Popper.

But that's quite a lot!

Philosophically, one is allowed to question whether our senses have any connection at all with reality. But in our everyday lives I would say that each of us takes it for granted that such a reality exists, and that our senses give us information about it, even if it may not always be complete or fully reliable information. I accept this premise, too, and I think most scientists, by virtue of their profession, do the same.

In short, when the physical evidence indicates that relativity is more accurate than classical mechanics, I would say this is not only a fact about our perceptions, but also a fact about reality. While we cannot prove absolutely that it is information about reality, I think we can sensibly assume that it is.

See, this is where I've been totally misunderstood in this thread (again, largely by my fault). I was not trying to claim that mathematics is a better window into reality than the natural sciences, or that abstract thought is a better source of information about reality than physical evidence, much less a sure source. What I was suggesting was that they are both sources of information about reality, and not just physical evidence. It was with this in mind that I insisted, a bit provocatively, that mathematics is also a science.

Along the way, another issue came along, that of objectivity. You said what interested you about the discussion were the possible implications for quantum mechanics, and then Ken wrote in a reply to you:

I know you've discussed this before in other threads. But if Ken has been trying to use this kind of reasoning to choose between the various interpretations of quantum mechanics, I think he's mistaken. He's trying to use philosophy to answer a question in physics, and that cannot work.

What I mean by this is that Ken seems to think that if only everyone would have a better understanding of what science really is, and what objectivity really means, the correct interpretation of QM would become clear, or at least that some interpretations could be immedietaly discarded as unscientific. But this conflicts with everything I've ever heard about the issue. As far as I know, the various interpretations of QM are all valid, in the sense that they are compatible with the predictions made by QM equations. They are of course incompatible with each other, so they can't all describe reality (unless reality truly is illogical, as Ken sometimes alleges, but I'll put that possiblity aside). If reality is logical, or consistent if you prefer, then only one of the interpretations can be right. Yet physicists have not been able so far to come up with any experiment that would test the interpretations against each other. Maybe one day they will think of one, but for the time being the matter is undecided. Ken is apparently trying to solve it by appealing to the "true meaning" of science, an approach which I think is doomed to failure, because appealing to true meanings is not science, it's philosophy. Only an experiment, or a series of experiments, can solve such a problem. The problem with scientists despising philosophy is that it makes them fail to realise when they themselves start to philosophise.

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"All your bias are belong to us." Ara Pacis
"A witty saying proves nothing." Voltaire

But that's another illogical position. To hold water, you would need to find posts of mine that made claims without offering any descriptive support, such as was your recent
...which was the sum total of that whole post. Or who can forget
Now, I'd be completely happy to let such absurd abuses of common discourse be ignored, since I really only care about the truth, but I cannot allow you to make the false argument that any of that is somehow "my own medicine". That simply isn't consistent with the evidence in this thread, and above all what we care about is what is in evidence.

Nevertheless, I believe I see a veiled apology in your remarks, and the truth is I don't take any of this personally enough to have any real objection. We can both embrace the opportunity for less rhetoric on the path to the truth.

I've been pretty clear on this, but I'm happy to repeat myself:

Yes, definitions are arbitrary choices, in the sense of the term "arbitrary" I have clarified numerous times (i.e., in the sense of being a choice that is subservient to some other objective, as opposed to being forced upon us. If the objective is arbitrary, then so is the definition that achieves it). This is the same sense that "postulates" are arbitrary (see my Wiki quote just before), they are conditional upon an objective that is free for the mathematician to decide. What I refered to as a "kind of game" is pure mathematics, i.e., the application of a set of rules on a set of axioms to determine the ramificatons. Playing chess is a fine example of that principle. The fact that sometimes the goal of the exercise is something different from entertainment is still up to the mathematician, yes.

However, it is certainly true that oftentimes the goal of the mathematician is not "pure mathematics", but rather is related to some application to reality, either present or potential. When that is true, the methods of science become relevant, and the pursuit is no longer pure mathematics, it is now in the overlap with science, as defined by what science is. That's the distinction I have been drawing, in particular to the importance of the concept of "objectivity" as the separation between the authority of the subject of reason and the authority of the object of reason.
Certainly. The best place to find examples are from the history of science. So when in the history of science did it get confused with mathematics? More often than not, actually. It happened when the Greeks thought the planets were perfect spheres, expressly because a sphere is a mathematical concept. It happened when people interpreted Newton's laws as saying reality was deterministic-- another mathematical concept being interpreted as the reality.

And even now it happens, for example in the "many-worlds interpretation" of quantum mechanics. As pure interpretation, there is nothing wrong with that, because an interpretation is nothing but a picture to use when applying a theory to make predictions or imagine that you understand something. But many current theorists don't stop there, they again confuse what is mathematical for what is real, and actually believe that quantum mechanics is somehow telling us that superpositions of many versions of our universe are accumulating to a vast degree constantly. They are of course welcome to any belief that doesn't hurt someone else, but they are mistaken to call it science-- it is again the same confusion I was talking about in the statement you bolded.

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If we understood everything going on in the head of a pin... we still wouldn't know not to step on the pointy end.

People think the problem with models is that they are limited by our minds, but the greater problem is that our minds are limited by our models.

I've just remembered that a while ago, in another thread, Argos posted a link to an article that described Popper's argument about the Uncertainty Principle and an experiment based on it. But the authors argued, on theoretical grounds, that test was not conclusive. I think it was the same test he describes in this link. Here's a copy of the article (pdf), and here's the Wikipedia page.

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"All your bias are belong to us." Ara Pacis
"A witty saying proves nothing." Voltaire

That is not what I have been trying to do at all. There is only one means for "choosing between interpretations", and that is an appeal to personal preference. Period. My objection to the many-worlds interpretation is that it if you google it, you will immediately encounter a handful of links in which that "interpretation" is not only framed as a theory, even worse, it is framed as a true description of the nature of reality. Try it.
That is precisely what I am not doing, indeed I am doing more than the opposite: I am pointing out why it cannot work to use philosophy, or mathematics for that matter, as an authority to answer a question in physics. In physics, observation of reality is always the master of our mathematical postulates and our philosophical principles-- never the other way around, despite the confusion I point out constantly in this forum.
That's not a terribly good summary of what I said, it would be a better summary of the opposite of what I've said. What is unscientific about the many-worlds interpretation has nothing at all to do with the interpretation itself, it has to do with the way people try to justify that interpretation as a way of deciding reality and an argument against the Copenhagen interpretation.
You will never find me making any statements to the contrary, indeed, you will find me meticulously distinguishing the proper meaning of an "interpretation" from the improper usage that is so easy to find.
It is already obvious that no interpretation "can describe reality" in a unique way, it is when one thinks this is the purpose of an interpretation that one has already stepped off the track.

See what I mean?

Had that been anything like my goal, I would definitely agree with you.

My goodness, that's precisely what I would say. You frame my arguments much better when you claim they are a point you are making to refute me.That barb might have had some application had your ideas about what I was saying borne the slightest resemblance to what I actually said. It's no matter-- Len Moran, to whom your comments are directed, seems to see what I am saying quite clearly.

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If we understood everything going on in the head of a pin... we still wouldn't know not to step on the pointy end.

People think the problem with models is that they are limited by our minds, but the greater problem is that our minds are limited by our models.

I hate to seem like I'm nitpicking, but this is quite an important issue. Popper's argument was not "inconclusive", it was simply wrong. He didn't know how to do the quantum mechanics correctly, so made an incorrect prediction for an interesting EPR-like experiment. The quantum mechanics is nontrivial, it takes people, like EPR, who actually know quantum mechanics to get it right.

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If we understood everything going on in the head of a pin... we still wouldn't know not to step on the pointy end.

People think the problem with models is that they are limited by our minds, but the greater problem is that our minds are limited by our models.

The full text of Galileo's Dialogue Concerning the Two Chief World Systems is available online. It's quite long, so I didn't have the time to read it, but I did a quick search for the words "subject" and "object". When the former appears, it's usually in the sense of "topic". It's never used in the sense of "observer". "Object" is never used in the dichotomous Cartesian sense, either, although it is used a few times in the sense of "thing" (Galileo also uses the word "objection" quite a lot in that dialogue... ).
However, I did find this interesting remark:

Funny how he associated mathematics with objective certainty, innit? (Salviati, slightly misspelled in this passage, was the character who represented Galileo in the dialogue.)

I also took a look at the Wikipedia article on the scientific method, which Ken dug up in his support. Nowhere is the subject-object dichotomy mentioned, or, in my opinion, even hinted at. On the other hand, regarding the collaborative aspect of science, which is what I alluded to when I spoke of consensus, there is of course the following:

The experiments of some imaginary Robinson Crusoe genius, alone in his own island-universe, would hardly be reproducible by other scientists/observers.

And, in the Wikipedia article about the history of the scientific method, I found the following principles, from Newton:

Not only is there no reference to the subject-object dichotomy, but it's interesting how the idea that scientific hypotheses may be superseded by others (the principle of falsification) was already articulated by him (point 4).

At this stage I feel that the discussion has gone far enough. Ken is being dogmatic, as he often becomes when the subject matter is mathematics, and I fear that nothing will change that. After all these pages of back and forth, I think there is already a sufficient sample of the arguments and counterarguments of both sides.

Oh, just one more interesting thing I found:

Is pure mathematics useless? Well, you just never know...

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"All your bias are belong to us." Ara Pacis
"A witty saying proves nothing." Voltaire

We covered this long ago on this thread. Galileo is credited as the father of the scientific method in some places, but nobody claims he gave birth to it fully formed. He was still raising science out of the darkness of natural philosophy, where if the Greeks believed it it had to be true. The process went through considerable evolution into its modern form, taking particularly important strides in Newton's time (Newton was born the year Galileo died). Hence, few would expect Galileo to have delivered science in its complete modern form, and cherry picking his remarks prove little. I note that the paradigm he was dispelling with his observations, that of Ptolemy, was supremely mathematical in nature. Is that not a well known fact?More insightful readers may see quite a bit more than a "hint" of a subject/object dichotomy in the sentence in that link "Among other facets shared by the various fields of inquiry is the conviction that the process must be objective to reduce a biased interpretation of the results." If they don't see it yet, they might actually click on the hypertext "objective" in that sentence, which brings them to the very first sentence:
"Objectivity in science is the property of scientific measurement that can be tested independent from the individual scientist (the subject) who proposes them."
In other words, the core of avoiding bias is not looking for people to agree with you (never the least guarantee of avoiding bias), but instead, to seek an explicit separation of the result from the person doing it. As I've been saying.
Useful for anyone who doubted that collaboration plays a role in the advancement of science. Of course, that group would not include those, such as myself, to whom that is already quite obvious. We can pretend that was the issue here, but it was not-- it was the role of consensus in the concept of "objectivity". Many times have I pointed out that many human pursuits, that are not science and make no effort to be objective, involve collaborative effort. If Disinfo Agent insists on repeating earlier arguments, it falls on me to repeat the refutations.
But they would be reprodicible by him/her, and thereby, he/she would indeed be doing science-- obviously. Indeed that is how babies learn the beginnings of scientific thought long before they are even able to communicate with any "consensus" results that is claimed here to be so essential to science. Galileo's telescopic observations did not become science when someone else confirmed them, they just became a more concrete scientific result. They became science when Galileo himself repeated them over and over and studied their repeatable elements, separately from any variables he could attribute to himself.
Again, more than a hint of a subject/object dichotomy is indeed present. Newton stresses (twice) the importance of experiment over pure thought. How anyone intends on doing such an experiment without first making a subject/object dichotomy, I suspect, would be news to Newton.

Yet another rehash from earlier in the thread. This was all covered above-- no one on this thread, least of all me, ever claimed pure mathematics was useless. I don't even think chess is useless, or any other game. If it was discovered that some of the solutions to achieving checkmate in certain positions in chess had some connection with a group of elementary particles, would I be the least bit surprised? No, not the least bit. I said as much above in the bit about Hardy. What separates pure mathematics from science is the way it is carried out-- specifically, what counts as the authority to justify a conclusion. Not the happenstance appearance of usefulness, nope, never said that, never thought it.

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If we understood everything going on in the head of a pin... we still wouldn't know not to step on the pointy end.

People think the problem with models is that they are limited by our minds, but the greater problem is that our minds are limited by our models.

I think you may like this...
In 1602, he wrote an opener to a syllabus used for his students in his cosmography course:

...Accordingly, we say the subject of Cosmography is the world, or we mean the universe, as indicated by the word itself, which only means "description of the world." But of everything that might be considered about the world, only a part belongs to the Cosmographer, which is to reflect on the number and order of the parts of this world and the shape, size, and distance of these, and especially about their motions, leaving to the Natural Philosophers consideration of the qualities of the said parts of the world.

As to the method, the Cosmographer customarily proceeds in the reflections in four ways, the first of which embraces the appearances, or phenomena, and these are nothing but sensate observations we see every day, as for example the rising and setting of stars; the darkening now of the sun and again of the moon; the latter's showing herself now crescent, now at quarter, now full, and again completely dark; the moving of the planets with very different motions; and many other such appearances. In the second place there are hypotheses, and these are nothing but some suppositions relating to the structure of the celestial orbs such as correspond with the appearances, as it would be when, guided by what appears to us, we should assume the heavens to be spherical and to be moved circularly, sharing diverese motions, the earth to be stable, situated at the center. Third there follow geometrical demonstrations [math] with which, by means of some properties of circles and straight lines, the particular events that follow from the hypotheses are demonstrated. And finally, what has been demostrated by lines being calculated by arithmetical operations, is reduced to tables from which without trouble we may later at our pleasure find the arrangement [predictions] of the celestial bodies at any moment in time."

[My bolds, and I removed my own insertion of 'theorinos' as an augmentation so as not to disrupt this powerful passage. ]

There is reason to believe the above methodology of Galileo stems from Ptolemy who tried so hard to connect math to that which is observed. Unfortunately, Ptolemy did not develop this as a recommended method, nor did he ever seem to apply it to anything other than astronomy, though he was active in other fields such as geography. Ptolemy's opening in his Almagest stated that Aristotle, unlike Plato, divided theoretics into three classes: physical, mathematical, and thoelogical (metaphysical). Agreement from philosophers would only be found in the mathematical.

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Lighten up! This is a stellar board! Author: duh.

"The Sun, with all the planets revolving around it, and depending on it, can still ripen a bunch of grapes as though it had nothing else in the universe to do..." Author: Galileo supposedly.

Len may like this article: Transgressing the Boundaries: An Afterword, by Alan D. Sokal.

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"All your bias are belong to us." Ara Pacis
"A witty saying proves nothing." Voltaire

Yes, I agree completely with that article that mistaking objectivism for a "power structure" (like a majority consensus) has negative ramifications for science. Indeed, it is just that conviction that caused me to counter that argument so aggressively, I admit in a way that might not have always seemed cordial but I meant no personal affront. I just think it is crucial to bear in mind what objective thinking actually is, independent from any social structure such as consensus agreement of those in authority.

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If we understood everything going on in the head of a pin... we still wouldn't know not to step on the pointy end.

People think the problem with models is that they are limited by our minds, but the greater problem is that our minds are limited by our models.

Mathematics comes from the Greek "learning" which comes from the Egyptian "Maat, Truth Order and Justice". There are three claims for mathematics and God. God has power over mathematics, God is mathematics and God is obedient to mathematics. Kurt Godel shows limits on mathematics but mathematics is none the less a powerful tool.

All the comments about mis-using or mis-applying the tool are correct. Science has two aspects observation and reasoning. Mathematics is useful in reasoning. Especially in long trains of reasoning far from observations.

Observations are not omnipotent and are often erroneous.
Physicists desire that a displacement in the direction of force should be called "energy" en-ergy (energy in). When Quaternion mathematics and observation shows this is ex-ergy (energy out). There are times when mathematics is smarter than physicists and their observations.

A very important example of this is the story of quaternions. Here the laws of mathematics were set aside to follow the laws of "observation". Quaternion Mathematics says dropping a ball releases energy as the displacement and the force are in the same direction. Lifting a ball absorbs energy because the displacement and the force are in the opposite direction. Quaternion rules are observed in nature but physicists "vector mathematics" use a different rule: I^2=J^2=K^2=+1. This Vector Rule causes non-associativity:
(II)J=J but I(IJ)= -J.

Today, quaternion mathematics ,a very powerful mathematics, has been set aside by poor observations. 4D spacetime is quaternion but physicists insist on Minkowski "mathematics" for 4D spacetime. The Universe is Quaternionic, says mathematics, the operation of the Universe is non-commutative and supports a division algebra, i.e. Ax=b is true for physics.

Orson Scott Card says that all science is a game! (It's a long article. I suggest scrolling down to the section titled "Why Faith in Darwinism Is No Better".)

They may disguise these assumptions by speaking of "elegant" solutions, or "symmetry," but the fact is that scientists commonly expect the universe to make sense. And "making sense" is a very unscientific idea.

Science thus becomes a game – you are allowed to play only within the rules.

I disagree with almost everything he says about science, but it's still an interestingly different viewpoint.

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"All your bias are belong to us." Ara Pacis
"A witty saying proves nothing." Voltaire