The reason I even bother to engage in this esoteric and pointless discussion is one of pragmatism. I see an inherent danger in separating maths from science because by making it an esoteric subject and suggesting things like 'maths is content free' (true as it may be) people see it as pointless and irrelevant to their lives. The best way to engage people in maths is to demonstrate it to them through practical methods, and show them that maths is a practical and useful tool, not an esoteric exercise in nothingness. I see trouble today and on the horizon in the way youth is turned off from maths and science, because too many academic wankers have made these inaccessible subjects just to make themselves feel superior by demonstrating how tricky and complex their field is. Simplifying what these fields are and demonstrating how they are ubiquitous and all encompassing I see as an excellent path to grabbing peoples attention and bringing some interest back into these fields.

[i]but how is a thought experiment not exploring the logic of nature?

Because it is not based on logic. The contradictions were not internal to Newtonian mechanics, otherwise it would have been rejected from the beginning. The contradiction arose when properties of light were incorporated. The nature of the universe is not inherently logical, just consistent (from experience). The contradiction was not exactly of a logical nature, it was a contradiction of predictions. If two theories make different predictions, then you know at least one is wrong without doing an experiment. The logic involved was in the manipulation of the theries so that the contradictions were clear. As I indicated earlier with the s=ut+1/2at^2 example, such manipulation can be linguistic or mathematical. You are still stating the same theory. When Einstein manipulated the theories he was doing logic, which you could call maths. In this example the logic was trivial, however looking for the contradiction and knowing where to look was science and invpolved a creative process (I'm not sure if Einstein was the first to do this). Coming up with the new world view was also science and is where Einstein's creative (not logical) genius shone through. Putting it in the form of equations and manipulating them was maths. What you've got here is two tools being used at roughly the same time, though in a historical sense their use was problably separated in time. Part of your inability to separate the two tools is in the way you were taught the theory, which focusses on the outcome and ignores the historical and philosophical background.

Essentially since Einsteins theories could not be tested, it was as much an exercise in maths alone as triangulation is to determine distance and position.

No, using triangulation like that is based on a previously existing world view. When you triangulate something you are not doing science, you are doing maths. You are not contributing to our understanding of the world, you are making predictions from the current understanding. What Einstein did was change the meaning of force, distance and time to come up with a new world view. He did not manipulate the old world view with some mathematics.

Exactly my point, it is just s different way of saying the same thing, one that works because the logic of maths and the logic of science are quite often compatible.

No, it works because the mathematical manipulation doesn not change the science, just as phrasing something differently does not change the knowledge you are trying to communicate. It has nothing to do with 'compatible logic' behind the two fields.

You seem to see the world as compartmentalised into little boxes of pure definition, on the other hand I have a far more zen approach in that I see the oneness and interaction of everything.

You are confusing 'seeing the oneness' with being unable to see the differences. Even a monk who sees the oneness between a cat and a dog would not argue that you cannot always tell them apart consistently and that there were 'grey areas' between them.

I am saying they are sufficiently compatible to be able to define one in terms of the other in many instances

I am saying that it is not possible to do that. Ever. If you are talking about maths and science still that is.

In fact,when you say science and maths are both philosophy you essentially state exactly what it is I have been trying to get across in my convoluted way.

No I am not. I am saying that they are both forms of knowledge, or methods of aquiring knowledge. I am not saying they have the same logical foundation.

I just go a bit further and say they are both a part of the same BRANCH of philosophy.

Oh really, what branch would that be? The 'maths and science' branch? Usually putting to things on the same branch like that implies that they have something in common which nothing else does. What is it?

Exactly my point, the world is grey and muddled.

That is not what I was saying. I was clarifying the situation, not muddling it. I was pointing out the difference between what humans do which involves the use of multiple tools, and the tools themselves, which are distuinguishable.

Here you go again saying that science is only science if experiment is involved, this is not true, observation and logical deduction are as much a part of scientific study as empirical experiment.

No it isn't. That view of science has been long discarded. Google 'the problem of induction'.

You define science far too specifically, you even admitted that it is difficult to define rigorously.

I never admitted that. In fact I claim to have a very rigourous definition of it. It was maths I had the problem with, but that has been solved.

I see an inherent danger in separating maths from science because by making it an esoteric subject and suggesting things like 'maths is content free' (true as it may be) people see it as pointless and irrelevant to their lives.

You can't seriously think that people are going to start seeing maths as pointless because it is not dependant on exernal factors. To me, it adds to it's usefulness. What you are saying is that philosophy is pointless because you cannot differentiate these fields of study. But you can differentiate them and it aids greatly in nunderstanding them and the knowledge aquired through them.

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I don't think it, I witness it and have experienced it myself. First maths course I did in university stressed above all else the practical applications of maths, because this is what appeals to people. Try to think like a lay person for a second, yes of course on an academic level the fact that maths can be applied to anything is its greatest strength, but tell this to the average person and then ask them to simply go out and work it out will and does turn them away. People need to be SHOWN the practical usefulness of maths.

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I am saying they are ubiquitous and all encompassing, and thus relevant to all man's pursuits.

Tell me, what is the simpler definition of science: that science and maths are a description of the process of human acquisition of knowledge about nature, or your long winded specific definition that relies on any number of other concepts for it to be reconciled? Who has the simple definition? My definition does not fit maths well enough, but Science and maths are still inseparable concepts that arose together and can often be used to define one another.

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Yes, in fact I do, on a very pointless level that is not relevant, I am describing the oneness of everything I experience. "life is only a dream and we are the imagination of ourselves"

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Good, so my points soared straight over your head...

How many times do I have to say that my point is not that science and maths are the same, my point is that they are both part of the same process, and both use the same methods, often, not always. Each can also be used to define the other in some limited sense. You cannot look at the interrelated nature of science and maths and deny there is some correlation between the two. To do so is only to impose artificially defined constraints to natural processes.

Yes, in fact I do, on a very pointless level that is not relevant, I am describing the oneness of everything I experience.

I still don't believe you have trouble telling them apart. Saying they are part of something bigger does not mean you can't distuinguish them. I think you are interpretting this oneness thing wrong.

my point is that they are both part of the same process, and both use the same methods, often, not always

never, unless you mean a method like 'thinking' but such an argument would be pointless, like arguing for the sake of argument.

Each can also be used to define the other in some limited sense.

Not in any sense at all. Try to define one in terms of the other.

I am saying that intuition shows that there is more to the relationship between maths and science than your definitions allow for

Intuition based on experience can be wrong. It is an excuse for ignorance.

I want to wrap this up because we are going nowhere, this a philosophical argument not a tangible one and as such neither of us can actually be right or wrong. I found this quote from a guy called David Gross, a theoretical physicist: (http://precondition.blogspot.com/2005/10/whether-mathematics-is-science.html)

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This sums up my belief well, yes maths can be 'content free' but to say that because it is not concerned with the observable means it is not concerned with nature is a fallacy. Even if all maths does is describe what we see in our imaginations, I see that as a natural process none the less, there is nothing special and unnatural about the imagined, if you like, maths is the science of the imagination. But it is so much more than that, as this particular scientist says, any part of maths that does not describe an aspect of nature is not maths.

That is my view, I disagree with freediver and sense, nothing more matters. Of course freediver will come off all high and mighty about how wrong I am and how this means don't actually know anything about maths or science and so on. This is not a black and white topic, and definitions of these terms add nothing to the fields of study. As has already been said, computer programming is maths, but do we put it under the umbrella of maths? not at all, it is computing. Definitions are moot, and irrelevant to the subject matter, what is relevant is knowledge: maths, science, language, engineering, computing, this is all just human knowledge, there is nothing really that distinguishes any bit of knowledge from another. All human knowledge is some meaningful description of nature, since there is nothing but nature to describe,even if its is only our imaginations.