Starts out with the usual mythology, actually (hypothesis first, then theory, then experiment) but toward the end it explains what people don’t understand and how it really works.

1) There is no proof in a mathematical sense about the science. The only proof exists within the theory, nothing can be proven about the relation between the theory (the model) and the reality it models.

2) The way it’s tested is by its coherence with other science, and with observation and experiment, obviously

3) Incoherent (wrong) science is not formally disproven in practice. It is more commonly simply ignored by practitioners.

Nowadays of course, ignored science is pitched to the press. And therein lies our problem.

You cannot know that your theory about scientific theory is even "self consistent" without the ability to perform a logical proof which your theory says can cannot be done. Even if you produce said proof, by your own theory, your theory is strictly self referential and has no necessary relationship to anything real. How then can it be about Science? In fact, how can it be about anything? Is it not pure religious dogma founded on faith?

You can indeed prove things about a given theory being consistent with other theories. That is mathematics and indeed self-referential.

But the proof of the whole pudding, like it or not, is empirical.

That's why it's not sterile. It's absolutely not self-referential; it's quite ludicrous to say so. The final say goes to observational and experimental data, and the whole method depends on assuming that everything is wrong all the time, and having maximum confidence in what is resistant to disproof and coherent with other similarly established theory.

There are branches of modern quantum physics that are rich in mathematics but so far from observational that some people consider them sterile in this way. But that does not apply to the vast range of science.

My objection to your formulation is that you presume mathematical proof has no foundation in the empirical. That it is somehow "pure" and not at all connected to reality. It looks to me from this perspective, your "theories" are just a bunch of random symbols chosen at whim and then defined so they meet the requirement of self consistent. Further, all the terms used in the definitions are also defined so that it can be said you have shown your "theories" are self consistent. In other words they are totally self referential filled with a maelstrom of circular reasoning. The process may produce a wonderful word salad that allows people who call themselves "scientists" to argue endlessly but little else. Rather like the endless scholastic discussions of how many angels can dance on the head of a pin: full of sound and fury but signifying nothing.

How is it that the empirical and the mathematically (aka logically) self consistent can be connected? Does not the empirical have to come first including the discovery of your so called mathematical logic? If not, tell me how you can create the universe of mathematics and your wonderful self consistent theories without any knowledge of anything: using a totally blank mind.

You are basically repeating yourself. I already answered this mostly.

The only thing you add, the idea that the discovery of mathematical logic is itself empirical, is a bit too philosophical for my taste, but okay, suppose I so stipulate. What have you gained by that? I don't see that this observation adds much of value to the epistemology of real science.

Do you have a point other than hairsplitting?

You may wish to not be philosophical but you have no choice. Your only choice is the philosophy you will follow: Plato and his progeny, Aristotle and his progeny, or a confused incomplete mixture of the two. Plato starts with ideal mental constricts that are intuitively known to be true. Aristotle starts with observation and applies logic to experience and experiment. Their progeny merely elaborated on the initial fundamental identifications. Some of the elaborations are useful and productive and some aren't.

As near as I can tell, you are starting in mid stream, accepting a huge body of what I call the maelstrom of circular reasoning that includes your world of math (effectively Plato's approach) and adding a magical word "empirical" to it. How do you connect the two and make sure that the huge body of circular reasoning is in fact connected to reality and not just merely deemed to be so (aka assuming what you must prove)? Especially since that huge body of accepted theory will seriously color your interpretation of your observations and the design of your experiments.

The point I am leading to is that the empirical leads to the theory. The logical check (also derived empirically) for consistency of the body of theory must start at the first perception made by an empty mind. Experience, experiment, and theory be slowly built and tightly integrated bottom up without contradiction. Then a reverse process of logical/conceptual reduction to the facts of reality is a cross check for actual consistency with reality. Anything less is building sand castles in the air and trying to rent the rooms to people calling themselves "scientists".

There is NO dichotomy between the so called analytical and the so called synthetic. They are of the same source, substance, and inseparable process. Proof IS the incremental bottom up integration of a totally consistent body of theory, experience, and experiment in a fully specified context. It is by this method one ties the empirical and the theoretical to reality. No magic, no mysticism, and no arbitrary deeming.

Tamino has a good take on this:

http://tamino.wordpress.com/2013/08/02/michael-mann-understands-science/

Like the intersection of science and religion, science and philosophy are part of our common humanity but not identical.

http://bbickmore.wordpress.com/2013/08/02/heartland-knows-squat-about-science/

http://tamino.wordpress.com/2013/08/02/michael-mann-understands-science/

older but on topic:

http://www.postnormaltimes.net/wpblog/perry-actually-says-something-correct-but-deeply-misleading/

on the other hand

http://wattsupwiththat.com/2013/08/01/bizarre-claim-mann-on-mathematics-alchohol-and-proof/#comment-1377892

Platonism in the Philosophy of Mathematicshttp://plato.stanford.edu/entries/platonism-mathematics/

is the mainstream view these days, at least among mathematicians. However note that

platonism must be distinguished from the view of the historical Plato.Also note that accepting platonism means one accepts a portion of reality which, in some sense, lies outside the physical universe.Eugene Wigner wrote

The Unreasonable Effectiveness of Mathematics in the Natural Scienceshttp://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

on a relationship between mathematics and physics. I'm going to read it now and so should you if you have not.

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"You cannot know that your theory about scientific theory is even “self consistent” without the ability to perform a logical proof which your theory says can cannot be done. Even if you produce said proof, by your own theory, your theory is strictly self referential and has no necessary relationship to anything real. How then can it be about Science? In fact, how can it be about anything? Is it not pure religious dogma founded on faith?"

Einstein: "Physics constitute a logical system of thought which is in a state of evolution, whose basis cannot be distilled, as it were, from experience by an inductive method, but can only be arrived at by free invention. The justification (truth content) of the system rests in the verification of the derived propositions by sense experiences. The skeptic will say: ‘it may well be true that this system of equations is reasonable from a logical standpoint. But it does not prove that it corresponds to nature’. You are right, dear skeptic. Experience alone can decide on truth."