Going back to Ken's latest post, I won't do a "point-by-point rebuttal" in the style of discussion boards, but will attempt to address a few excerpts from your post which I think capture the the crux of our diagreements and misunderstandings.

Yes. The problem here is precisely that we are not using those two important words "objective" and "observation" in the same way. Let me address the former first.

What is objectivity? I would explain it as follows. When some activity is such that, given enough time, all rational and sufficiently informed people eventually converge upon an agreement about it, it is objective. This is what happens with the natural sciences. There was much disagreement at first about, say, global warming -- or continental drift, or quantum mechanics, to use less popularly controversial examples --, but eventually most scientists came to accept all three theories. There are, of course, various people who doubt each of them. But when we investigate the doubters closely we typically come to the conclusion that they are either not sufficiently informed about the theory in question (in which I include those who do not make the necessary effort to understand it), or are being swayed by irrational impulses.

Another important part of objectivity is of course that the debate must be mediated by some kind of evidence: climate, ice, and gas measurements and estimates in the case of global warming, geological and paleontological traces in the case of continental drift, sophisticated laboratory experiments in the case of quantum mechanics. That's the standard of proof we abide by in science. I would say that this is the basic formula of objectivity: rational consensus mediated through evidence. I hope you will agree.

Where we part ways is in what we allow to constitute "evidence". You read this word and are only able to think of physical evidence -- the "observations" you mentioned. But I argue that logic provides us with its own kind of evidence. Good ideas in science are logically consistent, and consistent with other good ideas. Inconsistent ideas are invariably bad; we don't even bother to test those out empirically, most of the time (see a recent example with some resemblance here). I say that this is evidence, too; evidence of another kind.

As I wrote before, I disagree with that philosophy. Against it, I would give two arguments, one of a rationalist nature, and the other of an empiricist nature.

1/ The rationalist counterargument is that the constructions of pure mathematics are not an arbitrary game, that could just as well have been constructed according to different rules -- the definition of a game is precisely that its rules are a matter of convention (2 + 2 = 5?) --, but rather follow rules of logic which rational beings cannot break without denying their own rationality. These rules are not merely a subjective part of how our brains are wired, we find them in the very architecture of the universe. You are free to disagree, but then the burden is on you to show how an illogical universe could exist.

2/ The empiricist counterargument is that, in separating "pure mathematics" from applied mathematics, you are making an artificial distinction not supported by the empirical evidence found in the history of mathematics. You've mentioned Hardy a few times, no doubt because he famously made statements such as:

"I have never done anything 'useful'. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world."

The irony, which may not have been totally lost in you, although I think it's a double irony, is that what Hardy thought was pure, useless mathematics (his work on prime numbers and number theory) turned out to have beneficial practical applications. It's quite important for computer science and cryptography, because of encryption algorithms.

In fact, it can be soundly argued, with an empirical historical basis, that the whole of mathematics, no matter how abstract, is simply an extrapolation of the physical world. Mathematics started out with arithmetic (counting things) and geometry (making measurements). More recently, new fields in mathematics often arose to solve problems from the natural sciences, especially from physics, though by no means exclusively. New abstract mathematics is typically either invented to solve specific empirical problems (the whole of calculus, just for starters), or an extension of solutions to empirical problems. So, there is always a connection between the "purest" of mathematics and physical observations, no matter how distant. And those ultimate physical constraints of mathematics prevent it from flying away into pure arbitrariness.

Some further, secondary remarks follow below.

"Standard definition of science"? I wasn't aware that there was one. Is there an SI for philosophy?

I have never seen pure mathematics disagree with the empiricial sciences, nor do I see how that could ever happen. You have defined them to be activities with completely separate domains of applicability. To have them disagree would be like having science and pure religion disagree. Still, if you can show some examples of it I'm all ears.

On the flip side, the opposite conclusion -- that reality is logically inconsistent -- is not supported by any empirical evidence. I'd say it's a tie between the two views.

Aren't some animals incapable of swimming, even with training? Anyway, just because you need to train and refine a skill, it doesn't mean the ability does not exist in you in a potential form.

Ah, Occam's Razor. But didn't you agree, in a previous discussion, that Occam's Razor is not a valid principle of science?

As you have noted yourself, truth does not equal provability in science. Gödel showed that this is true not only in the natural sciences, but also in certain mathematical systems -- but so what? Not being able to prove its laws and theories deductively has not prevented physics from being a science; why should it prevent mathematics?

To be perfectly clear, when I use the word "logic" in this discussion I am speaking of very basic rules of internal consistency, such as the principle of non-contradiction. These are notions which have been with man for as far back as I can tell. I do not concern myself at the moment with the heavily formalised branch of mathematics, invented in the 19th century, known as "mathematical logic" (although there is undeniably a connection between the two); just everyday simple logic. So, appeals to Gödel and his scarecrows seem quite beside the point. I have never tried to claim that the whole of mathematics can be syllogistically derived from a single set of axioms.
And, even if I had, Gödel himself said:

"It might seem at first that the set-theoretical paradoxes would doom to failure such an undertaking, but closer examination shows that they cause no trouble at all. They are a very serious problem, not for mathematics, however, but rather for logic and epistemology. As far as sets occur in mathematics (at least in the mathematics of today, including all of Cantor's set theory), they are sets of integers, or of rational numbers, (i.e., of pairs of integers), or of real numbers (i.e., sets of rational numbers), or of functions of real numbers (i.e., of sets of pairs of real numbers), etc. When theorems about all sets (or the existence of sets in general) are asserted, they can always be interpreted without any difficulty to mean that they hold for sets of integers as well as for sets of sets of integers, etc. (respectively, that there either exist sets of integers, or sets of sets of integers, or... etc., which have the asserted property). This concept of set, however, according to which a set is something obtainable from the integers (or some other well-defined objects) by iterated application of the operation 'set of', not something obtained by dividing the totality of all existing things into two categories, has never led to any antinomy whatsoever; that is, the perfectly 'naive' and uncritical working with this concept of set has so far proved completely self-consistent." quoted here

So perhaps his famed and apparently much misunderstood Incompleteness Theorem is not so worrisome for actual mathematical work, as opposed to Grand Axiomatic Systems of Everything, which were never more than a 19th century utopia anyway. He certainly seemed to think so.

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