Alex: as Hegel comments on Kant's categorial imperative, order cannot be imposed on where there is no order, what 'ought' to be is no truth if all it is is 'ought', therefore, dialectics must be investigated and proved in different epochs/levels in history and in Nature.
Hear Hear! Absolutely! And this is close to what I understand as the meaning of the "inversion" of Hegel by Marx, methodologically.
Alex: "... to have seen and participated in a "logic of events" and felt its power and necessity........" Exactly. In Hegel's word dialectics is the 'the Road of Despair'.
Alex is quite right, .. that is, so far as the way I presented the issue. Asked to prove the objectivity of the dialectic, I went over to an over-objectivist position. Hegel describes Objectivism as "superstition and slavish fear", and over-emphasis on objectivity is neither Hegelian nor Marxist.
Alex: "Kant: mathematical knowledge is possible because our intuitions of space and time are a priori"
Kant's proposal of a priori space-time intuition was exploded by Einstein and put finally to rest by Vygotsky/Piaget. What was supposed for 2,000 years to be both true and given a priori was shown by Einstein to be a relative truth when compared with empirical measurement and logically flawed. Vygotsky/Piaget showed that while it was legitimate to regard Euclidean space/time conceptions as "intuitive", these intuitions were not a priori but learnt just like one learns one's Mother tongue.
Alex: "H2O is still H2O in solid, liquid or steam form. The heat applied, affect the spacing of the molecules in water, not the molecular structure of water, so I don't understanding why there is a change in 'quality'?"
Exactly, both water and steam are H2O; difference which is also identity is already Opposition. But water and steam are two qualitatively different substances (to say they are "states of water" is not more than to say they are "states of matter"). Each has its own "laws", and there is no "half-vapour-half-liquid" (well, there is, but not H2O, except under extreme pressure). H2O at 99 C is quantitively different from H2O at 98C, ie both are water, just one is more hot than the other. H2O at 100C is NOT water, it is steam. The temperature has incrementally changed in proportion to the specific heat of water, for each quantitative addition of heat. Suddenly there is no quantitative change in temperature, but a qualitative change, water becomes NOT-water. This process is universal. There is no quantitative change in Nature that does not at some point become qualitative change.
Julio: theorems turn out to be, as E. Wigner says, "unreasonably effective in science" and industry!! Why? IMO, they wouldn't be so effective if the axioms and logical rules weren't somewhat objective in content. But then, shouldn't we take the mathematicians' apparent disregard for the objectivity of their axioms at face value? Are the foundations of mathematics objective despite the claims of the most outstanding mathematicians?
I am satisfied by the historical experience that mathematics works best if matheamticians make little or no effort to frame axioms and definitions as analogues of specific objective relations. It is a fact that there is a continual mutual material interaction between mathematics and the sciences where the results of mathematics find application, but this by no means implies a requirement for definitions and axioms to have a physical "meaning" "at birth".
The whole development of science takes sciences away from being sciences of phenomena to sciences of essence and notion, in any case. I mean, where is the "empirical science" in modern physics? Schrödinger's wave equation has a physical meaning, representing a quantum-object as a matrix of wave and particle properties, but there is nothing "empirical" about the wave equation, far less the mathematics that Schrödinger used (he did not invent it, he just applied it, like Einstein applied mathematics invented by Riemann decades earlier).
Truth is a property of an idea that refers to the extent to which it adequately reflects a specific object. Hegel takes great joy in rubbishing formal sciences which begin with arbitrary axioms and definitions, but mathematics does not begin with axioms and definitions any more than any other genuine science. Axioms are the stuff of mathematics, it studies them, like anthropoly sudies peoples and zoology nimals.
I think mathematicians may wonder sometimes about how it doesn't seem to matter how "outrageous" a mathematical-object they "invent", anything that generates "interesting mathematics" sooner or later turns out to be the reflection of something in Nature. I'm inclined to think that this phenomenon is a bit like the concept of "ecological niche" - "if it's got legs, it'll run".
But what is definitely objective about mathematics is derived from the strictness of the requirement to consistently folllow through a set of "rules", whatever those rules may be, even deducing theorems by playing around with the "laws of logic", just so long as the "logic" applied, is applied consistently.
PS: Sorry for getting pushy. I was genuinely worried! And thanks heaps for the responses. I'll be more patient now!