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 area 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*.

*Andy*

PS: Sorry for getting pushy. I was genuinely worried! And thanks heaps for the responses. I'll be more patient now!