I'm not questioning the utility of the calculus as a predictive tool but merely its ability to precisely model a physical system. All predictions in physics are probabilistic only, and true only to a finite order of probability, and this is the case on both the subatomic and the cosmological scales as well as on all scales between. The reason for this is to do with the nature of determinism in non-linear dynamic systems, which is what our universe is. Essentially what the calculus is able to do is linearise the non-linear nature of such systems and make very accurate predictions but in an ontological sense such systems can only be modelled by using the mathematical tools of fractal geometry. Note that I do not claim that such modelling would yield more accurate predictions because in a non-linear dynamic system the future is quite literally unknowable, even in principle. Just like the real world, in fact.JSS wrote:They had to have been referring to something specific. Calculus yields perfectly correct answers for demonstratively physical shapes, forms, timings, pressures, and so on. As far as I know, calculus has never been found to yield an incorrect answer. And the reasoning behind it is pretty straightforward and logical.
Do you have an example of exactly what kind of situation they were thinking of wherein calculus would not yield a proper physical model or prediction?
It was these tools which Poincare was seeking to develop in order to model relativity theory when the grim reaper tragically snatched him away from science. However he lived long enough to see Minkowski's abomination and unhesitatingly declared it a load of bollocks.
It wasn't the only issue but it was a very significant one. Almost all the pioneers of early 20th century physics were either Germans or German speakers and the cultural influence of this in the way that physics evolved are not trivial. I reckon having a screw or two loose was a cultural contagion which infected the Germanic world for at least half a century and this was to be reflected in the insidious doctrine of logical positivism which essentially became the underpinning ideology of physics. This doctrine is still alive and well although some of the more enlightened theorists of the modern era are no longer as brainwashed by it as their predecessors. The tide is slowly turning.JSS wrote:I'm pretty sure being German wasn't the issue (Minkowski obviously had a screw or two loose).
He had no such arguments. Einstein's genius always lay in his instincts and never in his physics and certainly not in his mathematics. He eventually became an adequate mathematician but he never formulated his own ideas on relativity mathematically and he was never entirely satisfied with the way in which this had been done by others. In fact he only ever accepted the Minkowski model on sufferance and with deep reservations but he never succeeded in putting his finger on the problem with it. Poincare saw it right away because he was tackling the problem of relativistic gravitational motion from the entirely different angle of the "three-body problem" which had been around and ignored since Newton. This had come to be seen as an irresolvable question but this was only because the right mathematical tools to resolve it had yet to be invented. He had begun to develop such tools but it's fair to say that he hadn't got very far with this work when he unfortunately died when he was at the peak of his powers. However they were further developed by others throughout the 20th century and non-linear dynamics systems theory is now used to model ALL naturally occurring processes except those of physics. It is no coincidence that all the sciences except physics make sense.JSS wrote:And I am not familiar with arguments that Einstein had against calculus...?
Einstein revealed much in his pithy phrases and epithets and we often need to read between the lines to try and figure out what was going on in his remarkable mind. For instance he knew with an absolute certainty that the spooky action at a distance of QM was a complete and adequate proof that Minkowski was WRONG but he never managed to figure out where the fuck-up had occurred. Unfortunately the elephant in the room was hidden within his own GR theory but that's a story for another day.