Before Dirac, none considered Instantaneous Action definitionally. Instantaneous action was always a ratio with time, and to avoid infinite values the units of time were changed, uniformity was posited and empirical measurements were used rather than theoretical. A Good deal of common sense was exercised to keep mathematical formalisms from introducing non pragmatic solutions.
That all changed when new philosophers, spoon fed on the belief that mathematics can speak to the attentive about reality in a way no other reasoning can, started to believe their equations and identities and formulations were reality, not mere models of human experience of reality.
When one is brought up to suppose a uniform development of theoretical ideas, that the average is good enough . it becomes difficult to deal with actual data from sensitive measuring tools. Statistical methods were developed and applied for dealing with large volumes of data. consequently no simple direct relationships and formulae could be derived with theoretical hypothesis.
Eventually theoretical considerations became overwhelmed by the massive uncertainty in interpreting the data. The solution was to go probabilistic with the statisitical data. Now np one could be certain about anything!
The rise of computing machines able to cope with masses of data and apply the new probabilisitic methods restored some control to a nervous and jittery scientific community. But the answer was not to go probabilistic, but to go fractal, and to go into Aperiodic behaviours, notably called “chaos”. And the one trick that was missed, explpoted by Dirac was to use the envelope around space filling curves to define the value of whar is inside the envelope. The boundary condition became essential to describe behaviours within a boundary.
The issues that come together to support the modern structure of mathematical physics are wide ranging, but not always apt. De Moivre himself developed probability to a high degree in competition with Strenger, and based on refinements of Cardanos work in combinatorial sequences and structures. De Moivre saw a connection with the sines because producing accurate Sine tables was the major work of the times, both for commercial navigation and astronomical navigation.
Few realise that all the work on polynomials and probabilities ultimately have one root, the unit circle. De Moivre had a considerable advantage due to his mentor(Newton) and the development of the Cotes De Moivre Theorems, which evenso Cotes appreciated the importance of more than he.
Thus ultimately probability is defined on closed conditions, and to extend it to open conditions is an artifice of infinity!
De Moivre was Newton’s student. Thus , like Newton he was Archimedean. He did not accept unending quantities. Thus he expected and utilised approximations , finite quantifications. His argument was simple, but based on Euclids algorithm.,should a process be perisos, that is approximate, exhaustion of the process is acceptable if handled correctly. Certain elements of the procedure can be left off. These parts do not vanish, but combinatorially they are too small to make much difference, so they are simply not combined.
We can then consider the perisos result as the approximate unit for ” measurement, an approximate metron. This unit Is used in all subsequent synthesis, and the shortfall is rounded away. It is the failure to remember that the model is approximate that generates spurious small scale effects!
The calculus of continuous and infinite processes can also generate spurious effects. It is precisely when a ” model” is mistaken for reality when this has it’s most devastating effect.
The rhetorical paraphernalia, or terminology , is often mistaken for procedural combinatorics. Thus he mnemonic value of the notation is mistaken as an evaluation procedure. Many such mnemonics are not evaluatesble. Instead, some jiggerry pokery is done, and an identification is made that is evaluative, and that is used instead of the terminology. Dirac’s function is a simple example of this.
Drac’s delta function.
The algebraisation of astrological combinatorics does not arise , as we are told from generalisation, or going from the particular to the general. It arises from rhetorical style, in which spaciometric forms/ideas and relations are described terminologically, symbolically or translatably into another language, as in a code. Any method is already general, and not a particular instance. When I apply a method of combining forms in space that method is as general as it will ever actually get!
Now the habit of applying numerical mnemonics to these methods is not in fact a habit of giving a particular instance. It is simply reiterating the general relationship using a set of sequenced symbols. This set of sequences whether” numerical” or ” alphabetic” provides inversion, to be sure, but it give illustration to the already general method of combination, and disguises, encodes this method in a format that may or may not represent it. To call its representation a particular is misleading, unless such a reference be fully put as a ” particular encoding” using a ” particular” encoding sequence.
Further, should one” decode “the sequence, there is no meaningful information contained within it, because it is not an encoding of information but an application of a method that is already general.
These combinatorial methods or procedures are called algorithms, and of themselves encode no information. They are instructions in the sense of mnemonics of actual behaviours the recipient is expected to do. As such, they are rhetorical, and may be rewritten in any rhetorical style as art, sculpture, dance, speech etc. In biological systems of procedures they may be “written” as pure sequences of actions, resulting from mechanical/ helical/ electromagnetically interactions.
Thus the use of these rhetorical forms is a programming instructional language which we have now developed extensively into omputer programming languages with the miraculous effect of creating interactive technologies from the elements of our experiential continuum.
Let me no longer confuse ” mathematics” as being anything other thn an ancient computer programming language by which detailed process instructions are conveyed to an operator to perform.
The general method of quantification exposited explicitly by Newton, but implicitly utilised by all natural philosophers especially astrological and mechanical, is to use Spaciometry to represent experiences. Dynamic spatial events are represented by dynamic spatial models, and based upon the dynamic, metronomical response we often call counting. Static spatial events are based on static spaciometries, but the real observable precursor is that these are dynamic equilibria!
Thus nothing is truly, essentially static, all is dynamic and a consequence of an interplay, an interaction of pressures and forces inducing and directing dynamically all motion.
Newton’s Principia acknowledges this, because Newton wrote it in the light of his methods of fluents, a dynamical Spaciometry. Many indeed try to locate a Geometry as a prior Art, but in fact Newton only gives Mechanics as a prior art. Geometry, as Justus Grassmann, Schelling, and others found, was an entirely made up subject, drawing on mechanical principles and attempting to adhere them, unsuccessfully to Euclid’s Stoikeioon.
It bears repeating: Euclids Stoikeioon is not a work of geometry, but an introductory course in Platonic philosophy and the ” Theory of Ideas/Forms” that Plato and Socrates put forward as a metaphysical foundation to all their philosophising.
So, as Herakleitos opined, Panta Rhei, and motion, music, rhythm and dance poetry rhyme,etc are the essential rhetorical styles needed to acquire the wisdom of the Musai. Why mathematics should assume this role is a perverse set of historical circumstances which shall not detain me here( read my blog posts).
Using Spaciometry, dynamic Spaciometry in this way enables one at once to record spatial dynamics graphically. But Leibniz wanted to record it using any kind of symbol, that s algebraically. It was Hermann Grassmann who provided the Rosetta stone to translate between graphical Spaciometry and algebraic Spaciometry. Not many people would now thank him! However it truly is one of the most remarkable analytical methods to date.
Peano recognised this as a young man, and translated Die Ausdehnungslehre 1844 into his own, redacted form in Italian. This lead to an unexpected development. While Prussian society under Gauss retarded Grassmann, the educational reforms similarly hampered any progress with his work. The Prussian empire had a major task on its hands to equip its infrastructure with the new insights and technologies and with homegrown ingenuity! The Grassmann’s felt this was a major priority.
It was only some years later that Robert, Hermann’s brother, prevailed on him to revamp and republish through his own ( Roberts) printing company. As editor Robert infused the new version with his own ideas! (1862). It took Hermann by surprise and elicited some dismay, but it did popularise his ideas and lead to a renaissance of interest in his 1844 self published work.
Although the 2 are named similarly, they represent different ideals. Robert wanted to promote his father’s work through his work and that of the family name. Hermann wrote out of sheer genius and passion, and “tore up the ground!” Consequently Robert knew it was too far out of line to be accepted, and sought to tone it down to a more acceptable mathematical form.
I can recommend reading the 1844 version in German. All translations of it are not faithful to Herrmann, but rather are more swayed by Robert. Hermann’s ideas/forms as Peano realised, are truly radical, and are the basis of Peano space filling curves , n dimensional spaces etc.
The early group and ring theoretical ideas of Justus Grassmann are carried through, but corrected by Herrmann. It is this group theoretical structuring, ring theoretical and field theoretical structure which takes the place of early combinatorial theory in Hermann’s works.
We find David Hilbert, Klein, Cayley all exploring and writing on the same theme. In Britain, A N Whitehead and Russell were influenced heavily, if not converts. The impact of the Grassmann analyses have been far reaching and profound and rival the impact of their contemporary Gauss. What I have come to realise is that the two camps were writing for opposite ends of the Audience: Gauss for Academia, Grassmann for primary efucation. In that regard only Hermann’s work could have made the crossover.