Why infinitesimals and limits are two aspects of the same thing

[dasnulium]

Next to Nothing - a Single Paradigm: Abstract

I here tackle the most enduring controversy in mathematics, namely the question of what is the correct foundation for calculus. This has been taken to be either infinitesimals or limits at different times in history. I give a novel proof (the nilsquare-limit theorem) that these concepts are two aspects of the same thing – indefinite precision. This contrasts with the prevalent opinion that the two methodologies are incompatible. The infinitesimals considered are nilpotent – a property uncontroversially possessed by infinitesimals before the 20th century. These are also the infinitesimals of smooth infinitesimal analysis (SIA) which I contrast with the more widely known discipline of non-standard analysis (NSA). I argue that these schools are equivalent in effect but that the former is more convenient. I give a graphical demonstration of the proof and a corollary which explicates the old idea of ‘degrees of smallness’; and then use the new perspective offered by the proof to reinterpret the history of calculus, placing particular emphasis on Leibniz’s efforts to justify his notation for calculus and Lagrange’s later efforts to do the same. I mention the ancient antecedents of calculus (the Methods of Exhaustion and of Mechanical Theorems) in context. I then cover the crisis of foundations in mathematics in the late 19th and early 20th centuries with emphasis on the role (or lack thereof) of Cauchy, pathological functions, and the philosophies of Cantor and formalism (also mentioning their antitheses – namely intuitionism and constructivism).

In conclusion I clarify the role of series in calculus, discuss how to reconcile the new paradigm with the law of excluded middle (LEM), and explain the close connection between this approach and finite difference calculus (FDC). In the Continuation I respond to a criticism of Version 1 by explaining how ‘microlinearity’ originally justified calculus, I elaborate on an ‘increment free’ approach to calculus pioneered by Carathéodory, and I address a related issue – namely how the absence of a properly algebraic approach to calculus for most of the 20th century led to widespread confusion about how calculus actually works i.e. I explain Leibniz’s higher derivative notation and discuss a mistaken attempt to reformulate it. I then give an example of the use of differentials in ratios with an illustration. I finally conclude by appealing that the philosophical rift between most mathematicians and the rest of science be remedied. Other topics covered include: the attempts of the formalists to free mathematics from contradiction while also admitting the Axiom of Choice (ref 23); the need for FDC together with a simple numerical example (refs 24 to 26); and, one of the consequences of the period of hegemony enjoyed by formalism – namely the independent rediscovery of various aspects of its antithetical philosophies by various researchers (ref 37).

For full essay see: https://www.keepandshare.com/doc6/37803/nilsquarelimitv2-pdf-348k?da=y

https://vixra.org/abs/1901.0134

Edited December 7, 2021 by dasnulium
Final edit Abstract given

[LaurieAG]

Hi Dasnulium, welcome to Science Forums.

The following links to a free digital copy of "Elementary Calculus: An Infinitesimal Approach", On-line Edition. Copyright © 2000 by H. Jerome Keisler, revised March 2021. I used the first edition at university in 1976.

https://people.math.wisc.edu/~keisler/calc.html

[HallsofIvy]

Before you can say "the type of infinitesmals it uses are critically different from original infinitesmals" you have to say what the definition of "original infinitesmals" was and that was never done!

[dasnulium]

LaurieAG: I checked out that textbook - it takes the non-standard analysis (NSA) approach, which to me doesn't seem very helpful (especially for physics).

HallsofIvy: The post is the Abstract from the full essay - edited to make it clearer. From the Introduction: "Original infinitesimals always had one crucial property missing from those of NSA, namely they were nilpotent i.e. their higher powers were set to zero as they arose in derivations. Since this property is, for our purposes, equivalent to being ‘nilsquare’ I mostly use that term here."

[LaurieAG]

11 hours ago, dasnulium said:

LaurieAG: I checked out that textbook - it takes the non-standard analysis (NSA) approach, which to me doesn't seem very helpful (especially for physics).

I just thought it was interesting because it outlined the structural differences between Improper integrals that converge at infinite limits, Indefinite integrals that don't converge at infinite limits and finally Indefinite integrals that don't converge at infinite limits but as they are structurally cyclic sub functions of a higher level function they can be considered to behave like Improper Integrals.

https://arxiv.org/abs/physics/9807044v2

Quote

Though the general theory of relativity was completed in 1915, there remained unresolved problems. In particular, the principle of local energy conservation was a vexing issue. In the general theory, energy is not conserved locally as it is in classical field theories - Newtonian gravity, electromagnetism, hydrodynamics, etc.. Energy conservation in the general theory
has been perplexing many people for decades. In the early days, Hilbert wrote about this problem as ‘the failure of the energy theorem’. In a correspondence with Klein [3], he asserted that this ‘failure’ is a characteristic feature of the general theory, and that instead of ‘proper energy theorems’ one had ‘improper energy theorems’ in such a theory. This conjecture was clarified, quantified and proved correct by Emmy Noether.

 

[dasnulium]

Interesting paper on Noether. John L Bell talks about the use of smooth infinitesimal analysis (SIA) in General Relativity in the last few pages here: https://publish.uwo.ca/~jbell/invitation to SIA.pdf