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+#+TITLE: Two Millenia Pretending There's Nothing Between Points and Lines
+#+DATE: <2023-12-12 Tue 03:46>
+#+TAGS: Mathematics, Infinitesimals, Geometry, Analysis, Philosophy
+
+Euclid said "there are points (without direction or magnitude), quantites (without direction, but with magnitude) and line segments (with direction and magnitude)" [fn:1]. A basic contemplation on cases suggests an omission: what might have direction but no magnitude?
+
+They're called infinitesimals. And mathematicians have been pretending to not believe in them for 2000 years.
+
+* Nilsquares, Rigid Rods, and Microstraightness (Macrogayness?)
+
+Calculus, fundamentally, is about continuous change. And the very nature of "continuous" itself has always been turgid: only the hallucinogenic stimulant ZFC has been able to, very recently, convince people that continua consist of individual points.
+
+Arguably, the seeds of calculus lie in the work of the ancient Greek mathematicians Archimedes, Antiphon, and Bryson. They, following Aristotle's deflation of Zeno's various paradoxes of motion, viewed continua as successions of overlapping instants, each of which was /not/ a dimensionless quantity but retained the central /directionality/ of the continuum as a whole. Archimedes routinely viewed volumes as sums of (lots of) /indivisibles/, plane-like sections, in order to derive expressions for those volumes in terms of linear dimensions. The other two viewed curves as sums of (lots of) discrete, short segments0—rigid rods just long enough to have a slope but no appreciable length.
+
+Newton viewed time-dependent quantities (fluents) as having rates of change (fluxions) at each instant computable by evaluating the quantity perturbed by an infinitely small quantity and ignoring "doubly infinitely small" powers of that quantity in the result. Leibniz similarly employed infinitesimally small quantities, but in a more geometric context TODO: Leibniz's visual proof of the FTC
+
+ Euler's textbook on calculus, which is remarkably similar in order and content to modern presentations, has the following "proof" of Newton's nilsquare property of $dx$, quite humorous to the modern eye [fn:2]: algebra tells us that if we can divide by zero then $\frac{0}{0} = a$ for any $a ∈ \mathbb{R}$. If we let $dx = 0$, then this rearranges to $a \cdot dx = 0$; he concieved of this as two different ways of writing zero save for the fact one is permitted to apply all usual rules of algebra to these terms, and so $\frac{dx}{dx} = 1$. This allowed him to say
+ $$1 = 1 + 0 = 1 + dx = \frac{dx + (dx)^2}{dx} \Leftrightarrow dx = dx + (dx)^2 \Leftrightarrow (dx)^2 = 0.$$
+
+ Weirstrass's limit approach to analysis, and the set-theoretic constructions of the continuum courtesy Cauchy and Dedekind, claim to save us from the absurdities apparent here (some of which Berkeley had pointed out contemporary to Newton). However, these approaches still pervade pedagogy and application. Anyone taking calculus will learn the "washer method," useful in computing volumes. Every calculus book has a picture of a little tangent line segment, with $dx$ its $x$ component and $dy$ its $y$ component. Everyone thinks in infinitesimals before limits. And anyone with knowledge of physics, even from memes, will know that infinitesimal derivations are the rule rather than the exception! A course in classical mechanics wouldn't be complete unless the professor a) doesn't explain what a variation is, b) assumes you know properties like $(\delta q)^2 = 0$ and the multivariable chain rule for variations anyway, and c) invokes the fundamental lemma of the calculus of variations as if it's the most unobjectionable of axioms. In statistical mechanics, the basic principles relating state variables are still stated in infinitesimal form. And for engineers, limits have long since exited the mind; it's $dx$ (secretly, $\Delta x$) all the way down .
+
+ What we observe is a vast rift between theory and practice, between formalism and intuition. And that should strike any mathematician as an opportunity.
+
+* Abstract Nonsense to the Rescue
+
+Two approaches evolved, roughly coeval, to close this gap. The first is a logical method, which views infinitesimals as reciprocals of transfinite cardinals in a field of "hyperreal" numbers; it is provably syntax sugar for limits [fn:3]. It requires such terrifying phrases as "nonstandard models of Peano arithmetic" and "ultrafilter" to understand and apply, but successfully resurrects infinitesimals, and was used to prove some longstanding problems in analysis [fn:4]. The second fell from the clouds of category theory, is much more convenient to apply, has nice geometric semantics, explains why classical infinitesimal theories ran into problems, and also solves an analogous formalism-intuition gap in differential geometry.
+
+Smooth infinitesimal analysis, this new hotness is named. And the rules are straightforward to state [fn:5]:
+
+1. Reasoning about continua is constructive.
+2. We have a continuum $R$ forming a field (the elements are thought of as (possibly infinitesimal) segments on the continuum).
+3. There is a strict order $<$ on elements of $R$ that is compatible with $+$ and $\cdot$; all elements of $R$ are strictly between 0 and 1.
+4. Square roots are computable.
+5. (Microstraightness) For any function $C$ and any point $P$ on it, there is a (small) nondegenerate segment of $C$—a microsegment—around $P$ which is straight, that is, $C$ is /microstraight/ around $P$.
+6. (Integration) For any
+
+
+
+* Footnotes
+
+[fn:5] Extracted from  John Bell's /Primer of Smooth Infinitesimal analysis/.
+[fn:4] Bernstein and Robinson 1966 made a ginormous leap on the invariant subspace problem, proving it for the case of polynomially compact operators.
+[fn:3] Robinson 1960.
+[fn:2] This was introduced to me by Michael Penn, who cites A. Ferzola 1994.
+[fn:1] Not a quote; I made this up. But he basically said that.