Nonstandard Mathematics @NonstandardMath
Spreading awareness about the so-called nonstandard approach to mathematical analysis. Joined October 2024-
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@Anthony_Bonato Replace "electronic calculators" with "AI models" and it becomes serious very quickly.
It is consistent with the standard treatment of the foundations of arithmetic to assume a nonstandard number in N.
@AnalysisFact They are fractions and also limits of fractions. These do not interfere with each other. Treating them like fractions always, not just sometimes, works, provided you know what you are doing.
@Anthony_Bonato Following Cauchy, a differential is a difference of precisely first degree in the difference in the independent variable, in the value of a function. In particular, it need not, though it can, be infinitesimal.
Mathematicians (especially analysts) choose their symbols very suggestively, and then turn around to say that those symbols don't mean what they seem to mean...
Naturally, we love choice too, given that constructions of our world based on the approach of ultrafilters depends on it. However, there are other approaches, postulational ones too...
A few things in math that are false if the axiom of choice is false: -every vector space has a basis -the Cartesian product of nonempty sets is nonempty -every field has an algebraic closure -every nontrivial ring has a maximal ideal. Without choice, math is kind of broken
This is fictional. Cauchy gave the first well-recognised rigorous definition of continuity, and guess what -- it wasn't in terms of epsilontics; it wasn't even a pointwise definition. Epsilontics came into their own with Weierstrass's work, LATER! So again, this is pure fiction.
Some double entendres...
Contrary to popular belief, Cauchy is _not_ responsible for the epsilon-delta approach to analysis (that is, on an Archimedean continuum). This was basically the work of Weierstrass.
"If the number line is a true continuum, then it is not Archimedean." -Aristotle
Actually n is a hypernatural
I officially declare that n is a natural number
Construct it as the quotient ring of finite hyperrationals by the ideal of infinitesimal hyperrationals 😎
...insofar as u and v have differentials, for example, knowing how to accordingly interpret such expressions when needed depending on the nature of u.
Then all the rules we have discussed so far are valid indiscriminately; we do not for example need to say "Let u, v be functions of an independent variable x." Whatever kind of variables u and v are, we always have d(u + v) = du + dv, ...
The chain rule allows indiscriminate application of the same differentiation rules to variables (without a priori specifying which is independent).
...(primitive) variables and the variables that depend on them with regard to the calculus of differentials, which we now see more clearly exhibited by the derivation of the chain rule.
We see that it is almost trivial, but apart from its use in computation, it is even more important for us because it makes the distinction between independent and dependent variables clearer. Now we have stated emphatically in the past the distinction between independent...
Sometimes, a function y depends on a variable x which in turn depends on an independent variable t. The following rule shows how to differentiate y with respect to t.
Has anyone really seen a real number, for that matter, as opposed to, say, a hyperreal?
Has anyone actually seen a complex number in real life?
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