Ever since the first vestiges of calculus were developed, mathematicians have been trying to make logically precise the concept of infinitely small and infinitely large numbers. Our intuitions would go places our logic couldn’t follow, conceptually journeying out into the various orders of infinity and landing back in the finite realm with a correct answer, but we could never say precisely how without being mired in contradictions up to our eyeballs. Finally a logically precise system for talking about these kinds of operations was developed, but it was done by ignoring the way our intuition seems to go about understanding it, and going at it from an entirely different perspective: the limit was born. Limits involve no infinities… instead classifying the behavior of functions arbitrarily near values (or at arbitrarily large ones). While this approach worked, it slaughtered our intuition on the subject of calculus, and nearly all good mathematicians continue to think in terms of infinities and infinitesimals except when they have to nail something down precisely, at which point they set about the cumbersome task of translating their elegant intuitions into cumbersome limit-style statements.
I recently ran across some interesting theory that could change all that. The concept of hyperreal numbers (the set is denoted *R) and hypercomplex numbers (*C) (note that the term “hypercomplex numbers” is also used to refer higher-dimensional sets of numbers of which the complex numbers are a subset, such as quaternions, but that’s not how I’m using it today) is a logically precise and self-consistent formulation of infinitesimals and infinities that allows for the construction of calculus without the use of limits, in what seems a much more intuitive way. The basic concept is to suppose some number ε exists such that for integer n greater than zero, 0 < ε < 1/n. One of the axioms of the construction of the reals prevents this, specifically that any non-empty set with an upper bound must have a least upper bound (well in this case it’s a matter of lower bounds), however using model theory one can prove that assuming the existence of ε does not change the truth of any first order logical statement about the reals. Second order statements on the other hand… are fair game.
The way I think about *R is kind of like base-∞ numbers with real digits, or equivalently, polynomials in ε, though the polynomial model gives somewhat the wrong concept because ε is closer to zero than any real number, and 1/ε is larger than any real number, and thus where a polynomial in a normal variable might be equal to a different polynomial at a specific value for the variable, polynomials in ε are truly separate numbers. Note that these polynomial representations are not always finite in length. For example, ε/(1+ε)=ε-ε²+ε³-…
Now you might be thinking “hold on a second here: ε hasn’t been well enough defined!” In fact it seems the set of values that fit the criteria for ε is infinite. That’s true, but it doesn’t mean ε hasn’t been well enough defined, as it turns out that all possible values of ε are perfectly symmetrical, much as it doesn’t matter which solution to x²=-1 you name i, as there are no absolute distinguishing factors among them, only relative ones.
To illustrate the issues regarding second-order logic, suppose you define an operator a ≈ b which tests if |a-b| is infinitesimal. It is trivial to show that if a ≈ b and b ≈ c then a ≈ c, however if a(n) ≈ a(n+1) for any hyperinteger n then it does not follow that a(0) ≈ a(1/ε). These kinds of things are important to get your mind around in order to avoid making logical mistakes with hyperreal numbers.
The benefits to intuitive thought are really quite amazing. For example, differentiation is as simple as taking st((f(x+ε)-f(x))/ε) (where st(x) is the real number infinitesimally close to x). Integration is accomplished by adding together an infinite number of real values to get an infinite value, then multiplying by an infinitesimal value (or if you prefer, adding together an infinite number of infinitesimals). The dirac delta is a true function on the hyperreals, as opposed to a concept which only exists as a limit of functions, or a measure.
I’m currently considering the kinds of improvements that could be made to differential geometry that could be made through a similar kind of transformation.
concept.mull_over until concept.understand? 