Wikipedia:Reference desk/Archives/Mathematics/2021 July 5

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July 5

In search of a finit Calculus?

I have a question: how do followers of finitism actually define calculus?
There should be a problem with the fact that there is such a small symbol there, right?--82.82.76.143 (talk) 17:02, 5 July 2021 (UTC)

There is no precise definition of finitism in the form of a set of rules of reasoning that are accepted by all mathematicians who consider themselves finitists. The essential step in setting up calculus is the notion of limit. But you also need a notion of a real number. One possible approach is that of constructivism. We can define a real number as a machine that, when presented with a positive rational number ε, will produce two rational numbers a_ε and b_ε such that 0 ≤ b_ε − a_ε ≤ 2ε. Moreover, the machine is such that its guaranteed that, if ε < δ, then a_δ ≤ a_ε ≤ b_ε ≤ b_δ. Equivalently, using interval notation, [a_ε, b_ε] ⊆ [a_δ, b_δ]. The intuition is that the real number is "caught" in intervals that can be shrunk as much as you desire. This is not the whole story; we also need to define a notion of equivalence between such interval machines. For example, the constant machine for which [a_ε, b_ε] = [0, 0] for all ε > 0 is equivalent to one with [a_ε, b_ε] = [−ε, ε]. Anyway, limits can now be defined as (classes of equivalent) interval-producing machines. This is not essentially different from the concept of a Cauchy sequence, but (IMO) more intuitive. Obviously, no machine can be constructed for Chaitin's constant, but a constructivist will probably not agree that such a number exists.  --Lambiam 22:17, 5 July 2021 (UTC)

(edit conflict) N J Wildberger offers a YouTube course on what he calls "algebraic calculus". He's an avid finitist, so I assume this version of calculus follows finitist principles. But keep in mind that a lot of effort went into defining limits, continuity and derivatives without resorting to infinitesimals or "actual" infinities, a project that was completed in the late 19th century. This is turn was driven by criticisms of many people, notably George Berkeley in The Analyst, who found the notion of infinitesimals rather far fetched. Berkely's work, in turn, was a tu quoque attack on criticisms of his earlier work as a Christian apologist. --RDBury (talk) 22:24, 5 July 2021 (UTC)

Calculus as used in practice is fully consistent with having a finitist foundation. Calculus is formulated on a continuum, but in practice we don't deal with the typical functions that exist on the continuum, which are almost all infinitely irregular. Almost all functions are everywhere discontinuous. Of the infinitesimally small fraction of continuous functions, almost all are nowhere differentiable, and so on for higher derivatives. But if you learn calculus, you'll spend a lot of time learning computational techniques that deal with the infinitesimally small subset of all functions that are nicely behaved like continuously differentiable functions, analytic functions, etc. Such functions can in most cases be specified using only a finite amount of information while a typical function requires an infinite amount of information to be specified.

We can then set up a finitistic calculus that's suitable for describing regular function by working on a lattice and then move toward the continuum limit using coarse graining. In the continuum limit you'll then automatically end up with functions that have a cut-off on the small scale fluctuations. The continuum then doesn't exist as a thing in its own right, there then only exists a continuum limit, quite analogous to how infinity doesn't exist but limits to infinity do exist. This is also how Nature works. In quantum field theory one needs to define a path integral, which is an integral over all possible field configurations. But this is an ill defined object. Physicists have been able to find workaround methods that work well in practice but don't really solve the mathematical issues with fundamental indefinability of the object. However, if we take the view that the continuum doesn't exist in the first place and should always be considered as a limit of a lattice, then there is now problem. As pointed out by 't Hooft in page 12 of this book:

"Often, authors forget to mention the first, very important, step in this logical procedure: replace the classical field theory one wishes to quantize by a strictly finite theory. Assuming that physical structures smaller than a certain size will not be important forour considerations, we replace the continuum of three-dimensional space by a discrete butdense lattice of points. In the differential equations, we replace all derivatives ∂/∂xi by finite ratios of differences: ∆/∆xi where ∆φ stands for φ(x + ∆x) − φ(x) . In Fourier space, this means that wave numbers k are limited to a finite range (the Brillouin zone), so that integrations over k can never diverge. If this lattice is sufficiently dense, the solutions we are interested in will hardly depend on the details of this lattice, and so, the classical system will resume Lorentz invariance and the speed of light will be the practical limit for the velocity of perturbances. If necessary, we can also impose periodic boundary conditions in 3-space, and in that case our system is completely finite. Finite systems of this sort allow for ‘quantization’ in the old-fashioned sense: replace the Poisson brackets by commutators."

Count Iblis (talk) 13:03, 6 July 2021 (UTC)