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Please excuse my inventing new words in the title. I've been studying algebraic geometry, and one of my favourite parts of the subject (at least, before schemes) is the way that one can describe, or even define, the tangent space at a point $P$ of a variety $V$ purely algebraically, as the (dual space of) $\mathfrak{m}/\mathfrak{m}^2$, where $\mathfrak{m}$ is the unique maximal ideal of the local ring $\mathcal{O}_{V,P}$.

Are there other instances of concepts which are traditionally analytic in nature, but have been given alternative algebraic descriptions?

My search yielded little beyond differential algebra, which seems to be an underdeveloped subject of study, but I would think that if one wanted to "re-imagine" analysis using abstract algebra, this would be a good place to start - taking the interesting algebraic properties of derivative (such as the product rule) and building them into the definition of some new algebraic object. Algebraic analysis is apparently also a subject of study, but information is scarce.

Conversely, are there instances of algebraic concepts being re-cast only in terms of analysis?

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    $\begingroup$ Would the concept of a derivation count? en.wikipedia.org/wiki/Derivation_(differential_algebra) $\endgroup$ Nov 3, 2016 at 20:07
  • $\begingroup$ There are a few (complimentary) ways to treat calculus algebraically; real analysis can be used in the background to, for example, justify the nilsquare rule. I don't know how you missed that debate. $\endgroup$
    – user301988
    Nov 3, 2016 at 20:12
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    $\begingroup$ You didn't invent a word! jstor.org/stable/1970602 $\endgroup$
    – Nefertiti
    Nov 4, 2016 at 9:50

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In commutative algebra and algebraic geometry we have purely algebraic versions of some of the analytical objects such as module derivatives (sections of tangent bundle), module of Kahler differentials (sections of cotangent bundle), differential operators etc.

But the point is to use this objects as tools to study some properties of rings and schemes. For example we can use Kahler differentials to study smooth or etale maps. We need them not because it is possible to define them algebraically, but because they are actually useful.

Another important point is that rings and schemes can be singular, this is a phenomenon that we don't see in elementary differential geometry. For this reason we have (co)tangent complex (not just a (co)tangent space), completions in the definition of de Rham cohomology etc

As for algebraic concepts re-cast in terms of analysis, Koszul connection and its curvature is the first object that comes to mind. It is used in differential geometry, but it is a completely algebraic object and can be defined for any module over a ring or sheaf of modules over a scheme. But this notion is not as useful in algebraic geometry as in differential geometry.

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Plenty of things can be done in algebra. Completion is typically an analytical thing but it can be done through algebra. Let $A$ be some algebraic structure and for all $i\in\Bbb N$ we have that $S_i$ is a substructure such that we can make a quotient $A/S_i$, or if we go more exotic we would deal with congruence relations, such that $S_i\supseteq S_{i+1}$.

Anyhow using this we can deal with cauchy sequences by defining that, using additive notation, a sequence $(x_i)$ is cauchy if for a given $k$ there always exists an $N$ such that for $i,j>N$ we have that $x_i-x_j\in S_k$. Through this we can complete rings, groups, algebras, modules and much else. This is a simple way of getting the $p$-adic numbers. This is also reversable as the sequence of substructures induces a natural metric.

The derivative has been mentioned before is another.

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This might be a bit simple minded (compared to Alex' answer), but:

You can define what a convergent sequence and its limit is by purely lattice-theoretic means, namely by using $\limsup$ and $\liminf$, but defining them via countable suprema and infima (not involving limits).

You can define the value (possibly $\infty$) of a series of a nonnegative sequence $(x_i)$ by lattice-theoretic means via: $$\sum_{i\in \mathbb N} x_i := \bigvee \{\sum_{i\in I} x_i : I\subseteq \mathbb N, |I|<\infty\}$$ (where $\bigvee = \sup$, we are using countable suprema),

This is not enough to explain series in general, but enough for absolutely converging ones:

$$\sum_{i\in \mathbb N} x_i = \sum_{i\in \mathbb N} x_i^+ - \sum_{i\in \mathbb N} x_i^-$$

where $x_i^+$ and $x_i^+$ are the "positive-" and "negative" parts respectively.

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