Answer by Tom Copeland for When has discrete understanding preceded continuous?
One of Ramanujan's favorite insights on how the discrete informs the continuous is described in my reply to the MO-Q "Ramanujan's Master Formula: A proof and relation to umbral calculus."Surmising the...
View ArticleAnswer by Tom Copeland for When has discrete understanding preceded continuous?
The Riemann zeta function is a prime example.Far into antiquity are investigations of partial sums of integral powers of the natural numbers, leading eventually to the Bernoulli polynomials and the...
View ArticleAnswer by Tom Copeland for When has discrete understanding preceded continuous?
From SKETCHES OF KDV by E. Arbarello:Studying a dynamical system consisting of a finite number of particles of unitary mass distributed along a line segment with forces acting on adjacent pairs, Fermi,...
View ArticleAnswer by ThiKu for When has discrete understanding preceded continuous?
Benjamini-Schramm convergence was originally defined for graphs, later extended to metric spaces with applications especially in the setting of locally symmetric spaces. See...
View ArticleAnswer by SF. for When has discrete understanding preceded continuous?
Pretty much every Analytic Continuation from the integer, or other discrete, into real (or other continuous) domain is an example of this.The Gamma function extending the factorial into real domain is...
View ArticleAnswer by Yly for When has discrete understanding preceded continuous?
Quantum field theory is a very cogent example of contemporary interest: Path integrals and quantum fields on a lattice are easy to define (and lattice quantum chromodynamics is a big industry...
View ArticleAnswer by Sherif F. for When has discrete understanding preceded continuous?
The main line of inquiry in information theory concerns itself with theoretical limits on data compression for communication and storage, as well as theoretical limits on transmission rates for...
View ArticleAnswer by Tom Copeland for When has discrete understanding preceded continuous?
Fredholm integral equations of the first kind can be viewed as continuous analogues of matrix multiplication of a vector. The use of the terminology for matrices, such as the trace and determinant, for...
View ArticleAnswer by user78249 for When has discrete understanding preceded continuous?
I can think of two good examples. The first is rather straight forward. The hyper-operators. Namely,$$a \uparrow^n b : \mathbb{N}^3 \to \mathbb{N}$$$$a \uparrow^0 b = a \cdot b$$$$a \uparrow^n 1 =...
View ArticleAnswer by Lee Mosher for When has discrete understanding preceded continuous?
Group theory might be a good example. The first examples of groups were in a discrete setting, namely Galois groups of number fields, which were first understood as permutation groups of the finite set...
View ArticleAnswer by CoffeeCat for When has discrete understanding preceded continuous?
A recent example would be the generalization of graphs to continuous objects graphons, symmetric measurable functions on a square. Topological properties of spaces of these objects (e.g., compactness)...
View ArticleAnswer by amakelov for When has discrete understanding preceded continuous?
To add to the theoretical computer science angle: the study of graph algorithms has traditionally been done in a combinatorial setting, because that is way more natural at first of course. But recently...
View ArticleAnswer by Eelco Hoogendoorn for When has discrete understanding preceded...
In general you seem to be right. Since the invention of Calculus,"continuous" became easier, and usually is investigated before the discrete. One example of the opposite is the differential equations...
View ArticleAnswer by Alexandre Eremenko for When has discrete understanding preceded...
Another example is coin tossing and other probabilistic models. Coin tossing was studied long before Brownian motion. Discrete probability precedes continuous probability.
View ArticleAnswer by Alexandre Eremenko for When has discrete understanding preceded...
In general you seem to be right. Since the invention of Calculus, "continuous" became easier, and usually is investigated before the discrete. One example of the opposite is the differential equations...
View ArticleAnswer by Tom Copeland for When has discrete understanding preceded continuous?
Integers to rationals to reals.
View ArticleAnswer by Scott Aaronson for When has discrete understanding preceded...
Theoretical computer science offers lots of examples of what you want, since people almost always think first about the discrete setting, even if the continuous setting also turns out to be important....
View ArticleAnswer by Fedor Petrov for When has discrete understanding preceded continuous?
Discrete (more specifically, finite field version) Kakeya conjecture is solved by Zeev Dvir using polynomial method, while original continuous problem is wide open. If you let me speculate on the...
View ArticleAnswer by Steve Huntsman for When has discrete understanding preceded...
Probability and stochastic processes
View ArticleAnswer by Tom Copeland for When has discrete understanding preceded continuous?
Investigations of harmonic motion and heat flow (by Huygens, the Bernoullis, Euler, Fourier, et al.) as discussed in Sections 1.2, 3, and 5 of "The acoustic origins of harmonic analysis" by Olivier...
View ArticleAnswer by Sridhar Ramesh for When has discrete understanding preceded...
Surely summation of finite discrete series was well-understood, conceptually, long before integration. And I would not be surprised to see similarly for many other ideas of the differential/integral...
View ArticleAnswer by John Stillwell for When has discrete understanding preceded...
I would say that a lot of topology was discrete before it was continuous. The Euler characteristic was first observed (in 1752) as an invariant ofpolyhedra. Around 1900 Poincaré first calculated Betti...
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When has discrete understanding preceded continuous?
From my limited perspective, it appears that the understandingof a mathematical phenomenon has usually been achieved,historically, in a continuous settingbefore it was fully explored in a discrete...
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