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We all know that models play a major role in scientific practice. (By "model" here I mean any of various kinds of entities that function as representations or descriptions of real-world phenomena. This includes pictures, diagrams, equations, concrete physical objects, fictional or imaginary systems, etc.) Many models are valuable because they're simpler than their target systems, but they also generate useful intuition, understanding, predictions or explanations about the nature or behavior of those systems.

I'm sure that mathematicians use models in similar ways, for similar reasons. But there's virtually no academic literature (that I'm aware of) about the kinds of models found in pure mathematics, how and why they're used, how modeling practices in math compare to those in the empirical sciences, and so on. (By contrast, philosophers have written a massive amount about models in science.) As a philosopher interested in mathematical practice, this is something I'd like to understand better.

So my question is: What are some cases of mathematicians using models to better understand, predict or explain mathematical phenomena?

A few clarifications about what I'm after:

  1. I'm only asking about models in pure math. That is, the models in question should represent a mathematical object, fact or state of affairs, not an empirical one.
  2. I'm not necessarily or even primarily interested in cases involving model theory. My notion of model is broader and more informal: roughly, any thing M that can be used to give us a better handle on a system of interest S, apart from whether M satisfies some set of sentences in some formal language associated with S.
  3. The models can be (but don't have to be) mathematical objects themselves.
  4. I have no particular preference for elementary vs. sophisticated examples. Happy to see any good clear cases.
  5. It would be nice to see a published source where a mathematician explicitly describes their methods as involving a kind of modeling, but this isn't necessary.
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    $\begingroup$ Possible duplicate of mathoverflow.net/questions/1354/… $\endgroup$ – Iosif Pinelis Dec 15 '19 at 21:02
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    $\begingroup$ 1/2 There is a crucial distinction here: scientists might accept the predictions of a mathematical model as a source of knowledge about the target system, but mathematicians do not. Number theorists can prove that a random variable distributed according to Cramer's random model for primes obeys the Riemann hypothesis a.s. - this helps us gain intuition and understanding, and it's an interesting piece of mathematical knowledge in its own right, but mathematicians do not accept this as providing us with new knowledge about the prime numbers themselves. $\endgroup$ – Paul Siegel Dec 16 '19 at 4:35
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    $\begingroup$ 2/2 In contrast, if a physicist calculates the trajectory of a classical system by (correctly) applying the laws of Newtonian mechanics to it, physicists would likely accept the calculation as knowledge about the system. This is probably the reason why mathematicians don't write as much about models - scientists have to worry about the mix of theory and experiment which suffices to produce knowledge about a system, whereas the only standard of truth that mathematicians accept is formal proof at the level of the target system. $\endgroup$ – Paul Siegel Dec 16 '19 at 4:43
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    $\begingroup$ @marshalcraft I am confused by your comment - what do you think "model theory" means? It has a specific meaning in pure mathematics which does not seem very relevant here $\endgroup$ – Yemon Choi Dec 16 '19 at 14:50
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    $\begingroup$ Surprised no one has mentioned the act of discretizing a continuous problem or the reverse direction of making continuous a discrete problem. Both can be useful at times. $\endgroup$ – Josiah Park Dec 16 '19 at 23:36

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This one is pretty basic: linearization, e.g. replacing a nonlinear dynamic with a linear one allows to study stability and such.

Also, linearization is behind Newton's method and Taylor polynomials give a hierarchy of models.

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I think the most common occurrence of modeling in pure mathematics is the modeling of some specific complicated object that we do not fully understand as being random in some sense. Someone else has mentioned Cramer's model. Another number-theoretic example is the modeling of the zeros of the Riemann zeta function by a Gaussian unitary ensemble. Properties of cryptographic functions are also often heuristically predictable by pretending that they are random functions.

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    $\begingroup$ Cohen-Lenstra heuristics in number theory also follow these examples ("complicated object from number theory behaves, at least broadly speaking, as randomly as possible subject to obvious constraints") $\endgroup$ – Sam Hopkins Dec 16 '19 at 3:12
  • $\begingroup$ Thanks! I actually almost included a "modeling a non-random object with a random object" example in the question, because it was one of the few cases I could think of off the top of my head, but I didn't want to prejudice the answers in any particular direction. Good to know that wasn't just a figment of my imagination. $\endgroup$ – WBD Dec 16 '19 at 17:16
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Cramer’s model for primes. See, e.g. https://terrytao.wordpress.com/tag/cramers-random-model/

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Some of Terry Tao's work on the Navier-Stokes equation is in this spirit. He shows that an appropriately averaged analog of the equation has solutions that blow up in finite time. The analog provides what Tao calls a model of the actual equations, as Netz and Pavlovic did for the Euler equations. Tao's argument shows that many otherwise promising routes to proving regularity of the solutions will not work.

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Diagrammatic calculi provide very useful models in a number of areas, e.g.

  • physics (Feynman diagrams etc),
  • category theory (symmetric monoidal categories and all that),
  • knot theory (Reidemeister moves),
  • quantum groups (Yang-Baxter)
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  • $\begingroup$ Right, yes! Diagrams are maybe the obvious kind of modeling tool in pure math, but still extremely interesting and important. $\endgroup$ – WBD Dec 16 '19 at 17:36
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A prominent contemporary example is finite field models in arithmetic combinatorics (see e.g. the survey http://www.juliawolf.org/research/preprints/ffsurveyweb.pdf) and related areas like harmonic analysis (e.g. the finite field Kakeya problem).

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A couple of random models in Number Theory have been mentioned (Cramer's model, random matrix models), but there are also geometric models/analogies. Here are two:

  • while the goal is typically to prove statements about number fields, people have often attempted to prove things first for function fields, which are in some respects "easier" because you relate your problems to geometry: specific examples are Lafforgue's work the local Langlands correspondence and work of Ngo and others on the fundamental lemma (in the latter case, model theory is used to translate the results for function fields to number fields)

  • Kurt Hensel's invention of the $p$-adic numbers arose from analogies with coordinate rings in algebraic geometry

In general, it has long been the case that algebraic geometry and number theory served as "models" for each other, in that results that hold in one area are expected to have analogues in the other, and consequently many of the techniques developed in one area have inspired techniques in the other area. One book that tries to make these analogies clear is Dino Lorenzini's An Invitation to Arithmetic Geometry.

Of course analogies between different areas or settings in mathematics are widely used. Another which comes to mind immediately is the Lefschetz principle, and then Harish-Chandra's Lefschetz principle, which says that whatever is true for representations of real groups should be true for representations of $p$-adic groups, and this was used as a guide to develop the theory of representations of $p$-adic groups.

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In Operator Theory, modelling linear operators as matrices is one of the most basic tools of the trade. Both at the most basic level (that is, Linear Algebra) and at more advanced levels, like for instance modelling an operator with an invariant subspace as a $2\times 2$ block matrix $$ \begin{bmatrix} A&B\\ 0&C\end{bmatrix} $$ and variations of this.

Kind of unrelated to the above, a huge success in the theory of II$_1$-factors was Voiculescu's use of random matrices to asymptotically model the generators of the free groups within the von Neumann algebras they generate.

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  • $\begingroup$ Nice. Relatedly, I suppose random matrix models for elements of 2-1 factors might fit into this theme? $\endgroup$ – Yemon Choi Dec 16 '19 at 14:52
  • $\begingroup$ Absolutely. I'll mention that. $\endgroup$ – Martin Argerami Dec 16 '19 at 15:21
  • $\begingroup$ Thank you! Somewhat relatedly, on the philosophy end, Alex Paseau has a neat paper about whether matrices are two-dimensional arrays or whether the latter just represent the former. It's amazingly titled "Philosophy of the Matrix." academic.oup.com/philmat/article/25/2/246/2739361 $\endgroup$ – WBD Dec 16 '19 at 17:43
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    $\begingroup$ Well, one of the many steps that mathematics students need to take in the quest to acquire mathematical maturity is to recognize that anything with indices is just a function. That is, a sequence of real numbers is nothing but a function $a:\mathbb N\to\mathbb R$. In the same vein, an $n\times m$ real matrix is nothing but a function $A:\{1,\ldots,n\}\times\{1,\ldots,m\}\to\mathbb R$. And most if not all arguments with matrices can be done without writing rectangular arrays of numbers (although most of the time is the most practical and intuitive approach). $\endgroup$ – Martin Argerami Dec 16 '19 at 17:51
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    $\begingroup$ That was Alex's conclusion as well! $\endgroup$ – WBD Dec 19 '19 at 1:03
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"Internal" models abound in Mathematics, so it is possible that I have misunderstood the question.

  1. $\mathrm{SL}_2$ (or more generally $\mathrm{SL}_n$) as a model study for general semi-simple groups and their representations. Similarly, $\mathrm{GL}$ for the reductive case. Similarly, Grassmannians instead of general projective homogeneous spaces. The theory of roots and weights is used to extend "standard" results from Young tableaux etc to the more general case.

  2. Quadratic and cyclotomic fields as models for general number fields for questions in algebraic number theory. Of course, these are abelian, so sometimes have slightly special properties. Class field theory "lifts" results from these fields to more general fields.

  3. Hypersurfaces as a models for general algebraic varieties. To get even more special, one may look at Euler-Fermat hypersurfaces. This is how Weil was led to his famous conjectures which were later proved by Deligne.

  4. Kahler manifolds as models for smooth projective varieties and vice versa. Similarly for symplectic manifolds.

  5. Classical Fourier Series is a model for the development of Fourier Analysis on locally compact abelian groups.

  6. Classical function and sequence spaces (which are kind of function spaces!) are models for the general theory of Hilbert, Banach and Frechet spaces.

  7. Polynomials are models for the general theory of commutative rings. Matrix rings are models for non-commutative rings. Matrix groups and permutation groups are models for continuous and discrete groups.

One could go on! There are three (at least!) ways in which such models play a role.

  • As examples which motivate students and drive their intuition.

  • As test cases for various hypothesis about general behaviour.

  • As the fallback cases in which to try to prove one's results when the general case appears to be hard.

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    $\begingroup$ To add to this, "representation theory" broadly defined is concerned with understanding how the more general, 'abstract' objects can be realized as the 'concrete' models- e.g., representing an abstract group as a matrix group; or representing an abstract algebraic variety as a concrete variety embedded in projective space (Brill-Noether theory, etc.). $\endgroup$ – Sam Hopkins Dec 16 '19 at 19:59
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I think the idea of strategic players in a game is worth a mention somewhere. Of course, sometimes we mathematically model strategic scenarios in the real world, e.g. game theory or cryptography, but I think games are often used to model mathematical phenomena:

  • Alternating quantifiers are often expressed as a game between players: for any $\epsilon$ that you pick, I can respond with some $\delta$ such that... People often use this strategic terminology to help explain theorem statements and proof techniques.
  • Some optimization problems, especially saddle-point problems $\max_x \min_y f(x,y)$, have natural stories as games, e.g. one player picks $x$ to try to maximize $f$ after which the $y$-player tries to minimize.
  • I think in some areas like combinatorial game theory and information theory, even when original examples were models of the world (e.g. Nim or compressed communication), some research is really on purely mathematical phenomena modeled by the same strategic-play framework (e.g. maybe surreals or CHSH game from quantum communication).

Please feel free to edit with more examples.

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  • $\begingroup$ Ah, nice example! I was aware of some stuff of this sort from logic and model theory (back-and-forth and Ehrenfeucht–Fraïssé games and so on). Interesting to see a case where the model is (or can be construed as) a completely non-mathematical phenomenon. $\endgroup$ – WBD Dec 19 '19 at 0:40
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Perhaps geometric proofs of algebraic relations could be an early example of mathematical modeling. I am thinking of the geometric proof of Pythagoras theorem. We know that Euclidean geometry is equivalent to the algebra of real numbers, but quite often a diagram is helpful to build intuition and even arrive at a proof of an algebraic relation.

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E. Bolker describes his work on the Finite Radon Transform (1987) as originally a deliberate (and successful) effort to gain insight into the continuous namesake(s). Its influence can be seen in the “Bolker condition” of Guillemin & Sternberg (1979).

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  • $\begingroup$ This is great, thanks for the reference! $\endgroup$ – WBD Dec 16 '19 at 17:48
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Probabilistic graphical models use graphs to represent the conditional independence structure of a collection of random variables.

They are very useful for dealing with knowledge and uncertainty in artificial intelligence. The following also address conditional independence modelling:

  • Spohn's theory of natural conditional functions;
  • Zadeh's possibility theory;
  • Dempster-Shafer theory;
  • embedded multivalued dependency.
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The representation theory of quantum groups at roots of unity shares many similarities with the modular representation theory of finite classical groups (see e.g. http://ex.osaka-kyoiku.ac.jp/~fujii/2016/DL/file/conm082-982278.pdf), but is often somewhat simpler (for instance, it is still a "characteristic zero thing") and hence serves as a model of this more complicated theory.

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I am aware of two common practices that might be considered forms of modeling.

  1. Toy problems. Pure mathematicians will often approach a difficult problem by solving some much simpler “toy problem”, which is usually designed to shed insight on the more difficult problem without being as complex.

  2. Particular instances, examples, or evaluations. For example, when forming conjectures about algebraic structures of polynomials, it may be useful to reason about the evaluations of those polynomials on a particular (possibly arbitrary) input.

If you want to get really philosophical, it could be argued that algebra and topology each can be viewed as modeling approaches to problem solving. Algebra asks the question: If we think about this object according to the pattern of interactions of its constituents, what can we say about it? Topology asks the question: If we think about this object according to its shape and the shapes of its constituents, what can we say about it? So from a certain perspective these branches of mathematics could be considered ways of modeling too.

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The following are instances of "models" used in mathematics (probably each one in a different sense of the term "model"):

  • Liouville's theorem describing the local model for symplectic manifolds as $(\mathbb R^{2n},\sum_idx_i\wedge dy_i)$. Also Riemann's uniformization theorem falls under this umbrella conceptually.
  • A variety $X$ over the complex numbers may have an integer model $\mathscr{X}$, i.e. a $\mathbb Z$-scheme such that $X\simeq \mathrm{Spec}(\mathbb C)\times_{\mathrm{Spec}(\mathbb Z)}\mathscr X$.
  • A minimal model in the sense of the minimal model program in birational geometry.
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It occurs to me that Concrete Boolean Algebras are rather straightforward cases, along with the next step of representation employing vectors of 0-1's. That this leads to modeling via computational systems is another turn of the wheel.

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I am not sure if this complies with what you are asking for, but you can visualize many problems in mathematics with informal sketches and graphs in the $\mathbb{R}^2$ plane. I guess our intuition and reasoning tends to work better in two or three dimensions. A classic example would be to draw elements of some abstract vector space as vectors in $\mathbb{R}^2$.

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  • $\begingroup$ Yes, I think this informal stuff is also a kind of modeling. Maybe less fancy but not less useful. I've heard that some mathematicians spend a lot of time drawing potatoes! $\endgroup$ – WBD Dec 19 '19 at 0:50

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