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Because these are insights, I’m leaving off some important mathematical discoveries, like the properties of logarithms by Napier, and focus squarely on those things that upended people’s ways of thinking about things.Another example would be how Fibonacci gave us a lot of great works, and he also brought the Algorism and advocated for standard positional-notation to the Western world, but his work didn’t outright change how people thought of things as much as he helped solve many existing problems throw relatively existent means.My own unscientific, informal, subjective measuring stick that has no bearing on how often the result is used:⋆ - A very necessary result that made people go “OMG”⋆⋆ - Blew people’s minds⋆⋆⋆ - Led people to reconsider previous philosophically held beliefs⋆⋆⋆⋆ - Created new mathematical fields⋆⋆⋆⋆⋆ - Resulted in absolute crisis, joining many mathematicians togetherIrrational Numbers (⋆⋆⋆)Pythagoras-Hippasus (~500 BC)The Greeks (and many others) were attempting to formalize mathematics through the Laws of Logic. They thought they understood what a number was.These Greeks were very interested in ratios and believed that they could define a number as any ratio of two values, or as a fraction of a numerator and denominator:[math]\frac{n}{d}[/math]Then, they were absolutely dumbstruck by the realization that [math]\sqrt{2}[/math] just couldn’t be put into such a fraction. There were so many records of what are called Pythagorean Triples at that time, and there was a belief that anything can eventually be made as a ratio.This was the beginning of philosopher-mathematician’s rethinking what numbers actually are.Formal Logic’s Notion of Law of the Excluded Middle (⋆⋆)Aristotle, et al (~350 BC)Also known as the “Mathematics = Applied Logic” premise.I cannot give Aristotle full credit for formal logic. Formal Logic is seen in prehistory of China, of India, and the Maya and predates history.The only reason I place his name on this is because his formalization of The Law of the Excluded Middle in “Of Interpretations”, had so much influence, and continues to have so much influence, on mathematical logic.It formally states that any statement can only either be true or false. It’s important to note that this fits with much of mathematical thinking, but it is not the full story, and there are numerous other logical systems that reject the Law of the Excluded Middle.On its face, his compiled works, which collected a lot of what was known around the world, as connected by the mercantile exchange, gave us a formalism, which are the beginnings of placing mathematics in an axiomatic format.Axiomatic Geometry (⋆⋆)Euclid (~300 BC)Speaking of someone collecting the worlds works together, Euclid’s work placed geometry on a formal background.It also blew our minds trying to understand why the Parallel Axiom was necessary to begin with. It wasn’t understood why the Parallel Axiom had to exist for things to work. People tried for millennia to prove that it came about because of everything else he proposed, and they never could.Formal Arithmetic with Negative Numbers (⋆⋆⋆⋆)Predates Liu Hui (~200 AD)The idea of having numbers that cannot be “counted” was an anathema to early philosopher-mathematicians who could not understand how negative numbers related to any concept of “reality” as they saw it.The idea of using “direction” in computation was the key understanding, and this was one of the first realizations that “numbers encode” that which we observe. The concept of “encoding” ideas still didn’t catch on fully for millennia, and held philosopher-mathematicians back as they suffered from attempts to place numbers into a special class of “real things”.The Distinct Symbol for Zero (⋆⋆)Aryabhata (~500 AD)I will add here that the Hindu numeral system was already available here, and so were positional notations. I add that because it becomes important later.The importance of having the distinct symbol for zero was the ability to start doing mathematics with it as well, handling those cases where zero was used.One problem here, is that we cannot properly answer the question of whether 0 is positive or negative. It is either both or neither, depending on your culture.By itself, this would have been a great moment to start questioning when the “Law of the Excluded Middle” can be used.Axiomatic Treatment of Algebra (⋆⋆⋆)Muhammad ibn Musa al-Khwarizmi (~800 AD)Al-Khwarizmi gave us formal algebra, and was an advocate for the algorism, using zeroes as placeholders and bringing the place-value system to us.I cannot give him exact credit for either the introduction of the place-value algorism (that would have come from India) we use today nor for full treatment of formal algebra (coming from all over), but he too was a collector bringing them together and provided the idea of “a set of rules for computational procedure”.In other words, it formalized the separation of the answer from the method, such that multiple methods could arrive at the same answer.Equal Sign (⋆⋆⋆)Robert Recorde (1557)The creation of our modern notation for equality, addition, multiplication, and roots; however, mostly he gets credit for the equal sign.This is by far the biggest example of notation having an affect on how people think. The equal sign provided numerous ways of looking at things like:DefinitionReplacement (substitution)IdentityGeometry is Algebra (⋆⋆⋆⋆)Descartes-Fermat (~1600)Formally Analytic Geometry states that Geometry is algebra, taking place on an affine plane.Informally: Algebra and geometry are basically the sameWe get 1D, 2D, and 3D representations of geometric figures on an origin, allowing us to work through any geometric object that we can create a formula for.This is a huge realization that continues to this day to give us computer graphics, etc.It provided an idea for “points” (ideas which retained a notion of ‘origin’) and “vectors” (ideas which ignored the notion of ‘origin’). This made us dig deeper into “affine” and “vector” spaces.Formal Treatment of Infinite Processes (⋆)Jacob Bernoulli (~1700) [Note: there are many Bernoullis]Mathematical Induction, the process of Proof by Infinite Descent, was developed by Bernoulli. This took nearly 160 years to arrive at the fruition of Peano-Dedekind, providing proofs for WHY arithmetic is commutative and associative, etc.Formal Treatment of Change and Accumulation (⋆⋆)Newton-Leibniz (~1700)This realization was that we can mathematically talk about infinities, infinitesimals, and changes. Suddenly, this opened up the entire world of physics (especially dynamics).Consider that this is basically like introducing a new language, wherein the ability to talk about these things was muted before this introduction.One additional thing it gave us was the idea of a transformation (the derivative) which uses a formula as an input and a formula as an output. This could be said to be “function input” and “function output”. This transformation is also function-like itself.It also pointed us deeper into understanding the difference between affine spaces and vector spaces, since derivatives and integrals related particular affine and vector spaces to each other.Complex Numbers are Geometric (⋆⋆⋆)Bernoulli-Cotes-Euler (~1700)That is, the discovery of what we call Euler’s Formula:[math]e^{ix}=\cos z + i \sin z[/math]Pretty much combined:TrigonometryComplex NumbersCalculusIn one short-and-sweet formula. It made us realize the importance of Complex Numbers and gave us definitions for sin and cos that made all of the trigonometric identities come about.The 2nd big realization is that if these ideas are combined, what we know as Linear Algebra (to be developed later) has an intimate relationship with Infinitesimal Calculus.One more note here about this equation is how it demonstrates that there are complex functions that are periodic, and their inverse would therefore have numerous solutions. This actually provides a segway into modern topology.Mathematical Axioms are not “Self-Evident Truths” (⋆⋆⋆⋆⋆)Gauss-Schweikart (~1815)In other words, the “Mathematics = Rules of a Games and the Study of the Possible Plays”With the discovery of Non-Euclidean Geometry (the realization that we can deny Euclid’s Parallel Axiom and get a consistent system), we obtained spherical and hyperbolic geometries.However, the idea, in general, that we can just deny a particular axiom, and get new results (assuming the new system is consistent), rocked the world of mathematics, and continues to do so.This created the world of “structuralism” wherein mathematical structures were defined by their axioms, instead of some absolutist notion of “perfect mathematics” that was a kind of holdover from Platonism.This meant: new axioms gives us new mathematical structures. Hence: structuralism.It is also at this time, due to the works of Newton on Physics and the structuralism view, and the continual advances of technology, the philosophical era of Philosophical Modernism can be said to have “hit its stride”, promoting the myth of “endless progress”, and the beginning to topple the usage of “absolutism” in every corner of western philosophy.There are Many Equivalent Representations of Functions (⋆⋆)Fourier-Taylor-Maclaurin (1822)I give Taylor and Maclaurin credit, but Fourier gets the top rung.Taylor and Maclaurin demonstrated that any analytic function can be seen as an infinite-degree polynomial (a polynomial series). Furthermore, as it is refined further, even some jumps can be described via some limiting process, and we end up with generalizations of functions called “distributions”.Fourier’s realization is that, using Euler’s Formula, infinite sin and cos terms can also be used.He used this to come up with solutions of the evolution of the Heat Equation.2020–07–07: I don’t think that I expressed the importance of this to our current world. All movies, images, and music are encoded using the mathematics that Fourier defined here, and in fact, the Fast Fourier Transform and other fast transforms are used all over in signal processing, control, and compression.Polygons are Polynomials (⋆⋆)Galois (~1830)Galois TheoryThere is so much in Galois Theory, giving us notions of permutations as groups, fields (field extensions), quotient groups, etc.The idea that we can take a axiomatic system, such as that of the Rational Numbers, and just embed a [math]\sqrt{2}[/math] and make a new system is pretty much the epitome of an 1800s mathematician making use of the idea of structuralism.Furthermore, the definition of groups and fields, which are objects that are ubiquitous in mathematics just kept growing into monoids, magmas, rings, etc.We took Galois’ work and kept going with it, extending and contracting axiomatic structures as we went.All this by laying the foundations for what would soon be understood as “ways to rotate, flip, and otherwise transform polygons, such that they ended up the same”.With Abel and Ruffini, it demonstrates why we have some quintics that require exact solutions that require more than just:AdditionSubtractionMultiplicationDivisionRootsSuch as why trigonometry may be useful instead, and why some numbers are “Not Constructible”.2020–07–07: I want to add that Galois’ formal treatment of permutations and Group Theory is the precise reason that we can encrypt and decrypt messages over the internet. Effectively, privacy wouldn’t be possible without Galois’ work.A Formal Treatment of Hierarchies (⋆)Boole-Peirce-Dedekind (~1850)Why not just create yet another field of mathematics? Galois did it, why don’t we?The idea that we could formally discuss hierarchies and place them in relation to logic and create an field-like object (an algebra) from them was very Galois-like at the time.Formally, the construction of Classical Logic, called Boolean Logic, came out of this too, but formally, it was demonstrated to be an Order Relation.What does implication mean? If A implies B, then it means that A is at most as true as B.This was the beginning of the treatment of GCD and LCM as an algebra as well, and so it kind of birthed new thoughts on Number Theory as an algebraic process.The bigger insight was to take all of this and realize how closely relations (such as arrows on a graph) are.2020–07–07: Computers, as we know it, wouldn’t exist without this work. This work led Babbage and Ada to what would eventually become binary logic. Babbage recognized the importance of calculation in computing, and Ada recognized how to use it in a more set-like manner, and also recognized the value of “state transitions” to computers compute things more generally.A Formal Treatment of Linear Transformations (and Groups = Matrices) (⋆)Sylvester-Cayley (1850)Arrays were in use before, but Sylvester and Cayley came up with the algebraic treatment of matrices.Furthermore, Cayley shows us that matrices represent what Galois was doing with his permutations. This result, Cayley’s Theorem, is also ubiquitous in mathematics, but often in not-so-obvious ways. It basically tells us that, if you can make it into a finite group, you can make it into an algebra using matrices.2020–07–07: Computer graphics wouldn’t exist without Cayley’s treatment of matrices.Just about everything can be proven from set-based logic (⋆⋆⋆⋆⋆)Cantor-Frege-Russell (1874)The more consistent definition of sets would come from Russell, but taking Peano’s work, and continuing to move models, graph-relations, numbers, etc. all into sets, we end up with full-on constructivism.Constructivism tells us, “don’t just prove it exists, make one”.It, alone with intuitionism, was the end of the use of the Law of the Excluded Middle as the driving factor for mathematicians. Now, you’d think many mathematicians came from Missouri, the “Show Me” state.The importance was that everything mathematical was now in a formal logical framework, if we called Set Theory a Logic System. We could construct just about anything we wanted from it.There is a special shout-out to Kronecker, approximately at this time, who was so skeptical and rejected things that had to do with the infinite. This came, in contrast to Cantor, who demonstrated that there must be at least 2 types of infinity (what we now call countable and non-countable).This leads us to extreme constructivism (called “finitism”) and questions about the Axiom of Choice that Russell comes up with.Constructivism is important for Computer Programmers who cannot simply answer “yes” it exists as much as they must find the result and provide it. Likewise, this finitism is also important because it is ability feasibility, or the ability to provide such an answer, within a reasonable (finite or ultrafinite) amount of time.It’s less a philosophy of belief anymore as much as a philosophy of feasibility and efficiency. Likewise, ultrafinitism with constructivism means you get secure computers (that require work to break, but more work than a computer can do, unless it already knows part of the answer).Furthermore, this led to the formal definition of probabilities and led to the initial work of creating statistics via Linear Algebraic means.Introduction of Manifolds (⋆)Poincare-Reimann-Weyl (1912)So, you want to talk about doing mathematics on oddly shaped objects?Specifically, this gave rise to very important things, such as phase space used in industrial control and automation, but also to physics, by creating objects as geometric spaces on which to do math. This further generalizes Non-Euclidean Geometry, which allowed you do to math on the geometry of a sphere, or on the geometry of a hyperbola.2020–07–07: You may think that this only matters string theorists doing physics on 11-dimensional manifolds, but you’d be wrong. Typically, modern robotics also work in a configuration-space manifold of 6 or more dimensions. Surprisingly, manifolds and knots also connect to prime numbers, to error correction, to quantum computing, and even to statistics.Mathematical Symmetry = Physical Conservation Law (⋆⋆)Noether (1918)Probably the biggest result in applied mathematics, which demonstrated that physical conservation laws are derivable via the mathematical relations between phenomena.Although it can be formulated via Perturbation Theory (stochastic dynamical processes), or via Fields (see Galois), the idea of using it on a topological manifold made the physics usage very easy to spot.Recursiveness Presents Problems (⋆⋆⋆)Gödel-Russell-Turing (1931)Basically, the Barber Paradox shows up everywhere when you start doing this recursively (or in a “nested” manner).Russell’s result was to say, “don’t do that” (at least not with sets). However, Turing and Gödel knew that wasn’t always possible.Gödel finally gives us the full description of the problem in a way that tells us when our systems will be inconsistent.Basically, it says, pick 3:Effective / Decidable / Computable - we can get a resultRecursive (self-referential) - a concept is defined using itselfComplete - you can prove or disprove any statementConsistent - you cannot both prove and disprove the same statementPhilosophically, Gödel’s Theorem and Russell’s treatment of universes tells us an important fact: “Mathematics is inexhaustible” and mathematicians will therefore never be without a job. We knew this beforehand, but these ideas seemed to cement it.It does not imply that “mathematics is broken”. It actually, more specifically, tells us when it works best, and how to get there.2020–07–07: I don’t know how I missed this entire section here.Communication can be Mathematical (⋆⋆⋆⋆)Claude Shannon (1948)After formal treatments of sets and probabilities were made available, Claude Shannon made his historic paper “A Mathematical Theory of Information” and basically shocked the mathematical world by:Formally defining redundancy in informationFormally defining entropy, from thermodynamics, in terms of informationPractically dealing with the concept of noiseDefined “the bit” as an informational unit of measure, not just a binary digitThis allows us to compress things that cannot lose information (lossless compression). This is yet another event which led the way to giving us the Internet, as we now know it.This also led the way to the following:Hash codes, which summarize information into a small comparable tokenThe possibility of lossless compressionError correctionStalemates can be predicted (⋆⋆⋆⋆)Cournot-von Neumann-Nash (1950)Note: we aren’t talking about chessSets, probability, and now information are formalized, and there is a tight connection between communication and what mathematicians call “Games”. Nash equilibrium helped usher in Game Theory as another entire field of mathematical inquiry, and the advances of which connect to the lessons from Shannon. The idea that stochastic processes can end up with predictable equilibria, and that there are ways to predict them, suddenly gave us formal ways to:discuss evolutiondiscuss economicsdiscuss politicsdiscuss people’s behavior in linesMathematical Proofs are Computer Programs (⋆⋆⋆⋆⋆)Church-Turing-Curry-Howard (1969)First, we saw a formal description of what functions are computable mathematically (Church) and mechanically (Turing) and show that they were equivalent (Church-Turing Thesis. 1952) with help from Kleene.Then, we saw a formal correspondence between proofs that were mathematical in nature, and software “types”.In the end, it kind of fell to the realization that Modus Ponens (in logic) has the analogy to function application, comparing:[math]a \implies b, a \vdash b[/math][math]f\ a = b, a \vdash b[/math]Yes, Virginia, there are Fractional Dimensions (⋆⋆⋆⋆)Julia-Mandelbrot (1982)Technically, it’s not just the fractal dimensions that were so influential, but also the ability to discuss self-similarity (self-replication) in spaces at scales in which we’d not discussed them before.Suddenly, these things that showed everywhere in nature had some more formal descriptions for their patterns, and so the field of topology really deepened the understanding of what a dimension really is.An Object is Determined by its Relation to other Objects (⋆)Yoneda (1996)So, Yoneda’s Lemma takes Cayley’s Theorem about matrices and groups to an extreme generalization and gives us a distinct picture about how objects relate to their relationships.Software and Mathematics is Extraordinarily Linked (⋆)Brouwer-Heyting-Kolmogorov-Lambek-Voevodsky (1928–2012)Let’s see now:Logic is a CategoryCategories are Partially Ordered SetsSoftware Types are a kind of LogicSoftware Types are Topological (specifically, a Topos)Furthermore, ideas such as equivalence of structures (having an isomorphism between them) and analogy are now even more formally defined than they were.2020–07–07: Adding an additional linkage (or being more clear about the contributions of Voevodsky), but one that is a bit more difficult to explain. Starting with recognizing that the logic of Type Theory, as a formal system, separates things into terms, types, and contexts, whereas a type acts as a presheaf on a context where the morphisms are rewrites of equal terms, then we can use presheaf operations to extend the capability of types.What this means: This complicated viewpoint, which requires a LOT of fundamentals in topology, basically enables compilers the option to rewrite equivalent expressions eventually enabling compilers to optimize to the most efficient mathematically equivalent expressions.

Thank you for asking, Jeremiah. I frankly feel quite uncomfortable because here on Quora and elsewhere there are people who are way smarter than me, who are way more educated, who went farther, looked deeper and who are in a much better academic shape.My personal preferences on the essential contents of a mathematical library are just that - a double helix of my tastes and my academic upbringing which, with respect to those who were put through their paces by the system that existed at the time, was nothing special (to us) but which, with respect to most if not the rest of the world, was rather special.Necessity is the mother of invention - with the Iron Curtain firmly in its place, the Soviet government new all too well that grow-your-own is the only viable academic option. The buy-from-outside alternative was off the table. Obviously.Thus, for not even a blink of an eye on a historical time scale between the years of, say, 1917 and, say, 1997 for about 80 years there existed an idealized and an idealistic academic system whose financial burden was shouldered by the entire working population because while silently collecting the effective rate of about 80% in the income tax, it was the central government that unilaterally redirected the flow of the coin, as it saw fit, toward the academic endeavours.I, for example, was paid a stipend (steeh-pehn-deeh-yah) to study in college from such funds, after passing the series of written, 5-hour 4-problem, and verbal mathematics and physics entrance exams.Three tiers of stipends were available - base, for just cutting the mustard, elevated, for doing well, and top, for doing very well.The size of my stipend was very reasonable for mid 1980-ies Moscow - one could squeeze by on it autonomously, but personally I never cared for the coin. I do not even remember what I did with it - I think that I gave it all back to my parents.The two-year evening physics-and-mathematic-only prep school that I attended before the college was free of direct charge. All I had to do is show up.The contemporary wisdom was that the sole purpose of an educational system is to educate - not make money off of. In such a system the upper bound on what you can learn is not your coin but you.While the Soviet government did, somewhat liberally, sprinkle the material perks onto the academicians, judging by the direct interaction with my maternal grandmother and my paternal aunt’s husband, the Moscow Institute of Physics and Technology graduate, Yuliy Mikhailovich Gershenzon, god bless his soul, these people did not care much for it - to onlookers they were the naive fools, the idealists whose hearts and minds ignored the silly minutiae of the clothes they wore and the cars they drove. Moving the scientific needle was the only thing that mattered.Falling asleep, I remember my grandma on many an evening still writing something busily at her table well into the night, surrounded by the stacks of papers covered with mysterious symbols.Yuliy Mikhailovich would sometimes lock himself in his room and, as my aunt used to lovingly put it, solve his problems all day long, sigh. Some people here in US may actually know Yuliy Mikhailovich through his collaboration with certain local academic institutions.Alas, because it was too idealistic and because our primitive world just does not work that way, that system was not to be. However, being brought up by the people who choose to do something only because they love doing it and because at no point does the concern for the coin enters the equation leaves its imprint. I guess.My, as the saying of the time went, theoretical minimum of a mathematical library would consist of two distinct parts:the works of the capable individuals who came before us andthe digested and the sanitized versions of the same wrapped and distilled into the format of a modern academic textbookIn the first bookcase you would find the works of:Euclid, Archimedes and NicomachusNewton and HuygensLaplace, Lagrange, Legendre and CauchyWeierstrass, Dedekind and RiemannFermat, Fourier, Gauss, EulerHilbert, Tarski, Godel, von Neumannand the likes - you get the idea.While all the names above should be familiar to the readers, Alfred Tarski has a nice text “Introduction to Logic and to the Methodology of the Deductive Sciences” that he put together sometime in the late 1930-ies or early 1940-ies. The first edition of that book was published in the USSR in Russian in the late 1940-ies but the Western audience likely saw it earlier than that.Though some of these works are tilted towards physics, they clearly show how new mathematics was born.The motivation here, among other things, is to 1) avoid costly mistakes, 2) practice, 3) prepare.1) My advice to a budding mathematician - read wide, very wide.If it is not evident from my answers, I like to tinker with the basics. I like to create things from scratch. Develop them into something that can stand on its own. To that end - a short story. Others, surely, lived through the same or a similar experience.While our standard first-semester (1986) real analysis curriculum churns through and hovers over the Riemann integral, an idea enters Roman’s mind that there has to be a more abstract and a more generalized way. Foolishly, without talking to anyone, Roman works feverishly, head-over-heels style for some months, mostly at nights due to his busy academic schedule. After Roman thinks that he has got something, he shows his work to his analysis professor to which she replies that, in its gist, give or take, this Dutch gentleman by the name of Stieltjes (1856–1894) beat Roman to it by only 92 years or so (in case the modern curricula do not cover it - here is the Stieltjes integral).I hope that you are laughing at this as much as I did but the point is - before you decide to develop something in mathematics do your homework and scan the relevant surroundings thoroughly. Or ask someone whose opinion you respect. Life is too short. There is absolutely no point in duplicating someone else’s effort.And I can not even begin to imagine how modern students do their work - the width and the depth of mathematics being what it humongous is nowadays.My guess would be that the analysis professors world over probably witness the reinvention of the Stieltjes integral or its variants on regular bases, :O)2) Aside from checking for duplicates, there is another important aspect to reading the first-hand and not-sanitized-yet accounts: catching the whiff of the birth, the genesis of a germ of a new idea is very exciting.Once that newness is predigested, it is usually enough for a motivated person to run with it for a while on her/his own. It is a dress rehearsal of the aaaaaa moment. The relevant notation and apparatus to follow.3) Reading wide early on can only help to and may prepare you to taking on such abstract disciplines as algebra and the theory of groups later on.In the second bookcase you will find all the usual suspects - multiple copies of the textbooks on:the theory ofnumbersgraphsprobabilitygroupsgamesinformationanalysisrealcomplexfunctionalcombinatorialFouriernumericalgeometriesEuclidean and nonanalyticdifferentialprojectivelinear algebradifferential and integral equationsmathematical logicnumerical methods(the equations of) mathematical physicsA sample supplemental book on Numerical Methods could be (the Big Red) “Numerical Recipes in C” by Press, Teukolsky, Vetterling and Flannery.I hope that I do not sound like a pompous buffoon but the expectation is that you should give all or most of the above an honest once-over at a reasonably deep introductory level by the time you are 23 to 25 years of age.Right around that age your specialization will begin to, more for some or less for others, crystallize which, eventually, will lead to the increase in the depth of study which, eventually, will lead to the narrowing of your horizon.On a separate shelf there will, of course, be G. Polya’s “How to Solve it” and “Mathematics and Plausible Reasoning”. Similar high-caliber books. Brain teasers (including the 3D ones), board games, puzzles, riddles and non standard technical problems - something that can push me out of my comfort zone fast.Instead of buying the board games I went ahead and implemented (in Java) some number of them from scratch, as advertised. I, thus, have a small virtual, if you will, library of mathematically-themed games available free of charge at my GitHub repo:Sudoku (on a standard 9x9 board)8 Queens (on a standard 8x8 board)Tic-Tac-Toe (with alpha-beta pruning)Fifteen Puzzle (on a standard 4x4 board)Towers of Hanoi (with a configurable number of disks)Peg Solitaire (on a 33-cell board)Numeric Knight’s Tour (on a 5x5 board)In the game of Tic-Tac-Toe you will be crossing blades with an algorithm - watch out!Strange things.Beyond the tensors, residues, rings, saddle points, metric spaces, cyclic permutations, convolutions, analytic continuations, spectral decompositions, generating functions, vectors, limits, functors, prime elements, Fourier transforms and non linear differential equations in partial derivatives there exists an aspect of Mathematics that is difficult to verbalize but I will make an attempt.In this Quora answer Senia gave a very good definition of what mathematics is - a study of structure divorced from context.To that strategic the what, I would add the, tactical, how - by improvising within strict technical constraints (of a proof).The way I see it, that tactical aspect is the intellectual bridge that connects at higher level Mathematics to certain performing, concretely - Music, and visual, concretely - Drawing/Painting/Sculpting, arts. Their sample items are all top notch, all done by capable and true masters of their craft. All being worthy achievements of a human mind or a human hand (moved by the said mind).Thus, my mathematical library will (and does) also contain the copies of the works by:Mozart and Tchaikovsky (music)G. Bridgman (drawing)(portraiture by) Velazquez, Rembrandt and I. Repin (painting)Rodin (sculpture)

The other answers all address your question, but the explanations in some of them confused me a little even though I am quite familiar with minor-7th-with-a-flat-5 chords. In case you had difficulty understanding some of that too, let me give you a guitar-centric explanation that might be easier for you to process.You have the shorthand notation right for the half-diminished 7th chord, m7♭5, but the answer is “no”. A 7+5 chord is an entirely different thing, a dominant 7th with an augmented 5th. Consider what is in these two chord types.m7♭5 : 1, ♭3, ♭5, ♭77+5 : 1, 3, ♯5, ♭7Those chords differ for two of the four tones. They would sound quite different if the had the same root tone. So, they are not the same thing at all and cannot be cited using the same nomenclature. Getting your brain all twisted up trying to make them the same won’t ever succeed. The point I am trying to make here is that chord analysis is best done by picking apart the standard nomenclature and mapping out the intervals in the chord numerically as I have done.This kind of thing only makes sense to me once I do it that way. I have trouble visualizing the relationships between the tones without picturing them on a number line. When I was learning this stuff at the beginning, I would draw the number lines on quadrille graph paper. At one point, I even made a little “slide rule” so I could figure out what tones were used in different chord types with any root tone. (This was long before ordinary people could own a computer, or even a hand-held calculator. Consumer electronics didn’t exist.) I had a page with the number lines for all the chord types I knew that slid up and down in back of a slot that showed one number line at a time. A second strip had two octaves of twelve-tone scales that slid horizontally to position the letter with the root tone below the ‘1’ dot on the chord type template. (No one does this kind of thing any more because it’s easier to ask random strangers on internet forums to figure things out for you. Yeah, you deserved that.)Now, I can look this kind of thing up with a web-based app, but most of the time it’s quicker and easier for me to use my easiest and most accessible number line visualization tool, a strung guitar string. Think about how easy it is to do. If you play often, you instantly know where all the scale intervals are relative to the open string which represents the root tone. The inlays tell you how many semitones above the root you are. You can play the tones in any chord type, one at a time, on the one string to get a feeling for what the chord type sounds like.It’s possible to understand the difference in sound intellectually by reading about it, but you know that you really don’t “get” it until you have heard these different chord types over and over with different root tones. This is how musicians playing any instrument learn about chords and harmony in a manner that allows them to make use of such knowledge. You must do this kind of thing a lot to fully understand chords and recognize them by ear.Another reason why I answered this question is that m7♭5 is a commonly-used chord for guitar-based music, especially jazz, funk and any other type of music that routinely employs the “crunchy” quality of the diminished 5th interval. What galls me is that frequently, fingering recipes for chords in songs that employ these chords are incomplete, leaving out one of the four tones in the chord. The first time I encountered this, I wanted to play the “real chord” to see if it sounded different than the fingering recipe provided. I think the chord was Em7♭5. This is how my investigation went.The recipe given was 3XX333. I scowled at this because it didn’t have the root tone, E. This fingering provided just three tones, G, B♭ and D, which the last time I checked was a simple Gm triad.Furthermore, 3XX333 amounted to G-X-X-B♭-D-G. That’s two G’s and no E. How in hell was this Em7♭5? And, why so many G’s?I looked at that fingering for a while and figured out that I could slip in an E tone two ways, open 6th string and 4th string at the 2nd fret (0X2XXX).Combining XXX333 with 0X2XXX gave me a real Em7♭5 chord with two E’s an octave apart, plus the G, B♭ and D on the first three strings, fingered as 0X2333.I found that if I just fingered XX2333 with demi-barré, I could hit that E bass tone on the 6th string.If I really needed that B♭ tone, I could hit it on the 5th string and still do the rest of the chord using 012333. It wasn’t easy, but possible.Most of the time, XX2333 was good enough. Besides that, with an E on the 4th string, the root was the lowest tone, so this was, indeed, a “kosher” Em7♭5.XX2333 sounded as good or better than 3XX333 in the song, so I used XX2333 exclusively.Occasionally, I would hit the open E on the 6th string for a nice, low bass tone, or G on the 3rd fret for minor 3rd bass tone to mimic the arrangement I had used as a model for my own arrangement.Being curious as to whether or not I had worked this out correctly, I put my XX2333 fingering into a “chord finder” app and discovered that this combination of intervals was an inversion of a minor 6th chord! That is, if you take the 3rd as the root, ♭3, ♭5, ♭7, 1=8, and lower the each interval by 3 semitones so the new root is ‘1’, you get 1, ♭3, 5, 6, which is a minor 6th chord!This made things a lot easier for me because I was seeing those m7♭5 chords a lot, and not so often a m6. Chord charts would usually list m6, but m7♭5 not nearly as often. So, what I ended up doing was learning the four easy ways to finger m6 chords. Whenever a lead sheet or staff score would call for a m7♭5, I would pick out the easiest equivalent m6 to play in that vicinity of the neck instead of any weird fingering suggested by whatever I was looking at.The sound of my substitute m6 chords is usually just as good as the suggested fingering for m7♭5 because most of the time, guitar chords are inversions of what a keyboard player would usually play. The reason is that a keyboard player compresses a chord into as narrow a space as possible because they want to be able to reach all the notes in a chord at the same time with one hand. Guitars have a range at the same fret position between the 1st and 6th strings of two octaves, and we play chords, up to six tones at a time over that range. For example, an E chord fingered as 022100 contains the intervals 1–5–1–3–5–1, over a range of two octaves. No piano player can do that with one hand. Of course, they usually have two hands and can thus play up to ten different tones simultaneously, but they usually don’t play chords with both hand, reserving those for the left hand while the right hand does most of the single-note melody playing.It’s not that one instrument is better than the other, but that the instruments make different kinds of polyphonic sound that vary as much because of the differences in the architecture as differences in the tonal qualities of guitar strings and piano strings. The upshot of this is that when a guitarist is asked to play music designed by and for pianists, the clever guitar player will contrive accommodations better suited to the guitar than precisely mimicking what is played on a piano, note-for-note.In case you want to know the portable m6 fingering patterns you can use as substitute chords for m7♭5, transpose these up the neck to play any m6 chord you want.Am6: XX2212 (No barré. This is just a regular Am with your pinky holding down the 1st string at the 2nd fret. The lowest m6 you can play with this pattern is A♭m6, a.k.a. G♯m6, or substituting for Fm7♭5, fingered as XX1101. However, that is so rarely found that it’s only something to pull out of your hat when a piano player gets cute and tells you to play a chord that he thinks you can’t figure out. Get this right once and he will never hector you again that way.)Dm6: X00201 (0 is barré position)Em6: 022020 (0 is barré position)Fm6: XX0111 (0 is played with index finger, 1’s with demi-barré using third finger. Save middle finger for playing 9th chords using B9 pattern X21222 because you will want to do that eventually. Hear me now and believe me when you need to play a true 9th chord instead of that fake “cowboy” D9 chord which is really just an Amadd11 fingered as X00210. They sound different.)While I explained how to figure out all the tones for m6 chord from an equivalent m7♭5 chord, that is piano-player thinking, which is more complex than it needs to be, and totally unnecessary for playing guitar. This is the simplified procedure I use.What is the root tone of the Xm7♭5 chord?What is tone Y = X+3 semitones?Which pattern is most convenient to play Ym6?How many frets (i.e., semitones) above the root tone of the pattern do I go to have Y as the root tone?For example, if X=E in Em7♭5, then X+3=G. G is close to F, only two semitones up. So, the easiest Gm6 I can play is XX0111 + XX2222 = XX2333. After you’ve done this a few times, it gets pretty easy.The beauty of this simplified procedure is that you don’t have to even think about what any of the other tones in the chord are. Once you have memorized the four patterns, all you have to do is count semitone intervals between the root tone of the pattern and the root of the m6 chord you want to play.