𝕏On Algebra: On Lie Algebra

"During the last decade the methods of algebraic topology have invaded extensively the domain of pure algebra, and initiated a number of internal revolutions. [...] The invasion of algebra has occurred on three fronts through the construction of cohomology theories for groups, Lie algebras, and associative algebras. The three subjects have been given independent but parallel developments." (Henri P Cartan & Samuel Eilenberg, "Homological Algebra", 1956)

"Algebraic deformation represents a method for the quantization of Lie groups and Lie algebras. Quantum groups are not groups, but Hopf algebras. [3]

"Lie groups describe finite symmetries or symmetries which smoothly depend on a finite number of real parameters. Lie algebras are the linearization of Lie groups at the unit element. The passage from Lie groups to Lie algebras simplifies considerably the approach. Lie algebras are frequently called infinitesimal symmetries." (Eberhard Zeidler, "Quantum Field Theory III: Gauge Theory", 2006)

"Representations of symmetries with the aid of linear operators (e.g., matrices) play a crucial role in modern physics. In particular, this concerns the linear representations of groups, Lie algebras, and quantum groups (Hopf algebras)" (Eberhard Zeidler, "Quantum Field Theory III: Gauge Theory", 2006)

"Solvable Lie algebras are close to both upper triangular matrices and commutative Lie algebras. In contrast to this, semisimple Lie algebras are as far as possible from being commutative. By Levi’s decomposition theorem, any Lie algebra is built out of a solvable and a semisimple one. The nontrivial prototype of a solvable Lie algebra is the Heisenberg algebra." (Eberhard Zeidler, "Quantum Field Theory III: Gauge Theory", 2006)

"The fundamental Levi decomposition of a Lie algebra is the prototype of a semidirect product of Lie algebras." (Eberhard Zeidler, "Quantum Field Theory III: Gauge Theory", 2006)

"There is a fundamental relationship between Lie groupoids and Lie algebroids which is similar to the relationship between Lie groups and Lie algebras. There are, however, significant differences in the conventions involved, and it important to be aware of these." (Mike Crampin & David Saunders, "Cartan Geometries and their Symmetries: A Lie Algebroid Approach", 2016)

𝕏On Algebra (2010-2019)

"Of all mathematical disciplines, algebra has changed the most. While earlier generations of geometers would recognize - if not immediately understand - much of modern geometry as an extension of the subject that they had studied, it is doubtful that earlier generations of algebraists would recognize most of modern algebra as in any way related to the subject to which they devoted their time." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)

"Quantum theory may be formulated using Hilbert spaces over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. Indeed, these three choices appear naturally in a number of axiomatic approaches. However, there are internal problems with real or quaternionic quantum theory. Here we argue that these problems can be resolved if we treat real, complex and quaternionic quantum theory as part of a unified structure. Dyson called this structure the ‘three-fold way’ […] This three-fold classification sheds light on the physics of time reversal symmetry, and it already plays an important role in particle physics." (John C Baez, "Division Algebras and Quantum Theory", 2011)

"It turns out pi is different. Not only is it incapable of being expressed as a fraction, but in fact pi fails to satisfy any algebraic relationship whatsoever. What does pi do? It doesn’t do anything. It is what it is. Numbers like this are called transcendental" (Latin for 'climbing beyond'). Transcendental numbers - and there are lots of them - are simply beyond the power of algebra to describe." (Paul Lockhart, "Measurement", 2012)

"It turns out π is different. Not only is it incapable of being expressed as a fraction, but in fact π fails to satisfy any algebraic relationship whatsoever. What does π do? It doesn’t do anything. It is what it is. Numbers like this are called transcendental" (Latin for 'climbing beyond'). Transcendental numbers - and there are lots of them - are simply beyond the power of algebra to describe." (Paul Lockhart, "Measurement", 2012)

"Mathematics is an art, and creative genius a mystery. Of course, technique helps - good painters understand light and shadow, good musicians have a thorough knowledge of functional harmony, and good mathematicians can untangle algebraic information - but a beautiful piece of mathematics is just as hard to make as a beautiful portrait or sonata." (Paul Lockhart, "Measurement", 2012)

"The standard view among most theoretical physicists, engineers and economists is that mathematical models are syntactic" (linguistic) items, identified with particular systems of equations or relational statements. From this perspective, the process of solving a designated system of" (algebraic, difference, differential, stochastic, etc.) equations of the target system, and interpreting the particular solutions directly in the context of predictions and explanations are primary, while the mathematical structures of associated state and orbit spaces, and quantity algebras – although conceptually important, are secondary." (Zoltan Domotor, "Mathematical Models in Philosophy of Science" [Mathematics of Complexity and Dynamical Systems, 2012])

"The tangling and untangling of numerical relationships is called algebra. […] The point of doing algebra is not to solve equations; it’s to allow us to move back and forth between several equivalent representations, depending on the situation at hand and depending on our taste. In this sense, all algebraic manipulation is psychological. The numbers are making themselves known to us in various ways, and each different representation has its own feel to it and can give us ideas that might not occur to us otherwise." (Paul Lockhart, "Measurement", 2012)

"Galois and Abel independently discovered the basic idea of symmetry. They were both coming at the problem from the algebra of polynomials, but what they each realized was that underlying the solution of polynomials was a fundamental problem of symmetry. The way that they understood symmetry was in terms of permutation groups. A permutation group is the most fundamental structure of symmetry. […] permutation groups are the master groups of symmetry: every kind of symmetry is encoded in the structure of the permutation group." (Mark C Chu-Carroll, "Good Math: A Geek’s Guide to the Beauty of Numbers, Logic, and Computation", 2013)

"You can ask the question about these ancient topics, such as perfect numbers and amicable numbers [...] and ask, are these good problems [...] studying them helped us develop all of elementary number theory and from elementary number theory we developed the rest of number theory, and also you can argue that from elementary number theory came algebra." (Carl B Pomerance, "Paul Erdős and the Rise of Statistical Thinking in Elementary Number Theory", [lecture] 2013)

"[…] regard it in fact as the great advantage of the mathematical technique that it allows us to describe, by means of algebraic equations, the general character of a pattern even where we are ignorant of the numerical values which will determine its particular manifestation." (Friedrich A von Hayek, "The Market and Other Orders", 2014)

"Algebraic geometry uses the geometric intuition which arises from looking at varieties over the complex and real case to deduce important results in arithmetic algebraic geometry where the complex number field is replaced by the field of rational numbers or various finite number fields." (Raymond O Wells Jr, "Differential and Complex Geometry: Origins, Abstractions and Embeddings", 2017)

"Much of the final resistance to complex numbers faded as it became clear that their behavior posed no threat to the rules and operations of algebra. On the contrary, quite often the complex realm opened paths that made already existing results easier to prove." (David Perkins, "φ, π, e & i", 2017)

"The primary aspects of the theory of complex manifolds are the geometric structure itself, its topological structure, coordinate systems, etc., and holomorphic functions and mappings and their properties. Algebraic geometry over the complex number field uses polynomial and rational functions of complex variables as the primary tools, but the underlying topological structures are similar to those that appear in complex manifold theory, and the nature of singularities in both the analytic and algebraic settings is also structurally very similar." (Raymond O Wells Jr, "Differential and Complex Geometry: Origins, Abstractions and Embeddings", 2017)

"When we use algebraic notation in statistical models, the problem becomes more complicated because we cannot 'observe' a probability and know its exact number. We can only estimate probabilities on the basis of observations." (David S Salsburg, "Errors, Blunders, and Lies: How to Tell the Difference", 2017)

"The calculus of causation consists of two languages: causal diagrams, to express what we know, and a symbolic language, resembling algebra, to express what we want to know. The causal diagrams are simply dot-and-arrow pictures that summarize our existing scientific knowledge. The dots represent quantities of interest, called 'variables', and the arrows represent known or suspected causal relationships between those variables - namely, which variable 'listens' to which others." (Judea Pearl & Dana Mackenzie, "The Book of Why: The new science of cause and effect", 2018)

𝕏On Algebra (2000-2009)

"Algebraic topology can be roughly defined as the study of techniques for forming algebraic images of topological spaces. Most often these algebraic images are groups, but more elaborate structures such as rings, modules, and algebras also arise. The mechanisms that create these images - the ‘lanterns’ of algebraic topology, one might say - are known formally as functors and have the characteristic feature that they form images not only of spaces but also of maps. Thus, continuous maps between spaces are projected onto homomorphisms between their algebraic images, so topologically related spaces have algebraically related images." (Allen Hatcher, "Algebraic Topology", 2001)

"Algebraic topology is most often concerned with properties of spaces that depend only on homotopy type, so local topological properties do not play much of a role." (Allen Hatcher, "Algebraic Topology", 2001)

"The geometry of Algebraic Topology is so pretty, it would seem a pity to slight it and miss all the intuition which it provides." (Allen Hatcher, "Algebraic Topology", 2002)

"Combinatorics could be described as the study of arrangements of objects according to specified rules. We want to know, first, whether a particular arrangement is possible at all, and, if so, in how many different ways it can be done. Algebraic and even probabilistic methods play an increasingly important role in answering these questions. If we have two sets of arrangements with the same cardinality, we might want to construct a natural bijection between them. We might also want to have an algorithm for constructing a particular arrangement or all arrangements, as well as for computing numerical characteristics of them; in particular, we can consider optimization problems related to such arrangements. Finally, we might be interested in an even deeper study, by investigating the structural properties of the arrangements. Methods from areas such as group theory and topology are useful here, by enabling us to study symmetries of the arrangements, as well as topological properties of certain spaces associated with them, which translate into combinatorial properties." (Cristian Lenart, "The Many Faces of Modern Combinatorics", 2003)

"Some areas of human knowledge ever since its origin had shaken our understanding of the universe from time to time. While this is more true about physics, it is true about mathematics as well. The birth of topology as analysis situs meaning rubbersheet geometry had a similar impact on our traditional knowledge of analysis. Indeed, topology had enough energy and vigour to give birth to a new culture of mathematical approach. Algebraic topology added a new dimension to that. Because quantum physicists and applied mathematicians had noted wonderful interpretations of many physical phenomena through algebraic topology, they took immense interest in the study of topology in the twentieth century." (D Chatterjee, "Topology: General & Algebraic", 2003) 

"The exquisite world of algebraic topology came into existence out of our attempts to solve topological problems by the use of algebraic tools and this revealed to us the nice interplay between algebra and topology which causes each to reinforce interpretations of the other there by breaking down the often artificial subdivision of mathematics into different branches and emphasizing the essential unity of all mathematics. The homology theory is the main branch of algebraic topology and plays the main role in the classification problems of topological spaces. There are various approaches to the study of this theory such as geometric approach, abstract approach and axiomatic approach. Since geometric approach appeals easily to our intuition, we shall start with geometric approach, the relevant development being called simplicial homology theory." (D Chatterjee, "Topology: General & Algebraic", 2003)

"Algebra is the offer made by the devil to the mathematician. The devil says: I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvelous machine." (Michael F Atiyah, 2004)

"We divide math up into separate areas (analysis, mechanics, algebra, geometry, electromagnetism, number theory, quantum mechanics, etc.) to clarify the study of each part; but the equally valuable activity of integrating the components into a working whole is all too often neglected. Without it, the stated aim of ‘taking something apart to see how it ticks’ degenerates imperceptibly into ‘taking it apart to ensure it never ticks again’." (Miles Reid & Balazs Szendröi, "Geometry and Topology", 2005)

"Algebra is what most people associate with mathematics. In a sense, this is justified. Mathematics is the study of abstract objects, numerical, logical, or geometrical, that follow a set of several carefully chosen axioms. And basic algebra is about the simplest meaningful thing that can satisfy the above definition of mathematics. There are only a dozen or so postulates, but that is enough to make the system beautifully symmetric." (Terence Tao, "Solving Mathematical Problems: A Personal Perspective", 2006)

"Analysis is also a heavily explored subject, and it is just as general as algebra: essentially, analysis is the study of functions and their properties. The more complicated the properties, the higher the analysis." (Terence Tao, "Solving Mathematical Problems: A Personal Perspective", 2006)

"Number theory may not necessarily be divine, but it still has an aura of mystique about it. Unlike algebra, which has as its backbone the laws of manipulating equations, number theory seems to derive its results from a source unknown." (Terence Tao, "Solving Mathematical Problems: A Personal Perspective", 2006)

"Geometric algebra provides the most powerful artefact for dealing with rotations and dilations. It generalizes the role of complex numbers in two dimensions, and quaternions in three dimensions, to a wider scheme for dealing with rotations in arbitrary dimensions in a simple and comprehensive manner." (Venzo de Sabbata & Bidyut K Datta, "Geometric Algebra and Applications to Physics", 2007)

"Poetry and code - and mathematics - make us read differently from other forms of writing. Written poetry makes the silent reader read three kinds of pattern at once; code moves the reader from a static to an active, interactive and looped domain; while algebraic topology allows us to read qualitative forms and their transformations." (Stephanie Strickland & Cynthia L Jaramillo, "Dovetailing Details Fly Apart - All over, again, in code, in poetry, in chreods", 2007)

"The role of conceptual modelling in information systems development during all these decades is seen as an approach for capturing fuzzy, ill-defined, informal 'real-world' descriptions and user requirements, and then transforming them to formal, in some sense complete, and consistent conceptual specifications." (Janis A Burbenko jr., "From Information Algebra to Enterprise Modelling and Ontologies", Conceptual Modelling in Information Systems Engineering, 2007)

"[...] transcendental numbers, those numbers that lie beyond those that arise through euclidean geometry and ordinary algebraic equations. [...] The transcendentals are the numbers that fill the huge void between the more familiar algebraic numbers and the collection of all decimal expansions: to use an astronomical comparison, the transcendentals are the dark matter of the number world." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"Algebraic symbols carry a universality of interpretation that allows them to be manipulated in a way that words cannot. Indeed, this was the key breakthrough that allowed mathematics to flourish in a way that was not possible until the advent of algebra. All higher mathematics relies on constant use of algebraic manipulation and would be impossible without it." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"An algebra is a mathematical structure consisting of a set of elements and a collection of operators on those elements - though the term is variously defined in different contexts in mathematics, so as to narrow the meaning" (e.g. an algebra is a vector space with a bilinear multiplication operation). " (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

"For mathematics to be applicable in any sense at all we need to be able to do something with it. In practice this nearly always means developing forms of calculation, and this imperative channels its practitioners into algebraic manipulations of one form or another and ultimately into producing numbers. To the modern mind, this might seem natural and inevitable." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"The curious switch, from initially perceiving an obstruction to a problem to eventually embodying this obstruction as a number or an algebraic object of some sort that we can effectively study, is repeated over and over again, in different contexts, throughout mathematics." (Timothy Gowers, "The Princeton Companion to Mathematics", 2008)

"The rationals therefore also form a countable set, as do the euclidean numbers, and indeed if we consider the set of all numbers that arise from the rationals through taking roots of any order, the collection produced is still countable. We can even go beyond this: the collection of all algebraic numbers, which are those that are solutions of ordinary polynomial equations∗ form a collection that can, in principle, be arrayed in an infinite list: that is to say it is possible, with a little more crafty argument, to describe a systematic listing that sweeps them all out." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"The simple algebraic numbers, like √2, seem closest in nature to the rationals, while we might expect that non-algebraic numbers, the transcedentals, to live apart and not to have close rational neighbors. Surprisingly, the opposite is true. On the one hand, it can be proved that any irrational number that can be well-approximated by rationals" (in a sense that can be made precise) must be transcendental. Indeed this affords one of the standard techniques for showing that a number is transcendental." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"What was clearly useful was the use of diagrams to prove certain results either in algebraic topology, homological algebra or algebraic geometry. It is clear that doing category theory, or simply applying category theory, implies manipulating diagrams: constructing the relevant diagrams, chasing arrows by going via various paths in diagrams and showing they are equal, etc. This practice suggests that diagram manipulation, or more generally diagrams, constitutes the natural syntax of category theory and the category-theoretic way of thinking. Thus, if one could develop a formal language based on diagrams and diagrams manipulation, one would have a natural syntactical framework for category theory. However, moving from the informal language of categories which includes diagrams and diagrammatic manipulations to a formal language based on diagrams and diagrammatic manipulations is not entirely obvious." (Jean-Pierre Marquis, "From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory", 2009)

Allen Hatcher - Collected Quotes

"A space having the homotopy type of a point is called contractible. This amounts to requiring that the identity map of the space be nullhomotopic, that is, homotopic to a constant map. In general, this is slightly weaker than saying the space deformation retracts to a point; ". (Allen Hatcher, "Algebraic Topology", 2001)

"Algebraic topology can be roughly defined as the study of techniques for forming algebraic images of topological spaces. Most often these algebraic images are groups, but more elaborate structures such as rings, modules, and algebras also arise. The mechanisms that create these images - the ‘lanterns’ of algebraic topology, one might say - are known formally as functors and have the characteristic feature that they form images not only of spaces but also of maps. Thus, continuous maps between spaces are projected onto homomorphisms between their algebraic images, so topologically related spaces have algebraically related images." (Allen Hatcher, "Algebraic Topology", 2001)

"Algebraic topology is most often concerned with properties of spaces that depend only on homotopy type, so local topological properties do not play much of a role." (Allen Hatcher, "Algebraic Topology", 2001)

"An interesting feature of homology that begins to emerge after one has worked with it for a while is that it is the basic properties of homology that are used most often, and not the actual definition itself. This suggests that an axiomatic approach to homology might be possible. This is indeed the case [...]. One could take the viewpoint that these rather algebraic axioms are all that really matters about homology groups, that the geometry involved in the definition of homology is secondary, needed only to show that the axiomatic theory is not vacuous. The extent to which one adopts this viewpoint is a matter of taste, and the route taken here of postponing the axioms until the theory is well-established is just one of several possible approaches." (Allen Hatcher, "Algebraic Topology", 2001)

"From the viewpoint of homotopy theory, cohomology is in some ways more basic than homology. [...] cohomology has a description in terms of homotopy classes of maps that is very similar to, and in a certain sense dual to, the definition of homotopy groups." (Allen Hatcher, "Algebraic Topology", 2001)

"Homotopy theory begins with the homotopy groups πn(X), which are the natural higher-dimensional analogs of the fundamental group. These higher homotopy groups have certain formal similarities with homology groups. For example, πn(X) turns out to be always abelian for n ≥ 2, and there are relative homotopy groups fitting into a long exact sequence just like the long exact sequence of homology groups. However, the higher homotopy groups are much harder to compute than either homology groups or the fundamental group, due to the fact that neither the excision property for homology nor van Kampen’s theorem for π1 holds for higher homotopy groups." (Allen Hatcher, "Algebraic Topology", 2001)

"Since all groups can be realized as fundamental groups of spaces, this opens the way for using topology to study algebraic properties of groups." (Allen Hatcher, "Algebraic Topology", 2001)

"The computability of homology groups does not come for free, unfortunately. The definition of homology groups is decidedly less transparent than the definition of homotopy groups, and once one gets beyond the definition, there is a certain amount of technical machinery to be set up before any real calculations and applications can be given." (Allen Hatcher, "Algebraic Topology", 2001)

"The operation of collapsing a subspace to a point usually has a drastic effect on homotopy type, but one might hope that if the subspace being collapsed already has the homotopy type of a point, then collapsing it to a point might not change the homotopy type of the whole space." (Allen Hatcher, "Algebraic Topology", 2001)

On Vectors (2010-)

"Today, most mathematicians have embraced the axiom of choice because of the order and simplicity it brings to mathematics in general. For example, the theorems that every vector space has a basis and every field has an algebraic closure hold only by virtue of the axiom of choice. Likewise, for the theorem that every sequentially continuous function is continuous. However, there are special places where the axiom of choice actually brings disorder. One is the theory of measure." (John Stillwell, "Roads to Infinity: The mathematics of truth and proof", 2010)

"Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. Most commonly applied to the mathematical study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values" (the bifurcation parameters) of a system causes a sudden"qualitative" or topological change in its behavior. Bifurcations can occur in both continuous systems" (described by ODEs, DDEs, or PDEs) and discrete systems" (described by maps)." (Tianshou Zhou, "Bifurcation", 2013)

"In the tensor algebra constructed upon some vector space (or over some module), the vectors (or the elements of the module) are considered as tensors of grade or rank one. The vectors that one finds in the vector calculus are called tangent vectors. They constitute so called tangent vector spaces (one at each point of the manifold), though it is difficult to realize in Euclidean spaces why should one call them tangent. This will later become evident. Elements of the space that results from tensor product of copies of a tangent vector space are called tangent tensors." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)

"Vector calculus is horrible for several reasons. One of them is that its curl is based on the vector product. So, we do not have a curl in other dimensions. Another reason is that it uses tangent vectors where it should use differential forms. One more is that it often uses more structure than needed to solve a problem, say a metric structure. Still another one is that one can do so little with it that it has to be complemented with all the other calculi that we also think of replacing with differential forms. The Kähler calculus - based on Clifford algebra of differential forms - replaces tangent vectors and tangent-valued operators with differential forms whose coefficients are respectively functions and operators." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)

"We say that a set of quantities transforms vectorially under a group of transformations of the bases of a vector space if, under an element of the group, they transform like the components of vectors, whether contravariant or covariant. We say that some quantity is a scalar if it is an invariant under the transformations of a group. A quantity (respectively, a set of quantities) may be scalar (respectively vectorial) under the transformation of a group, while not being so under the transformations of another group." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)

"Curvature is a central concept in differential geometry. There are conceptually different ways to define it, associated with different mathematical objects, the metric tensor, and the affine connection. In our case, however, the affine connection may be derived from the metric. The 'affine curvature' is associated with the notion of parallel transport of vectors as introduced by Levi-Civita. This is most simply illustrated in the case of a two- dimensional surface embedded in three- dimensional space. Let us take a closed curve on that surface and attach to a point on that curve a vector tangent to the surface. Let us now transport that vector along the curve, keeping it parallel to itself. When it comes back to its original position, it will coincide with the original vector if the surface is flat or deviate from it by a certain angle if the surface is curved. If one takes a small curve around a point on the surface, then the ratio of the angle between the original and the final vector and the area enclosed by the curve is the curvature at that point. The curvature at a point on a two-dimensional surface is a pure number." (Hanoch Gutfreund, "The Road to Relativity", 2015)

"Dynamics of a linear system are decomposable into multiple independent one-dimensional exponential dynamics, each of which takes place along the direction given by an eigenvector. A general trajectory from an arbitrary initial condition can be obtained by a simple linear superposition of those independent dynamics." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"For us, a model is a stochastic process, that is, a probability distribution over a sequence of random variables, perhaps indexed by a vector of parameters. For us, model uncertainty includes a suspicion that a model is incorrect." (Lars P Hansen & Thomas J Sargent, "Uncertainty within Economic Models", 2015)

"So a very useful way to think about i (√−1) is as an operator that produces a 90◦ rotation of any vector to which it is applied. Thus the two perpendicular number lines form the basis of what we know today as the complex plane. Unfortunately, since multiplication by √−1 is needed to get from the horizontal to the vertical number line, the numbers along the vertical number line are called 'imaginary'. We say 'unfortunately' because these numbers are every bit as real as the numbers along the horizontal number line. But the terminology is pervasive, so when you first learned about complex numbers, you probably learned that they consist of a 'real' and an 'imaginary' part." (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)

"When datasets are small, a parametric model may perform well because the strong assumptions made by the model - if correct - can help the model to avoid overfitting. However, as the size of the dataset grows, particularly if the decision boundary between the classes is very complex, it may make more sense to allow the data to inform the predictions more directly. Obviously the computational costs associated with nonparametric models and large datasets cannot be ignored. However, support vector machines are an example of a nonparametric model that, to a large extent, avoids this problem. As such, support vector machines are often a good choice in complex domains with lots of data." (John D Kelleher et al, "Fundamentals of Machine Learning for Predictive Data Analytics: Algorithms, Worked Examples, and Case Studies", 2015)

"The association of multiplication with vector rotation was one of the geometric interpretation's most important elements because it decisively connected the imaginaries with rotary motion. As we'll see, that was a big deal." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"What is essentially different in quantum mechanics is that it deals with complex quantities (e.g. wave functions and quantum state vectors) of a special kind, which cannot be split up into pure real and imaginary parts that can be treated independently. Furthermore, physical meaning is not attached directly to the complex quantities themselves, but to some other operation that produces real numbers" (e.g. the square modulus of the wave function or of the inner product between state vectors)." (Ricardo Karam, "Why are complex numbers needed in quantum mechanics? Some answers for the introductory level", American Journal of Physics Vol. 88" (1), 2020)

𝕏On Algebra: On Vectors (2000-2009)

"From a formal perspective, much about complex numbers and arithmetic seems arbitrary. From a purely algebraic point of view, i arises as a solution to the equation x^2+1=0. There is nothing geometric about this - no complex plane at all. Yet in the complex plane, the i-axis is 90° from the x-axis. Why? Complex numbers in the complex plane add like vectors. Why? Complex numbers have a weird rule of multiplication […]" (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being", 2000)

"One reason for the importance of Riemannian manifolds is that they are generalizations of Euclidean geometry - general enough but not too general. They are still close enough to Euclidean geometry to have a Laplace operator. This is the key to quantum mechanics, heat and waves. The various generalizations of Riemannian manifold [...] do not have a simple natural unambiguous choice of such an operator. [...] Another reason for the prominence of Riemannian manifolds is that the maximal compact subgroup of the general linear group is the orthogonal group. So the least restriction we can make on any geometric structure so that it 'rigidifies' always adds a Riemannian geometry. Moreover, any geometric structure will always permit such a 'rigidification'. [...] Similarly, if we were to pick out a submanifold of the tangent bundle of some manifold, distinguishing tangent vectors, in such a manner that in each tangent space, any two lines could be brought to one another, or any two planes, etc., then the maximal symmetry group we could come up with in a single tangent space which was not the whole general linear group would be the orthogonal group of a Riemannian metric. So Riemannian geometry is the 'least' structure, or most symmetrical one, we can pick, at first order." (Marcel Berger, "A Panoramic View of Riemannian Geometry", 2003)

"Figures that plot the payoff vectors are common in the literature, as are figures that show the convex hull of the payoffs. The latter are used for discussing mixed strategies and bargaining games, and require real values. A complete representation of the strategic form requires that the strategic choices represented in the matrix be recoverable, and that is the reason for using different line styles to represent each player’s inducement correspondences." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)

"[...] one of the fundamental intellectual breakthroughs in the historical understanding of just what i = √-1 means, physically, came with the insight that multiplication by a complex number is associated with a rotation in the complex plane. That is, multiplying the vector of a complex number by the complex exponential e^iθ rotates that vector counterclockwise through angle θ." (Paul J Nahin, "Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills", 2006)

"Physics reduces Moonshine to a duality between two different pictures of quantum field theory: the Hamiltonian one, which concretely gives us from representation theory the graded vector spaces, and another, due to Feynman, which manifestly gives us modularity. In particular, physics tells us that this modularity is a topological effect, and the group SL2(Z) directly arises in its familiar role as the modular group of the torus." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

"Unfortunately, if we were to use geometry to explore the concept of the square root of a negative number, we would be setting a boundary to our imagination that would be difficult to cross. To represent -1 using geometry would require us to draw a square with each side length being less than zero. To be asked to draw a square with side length less than zero sounds similar to the Zen Buddhists asking ‘What is the sound of one hand clapping?’" (Les Evans, "Complex Numbers and Vectors", 2006)

"Geometric algebra is, in fact, the largest possible associative division algebra that integrates all algebraic systems (viz., algebra of complex numbers, vector algebra, matrix algebra, quaternion algebra, etc.) into a coherent mathematical language that augments the powerful geometric intuition of the human mind with the precision of an algebraic system. Its potency lies in the fact that it develops all branches of theoretical physics, envisaging geometrical meaning to all operations and physical interpretation to mathematical elements, e.g., it integrates the ideas of axial vectors and pseudoscalars with vectors and scalars at its foundation. The spinor theory of rotations and rotational dynamics can be formulated in a coherent manner with the help of geometric algebra." (Venzo de Sabbata & Bidyut K Datta, "Geometric Algebra and Applications to Physics", 2007)

"Linear algebra is a very useful subject, and its basic concepts arose and were used in different areas of mathematics and its applications. It is therefore not surprising that the subject had its roots in such diverse fields as number theory (both elementary and algebraic), geometry, abstract algebra (groups, rings, fields, Galois theory), anal ysis (differential equations, integral equations, and functional analysis), and physics. Among the elementary concepts of linear algebra are linear equations, matrices, determinants, linear transformations, linear independence, dimension, bilinear forms, quadratic forms, and vector spaces. Since these concepts are closely interconnected, several usually appear in a given context (e.g., linear equations and matrices) and it is often impossible to disengage them." (Israel Kleiner, "A History of Abstract Algebra", 2007)

"An algebra is a mathematical structure consisting of a set of elements and a collection of operators on those elements - though the term is variously defined in different contexts in mathematics, so as to narrow the meaning" (e.g. an algebra is a vector space with a bilinear multiplication operation)." (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

On Vectors (-1949)

"The algebraically real part may receive [...] all values contained on the one scale of progression of number from negative to positive infinity; we shall call it therefore the scalar part, or simply the scalar of the quaternion, and shall form its symbol by prefixing, to the symbol of the quaternion, the characteristic Scal., or simply S., where no confusion seems likely to arise from using this last abbreviation. On the other hand, the algebraically imaginary part, being geometrically constructed by a straight line, or radius vector, which has, in general, for each determined quaternion, a determined length and determined direction in space, may be called the vector part, or simply the vector of the quaternion; andmay be denoted by prefixing the characteristic Vect., or V." (William R Hamilton, 1846)

"Vectors which are referred to unit of length I shall call Forces, using the word in a somewhat generalized sense, as we shall see. The operation of taking the integral of the resolved part of a force in the direction of a line for every element of that line, has always a physical meaning. In certain cases the result of the integration is independent of the path of the line between its extremities. The result is then called a Potential." (James Clerk-Maxwell, "Remarks on the Mathematical Classification of Physical Quantities", 1871)

"I do think [...] that you would find it would lose nothing by omitting the word 'vector' throughout. It adds nothing to the clearness or simplicity of the geometry, whether of two dimensions or three dimensions. Quaternions came from Hamilton after his really good work had been done; and, though beautifully ingenious, have been an unmixed evil to those who have touched them in any way, including Clerk Maxwell." (William T Kelvin, [Letter to Robert B Hayward] 1892)

"The merits or demerits of a pamphlet printed for private distribution a good many years ago do not constitute a subject of any great importance, but the assumptions implied in the sentence quoted are suggestive of certain reflections and inquiries which are of broader interest, and seem not untimely at a period when the methods and results of the various forms of multiple algebra are attracting so much attention. It seems to be assumed that a departure from quaternionic usage in the treatment of vectors is an enormity. If this assumption is true, it is an important truth; if not, it would be unfortunate if it should remain unchallenged, especially when supported by so high an authority. The criticism relates particularly to the notations, but I believe that there is a deeper question of notions underlying that of notations. Indeed, if my offence had been solely in the matter of notations, it would have been less accurate to describe my productions as a monstrosity, than to characterize its dress as uncouth." (Josiah W Gibbs, "The Rôle of Quaternions in the Algebra of Vectors, Nature vol. xliii, 1891)

"I do think [...] that you would find it would lose nothing by omitting the word 'vector' throughout. It adds nothing to the clearness or simplicity of the geometry, whether of two dimensions or three dimensions. Quaternions came from Hamilton after his really good work had been done; and, though beautifully ingenious, have been an unmixed evil to those who have touched them in any way, including Clerk Maxwell." (William T Kelvin, [Letter to Robert B Hayward] 1892) 

"[...] it is as unfair to call a vector a quaternion  as to call a man a quadruped." (Oliver Heaviside, 1892)

"The invention of quaternions must be regarded as a most remarkable feat of human ingenuity. Vector analysis, without quaternions, could have been found by any mathematician [...] but to find out quaternions required genius." (Oliver Heaviside, 1892)

"Symmetrical equations are good in their place, but "vector" is a useless survival, or offshoot, from quaternions, and has never been of the slightest use to any creature. Hertz wisely shunted it, but unwisely he adopted temporarily Heaviside’s nihilism. He even tended to nihilism in dynamics, as I warned you soon after his death. He would have grown out of all this, I believe, if he had lived. He certainly was the opposite pole of nature to a nihilist in his experimental work, and in his Doctorate Thesis on the impact of elastic bodies." (William T Kelvin, [footnote in Letter to George F FitzGerald] 1896)

"Our bra and ket vectors are complex quantities, since they can be multiplied by complex numbers and are then of the same nature as before, but they are complex quantities of a special kind which cannot be split up into real and pure imaginary parts. The usual method of getting the real part of a complex quantity, by taking half the sum of the quantity itself and its conjugate, cannot be applied since a bra and a ket vector are of different natures and cannot be added." (Paul Dirac, "The Principles of Quantum Mechanics", 1930)

"It is a curious fact in the history of mathematics that discoveries of the greatest importance were made simultaneously by different men of genius. The classical example is the […] development of the infinitesimal calculus by Newton and Leibniz. Another case is the development of vector calculus in Grassmann's Ausdehnungslehre and Hamilton's Calculus of Quaternions. In the same way we find analytic geometry simultaneously developed by Fermat and Descartes." (Julian L Coolidge, "A History of Geometrical Methods", 1940)

Venzo de Sabbata - Collected Quotes

"According to quantum theory, a fermion does not return to its initial state by a rotation of 2π, but it takes a rotation of 4π to restore its state of initial condition." (Venzo de Sabbata & Bidyut K Datta, "Geometric Algebra and Applications to Physics", 2007)

"Descartes’ algebra could be used to classify line segments by length only. The fundamental geometric notion of direction of a line segment finds no expression in ordinary algebra. The modification of algebra to have a fuller symbolic representation of geometric notions had to wait some 200 years after Descartes, when the concept of number was generalized by Herman Grassmann to  incorporate the geometric notion of direction as well as magnitude. With a proper symbolic expression for direction and dimension came the broader concept of directed numbers, now known as multivectors." (Venzo de Sabbata & Bidyut K Datta, "Geometric Algebra and Applications to Physics", 2007)

"Euclid sharply distinguished between number and magnitude, associating the former with the operation of counting and the latter with a line segment. So, for Euclid, only integers were numbers; even the notion of fractions as numbers had not yet been conceived of. He represented a whole number n by a line segment that was n times the chosen unit line segment. However, the opposite procedure of distinguishing all line segments by labeling them with numerals representing counting numbers was not possible. Obviously, this one-way correspondence of counting number with magnitude implies that the latter concept was more general than the former. The sharp distinction between counting number and magnitude, made by Euclid, was an impediment to the development of the concept of number." (Venzo de Sabbata & Bidyut K Datta, "Geometric Algebra and Applications to Physics", 2007)

"Geometric algebra combines the algebraic structure of Clifford algebra with the explicit geometric meaning of its mathematical elements at its foundation. So, formally, it is Clifford algebra endowed with geometrical information of and physical interpretation to all mathematical elements of the algebra. This intrusion of geometric consideration into the abstract system of Clifford algebra has enriched geometric algebra as a powerful mathematical theory." (Venzo de Sabbata & Bidyut K Datta, "Geometric Algebra and Applications to Physics", 2007)

"Geometric algebra is, in fact, the largest possible associative division algebra that integrates all algebraic systems (viz., algebra of complex numbers, vector algebra, matrix algebra, quaternion algebra, etc.) into a coherent mathematical language that augments the powerful geometric intuition of the human mind with the precision of an algebraic system. Its potency lies in the fact that it develops all branches of theoretical physics, envisaging geometrical meaning to all operations and physical interpretation to mathematical elements, e.g., it integrates the ideas of axial vectors and pseudoscalars with vectors and scalars at its foundation. The spinor theory of rotations and rotational dynamics can be formulated in a coherent manner with the help of geometric algebra." (Venzo de Sabbata & Bidyut K Datta, "Geometric Algebra and Applications to Physics", 2007)

"Geometric algebra provides the most powerful artefact for dealing with rotations and dilations. It generalizes the role of complex numbers in two dimensions, and quaternions in three dimensions, to a wider scheme for dealing with rotations in arbitrary dimensions in a simple and comprehensive manner." (Venzo de Sabbata & Bidyut K Datta, "Geometric Algebra and Applications to Physics", 2007)

"In 1884, just 40 years after the publication of Grassmann’s 'Algebra of Extension', Gibbs  developed his vector algebra following the ideas of Grassmann by replacing the concept of the outer product by a new kind of product known as vector product and interpreted as an axial vector in an ad-hoc manner. This, in fact, went against the run of natural development of directed numbers started by Grassmann and completely changed the course of its development in the other direction. Grassmann’s outer product reveals the fact that the Greek distinction between number and magnitude has real geometric significance. Greek magnitudes, in fact, added like scalars but multiplied like vectors, asserting the geometric notions of direction and dimension to multiplication of Greek magnitudes. This revealing feature is a reminiscence of the distinction, carefully made by Euclid, between multiplication of magnitudes and that of numbers. Thus, Herman Grassmann fully accomplished the algebraic formulation of the basic ideas of Greek geometry begun by Renė Descartes." (Venzo de Sabbata & Bidyut K Datta, "Geometric Algebra and Applications to Physics", 2007)

"Mathematical objects of geometric algebra have one kind of addition rule, different from Gibbs’ vector algebra, and one general kind of multiplicative rule, known as the geometric product. The importance of the geometric product of two vectors can be visualized in the fact that all other significant products can be obtained from it. The inner and outer products seem to complement one another by describing independent geometrical relations." (Venzo de Sabbata & Bidyut K Datta, "Geometric Algebra and Applications to Physics", 2007)

"Though from the very beginning algebra was associated with geometry, Descartes first developed it systematically in geometric language. Three steps are of fundamental importance in this development. First, he assumed that every line segment could be uniquely represented by a number that endowed the Greek notion of magnitude a symbolic form. Second, he labeled line segments by letters representing their numeral lengths. This resided in the fact that the basic arithmetic operations of addition and subtraction could be described in a completely analogous way as geometric operations on line segments. Third, in order to get rid of the apparent limitations of the Greek rule for geometric multiplication, he invented a rule for multiplying line segments, yielding a line segment in complete correspondence with the rule for multiplying numbers." (Venzo de Sabbata & Bidyut K Datta, "Geometric Algebra and Applications to Physics", 2007)

"Vieta used letters to denote numbers, whereas Descartes introduced letters to denote line segments. Vieta studied rules for manipulating numbers in an abstract manner, and Descartes accepted the existence of similar rules for manipulating line segments and greatly improved symbolism and algebraic technique. Thus, it seemed that numbers might be put into one-to-one correspondence with points on a geometric line, leading to a significant step in the evolution of the concept of number." (Venzo de Sabbata & Bidyut K Datta, "Geometric Algebra and Applications to Physics", 2007) 

𝕏On Algebra (1800-1849)

"It is to be desired, that the charges of paradox and mystery, said to be introduced into algebra by negative and impossible quantities, should be proposed distinctly, in a precise form, fit to be apprehended and made the subject of discussion." (Robert Woodhouse," On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)

"Every negative quantity standing by itself is a mere creature of the mind and [...] those which are met with in calculations are only mere algebraical forms, incapable of representing any thing real and effective." (Lazare Carnot, "Geometrie de Position, 1803)

"Metaphor [...] may be said to be the algebra of language." (Charles C Colton, "Lacon", 1820)

"We may always depend upon it that algebra, which cannot be translated into good English and sound common sense, is bad algebra." (William K Clifford, "The Common Sense of the Exact Sciences", 1823)

"The mathematicians have been very much absorbed with finding the general solution of algebraic equations, and several of them have tried to prove the impossibility of it. However, if I am not mistaken, they have not as yet succeeded. I therefore dare hope that the mathematicians will receive this memoir with good will, for its purpose is to fill this gap in the theory of algebraic equations." (Niels H Abel,"Memoir on algebraic equations, proving the impossibility of a solution of the general equation of the fifth degree", 1824)

"Unless my Algebra deceive me, Unity itself divided by Zero will give Infinity." (Thomas Carlyle, "Sartor Resartus", 1836)

"Arithmetic has for its object the properties of number in the abstract. In algebra, viewed as a science of operations, order is the predominating idea. The business of geometry is with the evolution of the properties of space, or of bodies viewed as existing in space." (James J Sylvester, "A Probationary Lecture on Geometry", 1844)

"The first thing to be attended to in reading any algebraical treatise, is the gaining a perfect understanding of the different processes there exhibited, and of their connection with one another. This cannot be attained by a mere reading of the book, however great the attention which may be given. It is impossible, in a mathematical work, to fill up every process in the manner in which it must be filled up in the mind of the student before he can be said to have completely mastered it. Many results must be given of which the details are suppressed, such are the additions, multiplications, extractions of the square root, etc., with which the investigations abound. These must not be taken on trust by the student, but must be worked by his own pen, which must never be out of his hand, while engaged in any algebraical process." (Augustus de Morgan,"On the Study and Difficulties of Mathematics", 1830)

"Instead of seeking to attain consistency and uniformity of system, as some modern writers have attempted, by banishing this thought of time from the higher Algebra, I seek to attain the same object, by systematically introducing it into the lower or earlier parts of the science." (William R Hamilton, "Theory of Conjugate Functions, or Algebraic Couples; with a Preliminary and Elementary Essay on Algebra as the Science of Pure Time", Transactions of the Royal Irish Academy, 1837

"And feeling that it must be possible to introduce Time into Algebra, in some such sense as Space is introduced into Geometry, I had at least an unpublished fancy, of which I have sometimes spoken with my friends, that by banishing this thought of time (or motion) from the higher Algebra to the utmost of my power, I might perhaps be enabled to return to it with a better chance of success." (William R Hamilton, [letter on quaternions John T. Graves] 1843)

"Those who can, in common algebra, find a square root of -1, will be at no loss to find a fourth dimension in space in which ABC may become ABCD: or, if they cannot find it, they have but to imagine it, and call it an impossible dimension, subject to all the laws of the three we find possible. And just as √-1 in common algebra, gives all its significant combinations true, so would it be with any number of dimensions of space which the speculator might choose to call into impossible existence." (Augustus De Morgan, "Trigonometry and Double Algebra", 1849)

On Allgebra (2000-)

"The equation e^πi =-1 says that the function w= e^z, when applied to the complex number πi as input, yields the real number -1 as the output, the value of w. In the complex plane, πi is the point [0,π) - π on the i-axis. The function w=e^z maps that point, which is in the z-plane, onto the point (-1, 0) - that is, -1 on the x-axis-in the w-plane. […] But its meaning is not given by the values computed for the function w=e^z. Its meaning is conceptual, not numerical. The importance of  e^πi =-1 lies in what it tells us about how various branches of mathematics are related to one another - how algebra is related to geometry, geometry to trigonometry, calculus to trigonometry, and how the arithmetic of complex numbers relates to all of them." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being", 2000)

"At present, high school education in some countries covers only little, if any, statistical thinking. Algebra, geometry, and calculus teach thinking in a world of certainty - not in the real world, which is uncertain. [...] Furthermore, in the medical and social sciences, data analysis is typically taught as a set of statistical rituals rather than a set of methods for statistical thinking." (Gerd Gigerenzer, "Calculated Risks: How to know when numbers deceive you", 2002)

“Algebra is the offer made by the devil to the mathematician. The devil says: I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvelous machine.” (Michael F Atiyah, 2004)

"Calculating with letters instead of numbers is a big step forward in everyone’s education. It is rightly appreciated as a step from the concrete to the abstract, from the particular to the general, from arithmetic to algebra; but it is not always recognized as a step from confusion to clarity. To appreciate the clarity of algebra, ask yourself: what are the rules for calculating with numbers?" (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)

"Poetry and code - and mathematics - make us read differently from other forms of writing. Written poetry makes the silent reader read three kinds of pattern at once; code moves the reader from a static to an active, interactive and looped domain; while algebraic topology allows us to read qualitative forms and their transformations." (Stephanie Strickland & Cynthia L Jaramillo, "Dovetailing Details Fly Apart - All over, again, in code, in poetry, in chreods", 2007)

"Linear algebra is a very useful subject, and its basic concepts arose and were used in different areas of mathematics and its applications. It is therefore not surprising that the subject had its roots in such diverse fields as number theory (both elementary and algebraic), geometry, abstract algebra (groups, rings, fields, Galois theory), anal ysis (differential equations, integral equations, and functional analysis), and physics. Among the elementary concepts of linear algebra are linear equations, matrices, determinants, linear transformations, linear independence, dimension, bilinear forms, quadratic forms, and vector spaces. Since these concepts are closely interconnected, several usually appear in a given context (e.g., linear equations and matrices) and it is often impossible to disengage them." (Israel Kleiner, "A History of Abstract Algebra", 2007)

"What was clearly useful was the use of diagrams to prove certain results either in algebraic topology, homological algebra or algebraic geometry. It is clear that doing category theory, or simply applying category theory, implies manipulating diagrams: constructing the relevant diagrams, chasing arrows by going via various paths in diagrams and showing they are equal, etc. This practice suggests that diagram manipulation, or more generally diagrams, constitutes the natural syntax of category theory and the category-theoretic way of thinking. Thus, if one could develop a formal language based on diagrams and diagrams manipulation, one would have a natural syntactical framework for category theory. However, moving from the informal language of categories which includes diagrams and diagrammatic manipulations to a formal language based on diagrams and diagrammatic manipulations is not entirely obvious." (Jean-Pierre Marquis, "From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory", 2009)

"The tangling and untangling of numerical relationships is called algebra. […] The point of doing algebra is not to solve equations; it’s to allow us to move back and forth between several equivalent representations, depending on the situation at hand and depending on our taste. In this sense, all algebraic manipulation is psychological. The numbers are making themselves known to us in various ways, and each different representation has its own feel to it and can give us ideas that might not occur to us otherwise." (Paul Lockhart, "Measurement", 2012)

"To mathematicians who study them, moduli schemes are just as real as the regular objects in the world. […] The key idea is that an ordinary object can be studied using the set of functions on the object. […] Secondly, you can do algebra with these functions - that is, you can add or multiply two such functions and get a third function. This step makes the set of these functions into a ring. […] Then the big leap comes: If you start with any ring - that is, any set of entities that can be added and multiplied subject to the usual rules, you simply and brashly declare that this creates a new kind of geometric object. The points of the object can be given by maps from the ring to the real numbers, as in the example of the pot. But they may also be given by maps to other fields. A field is a special sort of ring in which division is possible." (David Mumford, ["The Best Writing of Mathematics: 2012"] 2012

"Geometry had its origins in the interest of working with lines, figures, and solids that could be imagined in the mind. Algebra had its origins in problems involving number - number hinged by geometric conceptions of iconic figures." (Joseph Mazur, "Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers", 2014)

"Geometry had its origins in the interest of working with lines, figures, and solids that could be imagined in the mind. Algebra had its origins in problems involving number - number hinged by geometric conceptions of iconic figures." (Joseph Mazur, "Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers", 2014)

"Mathematics is a technique, a tool, albeit a sophisticated one. Theory is something different. Theory lies in the discovery, understanding, and explaining of phenomena present in the world. Mathematics facilitates this - enormously - but then so does computation. Naturally, there is a difference. Working with equations allows us to follow an argument step by step and reveals conditions a solution must adhere to, whereas computation does not. But computation - and this more than compensates - allows us to see phenomena that equilibrium mathematics does not. It allows us to rerun results under different conditions, exploring when structures appear and don’t appear, isolating underlying mechanisms, and simplifying again and again to extract the bones of a phenomenon. Computation in other words is an aid to thought, and it joins earlier aids in economics - algebra, calculus, statistics, topology, stochastic processes - each of which was resisted in its time. The computer is an exploratory lab for economics, and used skillfully, a powerful generator for theory." (W Brian Arthur, "Complexity and the Economy", 2015)

"The primary aspects of the theory of complex manifolds are the geometric structure itself, its topological structure, coordinate systems, etc., and holomorphic functions and mappings and their properties. Algebraic geometry over the complex number field uses polynomial and rational functions of complex variables as the primary tools, but the underlying topological structures are similar to those that appear in complex manifold theory, and the nature of singularities in both the analytic and algebraic settings is also structurally very similar." (Raymond O Wells Jr, "Differential and Complex Geometry: Origins, Abstractions and Embeddings", 2017)

"Homology translates this world of vague shapes into the rigorous world of algebra, a branch of mathematics that studies particular numerical structures and symmetries. Mathematicians study the properties of these algebraic structures in a field known as homological algebra. From the algebra they indirectly learn information about the original topological shape of the data. Homology comes in many varieties, all of which connect with algebra." (Kelsey Houston-Edwards, "How Mathematicians Use Homology to Make Sense of Topology", Quanta Magazine, 2021) [

𝕏On Algebra (1825-1849)

"Moreover, the whole method has the essential disadvantage that it occupies the mind with the distinction of a great number of cases that can be recognized only by inner intuition, and thus neutralizes an important part of that which algebra is supposed to accomplish, which is relieving the power of inner intuition. Finally, in such a treatment algebra loses a great part of the generality that it can obtain by the mutual connection of different problems, which becomes evident so easily when one uses isolated negative quantities. [...] Since imaginary quantities have to occur, science would certainly not win that much by avoiding negative quantities than it would lose in terms of clarity and generality." (Johann P W Stein,  "Die Elemente der Algebra: Erster Cursus", 1828)

"In symbolical algebra, the rules determine the meaning of the operations […] we might call them arbitrary assumptions, inasmuch as they are arbitrarily imposed upon a science of symbols and their combinations, which might be adapted to any other assumed system of consistent rules." (George Peacock, "Treatise of Algebra", 1830)

”The first thing to be attended to in reading any algebraical treatise, is the gaining a perfect understanding of the different processes there exhibited, and of their connection with one another. This cannot be attained by a mere reading of the book, however great the attention which may be given. It is impossible, in a mathematical work, to fill up every process in the manner in which it must be filled up in the mind of the student before he can be said to have completely mastered it. Many results must be given of which the details are suppressed, such are the additions, multiplications, extractions of the square root, etc., with which the investigations abound. These must not be taken on trust by the student, but must be worked by his own pen, which must never be out of his hand, while engaged in any algebraical process.” (Augustus de Morgan, “On the Study and Difficulties of Mathematics”, 1830)

"Algebra, as an art, can be of no use to any one in the business of life; certainly not as taught in the schools. I appeal to every man who has been through the school routine whether this be not the case. Taught as an art it is of little use in the higher mathematics, as those are made to feel who attempt to study the differential calculus without knowing more of the principles than is contained in books of rules." (Augustus de Morgan, "Elements of Algebra", 1837)

“The science of algebra, independently of any of its uses, has all the advantages which belong to mathematics in general as an object of study, and which it is not necessary to enumerate. Viewed either as a science of quantity, or as a language of symbols, it may be made of the greatest service to those who are sufficiently acquainted with arithmetic, and who have sufficient power of comprehension to enter fairly upon its difficulties.” (Augustus de Morgan, “Elements of Algebra”, 1837)

"These sciences, Geometry, Theoretical Arithmetic and Algebra, have no principles besides definitions and axioms, and no process of proof but deduction; this process, however, assuming a most remarkable character; and exhibiting a combination of simplicity and complexity, of rigour and generality, quite unparalleled in other subjects." (William Whewell, "The Philosophy of the Inductive Sciences", 1840)

“Arithmetic has for its object the properties of number in the abstract. In algebra, viewed as a science of operations, order is the predominating idea. The business of geometry is with the evolution of the properties of space, or of bodies viewed as existing in space.” (James J Sylvester, “A Probationary Lecture on Geometry”, 1844)

"Those who can, in common algebra, find a square root of -1, will be at no loss to find a fourth dimension in space in which ABC may become ABCD: or, if they cannot find it, they have but to imagine it, and call it an impossible dimension, subject to all the laws of the three we find possible. And just as √-1 in common algebra, gives all its significant combinations true, so would it be with any number of dimensions of space which the speculator might choose to call into impossible existence." (Augustus De Morgan, "Trigonometry and Double Algebra", 1849)

𝕏On Algebra (1975-1999)

"The philosophical emphasis here is: to solve a geometrical problem of a global nature, one first reduces it to a homotopy theory problem; this is in turn reduced to an algebraic problem and is solved as such. This path has historically been the most fruitful one in algebraic topology." (Brayton Gray, "Homotopy Theory", Pure and Applied Mathematics Vol 64, 1975)

“In mathematics itself abstract algebra plays a dual role: that of a unifying link between disparate parts of mathematics  and that of a research subject with a highly active life of its own.” (Israel N Herstein, ”Abstract Algebra”, 1986)

"Mathematics is more than doing calculations, more than solving equations, more than proving theorems, more than doing algebra, geometry or calculus, more than a way of thinking. Mathematics is the design of a snowflake, the curve of a palm frond, the shape of a building, the joy of a game, the frustration of a puzzle, the crest of a wave, the spiral of a spider's web. It is ancient and yet new. Mathematics is linked to so many ideas and aspects of the universe." (Theoni Pappas, "More Joy of Mathematics: Exploring Mathematics All Around You", 1986)

"Instead of a state of nature evolving according to a mathematical fomula, the evolution is given geometrically. The full advantage of the geometrical point of view is beginning to appear. The more traditional way of dealing with dynamics was with the use of algebraic expressions. But a description given by formulae would be cumbersome. That form of description wouldn't have led me to insights or to perceptive analysis. My background as a topologist, trained to bend objects like squares, helped to make it possible to see the horseshoe." (Steven Smale, "What is chaos?", 1990)

"The value of diagram techniques even at this rudimentary level should be clear by now: it is easier to visualize where simplifications may be found in a complicated network by searching for a reducible linkage than by examining a complicated algebraic expression."(Geoffrey E Stedman, "Diagram Techniques in Group Theory", 1990)

"Symmetries of a geometric object are traditionally described by its automorphism group, which often is an object of the same geometric class (a topological space, an algebraic variety, etc.). Of course, such symmetries are only a particular type of morphisms, so that Klein’s Erlanger program is, in principle, subsumed by the general categorical approach." (Yuri I Manin, "Topics in Noncommutative Geometry", 1991)

"In most fields of mathematics, we study some class of objects and also the functions appropriate to the objects of study. In linear algebra, the objects studied are vector spaces and the corresponding functions, called linear transformations and usually represented by matrices, are those which preserve the vector space structure. In geometry, we study isometries, such as rotations: functions that do not change the geometric properties of length, angle measure, area, and volume. In abstract algebra, the objects of study are groups and the appropriate functions are homomorphisms, which preserve the algebraic properties of the groups. The best possible homomorphism is an isomorphism; isomorphic groups are essentially identical algebraically." (L Christine Kinsey. "Topology of Surfaces", 1993)

"Algebra is a kind of symbolic reasoning that works with numbers without knowing their actual values. Proofs provide a once-and-for-all guarantee that certain lines of reasoning always work." (Ian Stewart, "The Magical Maze: Seeing the world through mathematical eyes", 1997)

"We believe that numeracy is about making meaning in mathematics and being critical about maths. This view of numeracy is very different from numeracy just being about numbers, and it is a big step from numeracy or everyday maths that meant doing some functional maths. It is about using mathematics in all its guises - space and shape, measurement, data and statistics, algebra, and of course, number - to make sense of the real world, and using maths critically and being critical of maths itself. It acknowledges that numeracy is a social activity. That is why we can say that numeracy is not less than maths but more. It is why we don’t need to call it critical numeracy being numerate is being critical." (Dave Tout & Beth Marr, "Changing practice: Adult numeracy professional development", 1997)

"Mathematics, in the common lay view, is a static discipline based on formulas taught in the school subjects of arithmetic, geometry, algebra, and calculus. But outside public view, mathematics continues to grow at a rapid rate, spreading into new fields and spawning new applications. The guide to this growth is not calculation and formulas but an open-ended search for pattern." (Lynn A Steen, "The Future of Mathematics Education", 1998)

"Algebraic topology studies properties of a narrower class of spaces, - basically the classical objects of mathematics: spaces given by systems of algebraic and functional equations, surfaces lying in Euclidean space, and other sets which in mathematics are called manifolds. Examining the narrower class of spaces permits deeper penetration into their structure. (Michael I Monastyrsky, "Riemann, Topology, and Physics", 1999)

𝕏On Algebra (1950-1974)

"Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras are geometric facts which are proved.” (Omar Khayyam [quoted by J.J. Winter and W. Arafat, “The Algebra of ‘Umar Khayyam’”, Journal of the Royal Asiatic Society of Bengal, Volume 41, 1950)

"The designer employing Boolean algebra is in possession of a list of theorems which may be used in simplifying the expression before him; but he may not know which ones to try first, or to which terms to apply them. He is thus forced to consider a very large number of alternative procedures in all but the most trivial cases. It is clear that a method which provides more insight into the structure of each problem is to be preferred." (Maurice Karnaugh, "The map method for synthesis of combinational logic circuits", Transactions of the American Institute of Electrical Engineers Pt 1 72 (9), 1953) 

"During the last decade the methods of algebraic topology have invaded extensively the domain of pure algebra, and initiated a number of internal revolutions. 

[...] The invasion of algebra has occurred on three fronts through the construction of cohomology theories for groups, Lie algebras, and associative algebras. The three subjects have been given independent but parallel developments." (Henri P Cartan & Samuel Eilenberg, "Homological Algebra", 1956)

“The word ‘imaginary’ is the great algebraical calamity, but it is too well established for mathematicians to eradicate. It should never have been used. Books on elementary algebra give a simple interpretation of imaginary numbers in terms of rotations. […] Although the interpretation by means of rotations proves nothing, it may suggest that there is no occasion for anyone to muddle himself into a state of mystic wonderment over nothing about the grossly misnamed ‘imaginaries’.” (Philip E B Jourdain, “The Nature of Mathematics” in [James R Newman, “The World of Mathematics” Vol. I, 1956])

”Behind these symbols lie the boldest, purest, coolest abstractions mankind has ever made. No schoolman speculating on essences and attributes ever approached anything like the abstractness of algebra.” (Susanne K Langer, “Philosophy in a New Key”, 1957)

"A logic machine is a device, electrical or mechanical, designed specifically for solving problems in formal logic. A logic diagram is a geometrical method for doing the same thing. […] A logic diagram is a two-dimensional geometric figure with spatial relations that are isomorphic with the structure of a logical statement. These spatial relations are usually of a topological character, which is not surprising in view of the fact that logic relations are the primitive relations underlying all deductive reasoning and topological properties are, in a sense, the most fundamental properties of spatial structures. Logic diagrams stand in the same relation to logical algebras as the graphs of curves stand in relation to their algebraic formulas; they are simply other ways of symbolizing the same basic structure." (Martin Gardner, "Logic Machines and Diagrams", 1958)

"It is possible, however, to express geometric facts by using algebraic methods. In other words, the properties of figures and the relationships between them can be described by means of numbers and equations. Conversely, algebraic expressions and equations can be represented or interpreted geometrically. When we translate from the language of geometry to the language of algebra, or vice versa, we are using analytical methods. Geometry when studied in this way is called analytic geometry." (William L Schaaf, "Analytic Geometry; a college course guide", 1962)

"Essentially, algebraic theories are an invariant notion of which the usual formalism with operations and equations may be regarded as 'presentation'." (F William Lawvere, "Functorial Semantics of Algebraic Theories", 1963)

"Categorical algebra has developed in recent years as an effective method of organizing parts of mathematics. Typically, this sort of organization uses notions such as that of the category G of all groups. [...] This raises the problem of finding some axiomatization of set theory - or of some foundational discipline like set theory - which will be adequate and appropriate to realizing this intent. This problem may turn out to have revolutionary implications vis-`a-vis the accepted views of the role of set theory." (Saunders Mac Lane, Categorical algebra and set-theoretic foundations, 1967)

"The history of arithmetic and algebra illustrates one of the striking and curious features of the history of mathematics. Ideas that seem remarkably simple once explained were thousands of years in the making." (Morris Kline, "Mathematics for the Nonmathematician", 1967)

“[…] algebra is the intellectual instrument which has been created for rendering clear the quantitative aspects of the world.” (Simone Weil, “The Organization of Thought”, 1974)

𝕏On Algebra (1925-1949)

"What had already been done for music by the end of the eighteenth century has at last been begun for the pictorial arts. Mathematics and physics furnished the means in the form of rules to be followed and to be broken. In the beginning it is wholesome to be concerned with the functions and to disregard the finished form. Studies in algebra, in geometry, in mechanics characterize teaching directed towards the essential and the functional, in contrast to apparent. One learns to look behind the façade, to grasp the root of things. One learns to recognize the undercurrents, the antecedents of the visible. One learns to dig down, to uncover, to find the cause, to analyze." (Paul Klee, "Bauhaus prospectus", 1929)

"The word is of Arabic origin. ‘Al’ is the Arabic article the, and ‘gebar’ is the verb to set, to restitute." (Tobias Dantzig & Joseph Mazur, “Number: The Language of Science”, 1930)

 “By the help of God and with His precious assistance I say that algebra is a scientific art. The objects with which it deals are absolute numbers and (geometrical) magnitudes which, though themselves unknown, are related to things which are known, whereby the determination of the unknown quantities is possible. Such a thing is either a quantity or a unique relation, which is only determined by careful examination. […] What one searches for in the algebraic art are the relations which lead from the known to the unknown, to discover which is the object of algebra as stated above.” (Omar Khayyam [quoted by Daoud Suleiman Kasir in “The Algebra of Omar Khayyam”, 1931)

"The steady progress of physics requires for its theoretical formulation a mathematics which get continually more advanced. […] it was expected that mathematics would get more and more complicated, but would rest on a permanent basis of axioms and definitions, while actually the modern physical developments have required a mathematics that continually shifts its foundation and gets more abstract. Non-Euclidean geometry and noncommutative algebra, which were at one time were considered to be purely fictions of the mind and pastimes of logical thinkers, have now been found to be very necessary for the description of general facts of the physical world. It seems likely that this process of increasing abstraction will continue in the future and the advance in physics is to be associated with continual modification and generalisation of the axioms at the base of mathematics rather than with a logical development of any one mathematical scheme on a fixed foundation." (Paul A M Dirac, "Quantities singularities in the electromagnetic field", Proceedings of the Royal Society of London, 1931)

"The theory of probability as a mathematical discipline can and should be developed from axioms in exactly the same way as geometry and algebra." (Andrey Kolmogorov, "Foundations of the Theory of Probability", 1933)

“Every moment of time dictated and determined the following moment, and was itself dictated and determined by the moment that came before it. Everything was calculable: everything happened because it must; the commandments were erased from the tables of the law; and in their place came the cosmic algebra: the equations of the mathematicians.” (George Bernard Shaw, “Too True to Be Good”, 1934)

“Algebra is applied to the clouds, the irradiation of the planet benefits the rose, and no thinker would dare to say that the perfume of the hawthorn is useless to the constellation.” (Victor Hugo, “Les Miserables”, 1938)

"Algebra tends to the study of the explicit structure of postulationally defined systems closed with respect to one or more rational operations." (George D Birkhoff, "Some Recent Advances in Algebra", The American Mathematical Monthly Vol. 46, 1939)

"The methods of algebra are essentially an extension of arithmetic. In other words, the numbers, symbols and operations used in algebra are the same as those used in arithmetic, only they are more general in character. This means (1) that letters as well as numbers are are used te to represent quantities, and (2) that numbers are regarded as having quality as well as quantity." (William L Schaaf, "Mathematics for Mechanics", 1942)

"The invariant character of a mathematical discipline can be formulated in these terms. Thus, in group theory all the basic constructions can be regarded as the definitions of co- or contravariant functors, so we may formulate the dictum: The subject of group theory is essentially the study of those constructions of groups which behave in a covariant or contravariant manner under induced homomorphisms." (Samuel Eilenberg & Saunders Mac Lane, "A general theory of natural equivalences", Transactions of the American Mathematical Society 58, 1945)

"The subject of group theory is essentially the study of those constructions of groups which behave in a covariant or contravariant manner under induced homomorphisms. More precisely, group theory studies functors defined on well specified categories of groups, with values in another such category." (Samuel Eilenberg & Saunders Mac Lane, "A general theory of natural equivalences", Transactions of the American Mathematical Society 58, 1945)

“Algebra reverses the relative importance of the factors in ordinary language. It is essentially a written language, and it endeavors to exemplify in its written structures the patterns which it is its purpose to convey. The pattern of the marks on paper is a particular instance of the pattern to be conveyed to thought. The algebraic method is our best approach to the expression of necessity, by reason of its reduction of accident to the ghost-like character of the real variable.” (Alfred N Whitehead, “Essays in Science and Philosophy”, 1948

𝕏On Algebra (1900-1924)

"A knowledge of statistics is like a knowledge of foreign languages or of algebra; it may prove of use at any time under any circumstances." (Sir Arthur L Bowley, "Elements of Statistics", 1901)

"All the modern higher mathematics is based on a calculus of operations, on laws of thought. All mathematics, from the first, was so in reality; but the evolvers of the modern higher calculus have known that it is so. Therefore elementary teachers who, at the present day, persist in thinking about algebra and arithmetic as dealing with laws of number, and about geometry as dealing with laws of surface and solid content, are doing the best that in them lies to put their pupils on the wrong track for reaching in the future any true understanding of the higher algebras. Algebras deal not with laws of number, but with such laws of the human thinking machinery as have been discovered in the course of investigations on numbers. Plane geometry deals with such laws of thought as were discovered by men intent on finding out how to measure surface; and solid geometry with such additional laws of thought as were discovered when men began to extend geometry into three dimensions." (Mary E Boole, "Lectures on the Logic of Arithmetic", 1903)

"But in the mathematical or pure sciences, - geometry, arithmetic, algebra, trigonometry, the calculus of variations or of curves, - we know at least that there is not, and cannot be, error in our first principles, and we may therefore fasten our whole attention upon the processes. As mere exercises in logic, therefore, these sciences, based as they all are on primary truths relating to space and number, have always been supposed to furnish the most exact discipline." (Joshua Fitch,"Lectures on Teaching", 1906)

"Today it is no longer questioned that the principles of the analysts are the more far-reaching. Indeed, the synthesists lack two things in order to engage in a general theory of algebraic configurations: these are on the one hand a definition of imaginary elements, on the other an interpretation of general algebraic concepts. Both of these have subsequently been developed in synthetic form, but to do this the essential principle of synthetic geometry had to be set aside. This principle which manifests itself so brilliantly in the theory of linear forms and the forms of the second degree, is the possibility of immediate proof by means of visualized constructions." (Felix Klein, "Riemannsche Flächen", 1906)

"The method of solving problems by honest confession of one’s ignorance is called Algebra." (Mary E Boole, "Philosophy and Fun of Algebra", 1909)

"In the beginning of algebra, even the most intelligent child finds, as a rule, very great difficulty. The use of letters is a mystery, which seems to have no purpose except mystification. It is almost impossible, at first, not to think that every letter stands for some particular number, if only the teacher would reveal what number it stands for." (Bertrand Russell, "Mysticism and Logic: And Other Essays", 1910)

"Geometry formerly was the chief borrower from arithmetic and algebra, but it has since repaid its obligation with abundant usury; and if I were asked to name, in one word, the pole-star round which the mathematical firmament revolves, the central idea which pervades as a hidden spirit the whole corpus of mathematical doctrine, I should point to Continuity as contained in our notions of space, and say, it is this, it is this!" (James J. Sylvester, Presidential Address to the British Association, [The Collected Mathematical Papers of James Joseph Sylvester Vol. 2, cca. 1904–1912])

"The ends to be attained [in mathematical teaching] are the knowledge of a body of geometrical truths to be used In the discovery of new truths, the power to draw correct inferences from given premises, the power to use algebraic processes as a means of finding results in practical problems, and the awakening of interest In the science of mathematics." (J Craig, "A Course of Study for the Preparation of Rural School Teachers", 1912)

"Time was when all the parts of the subject were dissevered, when algebra, geometry, and arithmetic either lived apart or kept up cold relations of acquaintance confined to occasional calls upon one another; but that is now at an end; they are drawn together and are constantly becoming more and more intimately related and connected by a thousand fresh ties, and we may confidently look forward to a time when they shall form but one body with one soul." (James J. Sylvester, Presidential Address to the British Association, [The Collected Mathematical Papers of James Joseph Sylvester Vol. 2, cca. 1904–1912])

"The ends to be attained [in mathematical teaching] are the knowledge of a body of geometrical truths to be used. In the discovery of new truths, the power to draw correct inferences from given premises, the power to use algebraic processes as a means of finding results in practical problems, and the awakening of interest In the science of mathematics." (J Craig, "A Course of Study for the Preparation of Rural School Teachers", 1912)

"This diagrammatic method has, however, serious inconveniences as a method for solving logical problems. It does not show how the data are exhibited by cancelling certain constituents, nor does it show how to combine the remaining constituents so as to obtain the consequences sought. In short, it serves only to exhibit one single step in the argument, namely the equation of the problem; it dispenses neither with the previous steps, i.e., 'throwing of the problem into an equation' and the transformation of the premises, nor with the subsequent steps, i.e., the combinations that lead to the various consequences. Hence it is of very little use, inasmuch as the constituents can be represented by algebraic symbols quite as well as by plane regions, and are much easier to deal with in this form." (Louis Couturat, "The Algebra of Logic", 1914)

𝕏On Algebra (1875-1899)

"The most striking characteristic of the written language of algebra and of the higher forms of the calculus is the sharpness of definition, by which we are enabled to reason upon the symbols by the mere laws of verbal logic, discharging our minds entirely of the meaning of the symbols, until we have reached a stage of the process where we desire to interpret our results. The ability to attend to the symbols, and to perform the verbal, visible changes in the position of them permitted by the logical rules of the science, without allowing the mind to be perplexed with the meaning of the symbols until the result is reached which you wish to interpret, is a fundamental part of what is called analytical power. Many students find themselves perplexed by a perpetual attempt to interpret not only the result, but each step of the process. They thus lose much of the benefit of the labor-saving machinery of the calculus and are, indeed, frequently incapacitated for using it." (Thomas Hill, "Uses of Mathesis", Bibliotheca Sacra Vol. 32 (127), 1875)

"Some definite interpretation of a linear algebra would, at first sight, appear indispensable to its successful application. But on the contrary, it is a singular fact, and one quite consonant with the principles of sound logic, that its first and general use is mostly to be expected from its want of significance. The interpretation is a trammel to the use. Symbols are essential to comprehensive argument." (Benjamin Peirce, "On the Uses and Transformations of Linear Algebra", 1875)

"When the formulas admit of intelligible interpretation, they are accessions to knowledge; but independently of their interpretation they are invaluable as symbolical expressions of thought. But the most noted instance is the symbol called the impossible or imaginary, known also as the square root of minus one, and which, from a shadow of meaning attached to it, may be more definitely distinguished as the symbol of semi-inversion. This symbol is restricted to a precise signification as the representative of perpendicularity in quaternions, and this wonderful algebra of space is intimately dependent upon the special use of the symbol for its symmetry, elegance, and power." (Benjamin Peirce, "On the Uses and Transformations of Linear Algebra", 1875)

"’Divide et impera’ is as true in algebra as in statecraft; but no less true and even more fertile is the maxim ‘auge et impera’. The more to do or to prove, the easier the doing or the proof." (James J Sylvester, "Proof of the Fundamental Theorem of Invariants", Philosophic Magazine, 1878)

“’Divide et impera’ is as true in algebra as in statecraft; but no less true and even more fertile is the maxim ‘auge et impera’. The more to do or to prove, the easier the doing or the proof.” (James J Sylvester, “Proof of the Fundamental Theorem of Invariants”, Philosophic Magazine, 1878)

“Algebra is but written geometry and geometry is but figured algebra.” (Sophie Germain, Mémoire sur les Surfaces Élastiques”, 1880)

"This symbol [v-1] is restricted to a precise signification as the representative of perpendicularity in quaternions, and this wonderful algebra of space is intimately dependent upon the special use of the symbol for its symmetry, elegance, and power. The immortal author of quaternions has shown that there are other significations which may attach to the symbol in other cases. But the strongest use of the symbol is to be found in its magical power of doubling the actual universe, and placing by its side an ideal universe, its exact counterpart, with which it can be compared and contrasted, and, by means of curiously connecting fibres, form with it an organic whole, from which modern analysis has developed her surpassing geometry." (Benjamin Peirce, "On the Uses and Transformations of Linear Algebras", American Journal of Mathematics Vol. 4, 1881)

“As is known, scientific physics dates its existence from the discovery of the differential calculus. Only when it was learned how to follow continuously the course of natural events, attempts, to construct by means of abstract conceptions the connection between phenomena, met with success. To do this two things are necessary: First, simple fundamental concepts with which to construct; second, some method by which to deduce, from the simple fundamental laws of the construction which relate to instants of time and points in space, laws for finite intervals and distances, which alone are accessible to observation (can be compared with experience).” (Bernhard Riemann, “Die partiellen Differentialgleichungen der mathematischen Physik”, 1882)

“We may always depend on it that algebra, which cannot be translated into good English and sound common sense, is bad algebra.”  (William K Clifford, “Common Sense of the Exact Sciences”, 1885)

“A satisfactory theory of the imaginary quantities of ordinary algebra, which is essentially a simple case of multiple algebra, with difficulty obtained recognition in the first third of this century. We must observe that this double algebra, as it has been called, was not sought for or invented; - it forced itself, unbidden, upon the attention of mathematicians, and with its rules already formed.
But the idea of double algebra, once received, although as it were unwillingly, must have suggested to many minds more or less distinctly the possibility of other multiple algebras, of higher orders, possessing interesting or useful properties.” (Josiah W Gibbs, “On multiple Algebra”, Proceedings of the American Association for the Advancement of Science Vol. 35, 1886)

"In science nothing capable of proof ought to be accepted without proof. Though this demand seems so reasonable yet I cannot regard it as having been met even in […] that part of logic which deals with the theory of numbers. In speaking of arithmetic (algebra, analysis) as a part of logic I mean to imply that I consider the number concept entirely independent of the notions of intuition of space and time, that I consider it an immediate result from the laws of thought." (Richard Dedekind, "Was sind und was sollen die Zahlen?", 1888)

"Mathematics in its pure form, as arithmetic, algebra, geometry, and the applications of the analytic method, as well as mathematics applied to matter and force, or statics and dynamics, furnishes the peculiar study that gives to us, whether as children or as men, the command of nature in this its quantitative aspect; mathematics furnishes the instrument, the tool of thought, which we wield in this realm.” (William T  Harris, “Psychologic Foundations of Education”, 1898) 

"The elements of plane geometry should precede algebra for every reason known to sound educational theory. It is more fundamental, it is more concrete, and it deals with things and their relations rather than with symbols." (Nicholas M Butler, "The Meaning of Education, and Other Essays and Addresses", 1898)