"√-1 is take for granted today. No serious mathematician would deny that it is a number. Yet it took centuries for √-1 to be officially admitted to the pantheon of numbers. For almost three centuries, it was controversial; mathematicians didn't know what to make of it; many of them worked with it successfully without admitting its existence. […] Primarily for cognitive reasons. Mathematicians simply could not make it fit their idea of what a number was supposed to be. A number was supposed to be a magnitude. √-1 is not a magnitude comparable to the magnitudes of real numbers. No tree can be √-1 units high. You cannot owe someone √-1 dollars. Numbers were supposed to be linearly ordered. √-1 is not linearly ordered with respect to other numbers." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being, 2000)
"By definition, a Kähler manifold is one with a complex structure (this means in particular that the coordinates changes are holomorphic for the complex coordinates) together with a Riemannian metric which has with this complex structure the best possible link, namely that multiplication of tangent vectors by unit complex numbers preserves the metric, but moreover the complex structure is invariant under parallel transport. This is equivalent to the condition that the holonomy group be included in the unitary group, hence equivalent also to ask for the existence of a 2-form of maximal rank and of zero covariant derivative."(Marcel Berger, "A Panoramic View of Riemannian Geometry", 2003)
"Descartes’ idea to use numbers to describe points in space involves the choice of a coordinate system or coordinate frame: an origin, together with axes and units of length along the axes. A recurring theme of all the different geometries [...] is the question of what a coordinate frame is, and what I can get out of it. While coordinates provide a convenient framework to discuss points, lines, and so on, it is a basic requirement that any meaningful statement in geometry is independent of the choice of coordinates. That is, coordinate frames are a humble technical aid in determining the truth, and are not allowed the dignity of having their own meaning." (Miles Reid & Balazs Szendröi, "Geometry and Topology", 2005)
"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)
"The correlation coefficient has two fabulously attractive characteristics. First, for math reasons that have been relegated to the appendix, it is a single number ranging from –1 to 1. A correlation of 1, often described as perfect correlation, means that every change in one variable is associated with an equivalent change in the other variable in the same direction. A correlation of –1, or perfect negative correlation, means that every change in one variable is associated with an equivalent change in the other variable in the opposite direction. The closer the correlation is to 1 or –1, the stronger the association. […] The second attractive feature of the correlation coefficient is that it has no units attached to it. […] The correlation coefficient does a seemingly miraculous thing: It collapses a complex mess of data measured in different units" (like our scatter plots of height and weight) into a single, elegant descriptive statistic." (Charles Wheelan, "Naked Statistics: Stripping the Dread from the Data", 2012)
"At any rate, long before the curvature of space was first detected, Beltrami’s construction of the hyperbolic plane showed that more than one kind of geometry is possible. Beltrami assumed that Euclidean space exists, and constructed a non-Euclidean plane inside it, with nonstandard definitions of 'line' and 'distance' (namely, line segments in the unit disk and pseudodistance). This shows that the geometry of Bolyai and Lobachevsky is logically as valid as the geometry of Euclid: if there is a space in which 'lines' and 'distance' behave as Euclid thought they do, then there is also a surface in which 'lines' and 'distance' behave as Bolyai and Lobachevsky thought they might." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics" 2nd Ed., 2018)
"The higher the dimension, in other words, the higher the number of possible interactions, and the more disproportionally difficult it is to understand the macro from the micro, the general from the simple units. This disproportionate increase of computational demands is called the curse of dimensionality." (Nassim N Taleb, "Skin in the Game: Hidden Asymmetries in Daily Life", 2018)
"Quaternions are not actual extensions of imaginary numbers, and they are not taking complex numbers into a multi-dimensional space on their own. Quaternion units are instances of some number-like object type, identified collectively, but they are not numbers (be it real or imaginary). In other words, they form a closed, internally consistent set of object instances; they can of course be plotted visually on a multi-dimensional space but this only is a visualization within their own definition." (Huseyin Ozel, "Redefining Imaginary and Complex Numbers, Defining Imaginary and Complex Objects", 2018)
"Consider for example the complex numbers x + iy, where you of course ask what is i = √ −1 when you first encounter this mathematical construction. But that uncomfortable feeling of what this strange imaginary unit really is fades away as you get more experienced and learn that C is a field of numbers that is extremely useful, to say the least. You no longer care what kind of object i is but are satisfied only to know that i^2 = −1, which is how you calculate with i." (Andreas Rosén,"Geometric Multivector Analysis: From Grassmann to Dirac", 2019)
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