"The equation e^πi+1 = 0 is true only by virtue of a large
number of profound connections across many fields. It is true because of what
it means! And it means what it means because of all those metaphors and blends
in the conceptual system of a mathematician who understands what it means. To
show why such an equation is true for conceptual reasons is to give what we
have called an idea analysis of the equation."
"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."
"The significance of e^πi+1 = 0 is thus a conceptual
significance. What is important is not just the numerical values of e, π, i, 1,
and 0 but their conceptual meaning. After all, e, π, i, 1, and 0 are not just
numbers like any other numbers. Unlike, say, 192,563,947.9853294867, these
numbers have conceptual meanings in a system of common, important nonmathematical
concepts, like change, acceleration, recurrence, and self-regulation.
They are not mere numbers; they are the arithmetizations of concepts. When they are placed in a formula, the formula incorporates the ideas the function expresses as well as the set of pairs of complex numbers it mathematically determines by virtue of those ideas." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being", 2000)
"We will now turn to e^πi+1 = 0. Our approach will be there
as it was here. e^πi+1 = 0 uses the conceptual structure of all the cases we
have discussed so far - trigonometry, the exponentials, and the complex
numbers. Moreover, it puts together all that conceptual structure. In other
words, all those metaphors and blends are simultaneously activated and jointly
give rise to inferences that they would not give rise to separately. Our job is
to see how e^πi+1 = 0 is a precise consequence that arises when the conceptual
structure of these three domains is combined to form a single conceptual blend." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the
Embodied Mind Brings Mathematics into Being", 2000)
"[…] the equation’s five seemingly unrelated numbers (e, i, π, 1, and 0) fit neatly together in the formula like contiguous puzzle pieces. One might think that a cosmic carpenter had jig-sawed them one day and mischievously left them conjoined on Euler’s desk as a tantalizing hint of the unfathomable connectedness of things.[…] when the three enigmatic numbers are combined in this form, e^iπ, they react together to carve out a wormhole that spirals through the infinite depths of number space to emerge smack dab in the heartland of integers." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)
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