"As mathematics gets more abstract, diagrams become more and more prominent as the ways that things fit together abstractly become both more subtle and more important. Moreover, the diagram often sums up the situation more succinctly than the explanation in words, [..]" (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)
"Calculus is the study of things that are changing. It is difficult to make theories about things that are always changing, and calculus accomplishes it by looking at infinitely small portions, and sticking together infinitely many of these infinitely small portions." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)
"Category theory studies relationships between things and builds on this in various ways: characterising things by what properties they have, finding the pond in which things are the biggest fish, putting things in context, expressing subtle notions of things being ‘more or less the same’." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)
"In category theory
there is always a tension between the idealism and the logistics. There are
many structures that naturally want to have infinite dimensions, but that is
too impractical, so we try and think about them in the context of just a finite
number of dimensions and struggle with the consequences of making these
logistics workable."
"Infinity is a Loch Ness Monster, capturing the imagination with its awe-inspiring size but elusive nature. Infinity is a dream, a vast fantasy world of endless time and space. Infinity is a dark forest with unexpected creatures, tangled thickets and sudden rays of light breaking through. Infinity is a loop that springs open to reveal an endless spiral." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)
"Informally, people say things are growing exponentially just to mean they’re growing a lot, which is sort of true, but the formal mathematical meaning is that it’s growing at the same proportional rate all the time." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)
"It just so happens
that π can be characterised precisely without any reference to decimals,
because it is simply the ratio of any circle’s circumference to its diameter.
Likewise can be characterised as the positive number which squares to 2. However,
most irrational numbers can’t be characterised in this way."
"Mathematical rigour is the thing that enables mathematicians to agree with one another about what is and isn’t correct, rather than just having arguments about competing theories and never coming to a conclusion. Mathematics is based on the rules of logic, the idea being that if you only use objects that behave strictly according to the rules of logic, then as long as you only strictly apply the rules of logic, no disagreements can ever arise."(Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)
"Mathematicians start by playing around with ideas to get a feel for what might be possible, good and bad."
"Mathematics can sometimes seem like a process of never getting anywhere, because every time you work out something new it just reveals all the other things you don’t know."
"Mathematics is particularly good at making things out of itself, like how higher-dimensional spaces are built up from lower-dimensional spaces. This is because mathematics deals with abstract ideas like space and dimensions and infinity, and is itself an abstract idea. […] Mathematics is abstract enough that we can always make more mathematics out of mathematics." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)
"Mathematics often develops by mathematicians feeling frustrated about being unable to do something in the existing world, so they invent a new world in which they can do it."
"Mathematics suffers a strange burden of being required to be useful." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)
"Mathematics starts with the process of stripping away the ambiguities and leaving only things that can be unambiguously manipulated according to logic. It continues by then manipulating those things according to logic to see what happens." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)
"One of the roles of mathematics is to explain phenomena in the world around us, especially phenomena that crop up in many different places. If a similar idea relates to many different situations, mathematics swoops in and tries to find an overarching theory that unifies those situations and enables us to better understand the things they have in common." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)
"Sometimes mathematical
advances happen by just looking at something in a slightly different way, which
doesn’t mean building something new or going somewhere different, it just means
changing your perspective and opening up huge new possibilities as a result.
This particular insight leads to calculus and hence the understanding of
anything curved, anything in motion, anything fluid or continuously changing."
"The Axiom of Choice says that it is possible to make an infinite number of arbitrary choices. […] Mathematicians don’t exactly care whether or not the Axiom of Choice holds over all, but they do care whether you have to use it in any given situation or not." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)
"The reason we need
irrational numbers in the first place is to fill in the 'gaps' that are doomed
to be in between all the rational numbers."
"This is how category theory arose, as a new piece of maths to study maths. In a way category theory is an ultimate abstraction. To study the world abstractly you use science; to study science abstractly you use maths; to study maths abstractly you use category theory. Each step is a further level of abstraction. But to study category theory abstractly you use category theory." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)
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