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Infinity (both the "cardinality of sets" kinds and the "limit of an increasing/decreasing function" kind) are generalizations, not abstractions; and both have p

Infinity (both the "cardinality of sets" kinds and the "limit of an increasing/decreasing function" kind) are generalizations, not abstractions; and both have proven extremely fruitful generalizations both theoretically and practically.

lupire 5y ago
What's the practical application of infinity?
- kolinko 5y ago
  Anything that deals with calculus has infinities underneath. Much of the modern technology wouldn’t exist if we wouldn’t know calculus. Also, check out Fourier transform and try to rewrite it without infinities - https://bookdown.org/vshahrez/lecture-notes/fourier-transforms.html https://bookdown.org/vshahrez/lecture-notes/fourier-transfor... . Without fourier transform we wouldn’t have mp3, and the whole field of signal processing if I’m not mistaken. Without signal processing we would not have long distance digital communication.
  - TchoBeer 5y ago
    You can make calculus and Fourier transforms without infinite sets; all you need is limits.
    - jlokier 5y ago
      Indeed you can. But you can also make calculus without limits, using explicit definitions of infinitesimals (called non-standard analysis or non-standard calculus), and in some ways this is simpler to use. You can approach it either way. As limits, or by extending the number system with particular consistent definitions of infinitesimals and infinities. In practice using infinitesimals is often what people actually do when using calculus anyway, even when mentioning the limit in a sloppy hand-wave in the background, so it's good that non-standard analysis shows that the simpler method people actually use is usually logically coherent and gives the same answers as limits.
- t0mek 5y ago
  The infinity is an important concept which appears in most areas of the maths and physics. The maths is full of ideas that may seem counterintuitive (eg. "imaginary unit" - the square root value of -1), but as long as they are precisely defined, they can be used in reasoning that gives practical results. And even if some math concepts can't be applied to anything practical yet, we never know what will be used to formulate another theory about the stock market behaviour or processes inside the black hole. Coming back to your question, here's an example: many math constants (pi, e) and functions (sin, cos) can be defined as infinite sums (series). They converge, getting closer and closer to the limit, but they never achieve it. Using these series it's possible to calculate (in a calculator, Excel, Quake or CAD) values of these functions with any required precision.
- jlokier 5y ago
  Calculus, differential equations, integrals, all rely on iterating towards infinite sums of infinitesimal values. It seems a bit dodgy at first but can be made rigourous by giving precise, consistent definitions of different infinite values and the operations between them. These are used extensively in many fields. This can be done by using limits "towards infinity / towards zero", but it can also be done more directly with precise defintions of infinities and infinitesimals: https://en.wikipedia.org/wiki/Nonstandard_calculus https://en.wikipedia.org/wiki/Nonstandard_calculus Infinite series, where an infinite sequence is summed (or multiplied, or ...). Many useful things fell out of reasoning about entire infinite sequences as if they are a single unit. I like this video about a method Newton found how to calculate pi efficiently, from playing with infinite series: https://www.youtube.com/watch?v=gMlf1ELvRzc https://www.youtube.com/watch?v=gMlf1ELvRzc Again it's possible to define everything about infinite series in terms of limits, but sometimes it's just a more useful thinking aid to think of infinite series as themselves. In a way, it's like dropping some unnecessary syntax sugar (the limits) as you realise you can still do useful things without it. And in fact it's possible to reason consistently about some kinds of divergent infinite series, where limits don't work but they are still logically consistent. Complex exponentials. The discovery and proof of Euler's formula, eⁱˣ=cos(x)+i⋅sin(x) comes out of reasoning about infinite series, showing that it's a consistent and useful definition of a complex exponential. Nowadays we use these exponentials to calculate all sorts of things in engineering and physics, we just take them for granted because they work and everything fits together consistently. Projective geometry, for example used in computer graphics. There is the concept of "point at infinity", which is actually represented by real values in a vector but in some respects you can think of it as representing "x/0" in the context of that geometry, and finding that further calculations using it still work out consistently to non-infinite results. Including the point at infinity as a value simplifies the system. Unlike with natural numbers where "infinity" is a complication that obeys different rules to numbers, the point at infinity in projective geometry removes edge cases and simplifies the rules. Elliptic curve cryptography also uses the "point at infinity" concept which arises from a geometrical or algebraic interpretation of the elliptic curve operations, and like in projective geometry it's a useful and consistent value to include, which is done because it simplifies the system to include it.

lupire 5y ago

What's the practical application of infinity?

kolinko 5y ago
Anything that deals with calculus has infinities underneath. Much of the modern technology wouldn’t exist if we wouldn’t know calculus. Also, check out Fourier transform and try to rewrite it without infinities - https://bookdown.org/vshahrez/lecture-notes/fourier-transforms.html https://bookdown.org/vshahrez/lecture-notes/fourier-transfor... . Without fourier transform we wouldn’t have mp3, and the whole field of signal processing if I’m not mistaken. Without signal processing we would not have long distance digital communication.
- TchoBeer 5y ago
  You can make calculus and Fourier transforms without infinite sets; all you need is limits.
  - jlokier 5y ago
    Indeed you can. But you can also make calculus without limits, using explicit definitions of infinitesimals (called non-standard analysis or non-standard calculus), and in some ways this is simpler to use. You can approach it either way. As limits, or by extending the number system with particular consistent definitions of infinitesimals and infinities. In practice using infinitesimals is often what people actually do when using calculus anyway, even when mentioning the limit in a sloppy hand-wave in the background, so it's good that non-standard analysis shows that the simpler method people actually use is usually logically coherent and gives the same answers as limits.
t0mek 5y ago
The infinity is an important concept which appears in most areas of the maths and physics. The maths is full of ideas that may seem counterintuitive (eg. "imaginary unit" - the square root value of -1), but as long as they are precisely defined, they can be used in reasoning that gives practical results. And even if some math concepts can't be applied to anything practical yet, we never know what will be used to formulate another theory about the stock market behaviour or processes inside the black hole. Coming back to your question, here's an example: many math constants (pi, e) and functions (sin, cos) can be defined as infinite sums (series). They converge, getting closer and closer to the limit, but they never achieve it. Using these series it's possible to calculate (in a calculator, Excel, Quake or CAD) values of these functions with any required precision.
jlokier 5y ago
Calculus, differential equations, integrals, all rely on iterating towards infinite sums of infinitesimal values. It seems a bit dodgy at first but can be made rigourous by giving precise, consistent definitions of different infinite values and the operations between them. These are used extensively in many fields. This can be done by using limits "towards infinity / towards zero", but it can also be done more directly with precise defintions of infinities and infinitesimals: https://en.wikipedia.org/wiki/Nonstandard_calculus https://en.wikipedia.org/wiki/Nonstandard_calculus Infinite series, where an infinite sequence is summed (or multiplied, or ...). Many useful things fell out of reasoning about entire infinite sequences as if they are a single unit. I like this video about a method Newton found how to calculate pi efficiently, from playing with infinite series: https://www.youtube.com/watch?v=gMlf1ELvRzc https://www.youtube.com/watch?v=gMlf1ELvRzc Again it's possible to define everything about infinite series in terms of limits, but sometimes it's just a more useful thinking aid to think of infinite series as themselves. In a way, it's like dropping some unnecessary syntax sugar (the limits) as you realise you can still do useful things without it. And in fact it's possible to reason consistently about some kinds of divergent infinite series, where limits don't work but they are still logically consistent. Complex exponentials. The discovery and proof of Euler's formula, eⁱˣ=cos(x)+i⋅sin(x) comes out of reasoning about infinite series, showing that it's a consistent and useful definition of a complex exponential. Nowadays we use these exponentials to calculate all sorts of things in engineering and physics, we just take them for granted because they work and everything fits together consistently. Projective geometry, for example used in computer graphics. There is the concept of "point at infinity", which is actually represented by real values in a vector but in some respects you can think of it as representing "x/0" in the context of that geometry, and finding that further calculations using it still work out consistently to non-infinite results. Including the point at infinity as a value simplifies the system. Unlike with natural numbers where "infinity" is a complication that obeys different rules to numbers, the point at infinity in projective geometry removes edge cases and simplifies the rules. Elliptic curve cryptography also uses the "point at infinity" concept which arises from a geometrical or algebraic interpretation of the elliptic curve operations, and like in projective geometry it's a useful and consistent value to include, which is done because it simplifies the system to include it.