Thank you for the overview. It seems as though this community will also look to IEEE-754 (the IEEE Standard for Floating-Point Arithmetic) for Reals and also Infinity.
Should Python raise exceptions for Integers or [Complex] Fractions involving Infinity, or should Python assume that IEEE-754 is the canonical source of truth about infinity? IEEE-754 (2019), a closed standard, costs $100 for the PDF. I'll Ctrl-F for 'infinity' in the Wikipedia article: https://en.wikipedia.org/wiki/IEEE_754 : Exception handling > The standard defines five exceptions, each of which returns a default > value and has a corresponding status flag that is raised when the exception > occurs.[e] No other exception handling is required, but additional > non-default alternatives are recommended (see § Alternate exception > handling). > The five possible exceptions are: > - Invalid operation: mathematically undefined, e.g., the square root of a > negative number. By default, returns qNaN. > - Division by zero: an operation on finite operands gives an exact > infinite result, e.g., 1/0 or log(0). By default, returns ±infinity. > - Overflow: a result is too large to be represented correctly (i.e., its > exponent with an unbounded exponent range would be larger than emax). By > default, returns ±infinity for the round-to-nearest modes (and follows the > rounding rules for the directed rounding modes). > - Underflow: a result is very small (outside the normal range) and is > inexact. By default, returns a subnormal or zero (following the rounding > rules). > Inexact: the exact (i.e., unrounded) result is not representable exactly. > By default, returns the correctly rounded result. It appears that IEEE-754 implemented as per the binary spec could not represent more complex assessments of inifnity: As with IEEE 754-1985, the biased-exponent field is filled with all 1 bits > to indicate either infinity (trailing significand field = 0) or a NaN > (trailing significand field ≠ 0). For NaNs, quiet NaNs and signaling NaNs > are distinguished by using the most significant bit of the trailing > significand field exclusively,[d] and the payload is carried in the > remaining bits. json5 extends the JSON to support IEEE-754 +-inf (and +-0, which can also be used to indicate 1D directionality sans magnitude). Presumably, in the IEEE-754 view of the world, from math import inf, nan assert type(inf) == float assert isinstance(inf, float) assert float("inf") == inf assert inf / inf == nan assert inf / 0 == inf # currently: ZeroDivisionError assert (x/0) < ((x+1e-10)/0) # where x>0 (x in Z+) # Not possible; Python is not a CAS https://en.wikipedia.org/wiki/List_of_computer_algebra_systems doesn't have a column for a "Surreal Numbers" or "non-Float handling of infinity". https://en.wikipedia.org/wiki/Quantum_field_theory#Infinities_and_renormalization deals with Infinities; which have curently been removed. inf**inf On Sun, Oct 11, 2020 at 9:07 PM Steven D'Aprano <st...@pearwood.info> wrote: > > On Sun, Oct 11, 2020 at 05:47:44PM -0400, Wes Turner wrote: > > > No, 2 times something is greater than something. Something over something > > is 1. > > Define "something". Define "times" (multiplication). Define "greater > than". Define "over" (division). > > And while you are at it, don't forget to define what you mean by > "infinity". Do you mean potential infinity, actual infinity, Absolute > infinity, aleph and beth numbers, omegas, or something else? > > https://www.cut-the-knot.org/WhatIs/WhatIsInfinity.shtml > > I am not being facetious. Getting your definitions right is vital if you > wish to avoid error, and to avoid miscommunication. Change the > definitions, and you change the meaning of everything said. > > > (1) In the so-called "real numbers", there is no such thing as infinity. > Since there is no such thing as infinity, infinity is not "something" > that can be multiplied or divided, or added or subtracted. In the Real > number system, there is no coherent way of doing arithmetic on > "infinity". "Two times infinity" is meaningless. > > In the real numbers, there's no sensible way of doing arithmetic with > "infinity" without leading to contradiction. > > Informally, infinity in the Real number system is a process that never > completes, so doing twice as much doesn't take any longer. > > > (2) Mathematicians have created at least two extensions to the Real > number line which do include at least one infinity. It is possible to > construct a coherent system that is not self-contradictory by including > either a pair of plus and minus infinity, or just a single unsigned > infinity: > > https://en.wikipedia.org/wiki/Extended_real_number_line > > https://en.wikipedia.org/wiki/Projectively_extended_real_line > > But in doing so, we have to give up certain "common sense" properties of > finite numbers. For example, with only a single infinity, infinity is > both greater than everything, and less than (more negative) than > everything. We lose a coherent definition of "greater than". > > Even in the extended number lines, two times infinity is just infinity, > and infinity divided by infinity is not coherent and cannot be defined > in any sensible way. > > The IEEE-754 standard, and consequently Python floats, closely models > the extended real number line. > > > (3) In the *cardinal numbers*, there is something called infinity. Or > rather, there are an *infinite number* of infinities, starting with the > smallest, aleph-0, which represents the cardinality of the integers, > i.e. what people usually mean when they think of infinity. > > Even in the cardinal numbers, two times infinity (aleph-0) is just > aleph-0; however you might be pleased to know that two to the power of > aleph-0 is aleph-1. > > Arithmetic with infinite cardinal numbers is strange. > > https://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel > > > (4) In other extensions of the real numbers, such as hyperreal and > surreal numbers, we can work with various different kinds of > infinities (and infinitesimals). > > For example, in the surreal numbers, we can do arithmetic on infinities, > and you will be gratified, I am sure, that twice infinity is different > from plain old infinity. In the language of the surreals: > > 2ω = ω + ω ≠ ω > > (That's an omega symbol, not ∞.) > > Unfortunately, the surreals are very different from the commonsense > world of the real numbers we know and love. For starters, they form a > tree, not a line. You cannot reach ω by starting at 0 and adding 1 > repeatedly. (ω is not the successor of any ordinal number.) Consequently > there are other infinite numbers like ω-1 that are less than infinity > but cannot be reached by counting upwards from zero but only by counting > down from infinity. > > And of course, in the surreal numbers, there are an infinity of > ever-growing infinities: not just ω+1 and 2ω but ω^2 and ω^ω and so on, > all of which are "bigger than infinity". > > https://en.wikipedia.org/wiki/Surreal_number > > All very fascinating I am sure, but I don't think that we should be > trying to emulate the surreal numbers as part of float. > > > > -- > Steve > _______________________________________________ > Python-ideas mailing list -- python-ideas@python.org > To unsubscribe send an email to python-ideas-le...@python.org > https://mail.python.org/mailman3/lists/python-ideas.python.org/ > Message archived at > https://mail.python.org/archives/list/python-ideas@python.org/message/MI4HG64LQGJ5TZYIOVNYNZKYVTEUHND4/ > Code of Conduct: http://python.org/psf/codeofconduct/ >
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