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In surreal numbers [1], the midpoint between 0 and infinity would be the simplest number greater than 0, which is { 0 | } = 1 [1] https://en.wikipedia.org/wiki
by tromp 6y ago
In surreal numbers [1], the midpoint between 0 and infinity would be the simplest number greater than 0, which is
{ 0 | } = 1
[1] https://en.wikipedia.org/wiki/Surreal_number https://en.wikipedia.org/wiki/Surreal_number
- npmisdown 6y agoTIL: apart from infinity (number, larger than any real number in absolute value) there are infinitesimal (number, less than any real number in absolute value and not a zero). Edit: also, those infinitesimals were the subject of political and religious controversies in 17th century Europe, including a ban on infinitesimals issued by clerics in Rome in 1632.
- zests 6y agoThose clerics were ahead of their time. They probably would have banned large cardinals as well (infinites so large we can't prove whether or not they exist.)
- teilo 6y agoYou win the internet, sir.
- user-the-name 6y agoI think a lot of cardinals around then were quite large, actually. https://www.alamy.com/portrait-of-cardinal-flavio-chigii-1631-1693-17th-century-cardinal-flavio-chigi-by-jacob-ferdinand-voet-image184942128.html https://www.alamy.com/portrait-of-cardinal-flavio-chigii-163... https://www.pinterest.com/pin/462252349233672807/ https://www.pinterest.com/pin/462252349233672807/
- colejohnson66 6y agoEven with the shortened URLs, I just knew this was going to be some reference to the Catholic church’s cardinals
- saagarjha 6y agothatsthejoke.jpg
- npmisdown 6y agoNot sure whether 'large cardinals' phrase in your message is intentional or accidental pun.
- greggyb 6y agoInfinitesimals and infinities are nice if you're doing some sort of bucketing logic. tiny = infinitesimal huge = infinity N = number to be bucketed tiny <= N < 10 --> bucket 1 10 <= N < 20 --> bucket 2 ... X <= N < huge --> last bucket This removes edge cases you need to test for if you're trying to bucket positive values. This may not be something you've had to do, but I've had reason to want this before on a few occasions.
- karmakaze 6y agoInfinitesimals do not exist in the standard real number system. 'tiny' seems to be more related to the smallest positive representable (normal or subnormal) IEEE 754 float/double type of value which is a real number.
- greggyb 6y agoInfinitesimals tautologically exist in any finite representation of numbers. For floats, it's the smallest representable positive number. For integers of any type, it's 1. As for how common they are, we learn about them in any introductory calculus course when defining derivatives. You come across the idea whenever discussing limits, if somewhat obliquely. If I learned about it in high school math, and again in "real" math courses at my university, I'd say it's pretty standard.
- alisonkisk 6y agoI've never heard of that, and your definition of 1 as infinitesimal is incompatible withbits properties (infinitesimal + infinitesimal + infinitesimal is greater than a non-infinitesimal 2?!) and I don't see a mention on Wikipedia, and it goes against the plain read it of "in-finite-simal". Also, you seem to be conflating "common" with "standard". "standard" is a mathematical term. Infinitesimal are handwavy in standard analysis (epsilon-delta are the rigorous alternative), but exist rigorously in nonstandard analysis. https://en.m.wikipedia.org/wiki/Infinitesimal https://en.m.wikipedia.org/wiki/Infinitesimal
- Sharlin 6y agoThe theory of infinitesimals is intimately connected to how analysis (differential and integral calculus) was first formulated. Leibniz and Newton understood that you could approximate instant rates of change, or areas under a curve, by taking smaller and smaller "slices" of a function, but they did not yet have the rigorously formalized notion of limits that modern analysis depends on. So they developed an arithmetic of infinitesimals, numbers greater than zero but less than any positive real number¹, with some rather ad-hoc properties to make them work out in calculations. Philosophical problems surrounding the perplexing concept of infinity were already hotly debated by the ancient Greeks. Aristotle made an ontological distinction between actual and potential infinities, and argued that actual infinity cannot exist but potential infinity can. This was also the consensus position of later scholars, and became a sticking point in the acceptance of calculus because infinitesimals (and infinite sums of them) were an example of the ontologically questionable actual infinities. As I mentioned before, standard modern analysis is based on limits, not infinitesimals, and requires no extension of real numbers. Indeed the limit definition of calculus only requires the concept of potential infinities, so philosophers should be able to rest easy! But infinitesimals still occur in our notation which is largely inherited from Leibniz, however. We say that the derivative of y(x) is dy/dx, or the antiderivative of y(x) is ∫ y(x) dx, and while acknowledging that dy and dx are not actual mathematical objects, just syntax, we still do arithmetic on them whenever it's convenient to do so! For example, when we make a change of variables in an integral, we can substitute x = f(t) for some f, and then say dx/dt = f'(t) and "multiply by dt" to get dx = f'(t) dt to figure out what we should put in the place of the "dx" in the integral. Actual infinitesimal numbers are not dead, either, they're used in a branch of analysis called nonstandard analysis which formalizes them in the logically rigorous manner that is now expected from mathematics. ________ ¹ Not that they had a rigorous theory of real numbers, either, that came in the 19th and early 20th century. In fact what we now understand as formal, axiomatized math didn't really exist before the 19th century at all!
- bollu 6y agoCan you tell me the hackenbush interpretation of this? sounds cool!
- coldtea 6y agoWill this do? https://www.youtube.com/watch?v=ixNUpUnwZpY https://www.youtube.com/watch?v=ixNUpUnwZpY
- evanb 6y agoOn a log scale 1 is also halfway between 0 and infinity.
- klodolph 6y agoOn a log scale, all numbers are equally distant from 0 and infinity. 1 is no more “halfway” than 10, or 13, or 1/95.
- DavidSJ 6y agoSome other senses in which 1 can be thought of as the "midpoint" between 0 and ∞ come from the many common ways to invertibly map an interval onto (0, ∞), in which the midpoint of that interval maps onto 1. For example: • tan maps (0, π/2) onto (0, ∞), and tan(π/4) = 1. • f(x) = x/(1 - x) maps (0, 1) onto (0, ∞), and f(1/2) = 1. • exp is an isomorphism from the reals under addition to the positive reals under multiplication. 0 is the natural "midpoint" of the reals under addition, and exp(0) = 1. (Closely related to that last one: when dealing only with positive numbers (and their limits 0 and ∞), it's natural to think of their group operation, multiplication, for which 1 is the identity, creating a symmetry between the numbers smaller than 1 and the numbers larger than 1 under reciprocation.)
- blm 6y agoI have a question If this was true: tan maps (0, π/2) onto (0, ∞), and tan(π/4) = 1 Wouldnt it imply that tan(π/8) would be halfway between 0 and 1 ie .5? By my calculations it is 0.414 or √2 - 1 ALso with: • f(x) = x/(1 - x) maps (0, 1) onto (0, ∞), and f(1/2) = 1. wouldnt this mean that f(0.25) is supposed to be half way between 0 and 1 or .5. However f(0.25) = 0.25/0.75 = 1/3
- yiyus 6y agoIt would only imply that if the tangent was a linear function of the angle, but it is not.
- jacksnipe 6y agoThis does not imply that tan(pi/8) is 0.5. There’s some overloaded language here that I’d like to unpack: The original issue at hand is to talk sensibly about “half way” between 0 and infinity. But in the standard way we think about distance between numbers, there’s obviously no way to do that — you can’t add and subtract real numbers to infinity! So, implicitly what’s happening here is that by talking about a map between a finite interval and (0,inf), we are equipping (0,inf) with a new, special definition of distance between numbers. This is called a metric. Usually, when we talk about the distance between x and y, we mean `d(x,y) = |x-y|` — this is called the Euclidean metric (in 1 dimension). Here, we’ve introduced a new metric on (0,inf): `d2(x,y) = |tan^-1(x) - tan^-1(y)|/(pi/8)` The half way point between 0 and 1 under the Euclidean metric is 0.5. The half way point between 0 and 1 under our fancy new metric d2 is ~0.414. TL;DR: You can generalize the notion of “distance between two numbers”, using distance functions called metrics. Under the typical metric, the half way point between 0 and 1 is 0.5. Under our cool new infinite-tan metric, the half way point between 0 and 1 is ~0.414.
- yiyus 6y agoThat's not right. 1 is different to half the sum of 0 and infinity. Another way to look at it is that infinity minus one is larger than one minus zero (which can be easily proven using the basic definitions used to define the surreals). Moreover, what is at the right side of {0|} is not infinity, but the empty set.
- Viliam1234 6y ago> Moreover, what is at the right side of {0|} is not infinity, but the empty set. But { 0 | ω } = 1, right?
- tromp 6y agoCorrect. As is { 0 | n } for any n > 1.
- alex_c 6y agoThis is also consistent with the old joke that developers count like cavemen: zero, one, many!
- tempodox 6y agoWell, if you map the contents of the unit circle (|z| < 1) through 1/z you can reach any point |z| > 1. So there really is no need to give unique names to anything beyond 1 as it's already contained in the proverbial nutshell of the unit circle via 1/z.
- lisper 6y agoYou don't even need the unit circle. The unit interval is enough, as is any set whose cardinality is aleph-1.
- alisonkisk 6y agoParent (who meant disc not circle) didn't give a dimension. An interval is just a 1 dimensional disc.
- recursive 6y agoIt's also the midpoint of the positive rationals, as modeled by the Stern-Brocot tree. https://www.cut-the-knot.org/blue/Stern.shtml https://www.cut-the-knot.org/blue/Stern.shtml
- Syzygies 6y agoExactly what I came to post. This is an important model for numbers. Traditionally one distinguishes between rational, algebraic, and transcendental numbers; e is transcendental. This tree is related to continued fraction expansions. Any number is a limit of a path in this tree, and one can talk about the pattern in the path from a CS automata theory point of view. Now e comes out one of the easier limits.
- Jweb_Guru 6y agoPerhaps I'm confused, but doesn't that site indicate that the midpoint of the positive rationals is 1.0? Or am I misreading it?
- recursive 6y agoYou are not misreading. That's correct.
- Jweb_Guru 6y agoBut the article is saying that in Ruby, it's 1.5, right? So it's not the same as here? Or were you referring to the general notion of a midpoint of an infinite range?
- recursive 6y agoRight again. The "also" is a reference to the parent comment. > In surreal numbers [1], the midpoint between 0 and infinity would be the simplest number greater than 0, which is { 0 | } = 1
- 6y ago