Infinity Is Not a Number - It's a Free Man | Hacker News Reader

Infinity Is Not a Number - It's a Free Man

nrich.maths.org

36 pointstokenadult30 comments

Infinity is a number if we want it to be: we can do math in whatever kind of topological space we want, including ones with defined ∞, such as the one point compactifications of ℝ or ℂ, the latter of which has a nice wiki page: http://en.wikipedia.org/wiki/Riemann_sphere

As wikipedia says, “arithmetic with infinity does not obey all of the usual rules of algebra, and so the extended complex numbers do not form a field.” But they’re quite handy for some uses.

In the opening paragraph she mentions that you may be convinced that 1 = 0, if you believe infinity is number. I don't think she showed the proofs there, so here goes:

Take equation: ∞ + 1 = x

Suppose that ∞ is a number.

Suppose ∞ ≠ x. Because ∞ + 1 = ∞, (same cardinality; you can make a bijection), you get ∞ = x. This is a contradiction, so x and infinity must be the same.

Suppose ∞ = x. Then the equation becomes x + 1 = x. Subtract x on both sides. 1 = 0. This too is a contradiction.

Therefore ∞ cannot be a number.

A shorter form:

Take equation: ∞ + 1 = ∞

If ∞ is a number you can subtract it from both sides giving 1 = 0

So ∞ cannot be a number.

But then 0 too is not a number:

Take equation: 0 * 2 = 0

If 0 is a number you can divide it into both sides giving 2 = 1

So 0 cannot be a number.

0 is special. You cannot multiply and divide by it unconditionally.

∞ is special. You cannot add and subtract it unconditionally.

You're getting to the point where you really need to use some number theory and define "number" more precisely. I would imagine you could construct a definition of number based on multiplication whereby 0 would no longer be a number... but only according to your new definition of number. There's nothing wrong with having a new definition of number, but there is something wrong with writing the statement in English "0 cannot be a number" without providing your definition of number. By the standard one that English usually means when the word "number" is used without mathematical qualification, yes it darned well is, just as infinity isn't. And, correspondingly, you can construct numerous number systems where there are various ordinals that have characteristics inline with what we expect from "infinity", but those aren't the usual infinity anymore, either.

When somebody asks whether infinity "is a number", I don't think it's really helpful to start dragging in constructs from number theory and defining new definitions of number for the apparently sole purpose of dazzling the poor question asker. Either just answer the question: "Is the simple, traditional infinity I learned about in high school an instance of the simple, traditional number I learned about in high school?" ("No."), or be more explicit about the fact that you're not talking about either the "infinity" or the "number" the asker is asking about. (And when I phrase it that way, my objection should be clear: You're not answering the question that was asked, and you're not telling them that's what you are doing. That's not educational by any useful metric.)

Sure, neither "traditional number" nor "traditional infinity" is particularly well defined, with pathological behavior readily demonstrable to even a high-school educated person... but not in such a way that affects this particular question.

Maybe this link will help:

http://scienceblogs.com/goodmath/2008/10/infinity_is_not_a_n...

All you managed to do is show why division by zero is not allowed. It doesn't show 0 is not a number.

0 is a number all right, and division by 0 isn't proving anything. My point is that maybe subtracting ∞ doesn't prove anything either.

Well, I suppose that's true if you limit yourself to the reals, but it doesn't hold if you're dealing with the surreal numbers :-)

Could you kindly give an example of addition in the surreal numbers? I've read a chapter in one of John Conway's books about the surreal numbers, but I don't recall the rules of arithmetic for those.

The whole thing is defined recursively. All you need to know is that a surreal number consists of two sets of surreal numbers, the left set and the right set, written {L|R}; and no member of the right set is less than any member of the left set. "Less than" is defined recursively - see http://en.wikipedia.org/wiki/Surreal_numbers#Order (it's easier to write using math symbols)

Some finite examples of addition (again, wiki has a better explanation of the rules than I could give b/c they can use math symbols http://en.wikipedia.org/wiki/Surreal_numbers#Addition):

{|} + {|} = {|} == 0 + 0 = 0 (zero has the empty set on both sides; everything is built from this)

{|} + {{ | } | } = {{ | } | } == 0 + 1 = 1

{{ | } | } + {|} = {{ | } | } == 1 + 0 = 1

{{ | } | } + {{ | } | } = {{{ | } | }, {{ | } | } | } == 1 + 1 = 2

{{{ | } | }, {{ | } | } | } + {{ | } | } = {{{{ | } | }, {{ | } | } | }, {{{ | } | }, {{ | } | } | }, {{{ | } | }, {{ | } | } | } | } == 2 + 1 = 3

As you can see, the "long" forms of surreal numbers quickly become unwieldy, which is why people use roman numerals to describe surreal numbers with the same properties as the corresponding real. When dealing with transfinite numbers it's not possible to use the long form. (And yes, I wrote a program to generate those sums for me. No way I'm doing that by hand!)

that is cool and informative - mind sharing the source for generating the sums?

http://pastebin.com/m3387c4ae

It's part of a game (in the Conway sense) framework I wrote up when I was learning about surreal numbers. Games are an even more general construct, where the left side is not required to be 'less than' the right side. Anyway, to use it, fire up GHCi and `:l game`. I've pre-defined -1, 0, and 1, but you can theoretically create any number with a finite representation (of course, the data structure and algorithms used are deliberately naive and incredibly inefficient and slow, so don't try anything crazy). For example, 5+3 requires 1156 characters to display, 5+4 requires 2556, and 5+5 requires 5118. This is because my code doesn't attempt to "reduce" any of the numbers - numbers can have multiple representations, and with finite representations you can just take the largest element of the left set and the smallest element of the right set and get the same number. My code doesn't do that. Anyway, there's also a fromInteger definition so you can mix Numbers and integers in computations.

Examples:

    Prelude> :l game
    [1 of 1] Compiling Game             ( game.hs, interpreted )
    Ok, modules loaded: Game.
    *Game> zero
    { | }
    *Game> 5 :: Game
    {{{{{{ | } | } | } | } | } | }
    *Game> let half = Game [zero] [one]
    *Game> half
    {{ | } | {{ | } | }}
    *Game> half < one
    True
    *Game> half > 1
    False
    *Game> half + half == 1
    True
    *Game> half * 1 == half
    True
    *Game> half * (one + one) > 1
    False

The best resources for learning are both by Conway: "Winning Ways for your Mathematical Plays" (WW) and "On Numbers and Games" (ONAG). WW is more accessible and introduces you to numbers via games, whereas ONAG is more of a hard-core math book that explains games via numbers. WW is probably better if you don't have a strong math background, ONAG if you do.

Surreal numbers where first described in Conway's "On Number's and Games." Since then there have been numerous other articles and books about them including a very nice one by Donald Knuth called "Surreal Numbers".

x = y / * x

x2 = xy / -y2

x2 - y2 = xy - y2 / *1/(x-y)

x + y = y

Since x = y,

2y = y

2 = 1

1 = 0

x = y / * x

x2 = xy / -y2

x2 - y2 = xy - y2 / *1/(x-y)

Error: Divide by zero.

;)

Ok, I understand the math part, don't quite get the 'Free Man' part. Is it a reference to something? Enlighten a poor soul.

http://en.wikiquote.org/wiki/The_Prisoner

Ah, that clears it up. Thank you, good Sir!

Be seeing you.

It depends, I think. In some circumstances ∞ is just a special number, as 0 is a special number.

Consider the operation (||) of parallel coupling of resistors

  1Ω || 1Ω = 1/2Ω   cf.   1Ω + 1Ω = 2Ω 
  1Ω || 0Ω =   0Ω   cf.   1Ω + 0Ω = 1Ω
  1Ω || ∞Ω =   1Ω   cf.   1Ω + ∞Ω = ∞Ω

For the operator || ∞ is the identity element as 0 is for operator +

Do we really consider infinite resistance a numerical value of (a measurement of) resistance?

No we don't consider ∞ a numerical value.

The big question is: Is it a value nonetheless?

Edit: Infinite resistance is as well defined as zero resistance.

Infinite resistance equals zero conductance.

Infinite conductance equals zero resistance.

Edit2: So if you measure the conductance of an infinite resistor you get the numerical value 0.

I think this reasoning ends up being something like dividing by zero. A measurement of "infinity" can't be in any definable place on interval scale.

Well then a measurement of zero can't be in any definable place on interval scale either.

Above you wrote,

No we don't consider ∞ a numerical value.

So you are agreeing with the title and with the substance of the submitted article, which I submitted for truth and for admiration of how it was written.

As for an interval scale

http://en.wikipedia.org/wiki/Level_of_measurement#Interval_s...

the zero point is well defined as the midpoint between -1 and 1, that is halfway between the values defined by those numbers. (It is commonly noted that the numerical designation of any point on such a scale is arbitrary, and the familiar examples are the Fahrenheit and Celsius scales of temperature, each with a different zero point, which in both cases is equidistant between -1 and 1.) No comparable definiteness of position is enjoyed by "infinity."

Psychologist Joel Michell makes the very good point that his fellow psychologists are probably abusing language by referring to interval scales as "measurements,"

http://www.questia.com/library/book/an-introduction-to-the-l...

but for our purposes here it is enough to note that you, I, and the whole world notice that zero has a place on an interval scale if such a place is arbitrarily set, but infinity does not have such a place.

I think infinity is well enough defined - for measurements at least - on the reciprocal scale:

   -4   -3   -2   -1    0    1    2    3    4
  --|----|----|----|----|----|----|----|----|--

  -1/4 -1/3 -1/2  -1    ∞    1   1/2  1/3  1/4
  --|----|----|----|----|----|----|----|----|--

I certainly didn't intend to devaluate the submitted article. Just wanted to point at some circumstances where infinity in fact is considered a distinct and definable value; by the example of resistance being the reciprocal of conductance.

That's still a division by zero error. Division by zero is no more valid in engineering than it is in mathematics. While I was out on a walk in my neighborhood I thought about the practical engineering problem you first mentioned above: whatever kind of resistor you are talking about, I think a lot less than infinite electricity could jump the gap filled by any finitely sized resistor, even if you initially regard it as having infinite resistance for purposes of analyzing the circuit. But I will leave further discussion of this point to readers here who have formal training and experience with electrical engineering, as my childhood best friend does. I'm not aware that he considers infinity a number or that he puts it anywhere on the number line.

http://mathforum.org/dr.math/faq/faq.divideby0.html

http://mathworld.wolfram.com/DivisionbyZero.html

http://en.wikipedia.org/wiki/Division_by_zero

It's excellently written, but I wish I could turn off the analogies. For me, analogies only muddle the point.

Infinity is the substrate on which numbers are defined. It's the space that numbers are carved out of.

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