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Arthur T Knackerbracket has found the following story:
With a surprising new proof, two young mathematicians have found a bridge across the finite-infinite divide, helping at the same time to map this strange boundary.
The boundary does not pass between some huge finite number and the next, infinitely large one. Rather, it separates two kinds of mathematical statements: "finitistic" ones, which can be proved without invoking the concept of infinity, and "infinitistic" ones, which rest on the assumption — not evident in nature — that infinite objects exist.
Mapping and understanding this division is "at the heart of mathematical logic," said Theodore Slaman, a professor of mathematics at the University of California, Berkeley. This endeavor leads directly to questions of mathematical objectivity, the meaning of infinity and the relationship between mathematics and physical reality.
More concretely, the new proof settles a question that has eluded top experts for two decades: the classification of a statement known as "Ramsey's theorem for pairs," or RT22. Whereas almost all theorems can be shown to be equivalent to one of a handful of major systems of logic — sets of starting assumptions that may or may not include infinity, and which span the finite-infinite divide — RT22 falls between these lines. "This is an extremely exceptional case," said Ulrich Kohlenbach, a professor of mathematics at the Technical University of Darmstadt in Germany. "That's why it's so interesting."
The abstract is available on arXiv — the full article is available as a pdf.
[Ed note: Not a new story but interesting and will hopefully spark some discussion.]
-- submitted from IRC
(Score: 5, Informative) by Wootery on Monday July 31 2017, @05:08PM (16 children)
No, that makes very little sense.
Division of a nonzero real number tends toward infinity as the denominator tends toward zero. It doesn't do to just invent a special-case.
Look at a chart that plots division. [wikimedia.org] Why should the quotient suddenly jump back down to 1? It's inelegant and arbitrary. Floating point arithmetic handles division by zero with positive and negative infinities, which make far more sense than just going with the numerator.
You'd also break injectivity. If a/b=c and you know the values of a and c, then you can figure out the value of b. With your redefinition of division, you break the way we treat division as the inverse function of multiplication - you're really only pretending that division by zero makes as much sense as any other denominator. Whenever the quotient is defined (i.e. wherever y does not equal zero), (x/y) * y = x. Not so in your system, where, say, (2 / 0) * 0 = 0.
Also you'd break the way division aligns with the intuitive idea of it. If six slices of pizza are shared evenly between zero people, how many slices of pizza does each person get? isn't meaningful, so it makes sense that 6/0 be undefined.
Lastly, you've misunderstood what 'division' even means. It's a word, which refers to a mathematical function. It would be awfully unhelpful to start using the same word to refer to a different function.
(Score: 2) by JoeMerchant on Tuesday August 01 2017, @03:10AM (10 children)
6/0 actually makes a great deal more sense than 0/0.
One could say, somewhat rationally: 6/0 = 3/0 + 3/0, and perhaps even 2 * 3/0 (though 2/0 * 3/0 might seem to be the same, depending on your order of execution...)
On the other hand 0/0 - might be 1, but is much fuzzier / less well defined than 2/2, or 1/2 + 1/2.
Then we have the question of "how big an infinity is it?" where quantities like (6/0)! might be considered much larger than sqrt(6/0). Can you then, meaningfully compare infinities? Give them meaningful finite ratios? Not always....
🌻🌻 [google.com]
(Score: 2) by maxwell demon on Tuesday August 01 2017, @06:27AM
But you cannot have 6/0 without 0/0: 6/0 - 6/0 = (6-6)/0 = 0/0.
But then, 6/0 = (2*3)/(2*0) = 3/0 = (3*1)/(3*0) = 1/0. Therefore from your equation one gets 1/0 + 1/0 = 1/0, and from a + a = a it follows that a = 0, so 1/0 = 0. Which implies that 0/0=0, too.
The Tao of math: The numbers you can count are not the real numbers.
(Score: 2) by Wootery on Tuesday August 01 2017, @08:20AM (8 children)
Neither makes any sense, because division by zero isn't defined.
One might be tempted to think that 0/0 = (1/0) * 0 = 0, but that doesn't work. [brilliant.org]
You can't say that, because division by zero is undefined. It doesn't matter if it strikes you as intuitive.
That doesn't work either. See my link above.
No, that question doesn't arise, because division by zero is undefined, and square root is only defined for finite values.
Yes, see Cantor's Diagonal Argument, but that's not relevant here. Again, division of one (finite) real number by any (finite) nonzero real number, always produces a finite result. It isn't meaningful to hypothesise about what division by zero 'would' produce.
I'm not an expert on infinities, but I believe they can never be given finite ratios. The cardinality of even numbers is the same as the cardinality of natural numbers, for instance. (And that's a different kind of infinity [wikipedia.org] than would be relevant here anyway. But, again, it's actually not relevant.)
(Score: 2) by JoeMerchant on Tuesday August 01 2017, @11:19AM (7 children)
The point is: some infinities are indeed "bigger" than others, and this does occasionally arise in physics as a thing to consider.
They are all "undefined" - but, if you care to delve into the realm of "undefined" there are some relationships that still exist. They aren't as neat and clean as rational (or irrational) mathematics, but they do exist.
🌻🌻 [google.com]
(Score: 1) by khallow on Tuesday August 01 2017, @12:09PM (2 children)
You're not going to find these "bigger" infinities by dividing by zero. For example, is (1/0)/0 bigger than 1/0? No, because (1/0)/0 = 1/(0*0) = 1/0.
(Score: 3, Interesting) by JoeMerchant on Tuesday August 01 2017, @12:34PM (1 child)
See, this is why traditional mathematicians just scream "UNDEFINED, STAY OUT!" because their tools are broken on that side of the undefined line. New tools need to be defined to start getting a handle on "the other side" of undefined.
Of course, the practical applications are quite esoteric and limited in real life, but there are some physicists who think they have reasons to go there...
🌻🌻 [google.com]
(Score: 1) by khallow on Tuesday August 01 2017, @12:49PM
Already happened in the 19th century with sequences and eventually big O notation. The problem was that we got too many choices and no obvious choice for how to do it. Cardinality has its own problems, but it at least has well defined ways to find higher infinities.
(Score: 2) by Wootery on Tuesday August 01 2017, @12:55PM (3 children)
To each paragraph: citation, please.
That's an outright contradiction. If you come up with some new function that extends the division operator, well, that's all you're doing. It's not meaningful to 'delve into the realm of the undefined'. That would by definition be the same sort of nonsense as how many angels can dance on the head of a pin?
(Score: 2) by JoeMerchant on Tuesday August 01 2017, @01:05PM (2 children)
Need better definition of angels, and what constitutes their dancing...
🌻🌻 [google.com]
(Score: 2) by Wootery on Tuesday August 01 2017, @01:20PM (1 child)
Just so.
I wasn't kidding about the citations though - if you have a good source on this stuff, do please post a link.
(Score: 2) by JoeMerchant on Tuesday August 01 2017, @01:40PM
Sorry, I have hazy recollection from physics class in the 1980s... the truth is out there, but I don't have a good map.
🌻🌻 [google.com]
(Score: 1, Redundant) by epitaxial on Tuesday August 01 2017, @11:36PM (4 children)
Say it again with me. Take pie and divide it into 0 pieces. What are you left with?
(Score: 1, Redundant) by Wootery on Wednesday August 02 2017, @08:47AM (2 children)
How many times do I have to say it? The question makes no sense. It's like asking How many angels can dance on the head of a pin?
I have already spelled out in length why your idea doesn't work.
(Score: 0) by Anonymous Coward on Friday August 04 2017, @12:12AM (1 child)
Sorry for the mismoderation, it's #547782 that's redundant.
(Score: 2) by Wootery on Friday August 04 2017, @08:33AM
Apparently not... did you not see epitaxial's comment?
(Score: 2) by Wootery on Friday August 04 2017, @08:35AM
Actually, I take back what I said about corresponding with the intuitive idea of division. We extend division to non-integer quotients, and we happily divide negative numbers, complex numbers, etc, but that's hardly intuitive in terms of pies.
None of those 'extensions' have the problems that your suggestion has, though.