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: 0) by Anonymous Coward on Monday July 31 2017, @01:58PM (27 children)
So, are we now allowed to divide by zero?
(Score: 0) by Anonymous Coward on Monday July 31 2017, @02:10PM
It's ± complex infinity.
(Score: 0, Disagree) by epitaxial on Monday July 31 2017, @03:08PM (19 children)
Division by zero should be the same as multiplying by one. Take something and divide it into 0 parts. You still have your original value.
(Score: 0) by Anonymous Coward on Monday July 31 2017, @03:58PM
completely wrong.
(Score: 0) by Anonymous Coward on Monday July 31 2017, @04:15PM
6 divided by 3 = How many groups of 3 are there in 6? 2
How many groups of 1.5 are there in 6? 4
How many groups of 0 are there in 6? ERROR
(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....
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(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.
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(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...
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(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...
Україна досі не є частиною Росії Слава Україні🌻 https://www.pravda.com.ua/eng/news/2023/06/24/7408365/
(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.
Україна досі не є частиною Росії Слава Україні🌻 https://www.pravda.com.ua/eng/news/2023/06/24/7408365/
(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.
(Score: 0) by Anonymous Coward on Monday July 31 2017, @03:14PM
You can, but you might put an eye out.
(Score: 2, Funny) by Anonymous Coward on Monday July 31 2017, @03:26PM
No, but the good news is dividing by 0.00000000000000000001 has proven perfectly safe on mice.
(Score: 2) by captain normal on Monday July 31 2017, @07:39PM (1 child)
Wrong question. Should ask, "are we now allowed to divide by infinity?"
"It is easier to fool someone than it is to convince them that they have been fooled" Mark Twain
(Score: 0) by Anonymous Coward on Monday July 31 2017, @11:13PM
That's if you are in Australia or New Zealand. The northern hemisphere still divides by zero.
(Score: 2) by jasassin on Tuesday August 01 2017, @05:24AM (1 child)
I just tried it on my calculator! The answer is E.
jasassin@gmail.com GPG Key ID: 0xE6462C68A9A3DB5A
(Score: 1, Funny) by Anonymous Coward on Tuesday August 01 2017, @05:54AM
A hex calculator, you lucky dog!
(Score: 1, Insightful) by Anonymous Coward on Monday July 31 2017, @02:01PM (3 children)
Might be a step closer to the Infinite Improbability Drive, as described in HHGTTG by Douglas Adams.
(Score: 3, Funny) by BsAtHome on Monday July 31 2017, @03:00PM
Sounds more like Bistromath to me...
(Score: 0) by Anonymous Coward on Monday July 31 2017, @07:00PM
It's one small step forward, one infinite leap away.
(Score: 3, Funny) by driverless on Tuesday August 01 2017, @05:34AM
Nahh, this is a case for... Mr.Billion [imgur.com]:
(Score: 0) by Anonymous Coward on Monday July 31 2017, @02:13PM (2 children)
e.g. $latexRT_2^2$
(Score: 0) by Anonymous Coward on Monday July 31 2017, @03:05PM (1 child)
If you refer to the arXiv abstract page, they use MathJax, which uses JavaScript to compile LaTeX formulas on the browser. So if you have JavaScript disabled, you'll only see the LaTeX source.
(Score: 0) by Anonymous Coward on Monday July 31 2017, @03:28PM
The same applies to the main link article.
(Score: 2, Interesting) by Anonymous Coward on Monday July 31 2017, @04:44PM (6 children)
In my mind this is fair. Not everything needs to be bleeding edge new, and we should have some science and technology stuff, lest we become an "all politics all the time" channel. I think this is a good example of a good article.
So that being said... can somebody more informed than me explain what this all means is laymen terms? As I read it, basically you can take any theorem and reduce it to some key assumptions... and some theorem requires the assumption that infinity exists and others do not. Is that effectively what it all means?
(Score: 5, Interesting) by maxwell demon on Monday July 31 2017, @05:57PM
I'm not a mathematician, so I'm not sure if I understood correctly; however what I think I understood is:
The theorems known before basically fell into one of two classes:
Now it turned out that this specific theorem is "in between" in the following way:
That means, despite not being provable without infinite sets, those who don't believe in the existence of infinite sets can use it to prove stuff about finite sets, as anything they can prove that way could lso be proved without infinite sets (although those proofs may be much more complicated).
What I wonder is, however, whether the proof of the second property itself requires use of infinite sets (in which case those not believing in them won't accept it anyway).
The Tao of math: The numbers you can count are not the real numbers.
(Score: 4, Interesting) by Anonymous Coward on Monday July 31 2017, @06:30PM (3 children)
I haven't read the paper, but basically this is the way mathematical systems work. You start with a bunch of facts (called axioms, which are true in the mathematical system by definition) and explore the consequences of these.
For a proposition P, we say that there is a proof of P in our mathematical system (alternately: we say our system proves P) if we can deduce P using a sequence of logical steps starting from the axioms. If our system proves two contradicting propositions, it is called inconsistent, otherwise it is called consistent. We want our systems to be consistent because otherwise they prove every proposition and are therefore useless. We assume our system is consistent (an interesting topic for another day).
In any sufficiently powerful axiom system (assuming it is consistent), there will be some propositions can be neither proved nor disproved. Such propositions are called independent of the axioms.
In the context of this article, these guys are presumably working in the axiom system known as Zermelo-Frankel Set Theory (ZFC), upon which most of modern mathematics is based. There are a lot of axioms in this system (actually an infinite number) but one is so important that it has a name: the axiom of infinity. This roughly says "There exists some set with an infinite number of elements". There are a couple different formal methods of expressing this which are basically all equivalent given the rest of the axioms of ZFC. From this one set we can then use the other axioms to prove the existence of many other infinite sets, such as the infinite ordinals and so on. One key fact is that the axiom of infinity is a proposition which is independent of the other axioms of ZFC -- that is, the existence (or non-existence) of infinite sets cannot be proved from the other axioms alone.
So now if we remove this axiom from ZFC then we get a new system (ZFC without the axiom of infinity) which cannot construct infinite sets so it has little to say about such objects. This system is less "powerful" than ZFC as it proves fewer propositions. A mathematician thus might be interested in asking what propositions can still be proved in this reduced system. In this case, there was a theorem (called "Ramsey's Theorem for Pairs") that is ostensibly a theorem about infinite sets but these guys managed to prove it in a system that doesn't presume the existence of any infinite sets. Put another way, this theorem is independent of the axiom of infinity. Neat!
(Score: 0) by Anonymous Coward on Monday July 31 2017, @07:23PM
(not GP)
Thanks. This was my understanding as well, but you put it in much simpler terms.
I also think the excitement comes from the possibility of now generating new proofs for other theorems, that are also independent of the axiom of infinity (unlike current proofs).
(Score: 2) by JoeMerchant on Tuesday August 01 2017, @12:40PM (1 child)
So, does that make the infinite sets in "Ramsey's Theorem for Pairs" a logical conclusion of ZFC without the axiom of infinity? Proving the existence of infinite sets without needing the axiom.
Would be fun to find another "provable axiom" to reduce out of the set of "required" ZFC axioms.
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(Score: 0) by Anonymous Coward on Tuesday August 01 2017, @04:20PM
Well, no, the axiom of infinity is known to be independent of the other axioms in ZFC so there is no way to prove the existence or non-existence of infinite sets in "ZFC without the axiom of infinity". (assuming that ZFC is consistent).
Anyway I decided to actually look at the paper and the system used is not in fact "ZFC without the axiom of infinity", but PRA (primitive recursive arithmetic), which is an axiom system for the natural numbers.
(Score: 2) by Marand on Monday July 31 2017, @11:43PM
I agree; even though I can't contribute much (or at all) to certain discussions like this, I still find them interesting. They also age well compared to current tech or politics news, so being "new" is less important. Keep the random interesting stuff coming, please.
(Score: 2) by Megahard on Tuesday August 01 2017, @04:59AM
(Score: 1, Interesting) by Anonymous Coward on Tuesday August 01 2017, @09:00PM
I am one of them ACs who bitch and moan about post quality here. A lot.
Please more posts like this. I'm no mathematician, but I learned something rather interesting here, and that from comments by those with some familiarty with the subject at hand.
Good post. More posts like this, please.
(Score: 2) by bart9h on Wednesday August 02 2017, @06:40PM
There is no such thing as a infinitely large number.
Furthermore, there is no such thing as a mathematical concept of a infinite number. Any given number is finite. 1/0 is not a number.
1/x approaches infinity as x approaches zero; but if x is equal to zero, 1/x is not equal to infinity.