6 ms·

Timothy Gowers on mathematical implication

24 points by cgopalan 16y ago

Jun8 16y ago
Now, when I saw the statement I though "Aha, but zero is both even and odd." As usual, Wikipedia has corrected me (http://en.wikipedia.org/wiki/Parity_of_zero http://en.wikipedia.org/wiki/Parity_of_zero). This article turned out to be a fascinating read, for example, the fact that zero is even leads to 1 having an even number of prime numbers, a slightly counterintuitive proposition, which turns out to be important in the definition of the Mobius function and ... there goes 15 minutes.
- JadeNB 16y ago
  > Now, when I saw the statement I though "Aha, but zero is both even and odd." As usual, Wikipedia has corrected me (http://en.wikipedia.org/wiki/Parity_of_zero http://en.wikipedia.org/wiki/Parity_of_zero). Where in the world do people get this idea? Every time that I teach a class where the number-theory content is elementary enough for this to come up, someone thinks either that it's both even and odd, or neither even nor odd. It's hard for me as a mathematician: a number is even if it's divisible by 2; 0 is divisible by 2; so 0 is even. (EDIT: Err, sorry, I was rebutting those who claim that it's neither. I should say instead: a number `n` is odd only if `n + 1` is even; 0 + 1 = 1 is not even; so 0 is not odd. Maybe this is a bit less convincing?) I honestly can't see the confusion—except to attribute it to a memory of a flustered grade-school teacher who was unprepared for a question and so had to recall his or her own flustered grade-school teacher ad infinitum. (Oh, or is it maybe a confusion with infinity, which, if we were to try to classify it, would have some claim to being both?)
  - tokenadult 16y ago
    Who says infinity is a number? http://scienceblogs.com/goodmath/2008/10/infinity_is_not_a_number.php http://scienceblogs.com/goodmath/2008/10/infinity_is_not_a_n... http://nrich.maths.org/2756 http://nrich.maths.org/2756 http://mathforum.org/dr.math/faq/faq.large.numbers.html http://mathforum.org/dr.math/faq/faq.large.numbers.html http://www.kuro5hin.org/story/2003/6/3/95744/71866 http://www.kuro5hin.org/story/2003/6/3/95744/71866 Once we are confident that infinity is not a number, why would we expect it to have parity?
    - JadeNB 16y ago
      > Who says infinity is a number? Not I: > Oh, or is it maybe a confusion with infinity, which, if we were to try to classify it, would have some claim to being both? … although the old maxim “What's a number?” “Whatever you treat as a number” (which I encountered in Linderholm's wonderful book (http://www.amazon.com/Mathematics-made-difficult-Carl-Linderholm/dp/0529045524 http://www.amazon.com/Mathematics-made-difficult-Carl-Linder...) ) means that the various flavours of infinity, considered as ordinals or cardinals, have some claim to being numbers; and it is often helpful in measure theory, and my own work, to treat even plain old infinity as being a number on at least as good a footing as any other (positive real). > Once we are confident that infinity is not a number, why would we expect it to have parity? Why not?
      - ezalor 16y ago
        Parity is a property of numbers and sets only. Or do you think infinity is a set?
        JadeNB 16y ago
        > Parity is a property of numbers and sets only. Why? Seriously, though: what do you mean by parity (EDIT: I mean even-ness) of a set? Almost assuredly, something like “it can be partitioned into two disjoint subsets that can be put into bijection”; but then you've just defined even-ness of cardinals (which are sets after all), and all infinite cardinals are even (in fact, satisfy 2κ = κ; EDIT: and odd; they also satisfy 2κ + 1 = κ).
        ezalor 16y ago
        You read my mind for "parity of a set". So Q.E.D.: parity is only applicable to reason about numbers (and infinity is not a number). See also surreal numbers.
        wnoise 16y ago
        > Parity is a property of numbers and sets only. It's a property of _integers_, or any system with a natural homorphism into the integers mod 2 under addition. It's certainly not a property of numbers in general. Is pi even or odd? How about sqrt(2) + i*e?
  - waqf 16y ago
    Perhaps they learn that "even" is equivalent to "divisible by 2", and then they go on to conjecture that "odd" is equivalent to "divisible by 3", or maybe "divisible by 1" or "divisible by at least one odd prime"?
    - JadeNB 16y ago
      Without meaning to be unduly cynical, I suspect that's attributing to the average person altogether too much mathematical inquisitiveness and willingness to conjecture; but I'd love it if your explanation, especially the last, were the correct one. EDIT: Incidentally, this reminds me of one of my favourite pedagogical techniques, which I was heartbroken to discover recently is not original to me. After discussing even and odd numbers, as a very basic introduction to modular arithmetic, I like to introduce and discuss ‘threeven’ and ‘throdd’ numbers, and discover that they don't behave quite as nicely—what to do, what to do ….
  - cyrus_ 16y ago
    If you can write is a 2n for some integer n, then it is even. If you can write it as 2n + 1 for some integer n, then it is odd. Therefore, zero is even and not odd.
  - kenjackson 16y ago
    I think the confusion is simple. They believe that an even number is 2n, but think about n over the naturals. Often they'll also be confused about -1.
    - JadeNB 16y ago
      > Often they'll also be confused about -1. Hmm, I've never encountered that—which is not to say you're wrong. I'll bet you'd have a hard time finding someone post-secondary school who wouldn't answer instantly that -2 was even, though.
      - kenjackson 16y ago
        You must not be from the US. Ask random adult on the street if 2,229 is even or odd. I think you might be surprised at how many people get that wrong. Not that I've ever done it, but I suspect that you'll get at least a couple in ten who don't know.
        JadeNB 16y ago
        Indeed I am from the US. I guess I would be surprised to hear people get that one wrong, but I'd be truly shocked to see someone get the classification of -2 wrong. (A disclaimer: The only people with whom I've had occasion to have this conversation have been students in (my) college mathematics classes, so I guess I should clarify that these are the people of whom I'm really thinking.)
    - ionfish 16y ago
      This wouldn't be a problem if people would stop thinking that the natural numbers start at 1.
      - kenjackson 16y ago
        That's probably the most popular way it is taught. And whole numbers starting at 0. Although oddly some people will learn in the complete opposite way. We could just throw away these short hands and just integers, positive integers, negative integers. Which actually led me to another potential reason for the confusion. Is zero positive or negative? OK, now is zero even or odd? I can imagine some just see zero with weird properties.
  - Jun8 16y ago
    Part of being a great teacher, I think, understanding where your students trip up and their wrong thought processes. Next time, you teach the course, perhaps you might want to ask that person their train of thought. I think it is based on mistaken beliefs like these : * 0 is not a number, it's the absence of a number. Therefore, standard number properties like parity does not apply. * All even numbers are twice another number, which is not true of 0. Also, very early on, people learn that 0 is "weird", e.g. you cannot divide by it. Also there's something that irks the common psyche about treating the statement "this set has zero elements" similarly to "this set has 5 elements". This doesn't show the stupidity of the general public (or some of your students), it's just that 0 is among those abstract mathematical concepts (like infinity) that you just have to get used to. Remember that it was invented much later than natural numbers. P.S. People also see other weird conventions (e.g. 1 is not a prime, 0!=1) and may think that this is another one.
- omaranto 16y ago
  The way I usually fix people who bizarrely think 0 is both even and goes like this: Me: OK, let's get started. Is 5 even or odd? Them: Odd. Me: And 4? Them: Even. Me: How about 3? Them: Odd Me: And 2? Them: OK, OK, I get it!
btilly 16y ago
There is a simple reason for the contradiction between math and the English language statements he makes. Namely that if we tweak English only slightly, the mathematically true but linguistically false statements become false. For instance consider If unicorns exist, then mice do not exist. Well in fact unicorns have been created by transplanting the horn buds of a goat so that they would grow into a single straight horn instead of being two horns. So in some sense unicorns exist, but yet mice still exist. Obviously the statement is false. Sure, you can argue, that they aren't really unicorns. Fine. Suppose that with genetic engineering we make it happen without the operation. Mice would still exist.
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mquander 16y ago
I don't really understand the crisis with which he opens this essay. It seems to me that the problem is trivial; when we use "A implies B" or "if A then B" in common conversation, we intend to mean that A causes B. Why? Because it's not usually very useful to bother saying it unless A causes B. Mystery solved, right? It seems like Tim likes to further classify things into "reasonable-sounding" and "unreasonable-sounding" implication statements, but I just don't follow the lines of that distinction at all.
- waqf 16y ago
  He doesn't mention this (I guess it goes without saying among mathematicians), but part of the problem is that in mathematics one fact doesn't generally seem to cause another: certainly there's no adequate causality that can trivially be verified by intuition or common sense. So mathematicians can only prove theorems by working with the definition of "implies" that makes no causality requirement.
  - JadeNB 16y ago
    > He doesn't mention this (I guess it goes without saying among mathematicians), but part of the problem is that in mathematics one fact doesn't generally seem to cause another: certainly there's no adequate causality that can trivially be verified by intuition or common sense. However, there is a sense (albeit informal) in which mathematicians recognise causality, and he alludes to that when he says that probably no one would say that the Riemann hypothesis implies Fermat's last theorem. I think that all mathematicians (or at least those who learned from Herstein!) would agree that Sylow's theorem on the existence of p-subgroups of all possible orders is equivalent to the weaker statement about the existence of p-Sylow subgroups; but no mathematician would mean by that merely that both are true.
- baddox 16y ago
  Indeed. He composes a huge article in an attempt to unite the plain English usage and the mathematical usage of if p then q when there is no reason to do so. Most people and especially those with basic mathematics education don't expect English and mathematical usage to be precisely compatible. I am however interested in this guy's next article, which will hopefully be a 8-page document regarding how, unlike in plain English, "x or y" in logic is still true if x and y.
trurl 16y ago
Implication is far less confusing if you are a constructivist.
- waqf 16y ago
  Go on then: as a constructivist, does it follow from "n is even and n is odd" that "n = 17"? Show your work. (Comment: I'm fairly sure I know the answer, but I don't think it's so trivial as to not make for an interesting discussion on HN. In particular, I'm not sure that it's really less or more confusing to the sufficiently uninitiated.)
  - trurl 16y ago
    I don't have time at the moment to give a proof in Coq or another constructive proof assistant, but it boils down to the fact that "n is even and n is odd implies n = 17" is a function that can never be executed.
ionfish 16y ago
This is known as the paradox of entailment and is part of the reason why people find material implication so weird sometimes. A desire for non-explosive negation is one motivation for using paraconsistent logic. http://plato.stanford.edu/entries/logic-paraconsistent/ http://plato.stanford.edu/entries/logic-paraconsistent/

24 points by cgopalan 16y ago

Jun8 16y ago

Now, when I saw the statement I though "Aha, but zero is both even and odd." As usual, Wikipedia has corrected me (http://en.wikipedia.org/wiki/Parity_of_zero http://en.wikipedia.org/wiki/Parity_of_zero). This article turned out to be a fascinating read, for example, the fact that zero is even leads to 1 having an even number of prime numbers, a slightly counterintuitive proposition, which turns out to be important in the definition of the Mobius function and ... there goes 15 minutes.

JadeNB 16y ago
> Now, when I saw the statement I though "Aha, but zero is both even and odd." As usual, Wikipedia has corrected me (http://en.wikipedia.org/wiki/Parity_of_zero http://en.wikipedia.org/wiki/Parity_of_zero). Where in the world do people get this idea? Every time that I teach a class where the number-theory content is elementary enough for this to come up, someone thinks either that it's both even and odd, or neither even nor odd. It's hard for me as a mathematician: a number is even if it's divisible by 2; 0 is divisible by 2; so 0 is even. (EDIT: Err, sorry, I was rebutting those who claim that it's neither. I should say instead: a number `n` is odd only if `n + 1` is even; 0 + 1 = 1 is not even; so 0 is not odd. Maybe this is a bit less convincing?) I honestly can't see the confusion—except to attribute it to a memory of a flustered grade-school teacher who was unprepared for a question and so had to recall his or her own flustered grade-school teacher ad infinitum. (Oh, or is it maybe a confusion with infinity, which, if we were to try to classify it, would have some claim to being both?)
- tokenadult 16y ago
  Who says infinity is a number? http://scienceblogs.com/goodmath/2008/10/infinity_is_not_a_number.php http://scienceblogs.com/goodmath/2008/10/infinity_is_not_a_n... http://nrich.maths.org/2756 http://nrich.maths.org/2756 http://mathforum.org/dr.math/faq/faq.large.numbers.html http://mathforum.org/dr.math/faq/faq.large.numbers.html http://www.kuro5hin.org/story/2003/6/3/95744/71866 http://www.kuro5hin.org/story/2003/6/3/95744/71866 Once we are confident that infinity is not a number, why would we expect it to have parity?
  - JadeNB 16y ago
    > Who says infinity is a number? Not I: > Oh, or is it maybe a confusion with infinity, which, if we were to try to classify it, would have some claim to being both? … although the old maxim “What's a number?” “Whatever you treat as a number” (which I encountered in Linderholm's wonderful book (http://www.amazon.com/Mathematics-made-difficult-Carl-Linderholm/dp/0529045524 http://www.amazon.com/Mathematics-made-difficult-Carl-Linder...) ) means that the various flavours of infinity, considered as ordinals or cardinals, have some claim to being numbers; and it is often helpful in measure theory, and my own work, to treat even plain old infinity as being a number on at least as good a footing as any other (positive real). > Once we are confident that infinity is not a number, why would we expect it to have parity? Why not?
    - ezalor 16y ago
      Parity is a property of numbers and sets only. Or do you think infinity is a set?
      - JadeNB 16y ago
        > Parity is a property of numbers and sets only. Why? Seriously, though: what do you mean by parity (EDIT: I mean even-ness) of a set? Almost assuredly, something like “it can be partitioned into two disjoint subsets that can be put into bijection”; but then you've just defined even-ness of cardinals (which are sets after all), and all infinite cardinals are even (in fact, satisfy 2κ = κ; EDIT: and odd; they also satisfy 2κ + 1 = κ).
        ezalor 16y ago
        You read my mind for "parity of a set". So Q.E.D.: parity is only applicable to reason about numbers (and infinity is not a number). See also surreal numbers.
      - wnoise 16y ago
        > Parity is a property of numbers and sets only. It's a property of _integers_, or any system with a natural homorphism into the integers mod 2 under addition. It's certainly not a property of numbers in general. Is pi even or odd? How about sqrt(2) + i*e?
- waqf 16y ago
  Perhaps they learn that "even" is equivalent to "divisible by 2", and then they go on to conjecture that "odd" is equivalent to "divisible by 3", or maybe "divisible by 1" or "divisible by at least one odd prime"?
  - JadeNB 16y ago
    Without meaning to be unduly cynical, I suspect that's attributing to the average person altogether too much mathematical inquisitiveness and willingness to conjecture; but I'd love it if your explanation, especially the last, were the correct one. EDIT: Incidentally, this reminds me of one of my favourite pedagogical techniques, which I was heartbroken to discover recently is not original to me. After discussing even and odd numbers, as a very basic introduction to modular arithmetic, I like to introduce and discuss ‘threeven’ and ‘throdd’ numbers, and discover that they don't behave quite as nicely—what to do, what to do ….
- cyrus_ 16y ago
  If you can write is a 2n for some integer n, then it is even. If you can write it as 2n + 1 for some integer n, then it is odd. Therefore, zero is even and not odd.
- kenjackson 16y ago
  I think the confusion is simple. They believe that an even number is 2n, but think about n over the naturals. Often they'll also be confused about -1.
  - JadeNB 16y ago
    > Often they'll also be confused about -1. Hmm, I've never encountered that—which is not to say you're wrong. I'll bet you'd have a hard time finding someone post-secondary school who wouldn't answer instantly that -2 was even, though.
    - kenjackson 16y ago
      You must not be from the US. Ask random adult on the street if 2,229 is even or odd. I think you might be surprised at how many people get that wrong. Not that I've ever done it, but I suspect that you'll get at least a couple in ten who don't know.
      - JadeNB 16y ago
        Indeed I am from the US. I guess I would be surprised to hear people get that one wrong, but I'd be truly shocked to see someone get the classification of -2 wrong. (A disclaimer: The only people with whom I've had occasion to have this conversation have been students in (my) college mathematics classes, so I guess I should clarify that these are the people of whom I'm really thinking.)
  - ionfish 16y ago
    This wouldn't be a problem if people would stop thinking that the natural numbers start at 1.
    - kenjackson 16y ago
      That's probably the most popular way it is taught. And whole numbers starting at 0. Although oddly some people will learn in the complete opposite way. We could just throw away these short hands and just integers, positive integers, negative integers. Which actually led me to another potential reason for the confusion. Is zero positive or negative? OK, now is zero even or odd? I can imagine some just see zero with weird properties.
- Jun8 16y ago
  Part of being a great teacher, I think, understanding where your students trip up and their wrong thought processes. Next time, you teach the course, perhaps you might want to ask that person their train of thought. I think it is based on mistaken beliefs like these : * 0 is not a number, it's the absence of a number. Therefore, standard number properties like parity does not apply. * All even numbers are twice another number, which is not true of 0. Also, very early on, people learn that 0 is "weird", e.g. you cannot divide by it. Also there's something that irks the common psyche about treating the statement "this set has zero elements" similarly to "this set has 5 elements". This doesn't show the stupidity of the general public (or some of your students), it's just that 0 is among those abstract mathematical concepts (like infinity) that you just have to get used to. Remember that it was invented much later than natural numbers. P.S. People also see other weird conventions (e.g. 1 is not a prime, 0!=1) and may think that this is another one.
omaranto 16y ago
The way I usually fix people who bizarrely think 0 is both even and goes like this: Me: OK, let's get started. Is 5 even or odd? Them: Odd. Me: And 4? Them: Even. Me: How about 3? Them: Odd Me: And 2? Them: OK, OK, I get it!

btilly 16y ago

There is a simple reason for the contradiction between math and the English language statements he makes. Namely that if we tweak English only slightly, the mathematically true but linguistically false statements become false. For instance consider If unicorns exist, then mice do not exist. Well in fact unicorns have been created by transplanting the horn buds of a goat so that they would grow into a single straight horn instead of being two horns. So in some sense unicorns exist, but yet mice still exist. Obviously the statement is false. Sure, you can argue, that they aren't really unicorns. Fine. Suppose that with genetic engineering we make it happen without the operation. Mice would still exist.

deleted 16y ago
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mquander 16y ago

I don't really understand the crisis with which he opens this essay. It seems to me that the problem is trivial; when we use "A implies B" or "if A then B" in common conversation, we intend to mean that A causes B. Why? Because it's not usually very useful to bother saying it unless A causes B. Mystery solved, right? It seems like Tim likes to further classify things into "reasonable-sounding" and "unreasonable-sounding" implication statements, but I just don't follow the lines of that distinction at all.

waqf 16y ago
He doesn't mention this (I guess it goes without saying among mathematicians), but part of the problem is that in mathematics one fact doesn't generally seem to cause another: certainly there's no adequate causality that can trivially be verified by intuition or common sense. So mathematicians can only prove theorems by working with the definition of "implies" that makes no causality requirement.
- JadeNB 16y ago
  > He doesn't mention this (I guess it goes without saying among mathematicians), but part of the problem is that in mathematics one fact doesn't generally seem to cause another: certainly there's no adequate causality that can trivially be verified by intuition or common sense. However, there is a sense (albeit informal) in which mathematicians recognise causality, and he alludes to that when he says that probably no one would say that the Riemann hypothesis implies Fermat's last theorem. I think that all mathematicians (or at least those who learned from Herstein!) would agree that Sylow's theorem on the existence of p-subgroups of all possible orders is equivalent to the weaker statement about the existence of p-Sylow subgroups; but no mathematician would mean by that merely that both are true.
baddox 16y ago
Indeed. He composes a huge article in an attempt to unite the plain English usage and the mathematical usage of if p then q when there is no reason to do so. Most people and especially those with basic mathematics education don't expect English and mathematical usage to be precisely compatible. I am however interested in this guy's next article, which will hopefully be a 8-page document regarding how, unlike in plain English, "x or y" in logic is still true if x and y.

trurl 16y ago

Implication is far less confusing if you are a constructivist.

waqf 16y ago
Go on then: as a constructivist, does it follow from "n is even and n is odd" that "n = 17"? Show your work. (Comment: I'm fairly sure I know the answer, but I don't think it's so trivial as to not make for an interesting discussion on HN. In particular, I'm not sure that it's really less or more confusing to the sufficiently uninitiated.)
- trurl 16y ago
  I don't have time at the moment to give a proof in Coq or another constructive proof assistant, but it boils down to the fact that "n is even and n is odd implies n = 17" is a function that can never be executed.

ionfish 16y ago

This is known as the paradox of entailment and is part of the reason why people find material implication so weird sometimes. A desire for non-explosive negation is one motivation for using paraconsistent logic. http://plato.stanford.edu/entries/logic-paraconsistent/ http://plato.stanford.edu/entries/logic-paraconsistent/