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Timothy Gowers on mathematical implication (cam.ac.uk)
24 points by cgopalan on Jan 19, 2011 | hide | past | web | favorite | 39 comments



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.


It is something of a problem in the teaching of the early stages of math, as opposed to arithmetic. Most people like to have motivating examples, most people like to see that what they are doing is relevant, and that it makes sense. Trying to tell people that F=>X is true, regardless of X, offends most people's sense of what's reasonable.

Consider. Suppose that in a tennis match each point is won or lost independently of all other points, and suppose further that the server has a 95% chance of winning each point. It seems reasonable to assume that the probability of winning the game from a winning position is always greater than the probability of winning the game from the beginning.

But this is not true! The probability of winning the game from 40:30 up is less than the probability of winning from Love All (99.986% vs 99.991%).

Some students, even those who are quite competent at probability and can follow the calculations, find that this offends their sensibilities and decide that it's all rubbish.

People get very upset if you contradict their common sense. To claim that the sentence:

  If n is an integer and both even and odd, then n=17.
is true gets some people really confused. The hurdle of understanding why it's true needs to be passed, otherwise most of math will remain unattainable.

(edited to add the actual probabilities)


More (but only a little) about the tennis paradox here:

http://news.ycombinator.com/item?id=2126890

Pretty sure no one cares, but I thought I'd add it for completeness.


There's a discussion of some of the points here:

http://news.ycombinator.com/item?id=2123265



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.


> 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.


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.


> 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.

Well it would sound pretty stupid to say "x XOR y" in regular conversation, which is what is usually meant. I'm trying to imagine how it would sound -- "ok, you can get a pony zhor an xbox?", "you can get ice cream ecks-or health insurance?" Also, not many people would probably talk to you.


That's my point. Mathematic terminology is expected to be different than plain English.


In my experience, this is not true when people are learning the basics of formal reasoning. More, people expect that mathematical logic will not contradict their strongly-held intuitions about logic and causality.


Implication is far less confusing if you are a constructivist.


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.)


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.


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).

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.


> 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).

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?)


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.



> 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-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?


Parity is a property of numbers and sets only. Or do you think infinity is a set?


> 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).


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.


> 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?


Infinity is certainly not a real number, nor a complex number. There are plenty of systems (such as various compactions of the real or complex numbers, like the Riemann sphere, or the surreal numbers) for which infinity is a member in good standing, and then there is no reason not to consider it a number.


Although it's always dangerous to say to a mathematician something like “But there's no sense in which that's true!”, still I would be reluctant to refer to elements of the Riemann sphere as numbers; the sphere seems fundamentally a geometric rather than algebraic object (of course, until you start thinking of it as the complex projective ‘line’, at which point it becomes both :-) ).

With that said, as I have argued elsethread, I certainly do think that there are occasions when infinity can be considered a number; the surreal numbers are an elegant example.


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.


This wouldn't be a problem if people would stop thinking that the natural numbers start at 1.


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.


    We could just throw away these short hands and just [use]
    integers, positive integers, negative integers.
This depends on who you consider "we" to be. If we're talking about computer programmers, in most languages integers are all you get: natural numbers aren't available (although the Haskell community want them [1]).

On the other hand, from the perspective of mathematics, the natural numbers are very different to the integers. For one thing, they're wellordered whereas the integers are not. Obviously a set-theoretic definition of the natural numbers (as the finite ordinals) will start at 0 since it's just identified with ø. Set-theoretic definitions of the integers are typically in terms of equivalence classes of pairs of natural numbers (or arbitrary elements from these equivalence classes if you want to work in ACA0) [2].

[1] http://hackage.haskell.org/trac/haskell-prime/wiki/Natural

[2] See SoSOA p.10 for details. http://www.math.psu.edu/simpson/sosoa/


> Is zero positive or negative? OK, now is zero even or odd? I can imagine some just see zero with weird properties.

I think that's the best explanation I've heard. Thanks! (Also you can't divide by it (although (http://www.bbc.co.uk/berkshire/content/articles/2006/12/06/d...) …), so who knows what trickery it's up to?)


> 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.


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.


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.)


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"?


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 ….


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.


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!


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.


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/




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