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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

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.
- mayank 16y ago
  > 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.
  - baddox 16y ago
    That's my point. Mathematic terminology is expected to be different than plain English.
    - RiderOfGiraffes 16y ago
      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.
RiderOfGiraffes 16y ago
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)
- RiderOfGiraffes 16y ago
  There's a discussion of some of the points here: http://news.ycombinator.com/item?id=2123265 http://news.ycombinator.com/item?id=2123265
  - RiderOfGiraffes 16y ago
    Oops - sorry - here: http://news.ycombinator.com/item?id=2123469 http://news.ycombinator.com/item?id=2123469
- RiderOfGiraffes 16y ago
  More (but only a little) about the tennis paradox here: http://news.ycombinator.com/item?id=2126890 http://news.ycombinator.com/item?id=2126890 Pretty sure no one cares, but I thought I'd add it for completeness.

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.

mayank 16y ago
> 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.
- baddox 16y ago
  That's my point. Mathematic terminology is expected to be different than plain English.
  - RiderOfGiraffes 16y ago
    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.

RiderOfGiraffes 16y ago

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)

RiderOfGiraffes 16y ago
There's a discussion of some of the points here: http://news.ycombinator.com/item?id=2123265 http://news.ycombinator.com/item?id=2123265
- RiderOfGiraffes 16y ago
  Oops - sorry - here: http://news.ycombinator.com/item?id=2123469 http://news.ycombinator.com/item?id=2123469
RiderOfGiraffes 16y ago
More (but only a little) about the tennis paradox here: http://news.ycombinator.com/item?id=2126890 http://news.ycombinator.com/item?id=2126890 Pretty sure no one cares, but I thought I'd add it for completeness.