You can work the language a little to make them analogous, but that’s not the point Gowers is making. Consider this instead:
“There are those who would believe that all equations have solutions, a view that leads to some intriguing new ideas. Consider the equation x + 1 = x. Inspecting the equation, we see that its solution must be a number which is equal to its successor. Numbers with this remarkable property are quite unlike the numbers we are familiar with. As such, they are surely worthy of further study.”
I imagine Gowers’s point to be that sometimes a contradiction does point to a way in which you can revise your assumptions to gain access to “intriguing new ideas”, but sometimes it just indicates that your assumptions are wrong.
“There are those who would believe that all equations have solutions, a view that leads to some intriguing new ideas. Consider the equation x + 1 = x. Inspecting the equation, we see that its solution must be a number which is equal to its successor. Numbers with this remarkable property are quite unlike the numbers we are familiar with. As such, they are surely worthy of further study.”
(Edited again: this example is wrong, and thanks to Kindly for pointing out why. CronoDAS gives a much better answer.)
Curiously enough, the Peano axioms don’t seem to say that S(n)!=n. Lo, a finite model of Peano:
X = {0, 1}
Where: 0+0=0; 0+1=1+0=1+1=1
And the usual equality operation.
In this model, x+1=1 has a solution, namely x=1. Not a very interesting model, but it serves to illustrate my point below.
sometimes a contradiction does point to a way in which you can revise your assumptions to gain access to “intriguing new ideas”, but sometimes it just indicates that your assumptions are wrong.
Contradiction in conclusions always indicates a contradiction in assumptions. And you can always use different assumptions to get different, and perhaps non contradictory, conclusions. The usefulness and interest of this varies, of course. But proof by contradiction remains valid even if it gives you an idea about other interesting assumptions you could explore.
And that’s why I feel it’s confusing and counterproductive to use ironic language in one example, and serious proof by contradiction in another, completely analogous example, to indicate that in one case you just said “meh, a contradiction, I was wrong” while in the other you invented a cool new theory with new assumptions. The essence of math is formal language and it doesn’t mix well with irony, the best of which is the kind that not all readers notice.
Yes. My goal wasn’t to argue with the quote but to improve its argument. The quote said:
But could mathematicians afford to use this literary device? How would a reader be able to tell the difference in intent between what I have just written and the following superficially similar passage?
And I said, it’s not just superficially similar, it’s exactly the same and there’s no relevant difference between the two that would guide us to use irony in one case and not in the other (or as readers, to perceive irony in one case and serious proof by contradiction in the other).
Your model violates the property that if S(m) = S(n), then m=n, because S(1) = S(0) yet 1 != 0. You might try to patch this by changing the model so it only has 0 as an element, but there is a further axiom that says that 0 is not the successor of any number.
Together, the two axioms used above can be used to show that the natural numbers 0, S(0), S(S(0)), etc. are all distinct. The axiom of induction can be used to show that these are all the natural numbers, so that we can’t have some extra “floating” integer x such that S(x) = x.
You can work the language a little to make them analogous, but that’s not the point Gowers is making. Consider this instead:
“There are those who would believe that all equations have solutions, a view that leads to some intriguing new ideas. Consider the equation x + 1 = x. Inspecting the equation, we see that its solution must be a number which is equal to its successor. Numbers with this remarkable property are quite unlike the numbers we are familiar with. As such, they are surely worthy of further study.”
I imagine Gowers’s point to be that sometimes a contradiction does point to a way in which you can revise your assumptions to gain access to “intriguing new ideas”, but sometimes it just indicates that your assumptions are wrong.
Yes, yes they are.
(Edited again: this example is wrong, and thanks to Kindly for pointing out why. CronoDAS gives a much better answer.)
Curiously enough, the Peano axioms don’t seem to say that S(n)!=n. Lo, a finite model of Peano:
X = {0, 1} Where: 0+0=0; 0+1=1+0=1+1=1 And the usual equality operation.
In this model, x+1=1 has a solution, namely x=1. Not a very interesting model, but it serves to illustrate my point below.
Contradiction in conclusions always indicates a contradiction in assumptions. And you can always use different assumptions to get different, and perhaps non contradictory, conclusions. The usefulness and interest of this varies, of course. But proof by contradiction remains valid even if it gives you an idea about other interesting assumptions you could explore.
And that’s why I feel it’s confusing and counterproductive to use ironic language in one example, and serious proof by contradiction in another, completely analogous example, to indicate that in one case you just said “meh, a contradiction, I was wrong” while in the other you invented a cool new theory with new assumptions. The essence of math is formal language and it doesn’t mix well with irony, the best of which is the kind that not all readers notice.
But that’s the entire point of the quote! That mathematicians cannot afford the use of irony!
Yes. My goal wasn’t to argue with the quote but to improve its argument. The quote said:
And I said, it’s not just superficially similar, it’s exactly the same and there’s no relevant difference between the two that would guide us to use irony in one case and not in the other (or as readers, to perceive irony in one case and serious proof by contradiction in the other).
Your model violates the property that if S(m) = S(n), then m=n, because S(1) = S(0) yet 1 != 0. You might try to patch this by changing the model so it only has 0 as an element, but there is a further axiom that says that 0 is not the successor of any number.
Together, the two axioms used above can be used to show that the natural numbers 0, S(0), S(S(0)), etc. are all distinct. The axiom of induction can be used to show that these are all the natural numbers, so that we can’t have some extra “floating” integer x such that S(x) = x.
Right. Thanks.