You’re right, and I somehow didn’t take that in from what you wrote before. But it’s still OK (as you pointed out) because we have a circle of possible points to add, which has continuum size, and fewer than that many points to avoid, some of which are the result of existing distances and some of which are intersections of that circle with bisecting lines between pairs of points that have already been chosen.

]]>Thanks for clarifying.

I may be missing something, but it seems to me that one must never duplicate any distance, whether it is existing, , or otherwise. The last type of distance may be duplicated if we add a new point such that for two existing points and .

]]>I agree with these comments. I should have said that when we add a point at distance from some point in the set, we make sure not only that it isn’t at an already used distance from one of the other points, but also that it isn’t at distance from one of the other points. But that adds only a small set (meaning of size strictly less than the continuum) of further points that need to be avoided.

Of course, that’s exactly what you are suggesting.

]]>Also, one might need to add another condition to the induction hypothesis saying that we add at most one point at each step. Otherwise, I don’t see how the cardinality of the deleted points can be controlled.

]]>[…]Just-do-it proofs « Gowers's Weblog[…]…

]]>Take a single, bi-infinite sequence $s_k$ with the property that the sequence approaches 1 as $k \to \infty$, and approaches 0 as $k \to -\infty$. For example, you can let $s_k = 2^k$ for negative $k$, and $s_k = 1 – 2^{-k}$ for positive $k$. Alternately, one can rig something up by rescaling the arctangent function.

Now, to get the desired matrix, let $a_{mn} = s_{m-n}$. As $m$ goes to infinity, the terms tend to 1, and as $n$ goes to infinity, the terms tend to 0. This solution has the added advantage of a lot of symmetry.

Or is this explicit way of “just doing it” not a good illustration of the inductive-construction philosophy?

]]>I have a question that I have been working on for a long time and somehow relates to a lot of articles I’ve read in your blog about Fermat’s little theorem.

Here’s the question,

Do all primes divides a Mersenne number? A Mersenne prime divides itself.

As a counterpoint 7, a Mersenne number itself does not divide any Fermat number, it seems so.

Thanks!

]]>There is a typo in example 5:

“How can we make the first column tend to 0, given that it must agree with the first row at a_{11}?”

It should say “first column tend to 1.”

*[Thanks very much for pointing that out — I’ve corrected it now.]*

———————

Studying the methods of solving problems, we perceive

another face of mathematics. Yes, mathematics has two

faces; it is the rigorous science of Euclid but it is also

something else. Mathematics presented in the Euclidean

way appears as a systematic, deductive science; but

mathematics in the making appears as an experimental,

inductive science. Both aspects axe as old as the science of

mathematics itself. But the second aspect is new in one

respect; mathematics “in statu nascendi,” in the process

of being invented, has never before been presented in

quite this manner to the student, or to the teacher

himself, or to the general public.

——— Polya —————

Regards!

]]>I thought we could use some words from Polya in the Introduction Page to the Tricki. I looked at the preface of “How to Solve It” and the paragraph I liked the most for us was this one:

<>

— George Polya, How To Solve It: A new aspect of mathematical method. 1944 —

If you like the idea, I plan to look also into his ‘more serious’ volumes “Mathematics and Plausible Reasoning” (vol1 “Induction and Analogy in Mathematics” and vol2 “Patterns of Plausible Inference”). In any case, I suspect that they can be a source for the Tricki to much greater extent than just quoting…

Btw, another “concept” (using the term I suggested in the hierarchy I sketched by email) would be the one of “reverse engineering”: If you know what you want your solution to look like, then try to arrive from it to your hypotheses (or to _some acceptable_ hypotheses as well) by a chain of double implications. If any extra assumption was made in the process, just add it to the hyphoteses.

]]>http://thetangentspace.com/wiki/index.php?title=Pigeonhole_constructions

along with a few other examples:

* a dense set in the plane with no two similar triangles

* an infinite matrix with no singular subsquares

* a graph exists for any countably infinite choice of positive degrees

On another note: if the questions are two elementary, please delete them. ]]>

You can well-order the reals using the well-ordering _theorem_:

http://en.wikipedia.org/wiki/Well-ordering_theorem

which is logically equivalent (in ZF set theory) to the axiom of choice. The ordering would have to be very different from the usual ordering of the reals, which is of course very poor (as opposed to well, hehe math humor).

“Normal” induction is usually meant to only apply to a countable set where each element has an immediate successor, such as the natural numbers. You proceed like this: if I know phi(x), then I also know phi(x+1). But this approach fails as soon as the set has an element that is not the immediate successor of another, even if it’s well-ordered.

Here’s a simple example of such a set: Let’s take N, the natural numbers, and add the single element B (for “Big number”) which is larger than any element of N. Clearly any nonempty subset still has a least element, so it’s still a well ordered set. But now normal induction fails sometimes — for example, let me define 1 \in N as odd, and then n+1 as (even,odd) iff n is (odd,even). Is B even or odd? I could write a normal inductive proof to show this is a good definition on N, but it fails to hold for B as well.

The wikipedia entry on transfinite induction splits induction into two types: successor case (as with the natural numbers, the normal way); or the limit case, of which B is an example. But you can also elegantly state transfinite induction as a single case, which planetmath gives:

http://planetmath.org/encyclopedia/TransfiniteInduction.html

]]>On both wikipedia and mathworld.wolfram.com the well-ordering principle is defined as “every nonempty set of positive integers contains a smallest number.” Maybe I am missing something here, how could that imply positive real numbers could be well ordered?

Another elementary question I have is: what is the difference between a normal induction and a transfinite induction? Is that in a normal induction, we prove something holds for a chain of sets a_1, a_2,…, a_n, then show it holds for set a_{n+1}, while in transfinite induction, a_n is replaced with the union of all sets before it? Thanks.

]]>My other thought on this is that I don’t see how a computer would “just write it down” but I do see how a computer could “just do it”.

]]>However, to make a serious point on this, there is a slight difference between Example 5 and the others: in Example 5 you construct an actual matrix with actual numbers. The others, while mildly constructive, give a procedure which, if followed, would produce the desired ‘whatever’ but no one is seriously going to follow those steps and the final answer is unguessable without doing so. Of course, this slight difference is probably too slight to mention: if one can see a solution straight away then one doesn’t need a trick but if one can’t then this is a good trick to use to get the solution. Still, bytes are cheap!

]]>One remark I intend to make is that establishing existence is easy if there are few enough constraints, and sometimes also easy if there are many constraints (in the latter case because you don’t have much choice about what to do), which leaves a rather interesting class of problems that are hard because they are somehow in between the two extremes. The problem of finding a Hadamard matrix of order is a good example. Another is the problem, also open, of finding a Hamilton cycle in the bipartite graph that consists of all subsets of size or of a set of size , with joined to if one is a proper subset of the other. There it is easy to get started just building a path, but after a while the constraints become overwhelming. And yet it seems that there are probably several examples, so one isn’t led to any particular construction.

This phenomenon partly explains another trick — that of proving a stronger result than you need, in order to reduce the size of the space in which you are searching. Of course, finding a suitable strengthening isn’t easy.

]]>Hmm, this brings up a more general suggestion for the Tricks Wiki – in addition to having examples and discussion of situations in which the trick shows some promise of working, one might also want to exhibit examples or general situations in which the trick is *not* likely to be effective. For instance, in this case, a “just do it” approach fails horribly for the task of constructing Hadamard matrices (orthogonal square matrices with entries +1 and -1) – the constraints are just too global and too algebraic to give one the “wiggle room” one needs for this sort of approach.

Also, I realize that this isn’t the point of the post, but *another* possible trick might be the trick that is actually used to compute that C, namely that of bounding a sequence by a geometric series. Similarly, certain integrals can be bounded by exponentials, which is useful for the (obviously related) problem of calculating a tail bound for the normal distribution. There are certainly other examples, although I don’t have them at my fingertips.

*Thanks — correction made. And your suggestion would indeed make a suitable Tricks Wiki article. And if you don’t have as many examples as you would like, it shouldn’t stop you as we will be encouraging submission of incomplete articles, provided they are clearly marked as such with a sentence such as “This article needs a few more examples,” or “I have several examples but I do not yet have a clear general description of the method.”*