That was interesting. For years I have felt that I am teaching my students an unrealistic view of the world. In mathematics all real numbers have an existence. Pi is a number which has an infinite, non-repeating decimal expansion. But this is not really meaningful. I forget the actual number, but if we accurately know Pi to less than 100 decimal places we can still calculate the diameter of the known universe to an accuracy of less than the diameter of an electron. For all practical purposes Pi = 3.14159265358979 is way more than accurate, so why do mathematicians say that is wrong if an exact value is required?

It is partly because we are all taught that ourselves and get into the habit of thinking that real numbers are real. But in reality they are no more real than imaginary numbers. Of course, at the other extreme, kids are taught to rely on calculators and that any number of decimal places is okay. They cannot understand why they get the wrong area by using Pi = 3.14 is they are asked for two decimal places of accuracy.

There are times when it is frustrating to be a mathematician!

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Still officially AGA 5d but I play so irregularly these days that I am probably only 3d or 4d over the board (but hopefully still 5d in terms of knowledge, theory and the ability to contribute).

Yeah, as an undergraduate discovering Gabriel's Horn disturbed me. But not as much as the Banach-Tarski paradox, and the realization that the axiom of choice and the way that mathematicians have handled infinity since Cantor has led to the rejection of Aristotle's fundamental distinction between actual infinities and potential infinities. While I mostly kept it to myself in graduate school...I have always sympathized with Kronecker and, especially, Poincare.

Think of a triangle as two different routes of getting from A to B.



It's true that B is four steps to the right of A and four steps above A. But if you take 4 steps to the right, those steps only take you to the right... they don't take you up at all. So now you have to take 4 steps up to get to B. That's not terribly efficient.

If you instead turn slightly and walk directly towards B then every step takes you both up and to the right. You're heading directly towards your goal instead of starting off heading parallel to it. Which means it's a shortcut. Of course it'll be shorter.

Try this experiment. Nail three nails into your Go board in the shape of a triangle (just kidding, don't actually do that!) Now take a piece of string and tie one end to A. And then run it along the bottom of the triangle and turn at the nail and run it up to B. Cut the string there. Now return all the string to A and stretch it directly to B. You'll see that the string that was needed to reach A going right and then up is much longer than you need when you go directly from A to B.

Now you've done it--encouraged me!

Perhaps you will find this old blog of mine of some small interest, also: http://www.britannica.com/blogs/2007/01 ... innocence/

BTW, I was Britannica's first blogger, as part of my duties as an editor.

Here's another one: Half of any positive number is still positive. The limit of ((((1/2)/2)/2)...) is 0. Therefore 0 is positive, and all the fun that entails.

The key idea these are all addressing is that the limit of a sequence is not necessarily a member of that sequence. It can be something else that the sequence never quite reaches, so it doesn't necessarily have the same properties as the sequence. So the length of every path in your sequence is 2, and the limit is indeed a straight line, but that straight line has length sqrt(2). It seems odd at first, but remember that any member of your sequence still has an infinite number of iterations in which to lose that extra ~.6

Your observation does demonstrate that any simple finite grid would have the property of a zig-zag being equal to two straight lines, so our universe is clearly not quantized in that fashion.

If you prefer a pixel universe, though, it could just be a more complicated one. After all, it's easy to simulate euclidean motion on a computer screen. All circles could be jagged, just too fine for us to detect =)

If one wants to look at infinite series...Consider any conditionally convergent infinite series. Then by Riemann's Rearrangement Theorem, the terms can be reordered in such a way to make the sum be any number. Do you get that? While 1 + 2 = 2 + 1 (commutative property), this does not hold true for conditionally convergent series. Allowing actual infinities generates (infinitely?) many paradoxes.

FYI, see links starting here http://mathworld.wolfram.com/Conditiona ... gence.html

That isn't really a paradox if you think about the definition of infinite series: It is the limit of a sequence of real numbers, where the sequence is defined by a particular partial sum formula.

An infinite series is not defined as the sum of all numbers in a particular sequence independent of permutations. If this were the definition of an infinite series, then the sum of a conditionally convergent sequence would be undefined.

This is not quite a paradox either. Volume and area have incomparable units. The Horn already contains a subset called the "surface"---the part of the horn that makes up the "surface" and has zero volume---that can be used to paint the surface. In mathematical idealizations, surfaces are infinitesimally thin. Real paint is not.

Mathematics is about an internal consistency of abstract ideas defined by humans, rather than physical reality. Our difficulties with accepting mathematics of infinities often arise when we confound the reality that generates some of the ideas for mathematics with mathematics itself, which is often an idealization. Platonic perfect circles help us think about real world imperfect objects similar to perfect circles, but we shouldn't be shocked that perfect circles of our imagination sometimes behave differently from the imperfect objects of reality that inspired them. People can understand this logically but still be emotionally discomforted by it. I wonder what part of our evolution makes it so.

(The Britannica blog posts were very interesting. I didn't realize that these existed)

Wrong way round there. Wikipedia gets it right.

Yes, I understand, and this was my real point. One needn't embrace Robinson's non-standard analysis to realize problems exist in applying math too literally to the real world. (No one can deconstruct and reassemble a real world object in the manner of Banach-Tarski.) Confusing models, mathematical or otherwise, for reality leads to paradoxes.Take painting Gabriel's Horn as an example, I know the argument about the volume of any paint application, but it is just a simple way of conveying an image and is not meant to be taken literally, while the actual mathematical volume and surface area are finite and infinite, respectively--and this imperfect match between model and reality, I suggest, is what leads to the perception of paradox or should I say cognitive dissonance. Of course, such observations are not original with me. Many smarter, more creative thinkers have explored some of the issues. (Owen Barfield, Saving the Appearances, is a relatively easy starting point if anyone is interested. But many other philosophers have tackled the subject.)

BTW, I am glad if you found any of our blogs of interest. I especially had fun writing How to Cheat at Chess (before I retired!): http://www.britannica.com/blogs/2006/11 ... -at-chess/.

That reminds me of the snowflake curve: an infinite length surrounding a finite area.

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Still officially AGA 5d but I play so irregularly these days that I am probably only 3d or 4d over the board (but hopefully still 5d in terms of knowledge, theory and the ability to contribute).

Damn, does my blog really say it backward? I was apparently in a hurry to go to lunch.

Edit: Or the copy editors changed it before uploading. (That was a common problem with English majors meddling with our math articles, though I may have been to blame for this error.)

Edit 2: You may notice I said it right concerning the paint analogy, so I think this may have been a copy editor meddling. This drove me crazy at EB!

Edit 3: Someone changed that sentence between 5/12/13 and today! I know because I decided to save a copy of most of my old blogs after I built myself a new computer. And as of 5/12/13 it said, "For example, Gabriel’s Horn (also known as Torricelli’s Trumpet) is formed by rotating the curve y = 1/x for x > 1 about the x-axis. Remarkably, the volume of the resulting three-dimensional figure is finite, while the area of the two-dimension surface of the horn is infinite." I cannot understand why anyone there would muck with my old blog.

This is just a result of the way we do maths. If pi, for example, had been a simple ratio then we wouldn't need to name it. By giving those irrationals that appear often in common usage we are able to work with them, often without even needing to know their exact values. The alternative would be to constantly rely on approximations.

Here's another one of modern math: 0.999... = 1. The one and only proof I accept meanwhile is that if 0.999... < 1 then you couldn't construct a legitimate number in between, because there is no number 0.000...1, and if there is no number between two numbers it'd be silly to say that 1 is bigger and if 1 is not bigger or smaller it must be even with 0.999... because there is only that left.

Here's another one: If there are infinite many natural numbers then are there natural numbers that are infinitely long? Answer: no, because by axiom every natural number must have an successor, but if a natural number would be infinitely long that one couldn't determine a successor, even in theory.

But the biggest struggle is that modern math often behaves "as if". They do "as if" a variable can stand for any value of a set, "as if" Cantor's diagonal number matches every number of an infinite!!! long list of real numbers, creating a number that can't be in that list, "as if" Gödel can still beat me with a Gödelnumber, although every time he constructs a Gödelnumber I instantly (and infinitely fast) fix my system from the previous Gödelnumber, so that I should not fall under Gödel incompleteness theorem. I do not like that. That's metaphysics and I doubt that math would be behind with "my approach", that is superstrict finitism: only what you or a machine can calculate for real is real. There'd be no proof that square root of 2 is no rational number, there'd be only a strong theory that there is none, because no one has ever found one, just like in normal science. Because my thinking is: Just as x/0 is suddenly non-defined in an otherwise seeminly perfect formal system of arithmetics who the hell are we to pretend we know that of infinitely many numbers there aren't some that would match sqaure root of 2? That's like pretending "as if" we are God which we are not.

@math guys: Would my kind of math really suck? Or could this be the future?

p.s. Another one: how do we know that pi=pi? Since pi is irrational it's infinitely long, so we have no point to compare. How are we so sure that pi=pi? If you say: well because the axiom is 'n=n' then I say: no problem, maybe pi is an inconsistent number, so pi is like sometimes 'n' and sometimes 'n+1' and then the axiom is not applicable at all.

No, it really is a deep mystery! For example, the Classical Greeks rejected irrational numbers for deeply felt philosophical reasons--mostly dealing with their abhorrence of infinity.

Of course, pi shows up for circles, though the Greeks thought that some exact ratio existed. And for f(x) = c*exp(x), the fact that f'(x) = f(x) (the derivative, or rate of change, of the exponential function is equal to the original function) has enormous consequences for modeling processes in the real world.

Perhaps the greatest mystery of all is why math can describe reality. One can take the stance that it is a useful fiction--that the order is in our heads rather than in reality--but this has always felt wrong to me. To put it another way, I don't believe that mathematicians create math, instead mathematicians discover math, which is absolute truth within each coherent system of math. (Basically, mathematical Platonism.) How or why any such mathematical universe is useful as a model for some realm in reality has never been satisfactorily answered. (At least in my humble opinion.)

That'd be my take. It remembers me of the guy that once wanted to prove creationism with the fact that an apple is perfectly rounded and sized for our hands, because God made it for us^^.

Although I retired from EB, I continue to teach math and stats part-time variously at Purdue University Calumet, Indiana University Northwest, and Ivy Tech--all near my home. In the past 30 years I have noticed a marked decline in basic arithmetic skills, which I attribute to the early introduction of calculators. (Of course, we now get the bottom quartile showing up in college owing to the lack of good manufacturing jobs.) Even more problematic than the lack of reliable arithmetic skills, however, seems to be a general decline in reading ability and critical thinking. This makes it very difficult to give students interesting applications to explore, and leaves little more than simple plug and chug problems--and many students will complain shrilly if they haven't seen an example the required the exact same steps for solving.

Except, if it is only in our heads, why do our models have any predictive value in the physical world? I, for one, believe that there exists a world outside my own head, which can be modeled somewhat inside my head. I think that this is very fundamentally different than the anthropocentric creationism beliefs. Or even the belief in "accidental" anthropocentrism. But now I'm lost and must go play Go!

Absolutely! I agree with every word of this. The quality of the students I try to teach calculus to now is abysmal compared to the first time I taught it 37 years ago. The problem is that now everyone has to go to college and they all think they deserve a grade A because they paid their fees. The truth is that 75% of them should not be there but there is nothing else for them to do. All the manufacturing jobs have gone overseas and anything except minimal wage requires a "degree".

In my opinion it is an organized effort to dumb down the populace so that they don't have the intelligence to understand what is happening to them. I once had a student, about 5 years ago, pull out a calculator out of habit and punch in 0x0.

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Still officially AGA 5d but I play so irregularly these days that I am probably only 3d or 4d over the board (but hopefully still 5d in terms of knowledge, theory and the ability to contribute).

And yet, it's a widely documented fact that on average we're collectively getting smarter. This is across all countries, developed and developing nations. See http://en.wikipedia.org/wiki/Flynn_effect. So your conspiracy theory of a deliberate dumbing down of the population doesn't square the evidence. If you are genuinely observing a weakening trend among your students, and you are not just applying a selective memory, what else can be the cause?

You have given one possible explanation, namely a lowering of standards required to enter the university, but that only makes sense if the university is generally accepting many more students than it used to, or there are many more universities than before, meaning the average student across them will be of a lower standard. Perhaps these are both true.

Another cause may be the metrics you apply are less relevant now. For instance, high levels of arithmetical numeracy is perhaps less important now (given computers/calculators) and instead we expect students to be better at abstract or scientific reasoning.

Photographers use that trick when posing their clients....