Why is the diagonal of a square not "2"

[Aidoneus]

... Gabriel’s Horn ... Remarkably, the volume of the resulting three-dimensional figure is infinite, while the area of the two-dimension surface of the horn is finite.

Wrong way round there. Wikipedia gets it right.

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.

[Coleslaw]

I'm surprised that so many "key numbers" like pi, e, gold number, root(2) are all irational numbers...
I think we are missing something. like if our world is just an imperfect "projection" of something else...

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.

[Pippen]

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.

[Aidoneus]

I'm surprised that so many "key numbers" like pi, e, gold number, root(2) are all irational numbers...
I think we are missing something. like if our world is just an imperfect "projection" of something else...

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.

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.)

[Pippen]

One can take the stance that it is a useful fiction--that the order is in our heads rather than in reality-

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^^.

[Aidoneus]

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

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.

[Aidoneus]

One can take the stance that it is a useful fiction--that the order is in our heads rather than in reality-

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^^.

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!

[DrStraw]

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

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.

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.

[quantumf]

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.

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.

[Nyanjilla]

So how long is the diagonal of a square if you rotate it by 45 degrees?

Stay-at-home Mum discovers amazing new diet technique: rotate through 45 degrees and instantly get slimmer!

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

[Bill Spight]

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.

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.

When I was in high school I ran across a few of my grandmother's high school textbooks. They were more difficult than my textbooks, college level, even. It seems like high school was harder in her day. But, according to the Flynn effect, people my age were "smarter" in academic terms than those of her day. How can we reconcile this apparent contradiction? Maybe it is a question of selection. In her day, most people did not go to high school. That is why in theory an IQ of 100 meant a "mental age" of 13. The apparent dumbing down can be attributed to a greater number of students going to high school in my day. Perhaps the same kind of selection effect is going on today. In relative terms the cohort of people who graduated from high school in my day but did not go to college would be capable today of getting an associate degree. And many of them are doing so. Add to that the economic pressure to get some kind of certification beyond the high school level and you can get students going on to college who can't hack it, even if they are better academically than the average high school graduate of their parents' time.

Thinking about this in terms of go, suppose that the Flynn effect means that the slice of the population who would have been 7 kyu 50 years ago would now play at the level of 6 kyus from that time. However, because of rank inflation, they would now be 5 kyus.

Edit: In my day the average college graduate had an IQ around 125. Translated into mental age, that gives a mental age of 16.25, a bit below the mental age of the average high school graduate of my grandmother's day. They were actually smarter than that, because IQ norms are adjusted to the current population. But I doubt if they were as smart as the average college graduate of my grandmother's day.

[DrStraw]

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.

That link does not say we are getting smarter. It say IQ scores are increasing. IQ scores measure a specific subset of those skills required to succeed. But beyond that, it is not really relevant because I was not referring to the average level of the entire populace, but to the average level of those attending university. This also is consistent with my claim of dumbing down: people's ability to perform certain tasks as tested by IQ tests does not imply those same people have the same ability to analyze the world around them - it merely means that they have been trained to pass standardized tests instead of think for themselves.

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.

Most definitely that is true. A few years ago the governor of Virginia mandated at all state colleges and universities increase enrollment by 14% in three years. Obviously, then, the top schools (UVa and VPI, etc.) get the best students and the second tier have to start at a lower level. The bottom tier school have to take what is left over and average standard must drop considerably. I have seen a drastic decline in just the last five years.

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.

The ability to do abstract thinking is what I was referring to. The overall ability of the general populace may have increased, although from my anecdotal evidence I would question that. However, when only 2% high school kids go on to college then their average IQ will be well in excess of 100. When well over 50% are going then the average IQ is going to be only a little above 100, even allowing for IQ inflation, and many of them are going to be below average IQ.

[Bill Spight]

One can take the stance that it is a useful fiction--that the order is in our heads rather than in reality-

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^^.

A friend once told me this story about his uncle, who lived in the American Bible Belt. His uncle picked up a hitchhiker one afternoon and after a while, lit up a cigarette. The hitchhiker said, "If God had wanted you to smoke, He would have given you a smokestack." The uncle pulled off the road and said, "If God had wanted you to ride, He would have given you wheels. Get out!"

[illluck]

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^^.

That's a new one. Ray Comfort had the argument that bananas are perfectly formed for human consumption and therefore God.

[Aidoneus]

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?

If small changes in IQ scores indicate anything, it is changes in poverty. This can be seen, for example, as differences among ethnic groups in the U.S. substantially disappear when adjusted for income. It is similar to the rise in longevity due to decreases in infant mortality.

When I was in high school I ran across a few of my grandmother's high school textbooks. They were more difficult than my textbooks, college level, even. It seems like high school was harder in her day.

Yes! Try having a modern student read the 6th grade McGuffey Reader. (Widely used in the U.S. in the 19th century.) Or maybe take this graduation test from 1912: http://www.huffingtonpost.com/2013/08/1 ... 44163.html

BTW, my father was born in 1895 and had only a sixth grade education. I have no doubt, however, that he could have joined me in Mensa.

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.

There is no maybe about the much higher percentage of the population going to college. As mechanization destroyed the viability of small farming--first in the U.S. and then around the world--the displaced populations relocated to urban areas and found new occupations in manufacturing. (Steinbeck may have romanticized his Okies, but not the problem of distribution of goods under mass production.) I pity young people as the mechanization of manufacturing has been accelerating their displacement since the 1970s; e.g., by the 1990s, U.S. Steel produced the same volume of steel with one-tenth of the old workforce. Forget outsourcing, the volume of U.S. manufacturing has not declined. Automation is behind the loss in manufacturing jobs, and few have considered the long-term implications. We are now getting many U.S. students into extreme debt for non-STEM degrees (science, technology, engineering, math)--degrees that have little marketable value, but which have done wonders for the bottom line of public and private institutions through government guaranteed loans and grants.

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.

I haven't seen any improvement in abstract reasoning among incoming students. But rather than share my anecdotes, I will share this link concerning the failure to develop critical thinking during college: http://www.mcclatchydc.com/2011/01/18/1 ... s-not.html.