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The Math Myth

econlog.econlib.org

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Re: The Math Myth

#311

Post by [deleted] »

MichaelBurge wrote: post #32
iopq wrote: post #18

I think society would be a lot better if BASIC math and statistics would be better understood. How many times do you see a study posted here with N=23 and people say "the sample size is too small" when it's clearly not? How many people ask for a card deck change to change their luck? How many times do people read a poll like 49% +/- 3% vs. 43% +/- 3% and conclude the two candidates are statistically tied? I could pro…

> I mean I wonder how many people even understand that 0.999... = 1? To be honest, I think it's unreasonable to expect anybody - even with a Ph.D in a field other than mathematics - to be able to even define the real numbers: My definition is probably very different from yours(I tend to say there's countably many real numbers).

[deleted]

Re: The Math Myth

#312

Post by aianus »

superuser2 wrote: post #173
treehau5 wrote: post #157

Earlier quoted context omitted.

For software engineers specifically -- every time you are coding formalized programmatic logic, you are using math. If you took a logics and proofs course, it would help you formalize logic better. Every time you write a "for" loop you are essentially using summations. There is a book called Concrete Mathematics and one of the primary authors is Donald Knuth, basically it's "Programmers math" and in my opinion, would…

>every time you are coding formalized programmatic logic, you are using math Almost no one (outside of those who were friends with math majors) are actually familiar with the nature of proof-based mathematics. When you hold out "programming is math" the general public, policymakers, admissions offices, kids who might want to be programmers, etc. don't make the association to Analysis and Abstract Algebra, they make i…

> This has the effect of selecting programmers for their skill, patience, and diligence in executing the algorithms they're given.

Our high school experiences couldn't have been more different.

In math class we were given the definition of a limit and derived all of the rest of high-school calculus collaboratively from there. Similarly for high-school linear algebra: couple of quick axioms and then all further theorems and algorithms derived from there.

Meanwhile in Chemistry class they'd give me a failing grade because I failed to memorize the exact charges of the various polyatomic ions and solving the rest of the problem using variables in place of the charges wasn't good enough.

Re: The Math Myth

#313

Post by banach2 »

nilkn wrote: post #225
banach2 wrote: post #195

Earlier quoted context omitted.

What? Why -0.9?

Because that's exactly what the argument above does. It simply subtracts away the decimal part of 9.999.... This is always wrong except in the case of infinitely many repeated digits, and the proof does not explain this. More rigorously, let 9.999{n} denote an expansion with n 9s after the decimal point, where n can also be infinity. The subtlety with the argument is that it needs X to be the same as everything after…

Okay, this is a little late but when you say 10X = 9.999... then X = 0.999... not because you have removed the 9 but because that's what X is; a tenth of 10X! So when you say in your argument 10X = 9.9 and therefore 10X - X = 9.9 - 0.9, that's wrong. 10X = 9.9 implies X = 0.99. No wiggle room. No need to consider infinites, just one movement of a decimal point.

[And hence 10X - X = 9.9 - 0.99 = 8.91 = 9X implying that X = 0.99]

Following, as delineated above, from there, you'll see there's no contradiction. It's not so easy to break arithmetic that easily without dividing by zero :)

Sorry for dragging on this meaningless thread.

Re: The Math Myth

#314

Post by astazangasta »

yummyfajitas wrote: post #308
astazangasta wrote: post #307

Earlier quoted context omitted.

Irrational numbers can be represented exactly; integrals allow us to perform exact calculation using them. In what comparable way does truth (e.g. the true process underlying gene expression) have a role in statistics? We neither measure the truth nor model it; it is absent.

Integrals themselves are, except in very special cases (piecewise functions with rational values), only definable as limits - specifically the limit of the Riemann sum. (You can also use measure theory, but measures themselves are only definable on sigma algebras, which in the non-finite case are also not explicitly constructable.) In what comparable way does truth (e.g. the true process underlying gene expression) h…

What I'm arguing is we have no idea what "true model" is, what we have is "presumed model" and "observation". In the example I gave, we can never know the true source of the data we have observed (biology), we can only test our observations against some constructed model. Biologists are using a model they know to be wrong because that is what all models are - we know them to be wrong, we just can't do otherwise, because the truth is not available to us.

I feel like I've made this same point about four times already, so if you aren't getting it, let's just stop here.

Re: The Math Myth

#315

Post by tree_of_item »

wolfgke wrote: post #309
mattmcknight wrote: post #253

Earlier quoted context omitted.

>Almost no one (outside of those who were friends with math majors) are actually familiar with the nature of proof-based mathematics Hmm...when I was a CS major, a discrete math course was required, and it was all proofs. I believe that is still common. I don't find the proof aspect that relevant to programming, but the concepts of discrete math, such as sets, graphs, definitely are. As far as things that aren't prog…

> I don't find the proof aspect that relevant to programming By Curry-Howard isomorphism proofs in some particular proof system are equivalent to computer programs.

To a subclass of computer programs; computation is more general than logic. There are lots of programs that can't be typed but still do useful work.

Re: The Math Myth

#316

Post by nilkn »

lisper wrote: post #232
kazinator wrote: post #178

Earlier quoted context omitted.

So you're saying a countable infinity can be occupied, so that there is no room for one more. You must then disbelieve concepts such as that the even integers can be put in 1:1 correspondence with all integers; i.e. that there are exactly as many even integers as there are integers.

> So you're saying a countable infinity can be occupied, so that there is no room for one more. No. I'm saying that a countable infinity doesn't have room for one more at the end . There is no end. You can obviously stick a 1 in the middle, but then you have a number strictly less than 0.999...

Of course you can. Consider the sequence of rational numbers in (0,1). Now consider the sequence of rational numbers in (0,1]. The latter has had an extra term appended.

(This is of course flawed, but I think it illustrates that the question isn't completely trivial. It requires us to carefully distinguish between the notion of an ordered set and a sequence and even then we'll have to deal with the fact that the rationals can be made into a sequence, but not with the same ordering.)

Re: The Math Myth

#317

Post by [deleted] »

banach2 wrote: post #313
nilkn wrote: post #225

Earlier quoted context omitted.

Because that's exactly what the argument above does. It simply subtracts away the decimal part of 9.999.... This is always wrong except in the case of infinitely many repeated digits, and the proof does not explain this. More rigorously, let 9.999{n} denote an expansion with n 9s after the decimal point, where n can also be infinity. The subtlety with the argument is that it needs X to be the same as everything after…

Okay, this is a little late but when you say 10X = 9.999... then X = 0.999... not because you have removed the 9 but because that's what X is; a tenth of 10X! So when you say in your argument 10X = 9.9 and therefore 10X - X = 9.9 - 0.9, that's wrong. 10X = 9.9 implies X = 0.99. No wiggle room. No need to consider infinites, just one movement of a decimal point. [And hence 10X - X = 9.9 - 0.99 = 8.91 = 9X implying tha…

[deleted]

Re: The Math Myth

#318

Post by lisper »

nilkn wrote: post #316
lisper wrote: post #232

Earlier quoted context omitted.

> So you're saying a countable infinity can be occupied, so that there is no room for one more. No. I'm saying that a countable infinity doesn't have room for one more at the end . There is no end. You can obviously stick a 1 in the middle, but then you have a number strictly less than 0.999...

Of course you can. Consider the sequence of rational numbers in (0,1). Now consider the sequence of rational numbers in (0,1]. The latter has had an extra term appended. (This is of course flawed, but I think it illustrates that the question isn't completely trivial. It requires us to carefully distinguish between the notion of an ordered set and a sequence and even then we'll have to deal with the fact that the rati…

No. There is no such thing as the sequence of rational numbers in an interval. The rationals have a total order, and they can be enumerated, but the total order of rationals does not define an enumeration of the rationals and hence the rationals are not a sequence. If you want to dispute this, tell me: what is the next rational after 1/2?

Re: The Math Myth

#319

Post by nilkn »

lisper wrote: post #318
nilkn wrote: post #316

Earlier quoted context omitted.

Of course you can. Consider the sequence of rational numbers in (0,1). Now consider the sequence of rational numbers in (0,1]. The latter has had an extra term appended. (This is of course flawed, but I think it illustrates that the question isn't completely trivial. It requires us to carefully distinguish between the notion of an ordered set and a sequence and even then we'll have to deal with the fact that the rati…

No. There is no such thing as the sequence of rational numbers in an interval. The rationals have a total order, and they can be enumerated, but the total order of rationals does not define an enumeration of the rationals and hence the rationals are not a sequence. If you want to dispute this, tell me: what is the next rational after 1/2?

I took some liberties with the word "the", but I'm pretty sure that you know that it's easy to construct an ordered sequence of all rational numbers using any of a number of standard diagonalization arguments. I'll skip the details you already know.

Eventually, yes, you're going to be able to poke a hole in my argument. It is definitely flawed. I don't know how long it'd take us to get there, but it doesn't really matter. But we're already at the point where this cannot be considered "basic", and that is my true point here.

Attempting to demonstrate that 0.999... = 1 while meticulously avoiding any rigorous definition of what 0.999... means is not very easy and will require you to fend off all sorts of potential jabs from various directions. It's much easier to just talk about infinite sums and be done with it.

Re: The Math Myth

#320

Post by lisper »

nilkn wrote: post #319
lisper wrote: post #318

Earlier quoted context omitted.

No. There is no such thing as the sequence of rational numbers in an interval. The rationals have a total order, and they can be enumerated, but the total order of rationals does not define an enumeration of the rationals and hence the rationals are not a sequence. If you want to dispute this, tell me: what is the next rational after 1/2?

I took some liberties with the word "the", but I'm pretty sure that you know that it's easy to construct an ordered sequence of all rational numbers using any of a number of standard diagonalization arguments. I'll skip the details you already know. Eventually, yes, you're going to be able to poke a hole in my argument. It is definitely flawed. I don't know how long it'd take us to get there, but it doesn't really ma…

> It's much easier to just talk about infinite sums and be done with it.

Indeed. But it's a lot less fun :-)

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