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The Math Myth

econlog.econlib.org

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Re: The Math Myth

#141

Post by tgb »

kazinator wrote: post #62
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…

0.999... is equal to 1 only if we assign a particular semantics to the "..." notation. Namely if "..." means "the limit of the decimal number to the left, as the repetitions of the last digit grow ever larger", then 0.999... is an alternative notation for 1 since that limit is 1. The actual number formed by repeating 9's an infinite number of times is not constructable. Whereas 1 is constructable. So they cannot be t…

You might be interested in reading about transfinite induction, which is an interesting and legitimate mathematical proof technique. However your 'induction' does not succeed in doing this. Transfinite induction requires two induction steps, you've only provided one of them. Not that you would actually use transfinite induction for anything resembling this, that's what calculus and analysis is for.

For example your strategy would show that the sequence of positive integers 1, 2, 3... has a finite limit! After all, each one is finite so by induction the limit is finite too. Except of course no one would expect this induction to apply to the limit as well as to the individual steps.

Re: The Math Myth

#142

Post by Ologn »

analog31 wrote: post #25

At my workplace, we have about 60 scientists and engineers. The author's observation is accurate, that most people never use math beyond Excel and 8th grade math. They also never use most of the theory that they learned in their science (including CS) and engineering educations. The typical career arc is to get through college, then sit down at a CAD workstation, or programming terminal, and forget all of your math a…

> When a math or theory problem arises, they take it to the resident "math person."

When an obvious math or theory problem arises.

I'm sure they run across problems that more advanced math can solve, or stumble into problems due to not knowing more advanced math, yet their lack of knowledge prevents them from even understanding they confront a math problem. Sometimes the problem is not what they know they don't know, but what they don't know they don't know.

Re: The Math Myth

#143

Post by tgb »

kazinator wrote: post #124
jhanschoo wrote: post #111

Earlier quoted context omitted.

> So they cannot be the same thing. That's because, philosophically, two objects must be identical in every property to be the same object, and constructability is a property. Mathematicians have proposed several different constructions of the reals. If you were familiar with, say, constructing the rationals from the integers or constructing the reals from the rationals, you would know that this is not a problem. One…

I completely agree with you, and understand all that. > L is the smallest number in E such that L is greater than or equal to all numbers in S. And note that this doesn't imply that L is in S! So here we have S = { 0, 0.x, 0.xx, 0.xxx, ... }. E is R, the set of reals. The smallest number in R greater than or equal to anything in S is 1. That doesn't mean 1 is in S. If 1 is not in S, then we're taking a notation based…

Yes, that is exactly what that notation means, just as 0.1111... is not any of the numbers 0.1, 0.11,...

Supremum of an increasing sequence of reals is equal to the limit of the sequence in all cases.

Re: The Math Myth

#144

Post by flogic »

nostrademons wrote: post #88

This largely matches my experience - as a software engineer, I spend probably In a market economy, basically all returns come from marginal gains. The vast majority of your lifetime income will come from a dozen or fewer opportunities that you happen to be in a position to take advantage of, whether it's a new job offer or a high-profile project you volunteer for or a startup that takes off. You will qualify for thos…

I think there are 3 fundamental issues that are coming together. The first is a discounting of the fact that programming and using Excel are in their own way doing higher level math. You're not doing the lower level grunt work but combining those techniques to do something greater. The second being that many problems aren't about mathematical truth so much as mop and bucket work. Designing your process for off hours communication isn't going to simplify into an elegant equation. However being really good at that sort of thing is important. The final thing is most people are lousy at recounting what they did/do. So it maybe that a significant number of people do use advanced math < 1% of the time but don't actually remember it.

Re: The Math Myth

#145

Post by anthk »

moron4hire wrote: post #49
sp527 wrote: post #44

Earlier quoted context omitted.

Hardly. We use this reduction because it's essential from a mathematical and axiomatic perspective and because limits are a fundamental construct. But more philosophically, a natural and nonterminating simulation of 0.999... (were it possible) would never strictly equal 1. You would wait an infinite amount of time for it to do so. This comes down to how you view the problem. I would never argue with this through the…

You are discarding an essential part of the representation. You are moving the goal post to make yourself right. Just because people don't always understand infinitesimals doesn't make them right. It's the reason they are wrong. There just plain don't know what they are talking about. You can't represent 0.2 exactly using IEEE floats, either, but that doesn't mean the representation 0.2 is not exactly equal to 1/5th.

Scheme has (exact->inexact x) and (inexact->exact x) for those issues.

Re: The Math Myth

#146

Post by jhanschoo »

kazinator wrote: post #124
jhanschoo wrote: post #111

Earlier quoted context omitted.

> So they cannot be the same thing. That's because, philosophically, two objects must be identical in every property to be the same object, and constructability is a property. Mathematicians have proposed several different constructions of the reals. If you were familiar with, say, constructing the rationals from the integers or constructing the reals from the rationals, you would know that this is not a problem. One…

I completely agree with you, and understand all that. > L is the smallest number in E such that L is greater than or equal to all numbers in S. And note that this doesn't imply that L is in S! So here we have S = { 0, 0.x, 0.xx, 0.xxx, ... }. E is R, the set of reals. The smallest number in R greater than or equal to anything in S is 1. That doesn't mean 1 is in S. If 1 is not in S, then we're taking a notation based…

I think I was too hasty in replying to your post and misread and misunderstood the point you were trying to make. No, I don't think there's a difference.

Re: The Math Myth

#147

Post by kazinator »

soVeryTired wrote: post #127
kazinator wrote: post #71

Earlier quoted context omitted.

https://en.wikipedia.org/wiki/Constructivism_(mathematics) Do you have remarks not regarding the semantics of "constructable"?

But the "example from real analysis" section in your link uses exactly the same limiting technique to construct e. That section shows that 0.999... is constructible.

First we have to agree on what 0.999 is, then we can call it constructible or not.

Numbers formed by repeated 9's appended to 0.9 are certainly Turing computable. Which has the meaning that we have a terminating algorithm which, given a natural number N, will compute the N-th digit of the infinite sequence 0.999...

This is the same way that pi is computable. Given an N, we can compute the N-th digit of pi in a finite number of steps.

We don't say that 3.141... = pi! Unless, by convention, we agree that this "..." syntax has the semantics of (for instance) denoting the limiting numeric value of the non-terminating algorithm for producing the digits of pi ad infinitum. That is to say, "insert here a process for calculating the remaining digits of pi, and take this whole expression then to be the limiting value to which that converges".

Re: The Math Myth

#148

Post by dantillberg »

minikites wrote: post #28
NhanH wrote: post #20

Earlier quoted context omitted.

To be fair, 0.9999... = 1 is not quite basic. You need to know things like infinitesimals, the distinction between value and representation of numbers etc.

Most people are happy to accept 0.33333... is the same as one third and from there it's a quick hop to 3*0.3333...

I didn't quite accept that until I challenged a professor in a 300-level university math class. This came up in class, and I spoke up and was like, "no, that can't be true, because 0.3333 repeated N times will never equal 1/3."

Up to that point, I had only a fuzzy notion in my mind of what 0.33333... even was or how it was defined. But the professor helped clear this up for me: that "infinitely-repeating decimals" was actually just shorthand for a limit definition, i.e. "0.333..." is defined to be the limit as N approaches infinity of (3/10^1)+(3/10^2)+...+(3/10^N).

Re: The Math Myth

#149

Post by mmmBacon »

analog31 wrote: post #25

At my workplace, we have about 60 scientists and engineers. The author's observation is accurate, that most people never use math beyond Excel and 8th grade math. They also never use most of the theory that they learned in their science (including CS) and engineering educations. The typical career arc is to get through college, then sit down at a CAD workstation, or programming terminal, and forget all of your math a…

Worked in HW design for 16 years and I've never seen a design that was done by trial and error. Usually lots and lots of simulation and calculation up front followed by a DOE (design of experiment) to provide insight in areas which are difficult or impossible to simulate well. It's most certainly not trial and error.

Re: The Math Myth

#150

Post by kazinator »

iopq wrote: post #128
kazinator wrote: post #62

Earlier quoted context omitted.

0.999... is equal to 1 only if we assign a particular semantics to the "..." notation. Namely if "..." means "the limit of the decimal number to the left, as the repetitions of the last digit grow ever larger", then 0.999... is an alternative notation for 1 since that limit is 1. The actual number formed by repeating 9's an infinite number of times is not constructable. Whereas 1 is constructable. So they cannot be t…

The problem with this argument is it assumes you can get to infinity 9s by adding them one-by-one. It's like adding sides to a regular polygon to get to a circle. No matter how many sides you add, you still have a polygon. You can only say that a circle is a polygon with infinity sides, but you'll never get there adding them one by one.

The argument is that you cannot get to infinity by adding one by one.

(If the argument assumed otherwise, it would only be temporarily, for the sake of setting up a reductio ad absurdum.)

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