Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

The Math Myth

econlog.econlib.org

Post reply on HN
Print view View original

111–120 of 328 posts

Re: The Math Myth

#111

Post by jhanschoo »

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…

> 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 of the more straightforward constructions of rationals define a rational number as an equivalence class of tuples of integers, corresponding to the numerator and the denominator. We often use one of these tuples to represent a rational number, e.g. 2/3 or 4/6. And since they are in the same equivalence class, either alone is sufficient to represent the same rational number. This is despite the fact that the tuple 2/3 and the tuple 4/6 are different mathematical objects.

As for the definition of 0.999..., it's not what you think it is. There is a specific definition for infinitely repeating digits that is different from representations without repeating digits.

That is, the positive real number that 0.x... represents for all digits x is defined as L, where L is the supremum of the set containing 0, 0.x, 0.xx, 0.xxx, 0.xxxx, ...

A supremum L in E of a set S subsetof E is defined as follows: L is the smallest number in E such that L is greater than or equal to all numbers in S. I shall not prove here that L is unique, but it is. With respect to the reals and the set containing 0, 0.x, 0.xx, 0.xxx, the supremum of that set in the reals is 1.

My knowledge of the construction of the reals comes from Chapter 1 of Water Rudin's Principles of Mathematical Analysis, ISBN 0-07-085613-3, which you may be interested to read, but beware that it's considered a difficult read for beginners.

Re: The Math Myth

#112

Post by agumonkey »

nabla9 wrote: post #61
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…

All mathematics is applied mathematics. Pure mathematics is just mathematics applied to mathematics. This is problematic, because the way mathematics is currently taught only small number of students actually grok it and make deep connections that enable them to build up on what they previously learned and learn more. Others have the constant feeling of things getting progressively harder to understand and use. I'm s…

Wouldn't that be metamathematics ?

Re: The Math Myth

#113

Post by kazinator »

yequalsx wrote: post #56
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.

Property of real numbers: between distinct real numbers is at least one other number. Now try to find a decimal representation of a number bigger than 0.9999999... but less than 1.0. You clearly can't. They must be equal. No need for infinitesimals.

> find a decimal representation of a number

0.99999999...1

This denotes the abstract idea that we take 0.999... (infinite number of 9's) and add another digit.

This is no more or less abstract than 0.999... to begin with.

0.999... is a unicorn, and 0.999...9 is a unicorn with a pink ribbon on the tip of its horn.

Re: The Math Myth

#114

Post by cs702 »

I think this essay asks the wrong question, and then reaches doubtful conclusions from it.

We should not be asking whether most individuals today use higher-level math in their daily lives, because the answer we get will depend on the degree of math literacy of the people with whom those individuals must interact every day. The level of discourse is often dictated by the 'lowest common denominators' -- that is, the people with the least math literacy.

For example, freshly minted engineers who are surrounded by math-illiterate work colleagues quickly learn that they must avoid higher-level math if they want to interact successfully with others at work. Over time, the level of discourse of these engineers gradually drops toward that of the work colleagues with the least math literacy.

A type of "Gresham's Law for math literacy" is at work.[1]

The question we should be asking instead is whether society would be better off if more people had greater math training and literacy. Would our debates be more informed and higher-quality? Would our decisions be smarter? Would there be more technological innovation and wealth creation? Would society as a whole be better off if more people were trained to think creatively and critically with the rigor of higher-level mathematics?

I suspect the answer is yes.

[1] Gresham's Law -- https://en.wikipedia.org/wiki/Gresham%27s_law -- states that "bad money drives out good." In this case, unsophisticated discourse drives out high-level discourse.

Re: The Math Myth

#115

Post by sotojuan »

Davidp00 wrote: post #89
yodsanklai wrote: post #67

Earlier quoted context omitted.

> It makes more sense for an intelligent person to take the lower overhead and more achievable approach to becoming a value creator (e.g. full stack engineer with a strong focus on product development) I would say that the surest way to make money for a mathematicaly-inclined person is to graduate in maths from a prestigious school and work in finance. At least, that's how I feel when I look at alumni from my school.…

>Those that went into finance make consistently much more than the others Hm, I would never have guessed that. Does anyone have any data on this? The top 1% sure, but the average and median also?

I have no data either but one point is that the "ceiling" is higher. Meaning, a non-management programmer has a rough salary ceiling. Few programmers are paid $400k and it's not a very realistic goal.

My mathematics -> finance friends (that are very good) virtually have little to no ceiling and have already doubled my salary.

Like @wrong_variable said though, it depends what jobs and if they make it through (it's pretty competitive), so you're getting survivor bias from me.

Re: The Math Myth

#116

Post by GFK_of_xmaspast »

dredmorbius wrote: post #54
iopq wrote: post #42

Earlier quoted context omitted.

If the difference between samples is VERY large, you don't need a very large sample size. In other words, we're trying to find the chance that the result we got was due to chance. Let's say you have numbers like these: A: 11, 11, 12, 12, 13, 13, 13, 13, 13, 13, 14, 15 B: 90, 92, 93, 94, 94, 95, 95, 96, 97, 99, 99, 101, 101 What is the chance that those two samples come from the same distribution? On the other hand, i…

That sounds intuitively reasonable. Is there a cononical reference argument that you're aware of?

See also "statistical power" : https://en.wikipedia.org/wiki/Statistical_power

Re: The Math Myth

#117

Post by [deleted] »

kazinator wrote: post #93
iopq wrote: post #48

Earlier quoted context omitted.

It's not a reduction. If you try to find where to put 0.999... on the number line, it has to go exactly where 1 is. For one thing, 1 - 0.999... = 0.000... because you never get to have any remainder since 0.999... is infinite. Or here's another proof: x = 0.999... 10x = 9.999... 10x - x = 9.999... - 0.999... 9x = 9.000... = 9 9x = 9 x = 1

Your "proofs" simply assumes that 0.999... is a notation denoting 1, without examining the underpinnings which might legitimize that. 9.999.. - 0.999 is 9 no matter how we define .999... just as long as two or more occurrences of the 0.999... notation all denote the same entity, and we understand that the syntax 9.999... is 9 + 0.999... For example, if we define 0.999... as "rubber duck" then 9.999... stands for 9 +…

[deleted]

Re: The Math Myth

#118

Post by nilkn »

JoeAltmaier wrote: post #85
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.

Its just algebra. As noted elsewhere in this thread, 10X = 9.999... so 10X - X = 9.999... - 0.999... = 9 = 9X. So X=1.

This proof is more subtle than it appears. Here's a bogus rewrite, for instance:

Let 10X = 9.9. Then 10X - X = 9.9 - 0.9 = 9 = 9X. Hence X = 1, but X is actually 0.99 in this case (not 0.9). You need 0.99 = 0.9 for this to work with the exact same structure as your version.

Your proof only works because appending a 9 to an infinite expansion of 9s does not actually add a 9. But at this point you're forced to establish meaning for an infinite expansion of 9s, at which point this is really not just algebra anymore.

Re: The Math Myth

#119

Post by Ologn »

Tyr42 wrote: post #31

As a mathematician, I would like to point out that there are a lot of different areas of math, and higher math isn't just learning more calculus. Graph Theory and Stats, for example. I have no idea what he's talking about with including Stats in up to 8th grade math. I've taken a few university classes on it, and I still don't feel like I have enough to be confident solving all but the simplest statistical problems.…

> graph theory

Then once you learn graph theory and about trees and graphs, you can learn about data structures like self-balance binary trees, dawgs, flow networks etc., then algorithms that run on those data structures like Dijkstra's algorithm or the Ford-Fulkerson algorithm.

Re: The Math Myth

#120

Post by maus42 »

ausjke wrote: post #87

This is not just Math. I learnt chemistry, physics, biology and all that from middle to high school. Now as a software engineer they're totally useless and I have long forgotten all those details that I spent months and years to memorize and master. Even reading a science-101 book in one day now will teach me more than what I can remember. Unless you plan to major in those fields, should we just take some introductio…

I would argue that it's valuable in itself to actually know what the humanity has found about the universe and how it works and how this method called science is supposed to work.

The only purpose of the continued existence of the mankind is only what we ourselves decide upon and make our purpose. What is the best of ourselves? Towards what end should we aspire to? The greater understanding of nature and ourselves and history and truth and beauty, or being marginally more effective in producing more shiny skinner boxes to enthral our neighbours? The man standing on the Moon, or a new fancy gadget that wibbles and wobbles a bit better than the previous version on wibble-wobbler?

Especially so in the western societies, where the ideal is that citizens vote and participate in the public life and make collectively decisions. Plato claimed that society should be ruled by philosopher-kings. There are many reasons why his utopia would not work out in real life without being utterly horrible, but one thing that we ourselves, who have the right to vote, try our best to be a worthy of the tiny bit of the crown of a philosopher-king that democracy grants us as our right.

Of course, not too many people seem to be interested in being curious about universe. Sometimes it makes me despair.

Post reply on HN