The Math Myth(econlog.econlib.org) |
The Math Myth(econlog.econlib.org) |
"People say that we should train people for factory jobs, but everyone I know is gainfully employed in agriculture, and we don't have any factories where I live."
As a college professor, let me assure the author that statistics and programming is not a standard part of an eighth graders program. In fact, the ones I teach have passed the 12th grade, and most are woefully unprepared in algebra, statistics, probability and programming.
For me, understanding slightly advanced math (the type discussed in Taleb's Fooled By Randomness) helped me realize that Financial academic math is complete B.S. (in its use of the Gaussian Dist. in non-Gaussian processes). Yes, that's how learning more math has benefited me: it helped me discover how math is used to support complete B.S.
> I find it difficult to find anyone who uses more than Excel and eighth grade level mathematics (=arithmetic, and a little bit of algebra, statistics and programming)
I think even that's a bit optimistic; I think people whose further studies or jobs don't require that level of mathematics forget it pretty quickly. For a base level of "everyday life," you probably only need basic arithmetic operations.
As anecdotal evidence, look to all those times that a relatively convoluted expression is posted on Facebook or Reddit and people argue for weeks about what the proper solution is. Of course there's plenty of people who get it right, but the wrongs range from a subtle misunderstanding of order of operations to a complete lack of knowledge about it.
This is a conjecture that desperately needs resolving with solid statistics and in-depth interviews.
I think more empirical data is needed. If I go on the street or the math is comparable to 5th grade (at the very best) and in a business setting might bump up to 8th grade.
Does that preclude there being opportunities to need/use/benefit from math? No...
I think it just means there are opportunities that nobody is taking advantage of. Left open and collecting dust.
The answer I came up with was N0e^-λt. Exponential decay. Set N0 to 1. I could set t in various ways, I decided to make it days, so today is 0, yesterday 1, the day before yesterday 2 etc. The lambda I tunes, right now it is 0.04 (or -0.04 times t). So the score added for each use decays as it ages, giving new additions a chance at the top.
Worked real well. Straight out of calculus. I never learned exactly what e was until college. Who knows what I would have done if I didn't know what exponential decay, e etc. was. I can't even think of an "eight-grade math" solution of the type this article mentions.
I had to hash a small list of small numbers once when I had the epiphany - Goedel numbering! I Google'd that and saw the solution was unoriginal, but I wouldn't have even saw those pages without knowing what to Google.
I was looking at a large NP hard problem many years ago and thought I could program a solution. After a complexity class later on, I realized the futility of that approach, in a direct manner any how.
I am not sure where the line is between math and CS. Graph theory underlies graphs and trees and the algorithms which run on them. Math functions and theory of computation underlie functions and methods. Statistics and probablity underlie ML. Geometry and matrix math and algebra underlie computer graphics. I don't get people here who say they program without needing post high school math.
Or seeing the garbage code out there maybe I do. Github is beset with people who make basic errors in mutual exclusion, critical section violations, lack of understanding of concurrency etc. I forget and make these mistakes myself sometimes. I hardly think there is a problem in over-education in these things. On the contrary, race conditions are spun out all over the software infrastructure by people writing code who don't have the needed math and CS understanding of what they're doing. Understanding mutual exclusion and critical sections and avoiding critical sections is not something picked up in an hour, a day, or even a week.
To convince others I often use examples. One is the discipline issues over the years. All BS. (1) totally made up. No data.
You need to have experienced that some things are complicated. And we need a lot of people to respect science and engineering, because they will be the ones taking decisions, and those decisions need to be good ones.
I use math all the time, especially when I help my kids with their homework because they don't understand their math assignments properly because the school can't hire anyone with a decent math understanding to teach because those people all took high paid jobs elsewhere..
looks like they [relaunched this year](https://en.wikipedia.org/wiki/Arthur_Andersen)
It isn't even correct on the employment front, though, because it is attempting to unimaginatively extrapolate from the current state of employment.
John Nagle's comment at https://news.ycombinator.com/item?id=12422307 probably expresses this better than I can, but if you're going to make an advance in a scientific or engineering field — any advance — you need math. If you're planning to spend your working life as a button-pusher, carrying out algorithms that other people have designed, or proceeding blindly by trial and error, you don't need math.
But those button-pusher and blind blunderer jobs will be automated in five, ten, or maybe twenty years. And the article's comment section suggests that even today they aren't nearly as common as the article asserts.
There are other categories of work, such as child care, elder care, sex work (which, defined broadly, includes trophy wives, Hollywood, and a substantial fraction of secretaries and maids), sales, and family counseling. So there will probably still be employment that doesn't require math as long as there are humans, even if it's not the kind of employment the article discusses.
But the bigger question is whether education should be directed at employment. Is being an employee what you aspire to in your life? It is very good to be useful to other people. Allowing other people to employ me has benefited me greatly, and that's true for most people I know. But being used by others is not the only or even the primary good in life.
Education is what makes us human. Education is a process of personal evolution from a dumb beast into a human being. Education gives us control over our impulses and prevents us from being suckered by predatory salespeople, politicians, lawyers, preachers, and others. At its best, education makes democracy possible despite such predators, although democracy is rarely possible because the people is nearly always sufficiently uneducated to vote it down unintentionally. Education begins before schooling and doesn't end when schooling ends, but I, like many people, have found that schooling can speed education up considerably.
And mathematics is fundamental to education in all of these senses. Even if mathematics isn't necessary for someone else to use you — which is all this "Math Myth" article tries to show — mathematics is necessary for you to judiciously choose when and how you will be used, and mathematics is necessary for citizenship.
Also to be measured in how sure I am about something being right.
It is fair to say that there is an old and strong belief that a person who has studied broadly, and deeply through, say, college, in math, physical, biological, medical, social, and computer science, and the humanities will have a significant advantage in much of the rest of life. Lacking a better name, here I call such study a broad education.
To argue this belief in the context of the OP, the OP seems to claim that for 90% or so of people, it is enough for them to stop their math education, and by extension all their education, after the eighth grade. But in life it is fairly easy to tell the difference between the OP's eighth grade education and a broad education as I described it. So, there is a difference. Maybe the difference is significant and the broad education an advantage and worthwhile.
One point not mentioned very often is that, whatever 90% of the students do, the broad education was hoping that some of the students would find some really good uses of some of the education well past the eighth grade. The educators could have that hope even without knowing just what the good uses might be.
I studied a lot of math and physics heavily, but not entirely, because I hoped that they would help me make money. Well, early in my career within 100 miles of the Washington Monument, that hope was fully correct. I used what I had and was learning more as fast as I could drinking from a fire hose. Of course that work was mostly for US national security; there the math and physics were crucial.
Yes, it does appear that away from the work of US national security, the math and physics are less commonly used.
Still, in US commercial work, there are significant applications of the math and physics. Examples:
(A) How to operate an oil refinery. In simple terms, here is a list, with prices, of crude oil can buy and put into the refinery and a list, with prices, of refined products get out of the refinery, so a question is what to buy, produce, and sell to make the most money? First cut, the problem is linear programming, and for a while there was good money in selling IBM mainframe computers just for that work. Of course, past the first cut, the problem is in non-linear optimization.
A practical challenge is: It's a good guess that the first refinery management that did well seeing and exploiting this opportunity was well paid for their insight. Since much of the crucial core of that work was some college and/or grad school applied math and numerical analysis, knowing some math could have been an advantage for the management trying to understand and make good decisions.
(B) Take a big hammer and hit the ground and send an acoustic pulse through the ground. That pulse is commonly partially reflected at the boundaries of layers of rock, sand, etc. So, the acoustic signal that comes back is a convolution of the original. Doing a deconvolution, can map the underground layers and get some good hints of where to drill for oil. The deconvolution is basically some Fourier theory, and the fast way to do the computations is the fast Fourier transform (FFT). After Cooley, Tukey, etc. invented the FFT, such acoustic processing had an explosion that is still active. So, again, oil prospecting management needed to see, understand, and actively exploit the FFT. For that, some math was no doubt an advantage.
There are more commercial applications of math and physics. Some of the applications have been valuable already, and likely some more will be valuable in the future. So, in looking for what might be valuable in business, some math and physics stands to be an advantage.
So, in part, with a broad education we are fishing for advantages in the future. We are not sure just what subjects will lead to what advantages in the future, but we are quite sure that there will be powerful, valuable new work where, for successful exploitation, some studies will be important.
Or, the OP is concentrating on what the 90% of the people actually are using now. Well, in a sense the education wants to concentrate on what is new no one is doing yet.
I would state this differently. Borrowing from the Pareto principle, one could conjecture that 80% of mathematically advanced work in the economy is performed by less than 20% of STEM graduates. The remaining 80% of STEM graduates do not get the economic opportunity to apply the skills which they trained for and end up doing less prominent work (e.g. middle management).
As the OP and others have pointed out, there is a lot of anecdotal evidence to support this conjecture.
But it is hardly surprising, and it is not limited to mathematical talent.
Take management, for example. Just because you studied business in school, does not mean that you will be an executive. I would guess that less than 20% of MBA graduates manage 80% of economic resources (senior executives, bankers, consultants, traders, etc) , while the remaining 80% of MBA graduates are left managing relatively small and inconsequential activities.
Similarly, I would bet that less than 20% of design school graduates do 80% of the design work in the economy. I bet that less than 20% of classical musicians perform 80% of orchestral music. Less than 20% of programmers implement 80% of software used. Less than 20% of athletes win 80% of medals. Less than 20% of science graduates produce 80% of scientific research. And so on.
OP's conclusion is that, in light of this dismal reality, students should not bother learning mathematics after the 8th-grade level (except for "those who need it"). Well, if we apply the same logic across all disciplines, then the OP should conclude that all forms of education should stop after the 8th-grade level for the vast majority of students (and only a minute fraction should need to pursue higher education). That is exactly what the state of education looks like in undeveloped feudal economies, and this was also the state of Western education until relatively recently. I don't think I need to expend a lot of effort convincing anyone that this a socially, economically and ethically terrible idea.
I'll also point out that there there are a couple false assumptions implicit in the OP's original, imprecisely worded conjecture. Firstly, advanced industrial mathematics is not the exclusive preserve of traditional engineering. The generalization that "engineering positions" use advanced math and "management/financial positions" use 8th-grade math, is obviously false. Many areas in finance require advanced mathematics (derivatives, trading, fixed income, etc). Much of actuarial science also depends on advanced mathematics. Marketing, management sciences and operations research are also steadily moving towards advanced analytics. Secondly, it is a false assumption that use of Excel implies that the underlying mathematics is limited to an 8th-grade level. For example, in finance, it is easy to find Excel add-ins for performing highly advanced mathematics (e.g. stochastic differential equation solvers for derivatives pricing).
At the level of societies, maybe. Can a poor society with lots of mathematicians "beat" a society with lots of wealth and infrastructure and comfortable niceties but whose individuals can only use Excel? Probably not, certainly not in < 1 generation. It certainly didn't turn out great for the soviets.
The analogy with sports fails miserably and the author seems to not understand this. Math is a brain skill and we do need to apply brain to understand a given situation in a better manner, to abstract away some things and focus on some other things. So, if you expect someone to better understand complex situations, then you need them to have some knowledge of higher math.
One may ask where do you encounter such situations? Insurance, debates of fiscal policies, debates about racial biases and social structures, anything to do with modern finance, language structures, medical decisions. Take your pick.
So, if you have to do anything complicated in such social areas too, you need to have some knowledge of higher math.
Skills in sports are not of such versatile nature, hence the analogy fails.
I don't think you have to go back to undeveloped feudal economies. Even a generation ago, the bulk of people in the US effectively did not receive more than an elementary education. Ironically, they were often better prepared than students today to actually enter the workforce after graduating high school, since vocational education was more in vogue, and so they spent more of the four years of their high school education learning practical skills, rather than the vague, college prep holding pattern that is the norm now.
Most people will spend very little time giving first aid, controlling a vehicle in dangerous weather, resolving serious relationship discussions, negotiating important deals, or doing cost/benefit analysis of large purchases.
However, in each of those cases, the tiny fraction of time where they do those is so important, it's still worth preparing for them. It may be that most people rarely use math, but when they do, they use it on important enough things to still warrant teaching them.
Need some examples? Public key cryptography, machine learning, physics simulations of the aerodynamic properties of an airplane. I could go on and on. Just because a vast majority of the population never has to think about how this stuff works does not imply that it is somehow useless. We would not be where we are today without all this mathematics.
At some point somebody had to be the person that figured out all these things so the other 99% of people can use it and not think about it ever. And I'd much rather be the person solving the interesting problems than the person using the tools without truly understanding them to solve more mundane (but still probably useful and important) tasks, but that is just my personal taste.
If you look at the requirements to be a software engineer for a company that makes video games these days, the mathematics needed is rigorous in the geometry aspect.
I'm not sure what kind of actuary this guy was interviewing but all the actuaries I know in the industry that are respected have used a significant amount of math in their career before reaching management.
I myself am no expert. I have a MS in applied Mathematics from a regular school and make over $150k in the Reinsurance industry.... I only have 4 years of experience. My superiors are definitely making 7 figures.
These days with emergence of predictive analytics which definitely using above 8th grade math, shows the relevance of advanced mathematics.
Because we can program computers, Of course you don't have to write these formulas/equations etc... Everyday but to initially design these systems, implement, revise, research and innovate, the skills are needed.
That's why at these top companies at the forefront of the industry have a very diverse international makeup of countries that excel in mathematics
In most projects, there are minority extra-brilliant people, whose talent/knowledge reflect on the entire outcome/product. So you always need people to handle more complexity. And as some others in the discussion have pointed out, often concepts at a level, become clearer when you grapple the next level of complexity.
Soviet society did not fail because they were better at Maths. It may have happened despite it. The right point to infer about that would be, brilliance in Maths is not a sufficient condition for society as a whole to excel. And that's a moot point, as there are so many other necessary conditions - food/shelter/being-alive/etc - leave aside politics.
Also the article ignores probability as a core life concept, by which you can understand so many things. I use it with my kids all the time.
Another thing which frustrates me recently is my inability to grasp modern physics. Without the relevant understanding of the complex maths, one can only get the vaguest idea, of what they(the physicists) are saying. This creates a huge intellectual gap in society.
Also the knowledge gap has another problem. If it gets too wide, then there will be a very-very few ultra elites who all know what they are saying (perhaps that's already the case, unfortunately). And the rest of us, only take their word on face value. I know one person can't know everything and this is an era of specialization. But still, I think Maths is a fundamental thing. And society would only gain when more number of people are proficient at it.
Mathematical communication skills will become more important in the information age with huge amount of quantitative data.
* A strong mathematical background does not guarantee success in science.
* There's a large amount of foundational theory and work which involves thinking in images and facts, not mathematics.
* Maths phobia deprives science of an immeasurable amount of talent.
* True maths talent is probably at least partially hereditary.
* Maths and conceptual work are complements, not replacements.
http://www.worldcat.org/title/letters-to-a-young-scientist/o...
Statistics and programming is way way higher than eighth grade from what I've seen.
But taking his premise above, then I think no one is arguing for the general public to learn more than the aforementioned eighth-grade maths. It's just that the majority of the population isn't any where near that. Specifically in statistics, programming, and a bit of logical reasoning, I might add (around modus tollens).
I might be mistaken here, but I've always thought that when someone talks about "higher maths" the public should learn, it is capped around calculus I, or some basic linear algebra. Which is like half a year more study over the list of the author.
On the other hand I tend to believe - possibly misguided - that a lot of my thinking is influenced by having gone through the math education. I may seldom need exponential functions but I know what is linear and exponential by heart. Consultants, engineers, architects and managers work with long levers and knowing how things scale up and down and when they don't matter. Understanding linear systems, frequency domain and where nonlinearity starts mattering informs quite a number of my decisions.
Math as a filter for hiring is questionable as imho. most grades. The skills that matter every day are mostly not analytical skills. Universities as they are set up are not well geared towards filling that educational need.
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 introduction courses instead?
Also I can testify that I rarely need use any math beyond 8th grade since graduate school as a software engineer, I mean those calculus, matrix theory, fuzzy logic, neural network, etc. Well I may pick up some AI stuff now, but it's more like a start-from-scratch-now as I forgot what I learnt then totally already.
So yes the education system can be optimized to be more efficient.
I used to work as an aerospace engineer, doing trajectory analysis. We used high school trig and algebra, as well as first semester calculus pretty much all the time. But, I don't think any of the other required math classes for my aerospace degree were ever used (3 semesters calc, 2 semesters linear algebra, diff. eq, IIRC). I did end up using a fair amount of stats, which was not required for my degree, but really should have been.
Statistics was the one ongoing use of math in my former roles as a manufacturing engineer. Beyond that I haven't used much math directly. However, I have replied upon my knowledge of engineering core courses to understand and solve problems. I needed to understand calculus-based math in order to understand that coursework. So math is important.
There is a saying among teachers that in K-3 you "learn to read" and from then on you "read to learn". The same principle holds for math. Math itself may not be the end goal for many degrees, but after you "learn to math", you "math to learn".
The vast majority of human beings will never do anything intellectually intensive post-college. Even those in STEM fields. Not that an undergrad degree is "intellectually intensive" anyway.
EDIT:
>The second argument is the one I always hear around the mathematics department: mathematics helps you to think clearly. I have a very low opinion of this self-serving nonsense. In sports there is the concept of the specificity of skills: if you want to improve your racquetball game, don't practice squash! I believe the same holds true for intellectual skills.
Dear God, I'm so happy to see this in writing. For a while, I was afraid that I was the only one who had realized this. This observation has several useful immediate corollaries. For one, it shows that those "brain training" games that some people like to play are a waste of time. Also, it shows that if you ever catch yourself saying "I'm working on my X to help with Y", it probably means that you're just afraid of the failure that will inevitably come when you initially begin to practice Y, and that fear can only be ameliorated if you just dive in and start doing Y.
Of course I know it's the elites among the upper echelons of the society who are more than happy to see and maintain such a situation, and unfortunately this article might well be another addition, a so-called academic/think-tank publication that serves their agenda. It can't get more obvious at the end of the article: "leave elite education to those who 'need' it! Keep the mass ignorant!" Yeah, sure, so that the children of the elites always stay powerful and the mass keep remaining ignorant. It doesn't matter for the massive power wielded by the US, the state terrorism employed by Uncle Sam, but it matters, a lot, for genuine empowerment of the people and true democracy, which people including the author here doubtlessly want to stop at all costs.
Is better to have strong basic skills, than high-level skills.
I was (supposedly) of the bests student of my college. However, terrible at math? Of course.
My grandmother was able to do arithmetic in his head like nothing, yet I even have trouble with sum and rest.
She only have 3 years of education after kindergarten.
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In the first class of calculus in the University, the teacher make us do a division between a largueish number and a small number, at hand. We was something like 50 people. I don't remember anyone was able to perform it in time and give the correct result (or if somebody was able, surely was a very small number, I don't remember it well).
At that moment I know that the whole point of learn calculus will be a disaster.
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People not need to learn advanced math. They need to have strong, fluent understanding of the very basic (imagine if a developer can't perform without look basic list manipulations).
Like Bruce Lee say:
"I fear not the man who has practiced 10,000 kicks once, but I fear the man who has practiced one kick 10,000 times."
Why? Basic arithmetic is almost entirely useless as a skill.
Why should I spend my time learning something which computers will always be able to do more reliably and faster than me?
And is possible to achieve knowledge of it, without a strong foundation of arithmetic?
Truly, I don't know, as I say I'm bad at math.
However, the point is that without strong basics the rest is a lost cause (IMHO). What is the basic, I'm open to know, however, I think arithmetic is part of it...
But even with excel spreadsheet, if you leave out the "why" you still end up with a boring class of formulas and UI work through tutorials that will leave you questioning relevance just the same. And your school paid MS how much?
If you're developing a game, or designing a building, or analyzing online sales, or trying to build a web site, you now have the context, but you probably don't have the education unless you took special courses in higher education which is practically the only place they teach with context. Maybe they just need to add more context to lower and general levels as well?
Anecdotally, it's fair to say good students manage to identify context beforehand and keep things relevant. It really helps with learning when you're driven by purpose, not obligation.
Can you use this result to go back and justify that we don't need math beyond 8th grade?
I beg to differ. I only can provide anecdote. As myself are not bad at math, I find myself use calculus and linear algebra all the time. In fact, I think differently. As another anecdote, my son, who I consider is not nearly good at math, he is in middle school and he uses trigonometry all the time.
You use what you have. Knowledge changes the way we think and work. With knowledge, you simply see the world differently.
But I left academia over a decade ago, and have never used any of that. I have, however, used a few things above 8th grade maths, such as linear algebra, regression analysis, and some basic numerical analysis. I used once a FFT.
All in all the author is spot on. And I believe an advanced degree in maths is not a legitimate requirement for anything but a handful of positions, and most of those aren't particularly desirable.
I still remember fondly my days studying Baby Rudin, though. Definitely one of the courses that made an impact in my education. But in hindsight, it's been as useful in my career as my study of Latin.
All of our former experiences help us shape intuition, the same intuition which we may use to make decisions in seemingly unrelated fields. I have therefore become somewhat careful in dismissing a skill, domain of knowledge or lesson for apparent lack of use. The human brain is a fascinating machine; even while we sleep ideas and solutions take shape and who's to say which information is used to form those mental artefacts?
I detest utilitarian arguments, that something is worth learning only because it is useful in my day-to-day. I haven't had the faintest use in my daily life for knowing anything about igneous rocks, sorghum, golgi bodies, Chandragupta Maurya, black holes, playing hockey. Yet, it would be singularly depressing to not know it or something to this level of detail.
Second, regarding the argument that only a select few will be interested in ascending the peak and that the rest are content in the plains. While that is true, it takes a whole community of people interested in an area for there to be a star. A Messi or Usain Bolt comes out of having a sporting culture, in addition to athletic and soccer academies of a high enough standard.
needing higher level math is absolutely unnecessary for 99% of the working population, so the argument that our economy will fall to pieces unless we force high level calculus and trig on every student seems suspect.
however, it is hurting america in an indirect way. we have an elitism going on that only students from top universities get dibs on the best and important positions out of college, and our graduates are becoming increasingly foreign, as american students aren't prepared to get 165+ math GRE scores.
My point is that it is important for a culture to aim high. There is a reason why Israel produces way more innovation than Saudi Arabia, although most of the population in both countries has no use for calculus or igneous rocks.
Trigonometry is useful in so many ways. It's even useful for projects around the house, let alone for a lot of careers. Last I checked it comes after 8th grade.
On the other end of the spectrum he concedes Harvard philosophy undergrads might want to read "The Road to Reality". Bullshit - No undergrad can understand all the math in this book and no one is proposing that they should. Reductio ad absurdum.
And don't forget gaming. Lots of young people these days dream about a career at a game studio and there are a lot more options if you have good math.
He mentions Sputnik but it's not the 1950's anymore. The number of careers that benefit from math will only continue to grow.
[1] https://www.quora.com/Working-at-Google-1/What-is-the-worst-...
This article misses the point. The purpose of teaching math is not to memorize equations and solving methods, but to teach to approach problems in different ways.
As a developer and now a PM I've used complex math at many different times. I'm not solving in paper, but actually using it to solve real world problems.
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.
It's a shame, because some of these things are valuable to know.
If you're felling a tree, trigonometry is pretty useful for trying to figure out whether you've got room to drop it without hitting your house. Also, when you're building that house, for how long you need to cut your rafters to get the right pitch on your roof.
You don't necessarily need to know chemistry to cook, but it doesn't hurt, and you'll understand why you need to use baking powder instead of baking edit: soda when you bake your biscuits.
There's a lot of basic physics and mechanics that can be incredibly useful if you're trying to lift heavy objects with less than adequate equipment.
I blame capitalism and it's tendency to create useless positions in hierarchical organizations.
If you think the gov makes all the useless and pointless paper pushing jobs, you have never been in a large company.
In fact, market capitalism creates all kinds of pointless soul sucking jobs like lawyers, police, and insurance brokers. This is because nobody trusts each other.
You can blame gov regulation, except most regulation is written by lobbyists for mega corps
The majority of people throughout history worked manual labor jobs that weren't intellectually intensive, but they were often quite necessary.
>If you think the gov makes all the useless and pointless paper pushing jobs, you have never been in a large company.
I'm currently employed at a large company doing mostly pointless work, so I'm not sure where you got this assumption from.
It's important for enough programmers, and their managers, to have enough of a foundation in CS theory that they can understand and believe when they've encountered an intractable, impossible, or research-level problem. Otherwise you're just going to spend time sputtering to your business people about why you can't solve the Traveling Salesman Problem before lunch and the Halting Problem by dinner.
Not true of the people I know or the people who work for me.
Perhaps its true for some-- the idea makes me quite sad.
What a waste.
OTOH, I have basically little problem with any other branch of mathematics. The more advanced the better, actually. In school it got easier as it got more advanced (although geometry was always easy no matter what level). Getting calculus in high school felt like I could suddenly breathe with full lungs. And I studied theoretical physics later at university. Apart from the basic reality that the math required is hard and needs a lot of work no matter how smart/gifted/whatever you are, I had no substantial problem with most of it. Abstract algebra, group theory, vector spaces and manifolds... oh the joys! Because none of it requires you to do any kind of mental arithmetical computation. This is not a unique experience, I once saw a TV report on a young successful astrophysicist with basically the same problem: Calculus on manifolds, GR field equations? Pfff... All day and every day. But, give her some numbers on a blackboard to multiply and she gets completely lost.
Also about the gov bit. I was talking to the larger HN readers than just you. Sounds like you know exactly what I am talking about.
I know very well what a computer is doing when I tell it to divide 188416 by 20148. That doesn't mean it's useful for me to be able to mentally do it.
Mental arithmetic is incredibly overvalued and useless as a skill, especially since it's largely uncorrelated with actual depth of mathematical understanding.
Basic mental arithmetic is probably the most used in day-to-day life. It feels weird having to whip out a calculator/phone every time you need to make a quick estimate.
That's equally useless.
Why would I try to figure out the result manually when I have a computer in my pocket at all times? If anything, doing things manually when we have computers seems weirder.
As for mental arithmetic, its main advantage is speed. Faster to mentally solve and continue along your train of thought/flow of conversation than to pull out a calculator. Physical actions will never be faster than speed of thought.
Its the impression that computers are easier, that gets me. Why not practice for 2 minutes and learn a better way? Instead of limping along with the computer all your life.
I have a phone with me at all times. It's hardly an inconvenience to use it.
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 those opportunities based on the skills you have that other people don't have. They will make money for the organization because of features or insights that your competitors lack. Your customers will buy it because it lets them do things that they couldn't otherwise do.
The stuff that you and everyone else spends 99% of your time doing is economically irrelevant. You probably still need to do it (though if you can program a computer to do it, you have a huge leg up on competitors), but it doesn't get you anywhere.
Ironically, this is one of those insights that a good understanding of math will give you. The common-sense understanding is that we should be teaching what the majority of people are doing; the data says that we should be teaching what the majority of people are not doing but desire the results of.
I wish these skills were emphasized more earlier in people's education, they are important skills for maximizing success IMO.
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 make your average developer who completes the text into an exceptional developer.
Sample quote: "Once you realize that computing is all about constructing, manipulating, and reasoning about abstractions, it becomes clear that an important prerequisite for writing (good) computer programs is the ability to handle abstractions in a precise manner."
A lot of people I meet don't understand logic and reasoning. A lot of people don't understand rigor. Many people have difficulty thinking systemically or have difficulty thinking about the 2nd and 3rd order effects of their actions. I mean, think about the way people drive and some of the things you've seen other drivers do! People also seem to be much more self centered, less civic minded, and less cognizant of details.
I wish these skills were emphasized more earlier in people's education, they are important skills for maximizing success IMO.
I think American kids would have a better understanding of 2nd and 3rd order effects if they were required to clean their own school bathrooms, like in Japan. (Disclosure: I had to clean my own school bathrooms, but did not go to school in Japan.)
The article speaks about Sputnik and one of the world's richest men, mathemetician and retired hedge fund manager Jim Simons notes that he is one of the first students to have benefited from the post-Sputnik STEM student grants. He also mentioned how many people in the US were getting doctorates in mathematics in the year Sputnik went up - I'm looking at the list of names right now and it is much less than 300. So the additional NSF funding etc. that paid for his doctorate resulted in the billions his hedge fund made.
>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
Is this really true? I'm still way too early in my career to know. But I'd have thought if you stick with the more "traditional" route, your income would be fairly consistent (as opposed to something similar to the startup lottery, for example).
Many people can tell you about their "big break". That one success that seemed to cascade into other successes. Whether this is a real phenomenon or simply an illusion of memory, I don't know. It's quite possible that if that big event hadn't taken place, there'd have been a similar one just a little later, taking the person on a different, but equally pleasant path. I suggest you not trust individual narratives, only systematic analysis.
This arguement reminds me of people who focus on a key, heroic play in sports, rather than all the good and bad that led up to it.
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 and theory within a few years or even months. Time that isn't spent doing CAD, is spent on testing, troubleshooting, dealing with vendors, and so forth. A few of them start prepping for management. It's becoming increasingly common for engineers to start their MBA training as soon as the company agrees to pay for it.
Truth be told, outside of a few life-support-critical applications, most design is done by trial and error. Very little real engineering gets done.
And the products we make, are designed to provide similar benefits in another profession. We are told by management: "Our customers don't want products that require them to think. They want something where a person with an 8th grade education can push a button and get an answer."
When a math or theory problem arises, they take it to the resident "math person." That's me. I'm glad that I spent the better part of my youth learning math and theory, because I'd be as suited to the CAD workstation as I'd have been to working at a loom, or a lathe, 100 years ago. For the most part, the people who emerge as "math people" are the ones who were interested in it as an end unto itself, in the first place. I didn't study math because I expected it to be necessary for a job. I studied math (and physics, programming, electronics, etc.) because I was interested in those things. They were for me an escape from preparing for my career.
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.
Before that, I'd worked on automatic theorem proving and proof of correctness. I still like Boyer-Moore theory. I recently revived the old 1970s-1992 Boyer-Moore theorem prover and put a working version on Github. It's fun to run that again; it's a thousand times faster than it was in the early 1980s.
If you do anything serious with graphics, you need to understand 4x4 matrix transformations throughly. I have the whole shelf of Graphics Gems books, and they're mostly math. At one point I rewrote many of the C code in C++, and got rid of their start-at-one arrays. (The original was Graphics Gems in FORTRAN, and the C version used a horrible hack to make arrays start at 1.)
I didn't know enough filter theory when we were doing the DARPA Grand Challenge. We had a lot of trouble integrating the GPS and AHRS data into a good position and orientation. We had about 3 degrees of heading noise, which kept messing up the map-making function. We really need 3D SLAM, but didn't know how.
Now I need more math to understand machine learning.
I'm also looking at designing a specialized switching power supply for the antique Teletypes I restore. You can get enough energy from a USB port to drive the big selector magnet if you use and store it properly. Fortunately I can get LTSpice to do most of the number crunching.
I think I've used all the math I was ever taught. And I'm not really into math.
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 probably keep going with just examples from statistics/probability/combinatorics. But there are other examples of people misunderstanding math.
I mean I wonder how many people even understand that 0.999... = 1?
[1] https://medium.com/@jeremyjkun/habits-of-highly-mathematical...
With others. Mathwise I am always right but others can't see it. So I have a deacartes moment with others
I do think that learning math does help you to think more clearly and to analyze problems in a more systematic matter.
Now, he does not define well what he means by "higher mathematics": I agree that (as with almost all learning) there is diminishing marginal utility in mathematics education. While I would argue that learning how to work with percentages and also basic calculus (to get a feeling for the difference between a change in position and a change in velocity, for example) increase your general problem solving skills by a lot, if you have been through all this then learning about Ricci flow will probably not do that much to your general problem solving anymore.
[1]: http://well.blogs.nytimes.com/2014/07/16/train-like-a-german...
The market of people who are genuinely passionate about complex subjects in math and science is saturated relative to available opportunities. 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) than waste time competing against the countless PhDs vacating academia.
I use a similar argument for avoiding the ML/Deep Learning hype train. At a large corp, that job should be left to people who've spent a lot of time mastering the subject. And if you're using ML heavily in an early stage company and don't have a PhD, you may very well be out of your depth competitively or wasting your time optimizing prematurely.
But even ignoring all of that: anyone who's either spent time on or interacted with a data science team understands how difficult it is to create value with ML as well as how intangible the value that's created can often be. I worked at a fairly well known company that told clients we have a data science team and could use ML, knowing full well that the team rarely if at all manages to generate meaningful insights, because dropping buzzwords is an essential branding tool.
Here's a better approach and the crux of why higher math is often superfluous: the best way to create value is to specialize in problem-solving first principles and remain amenable to either adopting new skills ad hoc or hiring to fill any skill deficiencies.
The caveat: if you're passionate about STEM and that's a higher priority than 'creating value' in a deterministic and practical way (and maybe it is and that's perfectly fine and even reasonable), then by all means indulge in it. But it's important to align your expectations about what you want to do with yourself with the way in which you spend your time. A lot of pain arises in misconceptions around the question of what we want and the reality of what we're doing.
3D rendering and animation and 2D browser transforms are math.
AI and ML have large math components.
Speech recognition is math.
Industrial electrical power distribution engineering is math.
Bridge and other kinds of structural engineering are math.
Analog circuit design is math. Once you get past the op-amp cookbook stage it can get quite complicated, especially if you need to handle RF issues.
Rocket science and aerospace design is math.
Supply chain process optimisation is math.
Traffic modelling is math.
Quant fintech is math.
Encryption and security are math.
At the absolute minimum these need geometry and trig/complex numbers. Many are impossible without differential equations/calc.
So this is one of the most idiotic comment pieces I've ever read. But unfortunately it proves that many people don't understand professional engineering at all, which makes it very hard for them to value it.
Even if the math is packaged and hidden (CAD etc) someone still has to write and check the software. If math isn't taught properly at school, the number of people capable of that shrinks.
Because these people are disproportionately valuable, that's a very bad policy indeed.
So, I think, even if you don't need that particular stuff in your work, it's still a good training.
Also, there has been a pushback against "rote learning" in the past couple decades. I believe that our minds need repetition in order to learn patterns and understand abstractions properly. Yes, you forget most of it, but without it, you won't learn it properly. I don't think you can be any good in any field without lot of time spent on boring and repetitive things (AKA "work").
That's interesting, I have similar experience. Maybe that just means we didn't learn the original material well enough?
https://micromath.wordpress.com/2011/05/17/time-lag-in-learn...
The article asserts that most modern professional jobs requires only "Excel" and 8th grade programming. In my experience, over-reliance on software like Excel rather than a basic competency in numerical programming is a hindrance to economic growth. Spreadsheet-based numerical programming is opaque and ill-suited to interoperation. This leads to subtle errors, duplication of work, difficulty of replication and silo-ing of meaningful results in the private sector, the public sector and academia.
I take the second point that the transferability of critical thinking skills developed by learning mathematics is unproven. Nonetheless, history is flush with anecdotal evidence of this hypothesis, and in the absence of empirical evidence, it seems unwise to reject that hypothesis out of hand.
EDIT: removed an assertion that the article was poorly argued.
Someone in the comments shared https://betterexplained.com/ - which I hadn't heard of and looks great.
I've also found QuantStart's guide[0] to be a particularly good resource, but bear in mind that is oriented toward learning quantitative finance (in which I have no particular interest per se).
[0] https://www.quantstart.com/articles/How-to-Learn-Advanced-Ma...
Linear algebra, computational complexity, type theory, Newtonian physics, circuit design (it's weird being a computer scientist a room full of electrical engineers and being the only person who knows Ohm's law off the top of his head and what it means for the project we're dealing with right now) all of it has been a constant companion for the last 15 years of my career. The more I can get my hands on, the better.
I know my colleagues in the past [0] haven't employed knowledge to the same degree that I have, but they have also typically given up and come to me to solve even fairly trivial problems in trigonometry or object oriented design. They don't "need" math because they don't care if the only work they work on is solved problems with easy copypasta solutions on StackOverflow.
My take away from this is not that math isn't "necessary" for work. To me, it is necessary because I could not be happy living the kind of mediocre, under achieving lifestyle that it takes to willfully ignore math. My takeaway from this is that most people are just bad at their jobs. If you want to be any good (and being this site is focused on startups, I think that is a fair assumption), you necessarily have to avoid doing what most everyone else does.
[0] I'm finally out of those sorts of environments.
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.
There's a lot you learn in High School. Functions is a big one that comes to mind. The idea that f(x) = x^2 + 2 or something, and you can compare it to another function g(x) is pretty important, but not really covered till the end of High School. Sure, if you have studied programming too, then you know what functions are, but that's not quite a good assumption to make for the general population.
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.
Mathematical thinking and problem solving are skills that need to be honed and kept up to date. You do that by learning new methods and tricks constantly. There are disciplines that require similar skills and have a positive cross-over to other skills. Computer science theory is very obvious application. Cryptography is another. The "tricks" in CS or crypto are not taught in school for everyone, yet having the background in math will undoubtably help getting into CS and crypto.
What I wish that mathematics education would get through to students is a better understanding on how mathematical methods are used in a lot of domains. I see too much of a divide between "math guys" and "non-math guys", with the latter group sometimes getting quite anti-intellectual when it comes to math (even if they seem smart otherwise). Even the author of this article has a very dismissive tone, if we just teach people how to apply 8th grade math and Excel, who will be the guys developing Excel and other tools?
Even if math education is learning new methods and tricks, they are not the skill that should be learned. It's the methodology of what it takes to master a new method - learning how to learn.
Just to give a counter point: I regularly use math skills, advanced calculus, arcane series formulations and spherical and hyperbolic trigonometry. A lot of these methods were not taught to me in formal education, but my education gave me the tools to tackle these advanced subjects on my own by reading text books and old research papers.
I will also note that I found this works best when surrounded by a few like-minded individuals with complementary skills to serve one another as "a second pair of eyes" and a sounding board for hypotheses and conclusions.
Learning to learn what? Nobody taught me how to learn how to learn.
I taught myself how to program. I read books, watch tutorials, and so on.
There's no systematic methodology to the whole thing.
Maybe until recently I found a guy who have a 'systematic methodology' for learning. I have yet to use it myself.
One could interpret a lack of use of mathematics as less of an indication it's not needed and more of an indication we're not maximising our efficiency.
You can go even further and say that the whole thing would be even better if I understood more about stochastic processes and could potentially model the spot market and make predictions ahead of time to simplify the control problem and get ahead of the price fluctuations. Saying all you need is Excel and 8th grade is in the words of one famous physicist "not even wrong".
If you're in an engineering discipline then the more math you know the better.
I've also been in analytics jobs where college educated people mistake correlation for causality. (It seemed so profound when I learned the concept only in how often it's abused)
I've seen people in customer support management make enormous judgment errors because they think don't comprehend the difference between a 500K account and a 1K account.
Requiring calculus of everyone may not solve this, but requiring a couple years of hard (beyond 8th grade) stats could help.
As for the CS/engineering/Math requirement for jobs - I think that's just a reaction to the weak rigor (on average) of so many other majors.
I consider him a better programmer than me, and he is honest about his shortcomings (like in this tweet), but I am often very surprised about what things he says he just learned - things any CS undergrad would know.
It's kind of like stories of programmers who were allowed to feed punchcards to the mainframe once a week - it's amazing what they accomplished with that limitation, but one thinks how much more productive they would have been with a more robust interaction.
Advanced mathematics are rarely used for any professional position (including software engineering), but that doesn't mean that technical degrees are irrelevant. In my experience, such filters (like an MIT degree in CS) are invaluable for two reasons:
1. Math does teach you to think logically, which is an invaluable skill in all careers and essential in some (software engineering, specifically). He claims that "transference of mathematical skills is unsettled," but in my experience that's totally untrue: try teaching programming to a bunch of math majors and a bunch of sociology majors, see how learns more easily. Of course, math is definitely not the only way to learn this—philosophy is also an excellent way to learn logical thinking, and I wish that more CS departments required some basic philosophy courses.
That being said, what would be a better way to teach logic skills directly? The sports analogy is pretty bogus because athletes typically spend the majority of their time practicing things besides full games of their sport.
2. For almost any professional field, having smarter employees is an advantage. Unfortunately, administering and/or requiring IQ tests is cumbersome and potentially illegal. A technical degree from a top university is a convenient proxy.
It was partly my fault because I was using Blender instead of a CAD system. Had to rotate something to align it to the base plane for 3D printing. But hey, I'm no engineer and it worked. And all for a door stopper with the company logo.
b) math and computer science degrees are used as a filtering criteria, by recruiters hiring for actuary/stats/finance and programming jobs
I disagree with what appear to be a conjecture, and the subsequent conclusion:
> Acceptance of the conjecture should have revolutionary
> educational implications .
> In particular, it undermines the legitimacy of requiring higher mathematics of all students.
> Such mathematics is actually needed by only a
> minute fraction of the workforce
This is the type of capability (together with knowing a broad universe of solved topics), that the graduates with CS and Math degrees should bring in into the workforce.
I do agree with the author's implication, that there is a 'placebo-style' filtering that's going on by most of the recruiter.
And it is unfortunate, because it brings into Computer Science, especially, a huge number of people who have neither the passion, no life-long perseverance to be current in the subject.
How much of any advanced learning do most people use in their day to day lives? All of it in the periphery would be my counter-conjecture.
I was always taught trade schools were for learning a particular skill. College was to equip you with the knowledge and ability to think logically required to have a better life.
But why should everything be tied to the demands of the workforce? At some level, saying "Don't bother learning calculus because you'll never need it" seems akin to saying, "Don't bother looking at the Mona Lisa, because you'll only ever have to read road signs."
Is there no intrinsic value to learning? No need to be connected to the cultures of the past (or the present)? Nothing to be gained by studying all those ideas that underlie those CAD programs?
The author traces the myth back to Sputnik. It sounds to me much more like every kid who's ever wept over their algebra homework and asked "When am I ever going to use this?"
Given the anti-intellectual and antiscience trends in US and Europe - where US seems to lead the way, it's at least one way of monitoring the situation.
As a practical skill, anything beyond 5th grade is rarely used, except if in rather specialized professions, but learning math most probably gives you tools for abstract reasoning, and probably changes how you look at the world.
The argument that mathematics skills in the US (or many other Western countries) are inadequate usually isn't a complaint that there aren't enough graduates familiar with advanced pure mathematical theorems. It's usually a complaint that after years of compulsory education the masses struggle to do basic arithmetic and understand basic statistics, and plenty of people whose jobs do entail working with figures or making calculations from time to time lack the "eighth grade level" numeracy to spot the figures in the Excel output table are out by two orders of magnitude because someone screwed up inputting the formula.
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 sure that most people (including engineers) would benefit from the ability think and really internalize concepts taught in high school level.
Pedagogical research is almost entirely directed towards small children. Psychologists have studied children and know the common hangups children have. What are common misconceptions, how to use them to make children learn. How they learn to count numbers past ten. Competent teacher can help small children to learn faster.
I think it's possible to teach most people to think _in math_ but it's much slower process.
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.
I think people don't even realise when all that extensive math they studied is even helping them. They may have forgotten or not used the specifics in real life, but if they originally understood it well then they probably gained an important subconscious intuition that shapes, optimises and/or narrows that design trial and error. And the best engineers have a great subconscious intuition for finding solutions.
Note: I studied Civil Engineering, so my math background was nearly all Trig and Calculus rather than the Discreet Math, Logic or Number Theory etc that a Computer Science degree might entail. I've forgotten nearly all of it and never really used it IRL, but I can still appreciate having a 'feel' for relationships between changing quantities etc that others without that math background don't seem to have. And having spent far longer working with software (self taught) than I ever spent with Civil Engineering, I often wonder what subconscious intuition of CS style math I'm missing that would help me.
This seems like saying "most people never use science beyond English". Excel is a language for expressing numerical calculations, but how complicated or 'advanced' those calculations (or the theory behind them) are is orthogonal to the tool used.
I loved studying all that stuff at school, pretty pictures or no. But as a coder the opportunities to use much of it in anger are really quite restricted.
India graduates good math and CS students from IIT, why is India not an engineering excellence hub?
Because they come to the US. US STEM grad schools are full of students from China and India.
I have interviewed many, many people and am usually quite impressed with the Russians interviewed. They usually do much better than American-born people. Maybe 1 out of 6 non-Russians are decent, the Russian batting average in my experience is 1 out of 2 or even 2 out of 3.
They do make good engineers, the problem for those countries is the good engineers move to the US where they can make money.
I think that computation should be part of elementary math, not to produce the next generation of career programmers, but because computation is actually how a lot of math is done. And it might change the curriculum -- having students think about more complex problems that they have the tools to solve, rather than learning algorithms by rote.
I speculate that people might have a better grasp of statistics if they could just play around with artificial distributions generated by a computer -- even just in a spreadsheet.
You're spot-on about needing to understand there's a distinction between a number and its decimal representation, though.
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).
I would expect that most people with a passing knowledge of basic calculus would be able to eventually understand the argument that not only does 0.999... = 1, but that the real numbers are uncountable. It might take some convincing, but the truths are provable and very well understood across the world.
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, if A averaged something like 12.5 and B averaged something like 12.4 it would require a huge sample size to prove that those two samples come from two different distributions.
or google: sample size calculator
The actual number formed by repeating 9's an infinite number of times is not constructable. Whereas 1 is constructable. 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.
As we add 9's, we are getting ever closer to 1, and the concept of a limit lets us "fast forward" to that value. If we agree that this "..." denotes the limit, rather than the non-constructable number implied by the notation's "face value", then the equality holds.
To actually regard 0.999... = 1 to hold without involving the limit shows an ignorance of (or denial of) the validity of induction. Because, look:
Base case: 0.9 is not equal to 1.
Inductive hypothesis: Adding another digit to a decimal fraction which is not equal to 1 produces a new decimal fraction which is also not equal to 1.
Therefore, by induction, no matter how many 9's we add, we do not get 1.
Induction is not somehow canceled by infinity; induction is how we understand that a property holds for infinity: a property such as "not equal to 1".
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.
Edit: sorry, didn't mean to ad hominem, but I'm going to leave my original comment there all the same. To make my post a bit more constructive, OP, what exactly is your definition of "constructible"? Because it doesn't relate to any mathematical concept I'm familiar with. Other than maybe "finitary".
It seems that you object to the commonly accepted meaning of "...".
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.
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.
In my engineering career, successfully solving technical problems has generally consisted of working out what basic techniques solve an approximation of the problem and leaving it at that. I would say first-year undergrad level rather than 8th-grade, but definitely not the most complex mathematics I've ever looked at. Apparently being able to put together any sort of solution from scratch is relatively rare.
I do think problem-solving could be better taught. And schoolkids should definitely learn more about finance and statistics. Going on, the OP's stance seems fairly objectionable, but it's hard to disagree that employers use success in maths-heavy degrees as a proxy for selecting who may be best at a technical job. It seems like a fairly good filter, but it probably leads to injustice in certain cases, and the credential-chasing and learning less-necessary things may be inefficient.
I used to do IT work at a company where playing chess was popular among the techs and I decided to improve my skill.
The ways to improve:
* Study tactics and strategy books
* Study grandmaster games
* Do two-to-mate or tactical chess problems
* Review your own games and look for mistakes
* Study openings (once you're more advanced)
If I spend months carefully studying the games of Kasparov, Fischer, Karpov etc., and my playing begins improving - how do I prove studying the greats carefully has improved my game? It might be "self-serving nonsense", as I can't draw a line between a move I do to some game I studied, like I can for an opening I memorized from a book. All I know is my general problem solving skills have somehow increased. I now see patterns I did not see before, and the correct path forward where before it was muddled, although I can't fully explain why. The only test is a sample comparison of those who do it versus a sample of those who don't.
For instance modern coaching has the kids run around with the ball a lot more than in a real 11-a-side match. That's because it matters a heck of a lot that the kids are comfortable on the ball, and only one of 22 people has the ball during the game.
The same goes for math. You may not have to solve PDEs very often, but if you never do it, you will be stuck when it comes time to do so. I recall writing a Bessel function for an option valuation routine once, and if I hadn't come across it in uni, I'd have been struggling with it.
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. People basically could specialize in finance or CS. Those that went into finance make consistently much more than the others.
I wish I knew that at the time. I thought banks were boring and unappealing places. But now I think finance is one of the rare field (if not the only) where you can earn a lot with a technical, non-managerial position.
Luck is also a huge factor. I know cases of International Olympiad gold Medalists, who didn't make it as portfolio managers. Do you really think you are smarter?
If you are mathematically inclined software engineer, I would avoid finance unless you are immediately hired into the quantitative role in the front office. In Silicon Valley you will get similar or better salary, more freedom, more respect and better culture. I worked on the both sides.
Hm, I would never have guessed that. Does anyone have any data on this?
The top 1% sure, but the average and median also?
My current job is on a data science team. I find it amusing that the business folks are able to sell our product, and then sigh to myself and do a little crying inside when I realize how it's possible.
Once you learn it with sufficient mathematical sophistication, it stops being magic and starts being a tool that works in some situations and not in others.
You are surrounded by people who believe in the magic and will buy anything whether it works or not. Equally frustrating is being surrounded by non-believers who don't accept that simple things are actually possible.
We are still on the upswing for now so there are more believers than not. But if another AI winter happens, be prepared for the mbas to reject applications of data science that make complete sense because "we tried that data science stuff and it doesn't work".
It's clear that the trend in software is towards higher abstraction, and that's what makes selecting what to learn so hard.
But maybe a heuristic: learn thing that could not be abstracted(unless AI appears), or at least choose a few abstractions up ahead.
One such thing is desinging and developing domain-specific-languages, which also looks to be an important tool in some systems.
Another thing is prototyping with very high-level tools, because it gains you experience in working with customers, extracting requirements, product management, etc.
That's the point -- the "math myth" is specifically in relation to math (and science) skills as being of particular importance as a comparative advantage over other countries (Russia, Germany, Japan, India, China) where these fields are perceived to be prioritised higher in education.
If there's low market demand for the skills, it's unlikely that countries with a higher supply of those skills can leverage them to surge ahead in comparative advantages.
That said, I don't think I agree with the conclusion. Most people who are good at something don't particularly know they are, and few can explain why they are, so just asking them is useless. Sure, math and science are probably of limited value in their distilled, pure forms, but my impression is that most successful people in computers and software do subconsciously draw on a fairly solid math/science basis on a daily basis, even if they never consciously sit down to apply science to a problem.
I took particular issue with the Accenture anecdote - we can all laugh at useless consultants all day long, but I've met more than a few who can pick inconsistencies (not validate hard math, but "why is that number so low when that is so high, and is revenue in X really only a third of Y" style things) out of a wall of numbers in what seems like an instant. You don't do that without a solid math foundation, even if you don't think you use it.
Isn't that math pre-level 8? It's just simple algebra or arithmetic. Interesting how little use trigonometry and calculus have.
btw. 12% of US millionaires are educators. http://www.barrons.com/articles/SB50001424053111904370004577...?
The Economics of Superstars Sherwin Rosen The American Economic Review Vol. 71, No. 5 (Dec., 1981), pp. 845-858 https://www.jstor.org/stable/1803469?
But now, since more and more people are pushed towards college, the value of a degree as a proxy for talent has been debased. On top of that, there happens to be some correlation between the actual economic value of the skills taught in a degree and its power as a proxy for intellectual talent. For example, a degree in engineering teaches economically useful skills, and is also less accessible to those of middling intelligence. But a degree in English, on top of teaching skills of little economic utility, also lacks a strong filter for intelligence. So more of its value was in its role as a proxy, and it took a bigger hit to it.
And that is why you see so many people with English degrees working as baristas. But the key point is that those are mostly people who would have been baristas anyway. The magical feather was fake; the employability, or lack thereof, was within them all along.
That doesn't necessarily mean that teaching math to people makes them better at programming. It could also mean that the kind of people who major in math (and then don't drop out in the first year) tend to be better at the kind of thinking used in programming than people who go into sociology.
It's hard to tell what causes what, and that's what the author is talking about when he says that the matter of skill transfer isn't settled.
I read somewhere that many of the most successful entrepreneurs in the US are foreign born. That doesn't necessarily mean that foreign born people are better than American born people (which would be unlikely). It probably just means that people who have tenacity to immigrate also have tenacity to do bigger things than average.
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).
A "proof" that just plays a trick on people's logically inconsistent assumptions to derive a result isn't very satisfying. You're not really uncovering anything fundamental through that proof, just playing games.
Edit: what I mean to say is there are reasonable things "..." could mean such that .99... and .33... are both not equal to 1 and 1/3, respectively. But there are none for which one pair is equal and the other not, as you rightly point out. So all your proof does is show that the other person's viewpoint is inconsistent, but it doesn't give any evidence for one of the two consistent viewpoints over the other.
What slips people up is that ignoring everything but the total pie consumed (taking the limit) is embedded in the definition of real numbers.
There's an analogous story with rationals: Suppose x1 = 1, y1 = 3, x2 = 2, and y2 = 6. If we plot them, (x1, y1) and (x2, y2) are clearly different points, but x1/y1 "equals" x2/y2 because they lie on the same line through the origin. We decide that we don't need to know about those individual points.
The only problem is that this not-finite value does not occur among the integers; thus the inductive procession doesn't actually have that value k as its target. It does not lie in its path, so to speak.
> It seems that you object to the commonly accepted meaning of "...".
I formulated it as a limit and stated that it denotes 1 in the case of 0.999...
How do I object?
> For example your strategy would show that the sequence of positive integers 1, 2, 3... has a finite limit!
How so? Induction is in fact the basis for the common proof that there is no highest integer: for any integer k, we can add 1 to find a larger integer. Every integer k has the property P(k) := "k is not the highest integer".
Just like people working at helpdesk say they are working in "IT".
Higher finance is filled with crazy maths, if you are interested in statistics and probability then finance is the best way to make a lot of money.
Its also not "gambling,etc" - its educated decision making. People in higher management need to make a lot of very important decisions all the time. 90% of the time they just use their "gut feeling". This is where people with strong technical knowledge in finance comes in.
Where to allocate money is a very hard question to answer, you can throw a random dart at your options or bring in complex ML models. The sky is the limit.
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.
Now, admittedly, I think the deep-learning folks are so glad to finally have their faith in neural networks vindicated that they've let themselves buy into their own propaganda, but that doesn't mean there's any actual magic!
My spell check initially game me meth instead of math. Proud methamatician
In any case, statisticians are not interested in learning the truth, either, and the goal of their enterprise is not access to it.
The entire purpose of statistics is to quantify the difference between our beliefs and actual truth.
That said, I prefer to say that only real numbers that can be emitted by computer programs exist.
[1] Use the Downward Lowenheim-Skolem theorem to get a countable set theory, and choose something like ZFC as your set theory.
In any case, if you want to talk about the computable reals, then call them computable reals. Don't say "reals" for "computable reals", even if you think the former don't exist. Unicorns don't exist either, but that doesn't justify defining "unicorn" as "a horse".
> 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 on condensing the names of the elements in S, like "0.x...", or 0.x with a bar over the x, and making that notation denote something not in S, namely 1.
This doesn't appear incompatible with the limit-based construction.
Are there instances in which this supremum approach toward continued decimals disagrees with the limit approach, in establishing the value?
Supremum of an increasing sequence of reals is equal to the limit of the sequence in all cases.
If we have an entity X and we use the function name(X) to inquire about its name property and obtain "Mohamed Ali". And then if elsewhere we use name(X) and it evaluates to "Cassius Clay", then name() is not a mathematical function.
You have to find a way to reason about an object and its properties so that they are functions. If it can have multiple names, you have to model that appropriately.
I would humbly submit that if your code is showing Cassius Clay is not the same person as Mohammed Ali, it is your code that would be in error rather than the universe.
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.
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 the decimal point (so that the result of the subtraction is just 9). This is never true for finite values of n, and the proof does not establish that it's true for an infinite value of n -- indeed, it can't do so without supplying a meaning in the first place.
Another way of phrasing it is that it assumes that if X = 0.999..., then 10X = 9.999..., where there are the "same number" of 9s after the decimal point in 10X as there are in X. This seems intuitive for an infinite repeating sequence of 9s, because "one less than infinity" is still infinity, but it's not very rigorous, and the argument as written certainly doesn't explain this.
Most people just look at the number of subjects in a study, but not the p-values
In Silicon Valley it's not uncommon to see new grads (not even PhD) getting 200K total first year compensation. I was comparing quantitative finance vs silicon valley as a career for mathematically inclined software engineers. You won't end up poor either way.
Are you sure about that? again, it's only anecdotal but I know at least 3 quants that didn't do a PhD (just a MS from reputable schools), including one that worked at GS as a new graduate. But there may very well be the exceptions.
In general, when many people hear Quant they think about quantitative portfolio manager - this is what I was talking about and where the "money" is at. It seems that IBs tend to use "quant" nowadays for a role that is very similar to Silicon Valley "data scientist".
(If the argument assumed otherwise, it would only be temporarily, for the sake of setting up a reductio ad absurdum.)
The number of nines is not in the set of natural numbers, yet you're trying to use induction by covering every natural number.
That simply isn't the basis for how 0.999... is regarded as 1.
Rather, 1 is the supremum ("least upper bound", LUB) of the countably infinite set { 0.9, 0.99, 0.999, ... } as a subset of the reals. We define that 0.9... denotes that supremum: i.e. that the ellipses suffixed to 0.9 denote the expansion of 0.9 into the set { 0.9, 0.99, 0.999, ... } followed by determining the LUB of subset among the reals.
No "infinite string of 9's whose length isn't a natural number" nonsense is involved.
It meant that I understood the topological idea of limits before I had to do proofs using just the epsilon (for sequence) or epsilon-delta (for functions) definition, and so could translate the logic of showing things about neighborhoods in to the terminology of (real analysis) limits.
Limits, in the abstract, are a fairly simple concept: in the case of sequences, for any neighborhood of the limit, the entire tail of the sequence (past some point) is contained in the neighborhood; in the case of functions, for any neighborhood of the limit at f(x), there's a neighborhood around x, such that every point in that neighborhood maps to the neighborhood around the limit.
There are several ways of defining real numbers, and one of them is the axiomatic definition. Real numbers are defined by a list of axioms they must satisfy. These would include, among others:
(1) If x and y are reals, then x + y = y + x. (2) If S is a nonempty subset of reals with an upper bound, then S has a least upper bound.
There is a crucial difference between these two. In (1) the variables x and y only quantify over reals, but in (2) the variable S quantifies over subsets of reals. We say that (1) is a first-order axiom and (2) is a second-order axiom. Actually, among all of the axioms of real numbers, (2) is the only one that is second-order. Therefore, it is natural to ask: can we rid of it?
No, we cannot. Löwenheim-Skolem theorem says that if we only have first-order axioms, then it is impossible to distinguish between countable and uncountable sets - even if we have an infinite number of first-order axioms. In particular, this means that if we try to define real numbers using only first-order axioms, then the definition cannot even capture the basic fact that there is an uncountable number of reals.
From here on, there are two roads you could take. If you're like me, then you just accept that real numbers cannot be defined using first-order axioms. By my standards, any definition that only uses first-order axioms cannot be a satisfactory definition of the real numbers.
But some people don't want to accept definitions that are not based on first-order axioms. And this is not as crazy as it might sound. First-order axioms are very nice from a theoretic point of view. For example, with first-order axioms it is absolutely clear what it means to prove something based on those axioms. With second-order axioms, the situation is a lot hairier.
On the other hand, for any finite value of N, the sum [1] is equal to 0.9 * (1-(0.1)^N) / (1-0.1) = 1 - (0.1)^N. This value goes to 1 as N goes to ∞.
More rigorously, for any positive value, ε, there is a value N, such that the value of the finite sum is within ε of 1.
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
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.
Do you have remarks not regarding the semantics of "constructable"?
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".
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 it to high school Algebra, Geometry, Algebra II/Trig, Precalculus, and (maybe, for the better kids at richer high schools) Calculus.
This has the effect of selecting programmers for their skill, patience, and diligence in executing the algorithms they're given. This is exactly backwards!
In my opinion, if you're looking for an analogy to a high school class, programming is chemistry. You're given a reasonable number of facts about the world, some lab and algorithmic procedures, and a problem. How to string together your intuitions, information, and procedures to arrive at an answer is entirely up to you.
I see the argument about chemistry, but I do think there's a very strong crossover of doing math and building software (in the large). In math, you begin with a set of base known statements. Whether starting from the axioms, or working within some system, you begin to build a vocabulary of the system, a feel of the way objects within the system interact, and an understanding of the limitations of the system you're currently in. Through the course of needing to get things done, you combine these various abstractions together to create more useful ones, constantly making your system a little bit more powerful. Often, taking steps back from your system and combining things into elegant ways gives insight into other patterns present in the system, and recasts solutions in slightly different lights with hints as to what they could have been, had you known what you know now. Then over the course of time, you begin to bring in new foreign systems, which you might not understand fully, and you use inelegantly. But again, working through issues, you build up that vocabulary and intuition to solve problems in your space, constantly putting tools and tricks into your belt on how things could fit together. Occasionally something will be insufficient in your tool belt, or a concept you weren't quite clear on will come up in your problems, causing you to go back, learn a thing, and come back armed ready to keep making progress.
To me, that all sounds like what I do when I'm building software.
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 programming go in relation to programming, contest math, with its focus on efficient and creative problem solving, is probably more similar to the daily experience of writing software than any course I took. The exception would be Math Modeling and other courses that actually required programming.
By Curry-Howard isomorphism proofs in some particular proof system are equivalent to computer programs.
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.
I have no dislike for logic, formal proofs, category theory and such: I can live in that world just fine. However, almost none of the problems I've ever had to dealt with in my 15 years programming made me wish for large amounts of that: I've used bits here and there, but going all in never made any sense.
The hard part of most tasks I had to do in professional programming is to gather enough information to even begin to write one of those formal specifications: Knowing what real business needs are, understanding how distributed systems fail, working around bugs, reading library code quickly and effectively, learning how to factor code in a way that would evolve well with changes in what the application needs to do... that's where the value is. 99% of that is talking to people and making educated guesses about the future: The implementation itself is trivial. With that, was able to get solid code working in languages that were ill equipped to handle any of the fun pieces of math that you'd need to write mathematically-minded programs.
If anything, the experience I've had working with people that were heavy on the math has been dismal. I worked with a rather well known dev who has published books, gives a lot of talks in FP circles, and is a lead dev in a huge mathematically driven library. We worked on on a rather novel distributed systems. He hid in a corner with a copy of Mathematica for a couple of months, and came back with a paper that proved a well performing solution to one of the things we had to do with our distributed system: Great!... Except, not so, because his proof only worked in a mythological environment where there is no latency between nodes. Under real conditions, the algorithm's properties led to unacceptable results, and he just hid behind the proof.
I don't have papers and patents behind me. He knows a lot more category theory than I do. But It took me a day to implement a completely unproven algorithm that had far better performance properties in a real case. This is not due to any special brilliance of mine, but because I sat down and approached the system like a scientist or an engineer would, instead of like a mathematician.
In my opinion, teaching the scientific method and the basics of engineering in other fields will serve your average programmer far better than being better at math ever will.
Teaser:
"When DEK taught Concrete Mathematics at Stanford for the first time he explained the somewhat strange title by saying that it was his attempt to teach a math course that was hard instead of soft. He announced that, contrary to the expectations of some of his colleagues, he was not going to teach the Theory of Aggregates, not Stone's Embedding Theorem, nor even the Stone-Cech compactification. (Several students from the civil engineering department got up and quietly left the room.)"
Formal mathematics training isn't necessary to be successful - many of us know many people who have picked up the intuition that those formally trained understand at a deep level & have been able to use it to great advantage. However, I argue that more training in areas of logic & reasoning would serve people well in all facets of their lives, including helping people be successful in software development over the long term. I know that I use my knowledge I gained from decades of mathematics to code strong abstractions faster than just about everyone else I have worked with so far (I've made mistakes too, but nothing terribly costly & unrepairable) - it helps me nearly every day I code & learn new things.
I saw a few people that had no problems with the math courses in CS but that never really got the programming part. (No names or details, people might be recognized.) But this is of course anecdotal.
(My surprise over this was similar to when I realized that rationality and intelligence didn't have as high correlation as I expected. I know some intelligent people which are Truthers and believe in other conspiracy theories. Others can't reason when their emotions are involved.)
> Great!... Except, not so, because his proof only worked in a mythological environment where there is no latency between nodes. Under real conditions, the algorithm's properties led to unacceptable results, and he just hid behind the proof.
Then he did a bad job. His failure does not undermine the power of taking a formal approach to a problem. He should have accounted for this latency in his initial model. What he did is the equivalent of someone working on an autonomous car and assuming that a noisy GPS signal is good enough to do control with.
We convince children that 1+1 =2 without delving into the Peano axioms. It's ok to not delve too deeply into the axiomatic structure of the reals.
There is nothing to do with completeness here; 0.999… = 1 is a statement about a series of rational numbers summing to a rational number, and the convergence of the series is established by the fact that it sums to the right-hand side, not by an abstract appeal to completeness.
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.
But you can't. All of the places where you might want to add a digit are already occupied by 9's.
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.
Since we are never at infinity, our observations are always finite, all this is saying is "we always deal with approximations of reality," i.e., statistics is never, in fact, about the "actual truth", it is about testing the set of our limited observations of reality against a model.
As a real-world example that I deal with every day, we frequently model the expected distribution of gene expression as a negative binomial. It doesn't matter how closely we observe the actual distribution of gene expression, it is never going to perfectly fit a negative binomial (it won't even get close), even though this is what we test our observation against, because the 'truth' is something different and far too complex for us to apprehend.
[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.
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 + "rubber duck", and 9.999... - 0.999... stands for 9 + "rubber duck" - "rubber duck" = 9.
0.999... * 10 = 9.999...
Because "one less 9 than infinity is still infinity" it's what really closes the loop on the proof.
I'm not going to deny the presence of some built-in preferences for formalized thinking among some subset of the general population, however the cognitive resources and preferences that would lead one to enjoy or excel at math, are about the same as those correlated with programming. e.g. a noticeable interest in understanding mechanical systems.
So given that set of preferences that would produce an interest in math/cs, if a person was raised by a mathematician parent, they'd likely be "good at math", but possibly still struggle with programming when they reach their CS classes in college. Meanwhile another person with the same set of preferences but raised by a hacker, might be good at programming by the time they reach college, but still have a hard time with math.
The issue is less about "can this CS person be good at math?" or vice versa, and more about "will this person be motivated to learn these complementary skill-sets?". Because it's not just a one-way street, mathematicians can also learn from thinking programatically, as shown by the growing popularity of automated theorem provers like Coq. And mathematics being helpful for programming is nowhere better epitomized than by the huge trend of nearly every popular language slowly adopting functional features (and even "optional" typing), because everyone is slowly discovering that mathematically sound programming principles actually are practical (just a bit more abstract and difficult to grasp at times).
Both of those examples should be pretty good evidence of these two fields being pretty helpful towards each other. The large show-stopping issue here making this relationship non-obvious, is education. If complex subjects that are already difficult get taught poorly, it's only natural that we don't expect people to make connections and think "oh hey, this [math/cs] concept I learned about might be helpful in this [totally unrelated] situation".
P.S.: as an anecdote, I just recently used my (minimal) category theory knowledge to change the way I take notes, so that I can smoothly use outlining tools (like org-mode) more like mindmaps when taking notes on complex topics.
I, and likely many others, would love to see a writeup of this if you ever felt inclined to write one.
Maybe those guys were just unmotivated? Or they were hardworking idiots which had gotten a good math background by sheer will and hard work? It didn't seem so.
The alternative is the existence of basic cognitive talents you need for coding which aren't needed for math? Like for us color blind, that really can't appreciate some art?
None chimed in with support, so I'd guess my sample was unusual; a lot of the guys here should have teaching experience from academia.
(Category theory makes for better Org Mode? I'm curious, too!)
What your real world example describes is something different entirely. That's just pragmatically choosing the wrong model due to computational or human tractability. That's not fundamental to statistics, that's just a cheat you made because it's good enough.
Statistics is about acknowledging that cheat and quantifying how much it hurts you; fundamentally such a thing is not possible if truth doesn't exist.
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.
I don't quite understand. You are arguing that statistics doesn't care about truth simply because some biologists are using a model they know to be wrong? That doesn't even make sense.
In applied math in general (which includes but is not necessarily limited to statistics), the following equation holds:
error = |true model - actual approximated model|
error <= |true model - best model in class X| + |best model in class X - actual approximated model|
Various mathematical techniques, or increasing sample size in a statistical scenario, can be used to reduce |best - actual|. Due to the triangle inequality, this brings you closer to truth.
Statistics is also concerned with expanding class X in such a way as to more easily reduce the model error.
I really feel like I'm missing something, because I truly can't comprehend what you are trying to argue.
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...
(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.)
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.
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.
Indeed. But it's a lot less fun :-)