Last week I needed an arctangent at work. Looked it up on Wikipedia and let Wolfram Alpha compute the result.
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.
Perhaps you're not aware of your math literacy? You knew, remembered, and understood what an arctangent is. That makes you tremendously more math literate than the typical non-STEM educated Joe or Jane.
Suggests the article, yeah. I honestly don't buy that - citation needed. After some time STEM-educated people might not always remember exactly how what they learned works, sure. I certainly don't, for one. But I'd be hard pressed to believe they don't remember it exists - i.e. or at least remember enough to google their way into rediscovering it and finding a shortcut to solve their problem.
Perhaps the biggest skill I got from university was knowing enough context to know which references (books, Wikipedia) I should consult, to solve a technical problem.
I think this reflects more our culture of compartmentalizing specific "math" and specific "science" topics and putting them on a pedestal.
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 made an account just to upvote this. I think it also leads to the possibility that even if the authors simplistic conjecture that ~10% of MIT graduates actually use math, potentially this number (10%) doesn't vary with tremendously with the number of people taught. I.e. If we teach 100 people math, 10 use it but if we teach 10,000, 1,000 will. I'm in a profession where using higher level math is highly, highly encouraged although avoidable. It is incredible to see CMU level grads not taking advantage of their education. I think this leads me to believe that the application of math is more of a personal choice (do you desire to be helpful and add vale) than skill-based.
I learned more useful things from my liberal arts class than my CS or Math classes. Granted, I had been programming for years at that point so a single public speaking class was more useful than my first 1-2 years of computer classes. The problem is not the material, the problem is 4 years is just not a lot of time vs 13 years of prior education.
Of course, the economy doesn't depend on masses having solid mathematical education (and knowledge), but the world would be a much, much better place if all kinds of “advanced” math was common knowledge and skill (and not just math). I am aware that that is currently a bit of a sci-fi scenario. Anyway, the author need not worry a thing — wishful thinking aside, as long as we live in a capitalist society, we're in no danger of large percentages of population being educated in any advanced subject. Or at all.
Indeed; just because a job can be done without mathematics, doesn't mean it can be done as well, or as fast, or with as much confidence in the solution.
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.
There is some truth to it - almost all tasks can be done without higher math skills in my job as consultant.
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.
Reading someone call for in-depth study in one sentence and saying they're already convinced of their own pet theory in the next because of anecdata (anecdatum?) has me puzzled - I can't decide if it's an indictment of the author or merely ironic evidence for his thesis.
I can tell from personal experience that I only properly understood simpler mathematics when I started learning more complicated one. For instance, in linear algebra, finite dimensional (euclidean) vector spaces became a cakewalk once we started talking about functional analysis.
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").
One of the commenters to the original article talks about this phenomenon specifically. Each math course you took taught you some ideas or techniques, but those techniques weren't really learned until you used them in the next course. For example, you didn't learn algebra well until you used it in calc 1; calc 1 skills are really cemented in difeq.
But the only way to maaster new material is to apply it to something more advanced, rather than just practice it. Rote takes you only so far; you can solve only problems you understand already. But learning new kinds of problems expands your understanding of the limits of the concepts and techniques you already know.
> 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)
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.
I think this gets it all wrong by considering mathematics to be a set of discrete tricks, like 8th grade arithmetic, algebra and statistics.
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.
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.
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.
Yes! This! I am 30+ years post Ph.D. and work in microscopy and image analysis. I have repeatedly needed to go into areas that I never anticipated and learn what I needed to solve the problem at hand. Happily, my graduate advisor taught our group to expect this to happen and to become self-directed learners and to embrace the process.
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.
I think society would be a lot better if BASIC math and statistics would be better understood.
How many times do you see a study posted here with N=23 and people say "the sample size is too small" when it's clearly not? How many people ask for a card deck change to change their luck? How many times do people read a poll like 49% +/- 3% vs. 43% +/- 3% and conclude the two candidates are statistically tied?
I could 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?
To be fair, 0.9999... = 1 is not quite basic. You need to know things like infinitesimals, the distinction between value and representation of numbers etc.
Hardly. We use this reduction because it's essential from a mathematical and axiomatic perspective and because limits are a fundamental construct. But more philosophically, a natural and nonterminating simulation of 0.999... (were it possible) would never strictly equal 1. You would wait an infinite amount of time for it to do so. This comes down to how you view the problem. I would never argue with this through the lens of abstract higher mathematics. I think about it from a computability perspective, which is one way in which you can discount the observation.
Your "proofs" simply assumes that 0.999... is a notation denoting 1, without examining the underpinnings which might legitimize that.
9.999.. - 0.999 is 9 no matter how we define .999... just as long as two or more occurrences of the 0.999... notation all denote the same entity, and we understand that the syntax 9.999... is 9 + 0.999...
For example, if we define 0.999... as "rubber duck" then 9.999... stands for 9 + "rubber duck", and 9.999... - 0.999... stands for 9 + "rubber duck" - "rubber duck" = 9.
You are discarding an essential part of the representation. You are moving the goal post to make yourself right. Just because people don't always understand infinitesimals doesn't make them right. It's the reason they are wrong. There just plain don't know what they are talking about.
You can't represent 0.2 exactly using IEEE floats, either, but that doesn't mean the representation 0.2 is not exactly equal to 1/5th.
Well, that's if they accept that an infinitely repeating number is actually EQUAL to the fraction. Some people just accept that a non-infinite number will never equal the fraction, but don't understand the concept of infinite repetition as an actual number (point on the number scale), not an estimate.
It's easy to imagine that eating 1 out of 3 slices of pie is "the same" in a very specific sense as eating 333333... out of 1000000... slices of the same pie (although that would be infinity out of infinity slices, which is meaningless).
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.
If people accept 0.33... = 1/3 but not 0.99... = 1, they are being logically inconsistent.
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.
I didn't quite accept that until I challenged a professor in a 300-level university math class. This came up in class, and I spoke up and was like, "no, that can't be true, because 0.3333 repeated N times will never equal 1/3."
Up to that point, I had only a fuzzy notion in my mind of what 0.33333... even was or how it was defined. But the professor helped clear this up for me: that "infinitely-repeating decimals" was actually just shorthand for a limit definition, i.e. "0.333..." is defined to be the limit as N approaches infinity of (3/10^1)+(3/10^2)+...+(3/10^N).
You don't need to understand anything about infinitesimals to understand 0.999... = 1. Perhaps you meant limits? The "standard" approach would be to point out that Σ_{i=1}^∞ 9/(10^i) = 1 (that is, the sum from i = 1 to infinity of 9/(10^i) is 1), and understanding an infinite summation requires the concept of a limit. (Of course, there are simpler proofs that use only basic algebra and intuition about decimals; a limit is just the most direct approach.)
You're spot-on about needing to understand there's a distinction between a number and its decimal representation, though.
Oddly enough, I never understood the epsilon-delta description of limits until I read David Foster Wallace's book on infinity. All through my degree in math I was taught about things without learning the historical context that created those things.
I'm always super happy that I stumbled in to taking topology before real analysis.
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.
Property of real numbers: between distinct real numbers is at least one other number. Now try to find a decimal representation of a number bigger than 0.9999999... but less than 1.0. You clearly can't. They must be equal. No need for infinitesimals.
So you're saying a countable infinity can be occupied, so that there is no room for one more.
You must then disbelieve concepts such as that the even integers can be put in 1:1 correspondence with all integers; i.e. that there are exactly as many even integers as there are integers.
You might be able to do something funny with a series whose terms are indexed by ordinals. The last `1` would then be indexed by `omega` (the first transfinite ordinal). I can't claim that the ordinary rules of arithmetic would make much sense, but it's certainly an idea that you can attempt to formalize.
Of course you can. Consider the sequence of rational numbers in (0,1). Now consider the sequence of rational numbers in (0,1]. The latter has had an extra term appended.
(This is of course flawed, but I think it illustrates that the question isn't completely trivial. It requires us to carefully distinguish between the notion of an ordered set and a sequence and even then we'll have to deal with the fact that the rationals can be made into a sequence, but not with the same ordering.)
No. There is no such thing as the sequence of rational numbers in an interval. The rationals have a total order, and they can be enumerated, but the total order of rationals does not define an enumeration of the rationals and hence the rationals are not a sequence. If you want to dispute this, tell me: what is the next rational after 1/2?
I took some liberties with the word "the", but I'm pretty sure that you know that it's easy to construct an ordered sequence of all rational numbers using any of a number of standard diagonalization arguments. I'll skip the details you already know.
Eventually, yes, you're going to be able to poke a hole in my argument. It is definitely flawed. I don't know how long it'd take us to get there, but it doesn't really matter. But we're already at the point where this cannot be considered "basic", and that is my true point here.
Attempting to demonstrate that 0.999... = 1 while meticulously avoiding any rigorous definition of what 0.999... means is not very easy and will require you to fend off all sorts of potential jabs from various directions. It's much easier to just talk about infinite sums and be done with it.
Okay, so it follows from completeness. Now prove that real numbers are complete (or provide a construction of the reals that uses completeness as an axiom) without using concepts foreign or confusing to someone with a middle school level exposure to math.
You're just proving their point. None of the things you talk about are even remotely obvious or “natural” to people we're talking about. You want them to ”choose axioms”? Axiom-a-whaaa? 1+1=2 does not need Peano axioms because it's cognitively fundamentally different than 0.999...=1.0. It is immediately obvious because dealing with simple arithmetic on natural numbers is in everybody's experience constantly. In other words we do not need to convince them. Clever untrained people would be able to get 0.999... thing quickly if you gave them some explanation and let them think a bit, if they cared, but properties of real numbers are far from obvious or basic.
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.
I think I came to the conclusion that 0.9999999... = 1 when I was 12 or 13. My reasoning was that for any n, we had 0 <= 1 - 0.999999... <= 10^(-n), so the difference had to be 0.
This proof is more subtle than it appears. Here's a bogus rewrite, for instance:
Let 10X = 9.9. Then 10X - X = 9.9 - 0.9 = 9 = 9X. Hence X = 1, but X is actually 0.99 in this case (not 0.9). You need 0.99 = 0.9 for this to work with the exact same structure as your version.
Your proof only works because appending a 9 to an infinite expansion of 9s does not actually add a 9. But at this point you're forced to establish meaning for an infinite expansion of 9s, at which point this is really not just algebra anymore.
Because that's exactly what the argument above does. It simply subtracts away the decimal part of 9.999....
This is always wrong except in the case of infinitely many repeated digits, and the proof does not explain this.
More rigorously, let 9.999{n} denote an expansion with n 9s after the decimal point, where n can also be infinity. The subtlety with the argument is that it needs X to be the same as everything after 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.
Yeah, yes and no. The premise is, that 0.9999... repeating CAN be represented as a decimal number, so that means it CAN be used in arithmetic. If we deny that 0.999... minus 0.999... is zero, then the premise is broken and the question is kind of moot.
Okay, this is a little late but when you say 10X = 9.999... then X = 0.999... not because you have removed the 9 but because that's what X is; a tenth of 10X! So when you say in your argument 10X = 9.9 and therefore 10X - X = 9.9 - 0.9, that's wrong. 10X = 9.9 implies X = 0.99. No wiggle room. No need to consider infinites, just one movement of a decimal point.
[And hence 10X - X = 9.9 - 0.99 = 8.91 = 9X implying that X = 0.99]
Following, as delineated above, from there, you'll see there's no contradiction. It's not so easy to break arithmetic that easily without dividing by zero :)
Depends on what effect size or power you are looking for and/or other factors like variance in the data (standard deviation), etc. There are some different ways to calculate sample sizes here:
https://en.wikipedia.org/wiki/Sample_size_determination
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.
> I mean I wonder how many people even understand that 0.999... = 1?
To be honest, I think it's unreasonable to expect anybody - even with a Ph.D in a field other than mathematics - to be able to even define the real numbers: My definition is probably very different from yours(I tend to say there's countably many real numbers).
This is not the easiest thing to explain briefly, but let's give it a shot anyway.
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.
Having a "different definition" of the reals doesn't make it correct, even if you tend to say it.
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.
There's nothing special about the definition. Use any construction you like, but do it over a countable model of set theory[1]. Now you have a countable set of real numbers, although there is no correspondence between it and the natural numbers within the model. The statement that 'there is an uncountable set' is also provable, although the set is countable. It's one of those Goedelian tricks, like the statement that is proven true but is unprovable.
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.
I think you are confusing the object-level and the meta-level (and incidentally confusing everyone who doesn't know advanced 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".
It's a definitional issue. "God created the integers. All else is the work of man." The theory of reals is a convenient abstraction defined by axioms. It's not created by construction.
Speaking of BASIC, I took a high school course in BASIC, in 1981, and it changed my life. For one thing, it changed my approach to math.
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.
I just wish they would teach the common curtesy of putting (page#/totalPages) on each slide of their power point presentation so I can decide whether I can make it through the deck or cut my wrists now.
0.999... is equal to 1 only if we assign a particular semantics to the "..." notation. Namely if "..." means "the limit of the decimal number to the left, as the repetitions of the last digit grow ever larger", then 0.999... is an alternative notation for 1 since that limit is 1.
The actual number formed by repeating 9's an infinite number of times is not constructable. Whereas 1 is constructable. So they cannot be 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".
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".
But the "example from real analysis" section in your link uses exactly the same limiting technique to construct e. That section shows that 0.999... is constructible.
First we have to agree on what 0.999 is, then we can call it constructible or not.
Numbers formed by repeated 9's appended to 0.9 are certainly Turing computable. Which has the meaning that we have a terminating algorithm which, given a natural number N, will compute the N-th digit of the infinite sequence 0.999...
This is the same way that pi is computable. Given an N, we can compute the N-th digit of pi in a finite number of steps.
We don't say that 3.141... = pi! Unless, by convention, we agree that this "..." syntax has the semantics of (for instance) denoting the limiting numeric value of the non-terminating algorithm for producing the digits of pi ad infinitum. That is to say, "insert here a process for calculating the remaining digits of pi, and take this whole expression then to be the limiting value to which that converges".
But that's exactly what the ellipsis means. What other interpretation is there? I might as well argue that 1 = 1 only if we agree on the convention that = means "equals".
So you're saying that "for all positive integers n : property(n)" means that property(k) might not hold for some k that is not a "finite value"?
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...
> So they cannot be the same thing. That's because, philosophically, two objects must be identical in every property to be the same object, and constructability is a property.
Mathematicians have proposed several different constructions of the reals. If you were familiar with, say, constructing the rationals from the integers or constructing the reals from the rationals, you would know that this is not a problem. One of the more straightforward constructions of rationals define a rational number as an equivalence class of tuples of integers, corresponding to the numerator and the denominator. We often use one of these tuples to represent a rational number, e.g. 2/3 or 4/6. And since they are in the same equivalence class, either alone is sufficient to represent the same rational number. This is despite the fact that the tuple 2/3 and the tuple 4/6 are different mathematical objects.
As for the definition of 0.999..., it's not what you think it is. There is a specific definition for infinitely repeating digits that is different from representations without repeating digits.
That is, the positive real number that 0.x... represents for all digits x is defined as L, where L is the supremum of the set containing 0, 0.x, 0.xx, 0.xxx, 0.xxxx, ...
A supremum L in E of a set S subsetof E is defined as follows: L is the smallest number in E such that L is greater than or equal to all numbers in S. I shall not prove here that L is unique, but it is. With respect to the reals and the set containing 0, 0.x, 0.xx, 0.xxx, the supremum of that set in the reals is 1.
My knowledge of the construction of the reals comes from Chapter 1 of Water Rudin's Principles of Mathematical Analysis, ISBN 0-07-085613-3, which you may be interested to read, but beware that it's considered a difficult read for beginners.
I completely agree with you, and understand all that.
> L is the smallest number in E such that L is greater than or equal to all numbers in S.
And note that this doesn't imply that L is in S!
So here we have S = { 0, 0.x, 0.xx, 0.xxx, ... }. E is R, the set of reals. The smallest number in R greater than or equal to anything in S is 1. That doesn't mean 1 is in S. If 1 is not in S, then we're taking a notation based 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?
I think I was too hasty in replying to your post and misread and misunderstood the point you were trying to make. No, I don't think there's a difference.
The problem with this argument is it assumes you can get to infinity 9s by adding them one-by-one.
It's like adding sides to a regular polygon to get to a circle. No matter how many sides you add, you still have a polygon. You can only say that a circle is a polygon with infinity sides, but you'll never get there adding them one by one.
You cannot simultaneously believe that "you never get there" (i.e. an infinite number of nines is not constructible) and believe that the syntax 0.999... denotes an actual infinite string of 9's which has a straightforward value (that happens to be precisely 1).
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.
You might be interested in reading about transfinite induction, which is an interesting and legitimate mathematical proof technique. However your 'induction' does not succeed in doing this. Transfinite induction requires two induction steps, you've only provided one of them. Not that you would actually use transfinite induction for anything resembling this, that's what calculus and analysis is for.
For example your strategy would show that the sequence of positive integers 1, 2, 3... has a finite limit! After all, each one is finite so by induction the limit is finite too. Except of course no one would expect this induction to apply to the limit as well as to the individual steps.
I'm not surprised that something called "induction" doesn't succeed in achieving what "transfinite induction" can do, otherwise the latter couldn't exist as a separate technique with its own name. :)
> 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".
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.
Or a name is not an attribute and .999... and 1 name the same thing.
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.
This has so much more to do with the lack of easily monetizable applications of complex mathematics. I'm sure a significant number of engineers and STEM professionals feel (as I do) that they're deliberately eschewing those subjects not for a lack of interest, but rather as a response to market demand.
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.
> I'm sure a significant number of engineers and STEM professionals feel (as I do) that they're deliberately eschewing those subjects not for a lack of interest, but rather as a response to market demand
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.
I've always thought trigonometry was one of the more useful things I did learn in school. Comes in handy a lot with carpentry and building projects. It's kind of cool when you realize that most of the shortcuts and rules-of-thumb that the old timers and carpenters use all the time can be derived from trigonometry, even if they never formally learned it that way.
What would you study instead to remain valuable in the next two decades? I've gotten pretty deep into engineering: worked on games, mobile, web, FP, legacy code revival, a half dozen popular languages, automated testing, and people management. I'm looking for something new to study to add flexibility and "luck surface area" to my career. I was thinking ML would be another area of valuable study. Do you have any alterative suggestions?
I think you're already doing the right thing in continually learning. No one can tell you whether or not studying ML will prove to be valuable with any certainty, so it might be a good bet to pursue it if it interests you and you have the spare time and especially if you're desperate to pivot your career towards it (then you're left with no options). I can't offer a definitive decisionmaking framework, except to say that if you want to monetize something, make sure you can (fairly obvious, but not always what happens with people in our industry).
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.
> It makes more sense for an intelligent person to take the lower overhead and more achievable approach to becoming a value creator (e.g. full stack engineer with a strong focus on product development)
I would say that the surest way to make money for a mathematicaly-inclined person is to graduate in maths from a prestigious school and work in finance.
At least, that's how I feel when I look at alumni from my school. 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.
Finance is fairly broad. People in middle management whose job is basically to deal with paper work many times say they are "Working in Finance".
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.
I have no data either but one point is that the "ceiling" is higher. Meaning, a non-management programmer has a rough salary ceiling. Few programmers are paid $400k and it's not a very realistic goal.
My mathematics -> finance friends (that are very good) virtually have little to no ceiling and have already doubled my salary.
Like @wrong_variable said though, it depends what jobs and if they make it through (it's pretty competitive), so you're getting survivor bias from me.
Only few people in finance really "make it" - and it mostly consists of portfolio managers (quantitative or else). "Superstar economy" analogy discussed in this thread have very strong effect in finance.
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.
> To even get considered for the good quant position you need a good phd in applied mathematics or statistics.
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.
I was editing the post while you replied. I realised that my post wasn't conveying any useful point.
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 you define "really make it" as making millions every year, yes that's rare. But if it's making 300k+ per year, there are multitudes of math-types doing that, and not much luck is involved.
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.
I wish everyone on r/MachineLearning and those preparing for data science careers would read this comment and heed the advice of those of us who have actually spent time on data science teams and have experienced all of this first hand.
The market for STEM professionals is superstar economy similar to the market of comedians, athletes, musicians and artists. The payout in the profession follows Zipf's distribution.
I want everyone to read and re-read your bit about data science and machine learning. Many times. I think even people in the software industry underestimate both how accurate and how difficult it is to employ statistics to produce something truly meaningful.
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.
To most people machine learning and data science are magic. They either believe in magic or they don't.
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".
Well, as someone who took a solid class in machine-learning, has an understanding of why it works, and regards it far more as science than magic, "we" who know it damn well committed a sin and sold our souls for funding if "we" just told everyone else it was magic.
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!
You know I might actually take a pay cut for a more mathematical role over my current situation 'creating value'. I would love for my work to be driven more by logic and data than big personalities and office politics.
I think the author is both right and wrong. Maths skills is a litmus test that reflects the scientific education in general among the population.
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.
I think the author is mostly attacking a straw man.
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.
How would this same conjecture apply to History, Literature, Biology, Physics, etc etc?
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.
I had the same thought. Follow the author's conclusions, and you'll end up rejecting college as such on the basis of its apparent lack of "utility."
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?"
At my workplace, we have about 60 scientists and engineers. The author's observation is accurate, that most people never use math beyond Excel and 8th grade math. They also never use most of the theory that they learned in their science (including CS) and engineering educations.
The typical career arc is to get through college, then sit down at a CAD workstation, or programming terminal, and forget all of your math 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.
I do complex mathematics for fun sometimes, it always surprises me when I start talking about the fractal images I generate and a large percentage of other engineers are wowed at how, well, complex it must all be. Then start to glaze over a bit...
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.
Maybe the best argument for teaching people math is to show them how complexity can be tamed via patterns. As in, if you sit down and think logically, you can find solutions or come up with approaches to even the gnarliest-looking problems. And things that sound very simple (e.g. the Fermat conjecture, which can be understood by 5-th graders) can have really complicated solutions. It should teach people how to use their slow thinking rather than their fast thinking and that they absolutely have the capacity to do it. I've similarly never met anyone who had to do 20 accurately timed hops in a row daily, but I don't hear anyone want to remove jumprope and P.E. Still, there should absolutely be a class on "real-world applied mathematics" where you're taught the mathematical principles behind loans, mortgages, taxes, etc.
All mathematics is applied mathematics. Pure mathematics is just mathematics applied to mathematics.
This is problematic, because the way mathematics is currently taught only small number of students actually grok it and make deep connections that enable them to build up on what they previously learned and learn more. Others have the constant feeling of things getting progressively harder to understand and use.
I'm 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 a math or theory problem arises, they take it to the resident "math person."
When an obvious math or theory problem arises.
I'm sure they run across problems that more advanced math can solve, or stumble into problems due to not knowing more advanced math, yet their lack of knowledge prevents them from even understanding they confront a math problem. Sometimes the problem is not what they know they don't know, but what they don't know they don't know.
This happens constantly to people who think they don't need CS to program. Something that should have been a 2 hour implementation of an algorithm invented in the 60s turns into a 2 week exercise in frustration.
Worked in HW design for 16 years and I've never seen a design that was done by trial and error. Usually lots and lots of simulation and calculation up front followed by a DOE (design of experiment) to provide insight in areas which are difficult or impossible to simulate well. It's most certainly not trial and error.
Designing and running a simulation sounds like a trial. Observing the results of the simulation then making a correction, seems to imply a sort of error.
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.
> 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.
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.
At the level of individuals, this article is complete bullshit. There is a WORLD of difference between an average mathematician/physicist/engineer and an average English literature graduate solving the same problem with the same tools. Those disciplines teach and/or reinforce A) critical thinking (question assumptions, look for counter-examples), B) decomposing problems to smaller problems, C) pattern recognition. The average software engineer would be somewhere in the middle, he's probably as good as a mathematician at A) and B), but probably not C). Even amongst reasonably mathematically educated people (say physicists), you see a difference in C) depending on the depth of their math education. There is a difference between people who know eg fractions, and people who know eg fractions AND integrals, and people who know eg fractions AND integrals AND group theory, in how often these individuals look at a situation and go "I've seen something like this before". Acquiring some areas of more advanced maths isn't a quantitative increase like doing bigger sums. In terms of the patterns that you see around you, it's a qualitative jump. It's like being able to use your eyes AND having IR goggles: you will see some aspects of the world very differently.
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.
As a mathematician, I would like to point out that there are a lot of different areas of math, and higher math isn't just learning more calculus. Graph Theory and Stats, for example.
I have no idea what he's talking about with including Stats in up to 8th grade math. I've taken a few university classes on it, and I still don't feel like I have enough to be confident solving all but the simplest statistical problems.
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.
But Carmack didn't finish college. He's using what he has. This tweet is him defending his math ignorance in a particular case.
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.
I don't buy the sports analogy with which he argues that it is "self-serving nonsense" if people state that mathematics education trains your general problem solving skills. His argument that soccer players should only play soccer seems not to be anchored in reality: Of course professional soccer players spend a lot of time in the weights room or go running to enhance their general strength and stamina [1]! They do not only train their bodies by playing soccer...
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.
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)
Some of these things are straightforward, some not. If I study openings and know e5 is a good response to e4, I know my study did that. I am following an opening I memorized from a book.
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.
Allegedly (association) football coaches seventy years ago would make players train without the ball all week, on the grounds that they would be keener to actually play football come Saturday. Of course they ended up under-skilled. My point is that one should find the combination of training that gets best results.
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.
You're right. In particular, when we train for something, we tend not to train things in the proportion that they occur in during the game.
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.
Perhaps the arc of my career is different but I've seen the opposite. I've been in Finance jobs where people who don't understand more advanced probability can't figure out how to price things. And even people with advanced knowledge make mistakes.
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 saw someone joke on Twitter that their anxiety level lately is the first derivative of the graph on the 538 2016 election forecast. So to get that I guess I needed to be able to see that in my head briefly. I think I didn't pick up that skill until calculus which for me at least was 11th grade.
I think one non-obvious benefit of a good mathematics education is that you have little choice but to develop a tolerance for and understanding of being wrong. See Jeremy Kun's blog post [1] for more, but my own experience has been that in e.g. discussing different ways to solve a problem or prove something almost every person eventually has an "oh, no, I see, I'm wrong and you're right" moment. Not that every mathematician is necessarily a font of humility and grace, but I think math offers more regular and irrefutable demonstrations of your own fallibility than many other fields, and this is good.
In my experience, a surprising number of people with a humanities education simply don't believe in "wrongness", but merely differences of opinion. They regard truth as peculiar abstraction used by mathematicians and hard scientists, not a phenomenon that actually exists. It's hard for someone to admit to being wrong if they don't even believe in the concept.
You obviously don't hang out with many statisticians, who also don't tend to believe in truth or being right or wrong; to them, they are just making models that can be tested to reliably approximate some observation. There is no pretense at truth, it's just a model.
The entire goal of statistics is to build approximations which converge to truth in some manner or another. E.g., a Bayesian posterior ideally converges to a delta measure on truth as N -> infty. A frequentist point estimate converges to truth (with P = 1) as N -> infty.
The entire purpose of statistics is to quantify the difference between our beliefs and actual truth.
>A frequentist point estimate converges to truth (with P = 1) as N -> infty.
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.
No, it's not. The fact that observation is a limit on our apprehension of the real world means the "truth" is fundamentally inaccessible, and is a useless concept to the statistician. All you can hope to do is test your model against the observation, you'll never, ever get access to the truth.
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.
Irrational numbers are also "fundamentally inaccessible" in the exact same (infinite limit) sense that statistical convergence is. It's not a useless concept at all, it's actually the fundamental concept.
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.
Irrational numbers can be represented exactly; integrals allow us to perform exact calculation using them. In what comparable way does truth (e.g. the true process underlying gene expression) have a role in statistics? We neither measure the truth nor model it; it is absent.
Integrals themselves are, except in very special cases (piecewise functions with rational values), only definable as limits - specifically the limit of the Riemann sum. (You can also use measure theory, but measures themselves are only definable on sigma algebras, which in the non-finite case are also not explicitly constructable.)
In what comparable way does truth (e.g. the true process underlying gene expression) 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|
We can use the triangle inequality to show:
error <= |true model - best model in class X| + |best model in class X - actual approximated model|
Presumably you all have decided that |true - best| is adequately small via scientific investigation. Or maybe not, maybe your workplace just doesn't care, I don't really know.
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.
What I'm arguing is we have no idea what "true model" is, what we have is "presumed model" and "observation". In the example I gave, we can never know the true source of the data we have observed (biology), we can only test our observations against some constructed model. Biologists are using a model they know to be wrong because that is what all models are - we know them to be wrong, we just can't do otherwise, because the truth is not available to us.
I feel like I've made this same point about four times already, so if you aren't getting it, let's just stop here.
I had a philosophy professor who believed evolution was 'just a theory', and routinely dismissed it. I don't think she was religious either. Then again, I had another philosophy teacher who taught logic, which is the most pure use of right/wrong that I can think of.
evolution IS 'just a theory'. like relativity and other scientific theories. it is just a coherent framework to explain observations. today it is our best theory, tomorrow it may not be. you cannot say that evolution is Truth. Truth exists only in mathematics, logics etc.
Reminds me of something one of my Economics professors said once: An Economist is just a mediocre Engineer, an Engineer is a a real bad Physicist, a Physicist nothing but a bad Mathematician, and a Mathematician the lowest form of Philosopher.
Biologist Edward O. Wilson makes a case for a similar, though not identical view, in his Letters to a Young Scientist. 2nd essay is "Mathematics". Distilled:
* 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.
The OP is a special case of the old, big question of what to teach.
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.
As a Math/Science teacher who has to make decisions every day about who and what to teach, I agree that "with a broad education we are fishing for advantages in the future". I see a similarity with playing sport. When we are young, we play various sports. Some will make a career out of a sport they are good at. Some will continue to play occasionally just for enjoyment it brings. Most will probably benefit health-wise from the experience. Similarly with Math/Science education. It may become a career, an occasional interest, or simply a memory that gives some quantitative insight into what goes on around us.
Seems kind of ironic that the math professor writing this article uses conjecture as evidence for recommendations to what people should be learning instead of actual statistics. He didn't even need 8th grade math to make his argument for fewer people needing to learn higher mathematics (though perhaps he's correct).
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[ 7.4 ms ] story [ 291 ms ] threadIt 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.
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.
https://youtu.be/iG9CE55wbtY?t=8m40s
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.
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.
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...
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.
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.
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.
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.
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?
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
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.
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.
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.
[1]: https://en.wikipedia.org/wiki/Geometric_series#Formula
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.
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.
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).
You're spot-on about needing to understand there's a distinction between a number and its decimal representation, though.
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.
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.
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...
https://en.wikipedia.org/wiki/Infinitesimal
(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.)
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 :-)
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.
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.
[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.
or google: sample size calculator
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.
Most people just look at the number of subjects in a study, but not the p-values
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).
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.
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.
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".
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.
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".
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".
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".
It seems that you object to the commonly accepted meaning of "...".
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?
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.
> 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.
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.
(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.
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.
> 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".
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.
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.
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.
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.
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.
Hm, I would never have guessed that. Does anyone have any data on this?
The top 1% sure, but the average and median also?
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.
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.
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".
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.
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?
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".
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!
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.
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?"
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.
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.
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.
Something like betterexplained.com who tries to develop people's math intuition ?
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.
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 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.
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.
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.
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.
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...
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:
Some of these things are straightforward, some not. If I study openings and know e5 is a good response to e4, I know my study did that. I am following an opening I memorized from a book.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.
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.
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'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.
[1] https://medium.com/@jeremyjkun/habits-of-highly-mathematical...
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.
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.
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:
We can use the triangle inequality to show: Presumably you all have decided that |true - best| is adequately small via scientific investigation. Or maybe not, maybe your workplace just doesn't care, I don't really know.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.
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.
With others. Mathwise I am always right but others can't see it. So I have a deacartes moment with others
My spell check initially game me meth instead of math. Proud methamatician
* 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...
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...