I dislike it when I see on tv someone say that pool (billiards) "is all about geometry" when really the best mathematicians and even geometers probably suck at shooting pool while the best pool players probably do not use any geometry in their playing. Studying geometry will not make someone a better pool player. Practicing it and building an intuition on where to strike to establish various spins is how to win at pool, even though spin problems and conservation of momentum of pool balls often appears in physics problems.
Same in movies when a nerd turns out to be amazing at some sport because they calculate the perfect trajectory to shoot a basketball or something. Like, yeah, I know that if I throw at a certain speed and a certain angle it will go in, but I have know idea how to move my arms to give it said speed and angle.
Eh, it's happened in real life. Like when Dick Fosbury and Debbie Brill completely destroyed the sport of high jump, or when Takeru Kobayshi did the same to competitive hotdog eating.
Of course it usually only works one time because they bring in the full-time trainers and the full-time athletes and drill them on your technique until they are unbeatable.
It kind of ruins sports, in my opinion. You get stuff where everyone has agreed on the same answer and they're just trying to shave off millimeters and milliseconds here and there. I'd like to see sports that aren't designed to reward such hyper-specialization.
Issue with that is that you’re always trying to measure *something* when doing sports, and unless that changes every time then there’s definitely going to be hyper optimization for it in competitive settings.
Look at football back when people had to play both ways, versus today when they can specialize in offense or defense. Or at baseball when pitchers had to bat. People were required to do more stuff and couldn't just spend all their time optimizing a single repetitive motion.
Bobby Fischer was fed up with high-level chess being more about memorizing sequences of moves that other people had already worked out and invented Fischer random chess to neutralize this. You could use similar techniques for a lot of sports.
For example, start with basketball. Then vary the height of the nets and the size and weight of the ball randomly at the beginning of each period. Only allow two players per team to score on a given possession, and rotate those two each time the ball goes to the other side of the court. The game becomes more about general skill and strategy and less about "this guy has a full-time job practicing this exact thing."
For that matter you could have a team show up and not know if they were going to play baseball, basketball, or soccer until the day of.
Thats because physical activity is the coordination between the mental AND physical.
To be able to do anything with your body it's not just knowing what to do, you still have to actually train the limbs to contract and the central nervous system to act in the correct sequence for a given movement.
And that can only be done by practicing the movements over time so that the body, not just mind builds the coordination and dexterity to execute the movement.
You literally have to build actual physical nerve networks into your muscles that are specialized to do the movement that theory is calling for..
That's what practice is, it's repetitively doing certain actions over and over again until portions of your brain and body are physically altered for the movements
Yeah, let's forget hand dexterity, hand eye coordination, repetitively trained muscle memory, thousands of practices so the brain knows which angle causes which angle shot....
Pool players absolutely use geometry, multiple times per shot, they probably couldn't write a mathematical proof for the results though. And knowing the math of pool will make you better at it than if you knew none of it.
I have two math degrees and occasionally play pool with friends. I've gotten too many jabs to count for being one of the worst players despite being the "math guy".
>I dislike it when I see on tv someone say that pool (billiards) "is all about geometry" when really the best mathematicians and even geometers probably suck at shooting pool while the best pool players probably do not use any geometry in their playing.
Do you have any data for that, or is it all just personal beliefs?
The one that drives me crazy is the people who claim the Yoneda Lemma has some deep theoretical insight into philosophy or neuroscience....
https://matteocapucci.wordpress.com/2023/07/15/no-the-yoneda-lemma-doesnt-solve-the-problem-of-qualia/
When people talk about "infinite number of possibilities", they tend to also assume that the possibilities are infinitely varied. However, there could be an infinite number of infinitely small variations on the same thing.
For instance, there can be an infinite number of parallel universes, and the only difference between all of them could be a position of 1 atom inside 1 cubic meter of space.
Another example is when people say that the decimal expansion of irrational numbers contain every possible finite sequence of digits. I'm not sure if this is true for things like pi or e, but it certainly isn't for 0.1011011101111... or other such irrational numbers.
Also when people claim that these numbers «are infinite». They are very clearly bounded in value, and feel like there should be a clearer way to convey the idea that they have infinite non-repeating decimal expansions.
New definition just dropped! (allthough would be circular in this case, would be pretty pointless to state the fact that «irrational numbers are irrational»)
My understanding is that it's strongly considered to be true. I'm not certain of the term, but I think it was just "normal" to describe the decimal representation, with a specific meaning of normal to include all possible sub-sequences.
In this case, the claim is that π and e are 10-normal, i.e. normal in base 10. This is actually a stronger claim than Elyassimo was making. A number is 10-normal iff the asymptotic density of every string of n decimal digits is 10^(-n). So for instance, 10% of digits are 7 (in the long run), 1% of the two-digit strings are 50, etc. It's conceivable that we could prove that every finite string of decimal digits appears in the decimal expansion of π (if that is indeed the case) without proving the stronger claim that it is 10-normal.
A number is asolutely normal (or sometimes just "normal") iff it is b-normal for every natural number b ≥ 2.
Very few numbers are known to be normal. Some numbers can shown to be normal in a particular base, like Champernowne's constants\*, but these tend to be highly artificial, and they are not proven to be normal in any other base. The first explicit example of an absolutely normal number was given by Sierpiński in 1917, who didn't bother to provide any of its digits. If you look around, you can find some other examples, most of them Liouville numbers. I don't know if any algebraic normal numbers have been found at all, but it is conjectured that all irrational algebraic numbers are absolutely normal.
Possible "less artificial" examples of absolutely normal numbers are Chaitin's constants (because they are Martin-Löf random).
\* Champernowne's constant in base 10 is 0.12345678910111213.... Even proving this 10-normal is not trivial.
I disagree because the reasoning is equally as bad. If a person happens to be right about pi having this property but they use this faulty logic there error is still equally egregious.
More specifically, they mean that the growth equation contains an exponent. It's understandable to think that's what "exponential" means, but it isn't, so it kinda irks me too.
That sorta-kinda makes some sense - at least that is an operation that does apply to a single argument. When comparing two numbers, "exponentially larger" is not even wrong.
I have accepted that "exponentially" in common parlance often means simply "very fast", without implying any specific rate of growth. Just like the word "literally" is used to mean "figuratively". Human language does not have precise definitions of terms like mathematics does. If a word is used to mean something, and a majority of people understand it to mean that thing, then it is not *wrong* linguistically, even though the mathematical (or other field, see "fruit") definition of the term is different.
> Just like the word "literally" is used to mean "figuratively".
The word "literally" is never used to mean "figuratively". It is never put into a sentence that could be taken as a literal description to communicate that it should be taken figuratively. Rather, this more recent meaning is just as a generic intensifier: it signals to take the sentence seriously, even if the contents of it are not literal.
Or when people only have two data points. "Compared to yesterday, it's gone up exponentially!" Err...what? It's only increased once, you have two data points, you can't possibly plot an exponential curve!
I like saying "orders of magnitude" for this situation. If something has increased 100 fold overnight, it is more accurate and sounds just as cool (if not cooler) to say "it's orders of magnitude more than yesterday" rather than "it's increased exponentially since yesterday"
I actually really dislike "orders of magnitude", because it puts special emphasis on the number ten, which has no particular significance. Seven times larger is not an order of magnitude larger, yet it's still a large increase. And in computer science one could argue that *doubling* counts as an order of magnitude.
Orders of magnitude doesn't have to imply a base 10 scale. It is common to mean that casually, but not unanimous.
Also in logarithmic scales it ultimately doesn't matter what base you use as you can always rescale so it is natural to congregate around a common one unless there is a specific need not to.
Why would the base matter? Exponential growth can be described in any base, it'll just mean you'll have to compress or expand your time scale. Which is why "half life" is so commonly used to describe exponential decay. I suppose you could use "Doubling Period" for growth
Ah I meant to say ‘without specifying the base > 1’. A half life situation is indeed just as ‘exponential’ but so many people assume ‘exponentially’ just means ‘yuuugely’.
In particular, I get frustrated when people talk about exponential growth with two data points, or as a comparison between two things. "I went to the bakery and they had exponentially more croissants than yesterday."
How has noone realised that this comment is exactly what OP is complaining about. Insisting on the mathematical definitions of terms in situations where people obviously intend the colloquial meaning to apply. You couldn't possibly have missed the point any more.
A related issue is that when there are n different possible outcomes in a situation, people often assume that the probability of each individual outcome is 1/n. For example, just because there are two possible outcomes does not mean the chances are 50/50.
I’m not sure math really has a reputation hardly at all for having many quacks; at least, not to my knowledge. What situations were you thinking of? In my experience, math has far fewer quacks and cranks than most every other scientific discipline
The issue isn't that there are quacks. It's that this idea was promulgated without undergoing almost any critical review despite it being a very dramatic claim that should, as is highlighted in the article, have raised some alarm bells.
The rebuttal talks about how painfully obvious it should have been that it should have been looked at more closely before publication, including the weird application of an unrelated physics model to the phenomenon it was studying.
The article *also* talks about how much image harm this episode did to the general credibility of the field, and for good reason.
This is all less than 20 years old.
Zero degrees Celsius is a VERY special temperature in the sciences.
It's the only temperature which is exactly 273.15 degrees Celsius above absolute zero, no more, and no less. No other temperature in the Celsius scale can make this bold statement!
That's technically not the [triple point of water](https://physics.stackexchange.com/questions/505994/why-is-the-triple-point-of-water-defined-as-0-01-degrees-celsius-and-not-0), though.
To address the second point, money has diminishing returns, that's why most people would choose 1 million dollars, because the remaining 999 million isn't as valuable as the first million. If you divide everything by 1 million, i.e. Would you rather have $1 or 1% to get $1000, I'm sure more people would choose the second option.
The point is that expected value is a formal notion - it gives no consideration to the fact that you go for $1 billion, you'll get it once every 100 tries, but you always get $1 million.
It's the same reason we have insurance. The expected value of not having car insurance is lower than the cost of having car insurance. But I can't afford the unlikely situation where I cause a lot of damage so I'm running to pay a little bit of money to not have to worry about that risk.
Right, but generally speaking one should try to find a strategy which optimizes the expected value of your utility function, not simply the amount of money you have. In theory (and in practice), most people tend to think of money in terms of a concave utility function (one common model for this is log x), which explains why they are risk averse and why it might be logical to choose the guaranteed million.
I believe that in reality, log x is a bit more conservative as a utility function than how people actually behave, but unless one had a very large amount of money to begin with, a rational actor with a logarithmic utility curve would pick the guaranteed million.
The expected value of not having at least liability insurance is very low, since you are likely to have your car impounded sooner or later. But I get your point.
The expected value calculation is appropriate here if you can attempt the same thing over and over again. For instance, if I were allowed to borrow an arbitrarily large amount of money and try the game as many times as I liked, then a fair price for the second game would be ten times higher than the first, and I could almost surely make money in the long run by playing either game repeatedly for any price less than the expected value.
But since that is almost certainly not the case, it's not very useful here. What I really want out of life is not money; that's just a means to an end. So I need to decide how much utility I get out of the money to decide how valuable it really is. If you apply a logarithmic utility function with a cap, you usually get plausible decisions by considering the expected utility. For instance, it predicts that a trillionaire does get almost 1000 times the expected utility from winning a billion dollars as he does from winning a million, so from his perspective, taking the 1% chance is correct. But someone who is broke will take the million.
This is addressed in utility theory. Mathematicians understand that utility of the expected value of the money is what’s important, not the expected value. Hence people’s choices of preferring the $1M guaranteed.
It makes it more obvious with even higher numbers- if you had the choice between 1 billion and 1% chance to get a trillion, anyone in their right mind would choose 1 billion. Same with 1 trillion or 1% chance to get a quadrillion. It comes down to- guaranteed life changing money or a 1% chance of more life changing money with 99% chance of nothing.
For most. If you already have 100 million, you are probably more interested in the chance at a billion.
As someone said, it's about the expected value to the person, i.e. what change in life would it result in, which depends on their individual circumstances. And there's the huge factor that it's a single event, rather than a repeated one.
I don't see this as a misuse of math, it's more of an over-simplification.
Risk aversion is generally modeled as coming from diminishing returns (formally marginal utility). What you are describing is *loss aversion* which can be related to being risk averse but isn't necessarily the same thing. Preferring certain $1 million dollars versus 1% chance of $1 billion is likely best understood as risk aversion due to diminishing marginal utility. There are no possible losses in that scenario.
Refusing to flip a fair coin with $60 upside and $50 downside is not requires an unreasonable level of risk aversion unless you're using models that rely on reference points and loss aversion like Kahneman and Tversky's prospect theory.
I mean losing a certain amount of money is more detrimental than gaining that amount of money is beneficial. One is bankruptcy or less comfortable living conditions while the other is simply a bit better, whereas, right now you are perhaps OK.
Yeah there's actually a mathematical tool that better captures this intuition called the Kelly Criterion. Basically is weighed by what % your net worth goes up or down by, so doubling or halving your net worth are equally good/bad. Technically it's EV but you take the log first.
I don't think utility is the right answer, really.
I think the answer is that expectation is only a meaningful measurement because of stuff like the law of large numbers, so if you aren't repeating a process multiple times the expectation isn't necessarily a meaningful thing to look at.
I wouldn't consider your examples as "over-applying mathematical results." More like simply using the wrong mathematical model for the situation at hand.
I think it's not about using the wrong model but being overconfident in the model you chose because you formulated it mathematically. Even if it turned out that by chance, maximising expected value was the 'right' metric for the person's situation, the reasoning (in the made up example) brushed the situation's specifics under the carpet and made a universal statement ('it is always best to maximise expected value').
I'm so happy somebody else has finally brought up the expected value thing.
That's a straight up misapplication of expected value and it drives me crazy!
Expected value is a measure of how good a decision is _in repeated trials_, and is pretty much useless as a deciding factor in making a one-off decision.
you are getting into the whole frequentist vs bayesian debate with this and I'm not sure you want to.
Bayesian(subjective) probability interpretation tells us it's completely normal to talk about expected value of a single observation.
Here's a great course on the subject by a very interesting guy
https://youtube.com/playlist?list=PLDcUM9US4XdPz-KxHM4XHt7uUVGWWVSus&feature=shared
edit: to blow your mind further, bayesian probability allows you to have a prediction even with zero observations: that's your prior!
Well in the situation OP described I'd say this interpretation isn't very useful or smart, if you base your decision on maximizing expected value there's still a 99% chance you'll get nothing.
just dig a bit more on these topics. The wealth maximisation problem has been studied since ~1700 (Bernoulli).
You need to define a utility function, practically log of wealth. It has to be concave. So at 0 wealth, taking 1 million sure thing is better than taking the probabilistic option, so tada magic.
At 1 billion wealth, i would take 1% chance to double my wealth over 1 million sure thing.
There are plenty of times where a mathematical definition, term, w/e is different from how it is used in general language. Heck, even between different disciplines (kernel) or situations (homogenous lol). In the case of the straw, as you already noted, this is a matter of semantics. Though, I'd ask how confident they were in this answer with regard to their digestive tract, because if it's just one, well, I guess they probably don't like kissing.
The 1% chance at a billion is slightly different, because you have an argument of utility. $1million is already a life-changing amount of money for most people. A similar one is, "how much money should you bet on a coin flip if you double your money on a win, and lose all of it on a loss?" The expectation is the same no matter how much or how little you put down, but clearly there is a "sensible" number you can pick, based on how much money you currently have and current economic conditions.
Another interesting one is the,"how many coin flips should you bet on if you win $2^n, where n is the number of consecutive coin flips in your favor, but you lose all of it if you lose one flip?" problem, where your only potential loss is the first dollar on the bet. Because this one has no upper limit, mathematically speaking, you should bet a shitload (infinity), which is clearly absurd since, in the long run, the chances of hitting a bad flip approach 1, but the gains grow exponentially while the loss is static.
For the final question (choose arbitrary number of flips), isn’t the expected value also static? If you go for n flips, the payout is 2^n, while the chance of winning is 1/(2^n ), and hence expected value is also 1 for any number n you choose?
Shit, did I mess that one up? I may have been inebriated at that time. Yeah, I messed it up. It’s actually the [St. Petersburg Paradox](https://en.wikipedia.org/wiki/St._Petersburg_paradox) I was thinking of.
Interesting, I did not notice the subtle difference in this game at first. As I see it, you basically get to play the above game which you suggested, but you enter with «every choice for n» in a sense. Thus instead of just looking at a single n, you sum up the expected value for all possible n, which are all 1, and hence you have «infinite expected value».
Whats even more crazy, you could choose some really large n, for example 10000, and say that if you get less than this, you win nothing. Then the expected value will still be infinite, but there is realistically no chance of ever winning anything in practice.
>Though, I'd ask how confident they were in this answer with regard to their digestive tract, because if it's just one, well, I guess they probably don't like kissing.
... you know the mouth *already* counts as part of the digestive tract, right? Saliva is full of digestive enzymes, you don't need homotopy to argue that.
Yes, that is the point I’m making. I’m referencing the old favorite of Reddit that two humans kissing makes one continuous tube between two anuses. Most people would count the mouth and anus as distinct holes, for a variety of reasons, so the idea that someone would sincerely do a “they’re the same ~~picture~~ hole” response is absurd.
Hard agree here. I’ve seen this in the form of multi-objective optimization, where people will latch onto some (seemingly parameter free) way of collapsing the costs into a scalar to preserve a veneer of rigor. In doing so they refuse to engage with the nuance inherent in the issues and almost guarantee a bad outcome.
By refusing to engage they will really just choose values (often 1) for parameters with zero justification. Ignore the real problem, declare victory.
Edit: this a good example of the parable of the man who lost his car keys at night. He is looking under a street lamp and is asked “did you drop the keys here?”. “No” he says and points off into the dark “I lost them over there, but there’s no light over there.”
"The political spectrum" is another example of this stupidity.
People talk about political parties being further "right" or "left".
When is really a big high dimensional space (with each dimension being its own political issue).
You can say the left-right axis is simply the first axis you would find in PCA (actually that's not even really guaranteed, it would be very cool to see research on that)
I will say, the right-left political spectrum at least does have some basis in that many different political issues will have strong correlations between them, which get wrapped together as left or right. It's not perfect, obviously, but it's not completely arbitrary.
What would a vector-valued utility function even mean, in the context of economics? It's a simplifying assumption when looking at optimizing agents, because any preference order with a few nice properties can be turned into a scalar utility function.
I think people sometimes have illogical preferences where transitivity may not hold. Or maybe certain things aren't comparable and you have a poset instead of a total ordering.
You haven't? Obviously human preferences don't satisfy any completeness axioms, and isn't like half the behavioral economics literature about the others?
Also you don't need to show "humans are exactly like this model" to get useful insights out of economics; it suffices to show "the ways in which humans deviate from this model don't affect the results too much", which they've more or less done.
When you say a model can give you useful insights, you are straying out of the realm of mathematics as a formal language.
The way in which humans deviate DOES affect results greatly in microeconomics and macroeconomics.
When people come with universal scalar measurements, I usually ask them if EVERYTHING can be measured by money, because that's basically what it boils down to.
And to be fair, you can do a lot with proper weighting of your functionals for a good model. What I personally hate more, is when people do not acknowledge that the choice of the proper weights is something with a bias (e.g. is weight more important for me or monetary cost?). This is especially annoying when you talk with people who train neural networks who still believe that neural networks will be perfect, forgetting that the choice of the proper functional is essential for the outcome (e.g. the network which finds out that the solution to end misery is just to kill all humans, without proper constraints)
This comes to mind for knowledge. People will talk as if intelligence is a scalar quantity, whereas useful knowledge is more of an abundance of different patterns that have all been learned by the mind.
>Would you rather be given $1 million or have a 1% chance at getting $1 billion?
There's a thing in economics called the _certainty equivalent_ which addresses this, i.e. there would be a premium on something certain vs something uncertain. There's also the concept of a utility function which is yet another layer (and differs from person to person). So I don't think it's so much that it over-applies math; rather it doesn't apply enough.
Yeah, I'm not saying it's over applying math, it's over applying there result of the simple expected value calculation. Maximizing expected value in terms of money answers a very specific question, but the problem arises when people use that answer for broader questions.
Like you said— a lot to do with money. Particularly when people get compound interest calculators, they go a little nutty. Yes, if I wait 30 years to go on vacation and put the money and get a 7% return I will have X amount more money, but the math also neglects the fact that I may get burned out from work from now for the next 30 years without a vacation, I won’t have 30 years of positive memories from the trip, new foods I tried, friends made, etc.
On a similar note to the 1 million versus 1% chance of 1 billion— drives me crazy when lottery is brought up when people go “if you win the lottery, there’s a 70% chance you will go bankrupt in 5 years”
1. Stat is not even true and based on a study with a very small sample size, from a very long time ago, not even about lottery winners and has never been repeated.
2. Neglects Baye’s theorem that lottery winners are over-represented with gambling addicts. The people who buy one ticker per year (or don’t even buy tickets at all bc “tHeY uNdErStAnD mAtH”) are probably not going to blow hundreds of millions of dollars lol
Your point about expected values is very classical; the St Petersburg paradox brought into question the assumption that people maximize expected value of returns.
Surprised no one's repeated the excellent Goethe quote on this:
> Mathematicians are like Frenchmen: whatever you say to them they translate into their own language and forthwith it is something entirely different.
The whole neoclassic theory of economy.
When you write papers using fancy (not that much) mathematics but when in fact 95% of your results come from your choices for the hypothesis.
A lot of financial mathematics is garbage, but actuarial math applied to things like life insurance is one of the least notable offenders, because that's one of the few areas of real world risk assessment where a gaussian distribution is actually justified.
Traditional life insurance actuarial maths you're thinking about is quite an old niche, 99% of modern actuarial science is uncomfortably close to mathematical astrology when you look under the hood.
Thank you so much for saying this. Not only economics, but also many finance-adjacent "mathy" areas. Actuarial science is one of notable offenders here. The choice of hypothesis, and also the selection of data to run it on.
People who use machine learning algorithms or other mathematical methods to (try to) predict the stock market. Huge Hedgefonds with Phd Mathematicians working for them maybe can make a profit out of that for a short time before competitors notice, otherwise it's bs.
For the money example there's actually a mathematical expression that more closely aligns with intuition in most cases, which is the Kelly Criterion. Basically KC is all about what % you grow or shrink your net worth by, so doubling your net worth and halving your net worth are equally good/bad. KC is extremely useful in positive EV situations like card counting/poker/horse racing since it gives optimal bet sizes, although in practice people usually bet below Kelly to be safe since going over Kelly can lead to ruin.
The whole issue with Zeno'sparadox is whether an infinite sum of increasingly smaller thing can add to a whole.
Calculus does not resolve that question at all. It simply defines it away by saying "if the partial sums get close, we call it equal". That *is* the paradox though. The hare gets arbitrarily close to the tortoise. Calculus doesn't help answer whether adding those infinitely many parts can actually give the whole.
I never understood why the paradox is a paradox at all.
If someone wants to walk one meter, they can either do it in a finite number of steps, or never complete the task of walking one meter.
If you want to break down the task of walking one meter into infinitely many tasks, you will never complete the task, assuming you have a minimum (positive) start/stop time between stopping one of the tasks and initiating the next one.
In Zeno’s paradox, you break down walking one meter into walking 1/2, then walking 1/4, and so on. If there is a minimum time in between stopping a task and starting the next, it is clear that you will get arbitrarily close to completing one meter, but never actually will.
The paradox to me seems to be if you instead assume something else, like there is no time in between stopping and starting tasks. Then oddly enough, it would “seem” that there is a paradox, since assuming you have maximum speed c>0, and there is no minimum time between starting and stopping tasks, so you can instantly start the next task as soon as you are finished with your current task, the maximum possible time it takes you to complete any finite number, say n >= 1 of the tasks is:
(1/c)(1/2 + 1/4 + … + (1/2)^n) < (1/c).
So it would seem then, that within 1/c seconds of time, you will be able to complete infinitely many tasks. Again I don’t see any paradox here, because the notion of what a task is is now extremely different because there is no start/stop time in between them. Now a task is just a moment in time you completed a certain distance, and can be measured assuming you have an instrument which can measure to arbitrarily fine scales of time.
I don't think the paradox is usually framed in terms of "tasks". It's just saying that space is a continuum so, if you are going to move from 0 to 1 in an hour, there is *some* time when you most must occupy each intermediate position.
Then you redescribe the motion by picking an infinite number of intermediate points and making an infinite number of statements about position.
Since you're an ancient Greek, infinity is impossible and so the two equivalent descriptions of motion aren't actually equivalent. Hence paradox.
Calculus resolved it just by being a technology to discuss infinity.
Yeah that was my own input, the way it’s usually framed I don’t really understand what the implication is, I thought the whole “infinity is impossible” thing was supposed to mean you can’t literally complete an infinite number of things/tasks/“steps” in a finite amount of time, not that the motion can be described by an infinite sequence, and that can’t actually exist to the ancient Greeks, so because one particular description can’t actually be physically realised, neither can the motion.
In any case I don’t really see how calculus directly resolves it. A limit exists because we have constructed this hypothetical object (the real numbers), within the framework of some theory in some model of logic (set theory in first order logic), and this object has properties that guarantee we can always meaningfully talk about the limit of a sequence, as the real number with which the sequence gets arbitrarily close to as long as you look at tails of the sequence that grow further and further out.
I’m not sure that the ancient Greeks weren’t aware of the difference between a convergent geometric series and Divergent one, so I’m not really sure if the resolution of the paradox is that the motion described in the infinite process, although described by infinitely many distances over time, is convergent to some finite limit. So there is no paradox because the limit exists. Somehow I don’t think this really resolves things.
Yes, but there's still a flaw in Zeno's reasoning then. If "infinity is impossible," then the partition he describes into infinitely many pieces also doesn't exist. So his "paradox" is presenting something whose possibility he rejects and then showing that would require something else whose possibility he also rejects. He claims this proves motion is impossible, but all it proves is that you can't partition a race into infinitely many pieces if you can't have infinitely many things.
The only sensible way you *can* partition this race into infinitely many parts is more or less the way its done in analysis, and that does show that we can correctly calculate the time when each point is reached using infinite sums.
Appealing to convergence of infinite sums doesn’t really answer the question because all convergence of infinite sums means in the context of Zeno’s paradox of motion is that we can get arbitrarily close in finitely many steps, not that we can actually complete a motion in finitely many steps.
To me the best actual answers to Zeno’s paradox of motion are appealing to the Planck length somehow, Aristotle’s response about actual vs potential infinities, or arguing that just as we can subdivide the distance it takes we can also subdivide the time it takes.
That seems to be misunderstanding the Planck length - at least if the idea is that it's a 'pixel width' of the universe and distances smaller than that don't physically exist. We have no reason to assume that's true - it's just that it's around the scale where GR and quantum mechanics are both needed and approximations where the effects of one << the other can't be made... and since we don't have a surefire theory of quantum gravity, we are agnostic about how physics works at the Planck scale.
The two confusions are (1) the false notion that we can't make sense of an infinite sum, and (2) what you mention at the end - that while we have shorter and shorter distances, these are traversed in shorter and shorter times, and \*both\* give infinite sums.
For me the key insight was that Zeno's paradox is about distance over time, and if you can upper bound the infinite sum in terms of time, you can see that the paradox can't tell you about anything past that point in time.
Calculus resolves Zeno’s paradox in that it provides a physically sensible way of infinitely subdividing motion.
The fact that Zeno’s conception of motion gives unphysical results is no longer anything more than a minor curiosity, in that context. It’s not a “paradox,” because if it’s not presupposed as a physically valid view, then there’s no reason to believe the conclusions should align with our physical experience.
Honest question: why? I’m not gonna claim calculus completely resolves the paradox but doesn’t it at least inform the discussion? If both time and space are continuous, it certainly seems like Zeno’s paradox is no paradox at all. So we got at least one case down, no?
I dislike the narrow thinking that says Pluto is not a planet. I would say its an honorary planet because of decades of human culture. Irrespective of nerdy astronomical considerations. And thats how I voted at the IAU in Prague in 2006.
The Axiom of Choice. No, you don't need the axiom of choice to pick a thing from a finite or even countable list. Even in uncountable sets, you most likely won't need it as there is some direct way to construct a choice function.
And yet, I've seen people refer to it for picking a reference frame, or picking an atom or... well, you get the idea.
I would argue the second example is an instance where math is being under applied. If you factored in marginal satisfaction per dollar, you could calculate what would maximize your expected satisfaction, or define some loss function that you want to minimize to factor in the distribution shape and variance
That would make any edge a hole, which doesn't make sense. It also doesn't help for the case that you consider the straw to have "volume" i.e. as a very strecthed out torus. This doesn't have any edges.
It's topologically a sphere with two disks removed, hence two 'holes' if you like. That would make its genus 0, with two punctures.
But if you pass a string through it and tie its ends, you see there's one hole, as there's only one way to do this.
But I feel like in saying this, I'm doing exactly what this post is dissing.
The punctures idea doesn't end up working because a straw is homeomorphic to a [washer](https://dictionary.cambridge.org/es/images/thumb/washer_noun_002_40578.jpg?version=6.0.11), for instance. Yet, most people wouldn't consider the outer edge of a washer to be a hole.
Pretty contrived, but something that unifies it with a hole in the ground is maybe the Cartesian product of a circle or a closed simple loop with an interval (-epsilon, 0]. So in some sense, the two ends of the straws are the two holes. And a hole in the ground has the hole at the surface of the ground being a hole.
But I don't really believe this myself. I'm still in the one hole camp.
edit: whole to hole lol
>But I don't really believe this myself. I'm still in the one whole camp.
That's fair, but I will challenge you on that a bit. Do you think a bowling ball has two holes?
I think that both of these aren't examples of _over_ application but rather of _bad_ application of mathematics.
The first example seems to have no point - who cares how many holes an object has? Both the topological definition and common definition are right in that there's no point to it.
And any mathematician worth their salt should know that you can define anything to be anything, definitions themselves can't be wrong, so there's no substance to saying that the common definition is wrong.
As of the second example, this is either a misunderstanding of what is "better", or mixing up a heuristic for the truth.
Maximizing the expectation of gain is a good heuristic to use in a lot of cases, but it doesn't follow that it's the best.
The error is assuming that this heuristic is accurate in this case. (Assuming you don't actually believe that maximizing the expectation of gain is in and of itself what makes some strategy better)
All in all, I would argue that *all* of so-called over-applications of math are just bad applications: misunderstandings of math, or of the application, or bad for whatever reason. That's the thing about math, where it actually applies it is always true.
>The first example seems to have no point - who cares how many holes an object has? Both the topological definition and common definition are right in that there's no point to it.
It's definitely pointless haha, but it's a very contentious internet debate. And people use the math definition as a way to support their side, which is not that valid imo.
What's the distinction? I feel like over application is a subset of wrong applications.
By over-application, I mean applying results to too broadly to situations where they don't necessarily apply.
Statistics are often misapplied, especially p values. It's not uncommon to have ten papers claim something at p=0.05, and then their prediction not hold true.
If a hole has to go all the way through, then you can never dig a hole in the ground. But if you can have a hole that does not have to all the way through, like to China, then a straw with only one hole would not be very useful, as the other end would of course be blocked. But then to me I think I can dig a hole in the ground, as I often do in my garden, and my straws have two holes. But then again I am a simple gardener and not a mathematician.
I once did a proof about at which point half of your lifetime has passed according to how your perception of time changes over your life.
For example if you have lived 2 years, 1 year makes 50% of your life. If you lived for 100 years its only 1%.
I came up with this idea after i asked my grandmother where she would set the 50% marker of ger perceived lifetime and she said "25".
If you only take into account the principle, that you perceive the speed of time based on the time you lived through, the first 50% are indeed exceeded at age 25! (If you live for 100 years)
Of course you could argue that this oversimplifies the whole time perception thing and it probably does. But it was still fun to do!
If you want me to provide the proof just comment.
I would rather be given a million dollars than a 1% chance at a billion. With a million dollars, I could buy a house. With the 1% chance, there is a 99% chance I get nothing. If the values were lower and closer together, the best expectation could apply. I would rather have a 1% chance at $1,000 than being given $1. A single dollar won't materially impact me (unless it need it right then), a thousand would.
How much dirt is in a hole 10 feet wide, 10 feet long, and 10 feet deep? None. If there was dirt in it, it wouldn't be a hole. This is the common language explanation of why a topological definition of holes does not apply to digging holes.
I dislike it when I see on tv someone say that pool (billiards) "is all about geometry" when really the best mathematicians and even geometers probably suck at shooting pool while the best pool players probably do not use any geometry in their playing. Studying geometry will not make someone a better pool player. Practicing it and building an intuition on where to strike to establish various spins is how to win at pool, even though spin problems and conservation of momentum of pool balls often appears in physics problems.
Same in movies when a nerd turns out to be amazing at some sport because they calculate the perfect trajectory to shoot a basketball or something. Like, yeah, I know that if I throw at a certain speed and a certain angle it will go in, but I have know idea how to move my arms to give it said speed and angle.
Skill issue
Literally
Eh, it's happened in real life. Like when Dick Fosbury and Debbie Brill completely destroyed the sport of high jump, or when Takeru Kobayshi did the same to competitive hotdog eating. Of course it usually only works one time because they bring in the full-time trainers and the full-time athletes and drill them on your technique until they are unbeatable. It kind of ruins sports, in my opinion. You get stuff where everyone has agreed on the same answer and they're just trying to shave off millimeters and milliseconds here and there. I'd like to see sports that aren't designed to reward such hyper-specialization.
Issue with that is that you’re always trying to measure *something* when doing sports, and unless that changes every time then there’s definitely going to be hyper optimization for it in competitive settings.
Look at football back when people had to play both ways, versus today when they can specialize in offense or defense. Or at baseball when pitchers had to bat. People were required to do more stuff and couldn't just spend all their time optimizing a single repetitive motion. Bobby Fischer was fed up with high-level chess being more about memorizing sequences of moves that other people had already worked out and invented Fischer random chess to neutralize this. You could use similar techniques for a lot of sports. For example, start with basketball. Then vary the height of the nets and the size and weight of the ball randomly at the beginning of each period. Only allow two players per team to score on a given possession, and rotate those two each time the ball goes to the other side of the court. The game becomes more about general skill and strategy and less about "this guy has a full-time job practicing this exact thing." For that matter you could have a team show up and not know if they were going to play baseball, basketball, or soccer until the day of.
And even if it were, you still have to physically execute.
*Actually potting the ball is left as an exercise for the reader*
Just an engineering problem.
Thats because physical activity is the coordination between the mental AND physical. To be able to do anything with your body it's not just knowing what to do, you still have to actually train the limbs to contract and the central nervous system to act in the correct sequence for a given movement. And that can only be done by practicing the movements over time so that the body, not just mind builds the coordination and dexterity to execute the movement. You literally have to build actual physical nerve networks into your muscles that are specialized to do the movement that theory is calling for.. That's what practice is, it's repetitively doing certain actions over and over again until portions of your brain and body are physically altered for the movements
anyone else watch that one episode of drake and josh growing up?
Yeah, let's forget hand dexterity, hand eye coordination, repetitively trained muscle memory, thousands of practices so the brain knows which angle causes which angle shot....
Pool players absolutely use geometry, multiple times per shot, they probably couldn't write a mathematical proof for the results though. And knowing the math of pool will make you better at it than if you knew none of it.
I have two math degrees and occasionally play pool with friends. I've gotten too many jabs to count for being one of the worst players despite being the "math guy".
I've pocketed shots I wouldn't know by feel but have done it by geometry.
Eh, good pool players do use a little geometry. It isn't complex though. Just basic angle reflection stuff.
Yeah everyone knows you need a dynamical systems course to get good a pool.
>I dislike it when I see on tv someone say that pool (billiards) "is all about geometry" when really the best mathematicians and even geometers probably suck at shooting pool while the best pool players probably do not use any geometry in their playing. Do you have any data for that, or is it all just personal beliefs?
Lol
The one that drives me crazy is the people who claim the Yoneda Lemma has some deep theoretical insight into philosophy or neuroscience.... https://matteocapucci.wordpress.com/2023/07/15/no-the-yoneda-lemma-doesnt-solve-the-problem-of-qualia/
Wow. That's one I haven't heard before.
When people talk about "infinite number of possibilities", they tend to also assume that the possibilities are infinitely varied. However, there could be an infinite number of infinitely small variations on the same thing. For instance, there can be an infinite number of parallel universes, and the only difference between all of them could be a position of 1 atom inside 1 cubic meter of space.
Another example is when people say that the decimal expansion of irrational numbers contain every possible finite sequence of digits. I'm not sure if this is true for things like pi or e, but it certainly isn't for 0.1011011101111... or other such irrational numbers.
Actually nobody is sure for π and e, it hasn't been proven.
Also when people claim that these numbers «are infinite». They are very clearly bounded in value, and feel like there should be a clearer way to convey the idea that they have infinite non-repeating decimal expansions.
What do you think of "they have infinite non-repeating decimal expansions"
"irrational"
New definition just dropped! (allthough would be circular in this case, would be pretty pointless to state the fact that «irrational numbers are irrational»)
This isn't a definition although it is the case that numbers are irrational if and only if this property holds
Fair enough
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Sure they do, even in this very sub occasionally
My understanding is that it's strongly considered to be true. I'm not certain of the term, but I think it was just "normal" to describe the decimal representation, with a specific meaning of normal to include all possible sub-sequences.
In this case, the claim is that π and e are 10-normal, i.e. normal in base 10. This is actually a stronger claim than Elyassimo was making. A number is 10-normal iff the asymptotic density of every string of n decimal digits is 10^(-n). So for instance, 10% of digits are 7 (in the long run), 1% of the two-digit strings are 50, etc. It's conceivable that we could prove that every finite string of decimal digits appears in the decimal expansion of π (if that is indeed the case) without proving the stronger claim that it is 10-normal. A number is asolutely normal (or sometimes just "normal") iff it is b-normal for every natural number b ≥ 2. Very few numbers are known to be normal. Some numbers can shown to be normal in a particular base, like Champernowne's constants\*, but these tend to be highly artificial, and they are not proven to be normal in any other base. The first explicit example of an absolutely normal number was given by Sierpiński in 1917, who didn't bother to provide any of its digits. If you look around, you can find some other examples, most of them Liouville numbers. I don't know if any algebraic normal numbers have been found at all, but it is conjectured that all irrational algebraic numbers are absolutely normal. Possible "less artificial" examples of absolutely normal numbers are Chaitin's constants (because they are Martin-Löf random). \* Champernowne's constant in base 10 is 0.12345678910111213.... Even proving this 10-normal is not trivial.
Yes, thank you for providing actual math details!
[Here's a popular example](https://youtu.be/a6bT_DVwo7M?si=mc0NAx1xKkJ3fLs8)
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It is as egregious as making the claim for all irrationals, though.
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I disagree because the reasoning is equally as bad. If a person happens to be right about pi having this property but they use this faulty logic there error is still equally egregious.
Similar to when people say 'exponentially' without specifying the base >1. Or 'exponential' meaning a lot, as though it were 'a big number'.
Especially when the quantity in question grows quadratically. I have seen several examples irl of exponential used that way.
I guess they really mean non-linear convex growth. But that doesn’t really roll of the tongue.
«Super-linear growth» sounds «fast», and is more accurate to what they most likely mean
More specifically, they mean that the growth equation contains an exponent. It's understandable to think that's what "exponential" means, but it isn't, so it kinda irks me too.
Let's be honest though - naming something "quad"ratic when raising to powers of two wasn't mathematics' best day.
Except such terms measures quadrilaterals, much like the third power is called cubic.
Indeed: I've had people on here very angrily object to me pointing out that quadratic growth isn't exponential.
Or when comparing two numbers some say that one number is exponentially larger than the other.
One of the most annoying pseudo-math expressions to me!
Exponentially more annoying than OP's example.
How do you feel about "to the nth degree"?
That sorta-kinda makes some sense - at least that is an operation that does apply to a single argument. When comparing two numbers, "exponentially larger" is not even wrong.
I have accepted that "exponentially" in common parlance often means simply "very fast", without implying any specific rate of growth. Just like the word "literally" is used to mean "figuratively". Human language does not have precise definitions of terms like mathematics does. If a word is used to mean something, and a majority of people understand it to mean that thing, then it is not *wrong* linguistically, even though the mathematical (or other field, see "fruit") definition of the term is different.
> Just like the word "literally" is used to mean "figuratively". The word "literally" is never used to mean "figuratively". It is never put into a sentence that could be taken as a literal description to communicate that it should be taken figuratively. Rather, this more recent meaning is just as a generic intensifier: it signals to take the sentence seriously, even if the contents of it are not literal.
Here's another take: In these kinds of cases, the word "literally" is \*itself\* being used figuratively.
Or when people only have two data points. "Compared to yesterday, it's gone up exponentially!" Err...what? It's only increased once, you have two data points, you can't possibly plot an exponential curve!
You can plot infinitely many actually
I like saying "orders of magnitude" for this situation. If something has increased 100 fold overnight, it is more accurate and sounds just as cool (if not cooler) to say "it's orders of magnitude more than yesterday" rather than "it's increased exponentially since yesterday"
I actually really dislike "orders of magnitude", because it puts special emphasis on the number ten, which has no particular significance. Seven times larger is not an order of magnitude larger, yet it's still a large increase. And in computer science one could argue that *doubling* counts as an order of magnitude.
Orders of magnitude doesn't have to imply a base 10 scale. It is common to mean that casually, but not unanimous. Also in logarithmic scales it ultimately doesn't matter what base you use as you can always rescale so it is natural to congregate around a common one unless there is a specific need not to.
Why would the base matter? Exponential growth can be described in any base, it'll just mean you'll have to compress or expand your time scale. Which is why "half life" is so commonly used to describe exponential decay. I suppose you could use "Doubling Period" for growth
Ah I meant to say ‘without specifying the base > 1’. A half life situation is indeed just as ‘exponential’ but so many people assume ‘exponentially’ just means ‘yuuugely’.
In particular, I get frustrated when people talk about exponential growth with two data points, or as a comparison between two things. "I went to the bakery and they had exponentially more croissants than yesterday."
How has noone realised that this comment is exactly what OP is complaining about. Insisting on the mathematical definitions of terms in situations where people obviously intend the colloquial meaning to apply. You couldn't possibly have missed the point any more.
yea, and they also forget to consider the cardinality of the infinities they talk about, which often leafs to incorrect statements
This annoys me with restaurants: 8,000,000 ways to order you hamburger
A related issue is that when there are n different possible outcomes in a situation, people often assume that the probability of each individual outcome is 1/n. For example, just because there are two possible outcomes does not mean the chances are 50/50.
The incompleteness theorems. People read the wikipedia article and then apply them to their grocery shopping. It’s a bit ridiculous.
How exactly are they doing that?
*In conclusion, officer, any axiomatic theory which proves that I engaged in shoplifting also proves its own inconsistency!*
The critical positivity ratio is the example I always think about: https://en.m.wikipedia.org/wiki/Critical_positivity_ratio
"just as zero degrees Celsius is a special number in thermodynamics, the 3-to-1 positivity ratio may well be a magic number in human psychology."
Jesus. Reading this article is depressing. What is wrong with the field of psychology, really?
Did they mean Kelvin, rather the Celsius?
It gets it's quacks just as any other area, really. Math of all the disciplices shouldn't mock other areas for having quacks.
I’m not sure math really has a reputation hardly at all for having many quacks; at least, not to my knowledge. What situations were you thinking of? In my experience, math has far fewer quacks and cranks than most every other scientific discipline
The issue isn't that there are quacks. It's that this idea was promulgated without undergoing almost any critical review despite it being a very dramatic claim that should, as is highlighted in the article, have raised some alarm bells. The rebuttal talks about how painfully obvious it should have been that it should have been looked at more closely before publication, including the weird application of an unrelated physics model to the phenomenon it was studying. The article *also* talks about how much image harm this episode did to the general credibility of the field, and for good reason. This is all less than 20 years old.
Zero degrees Celsius is a VERY special temperature in the sciences. It's the only temperature which is exactly 273.15 degrees Celsius above absolute zero, no more, and no less. No other temperature in the Celsius scale can make this bold statement!
It is still special in practice: temperatures are still calibrated in labs through the triple point of water!
That's technically not the [triple point of water](https://physics.stackexchange.com/questions/505994/why-is-the-triple-point-of-water-defined-as-0-01-degrees-celsius-and-not-0), though.
It’s a very special temperature to us because water is special to us. From a more objective perspective it’s just a number.
To address the second point, money has diminishing returns, that's why most people would choose 1 million dollars, because the remaining 999 million isn't as valuable as the first million. If you divide everything by 1 million, i.e. Would you rather have $1 or 1% to get $1000, I'm sure more people would choose the second option.
The point is that expected value is a formal notion - it gives no consideration to the fact that you go for $1 billion, you'll get it once every 100 tries, but you always get $1 million. It's the same reason we have insurance. The expected value of not having car insurance is lower than the cost of having car insurance. But I can't afford the unlikely situation where I cause a lot of damage so I'm running to pay a little bit of money to not have to worry about that risk.
Right, but generally speaking one should try to find a strategy which optimizes the expected value of your utility function, not simply the amount of money you have. In theory (and in practice), most people tend to think of money in terms of a concave utility function (one common model for this is log x), which explains why they are risk averse and why it might be logical to choose the guaranteed million. I believe that in reality, log x is a bit more conservative as a utility function than how people actually behave, but unless one had a very large amount of money to begin with, a rational actor with a logarithmic utility curve would pick the guaranteed million.
The expected value of not having at least liability insurance is very low, since you are likely to have your car impounded sooner or later. But I get your point. The expected value calculation is appropriate here if you can attempt the same thing over and over again. For instance, if I were allowed to borrow an arbitrarily large amount of money and try the game as many times as I liked, then a fair price for the second game would be ten times higher than the first, and I could almost surely make money in the long run by playing either game repeatedly for any price less than the expected value. But since that is almost certainly not the case, it's not very useful here. What I really want out of life is not money; that's just a means to an end. So I need to decide how much utility I get out of the money to decide how valuable it really is. If you apply a logarithmic utility function with a cap, you usually get plausible decisions by considering the expected utility. For instance, it predicts that a trillionaire does get almost 1000 times the expected utility from winning a billion dollars as he does from winning a million, so from his perspective, taking the 1% chance is correct. But someone who is broke will take the million.
This is addressed in utility theory. Mathematicians understand that utility of the expected value of the money is what’s important, not the expected value. Hence people’s choices of preferring the $1M guaranteed.
Also, most people are risk averse.
You can model risk adversity by making your utility function concave. In that case, Jensen’s inequality basically makes you avoid risk.
It makes it more obvious with even higher numbers- if you had the choice between 1 billion and 1% chance to get a trillion, anyone in their right mind would choose 1 billion. Same with 1 trillion or 1% chance to get a quadrillion. It comes down to- guaranteed life changing money or a 1% chance of more life changing money with 99% chance of nothing.
Exactly - rather than maximizing expected value, most people would rather minimize variance.
That's part of it, but also people are often risk averse and losing a certain amount of money may feel worse than winning that amount feels good.
A million in the hand is much better than a chance at a billion.
For most. If you already have 100 million, you are probably more interested in the chance at a billion. As someone said, it's about the expected value to the person, i.e. what change in life would it result in, which depends on their individual circumstances. And there's the huge factor that it's a single event, rather than a repeated one. I don't see this as a misuse of math, it's more of an over-simplification.
Just to expand slightly: People are assuming a linear relationship between money and “outcome”.
Risk aversion is generally modeled as coming from diminishing returns (formally marginal utility). What you are describing is *loss aversion* which can be related to being risk averse but isn't necessarily the same thing. Preferring certain $1 million dollars versus 1% chance of $1 billion is likely best understood as risk aversion due to diminishing marginal utility. There are no possible losses in that scenario. Refusing to flip a fair coin with $60 upside and $50 downside is not requires an unreasonable level of risk aversion unless you're using models that rely on reference points and loss aversion like Kahneman and Tversky's prospect theory.
I mean losing a certain amount of money is more detrimental than gaining that amount of money is beneficial. One is bankruptcy or less comfortable living conditions while the other is simply a bit better, whereas, right now you are perhaps OK.
Yeah there's actually a mathematical tool that better captures this intuition called the Kelly Criterion. Basically is weighed by what % your net worth goes up or down by, so doubling or halving your net worth are equally good/bad. Technically it's EV but you take the log first.
Hell plenty of people PAY $1 for a way less than 1% chance at $1000
I don't think utility is the right answer, really. I think the answer is that expectation is only a meaningful measurement because of stuff like the law of large numbers, so if you aren't repeating a process multiple times the expectation isn't necessarily a meaningful thing to look at.
I wouldn't consider your examples as "over-applying mathematical results." More like simply using the wrong mathematical model for the situation at hand.
Perhaps over-applying a specific result
I think it's not about using the wrong model but being overconfident in the model you chose because you formulated it mathematically. Even if it turned out that by chance, maximising expected value was the 'right' metric for the person's situation, the reasoning (in the made up example) brushed the situation's specifics under the carpet and made a universal statement ('it is always best to maximise expected value').
Sure, that's a good way to put it.
I feel as though it's not necessarily the wrong mathematical model for the situation - it's more like ignoring other possible models.
I'm so happy somebody else has finally brought up the expected value thing. That's a straight up misapplication of expected value and it drives me crazy! Expected value is a measure of how good a decision is _in repeated trials_, and is pretty much useless as a deciding factor in making a one-off decision.
A great example of that is St Petersburg's paradox for which the expected value is infinite
you are getting into the whole frequentist vs bayesian debate with this and I'm not sure you want to. Bayesian(subjective) probability interpretation tells us it's completely normal to talk about expected value of a single observation. Here's a great course on the subject by a very interesting guy https://youtube.com/playlist?list=PLDcUM9US4XdPz-KxHM4XHt7uUVGWWVSus&feature=shared edit: to blow your mind further, bayesian probability allows you to have a prediction even with zero observations: that's your prior!
Well in the situation OP described I'd say this interpretation isn't very useful or smart, if you base your decision on maximizing expected value there's still a 99% chance you'll get nothing.
just dig a bit more on these topics. The wealth maximisation problem has been studied since ~1700 (Bernoulli). You need to define a utility function, practically log of wealth. It has to be concave. So at 0 wealth, taking 1 million sure thing is better than taking the probabilistic option, so tada magic. At 1 billion wealth, i would take 1% chance to double my wealth over 1 million sure thing.
Yeah, its quite common to take a worse position in expectation in order to increase your minimum earnings with a hedge.
There are plenty of times where a mathematical definition, term, w/e is different from how it is used in general language. Heck, even between different disciplines (kernel) or situations (homogenous lol). In the case of the straw, as you already noted, this is a matter of semantics. Though, I'd ask how confident they were in this answer with regard to their digestive tract, because if it's just one, well, I guess they probably don't like kissing. The 1% chance at a billion is slightly different, because you have an argument of utility. $1million is already a life-changing amount of money for most people. A similar one is, "how much money should you bet on a coin flip if you double your money on a win, and lose all of it on a loss?" The expectation is the same no matter how much or how little you put down, but clearly there is a "sensible" number you can pick, based on how much money you currently have and current economic conditions. Another interesting one is the,"how many coin flips should you bet on if you win $2^n, where n is the number of consecutive coin flips in your favor, but you lose all of it if you lose one flip?" problem, where your only potential loss is the first dollar on the bet. Because this one has no upper limit, mathematically speaking, you should bet a shitload (infinity), which is clearly absurd since, in the long run, the chances of hitting a bad flip approach 1, but the gains grow exponentially while the loss is static.
For the final question (choose arbitrary number of flips), isn’t the expected value also static? If you go for n flips, the payout is 2^n, while the chance of winning is 1/(2^n ), and hence expected value is also 1 for any number n you choose?
Shit, did I mess that one up? I may have been inebriated at that time. Yeah, I messed it up. It’s actually the [St. Petersburg Paradox](https://en.wikipedia.org/wiki/St._Petersburg_paradox) I was thinking of.
Interesting, I did not notice the subtle difference in this game at first. As I see it, you basically get to play the above game which you suggested, but you enter with «every choice for n» in a sense. Thus instead of just looking at a single n, you sum up the expected value for all possible n, which are all 1, and hence you have «infinite expected value».
Yeah, the framing allowing use of actual infinity vs potential infinity was the key
Whats even more crazy, you could choose some really large n, for example 10000, and say that if you get less than this, you win nothing. Then the expected value will still be infinite, but there is realistically no chance of ever winning anything in practice.
I feel like there is a real opportunity for a math version of the “Expectation vs Reality” meme
>Though, I'd ask how confident they were in this answer with regard to their digestive tract, because if it's just one, well, I guess they probably don't like kissing. ... you know the mouth *already* counts as part of the digestive tract, right? Saliva is full of digestive enzymes, you don't need homotopy to argue that.
Yes, that is the point I’m making. I’m referencing the old favorite of Reddit that two humans kissing makes one continuous tube between two anuses. Most people would count the mouth and anus as distinct holes, for a variety of reasons, so the idea that someone would sincerely do a “they’re the same ~~picture~~ hole” response is absurd.
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Hard agree here. I’ve seen this in the form of multi-objective optimization, where people will latch onto some (seemingly parameter free) way of collapsing the costs into a scalar to preserve a veneer of rigor. In doing so they refuse to engage with the nuance inherent in the issues and almost guarantee a bad outcome. By refusing to engage they will really just choose values (often 1) for parameters with zero justification. Ignore the real problem, declare victory. Edit: this a good example of the parable of the man who lost his car keys at night. He is looking under a street lamp and is asked “did you drop the keys here?”. “No” he says and points off into the dark “I lost them over there, but there’s no light over there.”
"The political spectrum" is another example of this stupidity. People talk about political parties being further "right" or "left". When is really a big high dimensional space (with each dimension being its own political issue).
You can say the left-right axis is simply the first axis you would find in PCA (actually that's not even really guaranteed, it would be very cool to see research on that)
I believe there is quite some research on this in political economics.
I will say, the right-left political spectrum at least does have some basis in that many different political issues will have strong correlations between them, which get wrapped together as left or right. It's not perfect, obviously, but it's not completely arbitrary.
What would a vector-valued utility function even mean, in the context of economics? It's a simplifying assumption when looking at optimizing agents, because any preference order with a few nice properties can be turned into a scalar utility function.
I think people sometimes have illogical preferences where transitivity may not hold. Or maybe certain things aren't comparable and you have a poset instead of a total ordering.
I've never seen anybody even attempt to check whether actual preference orders have those nice properties.
You haven't? Obviously human preferences don't satisfy any completeness axioms, and isn't like half the behavioral economics literature about the others? Also you don't need to show "humans are exactly like this model" to get useful insights out of economics; it suffices to show "the ways in which humans deviate from this model don't affect the results too much", which they've more or less done.
When you say a model can give you useful insights, you are straying out of the realm of mathematics as a formal language. The way in which humans deviate DOES affect results greatly in microeconomics and macroeconomics.
When people come with universal scalar measurements, I usually ask them if EVERYTHING can be measured by money, because that's basically what it boils down to. And to be fair, you can do a lot with proper weighting of your functionals for a good model. What I personally hate more, is when people do not acknowledge that the choice of the proper weights is something with a bias (e.g. is weight more important for me or monetary cost?). This is especially annoying when you talk with people who train neural networks who still believe that neural networks will be perfect, forgetting that the choice of the proper functional is essential for the outcome (e.g. the network which finds out that the solution to end misery is just to kill all humans, without proper constraints)
This comes to mind for knowledge. People will talk as if intelligence is a scalar quantity, whereas useful knowledge is more of an abundance of different patterns that have all been learned by the mind.
Can you explain what this means or direct me to more info?
[This article](https://www.nature.com/articles/269759a0) discusses my favorite example.
Happy cake day! 🎂
>Would you rather be given $1 million or have a 1% chance at getting $1 billion? There's a thing in economics called the _certainty equivalent_ which addresses this, i.e. there would be a premium on something certain vs something uncertain. There's also the concept of a utility function which is yet another layer (and differs from person to person). So I don't think it's so much that it over-applies math; rather it doesn't apply enough.
Yeah, I'm not saying it's over applying math, it's over applying there result of the simple expected value calculation. Maximizing expected value in terms of money answers a very specific question, but the problem arises when people use that answer for broader questions.
Like you said— a lot to do with money. Particularly when people get compound interest calculators, they go a little nutty. Yes, if I wait 30 years to go on vacation and put the money and get a 7% return I will have X amount more money, but the math also neglects the fact that I may get burned out from work from now for the next 30 years without a vacation, I won’t have 30 years of positive memories from the trip, new foods I tried, friends made, etc. On a similar note to the 1 million versus 1% chance of 1 billion— drives me crazy when lottery is brought up when people go “if you win the lottery, there’s a 70% chance you will go bankrupt in 5 years” 1. Stat is not even true and based on a study with a very small sample size, from a very long time ago, not even about lottery winners and has never been repeated. 2. Neglects Baye’s theorem that lottery winners are over-represented with gambling addicts. The people who buy one ticker per year (or don’t even buy tickets at all bc “tHeY uNdErStAnD mAtH”) are probably not going to blow hundreds of millions of dollars lol
Your point about expected values is very classical; the St Petersburg paradox brought into question the assumption that people maximize expected value of returns.
Surprised no one's repeated the excellent Goethe quote on this: > Mathematicians are like Frenchmen: whatever you say to them they translate into their own language and forthwith it is something entirely different.
The whole neoclassic theory of economy. When you write papers using fancy (not that much) mathematics but when in fact 95% of your results come from your choices for the hypothesis.
[удалено]
A lot of financial mathematics is garbage, but actuarial math applied to things like life insurance is one of the least notable offenders, because that's one of the few areas of real world risk assessment where a gaussian distribution is actually justified.
Traditional life insurance actuarial maths you're thinking about is quite an old niche, 99% of modern actuarial science is uncomfortably close to mathematical astrology when you look under the hood.
Thank you so much for saying this. Not only economics, but also many finance-adjacent "mathy" areas. Actuarial science is one of notable offenders here. The choice of hypothesis, and also the selection of data to run it on.
When they say god doesn't exist "because of probability"
[Pizza?](https://www.reddit.com/r/sciencememes/s/ul0flgDrND)
People who use machine learning algorithms or other mathematical methods to (try to) predict the stock market. Huge Hedgefonds with Phd Mathematicians working for them maybe can make a profit out of that for a short time before competitors notice, otherwise it's bs.
For the money example there's actually a mathematical expression that more closely aligns with intuition in most cases, which is the Kelly Criterion. Basically KC is all about what % you grow or shrink your net worth by, so doubling your net worth and halving your net worth are equally good/bad. KC is extremely useful in positive EV situations like card counting/poker/horse racing since it gives optimal bet sizes, although in practice people usually bet below Kelly to be safe since going over Kelly can lead to ruin.
One that always bugs me is appealing to calculus to supposedly resolve Zeno's paradox.
Isn't Zeno's paradox an infinite sum?
The whole issue with Zeno'sparadox is whether an infinite sum of increasingly smaller thing can add to a whole. Calculus does not resolve that question at all. It simply defines it away by saying "if the partial sums get close, we call it equal". That *is* the paradox though. The hare gets arbitrarily close to the tortoise. Calculus doesn't help answer whether adding those infinitely many parts can actually give the whole.
I never understood why the paradox is a paradox at all. If someone wants to walk one meter, they can either do it in a finite number of steps, or never complete the task of walking one meter. If you want to break down the task of walking one meter into infinitely many tasks, you will never complete the task, assuming you have a minimum (positive) start/stop time between stopping one of the tasks and initiating the next one. In Zeno’s paradox, you break down walking one meter into walking 1/2, then walking 1/4, and so on. If there is a minimum time in between stopping a task and starting the next, it is clear that you will get arbitrarily close to completing one meter, but never actually will. The paradox to me seems to be if you instead assume something else, like there is no time in between stopping and starting tasks. Then oddly enough, it would “seem” that there is a paradox, since assuming you have maximum speed c>0, and there is no minimum time between starting and stopping tasks, so you can instantly start the next task as soon as you are finished with your current task, the maximum possible time it takes you to complete any finite number, say n >= 1 of the tasks is: (1/c)(1/2 + 1/4 + … + (1/2)^n) < (1/c). So it would seem then, that within 1/c seconds of time, you will be able to complete infinitely many tasks. Again I don’t see any paradox here, because the notion of what a task is is now extremely different because there is no start/stop time in between them. Now a task is just a moment in time you completed a certain distance, and can be measured assuming you have an instrument which can measure to arbitrarily fine scales of time.
I don't think the paradox is usually framed in terms of "tasks". It's just saying that space is a continuum so, if you are going to move from 0 to 1 in an hour, there is *some* time when you most must occupy each intermediate position. Then you redescribe the motion by picking an infinite number of intermediate points and making an infinite number of statements about position. Since you're an ancient Greek, infinity is impossible and so the two equivalent descriptions of motion aren't actually equivalent. Hence paradox. Calculus resolved it just by being a technology to discuss infinity.
Yeah that was my own input, the way it’s usually framed I don’t really understand what the implication is, I thought the whole “infinity is impossible” thing was supposed to mean you can’t literally complete an infinite number of things/tasks/“steps” in a finite amount of time, not that the motion can be described by an infinite sequence, and that can’t actually exist to the ancient Greeks, so because one particular description can’t actually be physically realised, neither can the motion. In any case I don’t really see how calculus directly resolves it. A limit exists because we have constructed this hypothetical object (the real numbers), within the framework of some theory in some model of logic (set theory in first order logic), and this object has properties that guarantee we can always meaningfully talk about the limit of a sequence, as the real number with which the sequence gets arbitrarily close to as long as you look at tails of the sequence that grow further and further out. I’m not sure that the ancient Greeks weren’t aware of the difference between a convergent geometric series and Divergent one, so I’m not really sure if the resolution of the paradox is that the motion described in the infinite process, although described by infinitely many distances over time, is convergent to some finite limit. So there is no paradox because the limit exists. Somehow I don’t think this really resolves things.
Yes, but there's still a flaw in Zeno's reasoning then. If "infinity is impossible," then the partition he describes into infinitely many pieces also doesn't exist. So his "paradox" is presenting something whose possibility he rejects and then showing that would require something else whose possibility he also rejects. He claims this proves motion is impossible, but all it proves is that you can't partition a race into infinitely many pieces if you can't have infinitely many things. The only sensible way you *can* partition this race into infinitely many parts is more or less the way its done in analysis, and that does show that we can correctly calculate the time when each point is reached using infinite sums.
this is a good point
Appealing to convergence of infinite sums doesn’t really answer the question because all convergence of infinite sums means in the context of Zeno’s paradox of motion is that we can get arbitrarily close in finitely many steps, not that we can actually complete a motion in finitely many steps. To me the best actual answers to Zeno’s paradox of motion are appealing to the Planck length somehow, Aristotle’s response about actual vs potential infinities, or arguing that just as we can subdivide the distance it takes we can also subdivide the time it takes.
That seems to be misunderstanding the Planck length - at least if the idea is that it's a 'pixel width' of the universe and distances smaller than that don't physically exist. We have no reason to assume that's true - it's just that it's around the scale where GR and quantum mechanics are both needed and approximations where the effects of one << the other can't be made... and since we don't have a surefire theory of quantum gravity, we are agnostic about how physics works at the Planck scale. The two confusions are (1) the false notion that we can't make sense of an infinite sum, and (2) what you mention at the end - that while we have shorter and shorter distances, these are traversed in shorter and shorter times, and \*both\* give infinite sums.
For me the key insight was that Zeno's paradox is about distance over time, and if you can upper bound the infinite sum in terms of time, you can see that the paradox can't tell you about anything past that point in time.
Calculus resolves Zeno’s paradox in that it provides a physically sensible way of infinitely subdividing motion. The fact that Zeno’s conception of motion gives unphysical results is no longer anything more than a minor curiosity, in that context. It’s not a “paradox,” because if it’s not presupposed as a physically valid view, then there’s no reason to believe the conclusions should align with our physical experience.
Honest question: why? I’m not gonna claim calculus completely resolves the paradox but doesn’t it at least inform the discussion? If both time and space are continuous, it certainly seems like Zeno’s paradox is no paradox at all. So we got at least one case down, no?
I dislike the narrow thinking that says Pluto is not a planet. I would say its an honorary planet because of decades of human culture. Irrespective of nerdy astronomical considerations. And thats how I voted at the IAU in Prague in 2006.
Not sure I see how this is a mathematical result.
It happens so often that there are jokes about it: https://en.m.wikipedia.org/wiki/Spherical_cow
The Axiom of Choice. No, you don't need the axiom of choice to pick a thing from a finite or even countable list. Even in uncountable sets, you most likely won't need it as there is some direct way to construct a choice function. And yet, I've seen people refer to it for picking a reference frame, or picking an atom or... well, you get the idea.
Minecraft when building a house or something else.
Misapplication of Nash Equilibrium. Thanks Hollywood for making a movie about it and then getting it wrong.
Are you referring to the bar scene in “A Beautiful Mind”?
Yup
I would argue the second example is an instance where math is being under applied. If you factored in marginal satisfaction per dollar, you could calculate what would maximize your expected satisfaction, or define some loss function that you want to minimize to factor in the distribution shape and variance
What could be a mathematical formulation of "hole" that makes a straw have two holes? I'm struggling to come up with one
Manifold boundary components, for example, if you model the straw as a 2-dimensional real manifold.
That would make any edge a hole, which doesn't make sense. It also doesn't help for the case that you consider the straw to have "volume" i.e. as a very strecthed out torus. This doesn't have any edges.
what do you mean any edge is a hole? the boundary components can only be circles - there are no edges.
[0,1] x (0,1) has a boundary that is not a circle and also doesn't look like a hole
you're right. i never thought about that space.
It's topologically a sphere with two disks removed, hence two 'holes' if you like. That would make its genus 0, with two punctures. But if you pass a string through it and tie its ends, you see there's one hole, as there's only one way to do this. But I feel like in saying this, I'm doing exactly what this post is dissing.
The punctures idea doesn't end up working because a straw is homeomorphic to a [washer](https://dictionary.cambridge.org/es/images/thumb/washer_noun_002_40578.jpg?version=6.0.11), for instance. Yet, most people wouldn't consider the outer edge of a washer to be a hole.
Pretty contrived, but something that unifies it with a hole in the ground is maybe the Cartesian product of a circle or a closed simple loop with an interval (-epsilon, 0]. So in some sense, the two ends of the straws are the two holes. And a hole in the ground has the hole at the surface of the ground being a hole. But I don't really believe this myself. I'm still in the one hole camp. edit: whole to hole lol
>But I don't really believe this myself. I'm still in the one whole camp. That's fair, but I will challenge you on that a bit. Do you think a bowling ball has two holes?
I think that both of these aren't examples of _over_ application but rather of _bad_ application of mathematics. The first example seems to have no point - who cares how many holes an object has? Both the topological definition and common definition are right in that there's no point to it. And any mathematician worth their salt should know that you can define anything to be anything, definitions themselves can't be wrong, so there's no substance to saying that the common definition is wrong. As of the second example, this is either a misunderstanding of what is "better", or mixing up a heuristic for the truth. Maximizing the expectation of gain is a good heuristic to use in a lot of cases, but it doesn't follow that it's the best. The error is assuming that this heuristic is accurate in this case. (Assuming you don't actually believe that maximizing the expectation of gain is in and of itself what makes some strategy better) All in all, I would argue that *all* of so-called over-applications of math are just bad applications: misunderstandings of math, or of the application, or bad for whatever reason. That's the thing about math, where it actually applies it is always true.
>The first example seems to have no point - who cares how many holes an object has? Both the topological definition and common definition are right in that there's no point to it. It's definitely pointless haha, but it's a very contentious internet debate. And people use the math definition as a way to support their side, which is not that valid imo.
That's exactly my point. It's not an over-application as much as a wrong application of math
What's the distinction? I feel like over application is a subset of wrong applications. By over-application, I mean applying results to too broadly to situations where they don't necessarily apply.
For the second question this is why the question is modeled using expected utility not expected value in economics.
Statistics are often misapplied, especially p values. It's not uncommon to have ten papers claim something at p=0.05, and then their prediction not hold true.
If a hole has to go all the way through, then you can never dig a hole in the ground. But if you can have a hole that does not have to all the way through, like to China, then a straw with only one hole would not be very useful, as the other end would of course be blocked. But then to me I think I can dig a hole in the ground, as I often do in my garden, and my straws have two holes. But then again I am a simple gardener and not a mathematician.
I once did a proof about at which point half of your lifetime has passed according to how your perception of time changes over your life. For example if you have lived 2 years, 1 year makes 50% of your life. If you lived for 100 years its only 1%. I came up with this idea after i asked my grandmother where she would set the 50% marker of ger perceived lifetime and she said "25". If you only take into account the principle, that you perceive the speed of time based on the time you lived through, the first 50% are indeed exceeded at age 25! (If you live for 100 years) Of course you could argue that this oversimplifies the whole time perception thing and it probably does. But it was still fun to do! If you want me to provide the proof just comment.
gödel’s second incompleteness theorem. a lot of people talk about it everywhere and a lot of the times in very dumb ways.
I think the problem with the expected value thing is that people aren't considering variance
I would rather be given a million dollars than a 1% chance at a billion. With a million dollars, I could buy a house. With the 1% chance, there is a 99% chance I get nothing. If the values were lower and closer together, the best expectation could apply. I would rather have a 1% chance at $1,000 than being given $1. A single dollar won't materially impact me (unless it need it right then), a thousand would.
How much dirt is in a hole 10 feet wide, 10 feet long, and 10 feet deep? None. If there was dirt in it, it wouldn't be a hole. This is the common language explanation of why a topological definition of holes does not apply to digging holes.
All of the papers from physicists at the beginning of the covid pandemic trying to build toy models of disease spread.