Non Standard Analysis, Probabilities, Word Salad
reddit.com/r/math/comments/fd73hl/-/fjg5lmrPosted Mar 4, 2020, 2:33 PM UTC
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Btw, I feel like I should point out that a number of the rebuttals to this guy have serious problems as well. Most notably: The whole thing with nonstandard analysis and hyperreals.
Let's say you wanted to define infinitesimal probabilities for some reason. That is almost certainly not something you should do, but let's say you did. You wouldn't use the hyperreals, and you wouldn't be doing nonstandard analysis!
You can't sensibly measure anything with hyperreals because there's not even any such thing as the hyperreals. It's not actually a specific system of numbers, so it's not really any good for measuring anything with. Basically the hyperreals are really only useful as a device for proving things about the reals. Which is what nonstandard analysis really is -- a method of proof. Not the study of some new system of numbers that includes infinitesimals. The hyperreals don't really have any independent interest except for how they reflect on the reals.
There are any number of ways one could potentially adjoin infinitesimals to the real numbers, obviously, but choosing any of them other than the hyperreals, while still a mistake in context, would at least make some minimal amount of sense. (But it would still be a mistake in context. Dragging in infinitesimals seems to have only made the situation worse here...)
Seeing "infinitesimal" and saying "nonstandard analysis" is the same sort of mistake as those who see "infinite" and say "cardinal", and it really bothers me.
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I don't know the first thing about NSA, but I really appreciate this comment. Every time I see someone expressing confusion over something like 0.999... = 1 and then people jumping in and telling them NSA exists rather than just explaining why we don't, in standard analysis, do things that way, I feel like they're doing much more harm than good.
Seeing "infinitesimal" and saying "nonstandard analysis" is the same sort of mistake as those who see "infinite" and say "cardinal", and it really bothers me.
God, I know exactly what you mean.
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Every time I see someone expressing confusion over something like 0.999... = 1 and then people jumping in and telling them NSA exists rather than just explaining why we don't, in standard analysis, do things that way, I feel like they're doing much more harm than good.
Yes, definitely. Especially as people have some vague idea that introducing infinitesimals automatically makes such things work according to the intuition where 0.999... is not equal to 1. In reality, it doesn't really work that way, and introducing infinitesimals generally makes things way harder.
(The basic problem is that introducing infinitesimals causes lots of limits to no longer work, so things that are usually formalized via limits have to be interpreted in other ways instead.)
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I'm sorry, but what are you saying here? The hyperreals are not a new number system that includes infinitesimals? They absolutely are
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No (because um that's blatantly false), I'm saying nonstandard analysis is not the study of the hyperreals as an object in itself (they're not even uniquely defined!), but rather is the study of the reals by means of the hyperreals. Nonstandard analysis uses hyperreals, but they're not the actual object of interest.
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ok sure. that is I suppose the reason why we have the separate term "nonstandard analysis", instead of just speaking about analysis in the hyperreals. A more axiomatic/intrinsic real number point of view.
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I've read some stuff on math overflow and some people do use non standard analysis to do probability. See here.
I do agree that you need to be very careful though, and even then I'm not sure that the probability of hitting a countable set is anything other than 0 in any models with infinitesimals for various reasons. One is the potential for a transfer principal, another is it would mean that the isomorphisms of your probability space would be a bit odd, specifically you couldn't have isomorphisms that send null sets to the empty set.
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Isn't there already a well established way to take the "probability" of infinitesimal sets using measures? I also don't understand why nonstandard analysis is necessary to solve this problem.
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What is an "infinitesimal set"? I don't think you're saying anything that makes sense here.
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I guess just finite subsets of the real numbers. At least that's I interpreted infinitesimal to mean in this context.
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"Infinitesimal set" isn't a standard term. A finite subset of the real numbers has measure 0.
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Got excited at the prospect of explaining non-standard models to someone. Took one look at the thread and decided “Fuck that.”
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It's like he had this hole in his heart his entire life just waiting to be filled and non-standard analysis was the only thing that could help.
Unfortunately the hole was deep and things were dislocated and fractured in the fall.
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Non-standard models filled my hole if you know what I’m saying. ;)
My knowledge. I’m talking about holes in my knowledge.
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Singular in the first appearance, plural in the second.
I find your claims of innocence suspect.
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Well there are lots. I’m really not that smart. You can’t expect me to fill all my holes at once.
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Ikr, that’s why I’m here too :/
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Sorry I took so long. I had to calculate the end of pi first.
Here's an archived version of the linked page.
^^Quote ^^| ^^Source ^^| ^^Send ^^a ^^message
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Infinity isn't a number, right? It's a cardinality.
aaaaaaargh
(This is actually not nearly the most crazy problem here -- not like the entropy shit, this is a fairly usual style of confusion -- but still, aaaaaaaargh)
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Is it just cause it’s too reductive? Your problem is that cardinality is only one concept of infinity?
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That is... not how I would put it.
What I would say is that there are many different systems of infinite quantities, that do not necessarily fit together with one another, each of which is applicable to different things.
But, that's not the only problem here. There's also the problem that there's no such cardinal as "infinity". Part of the whole point of cardinals is that there are many infinite cardinals. If you are dealing with a quantity that can simply be called "infinity", you are not dealing with cardinals. "Infinity", by itself, is something that makes sense in many contexts (such as: the extended real line, which is what's useful for measure theory), but not as a cardinal.
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proofs by contradiction lose validity when considering uncountably infinite sets, because the total entropy cannot be defined
Yet another great godel vortex quote contender
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So much delicious word salad
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So far I haven't ended up doing anything with my math degree, but one thing I'm happy for is that it's meant I don't end up like this guy. This guy is just demonstrating confusion about what mathematics is. We don't sit around debating how we feel things should work. You set definitions and you follow them.
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When did you get your degree and what are you doing now? If I may ask?
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I got my degree four years ago and am doing programming.
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My understanding of the probability is the same as everyone's in that thread, measure 0 blah blah blah.
But wasn't there an r/badmathematics poster/mod before the shutdown that was a measure theorist who vehemently disagreed with the way people interpret that? I recall a long post claiming that interpretation was wrong and correcting it, but I understood very little of it ...
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Yes, that was sleepswithcrazy. She posted an extensive writeup about the notion of impossibility in probability theory, which you can read here. The basic gist is that the conventional wisdom, that probability zero events can actually occur, is just wrong, since sigma algebras with null sets are isomorphic to ones where the only nullset is the empty set. It's actually mathematically, philsophically, and physically impossible for a probability zero event to actually occur.
that viewpoint is a little fringe perhaps but it's not crackpot.
I thought about linking the current badmather to that article, but it didn't seem appropriate.
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It's actually mathematically, philsophically, and physically impossible for a probability zero event to actually occur.
that viewpoint is a little fringe perhaps but it's not crackpot.
Agreed. It seems absurd because it suggests that no event can occur, because any single outcome is a measure zero event.
And that's exactly correct: we can't carry out any experiment that samples from [0,1] to ever conclude what could "actually occur". The definition for these sets and their relative probabilities is exactly that: a mathematical definition. Probability theory assigns probabilities, "possible" is not a property it defines. It is not a prescription nor a description of actuality. It's a model.
There are no infinite sequences of heads and tails and there are no perfectly precise dart boards.
I thought about linking the current badmather to that article, but it didn't seem appropriate.
Thanks for linking it here I've been looking for it!
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It might've actually been a nice resolution for the OP. They were struggling with the claim "the probability of choosing a rational number is zero", and resolved it by resorting to infinitesimals or radical notions about the definition of a limit. But a different resolution would be: you can't actually choose a rational or any single real number in a random process. Instead just try to choose an interval.
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Yeah and she is right.
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[deleted]
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He's still at it today. And he's progressing into crankery territory with his persecution complex.
What's happening here is measure theory is being shoehorned in probability theory to the point that they've been conflated, even when the reason for using measure theory for probabilities no longer makes sense in probability.
It's real nifty that a bunch of authorities in your field have agreed to literally rewrite the rules of probability theory so that this makes sense, but it's not useful outside that context.
We can set our premises to whatever we want to make anything true. It's the problem with logic. It's why theology exists. Dogma is self-consistant, but useless.
I'm a heathen in the Church of Measure Theory, and I refuse to accept your Lord. So you're crucifying me. It's an old, boring story.
Poor heathen, getting bullied by us awful measure theory disciples.
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and now it's locked. edderofer was right to lock the thread, but that last comment was pretty golden. The dude is literally christ, dying for the sin of believing in infinitesimals
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Not sure if dunning kruger or mental illness, but it's a tad insane to think there is some big measure theory conspiracy when you can't even follow the proof of a countable set having 0 lebesgue measure.
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Wait until he finds the category th. videos (wow, there's more than one!)
We can set our premises to whatever we want to make anything true. It's the problem with logic.
MM, you have 3 dollars and I have 4 dollars. Therefore I will take your money and you will still have 3 dollars (in my account). So long!
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Iirc there's a numberphile video with a geometric(?) proof that is something like the rationals are points over the real line, points have zero area, the sum of all the points is zero, therefore the probability is zero.
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Sure, but for this you need the fact that you can add countably many disjoint probabilities to get the probability for the union. I'm not sure if he'd agree with that. He would probably also refuse to accept that the probability of hitting a point is 0.
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Lmao, I was in that thread talking with the guy.
I was wondering how a chain comment thread between me and him going 20 comments deep was getting up/down votes.
Glad the guy got banned from r/math
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R4: The probability if selecting from a countable subset of (0,1) is 0, OP disputes this. Some stuff lower down about how proof by contradiction is invalid with uncountable sets and some random stuff on entropy.
Edit: Should mention you have to go down a fair bit to get to the actual bad stuff. The linked comment if fine, just the start of it all.
Some highlights:
Proof by contradiction is conditioned on sets being finite... and something about entropy. I don't know:
Probability only works because testing it isn't possible
Undefined probability is better than 0 probability.
0.999... not equal to 1, I think? Also, I don't know what a measure is but you're all wrong.
In response to "The limit of a function f(x) as x tends towards infinity (or any value) is EQUAL to the value that the function approaches. " we come across the stunning realisation that limits are imaginary.
Non-standard analysis is ore rigorous, but remember, I don't know what a measure is so my opinion on the rigour of standard analysis is relevant:
He started off just trying to understand an admittedly counterintuitive mathematical fact (measure of rationals is zero and more generally nonempty events can have probability zero).
But once someone linked him to the wikipedia article on non-standard analysis, it was like something clicked in his head and it was all over.
I feel like the few legitimate uses for non-standard analysis aren't sufficient payment for having to listen to people who think hyperreals are like reals but hipper.
we should tell him about smooth infinitesimal analysis and the law of excluded middle
I once was one of those who though "these hyperreals are the future, maaaaan", but I thankfully kept my terrible understanding limited to one or two people in my friend group before ultimately having a realization that I in fact didn't know anything.
Are there any cases of non standard analysis proving something that could not otherwise been proven with analysis? I know there are some minor results which can be translated into proofs in analysis, and some stuff is more slick, but I've not heard of any serious uses for it anywhere. It just seems to make most things less intuitive.
Its neat that Newton and Leibniz got validated eventually. Are there actually other uses for hyperreals?
Hipperreals.
Mods: This is my R5, sorry for being lazy.
Here's another wacky one:
(Btw, he elsewhere misuses "finite" to mean something like "finitary", another common mistake I'll admit but ugh)
...how does a transcendentally irrational probability express itself in the quantum reality of the real world.
My eyes have developed the capacity to regret being attached to a sentient creature and my mouth feels like I just finished chewing partially set concrete.
The best part is, he doesn't know nonstandard analysis either! He learns about it in that thread and just assumes it confirms his beliefs.
Add to that:
Wow that descends into pure nonsense really fast. Like not misunderstanding. Just nonsense.
[deleted]
Huh?
Even in NSA, I'm not aware of any nonarbitrary definition of 0.(9) that distinguishes it from 1.
The problem with NSA is that many of the sets you can construct ordinarily are not actually valid hypersets, so 0.(9) interpreted as the sum of 10^-n over positive natural n is not actually even a hyperreal number. If you decided to interpret 0.(9) as 1 minus some infinitesimal, the infinitesimal you choose would have to be arbitrary, and you'd still have infinitely many elements between 0.(9) and 1, which sort of defeats the intuition behind what 0.(9) is supposed to be (the closest to 1 of everything below it, right?). If you instead interpret 0.(9) as the sum of 10^-n over positive hypernatural n, it's just 1 again.
So I don't see how 0.(9) != 1 could ever hold.
It's not? So what is it?
See my flair.