/sci/ - Science & Math

[Anonymous No.12605271]

Well, /sci/?

[Anonymous No.12605289]

>>12605271
>Almost all
absolutely nothing

[Anonymous No.12605291]

>>12605271
that the set of numbers in [0,1] that don't have P has measure zero. the measure is not specified but I would assume that the mathematician is talking about Lebesgue measure

[Anonymous No.12605298]

>>12605291
Ok and what does that tell you?
a) b) or c)

[Anonymous No.12605309]

>>12605298
It tells me that hw goes on >>>/wsr/

[Anonymous No.12605312]

>>12605309
What course do you think this is homework from lmao

[Anonymous No.12605313]

>>12605298
I'll go with b) anon

[Anonymous No.12605317]

>>12605313
WRONG. Think harder.

[Anonymous No.12605322]

>>12605271
c, why would i take his word for it? why didn't he present a proof?
>mathematician wouldn't tell me shit unless i asked for laymans, he'd show the work?

[Anonymous No.12605327]

>>12605322
Let's say he gives you a proof that the set of numbers in [0,1] that doesn't have P have measure 0 and you are convinced by his proof.
What does that tell you, in practical terms?

[Anonymous No.12605328]

>>12605291
This. So neither a, b or c. Strictly, if you had to pick one, c.

[Anonymous No.12605373]

>>12605327
that im fucking dumb..
i feel like its a simple question..
>0 Property P paradox?
>inverse real imaginary corollary?
is it syntactic implication?
excellent autist bait OP its fucked me up how dumb i am..

[Anonymous No.12605392]

>>12605289

[Anonymous No.12605404]

>>12605271
>Applying the statement "almost all" to an infinity set of numbers

I go with C

[Anonymous No.12605406]

>>12605327
also isnt the question as-phrased invalid.. if it isnt a,b or c that defies the phrasing doesnt it?

or is it as simple as a..?

[Anonymous No.12606289]

>>12605271
a has very little meaning mathematically. c is wrong, almost all implies the existence of a real number in the interval which doesn't have P, so c is wrong. I'll go with b by default, but based on what I've read in the thread so far I guess it's a trick question.

[Anonymous No.12606321]

>>12605271
I'm sorry Baker-sensei, I don't know. Can you please return to teaching us English like you're supposed to?

[Anonymous No.12606327]

>>12605271
C

[Anonymous No.12606336]

>>12606289
>almost all implies the existence of a real number in the interval which doesn't have P,
it doesn't.

[Anonymous No.12606342]

>>12605271
(a) is incorrect, insofar as if property P has the property mentioned, then a random number is not just likely to have the property, rather it has the property with probability 100%.
(b) is incorrect, since random number generators generate rational numbers, and it is possible that P is the property to be irrational.
(c) is obviously incorrect, e.g. among many other things you can deduce that the Lebesgue measure of the set of numbers that have property P is equal to 1, assuming it is Lesbegue measurable.

[Anonymous No.12606385]

>>12606342
>is incorrect, insofar as if property P has the property mentioned, then a random number is not just likely to have the property, rather it has the property with probability 100%.
Wrong. If you pick a random real number, you don't know shit about whether it does or does not have property P.
>since random number generators generate rational numbers,
You can make random number generators that generate irrational numbers.
>Lebesgue measure of the set of numbers that have property P is equal to 1
And in practical terms that means.... absolutely nothing.

[Anonymous No.12606511]

>>12605271
That you can enumerate the counterexamples.

[Anonymous No.12606590]

>Wrong. If you pick a random real number, you >don't know shit about whether it does or does >not have property P.
The statement that almost every number has property P by definition means that a random number has property P with probability exactly 1, no less. What did you think that it means?

>You can make random number generators >that generate irrational numbers.
The entry appears to concern actual random number generators, not hypothetical ones.

>Lebesgue measure of the set of numbers that have property P is equal to 1
>And in practical terms that means.... >absolutely nothing.
I see, the goalposts have moved a bit. Yes, you are right. Your goal in life is to be expert at shoveling dung in your uncles pigfarm, and whatever can or may be said about basic real analysis means absolutely nothing to you, got it. But this changes for students who have to understand enough of mathematics to pass exams, graduate, do research and qualify for tenured position in areas heavy with analysis such as machine learning and AI.

[Anonymous No.12606644]

>>12606590
>The statement that almost every number has property P by definition means that a random number has property P with probability exactly 1
I said "you pick a random real number", this means there is a process in which you pick a random number.
Measure theory said jack shit about such a process.
The "probability" is just math jargon misdirection, it has absolutely nothing with probability as you encounter it in real life.
>The entry appears to concern actual random number generators, not hypothetical ones.
Fair point.
>Your goal in life is to be expert at shoveling dung in your uncles pigfarm, and whatever can or may be said about basic real analysis means absolutely nothing to you, got it.
You misunderstand what I meant by practical. Obviously measure theory is not going to help you with farming or construction or whatever. By practical I meant something outside of the pure math formalism, something that can be applied, like you mentioned for example in machine learning and AI, or statistics. In all such real applications of mathematics, the "fact" that almost all numbers have property P has absolutely 0 implications, it's an empirically meaningless statement. There is no way to prove it nor disprove it by experiment, not even in probability (in the practical sense).

[Anonymous No.12606653]

>>12606511
Wrong in the practical sense (countability in pure math has nothing to do with actually being able to enumerate it) and wrong in the pure math sense (measure 0 doesn't imply countable, see Cantor's set).

[Anonymous No.12606767]

>>12606644
>I said "you pick a random real number", this >means there is a process in which you pick a >random number.
And you know such a process, but I don't, is your point? Does your process always pick the same random number every time, by any chance?

>Measure theory said jack shit about such a >process.

That is absolutely correct. I can tell that you know a lot about shit.

>The "probability" is just math jargon >misdirection, it has absolutely nothing with >probability as you encounter it in real life.

That is totally irrelevant to what is written on the nice looking lady's blackboard. You have an obvious disconnect to what it says there.

>The "fact" that almost all numbers have >property P has absolutely 0 implications, it's >an empirically meaningless statement. There >is no way to prove it nor disprove it by >experiment, not even in probability (in the >practical sense).
Your hobby is to move goalposts? Let us say that P(x) is the statement that x is irrational. Then P is true almost always. This implies that there exists a number x that is irrational. This is verified empirically(?) by verifying that the longest side of an isosceles right triangle with two sides of length 1/2 has irrational length.
But of course I have no actual idea what you mean by verifying experimentally the properties of abstract notions. It would not be a surprise if we could continue the conversation for years on end, and you would forever refuse to explain what you mean by that.

[Anonymous No.12606824]

>>12606767
>And you know such a process, but I don't, is your point?
Yeah, you just take an arbitrary number that suits your fancy. Let me do it right now. I pick 0.133555126222
>That is absolutely correct. I can tell that you know a lot about shit.
The point is that measure theory tells you absolutely nothing that would ever be empirically relevant.
>That is totally irrelevant to what is written on the nice looking lady's blackboard
It is relevant. A mathematician uses words like "almost every" that sound meaningful when in reality they tell you absolutely nothing beyond the purely, meaningless formalism of shuffling formulas around.
>Your hobby is to move goalposts?
I didn't move goalposts even once. You sound confused.
>Let us say that P(x) is the statement that x is irrational. Then P is true almost always. This implies that there exists a number x that is irrational. This is verified empirically(?) by verifying that the longest side of an isosceles right triangle with two sides of length 1/2 has irrational length.
This is a good example of the type of reasoning that is a complete nonsequitur. Just because some property P holds for almost all numbers has absolutely nothing to do with how likely you are to find an example of a number with property P, and your example is a pure coincidence. It's like saying "Jane said 13543 is quite an odd number and that empirically verifies the fact that 13543 is odd (not divisible by 2)". A total confusion of language.
>But of course I have no actual idea what you mean by verifying experimentally the properties of abstract notions
Like literally everything to do with statistics. Experiments like Buffon's needle that let you calculate pi. The claim that there are infinitely many twin primes translates to the empirical claim that an algorithm that looks for twin primes will always find another one. Riemann hypothesis translates to the claim that an algorithm that looks for nontrivial zeros will only find them on the line Re=1/2.

[Anonymous No.12606839]

>>12606824
The abstract claim that the area of a circle is 3.14... is empirically verifiable by trying to fit small rectangles into a circle, analogously for the circumference of a circle by measuring the length of a piece of string around the circle.
Empirical applicability and verifying experimentally mathematical facts used to be the whole point and is the whole reason why it's useful to natural sciences and other applied fields.

[Anonymous No.12607204]

>>12605271
>almost all
>finite number of numbers have the property P
>cardinality [0,1] = inf
C

[Anonymous No.12607368]

>>12605271
it tells you a
not b since it doesn't get close to 1, it is 1 depends on how you define close.
fucking autist

[Anonymous No.12607376]

>>12607368
Wrong. Think harder. Remember that mathematicians are deceitful with their words.

[Anonymous No.12607483]

>>12607376
then b too because then you can say that it gets arbitrarily close to 1 since it is 1
and it can also say C depending on P and maybe c because who the fuck cares about what the numbers in the interval [0,1] have as proprieties

[Anonymous No.12607499]

>>12605298
a.) Except 100% chance
b.) Except it's 1 immediately

[Anonymous No.12608774]

>>12607483
>then b too because then you can say that it gets arbitrarily close to 1 since it is 1
Wrong.
>>12607499
>a.) Except 100% chance
Wrong.
>b.) Except it's 1 immediately
Wrong.

[Anonymous No.12608784]

>>12605291
This.

[Anonymous No.12608795]

>>12608774
go fuck yourself with your bait thread

[Anonymous No.12608798]

>>12608795
Yikes.

[Anonymous No.12609045]

>>12605271
c, because i have no idea what you autistic mathrats are saying.

[Anonymous No.12609080]

>>12605271
a) and b) are false in general as they make no mention of the distribution of numbers picked / generated.
c) is false because it tells me a lot.

[Anonymous No.12609168]

>>12609080
>c) is false because it tells me a lot.
Like what?

[Anonymous No.12609180]

>>12605271
I'm going to go with C.
Because who cares.

Mathematicians need to focus on applicable problems. You're being as useless to the world around you as art majors. Worse, since art is something other people can in rare instances enjoy and appreciate.

[Anonymous No.12609194]

>>12609168
A few examples:
That there are real numbers with property P.
That a) and b) hold for any continuous probability distribution function.
That you are retarded for even asking this question.

[Anonymous No.12609200]

>>12609194
>That there are real numbers with property P.
Which means absolutely nothing in reality, since they could all be inaccessible to us in any way.
>That a) and b) hold for any continuous probability distribution function.
Which is again completely meaningless and impossible to test.

[Anonymous No.12609252]

>>12609200
I notice that you haven't even tried to address the third example.

[Anonymous No.12609301]

>>12609252
Well there's nothing to address there. You're just angry because you are confused.

[Anonymous No.12609396]

>>12605291
What's the difference btn measure and cardinality of sets?

[Anonymous No.12609440]

>>12605271
>real numbers

[Anonymous No.12609495]

>>12605271
>why of course it's a), what else would it be

[Anonymous No.12609531]

>>12609396
The cardinality of a set can be used to build a measure, but not all measures come from the cardinality of sets

[Anonymous No.12609547]

>>12607376
>>12605322
so bait thread like you said OP. its a shame theres no payoff at all

[Anonymous No.12609600]

>>12609531
So measure is more general size of sets?

[Anonymous No.12610250]

>>12609396
>>12609600
a measure is a function that takes a subset and returns a number which can be thought of as a "size" of the subset, and satisfies some intuitive properties (e.g. the empty set has measure zero, the union of two disjoint subsets has measure equal to the sum of the two measures, and more). you can use measures to introduce an alternative, more general theory of integration

[Anonymous No.12610313]

>>12605271
(d) OP is returded.

[Anonymous No.12610552]

>>12605271
Almost is often defined (as in in my analysis 3 course) as only sets of measure 0 don't fulfil the property, so none of the above

[Anonymous No.12611518]

>>12605271
Its c)
The correct terminology for b) is “almost everywhere”

[Anonymous No.12611538]

>>12610552
Correct in your reformulation but wrong amswer. The answer is c.
>>12611518
Based retard

[Anonymous No.12613316]

>>12605404
Does "almost all" of an infinite set behave differently depending on whether it is countable or uncountable?

[Anonymous No.12613650]

>>12605271
Learning about L^p spaces for the first time blew my mind. Functional Analysis is pretty fun

[Anonymous No.12613687]

>>12605271
Under mild assumptions is b), if you were more rigorous in defining the question and made a) and c) more interesting it could have been a fun quiz.
This is bait, isn't it?

[Anonymous No.12613709]

>>12613687
The answer is c)

[Anonymous No.12613850]

>>12605271
>IQ test.png
why is this board so obsessed with iq?

[Anonymous No.12614111]

>>12605271
b) if and only if the RNG can produce numbers outside of Q.

[Anonymous No.12614132]

>>12614111
of course I mean almost all numbers in the interval outside of Q, not just one.

[Anonymous No.12614292]

>>12614132
>almost all numbers in the interval outside of Q, not just one
Do you think such RNGs exist?

[Anonymous No.12614306]

>>12614292
nope.

[Anonymous No.12615924]

>>12605312
Week one of measure theory for babies

[Anonymous No.12615930]

>>12605404
You dumbfuck, almost all is measure theoretic lingo. It doesn’t have to do directly with the cardinality of the set.

[Anonymous No.12615965]

>>12615930
It does. "Almost all/every" on an infinite set has absolutely no empirical meaning/consequences. There's no way to test such a claim and it has absolutely no implications beyond the pure abstract formalism.

[Anonymous No.12617311]

>>12605271
Made for BBC.

[Anonymous No.12617315]

>>12605271
A

[Anonymous No.12617335]

>>12605271
chose the longer answer

[Anonymous No.12617495]

>>12617315
>>12617335
Wrong.

[Anonymous No.12617519]

>>12617335
i heard shechem needed re-salting. in other news they'll likely do it to themselves.