/sci/ - Science & Math » Thread #12663815

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day 1 in real analysis

Anonymous Wed 03 Feb 23:14:54 2021 No.12663815 View Reply Original Report

Quoted By: >>12663863 >>12664019 >>12667400 >>12668441 >>12668481

>professor asks class what a real number is as review

>entire class goes silent and no one is able give a definition other than "the real number line"

>professor defines them as an "infinite" set of cauchy sequences of rational numbers

>professor then says we shall take them as axioms from here on out and moves on

is /sci/ really okay with this ?

Anonymous

Anonymous Wed 03 Feb 2021 23:17:43 No.12663823 Report

Quoted By: >>12663828 >>12663835 >>12663864 >>12663875 >>12663880 >>12663901 >>12663923 >>12664080 >>12664499 >>12665563 >>12666296 >>12668346 >>12668622 >>12669024 >>12669041 >>12669052

A liberal jewish sophist schizo and infinitist professor was teaching a class on real analysis (a religion not a science)

"Before the class begins, you must get on your knees and worship Cantor and accept that he was the greatest mathematician the world has ever known, even greater than Euclid!”

At this moment, a brave, patriotic, finitist who had watched 1500 Wildberger videos on youtube and understood the logical flaws of Zermelo–Fraenkel set theory stood up and held up a right-angled isosceles triangle with side length 1.

"What length is the hypotenuse of this triangle?"

The arrogant professor smirked quite Jewishly and smugly replied, " $\sqrt{2}$ , you stupid schizo."

"Wrong. $\sqrt{2}$ does not exist. If, as you say, it exists... then write down the precise decimal result of $\sqrt{2}$ !"

The professor was visibly shaken, and dropped his chalk and copy of Baby Rudin. He stormed out of the room crying those schizo infinitist tears. There is no doubt that our professor, Ezekiel Shekelstein, wished he had critically analyzed the foundations of mathematics and rejected nonsensical notions like infinite sets instead of becoming a schizo infinitist. He wished so much that he could shoot himself, but the bullet would never reach his brain as first it had to cover half the distance, then 2/3, then 3/4 and so on in an unending sequence that he himself had so foolishly proclaimed the existence of!

The students applauded and all subscribed to Norman J. Wildberger and accepted him as the savior of math. A finite number of ancient Greek mathematicians walked into the room and proved the incommensurability of irrational line segments, and God himself showed up to testify on the finitude of the universe and all that comprises it.

The professor lost his tenure and was fired the next day. He died of the gay plague AIDS and was tossed into the lake of fire for an incomprehensibly large but finite number of instances.

Anonymous

Anonymous Wed 03 Feb 2021 23:19:31 No.12663828 Report

Quoted By:

>>12663823
Based

Anonymous

Anonymous Wed 03 Feb 2021 23:22:50 No.12663835 Report

Quoted By:

>>12663823
Based

Anonymous

Anonymous Wed 03 Feb 2021 23:29:08 No.12663863 Report

Quoted By: >>12663893 >>12663907

>>12663815
What is the least upper bound for a number constructed following Cantor's diagonal argument?

Anonymous

Anonymous Wed 03 Feb 2021 23:29:12 No.12663864 Report

Quoted By:

>>12663823
based and burgerpilled

Anonymous

Anonymous Wed 03 Feb 2021 23:32:31 No.12663875 Report

Quoted By:

>>12663823
fpbp

Anonymous

Anonymous Wed 03 Feb 2021 23:33:52 No.12663880 Report

Quoted By:

>>12663823
based and cumpilled

Anonymous

Anonymous Wed 03 Feb 2021 23:36:39 No.12663893 Report

Quoted By:

>>12663863
What are you talking about? Numbers don't have least upper bounds.

Anonymous

Anonymous Wed 03 Feb 2021 23:38:29 No.12663901 Report

Quoted By:

>>12663823
LOOOOOOOL

Anonymous

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Anonymous Wed 03 Feb 2021 23:39:34 No.12663907 Report

Quoted By: >>12663925

>>12663863
>unironically believes in "least upper bounds"
schizo time

Anonymous

Anonymous Wed 03 Feb 2021 23:42:17 No.12663915 Report

Quoted By: >>12663920

It's a mistake to care too much about the real number system's underpinnings.

It's literally the serious math major's version of tau vs pi. It's just as inane and pointless and it's something only people who don't know or do real math care about.

Move on with your life and focus on doing proper math.

Anonymous

Anonymous Wed 03 Feb 2021 23:43:09 No.12663920 Report

Quoted By: >>12663931

>>12663915
>It's a mistake to care too much about the real number system's underpinnings

people like you are the problem with modern mathematics

Anonymous

Anonymous Wed 03 Feb 2021 23:44:09 No.12663923 Report

Quoted By:

>>12663823
this

Anonymous

Anonymous Wed 03 Feb 2021 23:44:21 No.12663925 Report

Quoted By:

>>12663907
yes i do believe

Anonymous

Anonymous Wed 03 Feb 2021 23:45:33 No.12663931 Report

Quoted By: >>12663943 >>12663998

>>12663920
Nice reddit space. Take a good look at yourself and try to understand what you will actually gain from caring so much about this. It is sufficient to understand the nuance of countable dense sets within uncountable connected sets and move on.

Anonymous

Anonymous Wed 03 Feb 2021 23:48:08 No.12663943 Report

Quoted By: >>12663949

>>12663931
>uncountable connected sets
>infinite sets

HAHAHAAHAHAHAHAHAHAHAHA

Anonymous

Anonymous Wed 03 Feb 2021 23:50:22 No.12663949 Report

Quoted By:

>>12663943
This is what happens when people don't shit on the wildberger schizos.

Anonymous

Anonymous Thu 04 Feb 2021 00:08:09 No.12663998 Report

Quoted By:

>>12663931
nice scientific inquiry you got there

Anonymous

Anonymous Thu 04 Feb 2021 00:10:47 No.12664007 Report

Quoted By: >>12665786

>ITT schizoburgers get btfod and attempt to save face with poor quality trolling

Anonymous

Anonymous Thu 04 Feb 2021 00:14:02 No.12664019 Report

Quoted By:

>>12663815
Yeah

Anonymous

Anonymous Thu 04 Feb 2021 00:28:54 No.12664080 Report

Quoted By:

>>12663823
God bless you, anon

Anonymous

Anonymous Thu 04 Feb 2021 02:06:28 No.12664452 Report

Quoted By:

Am I the retarded one or is there no reason why you can't have an irrational length line segment? Where in the Bible does it say that you need to be able to express a finite number with a finite number followed by a period and then another finite number?

Anonymous

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Anonymous Thu 04 Feb 2021 02:23:25 No.12664499 Report

Quoted By: >>12664504

>>12663823
fucking b*sed

Anonymous

Anonymous Thu 04 Feb 2021 02:25:15 No.12664504 Report

Quoted By:

>>12664499
>>12664499
kek

Anonymous

Anonymous Thu 04 Feb 2021 09:12:25 No.12665563 Report

Quoted By: >>12665575 >>12665786

>>12663823
why does a number needs to be writeable to say it exists? as long as numbers are abstract entities I don't see the problem. sqrt of 2 beeing irrational just means you cannot measure the side and the diagonal of a square with the same unit.

Anonymous

Anonymous Thu 04 Feb 2021 09:16:54 No.12665575 Report

Quoted By: >>12665786

>>12665563
it doesn't. the post is making fun of Wildberger as his entire argument amounts to "you can't write down an irrational number".

Anonymous

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Anonymous Thu 04 Feb 2021 11:12:38 No.12665786 Report

Quoted By: >>12665789 >>12666284

>>12664007
delusional.

>>12665563
>sqrt of 2 beeing irrational just means you cannot measure the side and the diagonal of a square with the same unit.
No, that just means that length is a property that only some line segments have. In fact, if someone tells you to do something (e.g. get a line segment of length sqrt(2)) that requires an infinite amount of work to accomplish, that probably means that the task is impossible, so talking about the completion of the task ("the actual number sqrt(2)") is nonsensical. You can get as close as you want to the "true value" but the only thing you will ever have is the approximation, not the number sqrt(2) itself. The only reason you think it is a number in the first place is because ZFC assumes by fiat the ability to do an infinite amount of work (lol).

>>12665575
that's not his argument and you clearly haven't actually listened to what he said, but whatever makes you feel better. let me know what pi + e (exactly, not approximately) is without:
1)writing pi + e
2)writing some digits and then ending with some dots
3)telling me to go do an infinite amount of work (e.g. add this infinite sequence of rationals)
And if you say that you have an algorithm that can get you arbitrarily close to "pi + e" (assuming that's even a number in the first place), consider the fact that since all you can give me are approximations to "pi + e", then that's all that there really is; things like "pi" are inherently approximate quantities (and not actual concrete things like rational numbers).
>inb4 what about 1/3 = 0.(3)
The point is that I can convert from repeating decimals to rational numbers and do all my arithmetic there. So there is a representation that allows me to work with repeating decimals that can give me exact results in a finite amount of time. The same can't be said of """real""" numbers.

Anonymous

Anonymous Thu 04 Feb 2021 11:17:25 No.12665789 Report

Quoted By: >>12665792

>>12665786
>let me know what pi + e (exactly, not approximately) is without:
>1)writing pi + e
>2)writing some digits and then ending with some dots
>3)telling me to go do an infinite amount of work (e.g. add this infinite sequence of rationals)
>And if you say that you have an algorithm that can get you arbitrarily close to "pi + e" (assuming that's even a number in the first place), consider the fact that since all you can give me are approximations to "pi + e", then that's all that there really is; things like "pi" are inherently approximate quantities (and not actual concrete things like rational numbers).
pi + e = e + pi

Anonymous

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Anonymous Thu 04 Feb 2021 11:21:10 No.12665792 Report

Quoted By: >>12665797

>>12665789
>the absolute state of """real""" number """arithmetic"""

Anonymous

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Anonymous Thu 04 Feb 2021 11:24:18 No.12665797 Report

Quoted By: >>12665803

>>12665792
seethe more

Anonymous

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Anonymous Thu 04 Feb 2021 11:27:19 No.12665803 Report

Quoted By:

>>12665797
cope

Anonymous

Anonymous Thu 04 Feb 2021 14:09:17 No.12666176 Report

Quoted By:

>pi+e
You can estimate pi+e, but because an infinite amout of work is impossible, you can't know pi+e up without an error margin. This is how:
$\pi+e=\sum_{n=0}^{N}\left(\frac{4(-1)^{n}}{2n+1}+\frac{1}{n!}\right)+\epsilon$ . The bigger the number $N$ , the smaller the error $\epsilon$ . This error will never be truly zero, but if you manage to make it less than the 0.0000001% of the pi+e value, why bother? We stopped bothering about the error parameter of the speed of light when its measurement uncertainty reached that of 4 parts per billion.

Anonymous

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Anonymous Thu 04 Feb 2021 14:11:25 No.12666184 Report

Quoted By: >>12666380 >>12668979

>pi+e
You can estimate pi+e, but because an infinite amout of work is impossible, you can't know pi+e up without an error margin. This is how:
$\pi+e=\sum_{n=0}^{N}\left(\frac{4(-1)^{n}}{2n+1}+\frac{1}{n!}\right)+\epsilon$ . The bigger the number $N$ , the smaller the error $\epsilon$ . This error will never be truly zero, but if you manage to make it less than the 0.0000001% of the pi+e value, why bother? We stopped worrying about the error parameter of the speed of light when its measurement uncertainty reached that of 4 parts per billion.

Anonymous

Anonymous Thu 04 Feb 2021 15:08:43 No.12666284 Report

Quoted By: >>12666288 >>12666351 >>12667447

>>12665786
>unironic schizo posting
>still completely unable to come up with an argument other than "lol you can't write it down"
>1)writing pi + e
not true, it's really not particularly difficult to write down a Cauchy sequence for pi + e
>2)writing some digits and then ending with some dots
I don't have to be imprecise if I write down a Cauchy sequence for pi + e
>3)telling me to go do an infinite amount of work (e.g. add this infinite sequence of rationals)
Add the infinite sequence $\forall n \in N. \, a_n = 1/n$ and $\forall n \in N. \, b_n = 1+1/n$ it takes a finite amount of work to add these sequences. Their sum is $\forall n \in N. \, a_n + b_n = 1 + 2/n$ . I didn't do an infinite amount of work since it clearly took me a finite amount of time to do it. If you complain that I've not added the equivalence class of all sequences with the same limit, and that would take an infinite amount of work to do then I will complain that you can't add rational numbers either. Every rational number has an infinite equivalence class of tuples of a natural number and an integer, e.g. 1/2 = (1,2) = (2,4) = (3,6) etc. When you add the rationals 0 = (0,1) and 1 = (1,1) you need to add their equivalence class of tuples of a natural and integer, but oh sorry sweaty, you can't do an infinite amount of work.
>I can convert from repeating decimals to rational numbers and do all my arithmetic there. The same can't be said of """real""" numbers.
I can convert from reals as infinite sequences, symbols like pi, e and decimal approximations as the situation merits.

Anonymous

Anonymous Thu 04 Feb 2021 15:10:22 No.12666288 Report

Quoted By: >>12666299

>>12666284
>write down a Cauchy sequence
You need to write down the equivalence class of cauchy sequences for pi+e, because thats how you "define" a real number.

Anonymous

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Anonymous Thu 04 Feb 2021 15:13:22 No.12666296 Report

Quoted By:

>>12663823
>tossed into the lake of fire for an incomprehensibly large but finite number of instances.

Anonymous

Anonymous Thu 04 Feb 2021 15:16:00 No.12666299 Report

Quoted By:

>>12666288
Then write down any rational number. You can't, since they're an equivalence class of tuples of an integer and non-zero natural (p,q) defined by $(p,q) \sim (p',q') \iff p q' = q p'$ . Clearly this class is infinite, so you can't write it down.

Anonymous

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Anonymous Thu 04 Feb 2021 15:36:41 No.12666351 Report

Quoted By: >>12666354 >>12667464

>>12666284
There are plenty of good ways to define rational numbers that do not rely on infinite sets. Using infinite sets to define them only brings confusion obscures things.
One way is to simply define a rational number as a pair of integers a/b, b nonzero and define equality symbol in the context of rational numbers as a/b=c/d iff ad=bc.
Since this relation is perfectly doable both in principle and in practice, this is a perfectly good definition.
If you're uncomfortable with having different looking things "represent" the same thing, if you want all equal things to look the same, you can for example define a rational number as a pair of integers a/b such that a,b are coprime and b is positive. Since the condition of being coprime is easy to check using the Euclidean algorithm this is a perfectly fine finite definition.
For other definitions, just take a look at any computer language which implements rational numbers (of bounded complexity, but it's not hard to vary the bound). I promise you will not see "infinite sets" anywhere in the definition.
...

Anonymous

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Anonymous Thu 04 Feb 2021 15:38:05 No.12666354 Report

Quoted By: >>12666367

>>12666351
...
You claim to have manipulated "infinite sequence" when in fact you only manipulated a finite rational function expression which you suppose corresponds to the infinite sequence, or in some way defines it. But this is a problem, because you have not explained what you mean by an infinite sequence. Surely not all infinite sequences are given by such rational expressions (or are they?). For you to claim that you have in fact manipulated infinite sequences, you need to explain what is and isn't an infinite sequence, and how we access and manipulate them through our finite grasp. If your conception of a sequence is close to that of a modern mathematician, you will probably find this impossible to do, since actually you don't know what a sequence is. You want it to be as general as possible, but that prohibits you from constructing a sensible definition of what a sequence is, since you have no real conception what "as general as possible" means.
>I can convert from reals as infinite sequences, symbols like pi, e and decimal approximations as the situation merits.
Can you? A foundationally aware modern mathematician would say no. Tell me how you do it.

Anonymous

Anonymous Thu 04 Feb 2021 15:42:01 No.12666367 Report

Quoted By: >>12666409

>>12666354
An infinite sequence in this context is a function from the naturals to the rationals you fucking schizo

Anonymous

Anonymous Thu 04 Feb 2021 15:46:28 No.12666380 Report

Quoted By: >>12667532

>>12666184
Jokes aside, no human brain can fathom infinity. It's like it can't be a part of reality because it's impossible to experience it.
This is a false logic though, but still, if an infinity exists, it can't be perceived by a human mind

Anonymous

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Anonymous Thu 04 Feb 2021 15:55:45 No.12666409 Report

Quoted By: >>12666685 >>12667447

>>12666367
That's the heart of the whole infinitist contradiction. You don't have a good answer of what is a general function from naturals to rationals/equivalently a function from naturals to {0,1}/equivalently a 0-1 sequence.
Say I'm designing an algorithm which takes in the data of a function f and ouputs f(3). The most natural question is what type of data I accept that represents the most general function N->{0,1}. And you have no good answer. You live in the delusion of being able to perform infinite work and whatever construction you think up (that has to be captured by an actually intelligible data), you will be able to perform a diagonal argument in your head (when in reality it doesn't work because it requires performing infinite work) to produce "another sequence" that's somehow more general and doesn't fit into your data.
This actually means you will never have a definition of a sequence that satisfies you, since a definition is always a finite description.
The vagueness of the notion of a sequence is brought into light even more after the results of Cohen and Godel. Not only is the set-theoretic "definition" not good enough to capture what the most general sequence is (again by a meta-diagonalization infinitist argument), it's not even good enough to answer the most basic question in infinitist mathematics( and also in a sense the only question you can ask), namely how many such sequences are there? Do we have a bound on the cardinality of the sequences? And infinitist schizos have no answer to this most basic question, which is supposed to be infinitely easier than asking what a sequence is. The cardinality can be whatever you want! Aleph_1, Aleph_33, Aleph_(omega + 69), it doesn't matter. Isn't that cool? You can choose whatever size you want, and you won't reach a contradiction.
It's like Freudian psychoanalysis, you can make up whatever answer suits your fancy on the spot, and keep going. Nobody is going to call you out on your bullshit.

Anonymous

Anonymous Thu 04 Feb 2021 17:24:09 No.12666685 Report

Quoted By: >>12666688

>>12666409
I agree with you for the most part, but I disagree with how you get there.
You can dive much deeper in how much infinitist mathematics proves itself inadequate to capture their inane interpretations of it.
Diagonalization has nothing to do with it, there is no algorithmical complication to it. Even after applying the diagonalization procedure (diagonalize, prepend, repeat, ...) ad infinitum, you'll still be left with "countably many" sequences.

The failure of set theory and naive (so called "classical") logic and its spawn, model theory, is the realization that Platonism cannot be applied to them without having them lose what made you consider them Platonic in the first place.
You cannot consider each formal system as an island of truth, or an approximation of a "full semantics" second order theory of "true natural numbers" (conveniently instantiated in the mind of the infinitist mathematician), because it will collapse that very Platonic realm into an illogical, contradictory place within which nothing has a clear meaning, which is anathema to mathematics. Trying to save the concept and existence of infinite cardinalities, and with them uncountability, when they are clearly not well defined beyond the formal system within they are stated, leads to all sorts of contradictions and underdefinedness, as you noted, even before stepping outside the classical realm towards more constructive pastures, where the concept loses all meaning and usefulness.
...

Anonymous

Anonymous Thu 04 Feb 2021 17:25:11 No.12666688 Report

Quoted By:

>>12666685
If the Löwenheim-Skolem theorem, proving that what is uncountable to me can be countable to you and viceversa, isn't enough for these people to realize that uncountability is a relative, not absolute, notion, and an artifice of syntax, what will?
They will take quick refuge in squabbles over first-order vs second-order semantics, arguing that first-order theories can only approximate "true" uncountable infinities, and that all is well in the (fundamentally unknowable, undefinable) in the second-order theory, that is categorical. When you nevertheless give definable semantics to their second order fantasies, they will complain that you didn't give the "true" or "full" or "standard" model, i.e. that you did not complete the uncountable infinity. But so didn't they, so what is the difference between before and after they set up all this fantastical machinery, what good is all this complication for, if it does not meet its purpose of completely describing absolute infinity, and even worse, confirm the whole venture to be unfruitful? What did we gain beyond linguistic confusion?

With that in mind, diagonalization is no more an ontological reality, but an operational restriction that a theory, in the right circumstances, may not put its natural numbers in correspondence with its sequences lest it derives a contradiction by a very simple operational argument.
And from that we get to the real matter, that, as higher-order computability results show, things like sequences and functions, fundamentally operational in their nature, are dependent on the their representation in the formal system, and the available operations, so that so called "uncomputable reals" are nothing more than a party trick (as uncomputability is highly volatile).

Anonymous

Anonymous Thu 04 Feb 2021 21:00:19 No.12667398 Report

Quoted By: >>12667421 >>12667490

I missed the previous platonitard bashing threads, so I'll ask this here.
>What's going on with reals in Coq?
>How can they have computable reals with some Cauchy construction in the CoRN library?
>Is interval arithmetic a valid approach to real analysis?
>What does computable real analysis do differently?

Anonymous

Anonymous Thu 04 Feb 2021 21:01:49 No.12667400 Report

Quoted By:

>>12663815
these foundations of mathematics are all bullshit
a real number should be defined as a computable number, end of.

Anonymous

Anonymous Thu 04 Feb 2021 21:09:53 No.12667418 Report

Quoted By: >>12668578

What is the problem with a existence that is never explicit, such as with the irrationals? Is the forever hidden expression for this objects enough for their rejection?

Anonymous

Anonymous Thu 04 Feb 2021 21:10:47 No.12667421 Report

Quoted By: >>12667429

>>12667398
>What's going on with reals in Coq?
>How can they have computable reals with some Cauchy construction in the CoRN library?
What do you mean?
A sequence is a fuction from the natural numbers, so it can be a computable function.
Most of the disagreement between constructive and classical mathematics is a matter of nuance, you can interpret classical mathematics in a constructive setting.

Anonymous

Anonymous Thu 04 Feb 2021 21:13:22 No.12667429 Report

Quoted By: >>12667490 >>12667493

>>12667421
So the only problem is (Girard forgive me) s*mantics? Or is it that classical syntax manipulations are logical garbage?

Anonymous

Anonymous Thu 04 Feb 2021 21:18:19 No.12667447 Report

Quoted By: >>12667453 >>12667491

>>12666284
>finite amount of work to add these sequences.
Ah, another day, another instance where infinitards play fast and loose with the definition of "infinite sequences". How are you getting your infinite sequences? Your two options are either:
1) sequences generated by algorithms (like 1/n, infinite sums, etc.)
2) sequences generated by a (random) choice of rational numbers (i.e. each rational number is essentially picked independently and at random).
Unfortunately for you, mathematics today goes with definition 2, which means that general Cauchy sequences (not your 1/n baby example) require an infinite amount of work to do arithmetic with (let me know if you can add an "infinite number" of random rationals finitely; in fact, let me know if you can even procure and infinite sequence of random rationals). Which is why you play another classic game that infinitists play: whenever you are forced to give examples of your """theory""", you immediately go to simpler examples like 1/n. Constraining yourself to "infinite" sequences generated by rational functions is perfectly reasonable, since, as your correctly demonstrated, there is a reasonable way of determining equality between two such sequences and doing arithmetic with them. However, when you allow general algorithms in your number system, you also have to answer questions such as: how are you determining when two algorithms are equal (you can't which is why real number arithmetic in general is undecidable)? how are you going to define the laws of arithmetic? So basically, what >>12666409 said.
cont

Anonymous

Anonymous Thu 04 Feb 2021 21:19:23 No.12667453 Report

Quoted By: >>12667459

>>12667447
And perhaps most importantly, constraining yourself to computable numbers, by definition, does not give you uncomputable numbers so you haven't actually gotten the full "set" of "real numbers". This is because your general Cauchy sequence is an infinite sequence of random rational numbers; so you can say "to add two general Cauchy sequences a_n,b_n, just add them pairwise", but you're actually asking me to go do an infinite amount of work, which is retarded since no one can do that and the only reason you delude yourself into thinking you can is because ZFC just assumes you can and you don't question the stupidity of assuming such a thing.

>If you complain that I've not added the equivalence class of all sequences with the same limit,
Yes, I will complain about that, because (one of) your definitions of a real number is an equivalence class of Cauchy sequences. So to add pi + e and have it be another element in your number system (an equivalence class of Cauchy sequences for "pi + e"), you really have to add all the sequences in the equivalence class. I'm not the one who came up with that retarded definition, so if you subscribe to it, get to adding.
cont

Anonymous

Anonymous Thu 04 Feb 2021 21:20:25 No.12667459 Report

Quoted By: >>12667464

>>12667453
But let's face it, even if I just ask you to add one Cauchy sequence for "pi" and one Cauchy sequence for "e", you aren't actually going to give me the result, because the result can't actually be demonstrated (let me know if you can exactly specify an uncountable set of Cauchy sequences of rationals in a finite way without three dots or requiring me to "consider all..." or do an infinite amount of work). You're at best going to give me option 3 from the original post, which is an algorithm that allows me to get arbitrarily close to the "value" of "pi + e". Therefore, if I want the "true value" (assuming it even exists in the first place), I am going to have to do an infinite amount of work to go from your algorithm to an actual number. And if you say the algorithm itself is the result, then you can't determine the equality of algorithms or do arithmetic with them (see above), so if someone else comes along with their own algorithm for "pi + e", you have no way of determining whether your algorithm is the same as their's. This is why """arithmetic""" with real numbers is so vacuous and you are left just saying that "pi + e = pi + e", since there are an infinite number of perfectly valid ways to represent algorithms for "pi + e" and there is no canonicalization that quickly allows you to determine whether two such algorithms are equivalent (unlike reducing a rational number).
cont

Anonymous

Anonymous Thu 04 Feb 2021 21:21:36 No.12667464 Report

Quoted By:

>>12667459
>I will complain that you can't add rational numbers either. Every rational number has an infinite equivalence class of tuples...
Which is why representing rational numbers as "infinite" equivalence classes of tuples of integers is retarded. As >>12666351 said, there are many finitary ways of working with rational numbers; your definition of a rational number is not God-given so there is no reason why anyone has to immediately subscribe to it. I can just work with tuples of integers <a, b> and <c, d> where b,d != 0, and equality is satisfied when ad = bc. So yes, there can be multiple representations of the same rational number, but I have a completely finitary arithmetic and equality metric. Meanwhile, for two general Cauchy sequences (not your 1/n or other baby examples), you have to go do an infinite amount of work (if your Cauchy sequence is a random choice of rationals) or try determining the equality of algorithms, which is impossible (if your Cauchy sequence is algorithmically specified, in which case you don't even have your uncomputable real numbers anyways).

Anonymous

Anonymous Thu 04 Feb 2021 21:27:40 No.12667490 Report

Quoted By: >>12667676

>>12667429
>>12667398
bump
>Is rational interval arithmetic a valid approach to real analysis?
>What does computable real analysis do differently?
>Is the whole disagreement over syntax or semantics?

Anonymous

Anonymous Thu 04 Feb 2021 21:27:46 No.12667491 Report

Quoted By: >>12667576

>>12667447
>how are you determining when two algorithms are equal
Let $f : N \to Q$ and $g : N \to Q$ be rational-valued computable functions of the naturals, "algorithms" for infinite sequences if you will, f and g are equal iff $\forall \delta \in Q$ such that $\delta > 0$ , there exists an $m \in N$ such that
$\forall n \in N . n \geq m \implies | f (n) - g (n) | < \delta$
what's the problem?

Anonymous

Anonymous Thu 04 Feb 2021 21:27:53 No.12667493 Report

Quoted By: >>12667518

>>12667429
The matter evolved over time.
Semantically, yes, the problem is that classical mathematicians literally have no idea what the fuck they're talking about and need someone to tell them the meaning of the shit they produce, because even their own work tells them that it's not what they read it to be, yet they refuse to listen. They don't want to descend from Cantor's paradise.
Syntactically, I feel most "computer constructivists" today would respond that classical mathematics is a stilted language that misses and conflates things that shouldn't be conflated, so while one can be interpreted in the other and viceversa, one is more "equal" than the other. Constructive mathematics makes more distinctions that are more useful. If you've done numerical analysis, knowing constructive analysis would tell you straight away what you can and cannot do.
Not all old school constructivists would agree with that, and would argue that classical mathematics is devoid of all meaning and use, but the problem with those in my opinion is that they took a step forward and were content with that, a problem that's occurring all over again with this new camp. And you can shuffle paper around in any mathematical discipline, it's just easier in a field where it's just fools fooling themselves. For fully constructive waste paper bin fillers, look no further than "pure" category theorists, who are today's "pure" set theorists.

Anonymous

Anonymous Thu 04 Feb 2021 21:28:38 No.12667500 Report

Quoted By:

No one has the autority to dictate what a genuine, objective, true number is. Nor is it that serious mathematicians intend to do so. By means of tradition, creativity, heuristics and scientific values is that mathematical researchers come up with some logical concepts and not others. The intention is not to vanish all the other possible logical notions. The spirit is not that of controversy. Since the the Ionians, the most distinctive aspect of scientific knowledge is its susceptibility to criticism; that its validity is open for anyone to refute, check or rediscover.

Anonymous

Anonymous Thu 04 Feb 2021 21:28:53 No.12667503 Report

Quoted By:

Wildbergians should be banished from /x/, alongside antivaxxers and other schizos

Anonymous

Anonymous Thu 04 Feb 2021 21:29:55 No.12667506 Report

Quoted By:

Wildbergians should be banished to /x/, alongside antivaxxers and other schizos

Anonymous

Anonymous Thu 04 Feb 2021 21:32:49 No.12667518 Report

Quoted By: >>12667681

>>12667493
>look no further than "pure" category theorists, who are today's "pure" set theorists
I don't know much about it, but is it actually computationally meaningless? I thought it was just some language for expressing "standard" i.e. commonly encountered properties of structures and other things.

Anonymous

Anonymous Thu 04 Feb 2021 21:35:22 No.12667532 Report

Quoted By:

>>12666380
So what? We make meaningful decisions about time and we literally can't perceive it directly.
Being unable to perceive something directly has not stopped us from being able to make meaningful statements about that thing.

Anonymous

Anonymous Thu 04 Feb 2021 21:37:58 No.12667548 Report

Quoted By: >>12668578

>chad literal geniuses conduct abstract mathematics, derive usable results, change the world
>virgin pseuds cant imagine anything not concrete, REEEEEEE and cry, do nothing of note and contribute only a further understand of autism spectrum disorders to the body of human knowledge
Imagine getting fucking filtered by the abstract definition of a number.

Anonymous

Anonymous Thu 04 Feb 2021 21:45:08 No.12667576 Report

Quoted By: >>12667617 >>12667694

>>12667491
The problem is that the definition of Cauchy sequences is inherently infinitary. You are writing something down that you, in general, cannot do (or that requires an infinite amount work to actually accomplish). If you could actually apply this definition generally, you could solve a lot of open problems in mathematics almost immediately.
As an example, consider the sequence s_n, where s_n=2 if n is a twin prime, and s_n=1 otherwise. Does this sequence converge to 1? Obviously, s_n and the sequence t_n=1 are completely algorithmically specifiable, but if I could apply your definition of equality of sequences to this problem, I can solve the twin prime conjecture.
I obviously can't do that, so if you choose this as your equality metric, you are requiring me, in general, to do an infinite amount of work to determine whether two real numbers are equal; this again comes back to the fact that there is no canonicalization of algorithms that allows me to quickly determine their equality. And this is why real number equality is in general undecidable.
And finally, you've just "specified" equality. Good luck defining arithmetic and proving the laws of arithmetic without an infinitary definition.

Anonymous

Anonymous Thu 04 Feb 2021 21:52:55 No.12667617 Report

Quoted By: >>12667870

>>12667576
It's not possible, in general, to determine whether two real numbers are equal, but I don't understand why that should be such a problem.

Anonymous

Anonymous Thu 04 Feb 2021 22:04:31 No.12667676 Report

Quoted By: >>12667681

>>12667490
>Is rational interval arithmetic a valid approach to real analysis?
I don't know enough to say. If you're interested in that you should just learn the formalisms and see for yourself. You won't necessarily become an infinitist schizo if you learn classical mathematics, nor will you commit to be a constructivist if you read up on constructive analysis.

>What does computable real analysis do differently?
There's many approaches.
In a pure constructive setting, you can't construct any discontinuous functions from $\mathbb{R}$ . In a sense all constructive functions are continuous. Continuity is linked to computability. Check out Martin Escardó's work for examples.
You can write certain discontinuous functions if you're explicit. That means changing the representation of the reals from "the continuum" to (for example) its decomposition in absolute value and sign: $x : \mathbb{R}_s = \mathrm{sgn}(x) \cdot |x|$
So say you have your favourite construction of $\mathbb{R}^+$ , positive nonzero reals, then $\mathrm{abs} : \mathbb{R}^+ + \{0\} + \mathbb{R}^+ \to \{0\} + \mathbb{R}^+$ is the type of the absolute value function, and you can say that $\mathbb{R}_s = \mathbb{R}^+ + \{0\} + \mathbb{R}^+$ .
In general, when you cut the continuum, you have to carry around proof that it is cut, explicit in the types/sets. Implicit data in classical logic become explicit dependences constructively, that are needed to compute.
There's problems in constructive analysis, stemming from the fact that constructive logic is excellent at telling you what there is, but awful at describing limited possibilities, which classical logic is better at by accident. So you end up in situations where you conclude weak double negations even where you could've squeezed more information out of it.
...

Anonymous

Anonymous Thu 04 Feb 2021 22:05:33 No.12667681 Report

Quoted By: >>12667737

>>12667676
This is not enough reason to discard the constructive formalism, but it is a pain point. For that reason, constructive analysis is somewhat underdeveloped.
The solution could be found in linear logic, but linear logicians are too busy being gay about proof search.

>>12667518
It's not computationally meaningless, for the most part it's computationally trivial. It's a powerful tool when applied to something concrete, but it's easy to lose yourself in the abstractness, and see profundity in trivial manipulations.
At its core, computationally, category theory is about reassociating pairs and shuffling and matching arguments to functions.
The problem is when you try too hard to make your structure fit in categories, rather than having categories fit your structure. When the structure fits the formalism though, it's incredibly useful.

Anonymous

Anonymous Thu 04 Feb 2021 22:07:23 No.12667694 Report

Quoted By: >>12667870

>>12667576
To apply the definition of equality I propose to your example, you need only prove or disprove the twin-primes conjecture, you don't have to actually explicitly check each number. I don't expect it will take an infinite amount of work to prove the twin primes conjecture.

Anonymous

Anonymous Thu 04 Feb 2021 22:16:38 No.12667737 Report

Quoted By:

>>12667681
Thanks
>linear logicians are too busy being gay about proof search
that's the best part, also don't forget quantum schizo

Anonymous

Anonymous Thu 04 Feb 2021 22:43:38 No.12667870 Report

Quoted By:

>>12667617
Then you don't actually have a number system. If your definition of real numbers is algorithmic (so a real number to you is an algorithm), then you do not have a way of determining whether two elements in your number system (i.e. two algorithms) are equivalent, let alone do arithmetic on them. So if you can't tell me whether two arbitrary "numbers" are equivalent without doing an infinite amount of work, then it is dishonest and incorrect to say you have a number system. Which is why people don't even bother with the computability approach, since such a thing is too difficult to pin down completely precisely and instead they just rely on axiomatic approaches to do an infinite amount of work by fiat. At least by assuming the ability to "do" an infinite amount of work they "can" establish equality on real numbers.
However, I do have a lot of sympathy with a computability approach to irrationals, but the essence of my objection is that I'm criticizing something that doesn't actually exist. It would be really cool if we had a theory of algorithms that worked in a purely finitary way, so we could get (computable) irrational numbers, but we don't so you're talking about something hypothetical as if it already exists.
That is the essence of the objection here; as of today, you are essentially talking about something that is approximate (you have an algorithm that can produce an arbitrary number of digits of say "pi" for example) as if it were something precise, but considering the algorithm itself is intractable arithmetically (e.g. to determine equality).

>>12667694
Great, and I can keep coming up with more and more sequences for more and more problems. Collatz conjecture? Any arbitrary Turing machine? We're straying into philosophy now, but the number of mathematical problems is almost surely unbounded, so to have a concrete, algorithmically-defined """real""" number system, as of today, we just need to solve all open (and future) problems in mathematics.

Anonymous

Anonymous Thu 04 Feb 2021 23:15:52 No.12668038 Report

Quoted By:

Fake faggot shit cuck professor scum

Anonymous

Anonymous Fri 05 Feb 2021 00:34:51 No.12668346 Report

Quoted By:

>>12663823
>but the bullet would never reach his brain as first it had to cover half the distance, then 2/3, then 3/4 and so on in an unending sequence that he himself had so foolishly proclaimed the existence of!
kek

Anonymous

Anonymous Fri 05 Feb 2021 00:56:00 No.12668441 Report

Quoted By:

>>12663815
if you start by constructing the reals on the first day of class, without really spending the time to see what properties you're constructing this set for, it's a waste of time that imparts no intuition.
The proper way to do it is this:
>talk about how we want number systems and functions to help us sketch out mathematical situations
>introduce rationals
>introduce relevant field and ring properties that let us do basic inequalities
>introduce LUB property
>show how the rationals fail the LUB, meaning we can't meaningfully talk about function behavior with just the rational set (wildberger leave reee) using our intuition about limits
>end first class

this is of course assuming your students are comfortable with sequences, convergence, etc etc. If this is an intro to real analysis and not a proper real analysis class, then just start with some basic properties of sequences, recall basic set theory like demorgan's, and then start the above.
Construct the reals later - you can always go back to it in the coming weeks.

Anonymous

View Same Google iqdb SauceNAO Dedekind cuts.jpg, 87KiB, 932x661

Anonymous Fri 05 Feb 2021 01:06:27 No.12668481 Report

Quoted By: >>12668578

>>12663815
Dedekind Cuts Chad
vs
Cauchy Sequences Virgin.

Anonymous

Anonymous Fri 05 Feb 2021 01:28:13 No.12668578 Report

Quoted By:

>>12668481
Both are equally fake.
>>12667548
Nothing wrong with abstract definitions. The definition of rational numbers and their arithmetic are abstract, but they're perfectly fine. The problem is that the reals are undefined. If you have a definition of real numbers please provide it.
>>12667418
>Is the forever hidden expression for this objects enough for their rejection?
What objects? What are these irrationals that you talk about?

Anonymous

Anonymous Fri 05 Feb 2021 01:37:59 No.12668622 Report

Quoted By:

>>12663823
And that ex marine student’s name was.... Albert Einstein

Anonymous

Anonymous Fri 05 Feb 2021 03:24:50 No.12668979 Report

Quoted By:

>>12666184
isnt this whole argument pointless since all practical instances of any theory ever explored will contain a finite amount of human work in them and no more? and isnt that his point anyway? may as well add some terms to expand on whatever you're talking about to give him some "real" work to do while you continue musing finitely upon the theories of infinities

Anonymous

View Same Google iqdb SauceNAO 1611251864850.jpg, 89KiB, 720x960

Anonymous Fri 05 Feb 2021 03:41:23 No.12669024 Report

Quoted By:

>>12663823
based

Anonymous

Anonymous Fri 05 Feb 2021 03:48:59 No.12669041 Report

Quoted By:

>>12663823
>>He wished so much that he could shoot himself, but the bullet would never reach his brain as first it had to cover half the distance, then 2/3, then 3/4 and so on in an unending sequence that he himself had so foolishly proclaimed the existence of!

First time I keked on this board, thanks anon

Anonymous

Anonymous Fri 05 Feb 2021 03:53:55 No.12669052 Report

Quoted By:

>>12663823
>wall-of-text schizo
erry time

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