/sci/ - Science & Math

[Anonymous No.7975561]

The burger goes into complexity,

tl;dr
if you set up a language, say with "1,2,3...,+,*,^,(,)" and potentially hyperexponentiation [math]\Delta [/math], so that
[math] 10\Delta 3:=((10^{10})^{10})^{10} [/math] or whatever (or any other operation).

In the worst case, a 7 digit number number like 4523255 has complexity C(4523255)=7.
but some numbers, like 4782969=9^7 actually have smaller complexity C(4782969)=3.
For the most part, C(n) behaves like log_10(n), corresponding to the cases where you need to use digits in base 10 and there is no representation using +,*,^ or whatever you introduce.
Up to numbers around [math] 10\Delta 10 [/math], the complexity typically grows [math] 10\Delta 9 [/math], except of course for the occasional stuff like [math] 7\Delta 8 [/math], which is an absurdly large number that nevertheless has just [math] C(7\Delta 8)=3 [/math] in your language.

With the value [math]VMU:=10\Delta 4[/math] he argues is basically the length of symbols you can use to express a number (quote: if you manage to build a Galaxy scale computer where storing stuff with [math]10\Delta 4[/math] symbols can be achieved), then most numbers in the interval (on those kind of scales) around them are naturally off limits.

[Anonymous No.7975566]

>>7975561
I want a graduate to answer to him.

[Anonymous No.7975568]

forgot link
https://www.youtube.com/watch?v=EUvFXd1y1Ho

[Anonymous No.7975712]

>>7975566
The hard part is the illustrative examples. All of the examples that people will use will either require the axiom of choice or an infinite logical sentence, neither of which he deems acceptable.

>>7975561
That's actually pretty interesting. Is he actually looking for a hard number upon which to base ultrafinitist mathematics or is he just exploring the general area?

Also has he done any videos explaining how he handles induction?

[Anonymous No.7975744]

>>7975712
>illustrative examples
If we define reals as convergent sequences of rationals, then the following sequence :
[eqn] U_{n+1} = \frac{2}{U_n} [/eqn] with [eqn] U_0 = \frac{3}{2} [/eqn]
Converges but its limit is not a rational, it's [math] \sqrt{2} [/math]

[Anonymous No.7975769]

what point is he trying to make? we shouldnt care about big numbers? they dont exist?

[Anonymous No.7975773]

>>7975769

[Anonymous No.7975828]

>>7975769
That we need to re-examine the rigor with regards to the way we are formalizing our numbers. He has chosen to see how much math he can formalize by working with an insanely rigorous subset of numbers, in particular he's beginning with the finite set of integers that can be represented in the physical world and from that set constructing a finite subset of rationals. Finite because as you approximate closer and closer to an irrational, via rationals, you end up with larger and larger integers a,b where the given rational is a/b. Eventually said integers become too large to be representable in the physical universe.

Think about it, what justification do you have for making claims about numbers that you don't even have enough matter in the universe to represent in any way (written or otherwise)?

[Anonymous No.7975833]

>>7975744
Have you watched Wildburgers videos?

Anyway, what you say doesn't work: Take The sequence you defined and consider the New sequence
[math] V_n := U_n [/math] for n not 74
and
[math] V_{74} := -28 [/math].

This new seqeunce has the same limit but is different from the other "real number". You get to many terms in your theory.

People thus pull up an autistically large framework of set theory so they can define reals as equivalence classes of such sequences.
okay, then take this equivalence class definition of the reals.
Consider a sequence [math]W_n[/math] which for the first d=10^10^10^10^10^10^10^10 numbers are some random natural numbers and after [math]W_d[/math] you have [math]W_d:=U_d[/math]. Hence [math]W_n[/math] converges to your root two and thus is a representative of that equivalence class. However, if you had someone the sequence [math]W_n[/math] and ask him what number it represents, he can't tell you, because all that's accessible to him is random gibberisch. Wildy disregards those nonconstructive approaches for this reason, pointing out that your + operation can't be effectively defined. You need to access the infinite to do anything but numbers defined via small complexity expression, say
[math] \pi = 2 \sum_{k=0}^\infty \frac{ 2^k (k!)^2 } { (2k+1)! } [/math]

[Anonymous No.7975837]

>>7975744
Have you watched Wildburgers videos?

Anyway, what you say doesn't work:
Take the sequence you defined and consider the new sequence
[math] V_n := U_n [/math] for n not 74
and
[math] V_{74} := -28 [/math].

This new sequence has the same limit but is different from the other "real number". You get too many terms in your theory of real numbers.
People thus pull up the autistically large framework of set theory so they can define reals as equivalence classes of such sequences.
Okay, then take this equivalence class definition of the reals.

Consider a sequence [math]W_n[/math] which for the first d=10^10^10^10^10^10^10^10 numbers are some random natural numbers and after [math]W_d[/math] you have [math]W_k:=U_k[/math]. Hence [math]W_n[/math] converges to your root two and thus is a representative of that equivalence class. However, if you had someone the sequence [math]W_n[/math] and ask him what number it represents, he can't tell you, because all that's accessible to him is random gibberisch. Wildy disregards those nonconstructive approaches for this reason, pointing out that your + operation can't be effectively defined. You need to access the infinite to do anything but numbers defined via small complexity expression, say
[math] \pi = 2 \sum_{k=0}^\infty \frac{ 2^k (k!)^2 } { (2k+1)! } [/math]

[Anonymous No.7975842]

>>7975837
let me correct myself:
the + operation can be algorithmically defined, but you can't actually evaluate it if each equivalence class, like the one representing pi, say, contains uncountably infinite many elements that are random numbers and only after the 10^10^10^10^10^10's slot in the sequence exhibit a relation that will converge to pi.

[Anonymous No.7976834]

correct

[Anonymous No.7977628]

finbump

[The Fairy Queen No.7977657]

>>7975828
>what justification do you have for making claims about numbers that you don't even have enough matter in the universe to represent in any way (written or otherwise)?
Oh, that. No, there's no such thing as "no representation." I can as easily assign a name to such a number, called it fibrilator, and it can be larger than any formal system can constructively express even with excessive use of hyperoperators, universe sized computational arrays, and storage-in-time-loops, and we'll still be able to do math with it simply because we know it's a number. Merely by creating symbols we can represent numbers that exceed any and all manners of fully qualified comprehension. It's not necessary to know all the properties of a number (where "base ten stringification" is one type of property) if we want to study it. Math is really about studying numbers well beyond our comprehension/construction range. Abstract math doesn't benefit from ultrafinistist notions for this reason: It doesn't care about finiteness.

You could as well call ultrafinist numbers boring numbers and it would hold as much meaning in the world of abstract mathematics. Topology for example literally cannot benefit from ultrafinitist mathematics because the structures used in topology are so abstract that algebraic topology becomes a second-order reflection of the richly genuine topological definition supposed.

You could also say topology is the study of the fibrilator. (Or "any" fibrilator, if you prefer.)

[The Fairy Queen No.7977674]

>>7975837
>and ask him what number it represents, he can't tell you
He doesn't need to because you didn't ask the question to begin with. Supposing that infinitely complex questions can be asked and then asking for a non-infinite answer to that image of a question is basically infinitely deceptive. If we aren't allowed to construct infinite answers then don't ask us to answer questions with infinite precision.

[Anonymous No.7977710]

>>7975828


x=TREE(TREE(TREE(TREE(TREE(TREE(TREE(2)))))))

I bet you've sucked x dicks ;^)

But seriously, if our way of representing numbers is inadequate, why not just improve the way of representing them with things like the Ackerman function or knuth arrows.

[Anonymous No.7977763]

>>7977657
As dubious as the claim is that you can name a number that you can't even describe and still make assertions about it; even if you were to do this for every unthinkable number you mange to think up (lel) you will still only ever be able to produce a finite number of names to such numbers. Note that I'm not referring to time or your own human limitations here but to the way that the formal languages work for upon which we describe logics and axiomatic systems. Eventually the sentences you attempt to use will be too large to write or represent.

There are computational formalizations of topology where none of your claims hold any water.

With regards to the claims that ultrafinitishm is useless or that they don't hold much meaning. I disagree, as do others:
ftp://pier.botik.ru/rented/logic/papers/SAZONOV/lcc.ps

>>7977710
The problem isn't in describing some of these exceptional numbers, the problem is in coming up with a system for describing all of them. Also, the issue with your approaches is that in order to introduce them at the foundational level (i.e. logic) then you would need extra predicates and then your theory becomes far more complicated. I think Martin Lof's Intuitionistic Type Theory might actually resolve this issue in asserting a canonical number but it's not something I've looked into so I could be trivially mistaken.

There was also an interesting paper a few years back that talked about the problem with using these higher operations as definitions that you may be interested in:
https://web.math.princeton.edu/~nelson/papers/warn.pdf

[The Fairy Queen No.7977785]

>>7977763
>a number that you can't even describe
Oh nononono. I can *describe* it just fine. The point is that even if we can't describe all of its traits without a universal computational time loop, we can still talk about its properties. For example, it's either divisible by two or not. No matter how large the number is, it'll still have the Boolean trait of divisibility by two.

>>7977763
>a finite number of names
God I hoped you were gonna go there. Fibrilator+1.

You can't win an ultrafinity argument against a fairy.

>>7977763
>Eventually the sentences you attempt to use will be too large to write or represent.
Except they won't be because you don't have it in you to construct an interesting statement that exceeds the computational mass of the universe. Legitimately interesting numbers crop out some time after Fibrilator^^32.

>>7977763
>useless
I never said it was useless. It CAN be interesting if you can find the interesting parts.
>>7977763
>coming up with a system for describing all of them
Fibrilator. Did you think my number was just a pet name for a theory or something and not an actual number?
>>7977763
>warn.pdf
I'd love to give it a read some time, but I'm still at a severe technological disadvantage here. Suffice to say I'm not permitted access to external resource networks at the moment.

[Anonymous No.7977807]

you guys fail to see that mathematics are not meant to be connected back to the empirical world.
The purpose of the formalization of your thoughts is to go as far as possible form the empirical world.

Constructivists refuse to see this and whine that >muh 10^10^10^10^10^10^10 is too hard to grasp. They admit that they have never ever reflected on their desire to formalize their thoughts and dwell in their abstractions.

[Anonymous No.7977828]

>>7977785
>Oh nononono. I can *describe* it just fine.
At best you've described the set it belongs to. Can you list out it's digits? If I hand you two black boxes with a slot on the side and a button on top and I claim that
>Each of these black boxes will begin to print out the digits of two numbers of unfathomably large size as soon as you push the button.
>These numbers are different.
How could you verify that the two numbers are actually infinite? Doing so would require you to compare each number digit by digit. Sure you could claim that since the numbers are finite then this process would complete in finite time (as one does in computability theory) however, what if these numbers are so large that you can't ensure you've spotted a difference in digits by the time the universe ends?

You are only able to describe properties of subsets of very large numbers but never actually able to give properties for individual numbers of that size. Can you tell me if the number is prime? Can you tell me if it's a sum of squares?

>Legitimately interesting numbers crop out some time after Fibrilator^^32.
>The set of legitimately interesting numbers is finite.
Of course, this is why no one realizes that we don't actually ever use real numbers, just a finite subset of them.

>Fibrilator. Did you think my number was just a pet name for a theory or something and not an actual number?
It is A number. If you want to describe a different number not equal to fibrillator then you need a different name for it. You would need an infinite set of names to talk about an infinite set of numbers. Said names, given a finite alphabet, will eventually become too large for the universe.

[Anonymous No.7977832]

>>7977807
>claims to see something everyone else in the thread fails to see
>proceeds to post proof of obvious brain damage by claiming that ultrafinitism and constructivism are the same thing

[The Fairy Queen No.7977879]

>>7977828
>At best you've described the set it belongs to.
Correct! Thus far in the dialogue you've yet to have me have any need for anything past the set itself. I can begin describing actual properties at any time, but we both know that'll be pointless, yes?
>>7977828
>Can you list out it's digits?
Yes. It's 10 in base fibrilator. A perfect 10.

I'm a fairy. You're not going to win. Definitions are the basis of all representation and I control the set of all partial definitions of fibrilator. You won't win in an ultrafinity argument when the fairy controls the definition.
>>7977828
>How could you verify that the two numbers are actually infinite?
What two numbers? Is there an unfathomable amount of something in the universe that I haven't been informed about yet?
>>7977828
>so large that you can't ensure
Then your question will remain unspecified for all eternity. This isn't hard.
>>7977828
>give properties for individual numbers
I can but you seem to be arguing that there'd be no point in my doing so. Sure, on your end the number only exists as an abstraction of intractable size, but to accuse fibrilator of not being a single number is just semantics. It is a single number and your inability to find it in an infinitely large set is not my concern.
>>7977828
>Can you tell me if the number is prime?
>Can you tell me if it's a sum of squares?
I haven't decided yet. I'm waiting for the heat death of the universe so I can transform all matter in the universe into computronium for the sake of a small dispute on 4chan. Hopefully we'll be able to finish this conversation before the backwards regressing time loop consumes our biophysical substrates.

I probably won't pick a prime though. That'd be too poetic for my taste. Maybe we'll represent such a prime number as fib-p or something equally benign.
>>7977828
>we don't actually ever use real numbers
Speak for yourself. Personally I hope to learn topology some day, and for that I'll need abstract mathematics.

[The Fairy Queen No.7977889]

>>7977828
>you need a different name for it
No I don't. fibrilator+n. See, the magic here is that I can make up as many interesting numbers as I want using a single trans-computational number and you can only refute bits and pieces of the set. Even before the universe ends you'll find that it can only contain a graph of numbers surrounding n and fibrilator. n could as well be in league with fibrilator or it could just be 3. I'd only give a name to a number if I necessarily knew that number couldn't be represented in the computationally bonded set of universes, even in reference to one such named number. I only need one name.

>>7977828
>an infinite set of names to talk about an infinite set of numbers
Not so. I can literally shunt you at any time prior to the computational convergence of the universe and force you to acknowledge fibrilator as a set. I can and will shift the goalposts if I have to. There's no way you get our of this dialogue without infinite acceptance of the abstract nature of notation.

>>7977828
>given a finite alphabet
Sounds like a Turing statement.

[Anonymous No.7977930]

>>7977879
>>7977889
You are only describing a finite number of integers, anon. Learn some basic formal language theory.

You don't need real numbers to do topology. In fact you can do topology on finite sets. You can also do topology in category theory where you never actually talk about the set itself and only the properties it possesses (abstracting away the problem of infinity).

>you can't win
You haven't actually made any statement in support of an infinite number of integers. At the very best you've made a very weak argument for potential infinity and against actual infinity. There is literally nothing to fight against except inform you of basic mathematics that you clearly are not familiar with.

[Anonymous No.7979509]

bimp

[Anonymous No.7979825]

The ultrafinitist idea is just bizarre to me. Math isn't about quantities or strings of symbols; it's about relationships and structures. Numbers are a kind of structure which happens to have come from the way we talk about quantities and the symbol-strings that represent them.

The whole idea of not being able to take things to infinity because, basically, "you can't count that high" is ... almost nonsensical. I don't *have* to. I'm working with numbers, not amounts, and being able to talk about something and being able to convert it into a physical quantity are not the same thing.

There can't be a biggest number, because we know that every number has a successor. And I can prove this without ever needing to "prove it" by counting to it.

[Anonymous No.7979868]

>>7979825

The problem is that you're only working with a model of the numbers here, not an axiomatic system. In order to talk about numbers formally you have to give an axiomatic system where these numbers exist as a model. The Peano axioms are popular because every model of this system is isomorphic to every other model and one is able to easily formalize induction. Set theory is also popular because using sets you can model a great deal of math. The type theoretical and categorical approaches for giving models of the naturals mimic the approach used by the peano axioms.

The thing that wildberger is complaining about is further under the hood than the model level. He's claiming that we as humans can't actually justify these models because we can't actually formalize them without an infinite universe and infinite resources over infinite time. It's a very pedantic argument but at the same time it's posed at a level where we lack the formalism to really fight him on it.

[Anonymous No.7979957]

>>7977763

> As dubious as the claim is that you can name a number that you can't even describe and still make assertions about it

Are you serious?

> what is 2^2^...some large but finite amount of times ...^2

I can tell you that it's even, it's prime factors are all 2, etc... Please tell me this was a troll and that your wall of text was just an elaborate cover up. Otherwise you're truly the most vapid human I could have constructed.

[Anonymous No.7979968]

>>7979957
>vaguely (no precise number of exponentiation) gives a number with a closed form expression ignoring the fact that in the grand infinitude very few numbers have such an expression.
>uses this as the basis for an argument that you given an arbitrary unthinkable number, one can make assertions about it.

gb2/kindergarten, you clearly didn't learn how quantifiers work.

[Anonymous No.7979971]

>>7979957
The only reason you are able to make claims about this number is because you have actually made claims about certain sets of numbers and then are defining this number so that it belongs in those sets by construction.

[Anonymous No.7979978]

>>7975568
HE LITERALLY JUST READS THE DIGITS OF POWERS FOR MINUTES FOR FUCKS SAKE WHO GIVES A SHIT
AND YOU FUCKERS TAKE THIS SHILL SERIOUSLY
WHAT THE FUCK IS WRONG WITH ALL OF YOU

[Anonymous No.7979979]

>>7979971
ok change the goalpost that's fine

literally go back to ancient greece and learn some rhetoric skills you mudslime

[Anonymous No.7980027]

>>7979868
>every model of this system is isomorphic to every other model

This isn't true, and in fact it isn't true of any first order theory with an infinite model (upward Lowenheim-Skolem theorem). Peano arithmetic is aleph-naught categorical though (it has a unique countable model), which is probably what you meant. Sorry for being autistic.

[Anonymous No.7980048]

>>7980027
>Sorry for being autistic.
No, thank you actually, you're correct. I've made this mistake before.

[Anonymous No.7980057]

>>7975837

SCNR

[Anonymous No.7980158]

>>7975561
Is the complexity that he refers to equivalent to Kolmogorov complexity, but for algebras?