/sci/ - Science & Math

[Anonymous No.5084102]

Daily reminder that mathematics is merely a human language which captures distilled computational artifacts from the computational processes of one's mind emergent from the underlying computational nature of reality, set theory is garbage and should be superseded by category theory, infinite sets don't exist, and infinitesimals are merely heuristic placeholders for asymptotic bounds of non-halting recursive algorithms.

http://www.youtube.com/watch?v=CaE76mwI8Nk
http://arxiv.org/abs/1003.5831
http://en.wikipedia.org/wiki/Church%E2%80%93Turing%E2%80%93Deutsch_principle
http://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence
http://en.wikipedia.org/wiki/Curry-Howard_correspondence#Curry.E2.80.93Howard.E2.80.93Lambek_corresp
ondence
http://en.wikipedia.org/wiki/Transfer_principle

[Anonymous No.5084120]

I was under the impression that 'infinitesimals' were rigorously defined by the notions of p-forms. Is there's some other context outside of integration where you'd encounter them and this interpretation doesn't hold?

[Anonymous No.5084134]

Also, what do you mean by infinite sets don't 'exist'?

[Anonymous No.5084163]

http://en.wikipedia.org/wiki/Digital_physics#Criticism

[Anonymous No.5084174]

Really? We have a proposition for a somewhat interesting thread and people are going to just throw around wikipedia links?

[Anonymous No.5084183]

>>5084174
to be fair OP started it by cherry picking some random sources. the article I posted basically covers all these points. I'm not opposed to these ideas necessarily but i don't think op really made his point clear either

[Anonymous No.5084188]

>>5084183
luckily i sound really whiny in this post.
I did think the church-turing-deutsch principle was pretty cool though.
"Deutsch generally takes a "multiverse" view to the question of continuous vs. discrete. In short, he thinks that “within each universe all observable quantities are discrete, but the multiverse as a whole is a continuum. When the equations of quantum theory describe a continuous but not-directly-observable transition between two values of a discrete quantity, what they are telling us is that the transition does not take place entirely within one universe. So perhaps the price of continuous motion is not an infinity of consecutive actions, but an infinity of concurrent actions taking place across the multiverse.”


i have totally thought about multiverses being continuous before..i dont have any ability to justify that however.

[Anonymous No.5084190]

Holy shit, your writing style is awful, OP.

[Anonymous No.5084191]

>cs

who let that monkey in?

[Anonymous No.5084200]

>>5084102
>infinite sets don't exist

So there is a "biggest number"?
Are you fucking retarded?
When were you diagnosed with your particualr mental disorder? Do you get assistance from the government?

[Anonymous No.5084205]

>>5084174
>this thread
>interesting

NOPE

[Anonymous No.5084206]

>>5084183
Fair enough. So, anyone else know anything about this 'replacing set theory with category theory'thing?

[Anonymous No.5084513]

>>5084102
>implying math has to be restricted to the physically possible
>implying CS isn't math
>implying we can't think of models of computation stronger than Turing machines

[Anonymous No.5084597]

>>5084513
What do you mean computationally stronger? There's nothing you can't do with a Turing complete model. Even theoretical devices such as quantum computers are just non-deterministic Turing machines.

[Anonymous No.5084604]

>>5084597
Of course th-
>Even theoretical devices such as quantum computers are just non-deterministic Turing machines.
Oh, right, you don't know what you're talking about.

[Anonymous No.5084615]

>>5084604
Ok then, please tell me what can't be computed with a turing machine, besides anything beyond turing-recognizable.

[Anonymous No.5084626]

>>5084615
You said it, anything beyond Turing-recognizable: real numbers, non-Turing-computable functions, the halting problem for Turing machines.

And quantum computers are not non-deterministic Turing machines.

[Anonymous No.5084632]

>>5084626
Also, Turing machines can't generate truly random numbers, so a Turing machine with a truly random number generator can compute things TMs can't compute.

[Anonymous No.5084641]

>>5084636
Yes. Read >>5084513 again.

[Anonymous No.5084636]

>>5084626
you're talking about hypercomputation. These things are physically impossible.

[Anonymous No.5084645]

>>5084626
Also, how are quantum computers not NDTMs? Their greatest advantage is superposition, which is only good for solving np problems in p time.

[Anonymous No.5084647]

>>5084641
oh, nevermind, derp.

[Anonymous No.5084646]

>>5084645
>which is only good for solving np problems in p time.
http://en.wikipedia.org/wiki/BQP

[encephalon !!shxfCFb4+j/ No.5084649]

>>5084102
OP is chill as fuck.

[Anonymous No.5084659]

>>5084646
I'm not seeing what you're trying to prove. BQP is in the NP domain.

http://en.wikipedia.org/wiki/File:BQP_complexity_class_diagram.svg

[Anonymous No.5084664]

>>5084659
I assumed "NP problems" = "NP-complete problems"

[Anonymous No.5084773]

>>5084626
What you fail to understand is that humans can't solve or compute those either.

Human mathematicians can't calculate non-computable reals.
Human mathematicians can't calculate the result of a non-Turing computable function.
Human mathematicians can't solve computationaly undecidable problems.

What holds for Universal computing machines also holds for human minds.

And Quantum Computers are not non-deterministic Turing machines. You do not understand Quantum Computation, nor do you understand Quantum Mechanics or how probability is not the same as non-determinism. Everything in Quantum Mechanics is deterministic and reversible, wave-function collapse doesn't actually occur in nature, it's an emergent illusion of how we as intelligent agents are embedded across a dimension of time which is most likely just an entropic ordering of states. You are lost in an outdated misunderstanding of QM.

Everything that can be computed by a Quantum Computer can be computed by a classical Turing machine, it'll just be orders slower at doing it and require orders more computational resources.

http://en.wikipedia.org/wiki/Quantum_computer

>Given unlimited resources, a classical computer can simulate an arbitrary quantum algorithm so quantum computation does not violate the Church–Turing thesis.[6]

[Anonymous No.5084788]

>>5084664
NP-complete = (NP-hard) inter (NP)

[Anonymous No.5084792]

>>5084773
If mathematics is not limited to the physically possible, it is not limited to what human mathematicians can do either.

>And Quantum Computers are not non-deterministic Turing machines.
Yes, that's what I said. You meant to quote >>5084645?

[Anonymous No.5084817]

>>5084645
For a quantum computer to solve an NP-hard problem in polynomial time, the problem needs to have a structure that suits the superposition well, so that you can read the solution of the problem by reading the superposed output.

Let's think of it this way: you have a function from R^n to R^n and you're trying to get its inverse. If the function is linear, you can represent it as a matrix and it's pretty easy to do that by inverting the matrix. If you don't have some structure like that, then it's extremely hard. Quantum computers are to solving non-deterministic problems what matrix inversion is to inverting functions: it works very well when you work on structured problems, but doesn't make sense in other cases.

Think of a ND problem in which the control flow of the program that tests if a solution is okay doesn't look the same at all for different test cases. Superposition won't help you, because it's somehow a superposition of variable states, but the quantum computer still runs a single piece of code.

[Anonymous No.5084826]

>>5084641
Computers can also reason about the possibility of models of computation stronger than Turing machines, using heuristics and negation of categories, and proofs by contradiction or induction--the same way a human mathematician's brain does it.

Just because you can speculate about how there are computationally undecidable problems via a proof of Godel's incompleteness theorems doesn't mean computers can't either.

Humans can't magically guarantee that they'll be able to determine if an arbitrary program halts, for example, just as there is no general algorithm to prove it either.

They only way to do it is to begin simulating every possible input to the program and see if some combination of values might halt, or identify if there is a statement that causes the program to halt and work backwards inductively to see if it is ever reachable. IF the program actually halts, then you might eventually come across a solution, but if you get bored and give up or run out of paper to keep track of your progress, it's not proof that there isn't a solution.

Computers, as Turing machines capable of simulating any other Turing computable program, can do the same thing. They just might run out of memory or you might get bored waiting for a result and decide to kill the simulation.

[Anonymous No.5084862]

>>5084792
>If mathematics is not limited to the physically possible, it is not limited to what human mathematicians can do either.

You need to look at how the human brain actually reasons about mathematics that aren't physically possible--we use cognitive heuristics which are isomorphic to symbolic manipulation of finite categories. Are you suggesting we humans have magical souls that give us direct access to an an omniscient oracle that can solve such problems?

Just because we think we can reason about stuff outside of what's computable doesn't mean it actually exists. Understanding why certain problems are computationally undecidable isn't the same as being able to actually solve such undecidable problems. Understanding why certain problems are undecidable is entirely within the realm of computational decidability.

>Yes, that's what I said. You meant to quote >>5084645?
Yeah, sorry.

[Anonymous No.5084866]

>>5084826
>Humans can't magically guarantee that they'll be able to determine if an arbitrary program halts, for example, just as there is no general algorithm to prove it either.

Well, they can even guarantee that there are problems for which they won't be able to determine if it halts, by simply comparing the infinite number of problems with the finite number of problems that require a halting proof in n steps (which is finite) and taking n an extremely large integer that upper bounds the amount of proof steps that humans would be able to do before the universe dies.

[Anonymous No.5084930]

>>5084866
>Well, they can even guarantee that there are problems for which they won't be able to determine if it halts, by simply comparing the infinite number of problems with the finite number of problems that require a halting proof in n steps (which is finite) and taking n an extremely large integer that upper bounds the amount of proof steps that humans would be able to do before the universe dies.

Computers can do the same thing, once you understand that the human brain uses heuristics to represent the concept of infinity as a non-halting recursive or iterative process which infects or imbues everything it touches with similar properties, and works in terms of types and categories.

It's not hard to write a program which can prove your given proposition in finite terms by applying category theory and ring theory.

Our digital computers we currently have can't come up with the proposition yet on there own, but only because we haven't managed to figure out how to build our own human-level (or beyond human-level) artificial general intelligent agents yet. Biological evolution has had the benefit of having been around billions of years to do so.

But that doesn't mean that there are mathematics that is somehow outside of what's possible that is graspable by only humans. Once you understand how the human brain actually reasons about stuff, you'll see how such mathematics is just a fantasy--how infinite sets don't actually exist, not even inside human minds.

Mathematics should be reworked to something more in accord with how our brains actually work as (extremely complex) biological linear bounded automatons. Category theory and type theories trumps set theory. You can still use set theory notation, nothing wrong with that--but the semantics must change.

[Anonymous No.5084961]

And computation is just a subset of physics. All knowledge is empirical.

[Anonymous No.5084973]

>>5084206
It's very simple, there's an exposition of it right at the end of Saunders Mac Lane's book "Categories for the Working Mathematician

[Anonymous No.5084976]

>>5084102
That guy in the first video is an absolute moron by the way.

[Anonymous No.5084981]

>>5084930
Category theory doesn't come close to the way people actually think, and it wasn't built for that.
It provides a very natural language to talk about modern differential geometry, algebraic geometry and obviously algebraic topology.
Changing ZFC to an equivalent category-theoretic foundation would not change a thing.

[Anonymous No.5084993]

>>5084930
>It's not hard to write a program which can prove your given proposition in finite terms by applying category theory and ring theory.
>by applying category theory and ring theory
Applying them how?

[Anonymous No.5084998]

I was going to post that /b/ is being even more elitist than /sci/ today, but then I realised I was on /sci/.

[Anonymous No.5084999]

>computer "science"

[Anonymous No.5085005]

I am a mathematician. Do I need to care about any of this or can I just carry on doing maths?

(this is why people hate philosophers)

[Anonymous No.5085013]

>>5085005
By learning a bit of computer science, you will be able to program rigorous proofs instead of being dependent on hand-waving verbal arguments.

[Anonymous No.5085015]

>>5085013
>hand-waving verbal arguments

4/10

[Anonymous No.5085025]

> MFW OP is full of bullshit and anyone who does serious CS and got a degree before 1990 has a PhD in math, not CS.

[Anonymous No.5085034]

>>5085015
Someone is mad that mathematical rigor is pretend rigor.

[Anonymous No.5085038]

>>5085013
If you ever read William Thurston's (very recently deceased) wonderful essay "On Proof and Progress in Mathematics," you'd understand the difference between actual mathematics and 100% rigorous mechanical mathematics.
Quoting directly:
"When one considers how hard it is to write a computer program even approaching the intellectual scope of a good mathematical paper, and how much greater time and
effort have to be put into it to make it “almost” formally correct, it is preposterous
to claim that mathematics as we practice it is anywhere near formally correct.

Mathematics as we practice it is much more formally complete and precise than
other sciences, but it is much less formally complete and precise for its content
than computer programs. The difference has to do not just with the amount of
effort: the kind of effort is qualitatively different. In large computer programs,
a tremendous proportion of effort must be spent on myriad compatibility issues:
making sure that all definitions are consistent, developing “good” data structures
that have useful but not cumbersome generality, deciding on the “right” generality
for functions, etc.

The proportion of energy spent on the working part of a large
program, as distinguished from the bookkeeping part, is surprisingly small. Because of compatibility issues that almost inevitably escalate out of hand because
the “right” definitions change as generality and functionality are added, computer
programs usually need to be rewritten frequently, often from scratch.

Full essay (as a part of larger series of responses to Jaffe and Quinn)
http://arxiv.org/pdf/math/9404236v1.pdf

[Anonymous No.5085089]

>>5085038
>computer
programs usually need to be rewritten frequently, often from scratch
what is modular programming?

[Anonymous No.5085097]

>>5084999
> "computer" "science"

[Anonymous No.5085173]

>implying maths are not a form of art

[Anonymous No.5085198]

>marshviperXwins
>shitty /g/ memes on channel page
How many cocks do you suck a day OP?

[Anonymous No.5085220]

>>5085173

[Anonymous No.5085224]

>>5085173

>implying math is a plural word

[Anonymous No.5085228]

>>5085224

Hail Britannia!

[Anonymous No.5085236]

>>5085173
I'd say that art is more a form of math. Aesthetic qualia, of course, must be congruous to a human framework. Even an abstract art has to appeal to human understanding of chaotic math (please derail).

[Anonymous No.5085244]

>>5085236
what exactly is art?

what exactly is math? (or "are maths", for that matter?)

[Anonymous No.5085260]

>>5085244
Encyclopedia Britannica: Art is "the use of skill and imagination in the creation of aesthetic objects, environments, or experiences that can be shared with others". Wikipedia: Math is "the abstract study of topics encompassing quantity, structure, space, change, and others; it has no generally accepted definition." So math (and I guess art) means whatever we want them to mean.

[Anonymous No.5085266]

>>5085260

So art is applied math, and math is the study of art and other stuff?

[Anonymous No.5085272]

>>5085244
I can't say that there are objective definitions, but I'll sum it up briefly: art is a synthetic form that appeals to human understanding of qualia (I can't say what is art to a duck, or if the stars can make "art"), and math is an umbrella term for just about any sort of conclusion arrived at through logical processes (calculating, judging, reasoning, defining, and perceiving are all mathematical in a sense, but "feeling" is not).

[Anonymous No.5085285]

>>5085284

*yawn*

[Bedlam No.5085284]

What the fuck did you just fucking say about me, you little bitch? I'll have you know I graduated top of my class in the Navy Seals, and I've been involved in numerous secret raids on Al-Quaeda, and I have over 300 confirmed kills. I am trained in gorilla warfare and I'm the top sniper in the entire US armed forces. You are nothing to me but just another target. I will wipe you the fuck out with precision the likes of which has never been seen before on this Earth, mark my fucking words. You think you can get away with saying that shit to me over the Internet? Think again, fucker. As we speak I am contacting my secret network of spies across the USA and your IP is being traced right now so you better prepare for the storm, maggot. The storm that wipes out the pathetic little thing you call your life. You're fucking dead, kid. I can be anywhere, anytime, and I can kill you in over seven hundred ways, and that's just with my bare hands. Not only am I extensively trained in unarmed combat, but I have access to the entire arsenal of the United States Marine Corps and I will use it to its full extent to wipe your miserable ass off the face of the continent, you little shit. If only you could have known what unholy retribution your little "clever" comment was about to bring down upon you, maybe you would have held your fucking tongue. But you couldn't, you didn't, and now you're paying the price, you goddamn idiot. I will shit fury all over you and you will drown in it. You're fucking dead, kiddo.

[Anonymous No.5085292]

>>5085272
So at least some forms of art are applied math, because artists often represent their understanding of the world through their depictions of it. They judge what they see and they draw the lines representing it, and they define what they see in pictorial form. They sometimes need to know what comes in front of what or what a shape looks like, and they need to know, sometimes, the dimensions of an object, as well as its position.

[Anonymous No.5085297]

>>5085284
lolwut

[Bedlam No.5085298]

>*yawn*

Well, that goes for the entire bloody thread, doen't it?

>set theory is garbage and should be superseded by category theory

>superseded

They are just two mathematical theories covering entirely different concepts!

What is this pretentious rant about, after all?

[Anonymous No.5085303]

>>5085298
>replyng to an obvious troll thread

Lurk more, newfag.

[Anonymous No.5085306]

>>5085298
fap fap fap fap fapfapfapfapfappityfap

[Bedlam No.5085312]

Well, I'm not the only one who answered. So newfag my ass.

[Anonymous No.5085318]

>>5085306

[Anonymous No.5085324]

>>5085292
When I said forms, I meant more the metaphysical form of art. Something that appeals to human "feeling" or understanding of actual quale can be considered art. Try reading Zen and the Art of Motorcycle Maintenance, as it mentions some of the questions you are asking.

[Anonymous No.5085350]

>>5085324
so charisma is also art?

[Anonymous No.5085356]

>>5085324
Perhaps tossing around words and sentences is also art. In short, language is a form of art.

[Anonymous No.5085361]

>>5085350
Honestly, anything presented as art could be considered art. That's the problem with essentially undefinable terms.

[Anonymous No.5085374]

>>5085324
There's such a thing as a metaphysical form of art?

[Anonymous No.5085415]

>>5085374
http://en.wikipedia.org/wiki/Theory_of_Forms

In a metaphysical sense, art is a form (idea). I'm not talking about a form as in a variation, but a form as something that is felt but cannot be quantified or substantiated.

[Anonymous No.5085689]

>>5084930
I'm >>5084866 and I don't understand why this is an answer to my post. My argument is completely outside of the "turing vs human mind" debate: I'm only considering the minimum number of steps in a logical proof, and saying that this is unbounded and therefore some things cannot be proven in bounded time with bounded speed, therefore when anon says
>Humans can't magically guarantee that they'll be able to determine if an arbitrary program halts
it is obvious because there are infinitely many programs, thus there are programs whose halting problem requires a number of proof steps as high as you want. No need for Godel, Turing completeness or anything "complicated" here, it's enough to just look at the number of programs compared to the number of logical proofs that one can do using a finite amount of steps based on a logic defined by a finite number of axioms and rules.

[Anonymous No.5085765]

>>5085298
>
>They are just two mathematical theories covering entirely different concepts!
>What is this pretentious rant about, after all?
Wrong. You do not understand Category Theory.

[Anonymous No.5085769]

>>5085356
Mathematics is a language. Therefore it is also art.

[Anonymous No.5085811]

>>5084981
>Category theory doesn't come close to the way people actually think, and it wasn't built for that.

Wrong. You do obviously have little understanding of what cognitive science has uncovered. Categories and being able to relate one category to another is at the core of cognition. It's an abstraction or generalization of pattern recognition.

http://www.youtube.com/watch?v=WuZZ9RSrSWE
http://www.youtube.com/watch?v=n8m7lFQ3njk

>It provides a very natural language to talk about modern differential geometry, algebraic geometry and obviously algebraic topology.

You're being a little too specific. Generally, Category Theory allows you to relate one area of mathematics to another area of mathematics. It also allows you to relate say the Lambda calculus to a machine programming language, or declarative forms to imperative ones and vice versa.

>Changing ZFC to an equivalent category-theoretic foundation would not change a thing.

You seem to lack an understanding of formal semantics and linguistics. Changing the semantics of ZFC is everything. The syntax is irrelevant.

It's like the difference between believing in an imaginary sky-god (infinite sets) and in manning up and accepting the intrinsic beauty found in purely constructivist methods that your brain internally employs anyway without you realizing it in the act of believing in such fantasies as sky-gods and infinite sets.

[Anonymous No.5085850]

>>5084961
>And computation is just a subset of physics. All knowledge is empirical.
Computation, mathematics, and physics are all one and the same underlying thing.

http://arxiv.org/abs/physics/0001020
http://arxiv.org/abs/1003.5831
http://arxiv.org/abs/0704.0646

[Anonymous No.5085856]

Category theory is based in set theory.

[Anonymous No.5085964]

So I still don't get what is wrong with set theory, what category theory does to fix this issue, or how "infinite sets don't exist" is a sensible statement when just about everything mathematicians do concerns not reality but logical consequences based on definitions.

[Anonymous No.5085968]

>>5085964
OP doesn't know either. He's a computer scientist, not a mathematician.

[Anonymous No.5085982]

>>5085856
Wrong. You're confusing how you were taught the subject for it being based on it. It's not.

You're also confusing syntax for semantics.

http://en.wikipedia.org/wiki/Category_theory

>The subsequent development of category theory was powered first by the computational needs of homological algebra, and later by the axiomatic needs of algebraic geometry, the field most resistant to being grounded in either axiomatic set theory or the Russell-Whitehead view of united foundations. General category theory, an extension of universal algebra having many new features allowing for semantic flexibility and higher-order logic, came later; it is now applied throughout mathematics.
>Certain categories called topoi (singular topos) can even serve as an alternative to axiomatic set theory as a foundation of mathematics.

[Anonymous No.5086004]

>>5085964
What's wrong with set theory is that it's a made up fantasy using heuristics to warp the mind into thinking its real.

Mechanical computers do not work with infinite sets.
Digital computers do not work with infinite sets.
Quantum computers do not work with infinite sets.
Biological computers do not work with infinite sets.
Neurons do not work with infinite sets.
Human brains do not work with infinite sets.

There is not an infinite set of the natural numbers that exists in your head that you can pluck numbers from.

How then are numbers defined in reality and computation? Mathematical concepts such as groups, rings, number fields, and other algebraic topological structures serve computationally rigorous alternatives. Category Theory lets you move and operate between different forms of such structures.

[Anonymous No.5086006]

>>5085982
And how exactly do topoi change the infinite nature of mathematics?

[Anonymous No.5086021]

>>5086006
Mathematics isn't infinite, you're just pretending it's infinite using cognitive heuristics.

An ordering of say numbers such that there is never a highest number is not the same thing as there existing an infinite amount of numbers somewhere. It's just indicative of a non-halting recursive process.

Stop confusing the map for the territory.

[Anonymous No.5086032]

>>5086004

I don't understand what you mean by 'do not work with'. I have no knowledge whatsoever on theoretical computer science, so I'll take you on your word when you say that x computers do not deal with them. But human brains obviously are capable of some notion of infinite sets, or we wouldn't have set theory.

I also don't quite get how you'd describe points on a line or something like that (pick your favorite 3d object if you want it to correspond to reality) without a notion of an infinite set.

[Anonymous No.5086070]

>>5086032
Think of it this way.

For your brain to contain within it an infinite set of say the natural numbers, you would have have a bijection of the natural numbers to a physical state inside of your head. Your brain is composed of a finite number of atoms in a specific state configuration. There is also a finite maximum number of possible state configurations your brain can ever be in. It cannot model an infinite number of states, nor can it accurately model an infinite set.

Same with computers. There's a finite amount of memory within a computer with which to represent state.

Same with the Hubble volume of our Universe.

[Anonymous No.5086083]

>>5086070
You seriously think that the only way to model an infinite set is to list all the entries? Just define it recursively or iteratively - if you know anything about computer science you should be fine with recursive definitions and infinite data structures.

[Anonymous No.5086086]

>>5086070
Sure. But my brain can grasp the notion of an infinite set and work with that. I mean, wouldn't your argument also rule out something like irrational numbers? As far as I'm aware, computers compute with numbers like pi by simply calculating symbolically. It is not possible to calculate, say, pi+1 precisely as a decimal expansion. Yet this doesn't mean that the notion of real numbers is 'flawed'.

[Anonymous No.5086092]

>>5086032
>I also don't quite get how you'd describe points on a line or something like that (pick your favorite 3d object if you want it to correspond to reality) without a notion of an infinite set.

Computers in their various manifestations do not work with arbitrary real numbers. Floating-point numbers are a convenient computable-real notation--in other words, they're actually stored as finite rational numbers in memory. There is no such thing as real numbers in computation, only computable-reals/rational numbers.

http://en.wikipedia.org/wiki/Computable_number

In computation, we uses types, not sets. Types are defined by the finite collection of axiomatic functions and relations one can perform on objects belonging to a type. In mathematics, this is done via groups, rings, fields, etc. Not sets.

Learn about Type Theory and Algebraic Topology.

[Anonymous No.5086102]

>>5086083
Defining it recursively is not the same thing as actually instantiating an infinite set exists. The recursive algorithm or function never halts.

You would be sitting there, recursing into the next iteration and writing the results down on your chalkboard which never seems to have enough space until the Universe approached maximal entropy and protons started to decay and our dimension of time collapsed.

It's impossible for it to actually exist.

[Anonymous No.5086107]

>>5086102
I think the problem here is that you're using "exist" in a way that nobody else is.

Let's say I define the natural numbers with the Peano axioms. I would say that the infinite set of natural numbers exists, even though I only used a finite number of terms to define it. I don't think I need to instantiate a concept for it to exist.

Or, let's say that you're right, and there is only a finite number of natural numbers. What's the biggest one? Whilst you're at it, please tell us what the biggest prime is, and the last digit of pi.

[Anonymous No.5086110]

>>5086092
>In computation, we uses types, not sets
>mfw typeless lambda calculus

[Anonymous No.5086125]

>>5086021
Yes, and how do CATEGORIES change the infinite nature of it?

[Anonymous No.5086131]

>>5086086
>Sure. But my brain can grasp the notion of an infinite set and work with that.
No, what your brain is actually doing is using a symbolic heuristic for what is actually a non-halting recursive process, which has certain asymptotic properties. A non-halting recursive process defined with a finite number of statements is not an infinite set.

You're confusing your mental model for the real thing, like so many naive mathematicians do.

>wouldn't your argument also rule out something like irrational numbers

Transcendental numbers like PI are computable reals that are only ever approximated using a finite polynomial expansion. You can always algebraically work with PI symbolically and eliminate it from your computations, much in the same way a mathematician does, without loosing accuracy until you need to compute your final result to within a certain degree of error.

In quantum mechanics, the accuracy of a measurement is dependent upon the finite amount of energy you're willing to expend, such as the wave-length of the photons you're bouncing off a particle. What you always get is a computable-real.

http://en.wikipedia.org/wiki/Compton_wavelength#Limitation_on_measurement

Real numbers don't exist.

[Anonymous No.5086137]

>>5086125
Category Theory doesn't necessarily rely on infinite sets. Simple as that. There are no infinite sets, stop assuming there are and that therefore Category Theory must change the non-existent "infinite nature" of mathematics.

[Anonymous No.5086138]

>>5086021
>An ordering of say numbers such that there is never a highest number is not the same thing as there existing an infinite amount of numbers somewhere. It's just indicative of a non-halting recursive process.

I think every programmer ever would like to have a word with you about your notion of that not being exactly the same as producing an infinite set.

[Anonymous No.5086154]

ITT: A guy who doesn't have the faintest idea about computer science or mathematics

[Anonymous No.5086156]

>>5086137
You're right, it has infinite categories instead.
How is that fundamentally different?

[Anonymous No.5086158]

>>5086110
> typeless lambda calculus
Typeless lambda calculus doesn't require types as you expand your form, but if you ever interpret/compute the final form you can define a homomorphism to the semi-ring of natural numbers for example. The "typeless/typed" in lambda calculus refers to how forms are expanded, not to how your final result should be interpreted or calculated.

[Anonymous No.5086163]

>>5086156
It doesn't have infinite categories, only set theorists bring with them the baggage of infinity into Category Theory, and so continue to talk about infinite this and infinite that when what they're really talking about are non-halting recursive algorithms for generating categories.

[Anonymous No.5086168]

>>5086154
ITT: undergraduate mathematicians and comp sci students who think they already know everything, who have been brainwashed into believing in infinite set constructions and the existence of infinitesimals, entering a state of cognitive dissonance as they are faced with the truth that their precious infinities are a fantasy.

[Anonymous No.5086448]

>>5086163
Silly you, non-halting recursive algorithms don't exist. Every algorithm has to halt at the end of the universe. Welcome to ultrafinitism, enjoy your brain damage.

[Anonymous No.5088412]

>>5086448
What end of the universe?

[Anonymous No.5088424]

>>5088412
The one where there's no energy to do useful work anymore.

[Bedlam No.5088441]

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