/sci/ - Science & Math

>>15030638
That’s not the definition of a function. A function is merely a relation that assigns one element in the codomain to every element in the domain.

A function is a subset of SxT where each element of S appears exactly once. Nobody defines it as a "procedure" lol

>>15030679
are you literally retarded?

>>15030703
What's the problem with that? It's the definition of a function. Nobody defines a function as a "rule/procedure" except, apparently, sneedburger

>>15030679
>merely a relation that assigns one
how does it go about assigning? does it have arms and legs and a arc welding machine to attach your new entropic information to your previous information, is there a rule of thumb, could you break it down into smaller steps, like a procedural tutorial?

Haven't watched the video yet, but i'm already sympathetic. Mathematicians walk around like they have some free energy device, in the real world it takes energy to go from point A to point B

>>15030704
>>15030704
>Nobody defines a function as a "rule/procedure" except, apparently, sneedburger
Tao does in his book Analysis.

>>15030702
>A function is a subset of SxT where each element of S appears exactly once.
That's a notion of set, but there's many other definitions in other fields.
Most notably, the concept in computability theory is very different

https://en.wikipedia.org/wiki/Computable_function

I wouldn't so boldly make assertions like that if you only have only been taught a limited view of mathematics.

>>15030747
>>15030741
>akshually
yes yes but he's talking about sets isn't he?

>>15030638
yes it should be illegal to convert between bases

>>15030638
he has a point, the act of taking an ordered pair of elements to get another element in the set is not defined, it is taken for granted

>>15030702
it doesn't matter, that's why he wrote rule or procedure. In this case that rule implies a procedure so your point is moot, you don't get the problem, which seems to me the finitist criticism to the axiom of choice

A function is just a set of ordered pairs with some extra properties. All other defintions can take the L.

>>15030794
What is a set? How do I construct them and how can I tell if something is a set or not?

>>15030794
nooo you dont understand he has super secret finitist problems, heh.. i guess you dont get it

>>15030795
>What is a set
A primitive, i.e. an undefined object. Intuitively a set is a collection of objects.
>How do i construct them
With the axioms of ZFC, examples: axiom of pair, axiom of union, axiom of power set and axiom schema of replacement.
>how can I tell if something is a set or not?
This is the major problem which led to the development of axiomatic set theory. In axiomatic set theory we assume the existence of an empty set and an inductive set and then we define and then prove the existence of all other sets using the rest of the axioms.
So if you can construct it and prove its existence with ZFC then it is a set.

>>15030805
>axioms
ah yes, the literal unproven "truths"

>>15030740
>how does it go about assigning?
It depends on the function. Dumb question.

>in the real world
Math isn't physics.

>>15030840
Yes, in order to make logical deductions you need premises. What are Wildburger's premises?

>>15030857
>It depends on the function.
This is canonical hand waving, in the end bits must be arranged from state A to state B, a physical process, whether your brain synapses or breadboard, its all physical

>Math isn't physics.
Indeed, so they should stop selling their snake oil to the physicists as 'real'

>>15030840
In all of your favorite mathematical theories set theory is used to an extent and for if not there are new axioms used, so stop pretending like infinitude is at fault here

>>15030873
I have already created a model that helps bridge the divide between the people who pretend they have a free energy infinite calculator and those that try to build things in the real world, i call it MMP

>>15030840
they are supposed to be self evident

>>15030876
I think π is a mystical number because it's digit sequences don't repeat, there's no sequential pattern

>>15030878
Or just a set of assumptions that if shown to hold for your system makes another set or mathematics already known to be true.

>>15030883
by lemma 0: all numbers are mystical, otherwise they could not manifest on this mystical universe, so you are not incorrect

however pi also belongs to the metaphysical, because, unlike infinity and zero, travelling down its digits increases its precision

>>15030887
In wildbergers mathematics, landing on this definition is the primordial 'watering down' of the notion of the axiom

>>15030891
for redundency travelling down 0.000... doesn't increase precision, as its already perfect and travelling down inf.inf inf inf inf doesn't either as there is no concept of precision at all to increase. With pi every digit adds something to the party if one added a digit and zoomed by 10x there will always be a jitter in the particle, i mean point on the number line

>>15030873
>>15030883
>>15030703
>>15030741
The axiom scheme of replacement expands the definition of a function to that because of the OBVIOUS fucking realization that there are countably infinite rules yet an uncountably infinite amount of functions.
You are fucking retarded. Kill yourself now.

>>15030873
>Indeed, so they should stop selling their snake oil to the physicists as 'real'
nobody does that, physicists are free to do their physics without math. Let's see how far it gets them

>>15030638
ngl i kind of missed the NJ threads

>>15030935
>nobody does that
they, by the letter, call their number system 'the real number system' not even 'a realistic number system' 'THE REAL number system'

math
>from Ancient Greek μαθηματικός (mathēmatikós, “on the matter of that which is learned”),

Im sure physicists will continue to use math, but the newtonian and libnitian gematria under the 'real number' system, i'm not so sure

>>15030943
>maybe if I quote the ancient greek my retarded fintardism will seem more legit

>>15030943
>etymological arguments
do you also think a tree data structure is supposed to be a literal tree from a forest?

>>15030954
How much money do you think has been misallocated over time because devs thought 1.2 - 1.1 = 0.1

>>15030956
no real is an adjective a descriptor while a tree is a noun. when you add an adjective but it means the opposite that is double speak. Tree is a noun the opposite of a noun is way less explicit

real - unreal <- robust dichotomy
tree - untree <- one object against every other object

a lament to aweful - full of awe. understand -stand under and woman - womb man

>>15030873
>This is canonical hand waving
No.

>in the end bits must be arranged from state A to state B
Bits of what? You mean if I write down a function? The media in which math is done is not part of math. It's irrelevant. I'm talking about concepts and you're talking about paper.

>Indeed, so they should stop selling their snake oil to the physicists as 'real'
What are you talking about? Math is only "real" if it can be applied to real things in physics. So no one claims math by itself is "real."

>>15030893
Which of Wildburgers axioms are self- evident truths?

a function is a machine

>>15031091

there is an archive link is this but manages to keep the unrendred LaTeX but the system thinks its spam

https://web.maths.unsw.edu.au/~norman/views2.htm

under "Does mathematics require axioms?"

>Mathematics does not require `Axioms'. The job of a pure mathematician is not to build some elaborate castle in the sky, and to proclaim that it stands up on the strength of some arbitrarily chosen assumptions. The job is to investigate the mathematical reality of the world in which we live. For this, no assumptions are necessary. Careful observation is necessary, clear definitions are necessary, and correct use of language and logic are necessary. But at no point does one need to start invoking the existence of objects or procedures that we cannot see, specify, or implement.

Wildberger first challenges whether the axioms stated are even axioms by definition, than concludes that, even as definitions they are uncomprehendable and hide all sorts of logical inconsistencies. He than concludes that Euclids definition, that if anything they should be self evident truths any novice can intuitively grasp, as an appropriate location to leave the grammatical object 'axiom'

So i suppose if i had to shove axioms in wildbergers mouth it they would be the axioms of mathematical earnestness and humility
1) you've never heard, smelt, tasted, felt or seen an infinite object
2) you've never divided something into an infinitessimal amount either
3) precision at iteration is not a 1:1 substitute for exactness

>>15031311
Does Wildesperg deny that triangles or squares exist since we cannot construct them in real life?

>>15031343
you can construct them with compass and straightedge retard

>>15031343
music is perfectly in tune when in equal proportions 1:1, 3:2, 5:8, 13:21.
Naturally some triangles just resonante better with me

from paper
>We then exhibit the impressive mathematical power of P322(CR-Decimal8) by showing that P322 holds
its own as a computational device even against Madhava’s sine table from 3000 years later. This is a strong
argument that the essential purpose of P322 was indeed trigonometric: suggesting that an OB scribe unwit-
tingly created an effective trigonometric table 3000 years ahead of its time is an untenable position

>The novel approach to trigonometry and geometrical problems encapsulated by P322 resonates with
modern investigations centered around rational trigonometry both in the Euclidean and non-Euclidean set-
tings, including both hyperbolic and elliptic or spherical geometries (Wildberger, 2005, 2013, 2010). The
classical framing of trigonometry and geometry around circles and angular measurement is only one of a
spectrum of possible approaches, so perhaps we should view angular trigonometry as a social construct
originating from the needs of Seleucid astronomy rather than a necessary and intrinsic aspect of geome-
try.

>The paper is laid out as follows. In the first section we introduce P322 itself. Then we review contem-
porary geometric concepts: the roles of reciprocal slope, known as ukullû or indanum, properties of similar
triangles and the Diagonal rule. This is followed by a review of OB numerical procedures: the role of recip-
rocal tables, the square side rule, the square root algorithm and the trailing part algorithm. We then return
to P322 to discuss its method of construction, missing columns and unfilled rows.


this is a natural outcropping of exactness versus precision at iteration versus exactness.

If i were to put forward a competition we would make music triangles of these proportia, and than some other random ones and judge by beauty of hearing

>>15031372
>look down at your drawing with microscope
>it's not of a triangle or square but a more complex shape
So is the business here just to deny that there is such a thing as abstraction?

>>15031404
wildberger isn't a platonist obviously

>>15031404
just that there is a step between in the middle.

namely that there is an incommensurable portion of the length that even with a perfect microscope could never be measured

>>15031412
>pic
>A cauchy sequence of rational numbers DOES NOT define an irrational number
Why not?

That isn't how functions are defined.
A function is a set of pairs (x,y).
In other words, the object A is a function if and only if
>for all p <- A, there exist r,s such that p = (r,s)
OP, you're tilting at windmills.
Go to /lit/ and tell the fine folks there that you want some advice as if you were a character in a Don Quixote novel.

>>15031311
>Mathematics does not require `Axioms'.
>The job is to investigate the mathematical reality of the world in which we live. For this, no assumptions are necessary.
This is a delusion. He thinks his axioms are obvious and therefore not axioms. There is no number "found in reality." They are certainly useful concepts to apply to reality, but everything, including Wildburger's math, are just axioms that are not self-evident.

>>15031417

>>15030703
cometh the fedneck

>>15031428
is it not a better delusion than the existing one however? that calls itself real, that portains sqrt(2) is a well defined distance on a number line, that type casts infinity the same way as unity, as the nothingness?

This is a K-12 thread.
This is not how educated people talk about math.

>>15030740
>how does it go about assigning?
What the anon is trying miserably to communicate is that SOME authors (mostly in discrete math and abstract algebra in my experience, not so much authors in analysis) define a function as a kind of relation, and they in turn define a relation as a SET OF ORDERED PAIRS. Ergo, a function is a set of ordered pairs. There's a rule associated with the relation/function to determine whether a particular ordered pair is in the set, but the relation/function is NOT the rule itself -- it's simply the set of ordered pairs.
Calculus/analysis authors tend to adopt the less autistic and more intuitive definition of a function as the rule itself. This lets us picture the function as a machine and also aligns with how we think of functions in computer programming.

>>15031452
A CS professor (and a worthless communist) once told me I was "arrogant" for suggesting students learn the ordered pair version of the definition of a function.
it makes sense though to say you have to be an arrogant autist to learn the ordered pairs definition and prefer it over the "rule/procedure/formula" definition
Vast majority of people will wtf if you say
>f(x) = y isn't a function, it's just the rule that defines a function

>>15031458
Yeah, I think their autistic concern is that they want to be able to operate on abstract function-like ordered pairs that don't really have much of a rule to them other than "because I said that this x goes with this y." But it's a disservice to insist on this autistic definition in more concrete contexts, imo.

>>15030985
>How much money do you think has been misallocated over time because devs don't understand their tools
probably a lot, code monkies are notorious retards
every single one of them ought to take numerical analysis course, but that would require them to be capable of more than copy-pasting from stack overflow

>>15031435
Where's the flaw? This is just a bunch of handwaving.

>>15030956
>do you also think a tree data structure is supposed to be a literal tree from a forest?
thank you, didn't think id find anything of use on a finitard thread, but but from your comment the idea of data tree "rings" came to mind, which id says as a concept it's slightly amusing, maybe some fun stiff can come from the mockery of finitardery

>>15031448
>is it not a better delusion than the existing one however?
No, because plenty of the math Wildburger doesn't like is applicable to physics. It's completely arbitrary what he considers "real math" but he pretends it isn't.

>>15030985
you don't even know what the fuck floating point is you abnormongrel
https://yewtu.be/watch?v=PZRI1IfStY0

>>15031452
Ah, i see many thanks for that. I've spent alot of time programming on a language that doesnt have floating or really fixed point. There was this one project that used a complicated equation, and they landed up basically hardcoded every output for every input because the language didnt have the ability to perform the operators I feel like at a surface level this is a similar story here.

Though perhaps this is just a set of pairs, I'll have to investigate further the importance of this ordering.

for an example, the hole in the function over the zero, i would just stick null in there and call it a day, but what do these discrete and abstract algebra mathematicians have to say? Is this the source of the injective and bijection ideas? they have been helpful in my coding adventure

>>15031452
What if there is no simpler way to define the function than as ordered pairs? Sounds fine to me.

>>15031476
>Is this the source of the injective and bijection ideas
When we use terms like 'injective', 'surjective', 'bijective', 'one to one', 'onto', 'one to one correspondence', it doesn't really matter which definition of a function we're using.

>>15030985
you don't even know what the fuck floating point is you mong
https://www.youtube.com/watch?v=PZRI1IfStY0
https://www.youtube.com/watch?v=dQhj5RGtag0

>>15031468
I cannot argue against that, for are not many inventions created only to have mathematical justifications appended after?

One may find the arguement silly or full of vain glory, but my attestion almost purely rests on the idea that 440×2^(7/12) wounds worse than 440×3/2 . To which i extend perfect commensurability yields more fractially resonant materials as there is no endless decimals to temper out the constructive and destructive standing wave amplification/interference

>>15031465
the inf limit n -> inf s_n = A and that there is always a smaller epilson.

I dont want to dis cauchy to hard though, the idea of 'imploding' down n and finding derivative d/dn is at the heart of what motivated my own number theory MMP

>>15031501
>yields more fractially resonant materials
where is the empirical evidence of this?

>>15031512
my cello, if you play the octave in a perfect 2:1 ratio, it rings forever, if you dont it rings for less time

>>15031518
one doesnt just not stick a node on the edges of the box

>>15031522
the harmonics of standing wave amplification perhaps is more a sci answer https://www.youtube.com/watch?v=DovunOxlY1k

>>15031518
>perfect
your cello wasn't perfectly made and your hand positioning isn't perfect.
stop LARPing

>>15031507
>the inf limit n -> inf s_n = A and that there is always a smaller epilson.
How are these flaws? Just because Wildburger says so? Handwaving.

>>15031528
>>15031522
>>15031518
>>15031507
Real schizo hours

>>15031538
the resonance effect increases in the accuracy of the fingering

>>15031546
so?
hello??? is this a fucking bot????

>>15031540
the motivation,
these ancient builders were obsessed with this

golden ratio too, loved it

>>15031569
https://www.youtube.com/watch?v=bCYcS57eCqs

In other words 0.999... is a cello string unattached below the bridge. notes with ratios of endless decimals can never fullt rectify in the ear.

>>15030679
>>15030702
>>15030740
It sounds like OP is mixing up function and algorithm

>>15030638
Yes Norman, but can you define "a pedant?"

>>15031612
And let me say what point I'm making about Norman being stupid and his ideas being shit tier. He is fine taking it for granted that his audience is already familiar with the English language and the Latin alphabet used to write it, so why is he not fine granting the basics of literacy and the basics of numeracy? The line of inquiry he raises with this, "What is a rule/procedure?," idiocy must be continued until he asks, eventually, "What is an alphabet?," and before saying that actually writing has a big problem because the definition of letters is not very good. He grants that his audience already knows how to speak and read English but his criticisms are only that other people also grant that their mathematical audiences are already literate in the language of mathematics. Why does Norman agree that that the definition of the symbols in the Latin alphabet doesn't need to be revisited outside of basic grammar school but then dispute the prevailing opinion that, "What is a function?," does not need to be revisited outside of the theory of functions? What is the basis for this double standard? This is what separates Norman being a pedant from another scholar scrutinizing the fundamentals: the point Norman is raising leads to nothing useful. There is no definition for "rule/procedure" that wouldn't have other words in it allowing Norman to REEE, "Aha! But what does word X mean?!?!" Norman is a turd and his ideas, if they are even his, are turd ideas.

TLDR: Why is ok assuming his audience knows the alphabet he uses to write his bullshit but not ok with assuming they know what functions are, something most people have learned before they hit their teens.

>>15031507
>there is always a smaller epilson
That's the entire point of the epsilon. You get arbitrarily close to a number without ever reaching it.

>>15031760
>arbitrarily close
no such thing

>>15031639
You are mostly right and to the point.
Although, I can remark that I once bought an "Encyclopedia of Mathematics", a > 2000
page compendium, which turned out not to contain a single definition of "function".

In type theory we are indeed often vague about what exactly a function is, in a way that parallels set theory's vagueness about what a set is. In fact we are often more vague. But that can actually be useful. The more vague you are about what your words mean, the more situations your words apply to.

Let me give an example. Let's define a sequence of type A as a function from the type of natural numbers 0, 1, 2, ... to type A. Then we can define a real number as a sequence of nested closed rational intervals whose widths are eventually smaller than any given positive rational number (we also have to define what "equality" of real numbers is). If we take "function" to include only functions we can compute, then this definition of real number excludes the limits of sequences like the ones described in https://en.wikipedia.org/wiki/Specker_sequence , whereas a more encompassing meaning would include these numbers. But we can do a lot of math without caring about whether these things are real numbers. And we get a bunch of results that apply not just to the abstract notion of real numbers that include all sorts of numbers we can't compute, but also to the computable real numbers that we can model on an actual computer.

>>15031954
Yeah true. Not in the physical world anyways. It's pixelated, discrete, digital. Should be a big clue that the physical world is an output of info processing.

>>15030883
>it's digit sequences don't repeat
Unproven. It could loop at the (farthest discovered digit + 1)th digit, we'll never know until someone proves it's normal.

>>15031954
Okay. Give me the number [math]x \neq 1[/math] such that [math] \forall y \in \mathbb{R}, |1-x| < |1 - y|[/math], since you claim that this number exists.
Let's assume for your sake that you already gave me this mystical number. But I can just take the number [math]1 - \frac{|1 -x|}{2}[/math], which is both real and closer to 1, which means that your number didn't satisfy the first condition. By contradiction, such a number doesn't exist.

>>15031639
>TLDR: Why is ok assuming his audience knows the alphabet he uses to write his bullshit but not ok with assuming they know what functions are, something most people have learned before they hit their teens.
No, you are completely wrong about this. The word "function" has a completely different meaning in contemporary mathematics than it does in every day life, or for example in programming.

>>15033343
maybe replying to the wrong post? you can always get closer to a point but not touch it, there is no magic place where it becomes 'arbitrary'

>>15033379
ESL or highschool dropout?

>>15033385
just because you label a number system 'the reals' doesn't mean you can go calling them real. to do so will admit that your employing doublespeek which is an atrocity upon the english language
"both real and closer to 1"
the projection and self refutation is incredible

>>15031954
The point of this
>>15033277
by the way is that in the physical world, there are minimum planck units such as lengths and so you can get one pixel distance from something and that's the closest. There is no defined intervening space between pixels, no half pixel. and so no continuity. Apparent motion is caused by screen refresh and population of the next pixel over as opposed to traversing actually infinite points, which is of course illogical, despite attempts to make it seem logical by mapping abstract maths onto the physical world. No continuousness in the physical world. Continuousness and infinity are grounded in minds, which is not to denigrate concepts of mind as not 'real' just that mind is not physical. Of course, physical objects are also objects rendered in mind, but they happen to be discrete and not infinitely divisible.

>>15033396
so than, nature has a preferred continuum?

>>15033402
No. No continuums in nature. There is a resolution. One by one the continuousness/infinities have been rooted out starting with the resolution to the ultraviolet catastrophe. Pic related part 1

>>15033394
So it's both I suppose?
>>15033396
Retarded popsci tier physics knowledge.

>>15033402
part 2
paper
https://people.csail.mit.edu/nhm/looking-at-nature.pdf

>>15033421
No. What I said is true. There is no such thing as infinite anything in the physical world. No continuousness, no infinite divisibility. It's data. It's discrete. Digital. Whenever infinities aries, you know something is wrong. Start your research with the ultraviolet catastrophe. Picrels here will give you a lead
>>15033420
>>15033422
As stated here in pic rel
>>15033277
You have no reason to STILL be trying to think of physicality as having anything to do with any kind of infiniteness. Zeno was onto this over 2000 years ago. The universe began, so it is past finite, there is a refresh rate and so cycles and so minimum time, planck time, finite time, finite length, finite energy, on and on.

>>15030638
Procedures are pretty well defined, right?
I'll watch the video later today, but I'll be surprised if
"Step one: square the number"
Is ill defined

>>15033449
>number
Define.

>>15033277
This is disproven.

https://www.sciencedirect.com/science/article/pii/S0012365X13000873

>>15033286
>Unproven.
Oof. Pi being irrational, so non-repeating, is proven. Normal doesn't mean non-repeating, it means the digits are distributed uniformly.

>>15033943
What I find fascinating is that not all approximations of Pi will recue a normal distribution.

Does this make some approximations of Pi, more legit than others?

>>15030747
that is not their definition of "function", that is their definition of "computable function." Computability theorists study uncomputable functions far more than they study computable ones

>>15033808
This is irrelevant. This is pre-supposing that there is a bottoms up materialist event causation in the universe. In a reality where the physical world is an output of computation the causality is coming from 'outside' the virtual space of the reality. This is by the way why bell type correlations in terms of faster than light correlations can be faster than light by the way, ie instant. All points are equadistant from the processor, and so space is a virtual thing and processing is the causation, not anything from inside physicality. None of the stuff down at that level that that paper is talking about ever even has to be rendered. And stuff down at the planck scale certainly never has to be rendered, unless we can some time in the future get down there to make a measurement. All that has to be rendered is stuff at the resolution correspondant to the specs of the consciousnesses logged on to the reality, or things down to the resolution of instruments devised by the instruments that the consciousnesses invent to prob down to a fine resolution. See pic. You render (calculate and define values and present to an observer) only on an as needed basis and in a multi fidelity manner.

>>15033808
So our universe, what we call physicality is number 3. But it's not one where every value of ever particle and every microdynamical event etc has to be calculated at every second every where in the universe at all times. This is retarded. None-of the consciousnesses immersed in the reality can see down that far, nor can their instruments in many cases. And even when they do demand such resolution, you only render the effect of what WOULD be PROBABLE to be down at that resolution. So you just have to render clicks on a geiger counter. You don't have to render the thing that is supposed to be causing the click. Just the clicks. And with photons, you don't have to ever actually render a photon to create light, just the effect, any more than you have to render photons for light in a dream, which is another consciousness generated virtual reality. You render the EFFECT.

>>15033808
So in other words, your supposed proof is begging the question. The paper pre-supposes a certain type of physicality, an unproven one by the way, and one that could never be proven, since infinitely divisible space can't be proven empirically, being that by definition there would be no point to which you could get to and go 'see, here is infinite divisibility!'. There would always be further to divide. You can't demonstrate physical infinities empirically. Anyways, he presupposes a bunch of unproven things and than claims that if his idea of what digital physics models claim can't fit with his idea of what reality is, then it can't be true. It's wankery and strawmanning.

>>15033808
Also, from your paper, picrel, I see some more very sketchy shit
>Our present analysis is limited to classical point particles, which is not very realistic since actual particles are governed by quantum mechanics.
>We strongly suspect that a statement analogous to Theorem 28 will also hold for guantum particles; for example, anisotropies are visible for quantum particles on hexagonal lattices at large enough momenta, an effect known as trigonal warding
This is nothing but assumptions. Very weak and 'disproves' nothing.

>>15030638
A place for normies who don't wish for metric time.

>>15030741
Well duh, exactly for idiots like you faggots who can't into intuition and abstraction so people have to willy nilly explain definitions in a different way when not rigorously

>>15033808
This part even says that the model of physicality that is a no go theorem for digital physics models is only a pre-supposed MODEL and that the version of digital physics considered is only ONE very specific model of digital physics.

>>15033360
Who gives a heck about what a function means in "programming"? And I say this as a codemonkey.

It is mostly a matter of marketing, this is why you can have "closures" and "functions" and "lambdas" and "anonymous functions" - all of them very slightly different practical realizations of the same base mathematical concept - and then have people on stack overflow confidently pontificate for many posts on end on why Rust closures are *not* like Java lambdas.

>>15030638
Alright here's my crack at it
>Rule procedure = precise manipulation
>Precise as in unambiguous, repeatable
>Manipulation as in change of symbols