How would math change if we pretend that instead of inf (∞) there is a largest number named a Divinitillion (Symbol: †)?
>>16784242It would just be a change in notation. I'm assuming the divinitillion has the same properties that infinite has
>>16784242about as much as switching to using your mom's weight
>>16784242by definition, inf can't be a number.Infinity is an unbounded quantity greater than every real number.
>>16784243It's really not that simple. Here's how the proof goes:For each number as you go sequentially, you'll always be able to find a bigger number.But at a certain point you'll find a number compared to which there's no bigger number, which is God. That amount is divintillion.It's not that God is a number, it's more like God has number capabilities, just like how a gun can fire in semi-auto and full-auto mode. This realization solves many problems that the naive concept of infinity introduces, such as infinity not being a proper number, but that doesn't apply to divintilion because God can do whatever he wants.So no, infinity and divintilion don't have the same properties.
>>16784242>How would math change if we pretend that instead of inf (∞) there is a largest number named a Divinitillion (Symbol: †)?You will have changed the meaning of "number" in a way that breaks normal arithmetic and algebra.
>>16784268Demonstrate why he wouldn't be able to do 1 + 1 = 2 because † exists.
>>16784269As soon as you show me where I said or implied that.
>>16784260You are one of those One True Finite faith fanatics aren't you?You going to start talking about mathematical heretics and burning people at the stake next.I swear you crazy fuckers are a cult.
>>16784270Then what obscure arithmetic operation were you referring to that would break due to † existing?As far as I see † would not break anything.
>>16784272Those are finitist troll fans of Wildberger you fucking idiot. You should only start taking things seriously when you learn to differentiate between things.
>>16784260So far this argument stands undefeated.
>>16784255>by definition, inf can't be a number.wronghttps://en.wikipedia.org/wiki/Extended_real_number_line
>>16784274I didn't say "1+1=2" (whatever that means under the new system) wouldn't be doable. I said it would break normal arithmetic and algebra.
>>16784242 >>16784260i get the feeling that you saw the veritasium video and so you are taking the piss out of cantor, unkindly take a hike
>>16784279Well as long as you can't give a substantive example of whatever the hell you're trying to say (and 1+1 was just trying to help nudge you towards that end), you said nothing.
>>16784275Don't try to pin this on the Wild Berger you disingenuous pigfucker.Its obvious that you are trying to introduce your batshit insane quasi religious notions into mathematics.Fucking fanatic.
>>16784281>implying watching yt slopKindly fold yourself in half.
>>16784282>give a substantive example of whatever the hell you're trying to sayOk.1+1=2 involves addition between two numbers.1+†=? involves addition between two numbers. Unless you can define addition for the second case, you have not defined the first case, either. But whatever that plus is doing in the second case, sure as fuck isn't normal addition.
>>16784242Take your ridiculous finite faith bullshit and shove it up your ass. It might stop your grandfather's penis penetrating you next time.
>>16784284You bloody fuck you, mathematics already has all notions of quasi-religiousness already.It's literally just a psychological displacement of wanting to do magic.The attitude you have against me is literally mirroring how religion would try to excommunicate someone for not being pure enough.
>>16784287>1+†You give the number 1 to God. You can only do that in heaven, so we have no idea in this world.
>>16784289Well said Brother!You have the full support of THE ONE TRUE FINITE FAITHBy divine grace we shall prevail in our noble quest to reclaim the HOLY LAND of Mathematics for the GLORY of GOD!And cleanse this Earth of those SATAN SPAWNED INFINITY LOVING SODOMITES!DEUS VULT!
>>16784288>being homophobic in 2025>>>/pol/
>>16784292>You give the number 1 to GodYeah, just as I thought. You finite faith cultists are all the same. You use God as just an excuse to seize power and have sex orgies all the time.Well you prattle off your usual shit about "sharpening swords" and "burning heretics" but dont think for one moment you will let you spread your crackpot ideas without a fight.
>>16784292>You give the number 1 to God.Fuck off, nigger. God is full.
>>16784302You asked a retarded question and I gave you the answer you deserved.Normal numbers are static and how agency but since † is God and God can be a number when he wants to but he can do whatever he wants when he wants to. 1 + † result in anything, God can give you back 1 or † - 1 or † - 2, or maybe you just die, it's absolutely not deterministic because you just simply can't force God to abide by any rules, he makes the rules.
>>16784307>and how agencywith no agency
>>16784305SILENCE HERETIC!All numbers belong to GOD. GOD will accept any and all excess numbers. Even the black ones. For GOD loves all of his Creations. Except for those infinity freaks. They will be condemned to HELL. And demons with pointy sticks will jab their bottoms.
Still received no good reason why mathematics should be independent of God. >>16784260 stands.
>>16784307Jesus Christ. I cant even comprehend the depth of your delusions. What is next on your agenda?You going to start burning math text books? Banning the use of the infinity symbol? Burning people at the stake because they dared to say 0.999...equals 1?
>>16784319It's evident that you're projecting because I never mentioned any of that.Hmm curious but please tell what kind of burning are you projecting onto me, what would *YOU* like to burn huh?
>>16784324Because you obviously belong to that finite faith cult and that is exactly the sort of shit they say.All of you are a clear and present danger to a mathematical system built up over centuries through hard work and logic.
>>16784328>danger to a mathematical system built up over centuries through hard work and logicWhat danger? Are you fucking autistic or something? Do you think just because other people have a different view it weakens the platonic zeitgeist of your One Rigorous Mathematics?That's why I'm saying that it's evident that it's you who is the religious fanatic. That would be fine if you weren't hypocritical on top of that.
>>16784315Amen Brother!Now all that remains for us to do is gather our weapons and lay siege to the Massachusetts Institute of Technology.Let them taste HOLY VENGEANCE for the mathematical SINS they have committed!PRAISE GOD!
>>16784296>conflating pederasty with homosexuality
>>16784333You and your blood crazed zealots plan to overthrow modern mathematics and replace it with your own ridiculous notions. Which do not work. The next thing aircraft will be dropping out of the skies, roads will lead off cliffs, computers will be having nervous breakdowns, and the planets crust will collapse under the weight of your sheer stupidity.
>>16784339Don't worry we don't have access to the Demiurge's temple. Yet.
I knew there was a good reason to read this thread.
>>16784352Who tf asked you?
cool it with the antisemetism
Op here, calm down it's just math.THE RULES OF DIVINE MATH:We define the Universe:[math]\mathbb{N}_\Omega = \{0,1,2,\dots,\Omega\}[/math]† has no successorFor all finite n, n < †† is the maximum, there is no m with † < mAddition:If a, b < † then a + b <= †If a + b > †, the sum is undefinedNo addition with †Same for multiplication.Division:Ratios are the only way to use †n/† is a finitesimal, smaller than any 1/m, m eof N
>>16784255is 2 more real than 1? what does its size have to do with its realness?
>>16784409The first latexed Omega should be †
>>16784281Op here, not the other poster, what Veriwhat video are you talking about?
>>16784287See>>16784409
What if infinity was just replaced with TREE(3)? For all practical intents and purposes that's the same as "divintillion" and the same as infinity.
>>16784434If we can't have inf we can't have 0.999... = 1, which is, or some people like to pretend to, the pillar of our creation.I agree in general with your statent but the satanic inf fuckers will instantly say "HURR DURR why not THREE(4)???"Thats why we need to define the finite max number as a concept and not a specific number or value.
>>16784452Amen Brother.The Finite Faith is strong with you.
>>16784412Infinity is an unbounded quantity greater than every real number.assumption: inf is a numberconsequence: infinity is greater than itselfwhich is obviously bsso, inf isn't a number
>>16784434>For all practical intents and purposesengineering =/= mathematics
>>167844521/inf = 0
>>167844821/inf does not exist, only 1/† does, which is the smallest possible unit, called a Thod (†hod (god)).With this math becomes quantifiable. You can even quantify the smallest space and time a calculation will need according to the syntax and method used.
>>16784260>god is a machine gun with never-ending, yet somehow not infinite, ammoDude, you are 12. Tops.
>>16784493Well you are actually talking about MOAN ( The Mother of All Numbers ) and FuSS (Fucking Small Something ), but that's fine, anything that contributes to those..those...THOSE GOD CURSED INFINITY LOVING SODOMITES being cast down into the discrete flames of HELL for all of a finite length of time.Amen.
>>16784482inf+r=infr-inf=-infinf+inf=infinf*inf=infr/inf=0inf-inf undefinedinf/inf undefined1^inf undefined
>>16784260>(P^~P)>thus>QEvery. Single. Time.
>>16784496SILENCE YOU SATAN SPAWNED CATAMITE!The Devil has taken command of your soul.It is up to us to exercise the Heresy from you, and with HOLY FIRE we will accomplish this.
>>16784409How far have you followed this? Any interesting results to present?
>>16784493Thod has good taste in music.
>>16784505Your analogy, tadpole. I just ran with it.Save the butthurt for someone who cares.
>>16784409>i like the ordinals more than the cardinals>i am so numerically elite that i have preferences you plebs never even dreamed ofHonestly, I have never used a number larger than 103, so you might be onto something.
>>16784242>>16784409True Computer Scientists have always known this and designed for it.The only remaining open question is the Word Size of His Creative Registers.Do you have any insight into this, OP?
>>16784499†+r=undefinedr-†=undefined on N††+†=undefined†*†=undefined1/†=smallest possible unitn/†=n*1/††-†=0†/†=11^†=1
>>16784538u tried
>>16784452>If we can't have inf we can't have 0.999... = 1Not true. You can prove 0.999... = 1 for any arbitrarily small †
>>16784550Proof that 0.999... 1= 1:The smallest number is 1/† (Axiom) 0.999... = 1 iif there is a number smaller than 1/† between 0.999... and 1 - this can't be.
>>16784519We could look at data as a bunch of freely adjustable units but we do that already in some languages with ducktyping.It could help by making math better to understand for programmers, they could cheat less and leave us with less dirty work. They would need to give us more information and therefore clarity.>>16784506I'm thinking about some funny proofs that could make people blood boil but I'm bit busy currently. Maybe I feed the axioms into grok and see what he can do with it.
>>16784558>Maybe I feed the axioms into grok and see what he can do with it.Q.
>>16784554Did you type something wrong there?
To answer the original question. Nothing would change if we changed a symbol. Math is a language. How we choose to visualize the idea doesn’t matter.
Euler was the first person to sum a divergent series to finite results.
Are you the same schizo or are you just stealing his crackpot ideas?>>16775320Infinity will be replaced by MOAN. The Universe is finite. It is discrete. And so shall be mathematics. It is God's will.
The universe is finite.
That’s hilarious. Humans!! Lol. It is infinite
>>16784620Why?
The universe is infinite in all measures. It has always and will always exist.
Divinitillion + 1 = ?
>>16784650Itself
>>167846551?
You can add, subtract, multiply, divide, square, etc. The answer is always infinity.
>>16784242Wouldn't the largest number just be the total amount of the most factored partilce/wave in the universe?
>>16784655So you can't math with it...
It will always be infinity.
>>16784242Unless you can count to "†" there's no difference. Just a gold calf of infinity to worship.
>>16784242>1^3 + (-1)^3 = 0^3Is he retarded?
>>16784274all retards ignore the fact that reals form a commutative firld and adding any sort of infinity would break this criterion.
>>16784675You can count to it just like you can count to 100100100 (Knuths uparrow notation)
>>167848814chan doesnt display uparrows100 arrows 100 arrows 100
>>16784881Meaning you can't count to it. So † is just a manmade idolatry of infinity. Lame af
>>16785032You seem to have some pretty fundamental misunderstandings about all this.It was explicitly specified that divintilion can be found >>16784260but just like how you can't just order an audience with God, and since † is an aspect of God himself (for it to be greater than anything else), you must be chosen by God for him to reveal divintilion to you. The point isn't that whether you can or cannot in generall, it's that if you're guided by hubris and you arrogantly try to reach † on demand, it will not be revealed to you. It's pretty self-evident that first of all you must not be theologically stunted to understand divine mathematics.It's the total opposite of being manmade.
>>16785036>but just like how you can't just order an audience with God, and since † is an aspect of God himself (for it to be greater than anything else), you must be chosen by God for him to reveal divintilion to you.If you're going to base it on an Islamic relation to God, get the cross out of your mouth. It's offensive. Maybe use [math] \blacksquare [/math] instead.
>>16785048It can fit into an Islamic framework but this reasoning is not uniquely Islamic at all.
>>16785048The problem is that the concept of "one final biggest number that nobody knows" is best described using a "god" comparison. I called it a Divinitillion and chose the cross as a symbol, and people's minds got creative and they understood what I meant without a lot of explaining because most people understand the concept of a singular god and they will use it to understand the definition of a divinitillion.
>>16784416https://www.youtube.com/watch?v=_cr46G2K5Fo
>>16784254
>>16784242>Lecture 13: Arithmetic with a tophttps://bpb-us-w2.wpmucdn.com/blog.nus.edu.sg/dist/4/10956/files/2019/01/13_top-1dhjq28.pdf
>>16784255There are several ways to make infinity behave itself, eg projective geometry.
>>16784269What is the largest number + 1?
>>16785215Yeah this.There is usually someone who has already thought of it, and usually in far more thoroughly, in far more detail and rigor.
>>16785233By definition, x + 1 is S(x), and since † is God, S(†) means Jesus.
>>16785215So if † is the top then 0 is the bottom?
Doing arithmetic with infinity is easy.[code]function one(f, z) { return f(z);}function two(f, z) { return f(one(f, z));}function three(f, z) { return f(two(f, z));}function god(f, z) { return f(god(f, z));}function plus(a, b) { return (f, z) => a(f, b(f, z));}plus(three, two)((x)=>x+1, 0);plus(god, one)((x)=>x+1, 0);[/code]>inb4 javascript badJust use lambda calculus, then. Use the Y combinator to define god (†) as an infinite recursion. Why is this wrong, mathfags? I demand an answer.
>>16784500im glad that you are mentally ill
>>16784260>For each number as you go sequentially, you'll always be able to find a bigger number.>But at a certain point you'll find a number compared to which there's no bigger number, which is God. That amount is divintillion.How do you find this point? Hopefully not by going sequentially...
>>16785317Through revelation.
>>16785409Right, I see. So if you just keep going from one Current Thing to the next like an NPC, you never get anywhere. But through the power of schizophrenia you can break this pattern and find thrembo.
>>16785411Absolutely not, if you keep going just like that, mechanically, you'll never get anywhere. It's not about how long you go if you think the answer can ever be concieved in you.If you do mathematics in your limited mortal head, you will never in a trillion years experience infinity. You'll think you have experienced it, but you'll be cheated. It's such a sadness, that you think you've got the faintest idea of a divine concept. Get real.
>>16785275>>inb4 javascriptYou're doing integer maths with floats. All numbers in JS are floats. JS is basically the worst language you could have picked for mathematical computation.
>>16785423You're right. My bad. Keep everything the same, just replace the last two lines with these:const five = plus(three, two);const satan = plus(god, one);
>>16785425You did the ChatGPT thing where you say you fixed it but you didn't fix it.
>>16785429You did the /pol/ thing where you were born heavily dysgenic. I mean... holy shit, what a retard. I'm used to mouth breathers, but you really take the cake today. I'm seriously not convinced you're not a sub-GPT bot.
>>16784242well, the most precise value of pi would then be produced with circumscription with a string of length divinitillion
>>16785444God is an intelligible sphere, whose center is everywhere and whose circumference is nowhere.
Who can number the sand of the sea, and the drops of rain, and the days of eternity?3Who can find out the height of heaven, and the breadth of the earth, and the deep, and wisdom?4Wisdom hath been created before all things, and the understanding of prudence from everlasting.5The word of God most high is the fountain of wisdom; and her ways are everlasting commandments.6To whom hath the root of wisdom been revealed? or who hath known her wise counsels?7Unto whom hath the knowledge of wisdom been made manifest? and who hath understood her great experience?8There is one wise and greatly to be feared, the Lord sitting upon his throne.9He created her, and saw her, and numbered her, and poured her out upon all his works.
>>16785441https://tc39.es/ecma262/#sec-ecmascript-language-types-number-type>The Number type has exactly 18,437,736,874,454,810,627 (that is, 2**64 - 2**53 + 3) values, representing the double-precision floating point IEEE 754-2019 binary64 values as specified in the IEEE Standard for Binary Floating-Point ArithmeticUsing the const qualifier doesn't make your code correct.This mouth breather /pol/ack advises you to use BigInts next time.https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/BigInt
>>16785453That's very nice, you subwhite, subnigger, subanimal entity. Now count the number of times floats appear in that code after the correction.
>>16785455Oh, I seee, you removed the lambda or whatever.Now your code does nothing lolGreat fix
>>16785461You're legit 70 IQ.
The ONE TRUE FINITE FAITH is mightily disappointed that OP has chosen not to respond to any of our supporting posts.After defending OP against the Heathens one might think that is the very least OP could do.But what rankles most is OP's intransigence in acknowledging the origin of his crackpot ideas.Those crackpot ideas belong to us.That is MOAN and FuSS.The "Mother Of All Numbers", an immense number that marks the end of the number line and to which nothing can be added.And"Fucking Small Something", the smallest number possible, which is indivisible, and therefore whose physical manifestation represents the true "atom" of the Universe and the foundation upon which everything in the Universe is constructed.In receipt of these failures THE ONE TRUE FINITE FAITH has no other choice but to declare OP a slimy scheming stealing scumbag, as befits our proclivity towards alliteration, and declare a HOLY WAR OF RIGHTEOUS RETRIBUTION!No mercy.No prisonersLots of sharp pointy things pokes in OP's nether regions.BROTHERS AND SISTERS OF THE ONE TRUE FINITE FAITH!WE MARCH!DEUS VULT!DEUS VULT!DEUS VULT!
>>16785463No, you just have a severly skewed idea of what programming is.I tried to think of a way to show you why it doesn't work like that but I know you're simply too arroagnt to understand. I see what your intention was, but it's just hot garbage.Go on and keep writing convoluted implementations of reduction, but in the meanwhile I'll leave this quote here>An idiot admires complexity, a genius admires simplicity, for an idiot anything the more complicated it is the more he will admire it, if you make something so clusterfucked he can't understand it he's gonna think you're a god cause you made it so complicated nobody can understand it. That's how they write journals in Academics, they try to make it so complicated people think you're a genius
>>16785475>I tried to think of a way to show you why it doesn't work like that But you couldn't because you're 70 IQ and have no idea what's actually happening, even though it's basic. If you had an extra 20 IQ points to elevate you above the average nigger, it would have been sufficient for you to see that it actually works fine and does exactly what it's supposed to.
>>16784243No. ∞ + 1 = ∞ whereas † + 1 does not exist.
>>16784287√1 = 1√-1 = ?If you cannot define √-1 in R then you cannot define √1 in R either. See how this works?
>>16784338>Thinking there's a different between two synonyms
>>16785484I see the drool on your chin. Imagine thinking you did something other than to prove my point again.
>>16785489I accept your concession. There is a number system in which † is the biggest number. 1+1 = 2 and 2+1 = 3 and... all the up to 1 + († -1) = † are all.dedknee.
>>16785487>.@FBI
Here we are, THE ONE TRUE FINITE FAITH, laying siege to OP's basement. We await his surrender.If he is lucky he will get dry wood rather than green.
>>16785468Here is my reply you are a good boi
>>16785499How can you accept my concession if I already accepted yours? The square root wouldn't be properly defined without addressing the negative number case. If you're in R, it's addressed by saying "sorry, that's invalid". You could say the same about 1+†=? (that's invalid) but now you've broken normal arithmetic. How come plus suddenly works for some numbers but not others? It used to work for any two numbers, no exceptions.
OP has mental problems and needs to go back to /x/
>>16784242You would just reinvent aleph numbers and continue to cope with why you can never define †-1 or †+1, so you would just come up with differing degrees of † that are completely disconnected from finite real numbers.
>>16784260>But at a certain point you'll find a number compared to which there's no bigger numberNo, at a certain point, you won't have enough memory or physical space to retain or express the sequence.
>>16784274Division by † since division by 0 is already broken and multiplication by † would be broken since any addition to † is invalid, so division and multiplication would both get broken because they wouldn't apply to all x greater than 0 and calculus would get broken since division and multiplication are broken, then that breaks combinatorics and differential equations and everything that uses division, multiplication or calculus- derivatives or integrals.
>>16784409So in your scenario "god" is just the dead unreachable nonsense at the end of the universe while 0 is still the origin point and creative force of math and the universe?
>>16784493>which is the smallest possible unitNot according to >>16784409 that says 0 is the smallest possible unit.
>>16784538>1/†=smallest possible unit>†-†=0So what is †-†+10/†?>1/†=smallest possible unit>†/†=1Wouldn't that mean 1/† = 1/†^2?
>>16784624Because its 1 universe, by definition.
>>16786707fuck off with your retarded gibberish
>>16784881No because there is a finite number that comes right before and one that comes right after that number, but no number comes right before †.
Finitism isa philosophical stance holding that only finite entities, quantities, or concepts exist. In mathematics, this means that completed infinite totalities, such as the set of all natural numbers, are rejected in favor of finitely constructible objects. However, finitists do not necessarily reject the usefulness of infinity in mathematics, and the concept can take various forms, including cosmological finitism (the idea that the universe is finite) and theistic finitism (a view of God as finite)Key aspects of finitism Rejection of completed infinities: Finitists believe that the concept of a completed infinity, like the entire set of natural numbers or all real numbers, is not real or does not exist. Focus on finite construction: Mathematical methods that rely only on finitely constructible quantities are preferred by finitists. Epistemological security: A key motivation for finitism is the belief that finitistic mathematics provides a more secure and privileged foundation for knowledge. Application in other domains: The core idea that "everything is finite" can be applied to other areas, leading to distinct types of finitism.Types of finitism Mathematical finitism: The most common understanding of finitism, this approach concerns what can be considered to exist in mathematics. Cosmological finitism: A theory that posits a finite universe, rejecting the idea of an infinite one. Theistic finitism: A theological viewpoint, often associated with a God who has limitations, rather than an infinite one.
Motivations for finitism Limited resources: The observation that everything in the universe, including calculational resources, appears to be limited. Conceptual clarity: A belief that the concept of infinity is often misunderstood. Consistency: A desire to build mathematics on a more secure foundation by avoiding the paradoxes and problems associated with infinite sets. Key tenets Rejection of actual infinity: Finitists accept all natural numbers as real entities, but they do not accept the set of all natural numbers as a single, existing mathematical object. This means they reject transfinite numbers and concepts built on completed infinite totalities. Focus on construction: Mathematical objects and proofs must be constructed from natural numbers in a finite number of steps. Limitations on logic: Because statements cannot quantify over infinite domains, certain logical principles of classical mathematics—such as the law of the excluded middle—do not apply in all cases. A statement can be "undetermined" if its proof would require an infinite computation.
>>16785445That isn't what a sphere is, though, by definition a sphere has a center that is r units from the circumference in every dimension.
Intuition-based arguments: The finitist position is often motivated by the belief that mathematics should not produce counter-intuitive results, such as the Banach-Tarski paradox, which suggests a sphere can be disassembled and reassembled into two identical spheres. Finitism and other philosophiesFinitism vs. Mainstream Mathematics: The central divide is the existence of infinite objects. Most mathematicians are platonists who believe in the existence of infinite objects within a set-theoretical universe, and most of modern mathematics, including calculus and analysis, relies on these concepts. Finitists would argue that a calculus limit approaching infinity is not a single entity but a description of a finite process that continues indefinitely.Finitism vs. Intuitionism: While sometimes confused, these are distinct ideas. Intuitionism is a form of constructivism that accepts the concept of "potential infinity," meaning that an unlimited process can go on forever, even if it never finishes. Finitism is stricter, disallowing this concept entirely. Intuitionists also have a more permissive view of the existence of the mathematical continuum than finitists.Finitism vs. Ultrafinitism: Ultrafinitism is an even more extreme form of finitism. While finitists may accept all natural numbers as existing individually, ultrafinitists also doubt the existence of impossibly large finite numbers due to physical and computational limitations.
Historical context Pre-Cantor: Historically, mathematics was classically finitist, not dealing with completed infinite objects. Ancient Greek mathematicians, like Aristotle, distinguished between potential and actual infinity, arguing that actual infinity was impossible. Cantor's Set Theory: The work of Georg Cantor in the late 19th century, which explored transfinite numbers and infinite sets, challenged the traditionally finitist approach and sparked major debate. Hilbert's Program: In the 1920s, mathematician David Hilbert attempted to save classical mathematics from set theory paradoxes by proving its consistency using only "finitistic" means. The hope was that infinite objects could be treated as useful abstractions that wouldn't lead to contradictions. This program was famously undermined by Kurt Gödel's incompleteness theorems, which demonstrated that the consistency of a formal system like arithmetic could not be proven using only its own finitistic methods. Modern relevanceWhile finitism is not the standard approach, it has seen modern developments and discussions: Finitist interpretations: Some mathematicians today re-evaluate theorems proven with infinitary methods to find finitistic proofs, an area sometimes associated with reverse mathematics. Discrete geometry: Research in discrete geometry explores mathematical models that rely only on finite quantities, which has connections to physics, computer science, and models of spatial perception. Critiques of infinite concepts: Some mathematicians and physicists continue to question the conceptual coherence of certain uses of infinity, prompting new avenues of exploration.
>>16786715A sphere is a perfectly round, three-dimensional geometric object. In mathematics, it is defined as the set of all points in three-dimensional Euclidean space that are at a fixed, equal distance from a central point. This fixed distance is known as the radius (\(r\)), and the central point is the center of the sphere.
>>16786721Yes and that definition is not compatible with >>16785445, since as you said, the circumference is always a fixed point away from the center in a sphere.
>>16784260Think of it this way . The number 6 is like a revolver, being a gun holding that many bullets, which fire one at a time in quick successionLarger numbers are more like an ak47 where a magazine is filled with so many bullets, fired at a higher rate of fireNo mum I dont need to poop get out
"Maths constructivist" refers to two related ideas: constructive mathematics, a philosophical approach where a mathematical object must be explicitly constructed to prove its existence, and constructivist mathematics education, a teaching method where students actively build their own understanding through experience and problem-solving, rather than passively receiving information. The philosophy emphasizes building knowledge from prior experience, with the teacher acting as a facilitator. Constructive Mathematics (Philosophy) Existence via Construction: In this view, to prove that a mathematical object exists, you must provide an explicit method or algorithm to find or construct that object. Algorithmic Proofs:Proofs must be algorithmic, meaning they demonstrate a process for finding the object, not just a logical argument for its existence. Contrast with Classical Math:This differs from classical mathematics, which allows for non-constructive existence proofs and does not always require a method for finding the object. Constructivist Mathematics Education (Pedagogy) Active Learning: Students are not passive recipients of information but actively engage in activities, problem-solving, and hands-on experiences to build their own mathematical understanding. Building on Prior Knowledge:Learning is a cumulative process where new concepts are built upon a student's existing knowledge and experiences.
Constructivism as a philosophy of mathematicsThis area, also known as constructive mathematics, sets a stricter standard for mathematical proofs by rejecting methods that do not produce a verifiable object.Existence proofs: In classical mathematics, one can prove an object exists by showing that its non-existence leads to a contradiction (a reductio ad absurdum). In constructive mathematics, a proof of existence must provide a method for finding or building the object.Logical principles: Constructive mathematics rejects the universal validity of the Law of the Excluded Middle (for any proposition \(P\), either \(P\) or \(\neg P\) is true). A constructivist might argue that for an unsolved problem like Goldbach's conjecture, we cannot yet definitively claim it is either true or false.Key figures: Important figures in constructive mathematics include L.E.J. Brouwer, the founder of intuitionism, and Errett Bishop, who showed that a large portion of modern analysis could be developed constructively.Constructivism as a learning theory in mathematics educationFor educators, constructivism provides a framework for creating student-centered classrooms that foster deeper, more conceptual understanding of mathematics. This approach encourages students to be active and collaborative participants in their own learning.Key principlesConceptual vs. procedural knowledge: The focus is on understanding mathematical concepts (e.g., why differentiation works), not just on memorizing procedures (e.g., the steps to find a derivative).Active learning: Students actively engage with problems, often through hands-on activities and exploration, rather than passively listening to a lecture.Prior knowledge: Learning builds on students' existing understanding, using it as a foundation for new concepts. Teachers assess what students already know and design lessons to challenge their assumptions.
Influential theorists Jean Piaget (1896–1980): This Swiss psychologist is a key figure in individual or "radical" constructivism. He proposed that learners construct their own knowledge through internal mental processes of assimilation (integrating new information into existing ideas) and accommodation (changing existing ideas to fit new information). Lev Vygotsky (1896–1934): Vygotsky's sociocultural theory emphasizes that learning is an inherently social and cultural activity. He introduced the concept of the Zone of Proximal Development (ZPD), where a student can perform more advanced tasks with support from a teacher or a more capable peer. Jerome Bruner (1915–2016): Bruner's work integrated with constructivism, promoting "discovery learning" where students explore and find principles themselves. He also introduced the Concrete-Representational-Abstract (CRA) instructional model, a structured path from hands-on activities to symbolic representation. Practical teaching strategiesIn a constructivist maths classroom, a teacher acts as a facilitator rather than a lecturer, using various strategies to engage students. Problem-based learning: Students tackle complex, open-ended problems that do not have a single, straightforward answer. Mathematical modeling: Students apply mathematical concepts to investigate and solve real-world problems. The focus is on the process of reasoning, not just the final solution. Manipulatives and visual aids: Teachers use concrete tools (like blocks or counters) and visual representations (like drawings or graphs) to make abstract mathematical ideas tangible. Scaffolding: Teachers provide tailored support structures and gradually reduce assistance as students' confidence grows, aligning with Vygotsky's ZPD. Mathematical discourse: Open-ended questions encourage students to explain their reasoning, debate solutions, and ask questions.
An infinitesimal is something immeasurably or infinitely small, or an adjective describing such a quantity. In common usage, it means extremely tiny, minute, or negligible. In mathematics, an infinitesimal quantity was historically a number smaller than any real number but not zero, a concept crucial to the development of calculus but later superseded by the rigorous concept of limits. In general use (adjective) Meaning: Excessively or immeasurably small, minute, or tiny. Examples: There were infinitesimal traces of heavy metals in the soil, meaning they were so low as to be insignificant. An infinitesimal amount of a poison can be lethal. The improvement was so small as to be infinitesimally small. In mathematics (noun) Historical context: When calculus was first developed, infinitesimals were thought of as quantities that were less than any assignable quantity but not actually zero. Modern concept:While the idea of a single, non-zero infinitesimal number doesn't exist within the standard real number system (because there's always a rational number between 0 and any other non-zero number), the concept was essential for the development of differential calculus. Influence:The concept of the limit, a cornerstone of modern calculus, emerged partly from the need to rigorously define what was happening when quantities became "infinitesimally" small. Current usage:Today, a sequence that approaches zero is sometimes called an infinitesimal sequence.
>>16786680This is also true for inf, yet we use it everywhere>>16785265Yes>>16785233Undefined>>16786676t-1 is t-1. It's 1 left from t on the number line.
>>16786725>>16786715You're a complete retard.
>>16787338Not either of those anons but let's get back to the question. How would math change? It wouldn't.
Is finitism really a thing?A friend of mine asked me if there were mathematicians who didn't think irrational numbers existed. I sincerely told him that for the last hundred year, pretty much everyone was on the same wavelength there. My friend then forwarded me a video of a mathematician explaining to people how the square root of two doesn't exist, not as analysts understand it. I watched the video and had an acid relapse.And I've never done acid.This guy is basically dancing around his conviction that infinite objects don't exist. In other places he suggests that that's because they're not computable. I thought this was utterly absurd, because that would eliminate a lot of finite numbers as well. But I'm talking this over with friends in math and one of them tells me about finitism and how strict finitists don't believe in really big numbers either.Is this a thing? And if so, how do we do it without breaking math?For example, if we don't have irrationals, a lot of neat, but terribly discontinuous functions become continuous (using the definition we learned in real analysis). A trivial one is a function is zero for all x where x2 < 2 and 1 otherwise. The most recent friend showed me a function that was not just continuous but locally constant, unbounded and not uniformly continuous... and this was defined on the set [0,2] intersect Q. But the intermediate value theorem also stops working. The Extreme Value Theorem stops working, the theorem that tells us that continuous functions on compact sets attain their max. Do we even still have a useful notion of compactness?Do any modern mathematicians really do this? Is this a thing? And why? And don't complex numbers disappear using similar logic? What is the appeal here?
Finitism is not really about the nonexistence of irrationals. It is more about the finitude of expression and proof, and this really is not talking about this. It is discussing ultrafinitim, but in a very awkward and stunted way that makes one question the author's competence. I think there are much better sources.Just to give a brief background: ultrafinitism takes meaning very seriously. We give the meaning to mathematical statements by providing models for a given theory. Models are where we talk about truth and falsity, they are where we talk about existence and where the actual ontology lives.That's standard across math. Tarski's model theory is used all over modern mathematics. Ultrafinitists, though, will point out that we never actually have infinite models. In our world, if we have a finite time to identify objects and each object identified takes a finite time, we will only ever work with and identify a finite collection of things to build models with. Instead, what typically happens is that mathematicians will say "assume we have this infinite model" and work from there.The point of ultrafinitism isn't to say "you can't do that". Obviously we can and do all the time. It is just saying "if you don't actually have a model to work in, what you are doing does not have meaning - in the well-established definition of 'meaning' in mathematics". So ultrafinitists look to find out about the fraction of math that is not meaningless.So, in the real world, numbers do not go on forever. To current knowledge, there is a finite information content in the universe and even if there isn't, we only know how a finite fraction of the information content could be accessible. This also means that other parts of symbol manipulation, including proof complexity, may have an upper limit. Any completed infinite collection, even countable, is meaningless.
Check out Doron Zeilberger, an ultra-finitist. If I may, it seems the short version of his opinions is that computers are the ultimate mathematics machines, and human mathematics is baloney (if it does not play into his ultra-finite computable world.)Now, I'm also a realist, and I understand the abstractions of mathematics are not "real", but that doesn't mean we can't still tell a cool story.edit: err i got selfish and posted before reading all the responses. I see that my response is essentially included already.
>>167873720.999... is simply an integer in base ten where you replace as many zeros as you need with as many nines as you need.
There is a position beyond that, ultrafinitism, which believes that there's actually some largest number. I'm not sure any serious mathematicians actually believe that.Its very believable, its based on simple physics. There is naturally a number so large, that one could not hope to write down its successor. At that point Z "breaks" and so you might as well call that the largest number integer. That number is probably somewhere close to 2|information content of the universe| or 2|particles in the universe| or something like that. Beyond that we just don't have the bits to encode arbitrary numbers.The finitist says: "Infinity doesn't exist because I can never reach it no matter how many times I apply the successor function." The ultra-finitist goes a smidge futher: "Skewes' number floor(eee^79) is not a member of Z because I can never reach it because I cannot physically apply the successor function enough times to do so."The better question is if it is useful to have this point of view. What is Skewes' number if it isn't an integer? Alternately one could ask if Skewes' number was ever useful. The proof behind Skewes' number might be useful (especially as the bounds have now been brought down to the very real e727.95...) but certainly the actual expression he derived was singularly useless.To flip things on their heads entirely. Suppose that the ultra-finitists are correct and that one of them derives a proof that basic arithmetic was contradictory, and that the simplest expression for which arithmetic failed involved an expression of the form floor(eee^341132)+floor(pipipi^12132312321) upon which the universe returns EOVERFLOW. Would we suddenly abandon arithmetic as a failure? Stop teaching it in elementary school?Similarly if one showed that the Riemann Hypothesis was false but that the only non-trivial root that violated it had a real part that deviated from 1/2 by substantially less than 1/Skewes' number... what would that change?
>>16787378If Doron is here, he'll certainly agree that 1 - 0.9... = 0
I was under the impression that finitism was just "some ancient philosophical movement" in mathematics, only followed by one or two nowadays, so It sounded like a joke to me.But then I got curious and, after reading a bit, It seems to me that the only arguments against infinite mathematics that finitists seem to have are that "there are numbers so big that we couldn't computate in a lifetime" or the naive set theory paradoxes. The former doesn't seem like a serious argument, and the latter is not a problem now that mathematics relies on consistent axioms.Are there some (maybe arguably) good mathematical reason to deny the existence of ∞or is it just a philosophical attitude? The concept of unboundedness seems pretty natural to me, so what could be a reason to avoid it? Does this attitude even make any sense?In short, why today-finitists have a problem with ∞?
I didn't know that "finitism vs. infinitism" was such a polemic topic. Now I myself agree this question might look as primarily opinion-based. However, It was not my intention to open a debate about "which posture is better"; I was just meaning to ask about what specific mathematical reason (argued and not-primarily opinion based) do finitists have to reject the "infinitists" use of infinity.Based on the two excellent answers I've already had (thank you again :) It is my understanding that their main problem with the use of ∞is that it leads to mathematical results (like the Banach-Tarski paradox) which they don't recognize as true when looked through the glasses of our real-world experience.\After reading every answer and comment I've came to the conclussion that there are not strictly mathematical reason to avoid the use of infinite. That my specified question on the last edit has no answer, and that the motivation to stick to finitists or infinitists view of mathematics relies on how much one expects mathematics to describe each one's "real world".
>>16787386It's a money thing. You don't have any so you wouldn't understand.
It's not known that modern set theory is consistent; in fact, by the Incompleteness Theorem, we can't ever have a system of axioms that we can prove is consistent. Which means that the only condition we can rely on for determining whether a set of axioms is "right" is whether or not it produces absurd results.Under ZFC, we have different sizes of infinity - there are sets which are larger than the set of natural numbers in a precise sense. We also have a lot of weirdness involving the Axiom of Choice - for example, with the Axiom of Choice, a theorem of Banach and Tarski states that a hollow sphere can be disassembled into five pieces and then reassembled (without stretching, tearing, or otherwise deforming the pieces) into two spheres that are both identical to the first one in both size and shape. But the Axiom of Choice simply states that given a set of sets, we can "choose" one element from each set - which seems intuitively true.A finitist's perspective on ZFCis often that results like the hierarchy of infinite cardinals and the Banach-Tarski paradox are absurd - that they should count as contradictions, because they patently disagree with the intuitive picture of mathematics. The sensible conclusion is that one of the axioms of ZFC is wrong. Most of them are intuitively obvious, because we can demonstrate them with finite sets - the only one we can't is Infinity, which states that there exists an infinite set. So a finitist's conclusion is to reject the Axiom of Infinity. Without that axiom, ZFC becomes purely finitistic.
>>16787390>"which posture is better";Only an Indian would think in those terms. Visa denied.
Now, many finitists are happy to stop here. But some are bothered by the fact that we still have an infinite collection of natural numbers; the infinite still "exists", in a sense, and gives the opportunity for the above weirdnesses to arise in the same way. So some people (including some mathematicians) subscribe to ultrafinitism and insist that there are only finitely many numbers at all. One ultrafinitist mathematician I know defines the largest integer to be the largest integer that will ever be referenced by humans.Among mathematicians, ultrafinitists are much rarer than simple finitists. Finitists generally agree with you that "unboundedness" is a natural idea - it's essential, for example, in the definition of a limit. But they would go on to insist that this is just a formalism - that a limit, for example, is just a statement of eventual behavior, involving only finite numbers. So ∞isn't an object, it's just a shorthand. This is (at least to my mind) more mathematically defensible than ultrafinitism.
Are there some (maybe arguably) good mathematical reason to deny the existence of ∞ or is it just a philosophical attitude? The concept of unboundedness seems pretty natural to me, so what could be a reason to avoid it?This is inherently a subjective position, and a perfectly valid one: "I can think of this idea, so how can someone say this idea doesn't exist?"Of course it would be ludicrous to claim that the idea of infinity doesn't exist. You just thought of it (thought about it), didn't you?I'll be an ultrafinitist for a moment and explain the position.It's not that infinity doesn't exist as an idea, it's that you cannot point to an infinity anywhere in the real universe. Anything you point to is necessarily finite, or you couldn't point it out or demonstrate it.Mathematics is all (all) based on working with symbolization of real or abstract data. You're dealing with ideas, fundamentally, and ways of representing those ideas to resolve, communicate about, or pose problems—again, either real or abstract.Please don't be so attached to a single system for ideas and their symbolization that you fail to recognize that other ideas may exist outside of that scope.You criticize ultrafinitists for failing to include the concept of an infinity in their abstractions and symbolizations. Very well, why is it that your own mathematics fail to include the concept of "certainty"? Or "knowledge"? Or "co-existence" (the same number having two different values at the same time)? Or how about "time" itself, since that is not included in mathematics?If you can work with your mathematics and get results that work, or even just that you find interesting, fine. If I can work with my mathematics and get different results than you, but they work for me (produce a desired result when applied to the real world), great.But this is all more general, covering the broad sweep of differences of mathematical ideas.
Leopold Kronecker was one of the leading mathematicians at the end of the 19th century. Kronecker disagreed sharply with contemporary trends toward abstraction in mathematics, as pursued by Cantor, Dedekind, Weierstrass, and others. Specifically, Kronecker rejected the notion of completed infinity. Many of today's mathematicians are so used to set theory being "the foundation" that they have difficulty relating to Kronecker's point to begin with. Kronecker's ideas were close to but not identical to the modern constructivist ideas; thus Bishop for one arguably did accept actual/completed infinity, since early on in his book he speaks matter-of-factly about functions f from N to N. The difficulty we have of relating to Kronecker's viewpoint has to do with our training. Recently Yvon Gauthier tried to explore Kronecker's position; see in particular hisGauthier, Yvon. Towards an arithmetical logic. The arithmetical foundations of logic. Studies in Universal Logic. Birkhäuser/Springer, Cham, 2015andGauthier, Yvon. Kronecker in contemporary mathematics, general arithmetic as a foundational programme. Rep. Math. Logic No. 48 (2013), 37–65.Gauthier argues in particular that many applications Kronecker worked on can be handled without superfluous infinitary assumptions that merely clutter the picture.
If you believe that the natural world is all that exists, then that automatically makes you an ultra-finitist. There is simply no room in the physical world to store a monstrously large integer.The axiom of infinity is obviously false when taken literally in the physical world, so if the physical world is all that exists, then the axiom of infinity is false.This does not mean that one should reject ZFC or any of its theorems. While Con(ZFC) (ZFC is consistent) has not been proven in the usual meaning of proof, there are nevertheless very good reasons to believe Con(ZFC). It has been tested very thoroughly, both theoretically and practically, to the point that doubting Con(ZFC) is simply no longer a reasonable position to take.Con(ZFC) means that all predictions of ZFC that are testable in the real world ought to be believed. There is no problem with using real numbers and infinite sets to prove theorems that are then used in weather forecasts and other scientific computations. Con(ZFC) means that in testable predictions, we can trust math.That's why we should teach theorems derived from ZFC without any reservations. That also means: accept the Banach-Tarski theorem, different infinite cardinalities, etc. (not as "true" in the physical world, but as theorems of mathematics).Ultra-finitism is true in the real world. However, none of the theorems based on ZFC ought to be controversial.Scientific computations usually use approximate floating point numbers. It is hard to prove clean statements about such objects. It is much easier to first prove statements about infinite-precision "real" numbers (even though they don't actually correspond with anything in the "real" world). Infinite sets, and infinite-precision real numbers are so useful that it would be foolish not to work with them, and the highly probable Con(ZFC) means that math based on infinite objects can be trusted.
