#Mathematical Infinity

[omegaphilosophia]

The Philosophy of Infinity

The concept of infinity has fascinated philosophers, mathematicians, and theologians for centuries. In philosophy, infinity refers to something that is unbounded or limitless. It challenges our understanding of the world, pushing the boundaries of human cognition and logical reasoning. Here's an exploration of the philosophy of infinity:

1. Mathematical Infinity

In mathematics, infinity is used in various contexts, such as the idea of an endless sequence of numbers, the concept of limits, and the notion of infinite sets. The work of Georg Cantor in the 19th century revolutionized the understanding of infinity by showing that there are different sizes of infinity, known as cardinalities. For instance, the set of all natural numbers is countably infinite, while the set of all real numbers is uncountably infinite, meaning there are more real numbers than natural numbers.

2. Metaphysical Infinity

Metaphysically, infinity often refers to the nature of the universe, God, or other ultimate realities. Philosophers like Aristotle and Aquinas discussed infinity in relation to the nature of the divine. Aristotle distinguished between potential infinity (a process that could continue indefinitely) and actual infinity (a completed state of infinity). Thomas Aquinas later used the concept of infinity to argue for the existence of an infinite being, God.

3. Epistemological Considerations

Infinity also raises epistemological questions about the limits of human knowledge. Can we truly comprehend infinity, or is it beyond the grasp of finite minds? Some philosophers argue that infinity is a useful theoretical construct but cannot be fully understood or visualized. Others suggest that our understanding of infinity is shaped by our cognitive and perceptual limitations.

4. Ethical Implications

The concept of infinity has ethical implications, particularly in the context of moral philosophy. For instance, the idea of infinite value or worth can influence discussions about human dignity and the sanctity of life. If each human life is considered to have infinite value, it could have profound implications for ethical decision-making and the prioritization of resources.

5. Infinity in Cosmology

In cosmology, infinity pertains to the nature of the universe. Is the universe finite or infinite in extent and duration? The concept of the infinite universe has been a topic of debate among philosophers and scientists alike. Some theories in modern physics, such as the multiverse hypothesis, suggest that there could be an infinite number of universes.

6. Infinity in Literature and Art

Infinity also appears in literature and art, often symbolizing the sublime, the eternal, or the boundless. Artists and writers use infinity to evoke a sense of wonder and to explore themes of eternity, the divine, and the limits of human experience.

The philosophy of infinity is a rich and complex field that intersects with many areas of human thought, from mathematics and metaphysics to ethics and cosmology. Infinity challenges us to think beyond the finite and to grapple with concepts that stretch the limits of our understanding. Whether in the precise language of mathematics or the profound reflections of metaphysics, infinity continues to inspire and perplex, pushing us to explore the boundaries of knowledge and existence.

[noosphe-re]

I am not a plane Figure, but a Solid. You call me a Circle; but in reality I am not a Circle, but an infinite number of Circles, of size varying from a Point to a Circle of thirteen inches in diameter, one placed on the top of the other. When I cut through your Plane as I am now doing, I make in your Plane a section which you, very rightly, call a Circle. For even a Sphere – which is my proper name in my own country – if he manifest himself at all to an inhabitant of Flatland – must needs manifest himself as a Circle.

Edwin Abbott, Flatland: A Romance of Many Dimensions

[marieisnothere12]

∞ “math” bc im high on algebra (is this algebra?)

im high on math rn so hypothetically lets find yhr mean of ∞

so an example of mean is

(1 + 2 + 3) / 3

So,

(1 + 2 + 3…) / ∞

we have ∞/∞, which is 1. (The denominator is ∞ bc we have an infinite amt of numbers)

Howver, this is incorrect. We havent accounted for negatives. The mean would be 1 if ∞ was only positivr numbers (which its not)

(-1 + 1 - 2 + 2…) / ∞.

bc there is an equal amount of negatives amd positives, the mean of ∞ is 0.

also you can find thr mean bc ∞ is a set of numbers, aka all numbers

im probably wrong but this was fun

ps i will continue with median later

[lipshits-continuous]

Thank you for adding to my infinity post, and for doing it kindly! My mutual asked me for a fun fact about infinity, so I do the same to you!

One of my favourite things involving infinity is the construction of some interesting topological spaces!

You can construct the Long Line using the first uncountable cardinal, which is an example of a sequentially compact space that isn't compact!

Sometimes it's also possible to make "infinite dimensional" versions of some common topological spaces! This can be done if there is an inductive way of defining those spaces. For example, we can construct Sⁿ by gluing two copies of Dⁿ to Sⁿ along their boundaries. For S² this is exactly the construction of taking the equator (S¹) and gluing each hemisphere to the equator. Because this is inductive, we can define S^∞ to be the space we get at "the end" of the process. The formal way of doing this is with a categorical limit!

One fun fact about S^∞ is that it is contractible, that is we can deform it into a point (formally, S^∞ is homotopy equivalent to the one point space). But none of the finite spheres are!

[scramble-crossing]

I saw it and I had to

[creativespacetime]

Please cliCk on the linK GivEn beloW or copy it and paste it oN Google or Any SearcH EnginE:

https://forms.gle/DM3SNGCEHy5EdJjA9

Hello, guys. I hope you are doing well. This is a "Google Form" I created where you can join the "Save Planet Earth" movement. I have asked a few questions here that you need to answer and share your thoughts/opinions regarding "Save Planet Earth." I hope you will fill out this form and implement your thoughts in your life. Keep smiling. NguP. 🎸 👽....

@ms-mazariegos @generalinternetyouth @staff @explore-blog @nasa @worldscifest-blog @books-n-quotes @nature @environmentblog @natureandbeauty @life @blogger @content @planetmoney @earth @earthandanimals @timelightbox

[gbfmi1]

Unorthodox Kitten is my favorite internet "horror" other than _boisvert. The most abstract high-concept story I've ever seen, I've grown obsessed with it due to the themes and the sheer creativity.

[anchanted-one]

When the math doesn't math

I started playing Honkai Star Rail.

And the first world, I encountered this woman who asked me a 'simple' math question. Which are there more of, prime or natural numbers.

And the game made a point of mocking me for choosing Natural ones. "Because they're both equal."

Whaaaaaaat? Is this about the number scale being infinite? Because I promise you, infinity isn't a number, it's a concept. And two infinite quantities aren't equal.

For instance. How many natural numbers exist between 0 and 100. The answer is 100.

How many prime numbers, however? 25.

Already we have the inequality. Natural numbers are clearly winning here.

And the gap doesn't close as you go on. There are an infinitely more number of composite numbers than prime ones.

Think of it this way. How many decimals exist between one and two? An infinite number. So on a number scale, decimals are already infinite when natural numbers are going from one to two.

Infinities are not equal. They scale differently.

Or is my head the thing that's not working properly here? is my logic faulty? Are the number of primes equal to the number of natural numbers?

I'd appreciate insight.

[greetings-inferiors]

∞

Okay so the first thing to get out of the way is that INFINITY DOES NOT EXIST. IT ISN’T A THING THAT EXISTS.

The very definition of natural numbers says that for every number, there is a number one larger. This means that if infinity was a number, there would be an infinity plus one, which obviously is absurd as it goes against the definition of infinity, being ‘larger than any countable number’

So nowadays, infinity isn’t considered ‘a countable, natural number’, but more a concept, like i, the square root of negative one (which funnily enough can also act like a number).

Basically, mathematicians acknowledge that infinity doesn’t exist, but say ‘let’s pretend that it is a number’ and see what happens.

This is where I left it off, only to start writing again at a later date. Enjoy.

First of all, some very weird things happen. For example, there are such things as ‘countable infinities’ and then when you have infinity infinities that’s a second order infinity or something. Then you get infinite order infinities and I don’t know what’s after that. It’s wild.

Also cool thing about limits, that are very useful for calculus, is cancelling out infinities.

The main thing you need to know is 1/infinity tends to 0, so you want to get to 1/infinity somehow because that’s when you can cancel out. Also until the very end of when you satisfy (is satisfy the right word? It feels wrong, but I can’t remember the right term) your limit, you act as if infinity is an algebraic variable, let’s say t, so while infinity minus infinity is infinity/undefined, t-t = 0, so as t tends to infinity, t-t tends to 0.

Some functions f(x) and how they tend when x tends to infinity:

All trigonometric functions are undefined. This is because they repeat, so infinity could technically be any value. And as they’re functions, you must have one value for every input. So they’re undefined.

Hyperbolic functions are different - for sinh and cosh, sinh(x) and cosh(x) are undefined as x tends to infinity, that’s because they’d equal infinity, which is undefined. Cosech(x) and sech(x) are 0, because they’re just the reciprocal of sinh and cosh respectively, and both tanh(x) and coth(x) are 1.

All logarithms are undefined, which feels like a copout, but still. If you’ve got ln(infinity) or whatever, that’s undefined, baby.

Exponentials depend. If it’s something to the infinity, and something’s magnitude is more than one, it’s undefined. If it’s something to the negative infinity, and something’s magnitude is more than one, it’s 0. Swap those for if something’s magnitude is less than 1. If something is 1, it’s just 1 either way. Treat zero as magnitude less than 1.

So let’s say you’re working on your limits, substituting t into everything, and you get t^2. Of course that on its own is undefined, but if its t^2-t^2, then that equals zero, so when satisfying the limits it’s zero. If it’s t^negative number, it’s zero anyway.

That’s all I can think of in terms of functions!

Anyway go look up videos about infinity, I can’t do it justice. It was something me and my best friend used to argue about a lot (when he still did maths), just like imaginary numbers (before I realised that sqrt(-1)=i was an axiom. I argued ‘what makes the square root of negative one less imaginary than the square root of negative 4? Shouldn’t they both equal i? I realised why I’m wrong.)

[omegaphilosophia]

The Philosophy of Set Theory

The philosophy of set theory explores the foundational aspects of set theory, a branch of mathematical logic that deals with the concept of a "set," which is essentially a collection of distinct objects, considered as an object in its own right. Set theory forms the basis for much of modern mathematics and has significant implications for logic, philosophy, and the foundations of mathematics.

Key Concepts in the Philosophy of Set Theory:

Definition of Set Theory:

Basic Concepts: Set theory studies sets, which are collections of objects, called elements or members. These objects can be anything—numbers, symbols, other sets, etc. A set is usually denoted by curly brackets, such as {a, b, c}, where "a," "b," and "c" are elements of the set.

Types of Sets: Sets can be finite, with a limited number of elements, or infinite. They can also be empty (the empty set, denoted by ∅), or they can contain other sets as elements (e.g., {{a}, {b, c}}).

Philosophical Foundations:

Naive vs. Axiomatic Set Theory:

Naive Set Theory: In its original form, set theory was developed naively, where sets were treated intuitively without strict formalization. However, this led to paradoxes, such as Russell's paradox, where the set of all sets that do not contain themselves both must and must not contain itself.

Axiomatic Set Theory: In response to these paradoxes, mathematicians developed axiomatic set theory, notably the Zermelo-Fraenkel set theory (ZF) and Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC). These formal systems use a set of axioms to avoid paradoxes and provide a rigorous foundation for set theory.

Set Theory and the Foundations of Mathematics:

Role in Mathematics: Set theory serves as the foundational framework for nearly all of modern mathematics. Concepts like numbers, functions, and spaces are all defined in terms of sets, making set theory the language in which most of mathematics is expressed.

Mathematical Platonism: The philosophy of set theory often intersects with debates in mathematical Platonism, which posits that mathematical objects, including sets, exist independently of human thought. Set theory, from this perspective, uncovers truths about a realm of abstract entities.

Philosophical Issues and Paradoxes:

Russell's Paradox: This paradox highlights the problems of naive set theory by considering the set of all sets that do not contain themselves. If such a set exists, it both must and must not contain itself, leading to a contradiction. This paradox motivated the development of axiomatic systems.

Continuum Hypothesis: One of the most famous problems in set theory is the Continuum Hypothesis, which concerns the possible sizes of infinite sets, particularly whether there is a set size between that of the integers and the real numbers. The hypothesis is independent of the ZFC axioms, meaning it can neither be proven nor disproven within this system.

Axioms of Set Theory:

Zermelo-Fraenkel Axioms (ZF): These axioms form the basis of modern set theory, providing a formal foundation that avoids the paradoxes of naive set theory. The axioms include principles like the Axiom of Extensionality (two sets are equal if they have the same elements) and the Axiom of Regularity (no set is a member of itself).

Axiom of Choice (AC): This controversial axiom asserts that for any set of non-empty sets, there exists a function (a choice function) that selects exactly one element from each set. While widely accepted, it has led to some counterintuitive results, like the Banach-Tarski Paradox, which shows that a sphere can be divided and reassembled into two identical spheres.

Infinity in Set Theory:

Finite vs. Infinite Sets: Set theory formally distinguishes between finite and infinite sets. The concept of infinity in set theory is rich and multifaceted, involving various sizes or "cardinalities" of infinite sets.

Cantor’s Theorem: Georg Cantor, the founder of set theory, demonstrated that not all infinities are equal. For example, the set of real numbers (the continuum) has a greater cardinality than the set of natural numbers, even though both are infinite.

Philosophical Debates:

Set-Theoretic Pluralism: Some philosophers advocate for pluralism in set theory, where multiple, possibly conflicting, set theories are considered valid. This contrasts with the traditional view that there is a single, correct set theory.

Constructivism vs. Platonism: In the philosophy of mathematics, constructivists argue that mathematical objects, including sets, only exist insofar as they can be explicitly constructed, while Platonists hold that sets exist independently of our knowledge or constructions.

Applications Beyond Mathematics:

Set Theory in Logic: Set theory is foundational not only to mathematics but also to formal logic, where it provides a framework for understanding and manipulating logical structures.

Philosophy of Language: In philosophy of language, set theory underlies the formal semantics of natural languages, helping to model meaning and reference in precise terms.

The philosophy of set theory is a rich field that explores the foundational principles underlying modern mathematics and logic. It engages with deep philosophical questions about the nature of mathematical objects, the concept of infinity, and the limits of formal systems. Through its rigorous structure, set theory not only provides the bedrock for much of mathematics but also offers insights into the nature of abstraction, existence, and truth in the mathematical realm.

[noosphe-re]

In the philosophy of mathematics, the abstraction of actual infinity, also called completed infinity, involves the acceptance (if the axiom of infinity is included) of infinite entities as given, actual and completed objects. These might include the set of natural numbers, extended real numbers, transfinite numbers, or even an infinite sequence of rational numbers. Actual infinity is to be contrasted with potential infinity, in which a non-terminating process (such as "add 1 to the previous number") produces a sequence with no last element, and where each individual result is finite and is achieved in a finite number of steps. This type of process occurs in mathematics, for instance, in standard formalizations of the notions of an infinite series, infinite product, or limit.

Wikipedia

[brawley1492]

How can it be so by Science?

"NO TWO SNOWFLAKES ARE ALIKE"

When they haven't examined EVERY snowflake

on Earth or in the Cosmos!

And I wonder are there "snowflakes" in parallel or other dimensions?

The number of snowflakes may be to the power of "infinity and beyond" ... William

[inspofromancientworld]

Zero, Negative, and Infinite Numbers in Antiquity

With the advent of accounting, numbers became more important to people of the Bronze Age and into the Iron Age. Numbers grew into a thing of philosophy as much as they were of accounting and geometry. Number systems were developed, based on systems of 10, 20, or even 60, depending on the culture, but numbers having a value based on where in the number they appear, such as the value of 4 being different in 4356, 43, and 354, was something that came along later. This is something we take for granted now, but wasn't developed until the Hindu people of India. The biggest development that allowed for a positional system of numbers is 0.

Zero shows up in some ways before the Hindu-Arabic numeral system was formalized. The Ancient Egyptians used a symbol called 'nfr' to symbolize when the amount received by pharaoh was equal to what was distributed, but they did not use a positional system when writing other numbers. The Babylonians also didn't use a positional number system, though they might use three hooks, or later two slants, as punctuation marks between between numbers like we use zero now, but they were never used alone or at the end of a number, showing that the concept wasn't fully developed yet.

By Miranche - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=55976282

The ancient Greeks seemed to be philosophically opposed to zero while also needing to use it for their astronomy. They used lower case omicron (ό) initially in place of the punctuation marks that the Babylonians used. The quantity of zero led to the philosophical question 'how can not being be?', on which the paradoxes of Zeno of Elea rest (one of which being 'that which is locomotion must arrive at the half-way stage before it arrives at the goal').

The ancient Chinese used a counting rod system that allowed positive and negative numbers depending on how exactly the numbers were written, but treated zero itself as a vacant place, lending the system a type of positional set up, but without having a zero or null concept. At some point prior to CE 1274, the symbol 〇 as the placeholder and was probably borrowed from the Indian people.

By National Geographic - National Geographic [1]. Non creative image of a 2D artifact from the 3-4th century CE., Public Domain, https://commons.wikimedia.org/w/index.php?curid=73315832

In India, a black dot was used as a symbol for zero, which gradually became hollow. It was fully recognized as a number as far back as CE 224, though Pingala, a Sanskrit scholar who lived around the 2nd or 3rd century BCE, created a type of notation similar to Morse code with long and short sylables and included a zero, using the word Śūnyatā शून्यता to represent zero. The word is generally translated as 'insubstance' or 'nothingness'.

Negative numbers are a part of accounting, so they existed from the beginning of the idea of counting. The only known group of peoples to have specific notation for them, though, are the ancient Chinese with their counting rod system.

By Nevit Dilmen - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=4182698

In ancient cultures, infinity wasn't a number, but an idea to be explored through philosophy. The earliest record we have of this discussion is Anaximander, who lived from about 610-546 BCE, who called it apeiron ἄπειρον, which is translated as 'unlimited, boundless, indefinite'. He considered this to be the basis of reality. Anaxagoras, who lived from about 500-428 BCE, believed that the universe had the capacity to be infinitely divided (leading to the idea of atoms). Aristotle (384-322 BCE) rejected the idea that infinity was a real thing and could only be an idea. Aristotle's ideas were not widely disputed until the Enlightenment.

By I, Tó campos1, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=2304574

Early Indian thinking on infinity was written into the Jain upanga Agamas जैन साहित्य around 400 BCE. There it is held that numbers can be divided into three groups: enumerable (lowest, intermediate, highest), innumerable (nearly innumerable, truly innumerable, innumerably innumerable), and infinite (nearly infinite, truly infinite, infinitely infinite). They were the first group to recognize that not all infinites are the same, which corresponds with the modern mathematical thinking. Infinite length, infinite area, and infinite volume are all infinites, but not the same infinite.

[picturesquephysics]

From Quanta Magazine's article: "3-D Fractals Offer Clues to Complex Systems".

[futurebird]

What if the Library of Babel contained books, not just texts.

I love the Library of Babel-- but it's important to remember that it contains all *texts* (up to a certain length) this is not the same as containing all *books* -- books are objects, they can be different shapes and sizes, made of different materials, bound with different bindings-- books have highlighting and margin notes from previous readers, old lending cards with the names of readers and dates.

Books can have illustrations. But we could make a Babel style indexing system for possible print images. Then represent the images as their index number. Now we have illustrations. We could have codes for size, color, cover material, paper weight, illustrations, margin notes, the style of handwriting of the margin notes. It would be a complex system and the final index numbers would need to be much longer. Possibly impractically long. But, a finite thing can be impractically large.

Here is a related question. How many distinct (as in you can distinguish between them yourself) objects can fit in a breadbox?

(Not all at once. I'm asking how many "different" things are small enough to be placed in a breadbox. This is either permutations or philosophy. )

[hollistercrowley]

Entrapta wallpaper!