Wednesday, September 4, 2024

# Is Math Created Or Discovered

## Did Humans Invent Mathematics Or Is It A Fundamental Part Of Existence

Is mathematics discovered or invented?

Many people think that mathematics is a human invention. To this way of thinking, mathematics is like a language: it may describe real things in the world, but it doesn’t ‘exist’ outside the minds of the people who use it.

But the Pythagorean school of thought in ancient Greece held a different view. Its proponents believed reality is fundamentally mathematical.

More than 2,000 years later, philosophers and physicists are starting to take this idea seriously.

As I argue in a new paper, mathematics is an essential component of nature that gives structure to the physical world.

## What Did Plato Think

But if we are discovering something, what is it?

The ancient Greek philosopher Plato had an answer. He thought mathematics describes objects that really exist.

For Plato, these objects included numbers and geometric shapes. Today, we might add more complicated mathematical objects such as groups, categories, functions, fields, and rings to the list.

Plato also maintained that mathematical objects exist outside of space and time. But such a view only deepens the mystery of how mathematics explains anything.

Explanation involves showing how one thing in the world depends on another. If mathematical objects exist in a realm apart from the world we live in, they don’t seem capable of relating to anything physical.

## Math Does Not Have Natural Patterns That Is Discovered

The symbolism and mathematical natural patterns are patterns we see and have a consensus as humans that it’s correct. But is there really a mathematical pattern? Because we say it’s right and accurate and correct does mean in the universe that it actually is because in the world nothing is natural and is just because it is. The idea that mathematics was discovered is so false because humans created symbols relating items to mathematical properties, for example, 1+1=2 but if we apply real experiences in life, for example, say you are looking up at clouds another cloud nd you see 1 cloud and then another cloud adds to the first cloud. Still, there is only one cloud, not two separate clouds 1 cloud + 1 cloud = 1 cloud. So simply there is not mathematical natural pattern we discovered its just simply invented

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## Who Is Known As Queen Of Mathematics

Carl Friedrich Gauss one of the greatest mathematicians, is said to have claimed: Mathematics is the queen of the sciences and number theory is the queen of mathematics. The properties of primes play a crucial part in number theory. An intriguing question is how they are distributed among the other integers.

## Who Exactly Invented Math

Mathematics permeates every single area of our modern lives, but who do we have to thank for this important field?

Mathematics is at the center of our modern world, whether we’d like to admit it or not. Behind our smartphones, our cars, our computers, even the weather, math is quietly working to calculate the past, present, and future. Math is a scientific principle that seems to predate even science itself.

When you stop and think about it though, who was the first person to use math? After all, we know famous inventors of specific equations, but what about for math as a concept? This doesn’t seem like too far off of a proposition either given that modern realms of science have founders, like Max Planck, the father of quantum mechanics or Isaac Newton and calculus. So, who invented mathematics?

## When Did Math Invented

Beginning in the 6th century BC with the Pythagoreans, with Greek mathematics the Ancient Greeks began a systematic study of mathematics as a subject in its own right. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom, theorem, and proof.

## Most Believe Discovered: Arithmetic Heirarchy

There are questions in mathematics which cannot be phrases as the non-halting of a computer program, at least not without modification of the concept of “program”. These include

• The twin prime conjecture
• The transcedence of e+pi.

To check these questions, you need to run through cases, where at each point you have to check where a computer program halts. This means you need to know infinitely many programs halt. For example, to know there are infinitely many twin primes, you need to show that the program that looks for twin primes starting at each found pair will halt on the next found pair. For the transcendence question, you have to run through all polynomials, calculate the roots, and show that eventually they are different from e+pi.

These questions are at the next level of the arithmetic heirarchy. Their computational formulation is again more intuitive— they correspond to the halting problem for a computer which has access to the solution of the ordinary halting problem.

You can go up the arithmetic hierarchy, and the sentences which express the conjectures on the arithmetic hierarchy at any finite level are those of Peano Arithmetic.

There are those who believe that Peano Arithemtic is the proper foundations, and these arithemtically minded people will stop at the end of the arithemtic hierarchy. I suppose one could place Kronecker here:

• Leopold Kronecker: “God created the natural numbers, all else is the work of man.”

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## Definitely Discovered: Finite Stuff

9306781264114085423 x 39204667242145673 = ?

Then if I compute it, am I inventing it’s value, or discovering the value? The meaning of the word “invent” and “discover” are a little unclear, but usually one says discover when there are certain properties: does the value have independent unique qualities that we know ahead of time ? Is it possible to get two different answers and consider both correct? etc.

In this case, everyone would agree the value is discovered, since we actually can do the computation— and not a single person thinks that the answer is made up nonsense, or that it wouldn’t be the number of boxes in the rectangle with appropriate sides, etc.

There are many unsolved problems in this finite category, so it isn’t trivial:

• Is chess won for white, won for black, or a draw, in perfect play?
• What are the longest possible Piraha sentences with no proper names?
• What is the length of the shortest proof in ZF of the Prime Number Theorem? Approximately?
• What is the list of 50 crossing knots?

You can go on forever, as most interesting mathematical problems are interesting in the finite domain too.

## The Early Years Of Math

Is math discovered or invented? – Jeff Dekofsky

For the early years of math, cultures existed largely siloed into their own communities and geographical areas. This meant that each region developed its own means of doing math that slowly evolved to reflect the core principles of the mathematical laws of nature.

Each roughly 6000 years ago can be traced through a lineage of discovering addition, multiplication, and division.

Mesopotamian and Egyptian societies likely made the largest advancements in early mathematics simply due to their age of existence and their overall size and resources.

## Who Really Invented Math

Unlike a light bulb or a computer, mathematics isnt really an invention. Its really more of a discovery. Mathematics encompasses many different types of studies, so its discovery cant even be attributed to one person. Instead, mathematics developed slowly over thousands of years with the help of thousands of people!

## Inspiration Pure And Applied Mathematics And Aesthetics

Mathematics arises from many different kinds of problems. At first these were found in commerce, land measurement, architecture and later astronomy today, all sciences pose problems studied by mathematicians, and many problems arise within mathematics itself. For example, the physicistRichard Feynman invented the path integral formulation of quantum mechanics using a combination of mathematical reasoning and physical insight, and today’s string theory, a still-developing scientific theory which attempts to unify the four fundamental forces of nature, continues to inspire new mathematics.

Some mathematics is relevant only in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. A distinction is often made between pure mathematics and applied mathematics. However pure mathematics topics often turn out to have applications, e.g. number theory in cryptography.

As in most areas of study, the explosion of knowledge in the scientific age has led to specialization: there are now hundreds of specialized areas in mathematics and the latest Mathematics Subject Classification runs to 46 pages. Several areas of applied mathematics have merged with related traditions outside of mathematics and become disciplines in their own right, including statistics, operations research, and computer science.

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## Golden Ratio And Fibonacci Sequence

The golden ratio describes the most predictable patterns in the universe. It describes everything from atoms, the shapes of a hurricane, the face and the human body, to the dimensions of the galaxy. The golden ratio is when the ratio of parts and is equal to divided by the larger part . It has a value of about 1.618 and is depicted by the greek alphabet phi, . It is also known as the divine proportion.

The formula for the golden ratio

The golden ratio was derived from the Fibonacci sequence, named after the Italian mathematician Leonardo Fibonacci. For hundreds of years, the Fibonacci sequence has fascinated many mathematicians, scientists and artists. Each number in the sequence is the sum of the two numbers that precede it. So, the sequence will be: 0,1,1,2,3,5,8,13,21,34,55,.. and so on.

The Fibonacci sequence can be seen in various items around us, including seashells, animals, pyramids and other unexpected places.

Golden ratio in nature

Flower petals also follow the Fibonacci sequence. If you observe, the number of petals in a flower will be either one of the following: 3, 5, 8, 13, 21, 34 or 55. For example, a lily has 3 petals, cosmos has 8 petals, corn marigold has 13 petals, chicory and daisy have 21 petals and Michaelmas daisies have 55 petals. This supports the argument that mathematical functions existed in nature, and all we did was discover them!

Fibonacci sequence on an ox-eye sunflower

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## Definitely Invented: Set Of All Sets

This level is the highest of all, in the traditional ordering, and this is where people started at the end of the 19th century. The intuitive set

• The set of all sets
• The ordinal limit of all ordinals

These ideas were shown to be inconsistent by Cantor, using a simple argument . The paradoxes were popularized and sharpened by Russell, then resolved by Whitehead and Russell, Hilbert, Godel, and Zermelo, using axiomatic approaches that denied this object.

Everyone agrees this stuff is invented.

This is only a partial answer:

As a mathematician, I have been asked this sort of question from time to time. Like most other mathematicians, I tend to sort of evade the question, because it’s tricky. Usually, the question is put in the form, “Are you a platonist?”

The reference here is to Plato’s eternal form that we are able to recognize, and that allows us to recognize the world around us . When forced to continue, I usually respond “No.”

I think the fundamental problem with Platonism is summed up in Brian Davies’s paper, aptly titled “Let Platonism Die.” I also add – if a mathematical ‘discovery’ hasn’t yet been discovered, does it exist? A Platonist would say absolutely. An intuitionist would either say that it does not exist, or it exists only in the sense that some current or future mathematical system, devised and formulated vulgarly by humans, will lead to many more theorems – i.e. it exists only as an extension of what we have already created.

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## Was Math Invented Or Discovered Debate

This is true for all right-angled triangles on a level surface, so it’s a discovery. Showing it is true, however, requires the invention of a proof. And over the centuries, mathematicians have devised hundreds of different techniques capable of proving the theorem. In short, maths is both invented and discovered.

## Fourier Transform And Heisenberg Uncertainty Principle

The constant also appears as a critical spectral parameter in the Fourier transform. This is the integral transform, that takes a complex-valued integrable function f on the real line to the function defined as:

f . }=\int _^fe^\,dx.}

Although there are several different conventions for the Fourier transform and its inverse, any such convention must involve somewhere. The above is the most canonical definition, however, giving the unique unitary operator on L2 that is also an algebra homomorphism of L1 to L.

The Heisenberg uncertainty principle also contains the number . The uncertainty principle gives a sharp lower bound on the extent to which it is possible to localize a function both in space and in frequency: with our conventions for the Fourier transform,

|^\,dx\right)\left|^\,d\xi \right)\geq \left|^\,dx\right)^.}

The physical consequence, about the uncertainty in simultaneous position and momentum observations of a quantum mechanical system, is discussed below. The appearance of in the formulae of Fourier analysis is ultimately a consequence of the Stonevon Neumann theorem, asserting the uniqueness of the Schrödinger representation of the Heisenberg group.

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Their truth values are based on rules that humans created. Mathematics is thus an invented logic exercise, with no existence outside mankind’s conscious thought, a language of abstract relationships based on patterns discerned by brains, built to use those patterns to invent useful but artificial order from chaos.

## Was Maths Invented Or Discovered

Roger Penrose – Is Mathematics Invented or Discovered?

Asked by: Leah Victoria Smith, Hereford

The fact that 1 plus 1 equals 2, or that theres an infinite number of primes, are truths about reality that held even before mathematicians knew about them. As such, theyre discoveries but they were made using techniques invented by mathematicians. For example, according to Pythagoras theorem, the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides. This is true for all right-angled triangles on a level surface, so its a discovery.

Showing it is true, however, requires the invention of a proof. And over the centuries, mathematicians have devised hundreds of different techniques capable of proving the theorem. In short, maths is both invented and discovered.

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## Number Theory And Riemann Zeta Function

The Riemann zeta function is used in many areas of mathematics. When evaluated at s = 2 it can be written as

+ }}+}}+}}+\cdots }

Finding a simple solution for this infinite series was a famous problem in mathematics called the Basel problem. Leonhard Euler solved it in 1735 when he showed it was equal to 2/6. Euler’s result leads to the number theory result that the probability of two random numbers being relatively prime is equal to 6/2. This probability is based on the observation that the probability that any number is divisible by a prime p is 1/p Hence the probability that two numbers are both divisible by this prime is 1/p2, and the probability that at least one of them is not is 1 1/p2. For distinct primes, these divisibility events are mutually independent so the probability that two numbers are relatively prime is given by a product over all primes:

% . \prod _^\left& =\left^\\& =}}+}}+\cdots }}\\& =}=}}\approx 61\%.\end}}

This probability can be used in conjunction with a random number generator to approximate using a Monte Carlo approach.

The zeta function also satisfies Riemann’s functional equation, which involves as well as the gamma function:

. }.}

## How Old Is Mathematics

The tale of mathematics is as old as humanity. It has evolved from simple math, like counting cattle, to an intricate study of an object through abstract concepts that we know today. It was not until 600 BC, when civilizations settled and various occupations began, that math began its initial development. It was used to measure plots, calculate the taxation of individuals, etc. Later, in 500 BC, we saw the development of Roman numerals, which are still used to represent numbers.

Scientists believe that thousands of years ago, basic mathematical functions like addition and subtraction might have appeared at the same time, but in different places, like India, Egypt and Mesopotamia. Advanced math dates back to Greece over 2500 years ago, when mathematician Pythagoras proffered his famous equation. It was about the sides of a right-angle triangle, which we now study as the Pythagorean theorem.

Since then, more mathematicians started working on expanding their understanding of mathematics. Yet, no one could find the one true answer to the big question.

Pythagoras Theorem

## Modular Forms And Theta Functions

The constant is connected in a deep way with the theory of modular forms and theta functions. For example, the Chudnovsky algorithm involves in an essential way the j-invariant of an elliptic curve.

^+1}}\,dx=\pi .}

The Shannon entropy of the Cauchy distribution is equal to ln, which also involves .

The Cauchy distribution plays an important role in potential theory because it is the simplest Furstenberg measure, the classical Poisson kernel associated with a Brownian motion in a half-plane.Conjugate harmonic functions and so also the Hilbert transform are associated with the asymptotics of the Poisson kernel. The Hilbert transform H is the integral transform given by the Cauchy principal value of the singular integral

H . }\int _^}.}

The constant is the unique normalizing factor such that H defines a linear complex structure on the Hilbert space of square-integrable real-valued functions on the real line. The Hilbert transform, like the Fourier transform, can be characterized purely in terms of its transformation properties on the Hilbert space L2: up to a normalization factor, it is the unique bounded linear operator that commutes with positive dilations and anti-commutes with all reflections of the real line. The constant is the unique normalizing factor that makes this transformation unitary.