integralhistory

[integralhistory]

On Today’s episode

https://soundcloud.com/user-887302873/fermats-last-theorem-final

[integralhistory]

Fermat’s Last Theorem

Pierre de Fermat: [1]

Life:

French Mathematician, 1601 - 1665

Founder of the modern theory of numbers

Received his baccalaureate in law from the university of orleans

By trade he was a councillor in the parliament in Toulouse (essentially a lawyer)

Jump to Mathematics

In the custom of his day, he began to reconstruct ancient mathematical works, the most inspiring to his work was a reconstruction of the 3rd century greek work “Plane Loci of Apollonius”, loci here meaning “points”

Through his study of loci (points), he discovered that you can use algebra to describe geometry through using a coordinate system.

Mathematical Contributions

Study of curves:

Generalized the form for a  parabola: ay = x^2

Generalized the form for a hyperbola: xy = a^2 to a^n - 1y = x^n

Generalized the form of the archimedean spiral: r = aθ

The study of these curves lead him to an algorithm that is equivalent to differentiation (derivatives), and to integration, though it is not known if he discovered that these operations are the inverse of one another

Fermat’s only published work in his lifetime was: “De Linearum Curvarum cum Lineis Rectis Comparatione” (“Concerning the Comparison of Curved Lines with Straight Lines”), he proved that certain algebraic curves were rectifiable (able to find the length of a curve), disproving the long lasting dogma from Aristotle, that curves are not precisely rectifiable.

Fermat’s Last Theorem:

x^n + y^n = z^n, where x, y, z, and n are positive integers, has no solution if n is greater than 2.

In the margin of the notebook where he stated this theorem he wrote “. I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain”. No proof was ever found posthumously.

Progress towards a proof:[2]

First Case:

exponents which are relatively prime to , , and

Sophie Germain (French Physicist, Mathematician, and philosopher) proved this case for any prime P, where 2P +1 is also a prime

Adrien-Marie Legendre (French Mathematicia), proved this case for any prime P such that  , , , , or are also primes. Through these discoveries, Fermat’s last theorem was proven for n < 100

In 1849, Ernst Kummer (German Mathematician), proved the first case for all regular primes and composite numbers for which regular primes are factors

Second case:

The exponent divides exactly one of Coeffiecients X, Y, and Z

1852, Vandiver’s criteria: a criteria for determining if an irregular prime P is the exponent in fermat’s last theorem if the theorem holds

1700s.Leonhard Euler (prolific swiss mathematician), proved general case for n = 3

1600s, Fermat proved the general case for n = 4

1800s Peter Dirichlet(German Mathematician) and Joseph-Louis Lagrange (italian mathematician), proved the general case for n= 5

1800s Dirichlet also proved the general case for n = 14

Late 1800s the n=7  case was proven by Gabriel Lamé (french mathematician)

Hiccups:

1937, Ferdinand Lindemann published several invalid “proofs” of fermat’s last theorem

1988, Yoichi Miyaoka published another false proof of fermat’s last theorem

In 1987, A prize of 100000 German marks, known as the Wolfskehl Prize, was also offered for the first valid proof

Further Exploration:

1909, Arthur Wieferich(German mathematician), proved that if the equation in fermat’s last theorem is solved in integers with an odd prime P, then  then 2^(p-1) is equivalent to 1(mod p^2), these numbers are called Wieferich Primes

1909, Dmitry Semionovitch Mirimanoff(Russian Mathematician), similarly proved 3^(p-1) is equivalent to 1 (Mod p^2)

Equations of the form above also hold with Coefficients 5,7,11,13,and 17.

1941, All of these number theory discoveries led the smallest possible prime that may be a counter example for fermat’s last theorem to be:2532 253 747 889, in the first case of the theorem

1988,Granville and Monagan showed if there exists a prime p satisfying Fermat's Last Theorem, then q^(p-1) is equivalent to 1 (mod p^2), bringing the smallest prime counterexample up to 714 591 416 091 398

Andrew Wiles[5]

English Mathematician Born in 1953

At the age of ten, Wiles picked up a book on fermat’s last theorem, and was amazed that such a simple sounding theorem still remained an open question, and set out to solve it

He eventually realized through his teens that proof would not be straightforward, and put off the task for two decades, to acquire more mathematical skill

While set up as a research fellow at oxford, and then princeton, between the years of 1986 and 1993, he took up the task of constructing the proof

Taniyama-Shimura conjecture:[6]

A general theorem connecting topology ( the study of geometry without regard to area, concerned with stretching, twisting, and deformation of objects (not tearing)), and number theory

the conjecture says that every rational elliptic curve is a modular form

Semistable case:[7]

An elliptic curve is semistable if, for all such primes l, only two roots become congruent mod l

Proof of the theorem:

In 1993, Andrew Wiles came out of mathematical hiding and revealed what he had spent the past 7 years to find, by proving the semistable case of the Taniyama Shimura Conjecture, he essentially got a proof of Fermat’s Last theorem for free

Hiccups:

Several holes were found in Wiles’ initial proof,but were quickly resolved with the help of Former research student Richard Taylor in 1994, and re-published in 1995

Legacy of the theorem [2]

Fermat’s last theorem was an open question for 350 years.

Fermat’s last theorem has even entered pop culture:

In the episode “Homer^3” of the simpsons, the formula: appears on the blackboard in the background. This is a near-miss solution, only correct to the first nine decimal places

The Wizard of Evergreen Terrace mentions , which is only correct to the first ten decimal points

At the start of Star Trek: The Next Generation episode "The Royale," Captain Picard mentions that studying Fermat's Last Theorem is a relaxing process.

Fermat probably didn’t have a proof of the theorem, as the techniques used are far beyond what he had access to at the time

Works Cited:

Contributor:Carl B. Boyer, Article Title:Pierre de Fermat, Website Name:Encyclopædia Britannica,Publisher:Encyclopædia Britannica, inc.,Date Published:February 22, 2017,URL:https://www.britannica.com/biography/Pierre-de-Fermat.Access Date:July 08, 2017

Weisstein, Eric W. "Fermat's Last Theorem." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/FermatsLastTheorem.html

Weisstein, Eric W. "Regular Prime." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/RegularPrime.html

Weisstein, Eric W. "Bernoulli Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/BernoulliNumber.html

., .. .. "Andrew Wiles." Famous Mathematicians. Famous Mathematicians, 26 Sept. 2016. Web. 08 July 2017.

Weisstein, Eric W. "Taniyama-Shimura Conjecture." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Taniyama-ShimuraConjecture.html

Weisstein, Eric W. "Semistable." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Semistable.html

[integralhistory]

Kurt Godel Episode Copy

Episode Link:

https://soundcloud.com/user-887302873/kurt-godel

State of mathematics before the proof of the incompleteness theorem:

Principia Mathematica

Authors: Bertrand Russel, and Alfred Whitehead, both respected English Mathematicians

Alfred Whitehead

Established Cambridge academic, published “A Treatise on Universal Algebra”, which used boolean algebra, matricies, and quaternions (an extension of the complex numbers), to investigate topics of algebra and geometry [1]

Bertrand Russel

Discovered a very basic paradox in naive set theory, the set of all sets that do not contain themselves. “Russel’s Paradox” [1]

Resolved to fix this issue, and issues like them through the use of types. I.e. sets that contain other sets, and sets that contain normal elements, are of differing types [1]

A ten year labour of both authors.

build a solid foundation of mathematical logic based on a few axioms, and from those axioms, build out

As an example of rigour, it took 80 pages to prove that 1+1=2 [1]

The notation of principia mathematica was not describing numbers, but logic itself, as a result, whole pages go on while only using actual english words a handful of times

Incompleteness theorem:

Gödel's incompleteness theorem states that all consistent axiomatic formulations of number theory include undecidable propositions [2]

Any system which uses axioms that are consistent has statements that are unprovable

Requirements: [3]

A constant symbol 0 for zero.

A unary function symbol S for the successor operation (recursive function)

A symbol + for addition

A symbol x for multiplication

Logical conjunction ^

Logical disjunction ∨

Logical Negation ¬

Universal Qualifier ∀

Existential qualifier ∃

Binary equals =

Binary order <

Precedence left (

Precedence right )

Variable symbol x

Distinguishing symbol * (for use with variable)

Godel Numbering [3]:

Assign a distinct integer value to each of the required symbols above

Caveats:

The theory must be able to prove that for every number, there exists a godel number

given any Gödel number of a formula F(x) with one free variable x and any number m, there is a Gödel number of the formula F(m) obtained by replacing all occurrences of G(x) in G(F(x)) with G(m), and that this second Gödel number can be effectively obtained from the Gödel number of F as a function of m

G1 = f(x), G2 = f(m)

G2 = G(f(x)), subbing out G(x) with G(m)

For any godel number representing a well-formed mathematical starement F, one can recreate the original formula, make the substitution, and then find the Gödel number of the resulting formula. This is a uniform procedure.

Proof Construction[3]:

define q(n, G(F)), a relation between two numbers n and G(F), such that it corresponds to the statement "n is not the Gödel number of a proof of F(G(F))", where F(G(F)) can be understood as F with its own Gödel number as its argument.

decode the number G(F) into the formula F

Any proof of F(G(F)) can be encoded by a Gödel number n, such that q(n, G(F)) does not hold

If q(n, G(F)) holds for all natural numbers n, then there is no proof of F(G(F)). In other words, ∀y q(y, G(F)),corresponds to "there is no proof of F(G(F))".

define the formula P(x) = ∀y q(y, x),

Replace free variable x with the godel number for G(F)

P(G(F)) = ∀y q(y, G(F)) corresponds to "there is no proof of F(G(F))"

Consider the formula P(G(P)) = ∀y, q(y, G(P)). This formula concerning the number G(P) corresponds to "there is no proof of P(G(P))". We have here the self-referential feature that is crucial to the proof: A formula of the formal theory that somehow relates to its own provability within that formal theory. Very informally, P(G(P)) says: "I am not provable".

neither the formula P(G(P)), nor its negation ¬P(G(P)), is provable

Assume P(G(P)) = ∀y, q(y, G(P)) is provable. Let n be the Gödel number of a proof of P(G(P)). Then, as seen earlier, the formula ¬q(n, G(P)) is provable. Proving both ¬q(n, G(P)) and ∀y q(y, G(P)) violates the consistency of the formal theory. We therefore conclude that P(G(P)) is not provable.

Consider any number n. Suppose ¬q(n, G(P)) is provable. Then, n must be the Gödel number of a proof of P(G(P)). But we have just proved that P(G(P)) is not provable.

Layman’s terms

By assigning mathematically sound statements a godel number, we can create statements in tersely defined mathematical theories that act as contradictions.

In 1931 Godel did just that, by creating the mathematical equivalent of the liar’s paradox (this statement in a lie) inside of the walls of principia mathematica

Furthermore, Godel’s incompleteness theorem goes further than principia mathematica, and proves that for any axiomatic systems, there will always be examples of statements that are unprovable

Secondary incompleteness theorem[4]

A side effect of the first incompleteness theorem

A formal system containing arithmetic cannot prove its own consistency

In any system of arithmetic that is useful, there is no way to prove that the system contains no false statements

The only way to prove that a system is inconsistent, is if the system itself is inconsistent. Consistent systems have no way of proving themselves one way or the other[4]

Turn to philosophy[5]

Reasoning behind proving the incompleteness theorem

Mathematical platonism:

The belief that Mathematical sentences provide true descriptions about a collection of objects

These objects are non-physical, non-mental, and exists in a special “mathematical realm”

Kurt Godel hoped to prove that mathematical truths go beyond human created axiomatic systems

Kurt Godel responded to a typical argument to platonism, that “if mathematical truths were beyond human mental capacity, because if that were true, we would never have had any understanding of math in the first place”, by asserting that humans have an innate mathematical sense, mathematical intuition, that allows us to see numbers in the minds eye

These views were not widely accepted (obviously)

Works Cited:

“100 Years Since Principia Mathematica—Stephen Wolfram Blog.” 2017. Accessed June 27. http://blog.stephenwolfram.com/2010/11/100-years-since-principia-mathematica/.

Weisstein, Eric W. 2017. “Gödel’s Incompleteness Theorem -- from Wolfram MathWorld.” Wolfram Research, Inc. Accessed June 27. http://mathworld.wolfram.com/GoedelsIncompletenessTheorem.html.

Hofstadter, Douglas R. (1999) [1979], Gödel, Escher, Bach: An Eternal Golden Braid, Basic Books, ISBN 0-465-02656-7, retrieved 2016-03-02

“Incompleteness Theorem | Logic.” Encyclopedia Britannica. N.p., n.d. Web. 27 June 2017.---. Encyclopedia Britannica. N.p., n.d. Web. 27 June 2017.

Balaguer, Mark. "Kurt Godel." Encyclopædia Britannica. Encyclopædia Britannica, Inc., 24 Jan. 2017. Web. 27 June 2017.

[integralhistory]

Georg Cantor Episode Copy

Episode Link: 

https://soundcloud.com/user-887302873/episode-1-georg-cantor

Who was Georg Cantor:

German Mathematician (March 3, 1845 - January 6, 1918 )

Studied at the university of Zurich, and then the University of Berlin  studying under Karl Theodor Weierstrass, whose specialization of analysis probably had the greatest influence on him; Ernst Eduard Kummer, in higher arithmetic; and Leopold Kronecker in number theory

Pioneer of set theory, one of the most fundamental theories in mathematics

Set theory:

What is a set?

A collection of objects that share a particular property, can be finite or infinite

Used one-to-one correspondence to show that different infinite sets differ in length

Used one to one correspondence to show that the rational numbers, though infinite, are countable because they can be placed in a  one-to-one correspondence

Proved the uncountability of the real numbers through  what is now called “Cantor Diagonalization”

Furthermore, he proved that the set of all algebraic numbers (Real numbers that can be defined in a polynomial with integer coefficients) have a one to one correspondence with all integers, while the set of all transcendental numbers (numbers that cannot be expressed by algebraic means)  do not, and thus are larger in number

Despite Cantor’s pioneering work with set theory, his papers were refused for publication in Crelle’s Journal, by one of its referees, Leopold Kronecker

Transfinite Numbers:

What’s a transfinite number?

A way of denoting the cardinality (size) of an infinite set. Any set that has a one to one correspondence with the integers is given the cardinality Aleph null אo

The cardinality of the real numbers, called the continuum is given the cardinality Aleph c אc

Continuum hypothesis

Georg cantor developed a way of working with transfinite numbers that mirrors ordinary arithmetic, which led him to ask the question, are there any transfinite numbers between aleph null, and aleph c? Which is to say, does there exist any set of numbers with a size larger than that of a set with a one to one correspondence with the integers, yet a smaller size than the set of real numbers. Georg Cantor suspected not, and this open question is referred to as the continuum hypotheses

Cantor’s Fate:

Sadly, Mental Illness affected Georg Cantor, He suffered from depression,first hospitalized in 1884, and then again in 1889. Shortly after Cantor’s second hospitalization, his son died, and Cantor was left with little passion for mathematics. Cantor was hospitalized again in 1903. In 1904 Julius König  presented a paper at the Third International Congress of Mathematicians. His paper attempted to prove that the basic tenets of transfinite set theory were false. Since the paper had been read in front of his daughters and colleagues, Cantor perceived himself as having been publicly humiliated.[28]Although Ernst Zermelo demonstrated less than a day later that König's proof had failed. Cantor Retired in 1913, during WW1, suffering not only from depression, but malnutrition and poverty brought on from the war. He died in 1918 in the sanitarium where he spent the final year of his life

Post Cantor:

The Continuum Hypothesis

In 1900 David Hilbert, defender of Georg Cantor’s work, and founder of Metamathematics presented a list of 23 open questions to the international congress of mathematicians

In 1939 Kurt Godel proved that Zermelo-Fraenkel axioms, if consistent, do not disprove the contiuum hypothesis

In 1963 Paul Cohen showed the other side that, if ZF axioms are consistent, they don’t yield a proof of the coninuum hypothesis

In 1904, the Royal Society awarded Cantor its Sylvester Medal, the highest honor it can confer for work in mathematics  David Hilbert defended it from its critics by declaring:[14][15] “From his paradise that Cantor with us unfolded, we hold our breath in awe; knowing, we shall not be expelled.”