Alternative MethodsConventional Methods


March 29

Mathematicians Bridge the Divide Between Infinity and the Physical World

Julius Newman
Core Spirit member since Dec 24, 2020
Reading time 6 min.

WITH A SURPRISING new proof, two young mathematicians have found a bridge across the finite-infinite divide, helping at the same time to map this strange boundary.

The boundary does not pass between some huge finite number and the next, infinitely large one. Rather, it separates two kinds of mathematical statements: “finitistic” ones, which can be proved without invoking the concept of infinity, and “infinitistic” ones, which rest on the assumption—not evident in nature—that infinite objects exist.

Mapping and understanding this division is “at the heart of mathematical logic,” said Theodore Slaman, a professor of mathematics at the University of California, Berkeley. This endeavor leads directly to questions of mathematical objectivity, the meaning of infinity and the relationship between mathematics and physical reality.

More concretely, the new proof settles a question that has eluded top experts for two decades: the classification of a statement known as “Ramsey’s theorem for pairs,” or RT22. Whereas almost all theorems can be shown to be equivalent to one of a handful of major systems of logic—sets of starting assumptions that may or may not include infinity, and which span the finite-infinite divide—RT22 falls between these lines. “This is an extremely exceptional case,” said Ulrich Kohlenbach, a professor of mathematics at the Technical University of Darmstadt in Germany. “That’s why it’s so interesting.”

In the new proof, Keita Yokoyama, 34, a mathematician at the Japan Advanced Institute of Science and Technology, and Ludovic Patey, 27, a computer scientist from Paris Diderot University, pin down the logical strength of RT22—but not at a level most people expected. The theorem is ostensibly a statement about infinite objects. And yet, Yokoyama and Patey found that it is “finitistically reducible”: It’s equivalent in strength to a system of logic that does not invoke infinity. This result means that the infinite apparatus in RT22 can be wielded to prove new facts in finitistic mathematics, forming a surprising bridge between the finite and the infinite. “The result of Patey and Yokoyama is indeed a breakthrough,” said Andreas Weiermann of Ghent University in Belgium, whose own work on RT22 unlocked one step of the new proof.

Ramsey’s theorem for pairs is thought to be the most complicated statement involving infinity that is known to be finitistically reducible. It invites you to imagine having in hand an infinite set of objects, such as the set of all natural numbers. Each object in the set is paired with all other objects. You then color each pair of objects either red or blue according to some rule. (The rule might be: For any pair of numbers A < B, color the pair blue if B < 2A, and red otherwise.) When this is done, RT22 states that there will exist an infinite monochromatic subset: a set consisting of infinitely many numbers, such that all the pairs they make with all other numbers are the same color. (Yokoyama, working with Slaman, is now generalizing the proof so that it holds for any number of colors.)

The colorable, divisible infinite sets in RT22 are abstractions that have no analogue in the real world. And yet, Yokoyama and Patey’s proof shows that mathematicians are free to use this infinite apparatus to prove statements in finitistic mathematics—including the rules of numbers and arithmetic, which arguably underlie all the math that is required in science—without fear that the resulting theorems rest upon the logically shaky notion of infinity. That’s because all the finitistic consequences of RT22 are “true” with or without infinity; they are guaranteed to be provable in some other, purely finitistic way. RT22’s infinite structures “may make the proof easier to find,” explained Slaman, “but in the end you didn’t need them. You could give a kind of native proof—a proof.”

When Yokoyama set his sights on RT22 as a postdoctoral researcher four years ago, he expected things to turn out differently. “To be honest, I thought actually it’s not finitistically reducible,” he said.

This was partly because earlier work proved that Ramsey’s theorem for triples, or RT23, is not finitistically reducible: When you color trios of objects in an infinite set either red or blue (according to some rule), the infinite, monochrome subset of triples that RT23 says you’ll end up with is too complex an infinity to reduce to finitistic reasoning. That is, compared to the infinity in RT22, the one in RT23 is, so to speak, more hopelessly infinite.

Even as mathematicians, logicians and philosophers continue to parse the subtle implications of Patey and Yokoyama’s result, it is a triumph for the “partial realization of Hilbert’s program,” an approach to infinity championed by the mathematician Stephen Simpson of Vanderbilt University. The program replaces an earlier, unachievable plan of action by the great mathematician David Hilbert, who in 1921 commanded mathematicians to weave infinity completely into the fold of finitistic mathematics. Hilbert saw finitistic reducibility as the only remedy for the skepticism then surrounding the new mathematics of the infinite. As Simpson described that era, “There were questions about whether mathematics was going into a twilight zone.”

The Rise of Infinity

The philosophy of infinity that Aristotle set out in the fourth century B.C. reigned virtually unchallenged until 150 years ago. Aristotle accepted “potential infinity”—the promise of the number line (for example) to continue forever—as a perfectly reasonable concept in mathematics. But he rejected as meaningless the notion of “actual infinity,” in the sense of a complete set consisting of infinitely many elements.

Aristotle’s distinction suited mathematicians’ needs until the 19th century. Before then, “mathematics was essentially computational,” said Jeremy Avigad, a philosopher and mathematician at Carnegie Mellon University. Euclid, for instance, deduced the rules for constructing triangles and bisectors—useful for bridge building—and, much later, astronomers used the tools of “analysis” to calculate the motions of the planets. Actual infinity—impossible to compute by its very nature—was of little use. But the 19th century saw a shift away from calculation toward conceptual understanding. Mathematicians started to invent (or discover) abstractions—above all, infinite sets, pioneered in the 1870s by the German mathematician Georg Cantor. “People were trying to look for ways to go further,” Avigad said. Cantor’s set theory proved to be a powerful new mathematical system. But such abstract methods were controversial. “People were saying, if you’re giving arguments that don’t tell me how to calculate, that’s not math.”

And, troublingly, the assumption that infinite sets exist led Cantor directly to some nonintuitive discoveries. He found that infinite sets come in an infinite cascade of sizes—a tower of infinities with no connection to physical reality. What’s more, set theory yielded proofs of theorems that were hard to swallow, such as the 1924 Banach-Tarski paradox, which says that if you break a sphere into pieces, each composed of an infinitely dense scattering of points, you can put the pieces together in a different way to create two spheres that are the same size as the original. Hilbert and his contemporaries worried: Was infinitistic mathematics consistent? Was it true?

Amid fears that set theory contained an actual contradiction—a proof of 0 = 1, which would invalidate the whole construct—math faced an existential crisis. The question, as Simpson frames it, was, “To what extent is mathematics actually talking about anything real? talking about some abstract world that’s far from the real world around us? Or does mathematics ultimately have its roots in reality?”

Even though they questioned the value and consistency of infinitistic logic, Hilbert and his contemporaries did not wish to give up such abstractions—power tools of mathematical reasoning that in 1928 would enable the British philosopher and mathematician Frank Ramsey to chop up and color infinite sets at will. “No one shall expel us from the paradise which Cantor has created for us,” Hilbert said in a 1925 lecture. He hoped to stay in Cantor’s paradise and obtain proof that it stood on stable logical ground. Hilbert tasked mathematicians with proving that set theory and all of infinitistic mathematics is finitistically reducible, and therefore trustworthy. “We must know; we will know!” he said in a 1930 address in Königsberg—words later etched on his tomb.

However, the Austrian-American mathematician Kurt Gödel showed in 1931 that, in fact, we won’t. In a shocking result, Gödel proved that no system of logical axioms (or starting assumptions) can ever prove its own consistency; to prove that a system of logic is consistent, you always need another axiom outside of the system. This means there is no ultimate set of axioms—no theory of everything—in mathematics. When looking for a set of axioms that yield all true mathematical statements and never contradict themselves, you always need another axiom. Gödel’s theorem meant that Hilbert’s program was doomed: The axioms of finitistic mathematics cannot even prove their own consistency, let alone the consistency of set theory and the mathematics of the infinite.

by Wired.com

Comments

Leave your questions here

To write a comment you must
or
Services
Category filter
Concern filter
Type filter
Sort
 

All categories

Spiritual Healing
$220 USD
I will do spiritual transformation for you

I will do spiritual transformation. Life occurs as a series of cycles in a single human lifetime. Cycles exist in almost a spiral pattern. There is a flow and …

Wisdom Realms
Chinese Astrology
$10 USD
You are not ALONE

Do you think it’s your fault you haven’t found your partner yet? Or that you are the problem because the relationship is not working? Do you feel isolated, set…

Carmen Oana Oglan
NLP
$92 USD
Weightloss Course - Online 12 weeks at your pace

Surely you have noticed that some traditional diets are too restrictive and you can not follow them up to the end, others instead make you lose too much weight…

rossella tocco
Spiritual Healing
$55 USD
1 Hr Intuitive Nutrition Coaching Call

Dive in and discover your biggest areas of struggle and how to tap into the intuition for guidance.
We will talk about your history and your goals for your wel…

Jess Olson
Spiritual Healing
$10 USD
Praying for the Sick and Depressed

Are you sick? Are you really depressed? Do you feel that your life is worthless?
Come book a session with me and see the changes that will occur after that.
Th…

Luck Richards Omorede
5
Chakra Balancing
$10 USD
Soul To Soul Reading Session

Energetic Reading

Complete Chakra System Reading to identify energy blocks.
Techniques are provided to restore balance.

What Is An Energetic Reading?

Debo…

Deborah Lucero
Spiritual Healing
$222 USD
Healing from Trauma at a Quantum Soul Level

Are you burnt out from clinical therapy practices that require years of treatment with few results? Are you ready to release and transmute your trauma once and…

LunaRio Silver Deer
Career Coaching
$499 USD
Reset 1-0n1 Coaching and Mentoring Programme

Are you returning to work after a career break, or are you looking for a new job or career change?

This programme is for you.

I’ve developed the RESET coachi…

Carla Martins
Life Coaching
$399 USD
1-on-1 Coaching

The journey started with our discovery call, and You decided: ” I’m ready! Let’s do this!”.

We will engage in 1-on-1 coaching sessions for 90 minutes. In this…

Carla Martins
Emotional Stress Release
$800 USD
Retreat( India): Connect With Your Inner Wisdom 1st to 4th Sept

Join us for an immersive experience anchored amidst the beauty and tranquillity of Kasar Devi, Almora, India

1st Sept’22 ( Thursday) - 4th Sept 2022 ( Sunday …

Aikyam
Psychic Reading
$80 USD
I will do oracle and tarot card readings

I will do oracle & tarot card reading in any area of your life. You will get answer from the Angelic, Elemental, Faerie or Galactic realms. I will share wi…

Wisdom Realms
Natal Astrology
$15 USD
15 min reading of SOUL structure

[SOUL STRUCTURE]Each soul has own structure,and knowing it means to have unlimited access to inner energy. This readings are telling about your talents and str…

Ksenia Jdanova
Wellness Coaching
$10 USD
Ask a Question?

Supportive reflection of a current question that you would like clarity around. Using the SMART method to seek quidance in stepping onto your path of potential…

Beverly Lynn
Reiki
$150 USD
Reiki Services (distant healing session)

In this hour long distant healing session I will use long distance Reiki, crystals and essential oils to balance your Chakras. Balancing your Chakras allows en…

Erin Kirby
Tarot Reading
$10 USD
Intuitive Tarot Reading

Affordable Tarot reading to offer guidance on any given concern.

Elphie Shoppe
Telepsychology
$10 USD
Talk Therapy Sessions

Hello there: These times are dread. Do you need to talk with a safe professional about a troubling personal challenge? I am a professional listener and will li…

Eulalee Thompson
Tarot Reading
$10 USD
Tarot Reading

I offer tarot readings with a 100% accuracy rate. The tarot is meant to give you advice, to point you in the right direction or to provide you with clarity. Bu…

Zara Kalil
Angel Healing
$10 USD
Mental Health Advice & Tips

I can help you with past issues and give advice and tips on how to better your life. I can see through my third eye and receive messageson what you need to do …

Beings Of Light
Life Coaching
$10 USD
Discovery call

Are you interested in supporting your health with safe and effective holistic medicine? Not sure if you should switch diets, work with a health coach or explor…

Suzana Da Costa
Wellness Coaching
$50 USD
Mental Health and Holistic Lifestyle Coaching

Hi there! I’m Doug and I offer 1 on 1 Holistic Lifestyle and Mental Health Coaching sessions. Navtigating life with mental health issues of any sevarity is a h…

Doug Peterson
Wellness Coaching
$10 USD
Mental Health and Holistic Lifestyle Coaching Consultation

Hi there! I’m Doug and I offer 1 on 1 Holistic Lifestyle and Mental Health Coaching sessions. Navtigating life with mental health issues of any sevarity is a h…

Doug Peterson
Family Psychology
$10 USD
Psychic Reading

Expert in Love Relationship, Tarot Psychic Reading
I can give you more powerful insight of every part of your life. With the help of Crystal ball my Psychic re…

Aamir Bilal
Career Coaching
$195 USD
Holistic Lifestyle & Nutrition Coaching

ARE YOU TIRED? STRESSED OUT? STRUGGLING WITH YOUR WEIGHT? UNDER SLEPT?
LOST YOUR ZEST FOR LIFE?

If this sounds like you, then you don’t need a new diet… you n…

David Giddings
Bach flower therapy
$120 USD
Bach Flower Essence Consultation

Exploration of where you’re feeling stuck or limited in life and recommendations of Bach Flower Essences to help loosen and dissolve old patterns.

Suzana Da Costa