Adonis Diaries

Posts Tagged ‘theorem

 

Math forever? How can you start loving math?

Imagine you’re in a bar, or a club, and you start talking to a woman, and after a while this question comes up: “So, what do you do for work?”  (The most dreaded question I expect in a casual conversation)

And since you think your job is interesting, you say: “I’m a mathematician.” (Laughter)

Then 33.51% of women, in that moment, pretend to get an urgent call and leave.

And 64.69% of women desperately try to change the subject and leave.

Another 0.8%, probably your cousin, your girlfriend and your mom, know that you work in something weird but don’t remember what it is. (Laughter)

And then there’s  1% who remain engaged with the conversation.

And inevitably, during that conversation one of these two phrases come up:

A) “I was terrible at math, but it wasn’t my fault. It’s because the teacher was awful.” Or

B) “But what is math really for?”

I’ll now address Case B.

When someone asks you what math is for, they’re not asking you about applications of mathematical science.

They’re asking you, why did I have to study that bullshit I never used in my life again

That’s what they’re actually asking. So when mathematicians are asked what math is for, they tend to fall into two groups:

1. 54.51% of mathematicians will assume an attacking position, and

2. 44.77% of mathematicians will take a defensive position.

There’s a strange 0.8%, among which I include myself.

Who are the ones that attack? The attacking ones are mathematicians who would tell you this question makes no sense, because mathematics have a meaning all their own a beautiful edifice with its own logic — and that there’s no point in constantly searching for all possible applications.

What’s the use of poetry? What’s the use of love? What’s the use of life itself? What kind of question is that?

Hardy, for instance, was a model of this type of attack. And those who stand in defense tell you, Even if you don’t realize it, math is behind everything.” (Laughter)

Those guys, they always bring up bridges and computers. “If you don’t know math, your bridge will collapse.”

It’s true, computers are all about math. And now these guys have also started saying that behind information security and credit cards are prime numbers.

These are the answers your math teacher would give you if you asked him. He’s one of the defensive ones.

Okay, but who’s right then?

Those who say that math doesn’t need to have a purpose, or those who say that math is behind everything we do?

Actually, both are right.

But remember I told you I belong to that strange 0.8 percent claiming something else? So, go ahead, ask me what math is for. Audience: What is math for?

Eduardo Sáenz de Cabezón: Okay, 76.34% of you asked the question, 23.41 percent didn’t say anything, and the 0.8 percent — I’m not sure what those guys are doing.

Well, to my dear 76.31% — it’s true that math doesn’t need to serve a purpose, it’s true that it’s a beautiful structure, a logical one, probably one of the greatest collective efforts ever achieved in human history.

But it’s also true that there, where scientists and technicians are looking for mathematical theories that allow them to advance, they’re within the structure of math, which permeates everything.

It’s true that we have to go somewhat deeper, to see what’s behind science.

Science operates on intuition, creativity. Math controls intuition and tames creativity.

Almost everyone who hasn’t heard this before is surprised when they hear that if you take a 0.1 millimeter thick sheet of paper, the size we normally use, and, if it were big enough, fold it 50 times, its thickness would extend almost the distance from the Earth to the sun.

Your intuition tells you it’s impossible. Do the math and you’ll see it’s right. That’s what math is for.

It’s true that science, all types of science, only makes sense because it makes us better understand this beautiful world we live in.

And in doing that, it helps us avoid the pitfalls of this painful world we live in. There are sciences that help us in this way quite directly.

Oncological science, for example. And there are others we look at from afar, with envy sometimes, but knowing that we are what supports them.

All the basic sciences support them, including math. All that makes science, science is the rigor of math. And that rigor factors in because its results are eternal.

You probably said or were told at some point that diamonds are forever, right? That depends on your definition of forever!

A theorem — that really is forever. (Laughter) The Pythagorean theorem is still true even though Pythagoras is dead, I assure you it’s true. 

Even if the world collapsed the Pythagorean theorem would still be true. Wherever any two triangle sides and a good hypotenuse get together the Pythagorean theorem goes all out. It works like crazy.

Well, we mathematicians devote ourselves to come up with theorems. Eternal truths.

But it isn’t always easy to know the difference between an eternal truth, or theorem, and a mere conjecture. You need proof.

For example, let’s say I have a big, enormous, infinite field. I want to cover it with equal pieces, without leaving any gaps. I could use squares, right? I could use triangles. Not circles, those leave little gaps. Which is the best shape to use?

One that covers the same surface, but has a smaller border.

In the year 300, Pappus of Alexandria said the best is to use hexagons, just like bees do. But he didn’t prove it. The guy said, “Hexagons, great! Let’s go with hexagons!” He didn’t prove it, it remained a conjecture.

“Hexagons!” And the world, as you know, split into Pappists and anti-Pappists, until 1700 years later when in 1999, Thomas Hales proved that Pappus and the bees were right — the best shape to use was the hexagon. And that became a theorem, the honeycomb theorem, that will be true forever and ever, for longer than any diamond you may have. 

But what happens if we go to 3 dimensions?

If I want to fill the space with equal pieces, without leaving any gaps, I can use cubes, right? Not spheres, those leave little gaps. 

What is the best shape to use? Lord Kelvin, of the famous Kelvin degrees and all, said that the best was to use a truncated octahedron which, as you all know — (Laughter) — is this thing here!

Come on. Who doesn’t have a truncated octahedron at home? (Laughter) Even a plastic one.

“Honey, get the truncated octahedron, we’re having guests.” Everybody has one!  But Kelvin didn’t prove it. It remained a conjecture — Kelvin’s conjecture.

The world, as you know, then split into Kelvinists and anti-Kelvinists (Laughter) until a hundred or so years later, someone found a better structure.

Weaire and Phelan found this little thing over here, this structure to which they gave the very clever name “the Weaire-€“Phelan structure.”

It looks like a strange object, but it isn’t so strange, it also exists in nature. It’s very interesting that this structure, because of its geometric properties, was used to build the Aquatics Center for the Beijing Olympic Games.

There, Michael Phelps won eight gold medals, and became the best swimmer of all time. Well, until someone better comes along, right?

As may happen with the Weaire-€“Phelan structure. It’s the best until something better shows up.

But be careful, because this one really stands a chance that in a hundred or so years, or even if it’s in 1700 years, that someone proves it’s the best possible shape for the job. It will then become a theorem, a truth, forever and ever. For longer than any diamond.

So, if you want to tell someone that you will love them forever you can give them a diamond. But if you want to tell them that you’ll love them forever and ever, give them a theorem! (Laughter)

But hang on a minute! You’ll have to prove it, so your love doesn’t remain a conjecture.

Reine Azzi shared this link on FB

For the love of math (I sucked at it) in Spanish with English subtitles.

With humor and charm, mathematician Eduardo Sáenz de Cabezón answers a question that’s wracked the brains of bored students the world over: What is math for?
He shows the beauty of math as the backbone of science — and shows that…
ted.com|By Eduardo Sáenz de Cabezón

Idiosyncrasy in “conjectures”; (Dec. 21, 2009)

Idiosyncrasy or cultural bias relates to “common sense” behavior (for example, preferential priorities in choices of values, belief systems, and daily habits…) is not restricted among different societies: it can be found within one society, even within what can be defined as “homogeneous restricted communities” ethnically, religiously, common language, gender groups, or professional disciplines.

Most disciplines have mushroomed into cults.

A cult is any organization that creates its own nomenclature and definition of terms to be distinguished from the other cults in order to acquiring recognition as a “professional entity” or independent disciplines that should benefit from laws of special minorities (when mainly it is a matter of generating profit or doing business as usual).

These cults want to owe the non-initiated into believing that they have serious well-developed methods or excellent comprehension of a restricted area in sciences. The initiated on multidisciplinary knowledge recognize that the methods of any cult are old and even far less precise or developed; that the terms are not new and there are already analogous terms in other disciplines that are more accurate and far better defined.

Countless experiments have demonstrated various kinds of idiosyncrasies.  This article is oriented toward “cult” kinds of orders, organization, and professional discipline.  My first post is targeting the order of mathematicians; the next article will focus on experiments.

Mathematics, meaning “sure study” (wisekunde), has no reliable historical documentation. Most of mathematical concepts were written many decades or centuries after they were “floating around” among mathematicians.

Mathematics is confusing with its array of nomenclature. What are the differences among axiom, proposition, lemma, postulate, or conjecture?  What are the differences among the terms, theorem, questions, problems, hypothesis, corollary, and again conjecture?  For example, personally, I feel that axiom is mostly recurrent in geometry, lemma in probability, hypothesis in analytical procedures, and conjecture in algebraic deductive reasoning.

Hypothesis is in desuetude in mathematics. For example, Newton said “I am not making a hypothesis”.

Socrates made fun of this term by explaining how it was understood “I designate hypothesis what people doing geometry use to treating a question.  For example, when asked for their “expert opinion” they reply: “I still cannot confirm but I think that if I have a viable hypothesis for this problem and if it is the following hypothesis… then I think that we may draw a conclusion. If we have another hypothesis then another conclusion is more valid.”

Plato said: “As long as mathematics start from hypothesis instead of facts then we do not think that they have true comprehension, since they are not going back to fundamentals”

Hypothesis is still the main term used in experimental research. Theoretically, an experiment is not meant to accept a hypothesis as true or valid, but simply “Not to reject it” if the relationships among the manipulated variables are “statistically significant” to a pre-determined level, usually 5% in random errors.

Many pragmatic scientific researchers don’t care about the fine details in theoretical mathematical concepts and tend to adopt a hypothesis that was not rejected as law.  This is one case of idiosyncrasy when the researcher wants badly the “non-rejected” hypothesis to represent his view. Generally, an honest experimenter has to repeat the experiment or encourage someone else to generalize the results by studying more variables.

Conjecture means (throwing in together) and can be translated as conclusion or deduction; basically, it is an opinion or supposition based on insufficient proofs.

In the last century, conjectures were exposed in writing as promptly as possible instead of keeping them floating ideas, concepts, or probable theorems. This new behavior of writing conjectures was given the rationale that “plausible reasoning” is a set of suppositions thrown around as questions mathematicians guess they have answers to them, but are unable to demonstrate temporarily.

The term conjecture has been used so freely in the last decades that Andre Weil warned that “current mathematicians use the term conjecture when they fail after a few attempts to verify a concept, even if the problem is of no importance.”  David Kazhdan ironically warned that this practice of enunciating conjectures might turn out like a 5-year Soviet plan.

At first, a set of conjectures was meant to be the basic structure for a theorem or precise assertions that were temporarily used in the trading of logical discussions. Thus, conjectures permit the construction of rigorous deductions that are accessible to direct testing of their validity. A conjecture was a “research program” that move ahead in order to foresee the explored domain.

Consequently, conjecture is kind of extending a name and an address to a set of suppositions and analogies for a concept, long before tools and methods are created to approach directly the problem.

A “Problem” designates a mental task submitted to the audience or targeted for research or project; usually, the set of problems lead to demonstrating a general theorem. Many problems are in fact conjectures such as the problem of twin primary numbers that consists of proving the existence of an infinity of coupled numbers such that p-q = 2.

One of the explanations for using freely the term conjecture is the modern facility of mathematicians of discriminating aspects of uncertainty at the theoretical level. It is an acquired habit, an idiosyncrasy. Thus, for a mathematician to state a conjecture he must have solved many particular cases and recognize that a research program is needed to developing special tools for demonstrating the conjecture.  This is a tough restriction in this age where time is of essence among millions of mathematicians competing for prizes.


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October 2020
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