Posts Tagged ‘mathematicians’
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.
For the love of math (I sucked at it) in Spanish with English subtitles.
Efficiency has limits within cultural biases, and within mathematicians…
Posted on: December 11, 2009
Efficiency has limits within cultural bias; (Dec. 10, 2009)
Sciences that progressed so far have relied on mathematicians: many mathematical theories have proven to be efficacious in predicting, classifying, and explaining phenomena.
In general, fields of sciences that failed to interest mathematicians stagnated or were shelved for periods; maybe with the exception of psychology.
People wonder how a set of abstract symbols that are linked by precise game rules (called formal language) ends up predicting and explaining “reality” in many cases.
Biology has received recently a new invigorating shot: a few mathematicians got interested working for example on patterns of butterfly wings and mammalian furs using partial derivatives, but nothing of real value is expected to further interest in biology.
Economy, mainly for market equilibrium, applied methods adapted to dynamic systems, games, and topology. Computer sciences is catching up some interest.
“Significant mathematics” or those theories that offer classes of invariant relative to operations, transformations, and relationship almost always find applications in the real world: they generate new methods and tools such as theories of group and functions of a complex variable.
For example, the theory of knot was connected to many applied domains because of its rich manipulation of “mathematical objects” (such as numbers, functions, or structures) that remain invariant when the knot is deformed.
What is the main activity of a modern mathematician?
First of all, they do systematic organization of classes of “mathematical objects” that are equivalent to transformations. For example, surfaces to a homeomorphisms or plastic transformation and invariant in deterministic transformations.
There are several philosophical groups within mathematicians.
1. The Pythagorean mathematicians admit that natural numbers are the foundations of the material reality that is represented in geometric figures and forms. Their modern counterparts affirm that real physical structure (particles, fields, and space-time…) is identically mathematical. Math is the expression of reality and its symbolic language describes reality.
2. The “empirical mathematicians” construct models of empirical (experimental) results. They know in advance that their theories are linked to real phenomena.
3. The “Platonist mathematicians” conceive the universe of their ideas and concepts as independent of the world of phenomena. At best, the sensed world is but a pale reflection of their ideas. Their ideas were not invented but are as real though not directly sensed or perceived. Thus, a priori harmony between the sensed world and their world of ideas is their guiding rod in discovering significant theories.
4. There is a newer group of mathematicians who are not worried to getting “dirty” by experimenting (analyses methods), crunching numbers, and adapting to new tools such as computer and performing surgery on geometric forms.
This new brand of mathematicians do not care to be limited within the “Greek” cultural bias of doing mathematics: they are ready to try the Babylonian and Egyptian cultural way of doing math by computation, pacing lands, and experimenting with various branches in mathematics (for example, Pelerman who proved the conjecture of Poincaré with “unorthodox” techniques and Gromov who gave geometry a new life and believe computer to be a great tool for theories that do not involve probability).
Explaining phenomena leads to generalization (reducing a diversity of phenomena, even in disparate fields of sciences, to a few fundamental principles). Mathematics extend new concepts or strategies to resolving difficult problems that require collaboration of various branches in the discipline.
For example, the theory elaborated by Hermann Weyl in 1918 to unifying gravity and electromagnetism led to the theory of “jauge” (which is the cornerstone theory for quantum mechanics), though the initial theory failed to predict experimental results.
The cord and non-commutative geometry theories generated new horizons even before they verified empirical results.
Axioms and propositions used in different branches of mathematics can be combined to developing new concepts of sets, numbers, or spaces.
Historically, mathematics was never “empirically neutral”: theories required significant work of translation and adaptation of the theories so that formal descriptions of phenomena are validated.
Thus, mathematical formalism was acquired by bits and pieces from the empirical world. For example, the theory of general relativity was effective because it relied on the formal description of the invariant tensor calculus combined with the fundamental equation that is related to Poisson’s equations in classical potential.
The same process of adaptation was applied to quantum mechanics that relied on algebra of operators combined with Hilbert’s theory of space and then the atomic spectrum.
In order to comprehend the efficiency of mathematics, it is important to master the production of mental representations such as ideas, concepts, images, analogies, and metaphors that are susceptible to lending rich invariant.
Thus, the discovery of the empirical world is done both ways:
First, the learning processes of the senses and
Second, the acquisition processes of mathematical modeling.
Mathematical activities are extensions to our perception power and written in a symbolic formal language.
Natural sciences grabbed the interest of mathematicians because they managed to extract invariant in natural phenomena.
So far, mathematicians are wary to look into the invariant of the much complex human and social sciences. Maybe if they try to find analogies of invariant among the natural and human worlds then a great first incentive would enrich new theories applicable to fields of vast variability.
It appears that “significant mathematics” basically decodes how the brain perceives invariant in what the senses transmit to it as signals of the “real world”. For example, Stanislas Dehaene opened the way to comprehending how elementary mathematical capacity are generated from neuronal substrate.
I conjecture that, since individual experiences are what generate intuitive concepts, analogies, and various perspectives to viewing and solving problems, most of the useful mathematical theories were essentially founded on the vision and auditory perceptions.
New breakthrough in significant theories will emerge when mathematicians start mining the processes of the brain of the other senses (they are far more than the regular six senses). Obviously, the interested mathematician must have witnessed his developed senses and experimented with them as hobbies to work on decoding their valuable processes in order to construct this seemingly “coherent world”.
A wide range of interesting discoveries on human capabilities can be uncovered if mathematicians dare direct scientists and experimenters to additional varieties of parameters and variables in cognitive processes and brain perception that their theories predict.
Mathematicians have to expand their horizon: the cultural bias to what is Greek has its limits. It is time to take serious attempts at number crunching, complex computations, complex set of equations, and adapting to newer available tools.
Note: I am inclined to associate algebra to the deductive processes and generalization on the macro-level, while viewing analytic solutions in the realm of inferring by manipulating and controlling possibilities (singularities); it is sort of experimenting with rare events.
Adonis Diaries 