The problem with teaching intro logic is you just learn a bunch of rules but not the reasoning behind them.
—  Logic professor
A new quantum logic: Trigram Multiplication - I

(continued from here)

Here we begin to examine and compare the Cartesian ordered triads and the Taoist trigrams(1) of the Book of Changes. We’re in totally new territory here. We’ve entered the third dimension. Whoopee! No need to be frightened. Just think of the trigrams and triads of the eight quadrants as the analogues in the cube of the ordered pairs and bigrams of the two dimensional square. The only difference here is that there is one more dimension to deal with. We’re already well on our way to our goal of six.

Descartes uniquely identifies every point in 3-dimensional space with his ordered triads. Mandalic geometry sets its eye on a somewhat smaller and self-contained goal. It limits its consideration to the range from minus one (-1) to plus one (+1) in each of the three dimensions. And because it is based upon a quantized or discretized geometry its entire universe of discourse is the unit cube as it morphs through its eight different identities in the eight octants of three dimensional Cartesian geometry.(2)

Where Descartes identifies the eight unique vertices of the eightfold unit cube(3) with ordered triads (1,1,1; 1,1,-1; etc.) mandalic geometry identifies them with the eight trigrams of the Taoist I Ching. This is initially a matter of different notation but it grows into many other differences of great importance. Some of these have to do with the manner in which the human brain is better equipped to manipulate and interchange the Taoist symbols, a fact elaborated elsewhere in this blog. Along somewhat similar lines, the eight trigrams serve as excellent mnemonic devices, no mean accomplishment(4) and one which the Cartesian triads clearly fail to do.

(to be continued)

Image: Cubic crystal system, modified to demonstrate positioning of trigrams and Cartesian triads. Generalic, Eni. “Cubic crystal system.” Croatian-English Chemistry Dictionary & Glossary. 31 July 2014. KTF-Split.19 Aug. 2014. <http://glossary.periodni.com>.

(1) Also known as bagua.

The bagua are eight trigrams used in Taoist cosmology to represent the fundamental principles of reality, seen as a range of eight interrelated concepts. Wikipedia

The Wikipedia article, however, does not broach the hidden or implied mathematical interrelationships of the eight trigrams. It is those very geometric/logical interrelationships missing in the article that mandalic geometry is about. As a point of interest it should be noted that even the I Ching itself fails to disclose these logical interrelationships in a fully explicit and satisfying manner. They are there to be sure, but in implicit form only. Also note that though historically the trigrams have existed in two different arrangements, both described in the Wikipedia article cited, mandalic geometry is based exclusively on the one arrangement that is commensurable with Cartesian coordinates, namely the “Earlier Heaven” arrangement.

(2) The only scalar quantities mandalic geometry is concerned with are one (1) and a few scalar numbers found in nature like 2, 4, 8, pi, square root 2, square root 3, etc. It is the sign portion of vectors that is of more interest to mandalic geometry and which it is principally concerned with. The stated range of scalar 2 for mandalic geometry is much too modest. Though the range of mandalic geometry is only scalar 2 as described in a three dimensional system, the system of the I Ching on which it is based is actually a six dimensional one so it is destined to squeeze a lot more in those eight unit cubes than one might expect. But not just yet. Though if you are eager to know more about the subject right now you could check out earlier entries in this blog where a lot has already been said regarding these matters.

(3) This is not as far as I know an official geometric term. I am using it as a kind of shorthand for “the larger cube composed of eight small unit cubes occupying all eight octants of three dimensional Cartesian geometry and mutually tangent at the single origin point of the coordinate system.”

(4) Particularly as they do so also in their alter egos as components of the 64 hexagrams. Recognizing and remembering how to distinguish eight closely related forms is difficult enough, and sixty-four all the more so. A mathematical wizard might be able to accomplish the 8-form feat using the Cartesian triads. I doubt that the same could be done with the Cartesian equivalent hexads that would be required for the hexagrams, other than possibly by a savant.

Ok so I was thinking.

Bayes’ Theorem goes P(H|EX) = P(E|HX)P(H|X)/P(E|X).

Now, suppose someone wants to estimate my height. Their prior for my height, p(h|X), is given by the distribution of the heights in my country. Sounds reasonable as a prior.

Then someone goes and says, “Hey, his name is Pedro! We should take that as evidence and measure a population of Pedros!”

And if you measure like 50 Pedros and none of those are me, the estimate isn’t going to be very useful. But if I am in that population of measured Pedros, then that population is going to be significant improvement over the entire population of Brazil.

Yet the difference here might just be that in the former case we’re measuring 50 Pedros, and in the latter we’re measuring 51 Pedros.

But…

p(h|Pedro, X) = p(h|Pedro, name is relevant, X) P(name is relevant|Pedro, X) + p(h|Pedro, name is irrelevant, X) P(name is irrelevant|Pedro, X)

Name is relevant if and only if I am in the studied population of Pedros, in which case P(name|PX) ~ 1 and P(~name|PX) ~ 0, and vice-versa.

This looks like it means that all evidence E should be weighted when considering an H that is not directly linked to it by how much we expect that evidence to be relevant.

But where is that, in P(H|EX) = P(E|HX)P(H|X)/P(E)?

I mean, it is in there somewhere, two ways of arriving at a result have to always arrive at the same result, but um.

raginrayguns any ideas? Or anyone else for that matter?

Since its inception in 1936, the Fields Medal has been awarded to 52 of the most exceptional mathematicians in the world under the age of 40. For the first time, that award has gone to a woman: Maryam Mirzakhani, 37, an Iranian-born mathematician who works at Stanford.

She shared the prize — the highest honor in mathematics — with Martin Hairer, 38, of the University of Warwick, England; Manjul Bhargava, 40, of Princeton; and Arthur Avila, 35, of the National Center for Scientific Research, France.

According to The New York Times, 70% of doctoral degrees in math are awarded to males, making the award to Mirzakhani especially noteworthy. In the related field of physics, only two women have ever won the Nobel Prize. Only one has won in economics.

The Fields was presented by the International Congress of Mathematicians to this year’s four winners in a ceremony in Seoul on Wednesday.

Mirzakhani’s research focuses on “understanding the symmetry of curved surfaces, such as spheres, the surfaces of doughnuts and of hyperbolic objects,” according to a Stanford release. A text provided by the ICM further explains that she works on so-called Riemann surfaces and their deformations. The ICM praised her for “strong geometric intuition.”

A Huge First For Women: Female Mathematician Wins Fields Medal

An Isochrone curve is the curve for which the time taken by an object sliding without friction in uniform gravity to its lowest point is independent of its starting point. The curve is a cycloid, and the time is equal to π times the square root of the radius over the acceleration of gravity.

- A ball set on an Isocrone (or Tautochrone) curve will reach the bottom at the same length of time no matter where you place the ball, so long as there is no impeding friction.

[Gif] - Four balls slide down a cycloid curve from different positions, but they arrive at the bottom at the same time. The blue arrows show the points’ acceleration along the curve. On the top is the time-position diagram.

[source]

I think mathematicians do mathematics for reasons that are very similar to those of musicians playing music or any artist doing their art. It’s all about trying to contribute to a certain understanding of ourselves and of the world around us.
—  Princeton mathematician Manjul Bhargava, who has been awarded the 2014 Fields Medal, one of the most prestigious awards in mathematics. Read more about Bhargava and the award here and watch a video about him here.
Interview with Maryam Mirzakhani, the brilliant Iranian mathematician who was the first woman to win the Fields Medal
• Interviewer:What advice would you give lay persons who would
• like to know more about mathematics—what it is,
• what its role in our society has been and so on?
• What should they read? How should they proceed?
• Dr. Mirzakhani:This is a difficult question. I don’t think that everyone
• should become a mathematician, but I do believe that
• many students don’t give mathematics a real chance.
• I did poorly in math for a couple of years in middle
• school; I was just not interested in thinking about it.
• I can see that without being excited mathematics can
• look pointless and cold. The beauty of mathematics
• only shows itself to more patient followers.

Portrait of Émilie du Châtelet by Maurice Quentin de La Tour, mid 1700s. du Châtelet was a mathematician and physicist who translated and annotated Isaac Newton’s “Principia Mathematica” into French, which includes her derivation of conservation of energy .

New research sheds light on how children’s brains memorize facts

As children learn basic arithmetic, they gradually switch from solving problems by counting on their fingers to pulling facts from memory. The shift comes more easily for some kids than for others, but no one knows why.

Now, new brain-imaging research gives the first evidence drawn from a longitudinal study to explain how the brain reorganizes itself as children learn math facts. A precisely orchestrated group of brain changes, many involving the memory center known as the hippocampus, are essential to the transformation, according to a study from the Stanford University School of Medicine.

The results, published online Aug. 17 in Nature Neuroscience, explain brain reorganization during normal development of cognitive skills and will serve as a point of comparison for future studies of what goes awry in the brains of children with learning disabilities.

“We wanted to understand how children acquire new knowledge, and determine why some children learn to retrieve facts from memory better than others,” said Vinod Menon, PhD, the Rachael L. and Walter F. Nichols, MD, Professor and  professor of psychiatry and behavioral sciences, and the senior author of the study. “This work provides insight into the dynamic changes that occur over the course of cognitive development in each child.”

The study also adds to prior research into the differences between how children’s and adults’ brains solve math problems. Children use certain brain regions, including the hippocampus and the prefrontal cortex, very differently from adults when the two groups are solving the same types of math problems, the study showed.

“It was surprising to us that the hippocampal and prefrontal contributions to memory-based problem-solving during childhood don’t look anything like what we would have expected for the adult brain,” said postdoctoral scholar Shaozheng Qin, PhD, who is the paper’s lead author.

Charting the shifting strategy

In the study, 28 children solved simple math problems while receiving two functional magnetic resonance imaging brain scans; the scans were done about 1.2 years apart. The researchers also scanned 20 adolescents and 20 adults at a single time point. At the start of the study, the children were ages 7-9. The adolescents were 14-17 and the adults were 19-22. The participants had normal IQs. Because the study examined normal math learning, potential participants with math-related learning disabilities and attention deficit hyperactivity disorder were excluded. The children and adolescents were studying math in school; the researchers did not provide any math instruction.

During the study, as the children aged from an average of 8.2 to 9.4 years, they became faster and more accurate at solving math problems, and relied more on retrieving math facts from memory and less on counting. As these shifts in strategy took place, the researchers saw several changes in the children’s brains. The hippocampus, a region with many roles in shaping new memories, was activated more in children’s brains after one year. Regions involved in counting, including parts of the prefrontal and parietal cortex, were activated less.

The scientists also saw changes in the degree to which the hippocampus was connected to other parts of children’s brains, with several parts of the prefrontal, anterior temporal cortex and parietal cortex more strongly connected to the hippocampus after one year. Crucially, the stronger these connections, the greater was each individual child’s ability to retrieve math facts from memory, a finding that suggests a starting point for future studies of math-learning disabilities.

Although children were using their hippocampus more after a year, adolescents and adults made minimal use of their hippocampus while solving math problems. Instead, they pulled math facts from well-developed information stores in the neocortex.

Memory scaffold

“What this means is that the hippocampus is providing a scaffold for learning and consolidating facts into long-term memory in children,” said Menon, who is also the Rachel L. and Walter F. Nichols, MD, Professor at the medical school. Children’s brains are building a schema for mathematical knowledge. The hippocampus helps support other parts of the brain as adultlike neural connections for solving math problems are being constructed. “In adults this scaffold is not needed because memory for math facts has most likely been consolidated into the neocortex,” he said. Interestingly, the research also showed that, although the adult hippocampus is not as strongly engaged as in children, it seems to keep a backup copy of the math information that adults usually draw from the neocortex.

The researchers compared the level of variation in patterns of brain activity as children, adolescents and adults correctly solved math problems. The brain’s activity patterns were more stable in adolescents and adults than in children, suggesting that as the brain gets better at solving math problems its activity becomes more consistent.

The next step, Menon said, is to compare the new findings about normal math learning to what happens in children with math-learning disabilities.

“In children with math-learning disabilities, we know that the ability to retrieve facts fluently is a basic problem, and remains a bottleneck for them in high school and college,” he said. “Is it that the hippocampus can’t provide a reliable scaffold to build good representations of math facts in other parts of the brain during the early stages of learning, and so the child continues to use inefficient strategies to solve math problems? We want to test this.”

We’re delighted for Professor Maryam Mirzakhani, the first female recipient of the prestigious Fields Medal in mathematics. In this long overdue landmark, Professor Mirzakhani has been commended for her work in complex geometry.

The mathematics community is hopeful that this will encourage more girls and young women to pursue careers in the field.

The following articles by Maryam Mirzakhani, published in International Mathematics Research Notices, are free for a limited time:

Image: Maryam Mirzakhani by International Mathematical Union (IMU). Public domain via Wikimedia Commons.

Congratulations to Maryam Mirzakhani! She is the first woman be awarded the top prize in mathematics, the Fields Medal 2014. Want to read some of her work? Check out her articles here.

So why is Euler’s Identity so Beautiful?

The physicist Richard Feynman called the formula it is derived from “one of the most remarkable, almost astounding, formulas in all of mathematics”.

I’m sure you’ve seen all the symbols in the identity before, but isn’t it weird how they are not connected in any evident, but combine to give such a normal result?

The next part is an excerpt from Surein Aziz’s article the Beauty in Mathematics:

'So, why does this happen? You might think that it is down to some really complex idea — how do we even take a number to the power of i? Well, actually, it isn’t too difficult to see how Euler’s identity comes about - that is one thing that makes the identity so wonderful! But first you have to see Euler’s formula, which leads to his beautiful identity, in full generality:

Doesn’t look quite as nice and neat now, does it? But don’t be put off. To understand how this formula comes about, we need something called Taylor series. These are just a way of expressing functions such as sin (x) or cos (x) as infinite sums. They were discovered by the mathematician Brook Taylor (who was also part of the committee which adjudicated the argument between Isaac Newton and Gottfried Leibniz about who first invented the calculus).

The Taylor series for the function e^ x is:

where n! denotes the product

You can verify this Taylor series using a calculator: choose a number \$x\$ and see what value the calculator gives you for \$e^ x\$. Now use the calculator to work out the value of the sum

for as many terms as you like, that is for a number \$n\$ as high as you like. You will find that the result very nearly equals the result you got for \$e^ x\$ and the more terms you add to the sum, the closer the two results become. At some point the two results will be the same on your calculator, as their difference becomes too small for the calculator to detect. In “reality”, the two results are equal when you have added an infinite number of terms to your sum.

The Taylor series for the other two functions appearing in Euler’s formula are:

Again you can check this using your calculator, bearing in mind that the angle x is measured in radians, rather than degrees.

Now let’s multiply the variable x in the Taylor series for e^ x by the number i. We get:

But certain powers of i can be simplified – for example, i^2 = -1 by definition, and so i^3 = -i and i^4 = +1, and so on. So we can simplify the above to:

We can gather the terms involving i together to give:

Now notice that these two series are the same as the series for sin (x) and cos (x) from earlier, so we can substitute these in to get:

which is Euler’s formula!

All we have to do now is substitute x = pi. Since sin (pi ) = 0 and cos (pi ) = -1 we get:

so

So you see, after a sequence of fairly complex mathematics we arrive back where we started — at the (seemingly) simple numbers 1 and 0. That is what I think is so beautiful about this identity: it links very strange numbers with very ordinary and fundamental ones. Seeing why it works feels a bit like treading a little-known path through the mathematical jungle to reach a secret destination somewhere in the thick undergrowth.’

And what about Euler? (from storyofmathematics.com)

Today, Euler is considered one of the greatest mathematicians of all time. His interests covered almost all aspects of mathematics, from geometry to calculus to trigonometry to algebra to number theory, as well as optics, astronomy, cartography, mechanics, weights and measures and even the theory of music.

Much of the notation used by mathematicians today - including e, i, f(x), ∑, and the use of a, b and c as constants and x, y and z as unknowns - was either created, popularised or standardised by Euler. His efforts to standardise these and other symbols (including π and the trigonometric functions) helped to internationalise mathematics and to encourage collaboration on problems.

He even managed to combine several of these together in an amazing feat of mathematical alchemy to produce one of the most beautiful of all mathematical equations, eiπ = -1, sometimes known as Euler’s Identity. This equation combines arithmetic, calculus, trigonometry and complex analysis into what has been called “the most remarkable formula in mathematics”, “uncanny and sublime” and “filled with cosmic beauty”, among other descriptions. Another such discovery, often known simply as Euler’s Formula, is eix = cosx + isinx. In fact, in a recent poll of mathematicians, three of the top five most beautiful formulae of all time were Euler’s. He seemed to have an instinctive ability to demonstrate the deep relationships between trigonometry, exponentials and complex numbers.

@obywan

Things to know about Fibonacci and his Numbers -(by request)

Leonardo Pisano Bigollo (known as Fibonacci, and also Leonardo of Pisa, Leonardo Pisano, Leonardo Bonacci, Leonardo Fibonacci) —was an Italian mathematician, considered by some “the most talented Western mathematician of the Middle Ages.”

Fibonacci is best known for the spreading of the Hindu–Arabic numeral system which we use today in modern times - In his Liber Abaci (1202), Fibonacci introduced the modus Indorum (meaning method of the Indians), today known as Arabic numerals - which include the numbers 0 - 9 and was one of the earliest numerical systems to use zero as a place holder.  The book also advocated place value in early hindu-arabic numerals.

^ modern Arabic numerals

The Fibonacci sequence

The Fibonacci numbers were introduced in his Liber Abaci which posed, and solved a problem involving the growth of a population of rabbits based on idealized assumptions.

The solution, generation by generation, was a sequence of numbers later known as Fibonacci numbers. The number sequence was known to Indian mathematicians as early as the 6th century, but it was Fibonacci’s Liber Abaci that introduced it to the West.

the Fibonacci sequence is widely known for its interesting properties. the one you may be most familiar with is that every term is the addition of the previous two terms:

for example, the Fibonacci sequence is represented here in this famous pattern.

Your sequence begins with a square with side length of 1. Imagine this is one rabbit - if you pair one rabbit with no rabbits you will have no offspring. we then add a partner rabbit, so you have 1 and 1 paired together.

the number of offspring they produce is the sum of the previous two generation’s population, in this case because we start with only 1 and 1 rabbits we get 2 in the next generation.

at this point your sequence looks like 1,1,2,

your next population of offspring continues the same rule - the sum of the previous two populations of the rabbit generation. So in this case where X is our fourth population in the next generation (1,1,2,X). X is the sum of 1 and 2 - the previous two populations.

The Rule is Xn = Xn-1 + Xn-2

so we now have the sequence 1,1,2,3

and the 1,1,2,3,5

and the sequence can carry on to infinity:

1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584…

The characteristics of the Fibonacci sequence is commonly found in sunflower seeds and seashells as well as many other forms of nature, Art and Architecture.

The Golden Ratio

The Fibonacci numbers were first expressed in terms of the Golden ratio by Daniel Bernoulli  in 1724.

The Golden ratio is one of the few Famous Mathematical constants along with e, √2, and π. It is an Irrational number.

In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities.

If we continue divide a term in the Fibonacci sequence by its preceding term we eventually approach the Golden ratio designated by the Greek numeral ϕ (phi, lowercase φ):

1/1= 1.000

2/1= 2.000

3/2= 1.500

5/3= 1.333

…..

55/34= 1.617

89/55=  1.618

etc.

the Golden ratio is approximated to the decimal 1.618033988

with this we can show that each Fibonacci number can be written in terms of Phi.

^ The golden ratio fits coherently with the Fibonacci pattern (where the curve is the Golden ratio and the squares are the Fibonacci numbers.)

^ Fibonacci numbers can be found in many other mathematical discoveries, as it is the one of the most naturally occurring sequences in Mathematics. Fibonacci numbers can be found in the Pascal triangle when you add the numbers diagonally.

Finding the Nth term in the Fibonacci sequence

The Formula to find the Nth term in the Fibonacci sequence can be calculated with the Golden ratio:

sources - [1] [2]

In his short and magnificent 1940 book A Mathematician’s Apology, the English mathematician G.H. Hardy draws attention to the fact that a theorem cannot be undone. Mathematics is the only science that deals with truth, something that can be demonstrated by popping into any library: mathematical literature is evergreen, while texts on other sciences become rapidly obsolete. Two thousand years have not written a wrinkle on the Pythagorean theorem. Nobody studies Ptolemy’s solar system, except out of historical interest, but Euclid is still standing. Mathematics works by accumulation, not substitution.

From the moment our gazes first intersected
my pulse took off at an indeterminate instantaneous rate of change
and I already found myself trying to calculate
how long it would take to lay tangent to you.

Further back on my own number line,
I chose to wrap myself in worlds of whimsy,
lounge in lands of language.
as Harry Potter felt at Privet Drive,
and it was literature which taught me
that some infinities are bigger than others.

You, however,
were the first boy to show me the mathematical implications of such;
the first equation I became so determined to solve
that I bothered to double check my work.

You plotted precise points along my perimeter
and found values for variables I hadn’t even realized I’d needed to solve for.
We accelerated exponentially,
spiraling out like the Fibonacci Sequence –
every step we took forward, a sum of where we’d already been.
You were a cosine function, constant in your fluctuation,
and I longed to be sine
so our points of intersection could be infinite,
stretching out in all directions.

I wanted to integrate the space between your heartbeats,
calculate the area under the curve,
find the volume of the passion we shared as it spread into all four quadrants
and every time I was 95% confident that my feelings for you
you found enough statistically significant evidence
to reject my original hypothesis.

You snagged my from my sanctuary of similes
and showed me the elegance of evaluation,
the beauty of a balanced equation,
the world’s carefully crafted congruence –
permutations and combinations that make up the very improbability of life and of love.
You taught me what it felt like to have within my veins
the sensation
of a limit approaching infinity.

In the end, unfortunately,
fate chose not to make trigonometric functions of us.
Instead, we were like perpendicular lines:
destined to meet once
then grow farther apart.
Even so, I carry the lessons you graphed within me
much like you used to carry your calculator –
close to your heart, and always.

—  A Mathematical Love Letter, by Rachel R. Carroll

You’re a doctor? I don’t really know anything about medicine, but can you explain exactly how the endocrine system works in two minutes or less? You’re a writer? I had a terrible writing teacher in…