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Abstract Orderism Fractal 65 – Art by G. Stolyarov II

Abstract Orderism Fractal 65 – Art by G. Stolyarov II

Abstract Orderism Fractal 65 - by G. Stolyarov II

Abstract Orderism Fractal 65 – by G. Stolyarov II

Note: Left-click on this image to get a full view of this digital work of fractal art.

This fractal consists of intersecting luminous strands.

This digital artwork was created by Mr. Stolyarov in Apophysis, a free program that facilitates deliberate manipulation of randomly generated fractals into intelligible shapes.

This fractal is an extension of Mr. Stolyarov’s artistic style of Abstract Orderism, whose goal is the creation of abstract objects that are appealing by virtue of their geometric intricacy — a demonstration of the order that man can both discover in the universe and bring into existence through his own actions and applications of the laws of nature.

Fractal art is based on the idea of the spontaneous order – which is pivotal in economics, culture, and human civilization itself. Now, using computer technology, spontaneous orders can be harnessed in individual art works as well.

See the index of Mr. Stolyarov’s art works.

Abstract Orderism Fractal 64 – Art by G. Stolyarov II

Abstract Orderism Fractal 64 – Art by G. Stolyarov II

Abstract Orderism Fractal 64 - by G. Stolyarov II

Abstract Orderism Fractal 64 – by G. Stolyarov II

Note: Left-click on this image to get a full view of this digital work of fractal art.

This ornate rotational fractal conveys intricacy and nobility.

This digital artwork was created by Mr. Stolyarov in Apophysis, a free program that facilitates deliberate manipulation of randomly generated fractals into intelligible shapes.

This fractal is an extension of Mr. Stolyarov’s artistic style of Abstract Orderism, whose goal is the creation of abstract objects that are appealing by virtue of their geometric intricacy — a demonstration of the order that man can both discover in the universe and bring into existence through his own actions and applications of the laws of nature.

Fractal art is based on the idea of the spontaneous order – which is pivotal in economics, culture, and human civilization itself. Now, using computer technology, spontaneous orders can be harnessed in individual art works as well.

See the index of Mr. Stolyarov’s art works.

Abstract Orderism Fractal 63 – Art by G. Stolyarov II

Abstract Orderism Fractal 63 – Art by G. Stolyarov II

Abstract Orderism Fractal 60 - by G. Stolyarov II

Abstract Orderism Fractal 63 – by G. Stolyarov II

Note: Left-click on this image to get a full view of this digital work of fractal art.

This fractal conveys an impression of sharp, determined, multi-layered rotation.

This digital artwork was created by Mr. Stolyarov in Apophysis, a free program that facilitates deliberate manipulation of randomly generated fractals into intelligible shapes.

This fractal is an extension of Mr. Stolyarov’s artistic style of Abstract Orderism, whose goal is the creation of abstract objects that are appealing by virtue of their geometric intricacy — a demonstration of the order that man can both discover in the universe and bring into existence through his own actions and applications of the laws of nature.

Fractal art is based on the idea of the spontaneous order – which is pivotal in economics, culture, and human civilization itself. Now, using computer technology, spontaneous orders can be harnessed in individual art works as well.

See the index of Mr. Stolyarov’s art works.

Math Education Should Be Set Free – Article by Bradley Doucet

Math Education Should Be Set Free – Article by Bradley Doucet

The New Renaissance Hat
Bradley Doucet
February 12, 2015
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At different times in my life, I have earned my living tutoring high school math, helping struggling students struggle a little less with quadratic equations and trigonometric functions. I always excelled at math when I was in high school, and my temperament is well-suited to being patient with kids who are not understanding, and to figuring out why they’re not understanding. The experience of assisting a couple of hundred different students over the years has convinced me that just about anyone can learn to understand high school math. Some people simply need more time than others to become proficient with numbers and graphs and such.
 ***
Given my background, I read with interest The Globe and Mail’s write-up in January 2014 on what they are calling the Math Wars, “a battle that’s been brewing for years but heated up last month when this country dropped out of the top 10 in international math education standings.” Specifically, since the year 2000, Canada has fallen from 6th to 13th in the OECD’s Programme for International Student Assessment (PISA). Robert Craigen, a University of Manitoba mathematics professor, points out that this slippage coincides with the move away from teaching basic math skills and the adoption of discovery learning. In much of Canada today, this latest fad has children learning (or failing to learn) math by “investigating ideas through problem-solving, pattern discovery and open-ended exploration.”
 ***

Interestingly, when the Canadian provinces are included in the PISA rankings, Quebec is first among them, places 8th overall, and has lost practically no ground over the last dozen years. Why is Quebec suddenly ahead of the pack? Another Globe and Mail article from last month says that little work has been done on this question, but that “researchers have started focusing on Quebec’s intensive teacher training and curriculum, which balances traditional math drills with problem-solving approaches.” Basic math skills and problem solving sounds like a winning combination to me—and I bet the extra teacher training doesn’t hurt either.

Personally, I have long thought that math students should be allowed to progress at different rates. Currently, the brightest students shine out by scoring 90s and 100s while weaker students flounder with 60s and 70s and are forced to move on to more complex topics without having mastered more basic ones, almost ensuring their continued difficulties. With student-paced learning, the brightest students could still shine out by progressing more quickly, but weaker students would be given the time they need to master each topic before tackling harder problems. Everyone would get 90s and 100s; some would just get them sooner. Teaching would have to change, of course, in such a system. Maybe students would end up watching pre-recorded lessons, a la Khan Academy, and teachers could become more like flexible aides in the classroom, in addition to monitoring individual students to make sure they aren’t slacking off.

The Globe and Mail ended its editorial on Canada’s math woes last Thursday with a call to action: “If our students’ success in math really matters—and it does—it’s time to a have national policy discussion on how to move forward. Everything should be on the table, including curriculum reform. Let’s think big.” I can’t think of a worse idea. Even if you put me in charge of developing this national policy, it would still be a bad idea. After all, who’s to say if I’m correct in supposing that learning at your own pace is the way to go, that it would help everyone succeed and take away some of the anxiety many feel about math? Maybe it would be good for some, and less good for others. Maybe some people need the thrill of competing for top marks, while others would thrive in a less overtly competitive environment. Maybe people are different.

It’s bad enough that governments fix policies for entire provinces; the last thing we need is for everyone in the entire country to be doing the same thing. To the extent that there is a better way (or that there are better ways) to teach math, ways that we may not have even tried yet, the best means of discovering them is to allow different schools to teach math differently, to vary curriculum and teaching style and class size and whatever else they think might help. Let them compete for students, and let the best approaches win, and the worst approaches fall by the wayside, instead of having everyone follow the latest fad and doing irreparable damage to an entire cohort of kids.

It’s very hard to imagine this happening, though, in a system that is financed through taxation. Even though it’s ultimately the same people paying, whether directly as consumers or indirectly as taxpayers, people get into the mental habit of thinking that the government is paying, as if the government had a source of income other than the incomes of its people. And if the government is paying, then the government has to make sure it’s getting its money’s worth, and it’s only natural then that the government (i.e., politicians and bureaucrats) should set the curriculum and educational approach and make sure everyone is progressing at the same pace, in flagrant disregard of human diversity. It seems that we have a choice between “free” education and setting education free. Politicians and bureaucrats won’t give up control without a fight, though, which is a shame in the short term. But it may not matter in the longer term, as private initiatives like the Khan Academy make government schooling increasingly irrelevant.

I love math, and I furthermore believe that it is important for people to learn math. Mastery of math does indeed matter, which is precisely why we should think small and avoid the siren song of a “national policy discussion on how to move forward” on the educational front. Instead, we should let a thousand flowers bloom, and work with, not against, the natural diversity of humankind.

Bradley Doucet is Le Québécois Libre‘s English Editor and the author of the blog Spark This: Musings on Reason, Liberty, and Joy. A writer living in Montreal, he has studied philosophy and economics, and is currently completing a novel on the pursuit of happiness.
Fibonacci Rondo (Rondo #1), Op. 54 (2008) – Musical Composition and Video by G. Stolyarov II

Fibonacci Rondo (Rondo #1), Op. 54 (2008) – Musical Composition and Video by G. Stolyarov II

The New Renaissance Hat
G. Stolyarov II
November 7, 2014
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The Fibonacci Rondo, a 2008 composition by Mr. Stolyarov, was inspired by the Fibonacci Sequence of numbers, where each subsequent number is the sum of the two previous numbers. If the Fibonacci Sequence begins with 1 and 1, then the first six numbers of the sequence are 1, 1, 2, 3, 5, and 8.

The recurring theme of this composition – which occurs once at 0:32 and again at 1:30 represents musically the beginning of the Fibonacci Sequence and the process of its formation.

If we assign the value 1 to the note C, then we can assign the following values to other notes in relation to it:

2 = D

3 = E

5 = G

8 = C one octave above the “1” note.

Then, through two eighth notes, we can represent the numbers being added, while the following quarter note represents their result.

So two eighth-note C’s will be followed by a quarter-note D to represent “1 + 1 = 2.”

Then the eighth notes C and D, followed by a quarter-note E, represent “1 + 2 = 3.”

Then the eighth notes D and E, followed by a quarter-note G, represent “2 + 3 = 5.”

Then the eighth notes E and G, followed by a quarter-note C from the next octave, represent “3 + 5 = 8.”

Thereafter, the same pattern is applied to other harmonies – both major and minor – to ensure a melodic progression.

The timpani accompaniment in the second appearance of the theme relates this basic structure without any other notes added to reinforce the harmony. Quite a bit of harmonic reinforcement is added in the parts for all the other instruments, however.

This composition is written for a piano, two string sections, and timpani, and remastered using the Finale 2011 software. It probably could not be played by a human orchestra, as the 32nd notes in one of the string sections are simply too fast to be played by human musicians. The ability to reproduce music of this sort is yet another way in which computers have expanded the range of human creativity.

Download the MP3 file of this composition here.

See the index of Mr. Stolyarov’s compositions, all available for free download, here.

Remember to LIKE, FAVORITE, and SHARE this video in order to spread rational high culture to others.

Abstract Orderism Fractal 62 – Art by G. Stolyarov II

Abstract Orderism Fractal 62 – Art by G. Stolyarov II

Abstract Orderism Fractal 60 - by G. Stolyarov II

Abstract Orderism Fractal 62 – by G. Stolyarov II

Note: Left-click on this image to get a full view of this digital work of fractal art.

This colorful, intricate fractal resembles the ornaments on a peacock’s tail.

This digital artwork was created by Mr. Stolyarov in Apophysis, a free program that facilitates deliberate manipulation of randomly generated fractals into intelligible shapes.

This fractal is an extension of Mr. Stolyarov’s artistic style of Abstract Orderism, whose goal is the creation of abstract objects that are appealing by virtue of their geometric intricacy — a demonstration of the order that man can both discover in the universe and bring into existence through his own actions and applications of the laws of nature.

Fractal art is based on the idea of the spontaneous order – which is pivotal in economics, culture, and human civilization itself. Now, using computer technology, spontaneous orders can be harnessed in individual art works as well.

See the index of Mr. Stolyarov’s art works.

Abstract Orderism Fractal 61 – Art by G. Stolyarov II

Abstract Orderism Fractal 61 – Art by G. Stolyarov II

Abstract Orderism Fractal 60 - by G. Stolyarov II

Abstract Orderism Fractal 61 – by G. Stolyarov II

Note: Left-click on this image to get a full view of this digital work of fractal art.

This fractal consists of spiral upon radiant spiral, spreading forth in glorious illumination.

This digital artwork was created by Mr. Stolyarov in Apophysis, a free program that facilitates deliberate manipulation of randomly generated fractals into intelligible shapes.

This fractal is an extension of Mr. Stolyarov’s artistic style of Abstract Orderism, whose goal is the creation of abstract objects that are appealing by virtue of their geometric intricacy — a demonstration of the order that man can both discover in the universe and bring into existence through his own actions and applications of the laws of nature.

Fractal art is based on the idea of the spontaneous order – which is pivotal in economics, culture, and human civilization itself. Now, using computer technology, spontaneous orders can be harnessed in individual art works as well.

See the index of Mr. Stolyarov’s art works.

Abstract Orderism Fractal 60 – Art by G. Stolyarov II

Abstract Orderism Fractal 60 – Art by G. Stolyarov II

Abstract Orderism Fractal 60 - by G. Stolyarov II

Abstract Orderism Fractal 60 – by G. Stolyarov II

Note: Left-click on this image to get a full view of this digital work of fractal art.

This is a fractal of exquisitely intertwined spiral patterns, incorporating both rounded and angular elements.

This digital artwork was created by Mr. Stolyarov in Apophysis, a free program that facilitates deliberate manipulation of randomly generated fractals into intelligible shapes.

This fractal is an extension of Mr. Stolyarov’s artistic style of Abstract Orderism, whose goal is the creation of abstract objects that are appealing by virtue of their geometric intricacy — a demonstration of the order that man can both discover in the universe and bring into existence through his own actions and applications of the laws of nature.

Fractal art is based on the idea of the spontaneous order – which is pivotal in economics, culture, and human civilization itself. Now, using computer technology, spontaneous orders can be harnessed in individual art works as well.

See the index of Mr. Stolyarov’s art works.

Abstract Orderism Fractal 59 – Art by G. Stolyarov II

Abstract Orderism Fractal 59 – Art by G. Stolyarov II

Abstract Orderism Fractal 59 - by G. Stolyarov II

Abstract Orderism Fractal 59 – by G. Stolyarov II

Note: Left-click on this image to get a full view of this digital work of fractal art.

This fractal resembles a feather emerging out of a spiral.

This digital artwork was created by Mr. Stolyarov in Apophysis, a free program that facilitates deliberate manipulation of randomly generated fractals into intelligible shapes.

This fractal is an extension of Mr. Stolyarov’s artistic style of Abstract Orderism, whose goal is the creation of abstract objects that are appealing by virtue of their geometric intricacy — a demonstration of the order that man can both discover in the universe and bring into existence through his own actions and applications of the laws of nature.

Fractal art is based on the idea of the spontaneous order – which is pivotal in economics, culture, and human civilization itself. Now, using computer technology, spontaneous orders can be harnessed in individual art works as well.

See the index of Mr. Stolyarov’s art works.

Putting Randomness in Its Place (2010) – Article by G. Stolyarov II

Putting Randomness in Its Place (2010) – Article by G. Stolyarov II

The New Renaissance Hat
G. Stolyarov II
Originally Published February 11, 2010
as Part of Issue CCXXXV of The Rational Argumentator
Republished July 22, 2014
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Note from the Author: This essay was originally published as part of Issue CCXXXV of The Rational Argumentator on February 11, 2010, using the Yahoo! Voices publishing platform. Because of the imminent closure of Yahoo! Voices, the essay is now being made directly available on The Rational Argumentator.
~ G. Stolyarov II, July 22, 2014
***

A widespread misunderstanding of the meaning of the term “randomness” often results in false generalizations made regarding reality. In particular, the view of randomness as metaphysical, rather than epistemological, is responsible for numerous commonplace fallacies.

To see randomness as metaphysical is to see it as an inherent aspect of reality as such – as embedded inextricably in “the way things are.” Typically, people holding this view will take it in one of two directions. Some of them will see randomness pejoratively – thinking that there is no way reality could be like that: chaotic, undefined, unpredictable. Such individuals will typically posit that, because reality cannot be random, it must therefore be centrally planned by a super-intelligent entity, such as a deity.

Others, however, will use the metaphysical perception of randomness to deny evident and ubiquitously observable truths about our world: the facts that all entities obey certain natural laws, that these laws are accessible to human beings, and that they can inform our decision-making and actions. These individuals typically espouse metaphysical subjectivism – the idea that the nature of reality depends on the person observing it, or that all of existence is in such a chaotic flux that we cannot ever possibly make sense of it, so we might as well “construct” our own personal or cultural “reality.”

But it is the very metaphysical perception of randomness that is in error. Randomness is, rather, epistemological – a description of our state of knowledge of external reality, and not of external reality itself. To say that a phenomenon is random simply means that we do not (yet) have adequate knowledge to be able to explain it causally. Based on past observational experience or some knowledge of aspects inherent to that phenomenon, we might be able to assign probabilities – estimates of the likelihood that a particular event will occur, in the absence of more detailed knowledge about the specifics of the circumstances that might give rise to that event. In some areas of life, this is presently as far as humans can venture. Indeed, probabilistic thinking can be conceptually quite powerful – although imprecise – in analyzing large classes of phenomena which, individually, exhibit too many specific details for any single mind to grasp. Entire industries, such as insurance and investment, are founded on this premise. But we must not mistake a conceptual tool for an external fact; the probabilities are not “out there.” They are, rather, an attempt by human beings to interpret and anticipate external phenomena.

The recognition of randomness as epistemological can be of great aid both to those who believe in biological evolution and to advocates of the free market. Neither the laws of evolution, nor the laws of economics, of course, would fit any definition of “randomness.” Rather, they are impersonal, abstract principles that definitively describe the general outcomes of particular highly complex sets of interactions. They are unable to account for every fact of those interactions, however, and they are also not always able to predict precisely how or when the general outcome they anticipate will ensue. For instance, biological evolution cannot precisely predict which complex life forms will evolve and at what times, or which animals in a current ecosystem will ultimately proliferate, although traits that might enhance an animal’s survival and reproduction and traits that might hinder them can be identified. Likewise, economics – despite the protestations of some economists to the contrary – cannot predict the movements of stock prices or prices in general, although particular directional effects on prices from known technological breakthroughs or policy decisions can be anticipated.

Evolution is often accused of being incapable of producing intelligent life and speciation because of its “randomness.” For many advocates of “intelligent design,” it does not appear feasible that the complexity of life today could have arisen as a result of “chance” occurrences – such as genetic mutations – that nobody planned and for whose outcomes nobody vouched. However, each of these mutations – and the natural selection pressures to which they were subject – can only be described as random to the extent that we cannot precisely describe the circumstances under which they occurred. The more knowledge we have of the circumstances surrounding a particular mutation, the more it becomes perfectly sensible to us, and explicable as a product of causal, natural laws, not “sheer chance.” Such natural laws work both at the microscopic, molecular level where the proximate cause of the mutation occurred, and at the macroscopic, species-wide level, where organisms with the mutation interact with other organisms and with the inanimate environment to bring about a certain episode in the history of life.

So it is with economics; the interactions of the free market seem chaotic and unpredictable to many – who therefore disparage them as “random” and agitate for centralized power over all aspects of human life. But, in fact, the free market consists of millions of human actors in billions of situations, and each actor has definite purposes and motivations, as well as definite constraints against which he or she must make decisions. The “randomness” of behaviors on the market is only perceived because of the observer’s limited knowledge of the billions of circumstances that generate such behaviors. We can fathom our own lives and immediate environments, and it may become easier to understand the general principles behind complex economies when we recognize that each individual life has its own purposes and orders, although they may be orders which we find mistaken or purposes of which we disapprove. But the interaction of these individual microcosms is the free market; the more we understand about it, the more sensible it becomes to us, and the more valid conclusions we can draw regarding it.

The reason why evolution and economies cannot be predicted at a concrete level, although they can be understood, is the sheer complexity of the events and interactions involved – with each event or interaction possibly being of immense significance. Qualitative generalizations, analyses of attributes, and probabilistic thinking can answer some questions pertaining to these complex systems and can enable us to navigate them with some success. But these comprise our arsenal of tools for interpreting reality; they do not even begin to approach being the reality itself.

When we come to see randomness as a product of our limited knowledge, rather than of reality per se, we can begin to appreciate how much there is about reality that can be understood – rather than dismissed as impossible or inherently chaotic – and can broaden our knowledge and mastery of phenomena we might otherwise have seen as beyond our grasp.

Click here to read more articles in Issue CCXXXV of The Rational Argumentator.