AI News, Beautiful Data

Beautiful Data

We were concerned about the strange dichotomy by which people separated old media from new media to make their point about quality, ethics, and aesthetics.

Just like people were scoffing social media to be just doodles, scribbling, or worse, I now see people scornfully raising their eyebrows about the lack of structure, missing consistency, and other alleged flaws they imagine Big Data to carry.

Again what these people see, is just an evil new vice swamped over their mature businesses by unseasoned startups, however insanely well funded.

It was wrong in the 90s when the web started, it was wrong again in the 2000s regarding social media, and it will not become right this time.

The traditional concept of media becomes more and more directly intertwined with data, with data storytelling, data journalism, and their likes, indirectly because search, targeted advertising, content filtering, and other predictive technologies increasingly influence what we will find presented as media content.

The survey was done face to face, based on a cautiously drawn sample of 10.000 people per year.

my employer was also involved in direct marketing with a huge database of addresses, call centers, and logistics, we developed a method to use the highly curated market survey with its rather small sample to calibrate and enhance the “dirty”

Television ratings are measured by expensive panels in most markets, usually run and funded by joint industry committees like BARB in the UK or AGF in Germany.

We again found a way to infuse the TV panel data into the online data and could by that calculate the probabilities that the owner of a certain cookie would have had contact with a certain advertising campaign on TV or not.

So again, a small but highly curated and very specialized data set was used to greatly increase the value of the larger Big Data set.

Godfrey Harold Hardy Another example where small but highly curated data is crucial for data science, are data sets that contain scientific information, which otherwise is not inherent in the data.

Detection of relevant content with ngram ranking, or text comparison based on cosine vector distance are the most powerful tools to analyze texts even in unfamiliar languages or alphabets.

However, all the quantitative text mining procedures require the text to be preprocessed: All vocabulary with only grammatical function that would not add to the meaning has to be stripped off first.

It is also useful to bring the words to their root form (picture verbs into infinitive, nouns into nominative singular).

fractals as an art-thing where certainly more a fad, not well suited to turn into real art, generative art as such has since then become a strong branch in the Arts.

science is said to be useful if its development tends to accentuate the existing inequalities in the distribution of wealth, or more directly promotes the destruction of human life.

Ethical data is relevant for people’s lives: To control traffic, to make agriculture more sustainable, to supply energy, to help plan cities and administer the states.

Thus, it does not come as a surprise that we start to see the development of the communication platforms that are genuinely meant to support and at the same time to experiment with new forms of political participation, like Proxy-Voting or Liquid Democracy, which had been hardly conceivable without the infrastructure of the Web.

Since these new forms of presenting, debating, and voting for policies have been occurring just recently, we can expect that many other varieties will appear, new concepts to translate the internet paradigm into social decision making.

Nonetheless, I already see that using data for quantitative simulation is a good approach to approximate the complex dynamics of future data-driven political decision-making.

Politics as defined by Aristotle means to have the freedom to make decisions based on ethics and beliefs, and not driven by necessities, the latter is what he calls economics.

Godfrey Harold Hardy Now returning to Hardy’s quote from the beginning, when I was studying mathematics, I was puzzled by the strange aestheticism that many mathematicians would force upon their train of thoughts.

Kepler could dismiss the simple heliocentric model because Tycho Brahe had measured the movements of the planets to such accuracy that the model of circular orbits could no longer be maintained.

Einstein’s Special Theory of Relativity relies on the data of Michelson and Morley who had shown that light would travel at constant speed, no matter the angle to the direction of our earth’s travel around the sun it was measured.

Finally, while mathematics is turning partially into syntax, the core of physics at the same time unfolds in the strange blossoms of the most beautiful mathematics imaginable.

the entirety of the cosmos, and quantum physics on the smallest scale lies the alien world of black holes, string theory, and quantum gravity.

The scale of these phenomena, the fabric of space-time is likely defined by relating Planck’s constant to Newton’s constant and the speed of light is so unimaginably small –

However, I hope that we will see many examples of valuable data, of data that helps people, that creates experiences unseen, and that opens the doors to new worlds of our knowledge and imagination.

Slow Data

We were concerned about the strange dichotomy by which people separated old media from new media to make their point about quality, ethics, and aesthetics.

Just like people were scoffing social media to be just doodles, scribbling, or worse, I now see people scornfully raising their eyebrows about the lack of structure, missing consistency, and other alleged flaws they imagine Big Data to carry.

Again what these people see, is just an evil new vice swamped over their mature businesses by unseasoned startups, however insanely well funded.

It was wrong in the 90s when the web started, it was wrong again in the 2000s regarding social media, and it will not become right this time.

The traditional concept of media becomes more and more directly intertwined with data, with data storytelling, data journalism, and their likes, indirectly because search, targeted advertising, content filtering, and other predictive technologies increasingly influence what we will find presented as media content.

The survey was done face to face, based on a cautiously drawn sample of 10.000 people per year.

my employer was also involved in direct marketing with a huge database of addresses, call centers, and logistics, we developed a method to use the highly curated market survey with its rather small sample to calibrate and enhance the “dirty”

Television ratings are measured by expensive panels in most markets, usually run and funded by joint industry committees like BARB in the UK or AGF in Germany.

We again found a way to infuse the TV panel data into the online data and could by that calculate the probabilities that the owner of a certain cookie would have had contact with a certain advertising campaign on TV or not.

So again, a small but highly curated and very specialized data set was used to greatly increase the value of the larger Big Data set.

Godfrey Harold Hardy Another example where small but highly curated data is crucial for data science, are data sets that contain scientific information, which otherwise is not inherent in the data.

Detection of relevant content with ngram ranking, or text comparison based on cosine vector distance are the most powerful tools to analyze texts even in unfamiliar languages or alphabets.

However, all the quantitative text mining procedures require the text to be preprocessed: All vocabulary with only grammatical function that would not add to the meaning has to be stripped off first.

It is also useful to bring the words to their root form (picture verbs into infinitive, nouns into nominative singular).

fractals as an art-thing where certainly more a fad, not well suited to turn into real art, generative art as such has since then become a strong branch in the Arts.

science is said to be useful if its development tends to accentuate the existing inequalities in the distribution of wealth, or more directly promotes the destruction of human life.

Ethical data is relevant for people’s lives: To control traffic, to make agriculture more sustainable, to supply energy, to help plan cities and administer the states.

Thus, it does not come as a surprise that we start to see the development of the communication platforms that are genuinely meant to support and at the same time to experiment with new forms of political participation, like Proxy-Voting or Liquid Democracy, which had been hardly conceivable without the infrastructure of the Web.

Since these new forms of presenting, debating, and voting for policies have been occurring just recently, we can expect that many other varieties will appear, new concepts to translate the internet paradigm into social decision making.

Nonetheless, I already see that using data for quantitative simulation is a good approach to approximate the complex dynamics of future data-driven political decision-making.

Politics as defined by Aristotle means to have the freedom to make decisions based on ethics and beliefs, and not driven by necessities, the latter is what he calls economics.

Godfrey Harold Hardy Now returning to Hardy’s quote from the beginning, when I was studying mathematics, I was puzzled by the strange aestheticism that many mathematicians would force upon their train of thoughts.

Kepler could dismiss the simple heliocentric model because Tycho Brahe had measured the movements of the planets to such accuracy that the model of circular orbits could no longer be maintained.

Einstein’s Special Theory of Relativity relies on the data of Michelson and Morley who had shown that light would travel at constant speed, no matter the angle to the direction of our earth’s travel around the sun it was measured.

Finally, while mathematics is turning partially into syntax, the core of physics at the same time unfolds in the strange blossoms of the most beautiful mathematics imaginable.

the entirety of the cosmos, and quantum physics on the smallest scale lies the alien world of black holes, string theory, and quantum gravity.

The scale of these phenomena, the fabric of space-time is likely defined by relating Planck’s constant to Newton’s constant and the speed of light is so unimaginably small –

However, I hope that we will see many examples of valuable data, of data that helps people, that creates experiences unseen, and that opens the doors to new worlds of our knowledge and imagination.

Occupational employment projections to 2022

Total employment in the U.S. economy is projected to grow to 161 million, or 10.8 percent, over the 2012–2022 decade and add 15.6 million jobs to the 2012 employment level of 145.4 million.

In addition to projecting growth, BLS projects the number of job openings that will stem from the need to replace workers who change occupations or leave the labor force and tracks the typical level of education that is needed for entry-level positions in each occupation.

Subsequent sections provide more detail of the projections, including information about drivers of occupational growth and decline, employment by education, and growth or decline within each of 22 major occupational groups.

Additional information about occupations may be found in the Occupational Outlook Handbook.1 The Handbook contains 334 occupational profiles with information on typical job duties, work environment, education, training, licensure requirements, median pay, and the job outlook.

The projections process begins with high-level labor force and macroeconomic projections, makes use of an input–output framework to convert final demand into industry output, and ends with detailed projections that are released for 818 detailed occupations in 329 detailed industries.2 The Employment Projections program’s methodology page includes a detailed recounting of the entire process, including the final occupational-projections step.3 Current projections data cover the decade from 2012 to 2022.

Industry employment is allocated among occupations on the basis of distributions from the Occupational Employment Statistics (OES) survey.4 Projected occupational employment is based on projected industry employment.5 BLS projections are a measure of how employment in industries and occupations grow if the economy were to operate at its full potential a decade from now.

In addition to projecting occupational growth—that is, the number of new jobs expected—BLS provides estimates of the number of jobs that will need to be filled in each occupation as workers change occupations, retire, or leave the labor force and need to be replaced.

These projections of job openings from replacement needs, when combined with projected job openings from occupational growth, provide a more complete picture of the opportunities jobseekers will encounter in the coming decade than is provided by projected employment alone.

So if an occupation is projected to gain 1,000 new jobs, and 2,000 people who currently work in the occupation are expected to leave it over the next 10 years, then the total number of positions projected to be available to jobseekers is the sum of the two sources of openings, or 3,000.

Across the economy as a whole, job openings from replacement needs are projected to account for about twice as many openings as those from growth.6 This means that 2 out of every 3 job openings are expected to be for replacing workers who leave an occupation.

These education and training assignments are based on a review of available data, interviews with occupational experts and people who work in an occupation, and reading of specific job postings.

The education and training discussion in this article focuses mainly on the education category assignments for entering an occupation.7 The education categories BLS assigns to occupations are Education assignments provide insight on the formal education typical of entry-level jobs in a field and how they compare with other similar occupations.

The projected employment growth rates for these hospitals and the nurses working in them from 2012 to 2022 are 15.2 percent for private general medical and surgical hospitals and 16.6 percent for registered nurses working in those hospitals.

The projected employment decline from 2012 to 2022 is 21.5 percent for textile product mills and 21.2 percent for sewing machine operators in textile product mills.

Paralegals and legal assistants are projected to handle more job responsibilities that were previously assigned to other legal support staff, causing this occupation to have expected growth that is more than twice as fast as that of the legal services industry.

The faster growth is due in part to the changing job responsibilities that will cause paralegals and legal assistants, who accounted for 17.7 percent of the legal services industry in 2012, to account for 19.9 percent of the industry in 2022.

The second measure, percent change, is the relative number of jobs projected to be gained or lost and is calculated by dividing the projected numeric change by base-year employment.

In contrast, the nursing assistants occupation is much larger, and the same projected growth rate in that occupation is expected to result in 312,200 new jobs added to the economy by 2022.

With base-year employment of 1.4 million jobs, the maids and housekeeping cleaners occupation is projected to grow 12.8 percent during the 10-year period ending in 2022.

These occupations’ starting employment levels were unusually low, resulting in higher growth rates than historically seen in these occupations as they return to long-term trends by 2022.

In 2012, 49.0 million jobs were in occupations that typically require at least some postsecondary education for entry—education beyond a high school diploma or equivalent—compared with 96.4 million that require a high school diploma or less.

Although a growing proportion of jobs is projected to require a postsecondary education, occupations that typically require no more than a high school diploma or equivalent are expected to add more jobs, 8.8 million, than the 6.8 million jobs projected for those requiring at least some postsecondary education.

Although occupations requiring a high school diploma or less are projected to add more new jobs, the following table for wage and salary workers shows that these occupations usually have lower wages than do postsecondary occupations.

This growth is largely due to the concentration of these occupations in the fast-growing healthcare and social assistance industry, which is projected to add a combined 255,000 of the 448,500 new jobs in occupations requiring a master’s degree.

Another major area of projected growth for people with a high school diploma or equivalent is construction, which is expected to add almost 1 million jobs as construction regains jobs lost during the 2007–2009 recession.

This section discusses each of the 22 SOC major occupational groups and includes projected employment change, the factors expected to drive the change, information about the education that is typically needed to enter occupations within the group, and wage data.

Jobs in 2012: 8,861,500 Projected jobs in 2022: 9,498,000 Numeric change: 636,600 Percent change: 7.2 percent (slower than average) Job openings: 2,586,700 Large employers in 2012: ·Manufacturing: 681,400 ·Education: 661,500 General and operations managers is projected to add 244,100 new jobs over the 2012–2022 period, accounting for more than one-third of new jobs in management occupations.

However, the number of management jobs that typically require a high school diploma is expected to decrease over the decade, primarily because of the projected decline in employment of farmers, ranchers, and other agricultural managers.

Jobs in 2012: 7,167,600 Projected jobs in 2022: 8,065,700 Numeric change: 898,100 Percent change: 12.5 percent (as fast as average) Job openings: 2,351,500 Large employers in 2012: ·

Government: 1,050,000 The business and financial operations occupations group includes business operations specialists such as human resources specialists, fundraisers, and market research analysts, and financial specialists such as financial analysts, credit counselors, and tax preparers.

While both business operations specialists and financial specialists are projected to grow about as fast as average, business operations specialists is much larger and will account for nearly two-thirds of the 898,100 jobs added.

In 2012, about 3 in 4 jobs in business and financial operations were in occupations that typically require a bachelor’s degree for entry, but the faster growth rate of these occupations requiring a 4-year degree means they are projected to account for 87.6 percent of new business and financial operations jobs.

Jobs in 2012: 3,814,700 Projected jobs in 2022: 4,500,500 Numeric change: 685,800 Percent change: 18.0 percent (faster than average) Job openings: 1,308,500 Large employers in 2012: ·

Although projected growth for information security analysts, at 27,400 new jobs, is smaller than for software developers and programmers, the rate of growth for information security analysts is expected to be 36.5 percent, making this the fastest growing occupation in this group.

Through 2022, more than 3 in 4 new jobs occurring in this group are projected to be in occupations that typically require at least a bachelor’s degree, with the fastest projected growth among occupations that need a master’s degree.

(See table 5.) The median annual wage for computer and mathematical occupations in May 2012 was $76,270, more than twice the median annual wage for all wage and salary workers of $34,750 and the second highest of any major occupational group.

All of the occupations in this group pay above the median wage for all occupations, and two occupations, computer and information research scientists and mathematicians, had median wages of more than $100,000 per year.

Jobs in 2012: 2,474,500 Projected jobs in 2022: 2,654,000 Numeric change: 179,600 Percent change: 7.3 percent (slower than average) Job openings: 763,900 Large employers in 2012: ·

Occupations that typically require a bachelor’s degree accounted for about 7 out of 10 jobs in 2012, but they will account for more than 9 out of 10 projected new architectural and engineering jobs.

In May 2012, the four highest paying occupations in this group were all engineering jobs that typically require a bachelor’s degree: petroleum engineers ($130,280), nuclear engineers ($104,270), aerospace engineers ($103,720), and computer hardware engineers ($100,920).

Jobs in 2012: 1,249,100 Projected jobs in 2022: 1,374,800 Numeric change: 125,700 Percent change: 10.1 percent (as fast as average) Job openings: 488,200 Large employers in 2012: ·

Education: 208,900 All three scientist groups—life scientists, physical scientists, and social scientists and related workers—are expected to grow at an about-average rate over the 2012–2022 period.

Nearly 4 in 5 new jobs created among the life, physical, and social services occupations group will be in occupations that typically require a bachelor’s degree or higher, and more than 2 in 5 will be at the graduate degree level.

Jobs in 2012: 2,374,700 Projected jobs in 2022: 2,783,400 Numeric change: 408,800 Percent change: 17.2 percent (faster than average) Job openings: 962,900 Large employers in 2012: ·

Religious, grantmaking, civic, professional, and similar organizations: 496,500 Community and social services occupations consist of two groups: (1) counselors, social workers, and other community and social service specialists and (2) religious workers.

Strong growth will be led by substance abuse and behavioral disorder counselors at 31.4 percent, marriage and family therapists at 30.6 percent, and mental health counselors at 28.5 percent.

Jobs in 2012: 1,247,000 Projected jobs in 2022: 1,379,900 Numeric change: 132,900 Percent change: 10.7 percent (as fast as average) Job openings: 333,800 Large employers in 2012: ·

This faster growth is expected among paralegals and legal assistants as many law firms are shifting some duties that were traditionally performed by lawyers to these workers in an effort to reduce costs.

The median annual wage for legal occupations in May 2012 was $75,270, with every occupation in this group earning more than the median annual wage for all wage and salary workers ($34,750).

Jobs in 2012: 9,115,900 Projected jobs in 2022: 10,131,700 Numeric change: 1,015,800 Percent change: 11.1 percent (as fast as average) Job openings: 2,896,900 Large employers in 2012: ·

Health care and social assistance: 484,700 Over the next decade, employment of postsecondary teachers is expected to increase 16.6 percent, while employment of preschool, primary, secondary, and special education school teachers is projected to increase 10.4 percent.

Nearly half of the new education, training, and library jobs created will be in occupations that typically require a bachelor’s degree, and 95.7 percent of new jobs are projected to be in occupations that require some form of postsecondary education.

Ambulatory health care services: 2,340,900 Healthcare practitioners and technical occupations are projected to add more than 1.7 million new jobs from 2012 to 2022, the most of any major occupational group.

Unlike most occupational groups, more than half of new job openings for healthcare practitioners will be new jobs, rather than jobs arising from the need to replace workers who change occupations or leave the labor force.

Although registered nurses is projected to grow 19.4 percent, nearly double the 10.8-percent rate of growth projected for total employment, this large occupation is expected to grow at a slower rate than the other, smaller and often specialized nursing occupations.

Education requirements vary by occupation within this group, with 20 occupations typically requiring a doctoral or professional degree and 5 typically requiring a high school diploma.

(See table 12.) While all education categories are expected to experience employment growth, about 45.8 percent of new jobs in the group over the 2012–2022 decade will be found in occupations that typically require an associate’s degree.

Nursing and residential care facilities: 1,195,100 Employment of healthcare support occupations is projected to increase 28.1 percent from 2012 to 2022, the fastest growth of any major occupational group.

Home health aides is a major driver of this fast growth, as the occupation is projected to grow 48.5 percent and account for more than one-third of the jobs added within this occupational group between 2012 and 2022.

Of the 17 occupations in this group, the two highest earners were in occupations that typically require an associate’s degree: occupational therapy assistants had a median wage of $53,240, and physical therapist assistants had a median wage of $52,160.

Most new protective service jobs will be created in local government and investigation and security services, which are the two largest employers and are projected to account for 65.2 percent of new security guard jobs.

(See table 15.) Food preparation and serving, with its nearly 1 million new jobs, is the occupational group with the largest number of jobs being created for people without a high school diploma.

Self-employed workers: 718,400 Building and grounds cleaning and maintenance occupations includes janitors and cleaners as well as maids and housekeeping cleaners, both of which are expected to be in the top 20 occupations in terms of the number of jobs created.

The number of self-employed building and grounds cleaning and maintenance workers is expected to grow 12.4 percent, slightly slower than the 15.4 percent for wage and salary workers in services to buildings and dwellings.

Wholesale trade: 1,521,800 Sales and related workers, the second largest occupational group, are found in nearly every industry, although almost two-thirds are employed in the retail and wholesale trade industries.

Sales and related occupations included 4 of the 20 largest occupations in 2012, including the 2 largest, retail salespersons, with 4.4 million workers, and cashiers, with 3.3 million workers.

The other sales occupations among the 20 largest occupations are first-line supervisors of retail sales workers (1.6 million) and sales representatives, wholesale and manufacturing, except technical and scientific products (1.5 million).

This group of occupations is expected to grow slower than average, at 6.8 percent, but will still add 1.5 million new jobs by 2022, the second most behind healthcare practitioners and technical occupations.

Of the 20 occupations expected to add the most new jobs, 5 are office and administrative support occupations: Because this occupational group is growing slowly, the large majority of job openings, about 4.9 million, will result from employers replacing workers who change occupations or leave the labor force.

The three office and administrative support occupations with the highest median wages were postal service workers: postal service mail carriers ($56,490), postal service clerks ($53,090), and postal service mail sorters, processors, and processing machine operators ($53,090).

About 8 out of 9 jobs in farming, fishing, and forestry in 2012 were in occupations that did not typically require a high school education, while most of the rest require a high school diploma or equivalent.

In spite of the 531,200 additional jobs projected, opportunities in installation, maintenance, and repair largely will result from the need to replace workers who change occupations or leave the labor force.

(See table 22.) The fastest growing occupation within installation, maintenance, and repair is medical equipment repairers, which will add 12,800 jobs that typically require an associate’s degree.

The two highest paid occupations were electrical and electronics repairers, powerhouse, substation, and relay workers, with a median wage of $68,810 and electrical power-line installers and repairers, with a median wage of $63,250.

(See table 23.) The three occupations requiring a postsecondary education—semiconductor processors, first-line supervisors of production and operating workers, and prepress technicians and workers—are all projected to have declining employment.

The three highest paying jobs were in power production: nuclear power reactor operators ($74,990), power distributors and dispatchers ($71,690), and power plant operators ($66,130).

Over half of new transportation and material moving jobs are projected to be in the occupations laborers and freight, stock, and material movers, hand and heavy and tractor-trailer truck drivers.

Personal care aides differ from home health aides in that home health aides may provide some basic medical services, while personal care aides cannot provide any medical services.

Because projections for many of the construction occupations include recovery from lower employment levels resulting from the 2007–2009 recession, several of these occupations are expected to grow much faster than average from 2012 to 2022.

The projected loss of 139,100 postal jobs is due to more reliance on email and online bill pay services and to technological advances that allow for automatic mail sorting.

Mathematics

Mathematics (from Greek μάθημα máthēma, 'knowledge, study, learning') is the study of such topics as quantity,[1] structure,[2] space,[1] and change.[3][4][5] It has no generally accepted definition.[6][7] Mathematicians seek and use patterns[8][9] to formulate new conjectures;

Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.[10] Galileo Galilei (1564–1642) said, 'The universe cannot be read until we have learned the language and become familiar with the characters in which it is written.

As evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have also recognized how to count abstract quantities, like time – days, seasons, years.[18] Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic, algebra and geometry for taxation and other financial calculations, for building and construction, and for astronomy.[19] The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns and the recording of time.

Numeracy pre-dated writing and numeral systems have been many and diverse, with the first known written numerals created by Egyptians in Middle Kingdom texts such as the Rhind Mathematical Papyrus.[citation needed] Between 600 and 300 BC the Ancient Greeks began a systematic study of mathematics in its own right with Greek mathematics.[20] During the Golden Age of Islam, especially during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics: most of them include the contributions from Persian mathematicians such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī.

For example, Saint Augustine's warning that Christians should beware of mathematici, meaning astrologers, is sometimes mistranslated as a condemnation of mathematicians.[24] The apparent plural form in English, like the French plural form les mathématiques (and the less commonly used singular derivative la mathématique), goes back to the Latin neuter plural mathematica (Cicero), based on the Greek plural τα μαθηματικά (ta mathēmatiká), used by Aristotle (384–322 BC), and meaning roughly 'all things mathematical';

It is often shortened to maths or, in North America, math.[26] Aristotle defined mathematics as 'the science of quantity', and this definition prevailed until the 18th century.[27] Starting in the 19th century, when the study of mathematics increased in rigor and began to address abstract topics such as group theory and projective geometry, which have no clear-cut relation to quantity and measurement, mathematicians and philosophers began to propose a variety of new definitions.[28] Some of these definitions emphasize the deductive character of much of mathematics, some emphasize its abstractness, some emphasize certain topics within mathematics.

Today, no consensus on the definition of mathematics prevails, even among professionals.[6] There is not even consensus on whether mathematics is an art or a science.[7] A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable.[6] Some just say, 'Mathematics is what mathematicians do.'[6] Three leading types of definition of mathematics are called logicist, intuitionist, and formalist, each reflecting a different philosophical school of thought.[29] All have severe problems, none has widespread acceptance, and no reconciliation seems possible.[29] An early definition of mathematics in terms of logic was Benjamin Peirce's 'the science that draws necessary conclusions' (1870).[30] In the Principia Mathematica, Bertrand Russell and Alfred North Whitehead advanced the philosophical program known as logicism, and attempted to prove that all mathematical concepts, statements, and principles can be defined and proved entirely in terms of symbolic logic.

and as far as they are certain, they do not refer to reality.'[15] Many philosophers believe that mathematics is not experimentally falsifiable, and thus not a science according to the definition of Karl Popper.[34] However, in the 1930s Gödel's incompleteness theorems convinced many mathematicians[who?] that mathematics cannot be reduced to logic alone, and Karl Popper concluded that 'most mathematical theories are, like those of physics and biology, hypothetico-deductive: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently.'[35] Other thinkers, notably Imre Lakatos, have applied a version of falsificationism to mathematics itself.[36][37] An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality.

For example, the physicist Richard Feynman invented the path integral formulation of quantum mechanics using a combination of mathematical reasoning and physical insight, and today's string theory, a still-developing scientific theory which attempts to unify the four fundamental forces of nature, continues to inspire new mathematics.[39] Some mathematics is relevant only in the area that inspired it, and is applied to solve further problems in that area.

This remarkable fact, that even the 'purest' mathematics often turns out to have practical applications, is what Eugene Wigner has called 'the unreasonable effectiveness of mathematics'.[40] As in most areas of study, the explosion of knowledge in the scientific age has led to specialization: there are now hundreds of specialized areas in mathematics and the latest Mathematics Subject Classification runs to 46 pages.[41] Several areas of applied mathematics have merged with related traditions outside of mathematics and become disciplines in their own right, including statistics, operations research, and computer science.

He identified criteria such as significance, unexpectedness, inevitability, and economy as factors that contribute to a mathematical aesthetic.[42] Mathematicians often strive to find proofs that are particularly elegant, proofs from 'The Book' of God according to Paul Erdős.[43][44] The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions.

According to Barbara Oakley, this can be attributed to the fact that mathematical ideas are both more abstract and more encrypted than those of natural language.[47] Unlike natural language, where people can often equate a word (such as cow) with the physical object it corresponds to, mathematical symbols are abstract, lacking any physical analog.[48] Mathematical symbols are also more highly encrypted than regular words, meaning a single symbol can encode a number of different operations or ideas.[49] Mathematical language can be difficult to understand for beginners because even common terms, such as or and only, have a more precise meaning than they have in everyday speech, and other terms such as open and field refer to specific mathematical ideas, not covered by their laymen's meanings.

This is to avoid mistaken 'theorems', based on fallible intuitions, of which many instances have occurred in the history of the subject.[b] The level of rigor expected in mathematics has varied over time: the Greeks expected detailed arguments, but at the time of Isaac Newton the methods employed were less rigorous.

Since large computations are hard to verify, such proofs may not be sufficiently rigorous.[50] Axioms in traditional thought were 'self-evident truths', but that conception is problematic.[51] At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system.

Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.[52] Mathematics can, broadly speaking, be subdivided into the study of quantity, structure, space, and change (i.e.

In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations), to the empirical mathematics of the various sciences (applied mathematics), and more recently to the rigorous study of uncertainty.

As such, it is home to Gödel's incompleteness theorems which (informally) imply that any effective formal system that contains basic arithmetic, if sound (meaning that all theorems that can be proved are true), is necessarily incomplete (meaning that there are true theorems which cannot be proved in that system).

the estimated models and consequential predictions should be tested on new data.[c] Statistical theory studies decision problems such as minimizing the risk (expected loss) of a statistical action, such as using a procedure in, for example, parameter estimation, hypothesis testing, and selecting the best.

In these traditional areas of mathematical statistics, a statistical-decision problem is formulated by minimizing an objective function, like expected loss or cost, under specific constraints: For example, designing a survey often involves minimizing the cost of estimating a population mean with a given level of confidence.[56] Because of its use of optimization, the mathematical theory of statistics shares concerns with other decision sciences, such as operations research, control theory, and mathematical economics.[57] Computational mathematics proposes and studies methods for solving mathematical problems that are typically too large for human numerical capacity.

Brain Tricks - This Is How Your Brain Works

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