ABSTRACT: Mathematics plays a major part in our lives, whether we know it or not. It is used in many ways behinds the scenes and our modern civilizations couldn't get along without it. We will discuss examples, large and small, and also consider questions like why do we have 60 seconds in a minute, 60 minutes in an hour, and 360 degrees around a circle, rather than 10 or 100 in each? Why do we say we have three "square" meals a day or that something was a "square" deal? What is so good about the 10-base number system that we use? (Think about doing division or fractions in Roman numerals!) What is the difference between numbers and what we call them?

- Instructor is Donald R. Snow ( snowd@math.byu.edu
) of Provo and St. George, Utah -- Don's
**Dad, Eldon Stafford Snow, was in the first graduating class of Dixie College, 1913** **Appreciation to Doug Alder for arranging these Dixie State University Colleagues meetings, Marilyn Lamoreaux for the publicity and email lists, and Dave Mortensen for the technical help -- DSU News article at https://news.dixie.edu/2017/10/24/presidents-colleagues-to-host-presentation-on-maths-impact/**- The problem for discussion: What is mathematics, what is it good for, where did it come from, and how does it affect our lives "behind the scenes"?
- 60 secs, 60 mins, 360 degrees around a circle? "historical accident" because ancient Babylonians used a 60-base number system; 360 may have been used since close to 365 days in a year and has many divisors, so easier to do fractional parts
- 24 hours in a day? Ancient Egyptians seem to be the first to use 24 hours in a day, two 12-hour periods, but no one knows why or when this developed
- 7 days in a week? 7 is an important prime number, but no one knows why or when the 7-day week started; Hebrews were using 7 days in a week when Moses wrote the Pentateuch (First 5 books of Bible) about 1000 BC
- Square deals or square meals? Greek's thought each number represented something; 1 was reason; 2 was male; 3 was female; 5 was marriage since 5 = 2 + 3; 4 was 2 + 2 and 2 x 2, so was associated with justice and correctness, hence a square deal was a just deal and a square meal meant a good and complete meal
- Do you invent or discover mathematics? Did Beethoven invent or discover his 5th Symphony? Is mathematics the same everywhere in the universe?
- There is no good definition of mathematics; all leave something out or include too much; one definition is "math deals with numbers and shapes and
"**is a study of patterns** - Brief history of numbers from counting numbers to complex numbers
**Number naming systems used -- influences how we learn and think about numbers -- numeracy**- Many different number systems over the centuries -- groupings, Roman numerals, tally marks
- Our 10-base place-value system allows us to do arithmetic and teach children how to do it
- Our digit names are based on 12s, e.g. one, two, ... , ten, eleven, twelve, then thirteen = three & ten, fourteen = four & ten, etc.
- 10-base place-value system has only been in general use world-wide since about 1200 AD after Fibonacci of Pisa wrote a book showing its advantages in commerce
- Other societies think of numbers different than the way we do
- Some primitive societies count "1, 2, 3, 4, many"; one-to-one correspondence can still tell who has the most of something
- Greeks thought numbers were always measurements, so no zero or negatives, and "completing the square" in algebra was a geometric construction for the Greeks
- Hebrews didn't think of numbers as exact like we do; important to keep in mind when reading the Bible -- I Kings 7:23 and II Chron 4:2 -- round font was 10 cubits across and 30 cubits around, so pi = circumference / diameter = 30 / 10 = 3 for the Hebrews; not exact, but within about 5%
- Mathematicians dream up esoteric mental structures based on undefined terms, axioms and postulates, and then theorems, and then some scientist comes along and says, "Oh, those building blocks describe exactly this real-world thing that I'm studying.", so he/she uses all the theorems and results and they describe things about what they are studying; happens over and over in mathematics -- examples
- Non-Euclidean geometries
- Complex numbers in engineering, computers, and electronics
- Super Particular ratios in music harmonies -- example of my work with George Forsythe at Stanford
- Rates of change (calculus and differential equations) -- Newton's Laws of Motion, biological populations, ecology, economics, banking, finance
- Linear algebra and matrix theory -- quantum mechanics, relativity, political science
- Maps -- projections of a spherical world onto flat maps
- Calendars -- 40-50 different calendars in use in the world today
- Time zones -- started when railroad came across US; so no time zones when Mormon pioneers settled Utah or even St. George; each city had its own time
- Algorithms
- Algorithms are step-by-step procedures, "recipes" -- simpler ones can be done by hand, more complicated ones need computers
- The words "algorithm" and "algebra" come from "al-jabr" which
book by Arabic mathematician "al-Khwārizmī"**means "the reunion of broken parts";** - Neural networks -- being set up on computers, they "learn" by many yes-no examples; can now read handwriting about 95% accurate, facial recognition, self-driving cars
- Big data -- collections of data about each of us from Google, Facebook, Twitter, Instagram, etc. -- computers analyzing it to learn about us
- Power of mathematics is in its abstraction, since the same mathematical structure can describe many different real-world situations; mathematical modeling
- Functions and equations -- Stan Ulam's comment to me at University of Colorado about problems between professors in a Math Dept, lead me to a new approach in combinatorics
- Computer simulations -- numerical descriptions on computers, e.g. weather predictions
- Question to ponder: Why should we expect anything we do mentally to have any relationship at all to the real world?
- Birthday Problem -- What is probability that there are two people here today with the same birthday? Can find out in 2 minutes
- Euler paths and circuits -- trace a diagram without going over same line twice; same problem as finding a route with no backtracking -- example of my paper route in North Hollywood
- Hamilton paths and circuits -- Traveling Salesman Problem -- no solution other than comparing all possible paths and that's only possible for small number of cities
- Famous theorems in math -- Pythagorean Theorem, Fundamental Theorem of Arithmetic, Fundamental Theorem of Calculus, others
- Mathematics in religious scriptures -- Book of Mormon, Alma 11, Nephite monetary system, 1, 2, 4, 7 -- requires fewer coins than our 1, 5, 10 system and is the system used in McBee Keysort Cards
- Infinitely many sizes of infinities
- Many unsolved important math problems --
-- Clay Mathematics Institute offers cash prizes**"Any damn fool can ask a question nobody can answer."** - Mathematicians study abstract structures (mental structures) which are based on logic -- they get ideas or conjectures in various ways, then set out to prove or disprove them with logic
- Structures start with undefined objects, e.g. numbers, points, lines, circles, etc., and axioms or postulates of how those are related, e.g. two points determine a line; then theorems and more definitions and conjectures, then theorems, etc.
- Mathematical structures are called theories, proven sets of results,
**e.g. theory of differential equations, graph theory, matrix theory, theory of integral equations; the word "theory" in mathematics is used differently than in other sciences** - Some mathematical theories can be visualized, others can only be hinted at; some are beautiful and aesthetically pleasing; others are messy and ugly, but still valid
- Greeks started the idea of axioms and postulates -- Euclid's Geometry, textbook from Greece in 300 BC;
**most important textbook ever written; was used for 2000 years** - World-wide total of new mathematics developed doubles about every 10 years; thousands of new mathematical structures and theorems; more than a thousand math journals
- Computer are sometimes used to get conjectures and
**to find counterexamples; one counterexample disproves a conjecture, but even thousands of good examples don't prove a conjecture** - Computers are now being programmed to do logic; also recent advances to form neural networks for "deep learning"; so far no radically new mathematics this way
- An open question in math is, if something has been proven by a computer and a human can't do it, should it still be considered a proof?
- True examples by computer gives you more confidence to keep trying to find a proof, but doesn't prove it
- Wikipedia articles -- history of mathematics, number, Babylonian mathematics, Greek mathematics, Pythagorean Theorem, sexagesimal number system, 12-hour clock, week, month, year, calendar, lists of mathematics topics, lists of mathematics journals, algorithm, music and mathematics, Golden Ratio, art and mathematics
- The World of Mathematics by James R. Newman, 4 vols -- Download free pdf's from Internet Archive -- https://archive.org/ -- https://archive.org/search.php?query=%22The%20World%20Of%20Mathematics%22
- Geometry games on smartphones -- Euclidea -- https://www.euclidea.xyz/ -- and Pythagorea
- Euler's Konigsberg Bridge Problem -- MAA article -- https://www.maa.org/press/periodicals/convergence/leonard-eulers-solution-to-the-konigsberg-bridge-problem
- How Natural is Numeracy? --
-- https://aeon.co/essays/why-do-humans-have-numbers-are-they-cultural-or-innate**Aeon article** **What's Special About This Number -- http://www2.stetson.edu/~efriedma/numbers.html****The Story of Mathematics -- http://www.storyofmathematics.com/story.html**- Annenberg Media -- http://www.learner.org/resources/discipline-math.html -- Interactives
- Quanta Magazine and Scientific American -- https://www.quantamagazine.org/mathematics/ and https://www.scientificamerican.com/section/quanta-magazine/ -- free and online, save articles in POCKET to read later
- Wolfram Alpha -- https://www.wolframalpha.com/ -- can ask questions
- Kahn Academy -- https://www.khanacademy.org/ -- thousands of helpful videos to learn math
- Math is continually growing and will always be needed, so we need to learn how to keep learning and we need more students, including women, in STEM programs (science, technology, engineering, and math)
- Place-value number system was crucial for development of science and technology, but is there a better way still? Also, it makes you wonder what other fields have a better basic way to describe things that we don't know about yet.
- Hope this has given you a little better understanding of what mathematics is and its uses in our world

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