Scientific thinking is much more than math. Quantification is often necessary but in no way sufficient condition for a scientific discover.
Human values are the ones that can relate, connect different axiomatic systems, and make them work together. It is the framework of human values that dictates which selection of theorems from different axiomatic systems we will make.
Math and logic do not imply right away that scientific thinking takes place. Math and logic will equally well serve any non scientific thinking, like a dogmatic teaching. It is the assumptions that differentiate scientific from non-scientific direction of thinking. Scientific assumptions are the ones that wins. Math and logic just serve them well.
Mathematics is defined not by objects it counts, nor by reasons or logic why those objects are counted, but with concepts used to define mathematics axioms and to define proofs of mathematical theorems.
In real world mathematics application, it is you who guides quantification. Guided quantification is the core of free applied math thinking.
Numbers have properties of their own, independent of anything else. Hence, real world can only specify starting points for calculations, and perhaps the sequence of numerical operations, but it can not influence, or change, in any way, these intrinsic properties of numbers within the mathematical system. And, on the other hand, a mathematical system, or numbers' properties, can not tell to which particular real world example they may be applied or be relevant to.
Allow complete creative freedom to play with initial assumptions then use strict logic to find true consequences.
It is the interplay of imaginative assumptions that lead to discovery. Only after the nested assumptions interplay logic should kick in.
Feel free to assume, propose anything you can imagine and only after that use logic and maybe math, to explore validity of your assumptions.
Logic and (possibly) some quantification should only be good servants to your uninhibited, creative, free thinking and assumptions play.
Logic can tell you if your assumption, premise is wrong. But logic, then, cannot tell you what would be the correct assumption or premise.
You use logic to TEST your assumptions. Logic hardly can help you to discover the correct assumption at the first place.
Behind all math initial premises and starting numbers may be a real world story explaining why the premises are there.
You can assume anything then apply correct logic. Only consequences will prove if your assumptions were true/correct.
Intuition, common sense, and experience probably served as the first quantitative tools for price setting. - "Energy Risk" by D. Pilipovic.
A mathematical model of a process is a set of premises driven by world extraneous to math yet they can be derived directly from math axioms
Force, energy, speed, momentum, inertia are not part of math. If they were, then math theorems will be proved using them. It's not the case.
What you may have to tell your primary school students when explaining math and a concept of a number - http://t.co/PkI7VdQ
Math can't tell real world from fictional one! Look! If Harry Potter flies 10 m/s how many meters he will advance after flying 5 seconds?
The very moment you said "as many apples as oranges" you defined the concept of a pure number. Moreover, no need to name the number.
More magic than in a new Harry Potter movie - take a journey from real world math applications to pure math and back http://t.co/PkI7VdQ
Take a thought journey from real world math applications to pure math and back - and have that "wow!" moment http://t.co/PkI7VdQ
Labeling a number generation as "random" is not part of mathematics. It's an attempt to describe some number selection by ordinary language.
Everyone, especially primary and secondary school math teachers, may consider reading Paul Lockhart's "A Mathematician's Lament".
To develop all mathematics you do not need a single other science. No need for physics, quantum physics, genetics, quantum chemistry,...
Whole math can be developed inside heads of mathematicians, without any pencil, paper, given they have enough big memory.
Math for [insert the field of application here]
How math can be applied to so many different fields and how we can use math in real life http://t.co/17QRRxV
Math is not about following directions, it's about making new directions. - Paul Lockhart, "A Mathematician's Lament"..
If you asked yourself Can I revisit my math from primary and secondary school and finally understand what is it about? http://t.co/17QRRxV
In order to even begin to count something, you have to know legal system, exchange rules, physics laws, economic laws, how to measure, ..
While membership to a set is not defined within math, it has exotic names outside it: transaction, ownership, buy, sell, exchange, measure..
The very method we quantify something (like measurement) is not a part of mathematics! Set and membership to a set are undefined within math.
Why would you use real world example for a math concept when you can derive it directly from axioms? Both approaches should be demonstrated.
Logical truth values entered the irrational world of emotionality with the statement 'true love'.
Logical truth values entered the emotional world of irrationality with the statement 'true love'.
To me, two core concepts to know for aircraft design are combustion reaction energies (bond energies, fuel, oxygen) and airfoil physics.
More than 20 motivational examples to introduce rational numbers to kids. Pirates, scuba diving, text messaging, pets ..http://t.co/j7N6QAc
You don't deny student's hate towards math, nor try to change it directly. Instead, you accept it and integrate their hate in math puzzles.
Field of math application shapes the math development in the same way the landscape shapes and guides the roads going through them.
While pure math is like building roads just to build them, applied math is like building roads through landscapes you want to go through.
Math lectures sometimes look to students like putting misleading, cluttered, over-detailed traffic signs on easy to use roads and highways.
It is way better to first explain math concepts in terms of required operations and sets of numbers they apply to, and only after that.....to label them with historical, outdated, misleading, confusing names that contribute nothing to the concept's definition.
It is way better to first explain math concepts in terms of required operations and sets of numbers they apply to, and only after that...
To calculate racing track length you need limit concept. For racing car fuel usage you need rational numbers. Teach both at the same time.
Knowing how to implement a business rule in C++ can make you a living. Knowing what business rule you will implement can make you a fortune.
Math and physics concepts should be think of only by the ways they are calculated. Ordinary language names are confusing, often misleading.
We talk about selling, buying, getting, sending energy, but energy is not an object. It's a calculated value from measuring mass, time, distance.
Internal combustion engine principle for beginners. Combustion is, in essence, an electrical reaction. - http://t.co/qxk6PF5
Different contexts will give different meanings for the same sentence. #semiotics
From real world math applications to pure math and back! http://t.co/PkI7VdQ
For many students, math looks like a maze. Students are lost in one area of maze while real fun with math is in the other part of maze.
High school math programs are like labyrinth for students. Students should get a hot balloon and take a bird view look where they are.
Student hates math? Integrate his resistance points, reasoning, into the math problems. Student will realize that he dictates quantification.
Student hates math? Milton Erickson wrote about utilizing person's resistance to a subject to, actually, achieve goal person is resisting to.
Motivation & Math for students who hate math. Ask what is the percentage of time they would do math compared to what they like to do daily.
Here is one motivational math example. Ask your students in how many ways they HATE math.
Awareness - making visible new axiomatic system not known to exist before. Truths presented in order to take action i.e. derive theorems.
Aeronautical Engineering, Aerodynamics, Aircraft Design References http://t.co/nce2gAQ #aviation #aircraftdesign
Math and magazine design. Designer has to know how to fit actor's surface area to the page dimensions. Lower and upper bound...
Emotions and math? He wrote very emotional Acknowledgment in his new book on Advanced Calculus.
Applied math can not be solely credited to the achievements in the applied field. Field dictates what, when, why is to be calculated.
Grammar can not be credited for a beauty of a literary work. Many stupid things are said using perfect grammar, and vice versa.
Saying that math is backbone of things is like saying that grammar is backbone of every single novel, science paper, literary work created.
From ZFC axioms you can create all math. Yet, it is the world extraneous to math (often non-axiomatized) that dictates math development.
Axiomatizing one system strangely isolate reasoning world outside of that system, thus hiding the motivation logic for system's theorems.
Once you hear word "axioms" (in any system), look for logic extraneous to that system. That's where motivation for theorems is coming from.
Motivational Math. Introducing Math Through Car Racing Concepts. Stay tuned for a new, exciting article! http://t.co/4ibp7w8
Even if you manage to, somehow, quantify right and wrong, you still need their firm definitions to be sure you are not quantifying..apples.
Whenever, in a physics textbook, you see phrase "arbitrary" (magnetic, electric field...), it's a placeholder for a design driven value.
Many math proofs start with "Let's assume...". But, wait! Can you explain where that starting assumption comes from?? #mathed
Assumptions coming from non-axiomatized fields (physics, economics, finance) can wreak havoc when used in a strict axiomatic system (math).
If mathematicians are so proud of their axiomatic approach, why they deal at all with applied math in non axiomatized fields??
How we are allowed at all to go from non axiomatic world, physics, economics,finance, to so strictly defined axiomatic world of mathematics?
You don't learn math, then apply it. Newton didn't learn calculus first (there was none, he invented it!), then applied it to physics.
While a proof, in math, has to be very logical and precise, the genesis of it is usually described as art or unexplainable inspiration.
Strict, precise Newton's Law of Gravitation does not prevent you to enter into it a completely random number for mass or distance.
Length of a musical note as a mathematical property has way less significance than emotional perception of the sound (note) of that length.
Surface and volume integrals should be explained using tattoos. They are a good example for arbitrary surface and ink volume calculation.
Things You Always Wanted to Know About Math * But Were Afraid to Ask http://t.co/PkI7VdQ
From Real World Math Applications to Pure Math and Back http://t.co/PkI7VdQ
You spent all your school years dealing with continuous functions only to hear after they are very small number of all functions of interest.
When I hear "it's just continuous function"..bad! No, it's not "just"! It took hundreds years to come up with the definition of continuity.
I would ban phrases "it's simply...(that)", "it's just...(that)" in math. Please leave to student to judge is it complicated or simple.
It's useful to quantify, but, relationships that are quantified are OFTEN quite non-mathematical.
A math proof, once you do it, is probably the only thing in math you are not obliged to explain your teacher how you did it.
Logic used in math proofs is the same as logic used in law. But, in law, axioms are fluid, relative, changing. Law is doing best it can!
Using logic or not, people are making decisions each and every day..
Can you master math? I think, yes! http://t.co/1J31IRE
Explaining essential ideas of mathematics. Talk about Applied Mathematics, Mathematics and Real World, Math Education. http://t.co/aVTTz4v
Puzzled with math graphs? Wondered why they use them? Where the graphs come from anyway?? http://t.co/lGLUfHq
Math and Film. "They had tied up all mathematics of plots and substructures and sub-characters." -Johnny Depp, interview, Cineplex Magazine.
Overheard in student cafe: Math text often starts with 'Lets suppose..'. I don't want to suppose anything, especially something THAT complex.
You can quantify and calculate as much as you want, but if you don't think scientifically, mathematics can't help you.
There are many scientific discoveries that has nothing to do with quantification nor math.
Math may be necessary, but definitely it's not sufficient part to make progress in science.
Every proof should be constructed within known and accepted axiomatic system, being it physics, math, economics, law, engineering.
An explosion into unknown..http://explainingmath.blogspot.com
The posts are terrific. They engulf! http://explainingmath.blogspot.com
Usage of a math theorem is in NOT dependent AT ALL on the way theorem is proved. How you use a theorem has nothing to do with its proof.
To understand a math proof is way easier than to make a proof, in the same way it's easier to consume a movie than to make it, or to appreciate an art painting than to make it.
There is no straightforward path how to prove math theorems, as there is no law that can predict what number you will chose right...NOW.
Updated article why graphs are chosen to visually represent quantities in math, physics, economics etc...http://t.co/lGLUfHq
At some point teachers should stop explaining math concepts with real world examples because none of theorems are proved by using apples.
Mathematics can not be Queen of all sciences because you can't start only with math and develop other sciences. Science is there first.
Math without science, i.e. without science to tell WHAT is counted and WHY is just play with numbers (but elegant, logical, often exciting).
At some point teachers should stop explaining math concepts with real world examples becuase none of theorems are proved by using apples.
Even word "RANDOM" does not belong to math. It's outside math as is measurement, observation, guess etc. Math sees only numbers given to it.
How to Teach Your Kids and Yourself to Think More Freely About Math and Real World Math Applications ... http://t.co/1J31IRE
Teachers should clearly explain the difference between lingustic framework within which math tasks are described and pure math itself. #math
Since differential equation specifies only the difference between two quantities, it can not tell you with which quantities to start with.
Math can't tell you what to count. Math deals only with numbers and a result of any math task is a number, and a number only.
Hockey, Physics, Axioms and Where Innovations Come From..http://t.co/ZGwvm6y #hockey #physics #innovation
Imagine creating rules of the game what to do with numbers. Then, new theorems will be dictated by this game. Without game - no theorems.
In math it is YOU who creates territory and then investigate its properties and boundaries.
If you want popular introduction to rational, irrational numbers you may want to read "Essays on the Theory of Numbers" by Richard Dedekind.
How to approach numerical values in a physics formula. How to much better understand and use physics formuls...http://t.co/i2vNn6J #physics
I would strongly recommend "Essays on the Theory of Numbers" by Richard Dedekind. Detailed and non boring introduction to continuity.
Math is not prerequisite for real world applications. Newton did not learn calculus then applied it. He invented calculus!
What would you like to do, what you have a talent for, and what economy, i.e.market is looking for can hardly be all found in one job.
During schooling (don't mix that with education!) best thing you can do is to follow your own ideas and ask, then answer your own questions.
Motivation to Use Graphs in Math, Physics and to Know Arbitrary Surface Area Calculations, http://goo.gl/gPXFE
Quantitative finance, dragons, and math - http://goo.gl/XEDL3
Relationships between dragons lead to math development... http://goo.gl/XEDL3
Math can be motivated by real world or by fictional world. It can also be developed independently from both worlds. Student should know that.
Math can't distinguish between REAL world and FAIRY TALE! Check this. Two dragons ate 53kg of coal each. How much coal they ate together?
You can assign EXACT number to any RANDOM event! :-)
In many cases, relationships (between objects) that define WHAT has to be calculated are far more interesting than calculations themselves.
Value of calculus: when you find something interesting to calculate, it can help! The trick is to find something enjoyable to analyze.
You can understand calculus too! How to Calculate Surface Area of an Arbitrary Shape - Story of Pirate Island http://goo.gl/hbVEY
Math deals ONLY WITH NUMBERS, COUNTS. However, math can be used in real life once you start keeping track WHAT is counted and why.
It is seldom that an airfoil camber line can be expressed in simple geometric or algebraic forms. Important illustration of a math function!
Sure, mathematics can make you think better, especially if YOU set and define ORIGINAL problems, and not only solving what you are told to.
Airplanes, pirates, treasure hunt - how to introduce calculus ideas to primary school students. http://goo.gl/9uaPr
More about Setology and Countology here http://goo.gl/rAPWx #math #mathematics
Mathematics = Setology. A science about sets. :-)
Mathematics = Countology. It's a science about counts. Sort of better than Numerology :-) Count is a required action that gives a number!
How math can be applied to so many different fields and how we can use math in real life http://goo.gl/cJQxs
Students are afraid to PICK a number by themselves. They think each number has to be calculated, obtained in some complicated manner.
Stochastic process PICKS a number. Physical measurement PICKS a number. Picking a (closer and closer) number is ESSENCE of limit definition.
Physical or any process doesn't "generate" numbers. It PICKS numbers. Numbers are already generated, defined inside mathematics.
Graph is an invention of using length as a representation of ANY imaginable measurable quantity or number.
Introducing math function to students: first step should be to let students draw an arbitrary curve and show it represents PAIRED numbers.
"Our education plays a trick with us, leading us to believe things which are not correct." BBC Environment, Geometry, http://goo.gl/BaFKQ
Many are looking for real world examples of math. But, math can't tell what is real and what's not! 5 dragons plus 3 dragons = 8 dragons!
You apply math only AFTER you chose WHAT to count. Hence, choosing WHAT to count and WHY it is counted has nothing to do with math!
Have you ever wanted to know what are the fundamental ideas in calculus? http://goo.gl/aVDbv
But, you don't have to even say "Trust me, I am a lair.". You can just say "I am a lair.". It's already a paradox.
Journey to the Pirate's Island to learn calculus ideas...like a treasure hunt, sort of..http://goo.gl/hbVEY
Have you ever thought what's behind calculus ideas? Maybe this will show just that! http://goo.gl/hbVEY #math #calculus
How to introduce calculus concepts to primary school students: "How to Calculate Surface Area of a Pirate Island" http://goo.gl/hbVEY
After they master basic algebraic operations, primary school students should be encouraged to define new math problems by themselves. #math
Here is my illustration of the Pirates Island at night, which will be used to introduce integration to students, http://goo.gl/BipFE
My new draft post "How to Calculate Surface Area of a Pirate Island" introducing integration to primary school students http://goo.gl/rVY1D
Many students see math, if not whole formal education, as a tunnel from which they have to get out, eventually, to do what they want.
How the rational numbers should be introduced to kids, http://goo.gl/dhUmS #math #rationalnumbers
Once we realize that math deals only with sets and numbers and that math does not need real world for examples, we can accept desire ...mathematicians to explore properties of numbers and their relations, without even thinking is there any "real" world application.
Real world can give math some initial counts, numbers, even sequences of required operations. But that's it. Math takes off by itself after.
While a proof, in math, has to be very logical and precise, the genesis of it is usually described as art or unexplainable inspiration.
Math can't tell you why you added two numbers but once you added them math can tell you what properties they have compared to other numbers.
Limiting process, in mathematics, may not itself lead to exact value, but, it can serve to point to where that value is, or can be.
Irrational numbers cannot be represented as a ratio of two integers? But, they are still infinite sum of ratios of two integers. So......?
Students should be shown that all the other numbers, rational, real, imaginary, transcendent, irrational, are CONSTRUCTED from integers.