Mathematical Proof for Enthusiasts - What It Is And What It is Not

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With the term mathematical proof we want to indicate a logical proof, i.e. proof using logical inferences, in the field of mathematics. So, it should really be “a proof in the field of mathematics”. Also, we have to assume, and be fully aware, that proof must be “logical” anyway. There are really no illogical proofs. Proof that appears to be obtained (whatever that means) by any other way, other than using rules of logic, is not a proof at all. Assumptions and axioms need no proof. They are starting points and their truth values are assumed right at the start. You have to start from somewhere. If they are wrong assumptions, axioms, the results will show to be wrong. So, you will have to go back and fix your fundamental axioms.
When you have first encountered a need or a task for a mathematical proof, you may have asked yourself "Why do I need to prove that, it's so obvious!?".

We used to think that we need to prove something if it is not clear enough or when there are opposite views on the subject we are debating. Sometimes, things are not so obvious, and again, we need to prove it to some party.

In order to prove something we have to have an agreement which things we consider to be true at the first place, i.e. what are our initial, starting assumptions. That’s where the “debate” most likely will kick in. In most cases, debate is related to an effort to establish some axioms, i.e. initial truths, and only after that some new logical conclusions, or proofs will and can be done.

The major component of a mathematical proof is the domain of mathematical analysis. This domain has to be well established field of mathematics, and mathematics only. The proof is still a demonstration that something is true, but it has to be true within the system of assumptions established in mathematics. The true statement, the proof, has to (logically) follow from already established truths. In other words, when using the phrase "Prove something in math..." it means "Show that it follows from the set of axioms and other theorems (already proved!) in the domain of math..". Which axioms and theorems you will start the proof with is a matter of art, intuition, trial and error, or even true genius. You can not use apples, meters, pears, feelings, emotions, experimental setup, physical measurements, to say that something is true in math, to prove a mathematical theorem, no matter how important or central role those real world objects pr processes had in motivating the development of that part of mathematics. In other words, you can not use real world examples, concepts, things, objects, real world scenarios that, possibly, motivated theorems’ development, in mathematical proofs. Of course, you can use them as some sort of intuitive guidelines which axioms, or theorems, you will use to start the construction of a proof. You can use your intuition, feeling, experience, even emotions, to select starting points of a proof, to chose initial axioms or theorems in the proof steps, which, when combined later, will make a proof. But, you can not say that, intuitively, you know the theorem is true, and use that statement about your intuition, as an argument in a proof. You have to use mathematical axioms, already proved mathematical theorems (and of course logic) to prove the new theorems.

The initial, starting assumptions in mathematics are called fundamental axioms. Then, theorems are proved using these axioms. More theorems are proved by using the axioms and already proven theorems. Usually, it is emphasized that you use logical thinking, logic, to prove theorems. But, that's not sufficient. You have to use logic to prove anything, but what is important in math is that you use logic on mathematical axioms, and not on some assumptions and facts outside mathematics. The focus of your logical steps and logic constructs in mathematical proofs is constrained (but not in any negative way) to mathematical (and not to the other fields’) axioms and theorems.

Feeling that something is "obvious" in mathematics can still be a useful feeling. It can guide you towards new theorems. But, those new theorems still have to be proved using mathematical concepts only, and that has to be done by avoiding the words "obvious" and "intuition"! Stating that something is obvious in a theorem is not a proof.However, using own intuition to construct a proof or to formulate theorem is definitely useful.

Again, proving means to show that the statement is true by demonstrating it follows, by logical rules, from established truths in mathematics, as oppose to established truths and facts in other domains to which mathematics may be applied to.

As another example, we may say, in mathematical analysis, that something is "visually" obvious. Here "visual" is not part of mathematics, and can not be used as a part of the proof, but it can play important role in guiding us what may be true, and how to construct the proof.

Each and every proof in math is a new, uncharted territory. If you like to be artistic, original, to explore unknown, to be creative, then try to construct math proofs.

No one can teach you, i.e. there is no ready to use formula to follow, how to do proofs in mathematics. Math proof is the place where you can show your true, original thoughts.

[ set, set theory, concept of a set, sets in mathematics, real world, applied math, applied mathematics, axioms, math education, math proof, mathematical axioms, mathematical proof, mathematical theorems, mathematics, theorems, tutoring ]

Concept of a Set and of a Number

You can download all the important posts as  PDF book "Unlocking the Secrets of Quantitative Thinking".

For instance, let's take a look at the cars on a highway, apples on a table, coffee cups in a coffee shop, pears in a basket. Without our initiative, our thought action, will, our specific direction of thinking, objects will sit on table or in their space undisturbed and unanalyzed. They are apples, cars, coffee cups, pears. But then, on the other hand, we can think of them in any way we wish. We can think how we feel about them, are they edible, we can think about theory of color, their social value, utility value, psychological impressions they make. We can think of them in any way we want or find interesting or useful, or we can think of them for amusement too. They are objects in the way they are and they need not to be members of any set, i.e. we don’t need to count them.

Now, imagine that our discourse of thought is to start thinking of them in terms of groups or collections, what whatever reason. Remember, it's just came to our mind that we can think of objects in that way. The fact that the apples are on the table and it looks like they are in a group is just a coincidence. We want to form a collection of objects in our mind. Hence, apples on a table are not in a group, in a set yet. They are just spatially close to each other. Objects are still objects, with infinite number of conceptual contexts we can put them in.
Again, one of the ways to think about them is to put them in a group, for whatever reason we find! We do not need to collect into group only similar objects, like, only apples or only cars. Set membership is not always dictated by common properties of objects. Set membership is defined in the way we want to define it! For example, we can form set of all objects that has no common property! We can form a group of any kind of objects, if our criterion says so. We can even be just amused to group objects together in our mind. Hence, the set can be specified as “all objects we are amused to put together”. Like, one group of a few apples, a car, and several coffee cups. Or, a collection of apples only. Or, another collection of cars and coffee cups only. All in our mind, because, from many directions of thinking we have chosen the one in which we put objects together into a collection.

Without our initiative, our thought action, objects will float around by themselves, classified or not, and without being member of any set! Objects are only objects. It is us who grouped them into sets, in our minds. In reality, they are still objects, sitting on the table, driven around on highways, doing other function that are intrinsic to them or they are designed for, or they are analyzed in any other way or within another scientific field.

Since, as we have seen, we invented, discovered a direction of thinking which did not exist just a minute ago, to think of objects in a group, we may want to proceed further with our analysis.
Roughly speaking, with the group, collection of objects we have introduced a concept of a set. Note how arbitrary we even gave name to our new thought that resulted in grouping objects into collections. We had to label it somehow. Let's use the word set!

Now, if we give a bit more thought into set, we can see that set can have properties even independent of objects that make it. Of course, for us, in real world scenarios, and set applications, it is of high importance whether we counted apples or cars. We have to keep tracks what we have counted. However, there are properties of sets that can be used for any kind of counted objects. Number of elements in a set is such one property. If we play more with counts and number of elements in a set we can discover quite interesting things. Three objects plus six objects is always nine objects, no matter what we have counted!  The result 3 + 6 = 9 we can use in any set of objects imaginable, and it will always be true. Now, we can see that we can deal with numbers only, discover rules about them, in this case related to addition that can be used for any objects we may count.

Every real world example for mathematics can generate mathematical concepts, mainly sets, numbers, sets of numbers, pair of numbers. Once obtained, all these pure math concepts can be, and are, analyzed independently from real world and situations. They can be analyzed in their own world, without referencing any real world object or scenario they have been motivated with or that might have generate them, or any real world example they are abstracted from. How, then, conception of the math problems come into realization, if the real world scenarios are eliminated, filtered out? Roughly speaking, you will use word “IF” to construct starting points. Note that this word “IF” replaces real world scenarios by stipulating what count or math concept is “given” as the starting point.
But, it is to expect. Since a number 5 is an abstracted count that represents a number of any objects as long as there are 5 of them, we can not, by looking at number 5, tell which objects they represent. And we do not need to that since we investigate properties of sets and numbers between themselves, like their divisibility, which number is bigger, etc. All these pure number properties are valid for any objects we count and obtain that number! Quite amazing!

Moreover, even while you read a book in pure math like "Topology Fundamentals" or "Real Variable Analysis" or "Linear Algebra" you can be sure that every set, every number, every set of numbers mentioned in their axioms and theorems can represent abstracted quantity, common count, and abstracted number of millions different objects that can be counted, measured, quantified, and that have the same count denoted by the number you are dealing with. Hence you can learn math in the way of thinking only of pure numbers or sets, as a separate concepts from real world objects, knowing they are abstraction of so many different real world, countable objects or quantifiable processes (with the same, common count), or, you can use, reference, some real world examples as helper framework, so to speak, to illustrate some of pure mathematical relationships, numbers, and sets, while you will still be dealing, really, with pure numbers and sets.

There may be, also, a question, why it is important to discover properties of complements, unions, intersections, of sets, at all? These concepts look so simple, so obvious, how such a simple concepts can be applied to so many complex fields?

Let’s find out! Looking at sets, there is really only a few things you can do with them. You can create their unions, intersections, complements, and then find out their cardinalities, i.e. sizes of sets, how many elements are there in a set. There is nothing else there. Note how, in math, it is sufficient to declare sets that are different from each other, separate from each other. You don’t have to elaborate what are the sets of, in mathematics. You do not even need to use labels for sets, A, B, C,… It’s sufficient to imagine two (or more) different sets. In mathematics, there are no apples, meters, pears, cars, seconds, kilograms, etc. So, if we remove all the properties of these objects, what properties are left to work with sets then? Now, note one essential thing here! By working with sets only, by creating unions, complements, intersections of sets, you obtain their different cardinalities. And, in most cases, we are after these cardinalities in set theory, as one of the major properties of sets, and hence in mathematics. Roughly speaking, cardinality is the size of a set, but also, after some definition polishing, it represents a definition of a number too. Hence, if we get a good hold on union, complement, intersection constructions and identity when working with sets, we have a good hold on their cardinalities and hence counts and numbers. And, again, that's what we are after, in general, in mathematics!

As for real world examples, you may ask, how distant is set theory or pure mathematical, number theory from real world applications? Not distant at all. Remember the fact how we obtained a number? A number is an abstraction of all counted objects with the same count, of all sets of objects with the same number of elements (apples, cars, rockets, tables, coffee cups, etc). Hence, the result we have obtained by dealing with each pure, abstracted number can be immediately applied to real world by deciding what that count represents or what objects we will count that many times. Or, the other way is, even if we dealt with pure math, pure numbers all the time, we would've kept track what is counted, with which objects we have started with. There is only one number 5 in mathematics, but in real world applications we can assign number 5 to as many objects as we want. Hence, 5 apples, 5 cars, 5 rockets, 5 thoughts, 5 pencils, 5 engines. In real world math applications scenarios it matter what you have counted. But that fact and information, what you have counted (cars, rockets, engines, ..) is not part of math, as we have just seen. Math needs to know only about a specific number obtained. Number 5 obtain as a number of cars is the same as number 5 obtained from counting apples, from the mathematical point of view. But, it can and does represent sizes of two sets, cars and apples. For math, it is sufficient to write 5, 5 to tell there are two counts, but for us, it is practical to drag a description from the real world, cars, apples, to keep track what number 5 represents.

[ concept of a set, math, math tutoring, mathematics, number concept, number definition, numbers, set, set concept, sets and numbers, tutoring ]

Twitter - Insights About Creative Thinking, Science, Mathematics, Logic, Intution, Innovation.

You can download all the important posts as  PDF book "Unlocking the Secrets of Quantitative Thinking".

Science is way more than math and logic. Math and logic can be equally applied do dogmatic teachings as well. Specific assumptions are the ones that define science.

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. #math

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] . It only means that you decide WHAT is counted and why. Math axioms and theorems remain the same!!

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.

One Insight into Mathematics, Axioms, Logic, and Their Relations to Other Disciplines

You can download all the important posts as  PDF book "Unlocking the Secrets of Quantitative Thinking".

Mathematics is, in the sense of methodology, like any other scientific discipline, including law, economy, psychology, biology, physics, chemistry. Or, more precisely, all other disciplines should be very similar to mathematics, if they are to discover new truths and solutions. I will demonstrate what are the two major similarities and one major difference. What differs mathematics from ALL of these disciplines is that mathematics deals exclusively with numbers, counts. It is, sometimes, hard to imagine that mathematics is independent discipline, given how much we, as students, and later in career, are fed (and fed up!) with numerous examples, starting with apples, pears, meters, acceleration, force, atomic mass, light wavelength etc.. Perhaps surprisingly to many of us, mathematics is an INDEPENDENT discipline and can be developed completely outside any example, i.e. examples (physical processes, decision generated numbers, measures) are not required for development and research in mathematics. Again, mathematics deals with counts, numbers exclusively, and that's it.

Now, similarities.

First major similarity, and the reason why mathematics is called one of the most precise sciences, is that it uses strong logic methodology. But note, this logic is used as a tool of thinking to solve problems in math, and develop mathematics. Logic is a separate discipline that can, and should, UNIVERSALLY be applied to any other scientific discipline.

Second major similarity is that mathematicians succeeded to define initial truths in mathematics, truths to start with, starting postulates or AXIOMS. But note, while axiom might be considered a mathematical term, it's meaning can be applied to ANY other scientific discipline or any other creative direction of thinking. Axioms are everywhere, we use it every day, but we do not call them axioms. Usually, these are initial assumptions in our directional thinking to solve a problem or to come up with a creative answer for something. The same truths are present in law, biology, physics, of course, the way they are discovered are different for each discipline. But, this is how it should be done. Define initial assumptions, make sure they are correct and start develop the system you are interested in.

Now, note how mathematics used LOGIC and AXIOMATIC approach to deal with counts! It is not that counts triggered development of logic and axioms. It's vice versa. Logic and axioms were there before, and are used to develop and enhance mathematics and are and should be used to enhance and develop other scientific and other creative human disciplines.

You can read more:

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• Math and its relationship with real world
• How math can be applied to so many different fields?
• Where the graphs in mathematics and physics come from?
• Tweets about math, physics, and how to approach math calculations
• One insight about mathematical axioms, logic and their relation to other disciplines
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• Mathematics Axiomatic Frontier
• Why math can be an independent discipline?
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