Sunday, July 31, 2011

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

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 ]

Sunday, July 24, 2011

From Poker, Basketball, Financial Math to Pure Math and Back

After some initial counting and some thinking put into it, you may have asked yourself, what is there more to investigate about numbers? A number is a number, I have just seen that, a clean and dry concept, a quite straightforward count of objects I have been dealing with. Five apples, five pears, the number five is common to all of them. We have abstracted it, and together with other fellow numbers (three, four, seven, ...) it is a part of a number system we are familiar and we work with. We may have a feeling there are only integers present, and that, really, it is not clear where those mathematicians find so many exotic concepts, so many other numbers, like rational, irrational, and others. Moreover, you may think that, without some real objects to count or measure, there would be no mathematics, and that mathematics is, actually, always linked to a real world examples, that numbers are intrinsically linked to the real world, to the objects counted, measured, that they are inseparable, that a number, despite its "purity", somehow shares properties of the objects it represents the count of.

In this article I will discuss these thoughts, assumptions, maybe even misconceptions. But, no worries, you are on the right track by very action that you put a thought about math and numbers.

Before I go to the exciting world of basketball and poker, as an illustration, let me discuss a few statements. A famous mathematician, Leopold Kronecker, once said that there are positive integers only and that everything else is the work of man. I support that view and essentially many mathematicians do. Here is the flavor of that perspective. Negative numbers are positive numbers with a negative sign. Rational numbers are ratios if two integers. Real numbers (rational and irrational) are limiting values of rational numbers (which are in turn ratios of integers), a sequence of ratios that are smaller and smaller and there are more and more of them, that converge to one value. So, essentially, all these numbers are constructed from positive integers.

As for "purity" of a number here is a comment. Number has only one personality! Take number 5, for instance. It's the same number whether we count apples, pears, meters...That's why we need labels below, or beside, numbers, to remind us what is measured, what is counted, because by looking at the number only we can not conclude where the count comes from. I have written about this in my previous posts. When you write 5 + 3 = 8, you can apply this result to any number of objects with these counts. So, numbers do not hold or hide properties of the objects they are counts of. As a a matter of fact, you can just declare a number you will be working with, say number 5, and start using it with other numbers, adding it, subtracting it etc.. No need to explain if it is a count of anything. Pure math doesn't care about who or what generated numbers, it doesn't care where the numbers are coming from. It works with clear, pure numbers, and numbers only. It is a very important conclusion. You may think, that properties of numbers depends on the objects that have generated them, and there are no other properties of numbers other than describing them as a part of real world objects. But, it is not so. Properties of numbers don't depend on the objects or processes that have generated them. While you can have a rich description of objects and millions of colorful reasons why you have counted five objects, number five, once abstracted, has properties of its own. That's why it is abstracted at the first place, as a common property! When you read any textbook about pure math you will see that apples, pears, coins are not part of theorem proofs.

Now, you may ask, if we have eliminated any trace of objects that a number can represent a count of, what are the properties left to this abstracted number?  What are the numbers' properties? That's the focus of pure math research. Pure means that a concept of a number is not anymore linked to any object whose count it may represent. In pure math we do not discuss logic why we have counted apples, or why we have turned left on the road and then drive 10 km. Pure math is only interested in numbers provided to it. Among those properties of number are divisibility, which number is greater or smaller, what are the different sets of numbers that satisfy different equations or other puzzles, different sets of pairs of numbers and their relationships in terms of their relative differences. That's what pure math is about, and these are the properties a number has.

In applied math, of course, we do care what is counted! Otherwise, we wouldn't be in situation to "apply" our results. Applied math means that we keep track what we have counted or measured. Don't forget though, we still deal with pure numbers when doing calculations, they are just marked with labels, because we keep track by adding small letters beside numbers, which number represent which object. When you say 5 apples plus 3 apples is eight apples, you really do two steps. First step is you abstract number 5 from 5 apples, then, abstract number 3 from 3 apples, then use pure math to add 5 and 3 (5 + 3) and the result 8 you return back to the apples world. You say there are 8 apples. You do this almost unconsciously! You see the two way street here? When developing pure math we are interested in pure number only. Then, while applying math back to real world scenarios, that same number is associated with a specific object now, while we kept in mind that the number has been abstracted from that or many other objects at the first place. This is also the major advantage of mathematics as a discipline, when considering its applications. The advantage of math is that the results obtained by dealing with pure numbers only, can be applied to any kind of objects that has the same count! For instance, 5 + 3 = 8 as a pure math result is valid for any 5 objects and for any 3 objects we have decided to add together, be it apples, cars, pears, rockets.

While, as we have seen, pure math doesn't care where numbers come from, when applying math we do care very much. We care so much what we have counted that we have invented devices to keep track of these counted objects. We have dials that keep track of fuel, temperature, time, distance, speed. Imagine, we have devices which keep track of kind of counted objects so when we look at them and see number five, we will know what that number five represent the count of! Say you have four dials in front of you, and they show all number 5. It is the same number five, with the same properties, and we can say that the power of mathematics is derived from noticing that number 5 is the same for many object and abstracting that number 5 from them, then investigating its properties. Now, here, we went back! We used that universal number five, and keep track in our dials, which exact objects the number 5 represents the count of.

In applied math it is so important to keep what is counted that we have invented dials for those countable objects or measures. There are dials in car, for instance, for fuel consumption, speed, time, engine temperature, ambient temperature, fan speed, engine shaft speed. If it was not up to us, those numbers would float around, like, 5, 2300, 120, 35, 2.78 without knowing what they represent until we assigned them a proper dial units. This example shows the essence of difference between applied and pure math, and how much is up to our thinking and initiative, what are we going to do with the numbers and objects counted or measured.

Now, we may go to great extent to investigate all kinds of properties of all kinds of numbers and sets of numbers. Hence, linear algebra,calculus, real analysis. They are all useful and sometimes very elegant parts of mathematics. But, frequently, we do not need all those properties to apply or use pure math results in everyday situations. Excelling in some business endeavor frequently depends on actually knowing WHAT and WHY something is counted, while, at the same time, mathematics involved, can be quite simple. When I say business, I mean business in usual sense, like finance, trading, but also, I mean, for instance, as we will see soon, basketball, and even poker.

When playing basketball we also need to know some math, at least dealing with positive integers and zero. However, knowledge of basketball rules are way more important than math, in basketball domain.
Those basketball rules are mostly non mathematical, which doesn't make them at all less significant. Moreover, they are way more important ingredient, and more complex part, for that matter, of a basketball game, than adding the numbers. You can have knowledge of adding integers, but without knowing basketball rules, and know how to play basketball, you will not move anywhere in a basketball team or in the game. Moreover, basketball rules are actual axioms of a basketball game. And, every move in the basketball court, any 30 seconds strategy developed by one team or the other, corresponds to theorems of the basketball game! Any uninterrupted part of the game, without fouls or penalties, is an actual theorem proof, with basketball rules as axioms. We can say that basketball rules are those statements that define what belongs to a set "number of scored points"! You see here how we have whole book of basketball game rules that serve the purpose to define what belongs to a set (of scored points). Compare that to those boring, and sometimes, ridiculous examples, in many math texts, with apples, pears, watermelons (although they may illustrate the point at hand well). With ridiculing the importance of rules of what belongs to a set, belittling their significance and logic associated to obtain them, those authors, unintentionally, pull you away from an essential point of "applied" math - in order to define what belongs to a set, and then, count it (like points in basketball) you need to know areas other than math, and to develop, logic, creativity, even intuition in those areas, in order to decide what really belongs to a set and what needs to be counted. Because, accuracy of rules and logic to determine what belongs to a set dictates the set's cardinality, the size of the set, the number of its elements. And this is the number you will enter in all your calculations later! Note, also, that only knowing rules of basketball game doesn't make you a first class player, nor your team can be a winner just knowing the rules. You have to develop strategies using those rules. You have to play within those rules a winning game. The same is in math. Knowing the fundamental axioms of math will not make you a great mathematician per se. You have to play "winning game" inside math too, you have to show creativity in math as well, as you would in basketball game!

In business it is often more important to know where the numbers are coming from than to know in detail their properties. For instance, in poker. again, only integers are involved. You have to remember that the same number 5 can be any of the card suits, and, in addition, can belong to one or more players. Note how abstracting number 5 here and trying to develop pure math doesn't help us at all in the game. We have to go back, we have to use that abstracted number 5 and put it back to the objects it may have been abstracted from. You have to somehow distinguish that pure number 5, and associate it with different suit, different player. And strategy you develop, you do with many numbers 5, so to speak, but belonging to different sets, suits, players, game scenarios. Hence, being a successful poker player, among other things, you need to memorize, not exotic properties of integers and functions, but how the same number 5 (or other number) can belong to so many different places (players, suits, strategies, scenarios).

Another example is finance. Any contract you have signed, say for a credit card, is actual detailed definition what belongs to a set, i.e. whether $23,789.32 belongs to your account under the conditions outlined in the contract. Note how even your signature is a part of the definition what belongs to a set, i.e. are those    $23,789.32 really belong to your account. You see, math here is quite simple, it is a rational number   23.789,32, but what sets it belongs to is outside mathematics, it's a domain of financial definitions. Are you going to pay the bill of $23,789.32, or someone else, is a non mathematical question, while mathematics involved is quite simple, it's number 23,789.32.

Now, note, when you are paid for your basketball game, suddenly you have math from two domains put together! It may be that the number of points you scored are directly linked to a number of dollars you may be paid. Two domains, of sport and finance, are linked together via monetary compensation rules, which can have quite a bit of legal background too, and all these (non mathematical in nature!) rules dictate what number, of dollars, may be picked after the set of games.

A theorem in one system is usually an axiom in another. Which systems to link in this way is in the center of innovative thinking.

[ applied math, applied mathematics, math applications, poker, basketball, financial math, poker and math, math concepts, financial mathematics  ]

Friday, July 22, 2011

Concept of a Set and of a Number

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 ]

Monday, July 18, 2011

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

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 -

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

Take a thought journey from real world math applications to pure math and back - and have that "wow!" moment

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

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?

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 ..

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 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. -

Different contexts will give different meanings for the same sentence. #semiotics

From real world math applications to pure math and back!

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 #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!

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

From Real World Math Applications to Pure Math and Back

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!

Explaining essential ideas of mathematics. Talk about Applied Mathematics, Mathematics and Real World, Math Education.

Puzzled with math graphs? Wondered why they use them? Where the graphs come from anyway??

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..

The posts are terrific. They engulf!

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...

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 ...

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.. #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... #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, 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,

Quantitative finance, dragons, and math -

Relationships between dragons lead to math development...

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

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.

More about Setology and Countology here #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

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,

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?

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 a treasure hunt, sort of..

Have you ever thought what's behind calculus ideas? Maybe this will show just that! #math #calculus

How to introduce calculus concepts to primary school students: "How to Calculate Surface Area of a Pirate Island"

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,

My new draft post "How to Calculate Surface Area of a Pirate Island" introducing integration to primary school students

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, #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.

Sunday, July 3, 2011

Mathematics Through Car Racing Concepts. Physics, Car Design, and Driver's Inputs...

Car racing is a captivating sport for many of us. It is very interesting sport for kids as well and that fact can be used to motivate some important concepts in mathematics. Car racing track, with its irregular shape, dictated by urban projects requirements and geography, can be used to introduce calculus, integration, rational, irrational numbers, finding the length of the curve of arbitrary shape, finding the surface area enclosed by racing track, which is also of an arbitrary shape. Kids would be more interested in math if they can be shown the applications in things they are interested in.

Examples you can use. Speed of Formula 1 cars (256.78 km/hr), time of arrival, fuel consumption (72.59 L/km), engine temperature (985.23 C), laps counts (2.5), tire rubber temperature, pit time (58.5 sec), randomness of pit times (probability distribution, average, expectation), track length (10.25km), compare tire diameter, volume with the length of the track.

When talking about racing car related concepts, care should be taken to explain the existance of logic and functional relationship between objects that will be, possibly, quantified later. This approach is important when using any real world example for mathematics. For instance, it should be signified that business, social, geographical, financial analysis was done before a race track is built. Hence, business, social, geographical, financial analysis dictated a number that you will obtain later by measuring the length of the track. In this framework, student should be shown that there are non mathematical relationships that dictates the shape, size, volume of objects that can be quantified later.

Looking at a number of cars that participate in a race, a function, as a mathematical concept, can be introduced. Moreover, functions of several variables can be introduced considering that each car travels different path, with different speed, uses different amount of fuel, engines have different temperatures, drivers change gears different number of times per minute, pit stops are of different length, and drivers complete the race at different times. Winning a race can be shown to depend on several mathematical parameters too, plus the skill of the driver. Randomness that is present in some of these measurable parameters can be used to introduce probability concept, stochastic processes, statistics.

But, when the driver of a racing car drives the car, he is generating quantifiable entities. He is dealing with objects and events that has mutual quantitative relations in addition to other relationships, like mechanical for instance, or spatial, or temporal. He is generating numbers, giving initial and boundary conditions for a number of PDEs and IDEs in mechanics, thermodynamics. He may be not aware that he is generating numbers. He does not think too much mathematically while driving a car, nor thinks about physics laws that take place every moment. The drivers is involved in the specification of physical initial and boundary conditions for those physics laws. Pressing the gas, accelerator pedal, steering the wheel, using the breaks. The driver knows what the output will be for his inputs. There is of course, a feedback from the car measurements system, providing him with certain important numbers, like fuel gauge, speed dial, temperature, tyre pressure. Given the context of race, and driving the car, these numbers will influence his decision about physical inputs to the car system, like slowing down perhaps, or accelerating.

An engineer, car designer, engine designer, must predict the range, the envelope of values that the car will be subjected to. These values will be dictated by the driver later. However, the engineer who designed the engine and knows the racing car mechanics and functioning inside out may not be a good racing driver, and usually is not allowed to race. His intrinsic knowledge of car design, mechanics, and thermodynamics usually doesn’t help when it comes to fast thinking when to turn left or right on the racing track, when to accelerate or slow down, meaning engineers role is to design a car for specified range of possible characteristics. It is not the physics laws that win the race. It is the selection and sequence of initial and boundary conditions provided to the physics laws, or more preciously, to the car components that behave in accordance to physics laws. Physics laws are the same for all racing car drivers in the race. It is this selection of initial and boundary conditions that decide who will win the race. Note, also, how here physics and mathematics touches on social values. We value who comes first in the race, not the discovery of physics laws and the mathematics used during the design process. Good car design, brilliant engineers, physicist, and mathematicians can make an excellent car. They know very well the math and physics and engineering. But, there is no physic or mathematical law that will specify the initial and boundary conditions that will win the race. It is up to driver to input them into the car while racing, given the information he is getting on the track, during the race, about his own position, positions of other drivers, their speed, track characteristics, etc.. Moreover, same type of car, with same characteristics, driven at the same time by different drivers, in an race, will show the significance of the initial and boundary conditions selection to the PDEs and IDEs underpinning the engine performance and car functioning in general.

Every set of action while driving a racing car is to the race like an invention is to the laws of physics. Maxwell’s equations are great discovery, but they are necessary, yet not sufficient condition to make an invention or a novel engineering solution. It is the set of initial, boundary conditions, and specific configuration of elements that are part of the winning invention. And there are no laws, nor formulas, in mathematics or physics, that will let you produce inventions one after another.

Note also, how financial mathematics is linked, via logical connectives” IF…THEN” to the car racing. The winner is awarded say $500,000. So, IF your driver wins, THEN he will get $500,000. See how the context defines what will be said about truth values of events at the car racing. It is us, or rather, the Sports Governing body that specifies what will happen after that “THEN…”. That’s the axioms of the Car racing Award. It is not known what can be put there after “IF your driver wins THEN….”. It can be that he will get ice cream or, his car will be painted in red. Who knows? It is us who specify these logical statements and build a system from it.

It should be shown to students that numbers generated in a car race can come from different sources, and can be generated in different ways. They can be a product of driver's decisions, physics laws, random events, and should be shown that mathematics accepts all those numbers in the same way, as numbers only. It is us who keep track where the numbers come from, when, and why, as I have written about that in my previous posts.

[ applied math, applied mathematics, car racing, cars, cars movie, film cars, Honda, Honda Indy, math concepts, motivation, movie cars, racing sports cars ]