Contextual triviality. What Has to be Done When Someone Uses the Word "Logic".

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When someone uses the word "logic"he/she should immediately point out what is initially assumed to be true/false and what truth results are derived, i.e. what can be proved from those initial assumptions.Word logic is nice to use, it can be fancy, it can show you want to be precise in your communication or explanation. However, the axiomatic system should be known and understandable for all the parties that are part of the "logic" communication. Saying that something is logical doesn't mean it is obvious, and it doesn't mean it does not require a proof. If something appear to be "trivial", the logical context of axioms and premises should be clear to all participants who want to accept that "trivial" remark. Contextually "trivial" is OK.

The other day, while browsing mathematics section at Indigo bookstore, I have noticed a book "Logic for Mathematicians" by A. G. Hamilton, (Google books http://goo.gl/17kEj). I was pleased to see that the book reflects my view that logic is an independent discipline from mathematics and that mathematics is only one of the area of the application of logic. While Frege may have integrated both directions of thinking, I am glad that A. G. Hamilton "presented the subject matter without bias towards particular aspects, applications or developments, but an attempt has been made to place it in the context of mathematics and to emphasise the relevance of logic to the mathematician.".

To me, this is important because of my view (most of the posts in this blog) to differentiate clearly the worlds that define what is to be counted, measured etc, from the mathematical world that accepts pure numbers as starting points. Logic is used in both worlds.



[ logic, mathematical logic, math, math concepts, axioms, mathematics, teaching math, teaching mathematics, understanding mathematics, Frege, Hamilton, ]

A Few Hints How to Introduce Mathematical Concepts

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Teacher should postpone introduction of strange, exotic mathematical names and labels to, otherwise, most likely, easy to explain and easy to understand concepts in mathematics. Teacher should  first explain mathematical concepts in terms of required sequence of mathematical operations on numbers and on sets involved and then introduce labels or historically accepted names for them. Most of us will agree that many of those names are there for historical reasons, and often they are confusing, misleading, and even intimidating (if Borel, Hausdorff needed so much work to prove that theorem or formulate it, what chances do I have?). To me, it is sometimes better to refer to a theorem by a number first, like Theorem 1, Theorem 2, … and only later assign historical names or other labels to reference them.

Math lectures sometimes look to students like putting misleading, cluttered, over-detailed traffic signs on an easy to use roads and highways.

But, what are we really doing or what we want to do when we say "introducing math concepts through A or B or C real world examples"? Do math concepts need to be introduced through real world examples at all? No, they don't. They can be derived or formulated directly from axioms.  So, what would be the goal of "introducing math" through real world examples? To show that math concepts can be motivated by real life examples, but, at the same time the same math concepts can be derived inside math only, without referencing any real world domain. To me, the mandatory step of introducing some math concepts from real world example is to mandatory show that the same concept can be defined or derived from ZFC axioms. This would clearly show the border and connections, if you wish, at the same time, between applied math and pure math developed from ZFC axioms.

[introducing math, math ], math concepts, math education, mathematics, theorems ]

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

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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  ]

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

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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 ]

From Real World Math Applications to Pure Math and Back

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Can you go from pure math to real world math applications? The answer is no. You can not start with math only and then “apply” to real world scenario. It is, however, possible to use math in real world (even in fictional world, like in Harry Potter movies), practical applications, but the path and direction are different and needs to be clarified.

The reason you can not go from pure math to real world math application is in the fact and in the nature, definition (in a sense) of a number. A number is obtained as an abstraction, a common property for many objects. By this very definition, because it is abstracted from counted objects, because it is, now, a separate concept, representing a pure count, without any object associated to it, you can not tell, by looking at the number only, where it came from, what and if anything has been counted to obtain that number. In other words, a number does not carry any information about any object extraneous to mathematics! Hence, you can not say, just looking at the number only, or at the sequence of math operations on numbers, what its or their application, in real world, may be. Newton did not learn calculus first, then applied it to the gravitational problems! Quite the opposite happened. Newton was dealing with non mathematical objects and relationships like apple falling from the tree, Moon orbiting Earth, and other body motions. Unless they are quantified, these are not mathematical objects nor relationships. If they were, then you would see theorems in math books proved by apples, Moon, speed, etc. but it is not so. Math theorems are stipulated and proved using only mathematical objects, like numbers, sets, set of numbers, or other mathematical theorems and axioms. So, let make that clear, Newton first dealt with physical objects and only then he invented calculus. So, when someone tells you you will learn math then apply it, it is not quite true.

When you deal with a number, you deal with an abstracted common property, a separated concept abstracted from all the objects whose count it represents. The very moment you start adding 2 apples and 3 apples you are doing two distinct steps. First one is recognizing that the object to be counted is an apple. That recognition process, a categorization, that you are looking or holding an apple is outside mathematics, since you can be counting apples, pears, cars, books, cups. This recognition is a focus of research of cognitive science, psychology, biology, color research, even socal sciences. The second step (which is, actually, common to counting all those objects) is dealing with numbers 2 and 3. Even if you may not notice, when dealing with 2 and 3, you are dealing with counts that can represent not only apples, but millions of other objects that can be counted, or put in sets, then counted, to obtain 2 and 3. Hence, the result you obtained for 2 and 3 apples, i.e. 2 + 3 = 5, can be used in ANY other situation where you have 2 objects and 3 objects and you want to add their counts. You right away invented and used "pure" mathematics when counted these apples. It is this universality, the common numeric property of counted objects, that gives mathematics ability to be a separate discipline, to deal only with numbers. While to us, and to the field that uses quantification, is very important what, when, why and where something is counted or measured, to investigate only the counts' properties is the task of pure math. Pure math does not care where the numbers or counts are coming from. It is very similar when we create a set of any objects (of interest), but we are only interested how many elements are in the set and not which objects are part of the set (that information, which objects are elements of the set, we keep track of on a separate sheet of paper) ! Math knows and should know only about numbers and sets of numbers. Notice how math may be "motivated" by counting apples, but, the result obtained, i.e. 2 + 3 = 5, can be used when counting any other objects!

Also, 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?

Mathematics deals with numbers and with numbers only. It does not care where the numbers are coming from (but, in the field of applied math, we do care where the numbers are coming from). Now, you may ask, how we can apply mathematics at all, if the trace what is counted is lost in this abstraction, in this definition of pure number? Well, here is how.

It is true that pure mathematics deals with numbers only and, of course with mathematical operations on them. We have abstracted, separated a concept of number from all possible real world objects that might have been counted. Thus, when you say 5 + 3, you right away know that the answer will be 8. No real world objects are mentioned nor even thought of when we did this addition. We just selected two numbers, and decided to do addition (we could also decide to do subtraction or multiplication). Now, how then we can apply math to real world if we don’t have a trace of what is counted? There is a way! When we want to “apply” or rather, use, math in real world, we will drag the names of objects counted into the math! We will keep track of numbers obtained, to know where they come from, which objects’ counts they represent. How we do that? We will add  a small letter, or abbreviation, or a word, name, just beside the count to tell us what we have counted. For instance, we can write numbers, 3, 5, 7, 10 after we counted (or measured) something. In order to keep track what we have counted we will add small letters right beside the numbers, like 3m, 5m, 7seconds, 10apples. Now, very important thing. These added letters do not represent math. They are for us to keep track what is counted. Unfortunately, frequently, this is all mixed up and students are often told they are doing math even when they describe what are they counted, why (to give 10 apples), where (apples from the basket, he went 3m downhill, then 5 meters uphill,), when (7seconds ago, not after). All this reasoning, descriptions, units, abbreviations, meters, seconds, apples, ago, before, after, downhill, uphill,  DO NOT BELONG TO MATH. Why is that? Because, if you look in any pure mathematical textbook or book, you will clearly see that no theorems are stated or proved by mentioning apples, meters, seconds, pears, etc. All the theorems are proved strictly in terms of mathematical objects, numbers, sets, set of numbers, using other theorems and axioms. No outside objects or descriptions, like apples, cars, downhill, uphill, will ever enter a mathematical theorem or its proof.

Now, when we distinguished what is pure math and what is applied part of it, we can make more interesting and significant conclusions. Pure mathematics deals with numbers and numbers only. Since number 3, say, can represent an abstracted count of so many, many objects, wouldn’t be interesting to have its properties investigated? It looks like there is some value in the fact that one concept, a number, stands for counts for so many objects. We can compare number 3 with other numbers. We can say which numbers are greater than or less than other numbers. We can multiply them and see what numbers we are getting. And all the time we deal only with numbers. The value is, if we get some interesting result for a certain number or numbers, from our “pure” number investigation, we can use that result for all those examples in real world. That’s the value of applied math. But, in order to use it, say, in order to use 5 + 3 = 8, we have to make a match between pure math numbers and real world counts. Hence, IF we count say, CDs and IF we get 5CDs, and, again we count another set of CDs just mentioned. This is a simple example, but there are more complex mathematical results where we don’t need to reinvent the wheel each time we get the real world count, but instead we take advantage of ready to use mathematical result, procedure, theorem, solution.

Mathematics does not see the reasoning from other worlds. Math will see number 6 (given, or picked), math will see number 10 (given, or picked), and math will see the selected, required operation, addition (it could be subtraction or division too). Does it mean that these numbers, 10 and 6 came from thin air to mathematics? They came from counting apples, but, where are the apples then? The point is, remember, when we said that numbers, in pure math, are abstraction for all the objects they can represent count of. They are not from thin air, they exist in our math as our starting point. Our pure math has already numbers available for our use, 1, 2, 3,…10, …, etc.. How? We did not even need to find objects to count to obtain different numbers. We can start with 1, then add 1 to get 2, then add 1 to get 3, etc. That’s why we have numbers already available for us. We only pick them, and do the operations. Math, for that matter, doesn’t need to know if we counted apples, or pears, of chewing gums. It is enough for math to tell there is number 10 and number 6 and that we want to add them. It is us who will keep track why we counted (because Peter was hungry),  what we counted (apples, they are edible), when we counted (in the evening, when was the time for dinner).

How we are allowed, at all, to go from non-axiomatic worlds, physics, economics, finance, to, so strictly defined, axiomatic world of mathematics? Apparently, mathematics does not care whether the problems come from axiomatized system or not! And that tells us that mathematics can not correct logical steps or see the flaws in the system it proudly claims it models. Assumptions coming from non-axiomatized fields, like physics, economics, finance, and into a strict axiomatic system, like mathematics, can produce results that can wreak havoc back in the field where the mathematics is applied.

You may ask, at this point, how we can mix these two logic worlds. One world appear to be very fluid, the real world, with objects selected, of any type, and any kind of relationships. On the other hand we have mathematical logic world, where we only deal with numbers, or sets of numbers only, and with what appears to be quite precise rules, axioms, logic, and well defined sequences of math operations (of addition, subtraction, multiplication, division). Is there a logic that will merge and connect these two worlds? YES! You can mix these two worlds, but you have to be very careful with the World # 1, the real world's objects and scenarios. Your logic there has to be correct. Then, you can use logic used to link these two worlds, and it's  is the logic already familiar to you, but with new domains of application. How it is done? Here is how. Let’s say that logical statement from the first, real world, is labelled as “p” ,  and logical statement from the second, mathematical logic world is labelled as “q”, then we can create a new, logical statement IF “p” AND “q” THEN “s”, or in shorter notation, p ^ q => s. I am sure you are familiar with this logical statement. Here is example. Let p = “Peter is hungry” and let q = “there are 10 apples”. Then we can form a statement “IF Peter is hungry and IF there are 10 apples, THEN he will get 6 apples”.

Mathematics is defined not by objects it counts, nor by reasons or logic why those objects are counted, but with concepts used to define matheatics axioms and to define proofs of mathematical theorems. 

[ applied math, applied mathematics, learn math, math applications, math concepts ]

How to Teach Your Kids and Yourself to Think More Freely About Math and Real World Math Applications

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When I was a kid, I remember, I was often asked, during a math chat, “How many apples are there?” or “If you subtract 3 apples from 10 apples how many apples are left?”. Or, "Peter has 5 apples. How many apples he will have if he gives Susan 3 apples"? Then, there were many more examples of counting objects.

I am sure that many of us have had this kind of first encounters with mathematics.

When I was 4th grade, an interesting idea came to my mind. Why do we need to know concept of an apple in order to subtract 3 from 10? How and when we become aware that an object is an apple, and not, say pear? And what if Peter doesn't own an apple? Isn't the ownership a legal concept? Does Peter really own an apple? Then, if Susan is really hungry and Stephen is not, then giving Susan 3 apples is way more important than actually counting apples. Also, turning left and drive 2km is not the same as turning right and drive 2km. Yet, left and right were not part of math. They are not quantities, yet they are so important. Those were my thoughts when I was in 4th grade..

After a while, and after many examples involving apples, cherries, cars, marbles I have been shown, it became clear to me that 3 and 10 are separate, abstracted concepts of numbers, and hence they can be dealt with independently of any apples, oranges, pears, and other objects used as counting example. Teachers are constantly talking about universal nature of math and even about pure math. Yet, somehow, teachers were always sticking to real world examples when explaining math concepts. At the same time they were talking where the math can be applied, but miss to inform us what is it we are actually applying, i.e. pure math. Fortunately, I have realized that math can be independent discipline and that the math tasks and problems can be tied to the numbers only, without any real world examples. But, that was a trouble, when I looked how the math is taught. I  did not talk to teachers about that, they were very strict, and I was afraid to raise any issue. Yet, I was the best student in primary school, in my class, and one of the best in school, in math. And the issue was why would you expect from kids to know non mathematical concepts in order to add two numbers? Why kids should know social structures and social relationships, legal concepts and then learn math from those examples? Math can be taught without them. The best approach is to say that math can be motivated by real world affairs and also it can be developed from math axioms.

Then, there was another puzzle. The opposite (of knowing non mathematical concepts) was also true. As a kid, I realized that I didn't know how the automobile engine works, yet I can count engines. Then, in 7th grade when physics kicked in, we have been counting objects that we only vaguely knew what they are, like, electrons, atoms, waves, particles. To me, if math says that definition what actually belongs to a set, in order to be counted, must be clear and non ambiguous, it should be followed. Yet, we do not know what is electron, but it was apparently sufficient to stick to "it looks like it's a particle, but, it can be a wave too, and it may appear and disappear time after time". So, apparently we can define set "of objects we don't know exactly what they are" and have that count too.

Throughout “schooling” teachers (not all), most likely, failed to educate us what math really is. What is left from so called math education, for many of us, is a bitter, frustrating, scary feeling, tight test schedules, unresolved problems, and many confusion about math concepts. Moreover, teachers were running through material without giving any deeper insights.

And that view I questioned through primary school. Later, it became clear that math can be developed as a discipline without waiting for real world examples. However, it is true that certain directions in math were motivated by physics problems, like calculus, but still, calculus can be developed from scratch without physics input. You can not expect from kids to fully know concept of force, energy, electrical field, in order to introduce calculus. Calculus can be introduced way earlier, right after irrational numbers.

Well, I did have my personal “fight” with math, and I think I won! And this happened relatively early, around 4^th grade, when I realized that you can work with numbers independently from any real world examples, but I did question how we define sets, and can we really always define clearly what belongs to a set. Later, that proved to be related to Russell's paradox. That was, at least for me, key point when I started to like math, to appreciate its elegance, its independence as a discipline. After primary school I enrolled in Mathematical Gymnasium, then later, finished my Bachelor's Degree in Electrical Engineering (later redefined as Masters Degree, Bologna, around 2000), studied Astronomy along the way, and have exciting career in these fields.

Here, I would try to keep examples simple, intuitive, and I would always emphasize that particular mathematical concept can be defined and derived without real world examples, but can be motivated by them.

In this article I want to explain a method how to teach kids, or primary school students, or anyone who is interested to understand math better, to think more freely about math, and to see that math is a game, puzzle game sometimes, whose variations can be created by you as well. You do not need to wait for someone else to give you mathematical problems and tasks. I hope that, after you read this post, you will have an unstoppable desire to revisit other parts of math, and to go on your own journey which will also include YOU creating new mathematical problems to solve.

Let’s start. Counting objects is the core of mathematics. Let’s say we have two kids, Peter and Stephen, who wants to learn math. Usually the learning starts with showing some objects, say apples, and asking kids to count them. Then, it comes addition of apples, subtraction, and perhaps later multiplication and division. Now, after, say one hour or so, or after even a few days, Peter and Stephen will always expect to see objects then to count them. That’s where the word “given” in math comes from, or even word “suppose”. Peter or Stephen will always expect to have a problem in front of him, to be “given” that he needs to solve. Needless to say, kids may be, at this point, scared to asked where these problems come from? Who “gives” them? Here is the answer.

It would be nice to show to kids that 2 + 3 = 5 is independent of any objects counted. This can be done by going through a number of examples with apples, oranges, cars, CDs, pencils, even time, like hours and seconds, and then show separately that 2 + 3 equals 5 no matter what are you counting. You can reuse this numerical answer for any objects you may be counting in future. At this point kids should be aware of numbers as separate concepts. They have to be aware that numbers can exist without any real world examples. They have to be aware that they can do two things with numbers. Deal with them as pure numbers, like 2 + 3 = 5, and, obtain these numbers by counting objects. It is very important to explain this fact to kids.

And, now, here is the main step!

Get a bunch of objects together, say 30 apples, in a basket. Ask Stephen to go outside of room. Then, tell Peter to specify how many apples he wants to be taken of the basket and put on table. Suggest to Peter that the number he specified is completely arbitrary, it’s up to him to tell the number. Or, he may have his own logic to do that. Let’s say Peter chose number 5, for whatever reason. Then 5 apples will be taken from the basket and put on table. Ask Peter again to chose another number. Peter says 3 and he takes 3 apples from the basket to table, in another group.

Now, explain to Peter how his arbitrary, starting selection, a starting number of apples dictated how many pencils will be counted. It also dictated the two numbers Stephen will be dealing with. Also, explain to Peter that it is him who specified how many apples were to be counted. There was no pressure from anyone else, no “given” number of apples.  It was Peter who “gave” the numbers.

So, in mathematics, not only that you can count some objects and tell how many of those objects are there, but also you can use a pure number to start with, and tell how many of things is to be counted. This count, specified by you, can be a “given” starting point for someone else’s calculation! Please feel free to select any “starting” number again and play with mathematical operations, addition, subtraction etc, using other numbers by your choice. The numerical results you get is important knowledge, and can be reused in real world applications.

Let’s say, after a while, you have a few pages with numbers chosen by you, and a number of results obtained by multiplication, addition, etc. You may want to sort them, as you wish, maybe by magnitude, smaller numbers first, bigger numbers later. You may ask yourself, how this can be applied in real life? But you can not go from math back to real objects because, as we have seen, when you write 2 + 3 you do not know what is counted. Can be anything. However, there is a way for real world application! If you start with “IF I counted CARS, and get 2, and then IF 3 more cars came by” what is the result? Hence, once you start keeping track WHAT you have counted, and WHY, you can use your mathematical results to match the numbers. Note here that math does not see where the numbers 3 and two came from or why. You can count cars passing by, you can count cars on parking lot, or these can be numbers you tell car dealer to move cars in the garage. Math can’t see that. Math only see you provided to it number 2 and 3 to add. It is you that you have to carefully keep track what did you count and why and where the result will go.

Back to Peter and Stephen. Now, it is time to ask Stephen to specify how many apples will be taken out of the basket while Peter is out of the room. Now Peter will have “given” numbers to add, and Stephen will become aware that it is him who can specify these initial conditions for counting.

These exercises should be continued with measuring and specifying length. First Peter will be the one who will specify the distance between two objects in the room, by moving those objects apart. Then, Stephen will come in the room and measure the distance. Second exercise will be for Stephen and Peter, to specify the distance and then measure and move two objects to match that distance. It is of fundamental importance here to make Peter and Stephen aware that it’s them who specify the initial numbers, counts, and then the change is done in real world by their own specification! During these exercises kids will start thinking more freely about mathematics, and starting creating their own mathematical problems, instead of only dealing with what is “given” to them.

In next post I will write how we separate and connect two logical worlds. First one is how we create rules and keep track what and why is counted and the second world is the world of mathematics that will deal with these numbers. With the second world we are already familiar at this post.

Real World Math Applications

Peter may want to ask “Ok, I have done all this math calculations, and what I get is a number as the result. Where is the real world application?”

Here is the answer. Let’s say Peter obtained number 5 as counting pencils. What now? What possible application can be created using number 5 and pencils?

Here is how the real world examples can be created. Make this sentence:

“If I have 5 pencils AND IF 'something else A' THEN 'something else B .”

Note first “AND IF”! Following that it can be a true statement about anything you can imagine that can depend on the number of pencils counted. You have a true statement that you counted and obtained 5 pencils. Then, you may say, IF I have 5 pencils AND IF it’s sunny outside THEN I will decide to go out and draw landscapes. Note how the number of pencils influenced your decision to go outside and draw landscapes. Please note how open ended logical connective “AND IF ” can be related to any specified fact YOU have in mind! The very true and accurate calculation of 5 pencils can lead to an action that you can attach to this number! It is really up to you or up to your analysis what will depend on number five. It can be your creative new statement, new initiative or new decision that will start once you have 5 (and not 3 or 6) pencils in. That connective on the other side of "AND IF" can be another count, obtained from counting other objects too.

Now, teachers will rarely ask you to do this! Teachers will just stop after you calculated 5 (pencils or other objects) and leave you hanging there asking “what for?” and “what now?” I want to change that. I want to show you how the math calculations are actually applied, used, and utilized in the real world.

[ applied math, applied mathematics, learn math, math, math and real life, math concepts, math education, math tutoring, mathematics ]

Motivation to Use Graphs in Math, Physics and to Know Arbitrary Surface Area Calculations

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

Now, you may ask, are there any other reasons to learn calculus and integration, other than calculating the surface area of arbitrary surfaces of physical objects (see my previous post below). Yes, there are. The point is that mathematics uses graph as a tool a lot. Graphical representation of functions is the most common use. Also, mathematicians, physicists, economists like to represent a number, a count, as a length of a line in a coordinate system, where length matches the count in question. Then, since we use line to represent counts or numbers obtained, the product of two numbers will automatically be shown as surface, bounded by these lines. Then, volumes can come into picture too. It is just pure convenience, call it visual convenience, if you wish, that we have found lines and surfaces to represent our quantities. We are still interested in numbers which these graphical elements represent.


As you know you can count many, many things. Moreover, what you have also seen, you can play with numbers without knowing where they come from. For you know that, for instance, 2 + 3 is equal 5 no matter whether you counted apples, CDs, meters, kilograms, pears, cars, pencils, rockets, measure length etc. You can work with numbers, 2, 3, and 5 now independently of any objects from real world. And it is helpful to know in advance that 2 + 3 = 5, so you can use in any possible counting situation.

Now, let’s say you want to count how many cars are driving by each minute on the street. These can be the result of your measurements: 1st min 15 cars, 2^nd minute 23 cars, 3 rd minute 10 cars, 4^th minute 12 cars, 5^th minute 22 cars. Of course you can draw a table with the columns showing minutes and associated count of cars.

Time (mins)	Cars
1	15
2	23
3	10
4	12
5	22

That’s an excellent way to do it. Now, let’s say you want to represent count of cars not only by numbers written on paper, or by table,  but with marbles. This means that for 10 cars you will have 10 marbles. For 22 cars you will have 22 marbles. Maybe you did not have a paper at hand, so you needed to use marbles. Then you can use the marbles to count minutes. One marble for one minute, two marbles for two minutes etc. You realize that you can use marbles to represent the count of anything, apples, pears, oranges, rockets, cars, cars, etc. But, marbles doesn’t seem to be practical too much. You have to carry a lot of them, and, although they carry exact information about the counts of things they represent, they are too heavy to carry around. What else can be used to represent conveniently numbers of anything? It was discovered that length can be used to represent counts of other things. You draw the line of certain length and say that it represents a count of some objects. In our example we will draw a line of 10 mm to represent 10 cars, then line of 23 mm to represent 23 cars. The similar things can be done for time measurements. We will draw a line of say 1 cm for one minute then 2 cm for 2 minutes etc. We are using length of line to show the numbers, counts we have obtained. Please note, selecting line as a tool is our choice, because of convenience since its length represents different counts, numbers, quantities. Why it is convenient? Look at the graph we can make with lines. You can immediately see when a quantity is bigger than the other, you can see the trend between quantities and their other relationship, like minimum, maximum, etc. Of course all this things can be seen from a table too, but you have to go row by row, and do a lot of comparison. Table is one of the ways to represent pairs of numbers as we did here. Moreover, we can use either table or graph! But note how graph is more convenient, because it visually shows relationship that can be harder to spot in the table. Let’s repeat again, we used length of a line  (as we first did with marbles, or as you used fingers to count objects) as a universal way to represent counts of all other objects we are interested in any particular measurements or calculation. You also can line up numbers along the line, matching the length of line with a corresponding number. Note that you use line, length of line for pure convenience! There is no law that tells you need to use line, length of line to represent numbers and counts. It just happen to be a convenient visual way to use line to represent numbers.
There is nothing wrong using marbles or fingers, it's just seem to be impractical to communicate some ideas, however, these two methods of representing numbers are completely correct too. For instance while you think you can represent only integers with you fingers, you can actually represent any number! How? Say, you want to show number 3/5 with your fingers. You will show first 5 fingers and say this is how many times I divide number one. Then you show 3 fingers and say this is how many times I will multiply the quantity I've just obtained by dividing number one into 5 parts. The result of these two simple operations gives exactly 3/5 and it's communicated with fingers only!

Graph of a function serves the purpose to represent and show all pairs of numbers. Line on a graph should be seen as a continuous set of PAIRED NUMBERS.

Here are some examples of lines in action! Of course these are called graphs in mathematics, physics, engineering, economics, or in any other discipline where we have to show some quantitative relationship.

Fig 1. Examples of graphs

We are, perhaps, used to look at the graphs and focus on their shape, or on, even, their aesthetics values or characteristics. But, aesthetic value is of far less importance, as graphs are concerned. No matter how attractive graph looks, the line of it represents the PAIRED NUMBERS which are the lengths from each coordinate. That's the main purpose of graph. The shape of graph tells us how these pairs of numbers differ from one to another by their magnitudes.

So, using graph is about calculating surface area that can represent not only physical objects, like Pirate Island, airplane vertical stabilizer, or ship’s sail, but any surface that can come from graphical representation of various amounts, measured, counted, obtained by any way, and their quantitative relationships. We came up to the point that surface will represent something because we agreed, at the first place, that LENGTH represents something, that length of line we draw in graph corresponds to a number, count, obtained by measurement (or by any other mean).

Perhaps, it is a good place to describe steps that we use in these scenarios and how we can use thus acquired knowledge.

1. First we measure quantities of interest and note their relationships.
2. We represent the counts of these different quantities by lengths of line. We draw a graph.
3. Since we started dealing with lines, automatically then the surface bounded by these lines may represent a quantity of interest. Example is speed, v, and time, t. If we plot graph v(t), then automatically surface area under the graph will represent path taken.
4. We use our “method of rectangles” to calculate surface area, if required!

Note how we first focus, without considering the usage of graph, on relationships and measurements, mathematical treatment, of different objects. Then, we can use numbers by themselves, or use  table, or graph, for our further analysis. In graph we represent all those counts and numbers by lines of certain length, to show us, visually, quantitative relationships between measured objects.

Not also that lines, curves generated can be analyzed in their separate worlds of geometry or analytical geometry for example.

Note that coded color scheme can be introduced as the third dimension on the two dimensional graph. Volume integration is possible even here, but first you have to have a map, an association between a color and the number it represents.

[ graphs, math, math concepts, math education, math graphs, math, mathematics, physics concepts, tutoring calculus, schoo,l education, mathematics, physics, real world math, real world rational numbers ]

Math, Real World Examples, and How You Can Understand Math Even Better

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

You can read more "How math can be applied to so many different fields and how we can use math in real life".

As you can see, from the picture, 2 + 3 counts can come from many different fields.


You can then abstract counting and numbers 2 and 3 as common to all those fields. Moreover, you can deal with counts alone, not even thinking where they come from. You conclude that 2 + 3 = 5 no matter what you have been counting. That's why math can take off as a separate discipline, called pure math (not that applied math is dirty!). 

Many students are not puzzled so much with real world math applications. Students know it's there! Students appear to be puzzled how math can exist as a separate discipline. When and how did it take off from all those apples, pears, and other counting objects? How math can be a part of physics, economics, engineering, chemistry, finance, commerce, trading, accounting, and yet, mathematics can exists as a separate and independent discipline, from all of the fields it is applied to? I am answering this question in my blog here, and in this article as well.

When you see a person writing down 2 + 3 =...   you don't know what is she adding! But you know the result will be 5. That's pure math!.  You see, a person is writing down 2 + 3 =… Now,  let's say it again, you don’t know what that person has in mind, which objects she was counting. But, you know that the result will be 5! That’s, so called, pure math. You did a great job abstracting math from its real world applications. You may now want to continue to develop math! Make more examples! 5 + 3 = 8. 3 x 4 = 12. 8 - 5 = 3. You can say now, that if you have 7 and you add 8 you will get 15, just by dealing with numbers, no matter what objects and rules about them were involved. You realize that you can work with numbers only! And operations you have at your disposal, addition, subtraction, division, multiplication. be sure, there are no other operations in math. All other, so called, operations are just different sequences of these basic four, +, -, x, /.  That’s pure mathematics, as the mathematicians want to say. This is probably the most important step you have made in learning math.

As I mentioned, we can separate counts 2 and 3 into different area and play with them exclusively.

These "pure" math means playing only with numbers. It's just about making new counts, i.e. numbers,  (like 2 + 9 = 11 or 7 x 5 = 35) and doing operations on them, without worrying where they come from. Motivation to do operations is up to you since real world at this point is out of our interest! This play with numbers only, is very similar to playing with marbles and counting how many marbles you have after some operations, like, addition, subtractions etc. However, if you are interested WHAT you are counting, then you can associate objects names (the ones you are counting) to the actual numbers, like 5apples, 12cars, 15$, 7CDs, songs, steps, kilometers. You keep track what you are counting on one paper, but on another paper you do actual addition, subtraction etc only with numbers! Don't forget, the object names do not influence actual numerical operations. Object names are for you only, to keep track what you have been counting. It can be, again, compared with marbles. If you say, these 5 marbles represent 5 CDs you have counted, and then you add 3 more you can say that you have 8 marbles, or, since they represent CDs that you have 8 CDs. The trick here is that, when counting marbles, YOU can decide that they represent various things you are interested to count, like apples, cars, CDs, balls, etc. Marbles are in some sense very similar to numbers, they are universal method of representing the counts of different objects.

You can play with other numbers and math operations (addition, subtraction, division, multiplication) on them and be sure they will be common and be in intersection of other fields (outside math) too.

At one point you may want to play with numbers 3 and 7, like 3 + 7 = 10, and perhaps think what are the fields where these counts can come from. Note that you used those numbers, 3 and 7, without knowing what you have counted! That's pure math! Once you are aware that you can deal with numbers by themselves, as separate entities, as we just did, you enter the area of pure math research :-) Congratulations!

Now, let's take a look at that intersecting area where the numbers are. You can enlarge that area and add more numbers. Moreover, you can add any counts you want and investigate any operations on them as you wish. You can add, 1, 5, 7, 10, 25, 26, 27, ...102, 1237. They are all in that central area. But you can, now, look at that area independently from what intersects it. You can add more numbers, like 2/3, 4/5, 6/7, 1/3, 1/129. You see how you can create new numbers (which are called rational numbers, because they are ratios of integers). That is all math, pure math. Other fields that use math and counts can use your results instead of rediscovering them each time when they need to calculate something. These fields can be any of academic fields and the math they use is called "applied math". But, you see that mathematical results can be motivated by real world examples, but also they can be devices by you, when you just play with numbers. Essentially, you do not need real world examples to develop mathematics. One example is geometry of Lobachevsky that has been developed first and only then discovered and used by Einstein in his Theory of Relativity. On the other hand, Newton's problems in Physics, like calculating speed, led to development of calculus. But be aware, calculus could be developed even without looking at physical problems.

At this point I would like to show how mathematics can be an independent discipline and also can be, and is, used in real life. I will make a comparison between lines and some shapes and numbers. When you look at the building, and you want to draw it, you  will draw some lines, squares, rectangles, trying to mimic the shape of the building as truthfully as possible. Look at the lines you draw. You draw vertical, horizontal, and lines at any arbitrary angle. It appears that which line and where you will draw it will depend on the shape of the building. And that's true. Now, look away from the  building and get a new, blank sheet of paper. Draw a line on it. Draw horizontal lines, vertical, and at arbitrary angles. You are not required to draw a building. Just draw lines. You see how you abstracted lines, and, moreover you can play with them without taking care whether they represent anything in real world. Moreover, you can draw completely new building with your lines and call a company to actually build a new object from your drawings!

Similar things is with numbers. You can count real objects and get their numbers and deal with them. You can add, subtract, multiply numbers following the real world examples, like, "how many liters of gas I will use if I travel 125 km with the car that consumes 8.9L/km..." etc. When you calculate this or any other example, you deal only with numbers. You keep track of units, like what you have counted, aside. But, then, you can notice that 7 x 5 = 35 regardless what is counted! Similar thing happened with lines! You could draw lines without worrying if they represent any object in real world. Now, you can play with numbers, any number, and use ANY operations on them without worrying do they currently represent anything in real world. That's the essence of math. When you do applied math you still do pure calculations while keeping aside the units, what you have counted. But, in pure math you start with some numbers, it's up to you with which ones, without providing reasons why they are there, and do calculations on them. Try both scenarios and you will see the point! 


Here are more links you might like as well:

• Real world examples for rational numbers, for kids
• 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
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• How to understand the role of math in economics, physics, engineering, and in other fields

[ applied math, applied mathematics, education, math concepts, math education ]