tag:blogger.com,1999:blog-27456894984486080062017-02-08T20:57:12.720-08:00About Mathematics and Real World Mathematics ApplicationsUnlocking the secrets of quantitative reasoning. Rewiring your existing math knowledge into a new, powerful web of innovation generating ideas. Axioms Discovery, Axiomatic Frontiers, Directional Thinking, Specific Consequences, Logic, Intuition, Innovation, Invention. http://www.webensource.com/mathematicsBill Harfordhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.comBlogger83125tag:blogger.com,1999:blog-2745689498448608006.post-88287708176594568862015-07-02T06:07:00.001-07:002015-07-02T06:07:25.762-07:00One Derivation of Euler's Equation for Complex Numbers<div dir="ltr" style="text-align: left;" trbidi="on">One derivation of Euler's equation using series expansions for e^x, cos(f), sin(f).<br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-LYHUsSKdWIo/VZU3WrIaM1I/AAAAAAAABR8/oZE3sfGZLf8/s1600/One%2Bderivation%2Bof%2BEuler%2Bidentity.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="292" src="http://3.bp.blogspot.com/-LYHUsSKdWIo/VZU3WrIaM1I/AAAAAAAABR8/oZE3sfGZLf8/s640/One%2Bderivation%2Bof%2BEuler%2Bidentity.jpg" width="640" /></a></div><br />Reference <a href="http://www.ee.nmt.edu/~elosery/lectures/Quadrature_signals.pdf">http://www.ee.nmt.edu/~elosery/lectures/Quadrature_signals.pdf</a><br /><br /><br /><br /><br />[complex numbers, Euler, Euler's identity, Euler's equation]</div>Bill Harfordhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-59542933773453567892014-07-02T10:15:00.000-07:002014-07-02T10:15:10.441-07:00Why does zero factorial (0!) equal one, i.e. 0! = 1?<div dir="ltr" style="text-align: left;" trbidi="on">Here are the most useful links with answers why does zero factorial equal one, i.e. 0! = 1:<br /><br /><a href="http://www.quora.com/Mathematics/Why-does-zero-factorial-0-equal-one">http://www.quora.com/Mathematics/Why-does-zero-factorial-0-equal-one</a><br /><br /><a href="http://wiki.answers.com/Q/Why_is_zero_factorial_equal_to_one">http://wiki.answers.com/Q/Why_is_zero_factorial_equal_to_one</a><br /><br /><a href="http://en.wikipedia.org/wiki/Factorial">http://en.wikipedia.org/wiki/Factorial</a><br /><br /><a href="http://math.stackexchange.com/questions/25333/why-does-0-1">http://math.stackexchange.com/questions/25333/why-does-0-1</a><br /><br /><a href="https://ca.answers.yahoo.com/question/index?qid=20090116132324AACQIGU">https://ca.answers.yahoo.com/question/index?qid=20090116132324AACQIGU</a><br /><br /><a href="http://www.zero-factorial.com/whatis.html" target="_blank">http://www.zero-factorial.com/whatis.html</a><br /><br /><a href="http://statistics.about.com/od/ProbHelpandTutorials/a/Why-Does-Zero-Factorial-Equal-One.htm" target="_blank"> http://statistics.about.com/od/ProbHelpandTutorials/a/Why-Does-Zero-Factorial-Equal-One.htm</a><br /><br /><a href="http://mathforum.org/library/drmath/view/57128.html">http://mathforum.org/library/drmath/view/57128.html</a><br /><br /><br /><br />[factorial, factoriel, factorial function, zero factorial]</div>Bill Harfordhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com2tag:blogger.com,1999:blog-2745689498448608006.post-61598625156035080882014-06-11T10:09:00.000-07:002014-07-23T11:36:09.503-07:00What is Mathematics?<div dir="ltr" style="text-align: left;" trbidi="on">Here is the link for the essay <a href="http://www.maa.org/external_archive/devlin/LockhartsLament.pdf" target="_blank">" A Mathematician's Lament" by Paul Lockhart.</a><br /><br /><a href="http://books.google.ca/books/about/What_is_Mathematics_Really.html?id=R-qgdx2A5b0C" target="_blank">"What is Mathematics, Really?"</a> by Reuben Hersh, in <a href="http://books.google.ca/books/about/What_is_Mathematics_Really.html?id=R-qgdx2A5b0C" target="_blank">Google Books</a>.<br /><br /><a href="http://books.google.ca/books?id=_kYBqLc5QoQC&printsec=frontcover&dq=What+is+Mathematics,+Courant&hl=en&sa=X&ei=2si2U7TFOJKWqAbAxYHYCQ&ved=0CCUQ6AEwAA#v=onepage&q=What%20is%20Mathematics%2C%20Courant&f=false" target="_blank">"What is Mathematics? An Elementary Approach to Ideas and Methods",</a> Richard Courant, Herbert Robbins, in <a href="http://books.google.ca/books?id=_kYBqLc5QoQC&printsec=frontcover&dq=What+is+Mathematics,+Courant&hl=en&sa=X&ei=2si2U7TFOJKWqAbAxYHYCQ&ved=0CCUQ6AEwAA#v=onepage&q=What%20is%20Mathematics%2C%20Courant&f=false" target="_blank">Google Books.</a><br /><br /><a href="http://www.google.ca/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0CB4QFjAA&url=http%3A%2F%2Fwww.springer.com%2Fcda%2Fcontent%2Fdocument%2Fcda_downloaddocument%2F9780817683931-c2.pdf%3FSGWID%3D0-0-45-1353006-p174671167&ei=eNK6U5DBLtSgyASLuoKICw&usg=AFQjCNG15IuvO4sHWMJsK8DPzmrtscrD6A&sig2=ReKKeqjbTkt4oCXBKvoMwQ" target="_blank">Mathematical Intuition (Poincaré, Polya, Dewey)</a> by Reuben Hersh, University of New Mexico, TMME, vol8, nos.1&2, p .35<br /><br /><a href="http://books.google.ca/books?id=CbCDKLbm_-UC&pg=PA265&dq=Dehaene,+S.:+The+Number+Sense.+Oxford+University+Press,+New+York+%281997%29&hl=en&sa=X&ei=f9O6U8u9F4OgyAT5uYAY&ved=0CCwQ6AEwAA#v=onepage&q=Dehaene%2C%20S.%3A%20The%20Number%20Sense.%20Oxford%20University%20Press%2C%20New%20York%20%281997%29&f=false" target="_blank"> The Number Sense : How the Mind Creates Mathematics, </a>by Stanislas Dehaene, Google Books.<br /><br /><a href="https://archive.org/stream/psychologyofnumb00mcleuoft#page/n11/mode/2up" target="_blank"><cite class="rw">The Psychology of Number and its Applications to Methods of Teaching Arithmetic</cite></a> by James A. McLellan; John Dewey, at www.openlibrary.org.<br /><br /><a href="http://www.webensource.com/mathematics/#HowMathCanBeApplied" target="_blank">How Math Can Be Applied To So Many Different Fields</a>, <a href="http://www.webensource.com/mathematics/#Top" target="_blank">Mathematics and Quantitative Reasoning web site</a>, B. Harford <br /><br /></div>Bill Harfordhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-54975923612276296702013-02-28T13:24:00.003-08:002013-02-28T13:24:26.071-08:00From Basketball, Financial Math to Pure Math and Back<div dir="ltr" style="text-align: left;" trbidi="on"><!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument></xml><![endif]--><br /><!--[if !mso]><img src="//img2.blogblog.com/img/video_object.png" style="background-color: #b2b2b2; " class="BLOGGER-object-element tr_noresize tr_placeholder" id="ieooui" data-original-id="ieooui" /><style>st1\:*{behavior:url(#ieooui) } </style><![endif]--><!--[if gte mso 10]><style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman";} </style><![endif]--> <br />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, there are a few operations on it, I have just seen that, a clean and dry concept, a quite straightforward count of objects you have been dealing with. Five apples, seven pears, six pencils. The number five is common to all of them. We have abstracted it, and together with other fellow numbers (three, four, seven,, 128, 349, ...) it is a part of a number system we are familiar and we work with.<br /><br /> From our everyday encounters with mathematics, we may have a feeling there are only integers present in the world of math, and that it is not really clear where and how those mathematicians find so many exotic numerical concepts, so many other kinds of numbers, like rational, irrational, algebraic ... Moreover, you may even think that, without some real objects to count or to 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 quantification of things in the real world, to the objects counted, measured, that they are inseparable. You may think that a number, despite its "mathematical purity", somehow shares other, non mathematical properties, of the objects it represents the count of.<br /><br /> 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 want to put a thought about math and numbers.<br /> 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 only positive integers in the mathematical world, and that everything else, i.e. definition of other kinds of numbers, is the work of men. I support that view. <span style="mso-spacerun: yes;"> </span>Essentially, many mathematicians do as well. Here is the flavour of that perspective. Negative numbers are positive numbers with a negative sign. Rational numbers are ratios of two integers, m/n, (where n is not equal 0). Real numbers (rational and irrational) are limiting values of rational numbers’ sums and sequences (which are in turn ratios of integers), convergent sums of rational numbers, where rational numbers are smaller and smaller as there are more and more of them. As we can see that all these numbers are, fundamentally, constructed from positive integers.<br /><br /> 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, cars...That's why we need labels below, or beside, numbers, to remind us what is measured, what is counted. For real world math applications that’s absolutely necessary, because by looking at the number only, we can not conclude where the count comes from. When you write 5 + 3 = 8, you can apply this result to any number of objects with these matching counts. So, numbers do not hold or hide properties of the objects they are counts of. As 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, without any reference to a real life object. No need to explain if it is a count of anything. <b style="mso-bidi-font-weight: normal;">Pure math doesn't care about who or what generated numbers, it doesn't care where the numbers are coming from</b>. Math works with clear, pure numbers, and numbers only. It is a very important conclusion. You may think, that properties of numbers depend on the objects that have generated them, and there are no other intrinsic properties of numbers other than describing them as a part of real world objects. But, it is not so. While you can have a rich description of objects and millions of colourful reasons why you have counted five objects, the 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.<br /><br /> Now, you may ask, if we have eliminated any trace of objects that a number can represent a count of, that might have generated the number, what are the properties left to this abstracted number? What are the numbers' properties?<br /><br /> 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 or reasoning why we have counted apples, or why we have turned left on the road and then drive 10 km, and not turned right. Pure math is only interested in numbers provided to it. Among those properties of numbers 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, what are the prime numbers, how many of them are there, etc. That's what pure math is about, and these are the properties a number has.<br /><br /> 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 actual calculations, numbers 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 8 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 numbers 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 have 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, membranes, stars, kisses.<br /> While, as we have seen, pure math doesn't care where the numbers come from, when applying math we do care very much how the counts are generated, where the counts are coming from and where the calculation results will go. We even have invented mechanical, electrical, electronic devices to keep track of these counted objects. We have all kinds of dials that keep track of fuel consumption, temperature, time, distance, speed. Imagine that! We have devices which keep track of counted objects so when we look at them and see number five, or seven, or nine, we will know what that number represents the count of! Say, you have several dials in front of you, and they show all number 5. It is the same number 5, with the same numerical, mathematical, properties, but represents counts of different objects or measurements. We can say that the power of mathematics is derived from noticing that number 5 is the same for many objects and abstracting that number 5 from them, then investigating number 5 properties. After mathematical investigation we can go back, from pure number 5 to the real world!<br /><br /> There are dials in cars, 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, enjoying their own purity like, 5, 23, 120, 35, 2.78 without knowing what they represent until we assigned them a proper dial units. This example shows the essential 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. Pure math deals with numbers only, while in applied math we drag the names of objects, associated them with numbers. In other words, we keep track what is counted.<br /> Now, when dealing with pure numbers, we may go to a great extent to investigate all kinds of numerical, mathematical properties of all kinds of numbers and sets of numbers. Hence a spectrum of mathematical areas like linear algebra, calculus, real analysis, etc. These mathematical disciplines are all useful and there is, frequently, a beauty and elegance in their results. But, often, we do not need to apply or use all those mathematical properties, and pure math results, in everyday situations. Excelling in some business endeavour frequently depends on actually knowing <b style="mso-bidi-font-weight: normal;">what<span style="mso-bidi-font-weight: bold;"> and why something is counted</span></b>, while, at the same time, mathematics involved, can be quite simple. When I say business, I mean business in usual sense, like finance, trading, engineering, but also, I mean, for instance, as we will see soon, basketball, and even poker.<br /> Let’s go now into a basketball game. When playing <b>basketball </b>we also need to know some math, at least working with positive integers and zero. However, in the domain of basketball game, knowledge of basketball rules are way more important than math,<br /><br /> Those basketball rules are mostly non mathematical. Most of basketball rules do not deal with any kind of quantification, which doesn't make them at all less significant. Moreover, they are way more important ingredient, and represent more complex part, for that matter, of a basketball game, than adding the numbers. <br /> You can posses knowledge of adding integers, but without knowing basketball rules, and without knowing how to play basketball, you will not move anywhere in a basketball team or in a <span style="mso-spacerun: yes;"> </span>game. Moreover, <b>basketball rules are actual axioms of a basketball game. </b>And, every move in the basketball court, any 30 seconds strategy development by one team or the other, corresponds to theorems of a 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 just to define <b style="mso-bidi-font-weight: normal;">what belongs to a set </b>(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 its elements (like points in basketball) you need to know areas other than math, and to develop logic, creativity, even intuition in those non mathematical 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! That number has to be accurate!<br /><br /> 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 the "winning game" inside math too, as you would in basketball game. You have to show creativity in math as well, mostly in specifying theorems, and constructing their proofs!<br /><br /> In business, it is often more important to know where the numbers are coming from than to know in detail the numbers’ properties. For instance, in poker. again, only integers and rational numbers (in calculating probabilities) are involved ( we will skip stochastic processes and calculus for now). 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 to the real world rules, in this case world of poker,, we have to use that abstracted number 5 and put it back to the objects it may have been abstracted from, in this case cards and players. 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, can be associated, linked to different players, suits, strategies, scenarios.<br /> Let’s consider another example, in <b style="mso-bidi-font-weight: normal;">finance</b>. Any contract you have signed, for instance contract for a credit card, is actual detailed list of definitions what belongs to a certain set. For example, 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 just a matter of declaring a rational number 23.789,32, but what sets it belongs to is extraneous to mathematics, it's in the domain of financial definitions, even in the domain of required signatures. Are you, or someone else, is going to pay the bill of $23,789.32, is a non mathematical question (it’s even a legal matter), while mathematics involved is quite simple. It's a number 23,789.32. <br /> Note, when you are paid for your basketball game, suddenly you have math from two domains fused together! It may be that the number of points you scored are directly linked to a number of dollars you will be paid. Two domains, of <b style="mso-bidi-font-weight: normal;">sport and finance</b>, are linked together via <b style="mso-bidi-font-weight: normal;">monetary compensation rules</b>, 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 and assigned to you, as a basketball player, after the set of games.<br /> <br /> </div>Bill Harfordhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-35399848596772331292013-02-27T15:58:00.002-08:002013-02-27T15:58:12.111-08:00Mathematical Proof for Enthusiasts - What It Is And What It is Not<div dir="ltr" style="text-align: left;" trbidi="on"><!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument></xml><![endif]--><br /><!--[if gte mso 10]><style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman";} </style><![endif]--> <br /><div align="center" style="text-align: center;"></div>Important things you can learn from mathematics are not about counting only, but also about mathematics’ methods of discovering new truths about numbers.<br /><br /> With the term <b style="mso-bidi-font-weight: normal;">mathematical proof </b>we want to indicate a logical proof, i.e. proof using logical inference rules, in the field of mathematics, as oppose to other disciplines or area of human activity. Hence, 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.<br /><br /> 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. Hence, you will have to go back and fix your fundamental axioms.<br /> Often 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!?".<br /><br /> 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.<br /><br /> 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 obtained.<br /> 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 <b style="mso-bidi-font-weight: normal;">established in mathematics.</b> 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..".<br /><br /> Which axioms, premises, and theorems you will start the proof with is a <i style="mso-bidi-font-style: normal;">matter of art</i>, <i style="mso-bidi-font-style: normal;">intuition, trial and error, or even true genius</i>. 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 or 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 to which axioms, premises, or theorems you will use <i style="mso-bidi-font-style: normal;">to start the construction</i> of a proof. You can use your intuition, feeling, experience, even emotions, to select starting points of a proof, to chose initial axioms, premises, 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.<br /><br /> 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 <i style="mso-bidi-font-style: normal;">on mathematical axioms</i>, 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.<br /><br /> 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.<br /><br /> 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.<br /><br /> 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.<br /><br /> 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.<br /><br /> 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. <br /> </div>Bill Harfordhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-90084673844670663402013-02-27T13:55:00.003-08:002013-02-27T13:55:45.757-08:00More About the Concept of a Set and the Concept of a Number<div dir="ltr" style="text-align: left;" trbidi="on"><!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument></xml><![endif]--><br /><!--[if !mso]><img src="//img2.blogblog.com/img/video_object.png" style="background-color: #b2b2b2; " class="BLOGGER-object-element tr_noresize tr_placeholder" id="ieooui" data-original-id="ieooui" /><style>st1\:*{behavior:url(#ieooui) } </style><![endif]--><!--[if gte mso 10]><style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman";} </style><![endif]--> <br /><br /> For instance, let's take a look at the cars on a highway, apples on a table, coffee cups in a coffee shop, apples in the basket. Without our intellect initiative, our thought action, will, our specific direction of thinking, objects will sit on the table or in their space, physically undisturbed and conceptually unanalyzed. They are and will be 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.<br /><br /> 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.<br /><br /> 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.<br /><br /> 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. <b>It is us who grouped them into sets, in our minds. </b><span style="mso-bidi-font-weight: bold;">I</span>n 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.<br /><br /> Since, as we have seen, we invented, discovered a <b>direction of thinking</b>which did not exist just a minute ago, to think of objects in a group, we may want to proceed further with our analysis. <br /> 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!<br /> 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.<br /><br /> 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.<br /><br /> 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!<br /><br /> 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.<br /><br /> 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? <br /> 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 <b style="mso-bidi-font-weight: normal;">cardinalities</b>. And, in most cases, we are after <b style="mso-bidi-font-weight: normal;">these cardinalities</b> 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!<br /><br /> 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. <br /> </div>Bill Harfordhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-87333888245245077772013-02-24T15:02:00.001-08:002013-02-24T17:03:40.703-08:00Abstract Nature of Geomatrical Figures<div dir="ltr" style="text-align: left;" trbidi="on"><br />One of the fascinating points observations about a circle is that the circle is a pure abstraction. It does not exist really anywhere but in our minds as a perfect abstraction of all points equally distant from a one single point, the circle center. No perfect circle can be found in nature, only approximations of it, and each one will have some imperfections, yet the major theories are based of this unexcited in nature geometrical figure. The same can be said for triangle, square, and most of other geometric figures. <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-9omPEc0KQV8/USqbrbRXORI/AAAAAAAAA9U/zYmHOgvxpVQ/s1600/Circle+Triangle+Square+Lines+as+abstract+geomatric+figures.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="96" src="http://4.bp.blogspot.com/-9omPEc0KQV8/USqbrbRXORI/AAAAAAAAA9U/zYmHOgvxpVQ/s400/Circle+Triangle+Square+Lines+as+abstract+geomatric+figures.jpg" width="400" /></a></div><br />Extrapolating these thoughts to electrical engineering, for example three-phase power systems are built around electric fields that by construction are with phase difference of 120 degrees, no ideal voltage is produced that calculates exactly sin and cos functions for the circuitry analyses (this includes complex numbers, that translates to active and reactive power in electric power systems). <br /><br /><br /></div>Bill Harfordhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-83055083660868722312013-02-09T15:02:00.004-08:002013-02-17T09:59:00.962-08:00Function, map, pair in mathematics<div dir="ltr" style="text-align: left;" trbidi="on">As an illustration of a mathematical function concept a teacher can write arbitrary numbers, each on a separate, rectangular piece of paper, and then let the students pair them arbitrarily, on the table, and then investigate numerical, mathematical, properties of the those pairs of numbers. The properties may be what sequence the pairs they can be put in, or the order of numbers magnitudes in different pairs.<br /><br />In the same way we can pair different fruits with, say, CDs (for whatever reason!), we can pair numbers together, and even fruits with numbers! Fruits and numbers are paired when an exchange of fruits for money takes place in an open fruit market! Note how is a third agent present when we pair fruits and numbers. It is the exchange "agent" that motivates pairing and that gives sense to the pairing action.<br /><br />These examples should reinforce main concept that a function is a pair of numbers and not necessarily a formula that gives y for given x. Function is not always output for a given input. Function is not a formula. Function is a map or pairs of numbers.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-7i3z60SOHOU/URbVRP5uY6I/AAAAAAAAA7g/TbioNV0aqjI/s1600/Mathematical+Function,+map,+pairing.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="http://2.bp.blogspot.com/-7i3z60SOHOU/URbVRP5uY6I/AAAAAAAAA7g/TbioNV0aqjI/s400/Mathematical+Function,+map,+pairing.jpg" width="341" /></a></div>[ math, mathematics, mathematical function, function, map ] </div>Bill Harfordhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-303514396917891552012-09-12T12:33:00.005-07:002013-02-25T06:14:24.060-08:00Flow of Quantification Results - Pure and Applied Math<div dir="ltr" style="text-align: left;" trbidi="on"><br /><br /><div style="text-align: left;">When you specify how much of some objects you need or want to count, when you first have a pure number in mind and only after that you chose the objects to count, based on that number, you bridge the space and connect pure and applied math. The number you had in your mind belongs to pure math domain, while the quantifiable objects you decide to count, together with the chosen number, belong to the applied math domain.</div><div style="text-align: left;"><br /></div><div style="text-align: left;">Applying mathematics means, as per the illustration, filtering out the units, objects counted, and dealing with pure sets and counts, numbers only. With these numbers and quantitative relations you enter the world of pure math, obtain the results, by doing new calculations or by using already proven theorems, and then return back the result to the real world, reattaching the units on the way back.<br /></div><div style="text-align: left;"><a href="http://4.bp.blogspot.com/-GyssT7Dpbks/UFDjc2yJehI/AAAAAAAAA6Q/HW3cs9_6qNA/s1600/Pure+and+Applied+Mathematics+-+Flow+of+Quantification+Results.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="240" src="http://4.bp.blogspot.com/-GyssT7Dpbks/UFDjc2yJehI/AAAAAAAAA6Q/HW3cs9_6qNA/s400/Pure+and+Applied+Mathematics+-+Flow+of+Quantification+Results.jpg" width="400" /></a></div><div style="text-align: left;">But, it also means, that the logic, reasoning within pure mathematics, similar to the chain of political reasoning and decisions before a certain action is taken, is important when it is required to know exact quantity that will be used in the real world scenario. Accuracy of pure mathematical processing, calculations, proofs, theorem resuse, is a significant, a central factor to obtain a correct number and hence go ahead with some directive how much of some objects need to be counted. Of course, initial conditions, numbers entered the pure math mechanisms, are coming from quantification in the real world, and that link will be the major connection for units reattachment and decision what to count and at which magnitude.</div><div style="text-align: left;"><br /></div><div style="text-align: left;">Hence, the importance of theorems, theorem proofs, although they may seem too abstract and distant from real world application at the first sight, has central role in obtaining accurate results that will be used back in decision making in real world domain from which initial conditions originated.<br /><br />Quantification result, or the obtained number, need not to be used in mathematical domain only or to quantify something else. It can be used, as a true or false fact, or as an information, as a part of any decision making process of any domain, in any hybrid axiomatic system, or any logical system as a part of logical expression and in any logical connective. </div></div>Bill Harfordhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-91750769855272407262012-08-16T11:19:00.004-07:002012-09-17T18:26:31.247-07:00That famous Cauchy definition of the limit, and another view that explains the elusive concept.<div dir="ltr" style="text-align: left;" trbidi="on">If you can come arbitrarily close to a value, in the limit process, then that value is the limit. Of course, there has been always a question "but there is still that small error there, no matter how many elements we add, and no matter how close we are". It is true if you do not let n -> infinity. If n-> infinity then error goes to zero. But there is another nice thing about it. The "arbitrary close" statement guarantees that WHEN n -> to infinity that value will be the limit. It does not say it is the limit if you have finite number of values, no matter how big that number is. It says that, essentially, the fact that you can come "arbitrarily close", i.e. "close as much as you want" to that value, in that, and only in that case, it guarantees that, when n-> infinity, that value is the limit. No other statement will guarantee that. No other statement will guarantee that anything similar will happen when n-> infinity. That's the statement you want. <br /><br />[ applied math, definition of limit, limit in mathematics, concept of a limit, limit, limiting value, integral, differential, ]</div>Bill Harfordhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-80676578972908053582012-06-07T15:14:00.002-07:002012-06-07T15:14:48.178-07:00Set Theory, Units and Why We Can Multiply Apples and Oranges but We Cannot Add Them<div dir="ltr" style="text-align: left;" trbidi="on"><!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><!--[if !mso]><img src="//img2.blogblog.com/img/video_object.png" style="background-color: #b2b2b2; " class="BLOGGER-object-element tr_noresize tr_placeholder" id="ieooui" data-original-id="ieooui" /> <style>st1\:*{behavior:url(#ieooui) } </style> <![endif]--><!--[if gte mso 10]> <style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman";} </style> <![endif]--> <br /><div class="MsoNormal"><span style="font-size: 10.0pt;">You have, probably, been told that you can not add apples and oranges. Why is that? But, you may realized or may have been taught that you can multiply them. How is that possible? And this state of affairs may be following you through your school, education, and even (non-mathematical!) career. Here is the explanation.</span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 10.0pt;">Let’s say there are some apples on the table and let’s say we want to count them. We decide we want to count apples. And there we go. Suppose there are eight apples on the table, and we correctly count them, thus obtaining count of eight. Eight apples. The most important thing here is what we decide to count. We decided to count apples. And nothing else. It is the apples count we are interested in and not any other objects. That’s our definition what belongs to our set, specific set that (qwe decided!) will contain apples only. Now, if we see pencils on the tables, pears, oranges, books, they don’t match our definition, they are not apples, and hence they will not be added to our “set”. That’s the reason we “cannot” add oranges and apples. It is our decision that we want to count apples only, and our decisions if more apples are put on the table we will add them. </span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 10.0pt;">If we decide that our set, the things we want to count, will have other objects and that we want to have a total number of objects we are interested in, then we have to specify that in our definition. We have to say, now, that we have decided to count, as members of our set, say apples, books, and oranges. We may not be interested at all how many of each are there, we just want their total number. IN this case, we clearly can add apples, books, and oranges together, because it is our definition of<span style="mso-spacerun: yes;"> </span>what belongs to a set that determines elements and number of elements in that set. And, with this set definition, we clearly can add apples and oranges, and<span style="mso-spacerun: yes;"> </span>for that matter any object we decide will belong to our set of interest. </span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 10.0pt;">Let’s see again in which scenario apples and oranges can be added again. Suppose that, on our table, we have 8 apples, 5 books, 7 oranges, and 3 pencils. And suppose<span style="mso-spacerun: yes;"> </span>that we define the set as “count all fruits on the table”. IN that case we will not count books and pencils, but we will correctly add together apples and oranges, because they are fruits and that’s the definition of a<span style="mso-spacerun: yes;"> </span>set membership. Hence, our set will have 8 fruits (apples) and 7 fruits (oranges), giving the sum of 15 fruits.<span style="mso-spacerun: yes;"> </span></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 10.0pt;">The conclusion is that the set membership definition determines what will belong to a set, what kind of objects, and that this definition will determine which objects we can add together. Definition of the set membership is essential to determine which objects we can count together. </span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 10.0pt;">Ok, so, we clarified that, when the set definition says “count only apples” we can not add apples and oranges together. But, when you say “multiply apples and oranges” we can do that. Why?</span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 10.0pt;">The answer lies in the two step process we always do, but we may not be aware of that. And, in some language imprecision as well. You do not multiply apples and oranges, You multiply the <b style="mso-bidi-font-weight: normal;">numbers</b> obtained by counting apples and oranges. Let’s suppose you want to multiply 5 apples with 3 oranges. But then, let’s, for a moment, focus only on 5 apples. Or, even there is a basket of apples, say around 30, beside the table. You can say “I want 5 apples on the table”. You take five apples from the basket and put them on the table. Now, you can say, I want 3 times 5 apples on the table. Then you<span style="mso-spacerun: yes;"> </span>take, from the basket, groups of 5 apples, 3 times. You essentially took 3 x 5 = 15 apples from the basket. But, where that number 3 came from. Ok, you can say, and you will be right, it came from your head,<span style="mso-spacerun: yes;"> </span>you just imagined number 3 and decided to count 3 x 5 = 15 apples from the basket. So, you have this, 3 x 5apples = 15apples. But, notice! While you arbitrarily imagined that number 3, it can also come from counting another objects! You can say, you have counted people in the room, there were 3 of them and each of them will have to have 5 apples. Hence, you obtained number 3, this time not from your head, but from real counting of the people in the room. And, again you will have 15 apples on the table, from the basket. We can write that as 3 people x 5 apples = 15 [ people x apples ] . The “unit” here is [people x apples ] and essentially it tells us HOW we have obtained numbers used in the multiplication! By these “units” we keep track what we have counted. So, it is not at all that we have “multiplied people and apples”, but that we have multiplied numbers obtained by counting people and apples. If we use oranges and apples, and say, I want to put 5 apples beside each of 3 oranges, how many apples I will need to take out of the basket, the answer will be 3 x 5 = 15 [oranges x apples ].</span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 10.0pt;">Only numbers can be multiplied, added, divided, subtracted. Objects, concepts, like apples, oranges, people, cars, pencils, books, can not be ‘multiplied”, they can be <b style="mso-bidi-font-weight: normal;">counted</b> only. By counting them we obtain the numbers to work with. It is with these <span style="mso-spacerun: yes;"> </span>numbers only that we do mathematical operations. You know that<span style="mso-spacerun: yes;"> </span>4 + 3 = 7 no matter if we count apples, or oranges, or cars. It is an universal result. When you write down 4 + </span><span style="font-size: 10.0pt;">3, a</span><span style="font-size: 10.0pt;"> friend beside you doesn’t know are you counting in your head CDs, dollars, or apples, but he knows that the result will be 7 no matter what. </span></div></div>Bill Harfordhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-24585106028167225432012-05-29T13:22:00.092-07:002012-07-26T05:57:59.339-07:00The Links Between Different Axiomatic Systems and Cross-Axiomatic Ideas Generation<div dir="ltr" style="text-align: left;" trbidi="on">Combinatorial creativity needs to recognize the cross-axiomatic links between partially or fully axiomatized systems combined. Puncturing the axiomatic membranes around conceptually delineated ideas using cross-axiomatic probing needle is the key to creative thinking. <br /><br />The systems I talk about here need not to be mathematical at all. Axioms and theorems can be a part of any system that uses logic.<br /><br />Let's take a look at the drawing. Small red, blue, green squares are theorems. The diagram shows that each system, S1, S2, S3, S4, S5, is axiomatized. The system itself is developed from the set of axioms, as indicated by the rectangle where they reside. It can be seen how the theorems (small squares) are derived from the axioms and other theorems. Even a proof is indicated to show that connection. <br /><a href="http://4.bp.blogspot.com/-KgwCWUUxbgw/T8ZxP9UdZ_I/AAAAAAAAAxs/YdYQX1iq82g/s1600/Axiomatic+Systems+Links+and+Ideas+Generation+2.jpg" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-KgwCWUUxbgw/T8ZxP9UdZ_I/AAAAAAAAAxs/YdYQX1iq82g/s1600/Axiomatic+Systems+Links+and+Ideas+Generation+2.jpg" /></a> <br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />Now, notice how the theorems from system S1 influence the theorem conception, even definition, in the system S2. Example can be the processes in physics that motivated developments in mathematics, but there are many other examples between other fields as well. We can see that these small squares have two fold connections. Let's look at the system S2. One line comes from system S1, and another comes from S2's own axioms. That fact shows one of the most important thing, and it is that theorem in one system (S2) can be motivated by the other system, system extraneous to S2, in this case the system S1, but also that those theorems can be defined directly from the axioms in the system S2. The proof of these theorems, in system S2, can come later, and if needed, can be and must be constructed only from the axioms (and already proved theorems) in the system S2. That leads us to the second most important point. The theorems in one system, example here is S2, must an can be proved only using axioms and already proved theorems from the system S2, no matter how clear and inspiring motivation and illustration is coming from the system S1.<br /><br />But, in order to advance in the creative work, we don't have always time to prove each and every step or a conclusion. Most often we would accept that the theorems, say in system S2, give real, true, correct consequences when the results are applied to the system S1. Example of this can be mathematics applied in physics or engineering. As Reuben Hersh wrote, "controlling a rocket trip to the moon is not an exercise in mathematical rigor.". This connection and method to accept that a theorem is true in one system as long as it gives and has the correct and desired, true consequences in another, linked system, that uses the theorems, is essential for our uninhibited, creative, combinatorial thinking, the use of our intuition that proved so successful in many scientific discoveries and engineering inventions.<br /><br />When we take a break of a tough problem at hand we have been trying to solve for hours, and start an undemanding task, not related to the problem we have been solving, we give our unconscious mind time to process and frequently find solution, while working in the background. We daydream with systems we just loosely axiomatize or don't axiomatize at all. We work with assumptions that we perceive or assume are correct and true, and we probe them with other systems to check whether the results are correct or even possible. That may be core of the creative thinking.<br /><br />Which fields to select and include into this game is the most challenging thing and usually amounts to an invention, innovation, new idea generation, and unexpected success in the obtained results. This <a href="http://www.brainpickings.org/">combinatorial thinking </a>is the core of the generation of new ideas.<br /><br />The diagram B shows how the new, hybrid, cross-axiomatic systems are created, by our thinking, combinatorial process. Note how theorems from the S3 and S4, when combined can be a theorem in the system S5 and very often some of the axioms in the system S5. <br /><br />As I have mentioned, we usually don't need, and do not axiomatize systems we work with during our intuitive, creative thinking, and even during the design. Axiomatization can come later. Axioms, and proofs constructed from them, can eliminate any contradictions within the system, that may creep from our, potentially incorrect assumptions since we may have been tricked by nice motivations and examples coming from other connected systems.<br /><br />Here is another diagram, an example in, mostly, scientific fields. Picture shows that each field is encircled within its own system of axioms (discovered or not).<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-aMioZ3JQWNk/T8Zuu7dZ-zI/AAAAAAAAAxY/wuW5pcxIRwE/s1600/Mathematics+and+its+Axiomatic+Relations+With+Other+Fields.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="640" src="http://1.bp.blogspot.com/-aMioZ3JQWNk/T8Zuu7dZ-zI/AAAAAAAAAxY/wuW5pcxIRwE/s640/Mathematics+and+its+Axiomatic+Relations+With+Other+Fields.jpg" width="507" /></a></div><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />But, the links with other systems allow these other systems to "puncture" through the surrounding circle of axioms to get into the other fields, and motivate generation of theorems within it. Then, when, and if, a rigorous proof is needed, these theorems will be proved only with the axioms of that system (and not by theorems or axioms from the system that motivated them) and by already proved theorems.<br /><br />You have to keep open mind and not to be bounded by only one system, even if it has firm axiomatic framework for itself. To be creative, you have to see how it relates to other systems. For instance mathematics. There's a rich world of ideas right behind math axioms. Axioms deny you access to them, yet it's from these ideas mathematics axioms came into being.<br /><br />One may ask, how we can advance in discoveries, design, even everyday actions, without proving theorems of the systems we are working with on a daily basis.<br /><br />We can do that because we work with hybrid systems, where an assumption is accepted as true if a combination of the component systems (that contain those assumptions) gives truthful, real, useful, and correct consequences in other system that uses them or is connected via some functionality to them. The consequences we can imagine that can happen, given premises we have at hand and we deal with, allow us to avoid the immediate proofs for these premises and play more freely, without inhibition, with combinations of different fields, ideas, systems.<br /><br />Combinatorial creativity needs to recognize the cross-axiomatic links between partially or fully axiomatized systems combined. Puncturing the axiomatic membranes around conceptually delineated ideas using cross-axiomatic probing needle is the key to creative thinking.<br /><br />To me, for instance, a deliberation is cross-axiomatic attempt to draw plausible conclusions from partially axiomatized systems at hand <a href="http://t.co/S3783HnQ">http://t.co/S3783HnQ</a> </div>Bill Harfordhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-72799433753526075582012-05-07T14:42:00.003-07:002012-05-08T12:03:47.099-07:00Unlocked secrets of quantitative thinking in the palm of your hand<div dir="ltr" style="text-align: left;" trbidi="on">My updated booklet, 157 pages, hard copy, <a href="http://explainingmath.files.wordpress.com/2011/07/unlocking-the-secrets-of-quantitative-thinking-applied-and-pure-mathematics-april-26-2012-e.pdf">"Unlocking the secrets of Quantitative Thinking"</a>. You can also download the book in PDF file format from the right pane.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-L0YetVrsoGY/T6hBT6X4cuI/AAAAAAAAAuk/ct3T_TETYqI/s1600/Unlocking+the+Secrets+of+Quantitative+Thinking+-+Booklet.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="http://explainingmath.files.wordpress.com/2011/07/unlocking-the-secrets-of-quantitative-thinking-applied-and-pure-mathematics-april-26-2012-e.pdf" border="0" height="300" src="http://3.bp.blogspot.com/-L0YetVrsoGY/T6hBT6X4cuI/AAAAAAAAAuk/ct3T_TETYqI/s400/Unlocking+the+Secrets+of+Quantitative+Thinking+-+Booklet.jpg" title="" width="400" /></a></div><br /><br />You may want to check the list of <a href="http://explainingmath.blogspot.ca/2012/04/references-for-book-unlocking-secrets.html">references</a>.</div>Bill Harfordhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-86613599129616390512012-05-03T21:08:00.009-07:002012-05-07T15:44:19.602-07:00Axioms and Theorems in Relation to the Mathematical Models of Real World Processes<div dir="ltr" style="text-align: left;" trbidi="on">A mathematical model of a real world process is a set of numerical, quantitative premises driven and postulated by that real world environment, by its rules and by its logical systems extraneous to mathematics. Yet, these premises can be also derived directly from mathematical axioms. Moreover, while the premises are motivated by the real world processes and scenarios, the proofs of theorems, theorems built on these premises, are done and can be done only within the world of pure mathematics, using pure mathematical terms, concepts, axioms, and already proven theorems.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-DrDsqQ_8iXw/T6NXH-F35gI/AAAAAAAAAtA/hvoMWEVt6lY/s1600/Mathematics+and+Relationships+With+Other+Fields..jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="400" src="http://4.bp.blogspot.com/-DrDsqQ_8iXw/T6NXH-F35gI/AAAAAAAAAtA/hvoMWEVt6lY/s400/Mathematics+and+Relationships+With+Other+Fields..jpg" width="351" /></a></div><br /><br />When illustrating to students applied math, it should be shown which premises are introduced from, and by the field, of mathematics application, and, as the second step, how these premises can also be defined from the inside of pure mathematics, without any influence of, or reference to the real world process or environment. Then, it has to be shown that the proofs of the theorems that use those premises are completely within mathematics, i.e. no real world concepts are part of the proof.<br /><br />Successful assumptions will give predictable consequences. Axiomatizing that set of assumptions should ensure no contradictions in consequences. Usually, we are after a certain type, a particular set of consequences. We either know them, or investigate them, or we want to achieve them. Hence dynamics in our world of assumptions.<br /><br />Theoretical, pure logic doesn't care what are your actual assumptions. It just assume that something is true or false and go form there. Sure, results in that domain are very valuable. But, we are after the particular things and statements we assume or want to know if they are true or false. Not in general, but in particular domain. Any scientific field can be an example. Logic cannot tell us what are we going to chose and then assume its truth value. Usually it is the set of consequences we are after that will motivate the selection of initial assumptions. Then logic will help during the tests if there are any contradictions. If you are interested in specific consequences, in particular effects, investigate what causes those effects. When you have enough information about causes, make every attempt to axiomatize them. And, again, as mentioned, axiomatizing that set of causes should ensure no contradictions in consequences.<br /><br />Axiomatizing the set of causes should ensure no contradictions in effects (consequences). When tackling the topic of applied mathematics, it should be explained how the mathematical proofs contain no concepts or objects from the real world areas to which mathematics is applied to. That very explanation will shed light on the realtionship between mathematical axioms, theorems and the logical structures in the field of mathematical application (physics, engineering, chemistry, physiology, economics, trading, finance, commerce...). <br /><br /><br /></div>Bill Harfordhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-7584356733109451702012-04-30T17:06:00.011-07:002012-05-04T17:47:49.916-07:00Mathematics, Intuition, Real World Mathematics Applications. Random Notes.<div dir="ltr" style="text-align: left;" trbidi="on"><b>Edison on the role of theory in his inventions</b><br /><br />“I can always hire mathematicians but they can’t hire me.” - Thomas Edison. <a href="http://www.brainpickings.org/index.php/2012/03/28/the-idea-factory-bell-labs/%20">http://www.brainpickings.org/index.php/2012/03/28/the-idea-factory-bell-labs/ </a><br /><b><br /></b><br /><b>Economics and mathematics</b><br /><br /><b>John Maynard Keynes</b>, a student of Alfred Marshall, makes a statement in<br />his General Theory of Employment, Interest and Money (GT) which reflects<br />Marshall’s statical method: "To large a proportion of recent ‘mathematical’<br />economics are mere concoctions, <b>as imprecise as the initial assumptions they<br />rest on, </b>which allow the author to lose sight of the complexities and<br />interdependencies of the real world in a maze of pretensions and unhelpful<br />symbols".<b> </b><br /><br /><b>About proof</b><br /><br />The premise may be true because it gives true consequences. Let's prove it later.<br /><br />Daydreaming is a comfortable investigation of a number of premises that can give true consequences without a need for premises' proofs.<br /><br />If the consequence is true, for a certain premise, it is a nice sign to go ahead and prove the premise within its axiomatic system.<br /><br />Even if pure math is clear if contradictions, it can not guarantee that non-axiomatized field of applied math will have meaningful results.<br /><br />Proof, in any field, with possible exception of mathematics, is often consider convincing, even correct, if the consequences, effects under investigation seem to be reasonably predictable in the whole mash up of a number of non-axiomatized systems and their relationships, in the web of their cause, effect connections. But, that should not put mathematics in any special position, because the proofs in other fields, and the investigation methods, in the way they are, are the best what can be done at the time. If the cause is not axiomatized, the effect can be predicted in only of handful of special cases, which is what we have to deal most of the time anyway.</div>Bill Harfordhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-5522360353324901372012-04-30T05:28:00.014-07:002012-06-12T12:41:17.592-07:00References for the book "Unlocking the Secrets of Quantitative Thinking":<div dir="ltr" style="text-align: left;" trbidi="on"><br /><div class="MsoNormal">References for the book <a href="http://explainingmath.files.wordpress.com/2011/07/unlocking-the-secrets-of-quantitative-thinking-applied-and-pure-mathematics-april-26-2012-e.pdf">"Unlocking the Secrets of Quantitative Thinking"</a>: </div><div class="MsoNormal"><br /></div><ol start="1" style="margin-top: 0cm;" type="1"><li class="MsoNormal" style="mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;">“Probability: Elements of the Mathematical Theory”, C. R. Heathcote.</li><li class="MsoNormal" style="mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;">“What Is Mathematics, Really?”, R. Hersh.</li><li class="MsoNormal" style="mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;">“Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being”, G. Lakoff, R. Nunez.</li><li class="MsoNormal" style="mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;">“Introduction to Set Theory”, K. Hrbacek, T. Jech.</li><li class="MsoNormal" style="mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;">“Foundations of the Theory of Probability”, A. N. Kolmogorov.</li><li class="MsoNormal" style="mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;">“Stochastic Differential Equations: An Introduction with Applications”, B. K. Oksendal.</li><li class="MsoNormal" style="mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;">“Essays on the Theory of Numbers”, R. Dedekind.</li><li class="MsoNormal" style="mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;">“Fundamentals of Mathematics, Volume I”, S. H. Gould.</li><li class="MsoNormal" style="mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;">“The Electrical Engineering Handbook”, R. C. Dorf.</li><li class="MsoNormal" style="mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;">“The Algorithm Design Manual”, S. S. Skiena.</li><li class="MsoNormal" style="mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;">“The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number”, G. Frege, J. L. Austin.</li><li class="MsoNormal">"Introductory Applied Quantum and Statistical Mechanics", P. L. Hagelstein, S. D. Senturia, T. P. Orlando.</li><li class="MsoNormal">"Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables", M. Abramowitz, I. A. Stegun.</li><li class="MsoNormal">"The Feynman Lectures on Physics, Volumes I, II, III", Richard Feynman. </li><li class="MsoNormal">"Electromagnetics", J. D. Kraus.</li><li class="MsoNormal">"Textual Strategies", J. V. Harari.</li><li class="MsoNormal">"Fundamentals of Aerodynamics", J. D. Anderson Jr..</li><li class="MsoNormal">NACA Airfoil Sections, Report No. 460.</li><li class="MsoNormal">"The Limits of Interpretation", Umberto Eco.</li><li class="MsoNormal">"The Open Work", Umberto Eco.</li><li class="MsoNormal">"A Theory of Semiotics", Umberto Eco.</li><li class="MsoNormal">"Metaphors We Live By", G. Lakoff, M. Johnson.</li><li class="MsoNormal">"Pi: A Source Book", L. Berggren, J. Borwein, P. Borwein.</li><li class="MsoNormal">"Option, Futures, and Other Derivatives", John Hull. </li><li class="MsoNormal">"Energy Risk", D. Pilipovic. </li><li class="MsoNormal">"Commodities and Commodity Derivatives", H. Geman.</li><li class="MsoNormal">"The Mathematics of Financial Derivatives", P. Wilmott, S. Howison, J. Dewynne. </li><li class="MsoNormal">"Components of Nodal Prices in Electric Power Systems", L. Chen, H. Suzuki, T. Wachi, Y. Shimura.</li><li class="MsoNormal">"Film Form: Essays in Film Theory", S. Eisenstein.</li><li class="MsoNormal">"Introduction to Topology", T. W. Gamelin, R. E. Greene. </li><li class="MsoNormal">"Quantum Mechanics for Applied Physics and Engineering", A. T. Fromhold, Jr..</li><li class="MsoNormal">"Complex Variables and Laplace Transform for Engineers", W. R. LePage.</li><li class="MsoNormal">"Partial Differential Equations of Mathematical Physics and Integral Equations", R. B. Guenther, J. W. Lee.</li><li class="MsoNormal">"Point and Line to Plane", W. Kandinsky.</li><li class="MsoNormal">"Bauhaus", F. Whitford.</li><li class="MsoNormal">"The Power of the Center: A Study of Composition in the Visual Arts", R. Arnheim.</li><li class="MsoNormal">"Visual Forces, An Introduction to Design", B. Martinez, J. Block. </li><li class="MsoNormal">"Why great ideas come when you aren’t trying", <a href="http://www.nature.com/news/why-great-ideas-come-when-you-aren-t-trying-1.10678#/ref-link-1">Matt Kaplan, Benjamin Baird and Jonathan Schooler</a>,</li></ol></div>Bill Harfordhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-49949346428247884162012-04-29T10:18:00.003-07:002012-04-29T10:20:25.172-07:00One useful exercise to reduce student's frustration with mathematics<div dir="ltr" style="text-align: left;" trbidi="on">One useful exercise for kids learning math. Let them decide what to do with numbers.<br />Let them select the number by themselves from these two groups of numbers, as well as operations on them.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-LYGRB7_79xQ/T512BjcgJrI/AAAAAAAAAs0/o67QmNE5JVE/s1600/Introducing+Numbers+and+Math+Operations+on+Them+for+Kids.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://1.bp.blogspot.com/-LYGRB7_79xQ/T512BjcgJrI/AAAAAAAAAs0/o67QmNE5JVE/s320/Introducing+Numbers+and+Math+Operations+on+Them+for+Kids.jpg" width="257" /></a></div><br />Then, another exercise can be to ask student to measure, count something and find the number that represent that count. Adding another object to that count, and find the matching operation on the picture.<br /><br />These exercises will show kids that mathematics can be dealt independently of real world objects. It will also show that it is us who can chose the numbers and sequences of operations on them, or we can obtain numbers by measurements. It should be shown to student that math does not know how we obtained the numbers. For math it is important only with which numbers it is dealing with. It is us who keep track, aside from that diagram, what is counted and why. That can be called applied math.</div>Bill Harfordhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-28022284120283563932012-04-24T17:42:00.007-07:002012-04-24T18:00:30.572-07:00Mathematics and Real World Applications Links. World #1 and World #2 Approach<div dir="ltr" style="text-align: left;" trbidi="on"><div style="text-align: left;">In mathematics, it is all about numbers, sets of numbers, and the sequences of operations on them.</div><div style="text-align: left;"><br /></div><div style="text-align: left;">Axioms how to link real world scenarios to mathematical axioms and theorems, have to be defined. The closer you are to the point of complete axiomatization of the real world domains you want to apply math at, the better. Mathematics will reward you with meaningful results. Quantitative aspects of real world axioms, theorems and laws are usually theorems within mathematics.</div><div style="text-align: left;"><br /></div><div style="text-align: left;">How that can help you to learn mathematics and the domains where the mathematics is apply to, whatever that means? The answers is that you have to separate the two systems of axioms. The real world system, let's call it World #1 and the pure mathematical system (with its own axioms!), let's call it World #2. Note that there are World #1 axioms (ideally), World #2 axioms and axioms how to link the premises, and eventually theorems, from these two worlds.<br /><br />If you quantify objects and their relationships in the World #1, the quantities, and their relations, you obtained enter mathematics as initial or boundary conditions, or simply as numbers, set of numbers, pairs of numbers, as the starting points or starting premises. Note that these numerical starting points can be obtained inside math as well, without any extraneous motivation, i.e. within the realm of pure math only, sometimes directly from fundamental axioms. </div><div style="text-align: left;"><br /></div><div style="text-align: left;">The interpretation of numerical results is, apparently, up to you. It is you who will keep track of what is counted and why. The logic why you would do certain mathematical operations on the specified numbers, if coming from World #1, has to be firm and, ideally, has to be derived from an firm axiomatic system. Of course, math doesn't care if you axiomatized your real world domain or not. Mathematics will, without asking any questions, follow your instructions for calculations, and give you back results. Interpretation and usage of those results will depend on the correctness of your real world domain assumptions, accuracy of its logic, and completeness and correctness of the real world domain axioms.</div><div style="text-align: left;"><br /></div><div style="text-align: left;">[ mathematics, real life math applications, learning math, teaching math, math applications, real world math applications ] </div><div style="text-align: left;"><br /></div></div>Bill Harfordhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-46969682274924466302012-04-22T12:33:00.017-07:002012-05-03T06:06:24.844-07:00How Free or Constrained We Are in Applying Mathematics to Real and Fictional Worlds<div dir="ltr" style="text-align: left;" trbidi="on"><br /><div class="MsoNormal">There are several ways to “apply” mathematics, or more importantly, to obtain numbers and work with them. Here they are:</div><div class="MsoNormal"><br /></div><ol start="1" style="margin-top: 0cm;" type="1"><li class="MsoNormal" style="mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;">If you have 3 apples and you say that each one costs $2, how much money you will earn by selling all of them?</li><li class="MsoNormal" style="mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;">Measure the distance.</li><li class="MsoNormal" style="mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;">Physics laws, initial conditions, results of formula calculations.</li><li class="MsoNormal" style="mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;">Harry Potter or Hunger Games story.</li><li class="MsoNormal" style="mso-list: l0 level1 lfo1; tab-stops: list 36.0pt;">Mathematical axioms.</li></ol><div class="MsoNormal"><b style="mso-bidi-font-weight: normal;">The first</b> example arbitrary associates a number with an apple. No measurements or physical law is required. Economical exchange and the quantity to exchange are solely based on human values. The selection of the price is usually how trader perceives the value, and it can be subjective, yet that subjectivity is the only way to go when agreeing on an exchange price of goods.</div><div class="MsoNormal"><br /></div><div class="MsoNormal"><b style="mso-bidi-font-weight: normal;">The second</b> example is <b>selective counting</b>. The same way we define apples and want to count apples (and no other things), we decide we want to count how many of some unit length are in the given distance. We have in advance a unit length, say inches, or meters, and then a distance we want to measure. Note here that measurement is not a part of mathematics. Precision of a measurement is also outside mathematics. It is a method in the realm of physical world, how to count something, in this case length or distance. Measurement implies only that we agreed what and how to count, how to obtain numbers that will enter the numerical world of mathematics, often as pure starting points. Let’s say, we have 1m as a unit, and the length between two tables in a coffee shop. After the measurements we found that the distance between the tables is 2.3m</div><div class="MsoNormal"><br /></div><div class="MsoNormal">Let’s compare first and second example. First one has arbitrary numbers put together and multiplication selected as math operation due to need to sell the apples. Hence, <b>math will see</b>: 3, 2, multiply. 3 x 2 = 6. In the second example <b>math will see:</b> 1, and the count 2.3. That’s it. The difference in these two examples is that in the second one you are <b>constrained </b>by the physical distance you want to measure. You also specified the unit of length, 1m. Once these two things are specified, the measurement is not arbitrary. But, note, technically, it was arbitrary which units of length you have selected, and, in a sense, it is arbitrary which distance you want to measure. However, once this is established, selecting numbers is not arbitrary any more, it actually depends on the length and measurement unit.</div><div class="MsoNormal"><br /></div><div class="MsoNormal"><b style="mso-bidi-font-weight: normal;">The third example </b>is a firm <b>physics law</b>. A physics law specifies <b>what </b>needs to be counted and then, very important, the <b>relations </b>between these counts. Are they are to be added, divided, multiplied, etc.. Note how you, in a physics formula, you still deal with counts, but you keep track aside what are those counts of. Now, in physics law, we have even less arbitrary things. It is not arbitrary anymore what needs to be counted (time, force, mass, energy, distance) but also the mathematical relations are firmly established (addition, division, multiplication etc). Interesting things is, mathematics, again, will see these quantities as given as starting point <b style="mso-bidi-font-weight: normal;">only. </b>Specifying formula is extraneous to math.</div><div class="MsoNormal"><br /></div><div class="MsoNormal">Let’s look at Newton formula F = ma. Virtually, no numbers are given compared to apples and price. What is given then? You are told that if you count mass and count acceleration of a body, then multiply these counts, you will get the quantity of force that is acting on the body. So, where is the freedom here, and where is the law, or constrain? You are completely free to select, arbitrary if you wish, completely up to you, a mass of a body, and acceleration. Example is, you arbitrary chose a car to drive from a dealer’s parking lot, and arbitrary accelerate when on the road, to test it. Of course, when you see other drivers driving their cars, you will have to measure their mass and measure acceleration, i.e. not arbitrary any more, it’s given by other’s driver’s arbitrary selection to you. The formula now tells you that it is the multiplication you have to perform on these two numbers to obtain the force on the car. That’s the value of the formula. A genius is required to select what to count and then to establish, discover, the relationships between these counts. Of course, the very first thing is to want to count something, as oppose to look for some other things in order to explain certain behaviour.</div><div class="MsoNormal"><br /></div><div class="MsoNormal"><b style="mso-bidi-font-weight: normal;">The fourth</b> example, a Harry Potter story, signifies the fact that mathematics can not distinguish real from fictional world. Yes, math can be applied to real life and quantitative relations within physical world are important. But, math deals with numbers you supply to it, and with numbers only. It can not distinguish where these numbers are coming from. It is you who use the math and keep track where the numbers are coming from. Have you really counted, measured something, or just say you think that the number should be like that, math doesn’t care. If Harry Potter flies on his broom with the speed of 5 m/s, what is the distance he will advance after 7 seconds? The result is 5 x 7 = 35. He will fly over the distance of 35m. Note how math did not really care how you specified the numbers. Harry Potter’s broom or a rocket, or from the fictional world of Hunger Games, math does not know where the starting numbers and operations are coming from.</div><div class="MsoNormal"><br /></div><div class="MsoNormal"><b>The fifth</b> example tells you that, for math, it is sufficient, just to say, hey, here is the number 5, here is the number 7, do the multiplication and give me the result back. This is axiomatic approach and it is called pure math. Axioms of mathematics, more or less, tell you that the counts and operations are already available, you can pick them and define any sequence of operations on them. This is the fifth way you can obtain and play with numbers. No rockets, no apples, no currency, no physics laws, no length measurements are required to deal with numbers and hence to develop mathematics. Counts are there and you deal with them. One of the values of pure mathematics is that counts, numbers themselves and relations between numbers and sets of numbers, have some interesting properties, and results of that investigation can be used when you obtain numbers by any of the previous four ways, because the results will be applicable in each of them. Like, even if you don’t know what is counted, you will know that 3 + 5 = 8, in pure counts, pure numbers. It is a generally applicable result. For math, only the numbers you provide to it exists. You say here is the number 3, here is the number 5, add them. If this comes from any of the previous four examples, it is you, and not math, who will have to keep track what is counted and why you have chosen addition and not, say, division.<br /><br />You can download this post as an article in a <a href="http://explainingmath.files.wordpress.com/2011/07/how-free-or-constrained-we-are-in-applying-mathematics-in-e280a6.pdf">PDF file</a> format by clicking on the picture below or from <a href="http://explainingmath.files.wordpress.com/2011/07/how-free-or-constrained-we-are-in-applying-mathematics-in-e280a6.pdf">here</a>. <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://explainingmath.files.wordpress.com/2011/07/how-free-or-constrained-we-are-in-applying-mathematics-in-e280a6.pdf"><img border="0" height="400" src="http://2.bp.blogspot.com/-dbmu7QLKmGQ/T5SZ54GJq4I/AAAAAAAAApM/WRPQ8FbKFuM/s400/How+Free+or+Constrained+We+Are+in+Applying+Mathematics.jpg" width="337" /></a></div></div><div class="MsoNormal">[ Harry Potter, Hunger Games, applied math, applied mathematics, math and real life, real world math, examples of natural numbers, counting, number concept, ]</div><div class="MsoNormal"><br /></div></div>Bill Harfordhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-760055400593125532012-04-13T13:27:00.007-07:002012-04-30T15:34:39.629-07:00One More Example to Show What a Number Is and the Search for Truth in Various Disciplines<div dir="ltr" style="text-align: left;" trbidi="on">Here is one more interesting and quite good example to define a number, to show, in essence, what a number is.Let's say you have three apples on the table. Let's do the following <br /><ul style="text-align: left;"><li>Do as many steps as you have apples on the table.</li><li>Count or wait as many seconds as you have apples on the table.</li><li>Count is many pencils as you have apples on the table.</li><li>Do as many push ups as you have apples on the table.</li></ul>You see, in all these examples, the count is the same, obtained by counting apples. It's number 3, count of 3. In each of those examples you can say that you matched all those objects with apples, in one to one fashion, to make sure there is the same number of each. By this matching, you can determine that the set of objects has the same number of elements as the set of the apples on the table. Of course, you almost unconsciously used pure numbers, 1, 2, 3 to count other objects.You can see the universality of the concept of a count, number. Same count 3, number 3 is used to count truly any kind of objects.<br /><br />You can deal separately with pure number 3, without linkage to any of the objects it can represent the count of. That's pure math. Once you start keeping track what you count, applied math kicks in.<br /><br />Of course, you are always (as in any scientific discipline) interested to find the truth. Here, you may want to be interested to find truths about numbers. That's where logic enters, with its initial assumptions, axioms, theorems, proofs. You, essentially, always want to prove what is true in math. Mathematicians are after the proofs about counts. Mathematicians are after what is true about numbers, counts and their relations. Lawyers are after the proofs what is true with regards to law, moral, what is right or wrong, and with regards to other human values. Physicists are after the truths in physical world, where various forces, energy, motions are central focus in their investigation. Story writers and movie makers are after the true emotions and true moral messages their work will convey and show, even with fictitious plots, i.e. no matter whether the story is fictional or not, the message about human values, be it emotional, moral, must be real and true, and this message will be true if the story line is logically consistent with the story's framework, no matter how fictional that framework may be.<br /><br />Now, back to the first example, with apples, steps, seconds, pencils, pushups. Mathematics, while apparently common to all those cases, can not define the actual concepts it has counted. What differentiate an apple from a pushup, and a pushup from a pencil, and a pencil from a second is not part of mathematics, and mathematics is, more or less, not part at all of that analysis and those very important relationships. Moreover, it is these non mathematical relationships that define the various disciplines and it is these non mathematical relationships that very often dictate the direction of mathematical development. These relationships dictate what, when, where, and why will be counted, measured, if required at all.<br /><br />You can download this post as an article in <a href="http://explainingmath.files.wordpress.com/2011/07/one-more-example-to-show-what-a-number-is-and-the-search-fe280a6.pdf">PDF file</a> format by clicking on the picture below or from <a href="http://explainingmath.files.wordpress.com/2011/07/one-more-example-to-show-what-a-number-is-and-the-search-fe280a6.pdf">here</a>.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://explainingmath.files.wordpress.com/2011/07/one-more-example-to-show-what-a-number-is-and-the-search-fe280a6.pdf"><img border="0" height="400" src="http://2.bp.blogspot.com/-eOWJCR-kygQ/T5S-5-g4PzI/AAAAAAAAApU/LIu_trBabQQ/s400/One+More+Example+to+Show+What+a+Number+Is+and+the+Search+f%E2%80%A6.jpg" width="373" /></a></div><br /><br /><br />[ mathematics, math, math tutoring, philosophy, cognitive, cognition, learning math, learning mathematics, number, count, number definition ]<br /><br /></div>Bill Harfordhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-13935465467108790132012-02-27T21:03:00.001-08:002012-02-27T21:11:31.696-08:00Interrelations Between Deductive Systems and Inventive, Innovative Thinking<div dir="ltr" style="text-align: left;" trbidi="on">Looking at one deductive system, call it A, in the context of other deductive systems, can show how these extraneous systems motivates development of the system A. By being "in context" I mean that theorems in one system, which has presence in the logical, conceptual surrounding of the system A, hence providing context for it, are axioms or starting propositions for theorems in the system A. The systems need not to be mathematical only. What is important is that the systems are based on deductive reasoning, and that they are axiomatized as much as possible. I allow inductive reasoning, and definitely intuition as a method of discovery, but eventually, these both approaches will be morphed into a deductive structure and method. I wouldn't even differentiate inductive reasoning from deductive, but rather call it "dynamic deduction" or "deduction with self error correction".<br /><br />This kind of deductive systems linking, where contextual nesting and inclusion can go infinitely (i.e. any system that provides context for system A can itself has its own context, etc), is a core of inventive, innovative thinking.<br /><br /></div>Bill Harfordhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-22472271528102829242012-02-25T16:24:00.008-08:002012-07-22T11:54:42.540-07:00About Number Definition, Pure, and Applied Mathematics<div dir="ltr" style="text-align: left;" trbidi="on">If we agree that math is about counts, and counts only (as it is, since numbers, counts come from the cardinality of sets and set theory) then geometry doesn’t belong to mathematics. It is, by some authors (mentioned in "What is Mathematics Really", R. Hersh), considered impolite to have any geometric drawing in a mathematical text. Geometry has link to mathematics as the morning purchase of vegetables on the local market has. The geometry is only more convenient (perhaps!) in representing numbers and their relationships. Geometrical figures do only one thing to mathematics – by measuring the distances, angles, etc. we generate numbers, and sets of numbers. None of ZFC axioms refer to anything geometrical in the same way that ZFC axioms do not refer to the bunch of carrots at the local produce markets.<br /><div class="MsoNormal"><br /></div><div class="MsoNormal">Geometry can help to visualize certain mathematical relationships and results. But, the link between pure numbers and sets to the geometry is in essence arbitrary. Geometric interpretation of mathematical results are neither mandatory nor necessary.</div><div class="MsoNormal"><br /></div><div class="MsoNormal">For mathematics, it is completely arbitrary what or who generates numbers. The process of numbers selection, generation, numerical operations can be scientific, guessing, or a product of any dogmatic philosophy. Math couldn’t care less. As for geometry, the reason it has a strong presence in mathematics is just because of some of its practical applications. The reason why we can abstract real world into points, lines, planes, spheres is extraneous to mathematics. For whatever reason a line is drawn, and for that matter, what that line represents abstraction of, is not a part of mathematics. From math point of view we draw lines to generate numbers by measuring the lines’ lengths. Measuring process (with instruments, visually, or in any other way) again, is not part of mathematics. Math will see only the number you obtained. </div><div class="MsoNormal"><br /></div><div class="MsoNormal">For example, when we write 2 x 3 = 6 (without any explanation) will the reader know where 2 and 3 came from? Of course not. It can be from 2 baskets, each one having 3 apples. Or, it can be from 2 cars, where each car has 3 passengers. Why do you need a rectangle with sides 2 and 3 to explain you this mathematical result? You don’t need it.<br /><br />As much as apples, cars, are not part of mathematics, in the same way is not rectangle or any other geometrical figure. Geometry is perhaps interesting because it selects, generates certain sets of numbers that are of interest in everyday applications, like lines, squares, rectangles, circles. It is quantification of these figures and their measures that matter to mathematics, and not figures themselves. The thought process that takes place in defining a circle as an ideal abstraction of all real world attempts to make a circle (as well as a straight line abstraction of all straight directions) is a nice thing to think about, but that’s not part of mathematics. Once you “idealize” circle, math cares only about the numbers you provide by measuring them.<br /><br />Simply put, no geometric figure should be considered an element or part of pure mathematics because none of the theorems in math are proven using them. If seemingly geometry terms are used in proofs or appear to be a focus of study, like trigonometry or differential geometry, it is because the axioms of geometry are part of it, but, they are not part of mathematics. Mixing ZFC axioms and geometry axioms is like mixing ZFC axioms and axioms of any other system, including "marbles used in counting", carrots methods of purchase, quantitative finance rules, etc..<br /> <br />"The formalist makes a distinction between geometry as a deductive structure<br />and geometry as a descriptive science. Only the first is mathematical. The use of<br />pictures or diagrams or mental imagery is nonmathematical. In principle, they<br />are unnecessary. He may even regard them as inappropriate in a mathematics<br />text or a mathematics class." ("What is Mathematics Really" Rueben Hersh)<br /><br />Of course, it doesn't mean you should not use them to better communicate your ideas, investigate new directions in math or other sciences, or visualize a bit more difficult concepts in mathematics.But, you have to clearly differentiate between mathematics and these non mathematical objects and concepts.<br /><br />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. <br /></div><div class="MsoNormal">[ to be continued...] </div></div>Bill Harfordhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-12455286275025135642012-02-20T20:37:00.001-08:002012-07-02T11:33:11.342-07:00From Reuben Hersh's book "What is Mathematics Really"<div dir="ltr" style="text-align: left;" trbidi="on">From Reuben Hersh's book "What is Mathematics Really".<br /><br />Any proof has a starting point. So a mathematician must start with some<br />undefined terms, and some unproved statements. These are "assumptions" or<br />"axioms." In geometry we have undefined terms "point" and "line" and the<br />axiom "Through any two distinct points passes exactly one straight line." The<br />formalist points out that the logical import of this statement doesn't depend on<br />the mental picture we associate with it. Nothing keeps us from using other<br />words—"Any two distinct bleeps ook exactly one bloop." If we give interpretations<br />to the terms bleep, ook, and bloop, or the terms point, pass, and line, the<br />axioms may become true or false. To pure mathematics, any such interpretation<br />is irrelevant. It's concerned only with logical deductions from them.<br />Results deduced in this way are called theorems. You can't say a theorem is<br />true, any more than you can say an axiom is true. As a statement in pure mathematics,<br />it's neither true nor false, since it talks about undefined terms. All mathematics<br />can say is whether the theorem follows logically from the axioms.<br />Mathematical theorems have no content; they're not about anything. On the<br />other hand, they're absolutely free of doubt or error, because a rigorous proof<br />has no gaps or loopholes.</div>Bill Harfordhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-20874745462359270342012-02-19T12:39:00.002-08:002012-02-24T17:09:42.229-08:00The Definition of Number<div dir="ltr" style="text-align: left;" trbidi="on">After a number of years dealing with mathematics in your primary, secondary school, there still may be a question what the number is. Moreover, unless you are a professional mathematician, with PhD in your resume, I can safely assume that your frustration and fear of mathematics is still present.<br /><br />What is a number? Seemingly popular approach I am using here does not reduce the strength and significant clarity of the definition. Bear with me, and listen carefully :-) You may find out many interesting things!<br /><br />Here is the clearest approach to defining number. <b> </b><br /><br /><div style="text-align: center;"><b>Number is a count.</b></div><div style="text-align: center;"><br /></div><div style="text-align: left;">I will repeat again, number is a count. The purity and significance of this definition can not be emphasized more. While it is simple, it conveys many more important messages than other definitions and approaches you may have read about before. One of the most important message, in my view, of this definition is that it implicitly specifies what you can do with counts. Knowing what you can do with counts, you actually filter out all non mathematical concepts that may be mixed during "bad" mathematical lectures over the years. Also, thinking of numbers as counts, you define what pure mathematics is about! And that can help you answering the questions how math can be applied (about what "applied" means we will see later) in so many different fields, and what differentiate pure and applied math.</div><div style="text-align: left;"><br /></div><div style="text-align: left;">What you can do with counts is what mathematics is all about! So, what can you do with counts? You can add them, subtract them, divide, multiply. You can, then, do any number of these operations in any sequence you want. No apples, pears needed to do that! Count 5 is a universal count. It can come from counting apples, pears, cars, atoms, money, steps, seconds. That number 5, count 5 is a universal thing for all of them. While you can eat 5 apples, drive 5 cars, wait 5 seconds, with count 5 you can not do that. But you can add another number 5 to it! or deduct count of 3 from it. Or do any other "counting" operation! Note very important thing -> how you call these counts, i.e. are they integers, positive, negative, odd, even, rational, are just labels we attach and associate to the concept of a count! There is only count we are dealing it all the time.</div><div style="text-align: left;"><br /></div><div style="text-align: left;">Looking at count 5 only, i.e. number 5 only, you can not tell whether it came from apples, pears, or counting seconds. So, how you will differentiate count 5 of apples and count 5 of seconds if count, number is actually so universal concept? There is no other way than to keep track by yourself what you have counted. Technically, you will write a small letter beside the number, beside the count to remind you what it is a count of! Or you can remember that in your mind. Whatever works for you.</div><div style="text-align: left;"><br /></div><div style="text-align: left;">So, whenever you read about those exotic mathematical concepts, like matrices, determinants, integrals, equations, algebra, arithmetic, you will know one thing - it is all and only about pure counts we have just talked about. There is nothing else there. For instance, a matrix is a set of counts arranged in rectangular fashion on page. But, you do not need even that rectangle. You can just imagine in your mind the same set of counts and differentiate between them in any way you want. It just happened that it was convenient to write those numbers in a rectangular grid on paper! It was just convenience.</div><div style="text-align: left;"><br /></div><div style="text-align: left;">The logic, the reasoning in the world of counted objects is separate from the logic that deals only with counts. You can investigate properties of counts only, completely independent from the real world objects they might represent count of. This is the topic of mathematics. To find what is true about counts i.e. numbers. Hence the proof. But note here, we are really interested in counts' characteristics, no matter which objects have been counted! What these characteristics can be? We can have odd numbers, or even! We can have prime counts. Some counts can be divided by others, while some not. But, the numbers are numbers, it is us who give them names to keep track of some of their properties or just we want to deal with some numbers while leaving other numbers alone!. Naming numbers is not a mathematical operation. It just help us describe, label numbers, counts, we want to deal with.</div><div style="text-align: left;"><br /></div><div style="text-align: left;">We can compare the way we can deal with numbers to the way a sculptor deals with clay. It is only the clay that he works with and nothing else. Clay! But, what clay represents when it is shaped, what sculpture represents is not about clay! The motivation how the sculptor will twist, press, mold, shape clay is outside clay's world. Same in math! The reason why we add, subtract, divide, or even select numbers to deal with, frequently are outside mathematics! The motivation can come from us buying CDs or from an economist measuring supply and demand, or police measuring speed of the car.<br /><br />Now, back to sculpture again. The sculpture can represent anything. The similar thing is with numbers. In mathematics we are dealing with numbers only, the same way sculptor deals only with clay! But, if we want to interpret math results and use math in some other fields then we will have to keep track what we have counted, measured, keep track which objects are numbers, counts associated with! That would be called applied math. And, again, the numbers can represent count of many, many different things.<br /><br />In developing and understanding a subject, axioms come late. Then in the formal presentations, they come early. - Rueben Hersh. <a class=" twitter-hashtag pretty-link" data-query-source="hashtag_click" href="https://twitter.com/#%21/search/%23math" title="#math"><s>#</s><b>math</b></a><br /><br />The view that mathematics is in essence derivations from axioms is backward. In fact, it's wrong. - Rueben Hersh <a class="twitter-timeline-link" data-display-url="books.google.ca/books?id=R-qgd…" data-expanded-url="http://books.google.ca/books?id=R-qgdx2A5b0C&printsec=frontcover&dq=what+is+mathematics+really&hl=en&sa=X&ei=6O9BT5RIqdfRAcyqteMH&ved=0CDsQ6AEwAA#v=onepage&q=what%20is%20mathematics%20really&f=false" data-ultimate-url="http://books.google.ca/books?dq=what+is+mathematics+really&ei=6O9BT5RIqdfRAcyqteMH&hl=en&id=R-qgdx2A5b0C&printsec=frontcover&sa=X&ved=0CDsQ6AEwAA" href="http://t.co/7sdaUEv8" rel="nofollow" target="_blank" title="http://books.google.ca/books?id=R-qgdx2A5b0C&printsec=frontcover&dq=what+is+mathematics+really&hl=en&sa=X&ei=6O9BT5RIqdfRAcyqteMH&ved=0CDsQ6AEwAA#v=onepage&q=what%20is%20mathematics%20really&f=false">http://books.google.ca/books?id=R-qgdx2A5b0C&printsec=frontcover&dq=what+is+mathematics+really&hl=en&sa=X&ei=6O9BT5RIqdfRAcyqteMH&ved=0CDsQ6AEwAA#v=onepage&q=what%20is%20mathematics%20really&f=false</a> </div><div style="text-align: left;"><br /></div><div style="text-align: left;">[ number concept, concept of a number, number, count, numbers, counts, integers, rational numbers, concept, math, mathematics ]</div><div style="text-align: left;"><br /></div><div style="text-align: left;"><br /></div><div style="text-align: left;"><br /></div></div>Bill Harfordhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-137961116738104302012-02-11T09:59:00.000-08:002012-02-11T10:02:16.444-08:00Mathematical Intuition (Poincaré, Polya, Dewey), Reuben Hersh, University of New Mexico. Link to the paper.<div dir="ltr" style="text-align: left;" trbidi="on">Found an interesting paper on mathematics and intuition. Here is the summary.<b> </b><br /><br /><a href="http://explainingmath.files.wordpress.com/2011/07/mathematical-intuition-hersh.pdf"><b>Mathematical Intuition (Poincaré, Polya, Dewey)</b></a><br /><b><br />Reuben Hersh<br />University of New Mexico</b><br /><br /><a href="http://explainingmath.files.wordpress.com/2011/07/mathematical-intuition-hersh.pdf">http://explainingmath.files.wordpress.com/2011/07/mathematical-intuition-hersh.pdf</a><br /><br />Summary: Practical calculation of the limit of a sequence often violates the definition of convergence to a limit as taught in calculus. Together with examples from Euler, Polya and Poincare, this fact shows that in mathematics, as in science and in everyday life, we are often obligated to use knowledge that is derived, not rigorously or deductively, but simply by making the best use of available information — plausible reasoning. The “philosophy of mathematical practice” fits into the general framework of “warranted assertibility,” the pragmatist view of the logic of inquiry developed by John Dewey.<br /><br /></div>Bill Harfordhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0