tag:blogger.com,1999:blog-27456894984486080062024-02-08T09:33:25.333-08:00About 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. Aaron Powellhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.comBlogger93125tag:blogger.com,1999:blog-2745689498448608006.post-82622011634410448402022-06-11T10:20:00.004-07:002022-06-11T15:27:52.096-07:00The Best References in Set Theory<p></p><div class="separator" style="clear: both; text-align: left;">All set theory, hence, all mathematics, is developed from a single relation - membership, element of, belongs to. For instance, element a is a member of a set A. Then the axioms are postulated (based on element membership) and then using logic and inference rules the mathematics is developed. How to link math to another discipline or another discipline to math? The answer is that what is a theorem in one system can be an axiom to the linked system. Using implications (often material implications) you link math to another field like engineering, medicine, chemistry. You can call this method <a href="http://www.implicativetruthsynthesis.com">Postulate (premise) driven Implicative Truth Synthesis</a>. You create new truths (theorems) using implications. </div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEho-h0v9Ov1uoRBEl6r7H2IzeQKbBoKyD_RBdfU3e2mmxaPh3Oy_qcWrycP4LG9C-NGfwbFuRafJa28stXnOXuVb9OUr9o9-g4r3MTIjLtYh6mUQS8FK_Q5oFHbz0x6SCwJYbCnC1oyqW_EXz03ItMzqxjrsW0BQi71Z8c8Fugkj5ehMxTCCeSQyw/s704/Best_references_in_set_theory.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="659" data-original-width="704" height="503" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEho-h0v9Ov1uoRBEl6r7H2IzeQKbBoKyD_RBdfU3e2mmxaPh3Oy_qcWrycP4LG9C-NGfwbFuRafJa28stXnOXuVb9OUr9o9-g4r3MTIjLtYh6mUQS8FK_Q5oFHbz0x6SCwJYbCnC1oyqW_EXz03ItMzqxjrsW0BQi71Z8c8Fugkj5ehMxTCCeSQyw/w537-h503/Best_references_in_set_theory.png" width="537" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><br /></div><br /> <p></p>Aaron Powellhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-51896147205206072332022-03-26T13:24:00.005-07:002022-03-26T13:24:55.615-07:00Telepathy Architecture with Elon Musk's Neuralink Implants<p> Telepathy Architecture with Elon Musk's Neuralink Implants</p><p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj1UHbfnK_S7C8TG8ByskpTrex3DjfXRlP6gPsc_jKmFo0KiBe9rg88zFyDHTD_JwW2uOM6tvgoZDUmKwJ-F1Auz3FUQ4q-gyB03CC2YjbIK1yEHymlEKnMsaV1rlxwducgySgL9J7WcSw8PudW5_VQDzJUtEnqdzOH1TaD-BF6DuTb-vvYYkVe8A/s1119/Neuralink_Telepathy.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="797" data-original-width="1119" height="415" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj1UHbfnK_S7C8TG8ByskpTrex3DjfXRlP6gPsc_jKmFo0KiBe9rg88zFyDHTD_JwW2uOM6tvgoZDUmKwJ-F1Auz3FUQ4q-gyB03CC2YjbIK1yEHymlEKnMsaV1rlxwducgySgL9J7WcSw8PudW5_VQDzJUtEnqdzOH1TaD-BF6DuTb-vvYYkVe8A/w582-h415/Neuralink_Telepathy.png" width="582" /></a></div><br /> <p></p>Aaron Powellhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-62079117310945777612022-03-23T17:52:00.006-07:002022-03-23T17:58:24.059-07:00For Aspiring Software Developers - how to link diagram with coding<p>Now, you will know! See this illustrations. Taken from one book about Java 3D programming.</p><p>This illustration shows relationships between coding and a diagram. You should also consider STACK manipulation approach too, because it boils down to stack push, pop, operation, operand, memory.</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgxFKYUsIYu83vAX35zGQ1fonrBH9iQ0MRSGIv63i1ZKkz3Yn4GoInzbeeuzwwIS2yGaAFI4alKVIRapn5qIgcnPc_MOsxCvy6oA_8pG_RAaE_yOu5m3YcNZw7HU9zthHcY4mvDBfvub7No401hIkCRtNCYR1FqNdYdGQjOkOmZyc2MSFY_6W2lxg/s1920/relations_diagram_code.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="775" data-original-width="1920" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgxFKYUsIYu83vAX35zGQ1fonrBH9iQ0MRSGIv63i1ZKkz3Yn4GoInzbeeuzwwIS2yGaAFI4alKVIRapn5qIgcnPc_MOsxCvy6oA_8pG_RAaE_yOu5m3YcNZw7HU9zthHcY4mvDBfvub7No401hIkCRtNCYR1FqNdYdGQjOkOmZyc2MSFY_6W2lxg/w594-h240/relations_diagram_code.png" width="594" /></a></div><p><br /></p>Aaron Powellhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-9581508567529635562022-03-17T09:43:00.011-07:002022-03-26T12:51:16.426-07:00How to Better Approach New Concepts in Math or in Any Field<p> You have, at your disposal, three mind faculties:</p><p>- Thought Initiator© - for thought inception and creation</p><p>- Thought Pusher© , or Referencer© - for manipulating thoughts</p><p>- Thought Visitor© - to visit established thoughts</p><p>With these faculties you manipulate thoughts in accordance to your wishes and motivations. Motivation is a mind state. Emotion is an idea. In order to effectively use these three faculties you consider everything as an idea or representations in brain, or mind. This includes complex structures like emotions, other ideas, objects, sensory inputs. Each of these have their representation in brain, mind and can be used to control them in order to take some actions or come up with new ideas. </p><p>Many talk about awareness. This is often understood as some kind of meditation state. But there is more to it. In order to take full advantage of 'meditation' the awareness needs to be connected with these three faculties. With Thought Visitor© you visit various thoughts and state of minds (which are considered as a thought object, and idea) and see where they come from, when did you create them, and how they can be manipulated in your advantage. For instance, language is based on references. But, you form references using Thought Initiator© and Thought Pusher©. These are powerful tools of your mind and brain. A proof or demonstration of existence of Thought Initiator© can be the event when you send command to your nose to smell. This is Thought Initiator© ordering mind circuitry to smell. In the same way you can visit an idea, decipher from which axioms (postulates!) is derived from and manipulate it with your initiatives. Similarly, you identify two or more objects or concepts and using Thought Initiator© you form references and manipulate references or associations between these objects or concepts</p><p>With these tools at hand we can discuss how to better approach new concepts in math. Usually, how we learn math is by reading. We form references to concepts from words outlined in the text. But these words are borrowed from everyday life and often confuse us as we try to match their meaning with the referenced mathematical concept.</p><p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhS8scUQrcOg9ZULKagJ6rtUcpk6UpRstuWkIhB2q5lLC7ctjkFOUn4uoQ4ecGudzR_Uta-KFPvRIhG8TrDMoew9MGicF5Apb5zqZJmpd-X9zmbKm13MfYPfdFGCwlQu9G61ATI7m16bkEzN4xvHmP_J8v3DpllnVkx9OtXtTEuZ-2gXNdoMQbCdQ=s764" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="453" data-original-width="764" height="333" src="https://blogger.googleusercontent.com/img/a/AVvXsEhS8scUQrcOg9ZULKagJ6rtUcpk6UpRstuWkIhB2q5lLC7ctjkFOUn4uoQ4ecGudzR_Uta-KFPvRIhG8TrDMoew9MGicF5Apb5zqZJmpd-X9zmbKm13MfYPfdFGCwlQu9G61ATI7m16bkEzN4xvHmP_J8v3DpllnVkx9OtXtTEuZ-2gXNdoMQbCdQ=w561-h333" width="561" /></a></div><br /><p></p><p>Instead of going following path from term to concept as shown in the picture, you need to follow path of the blue line. This means mask (hide!) the word used as reference and form an object thought from the side of theorems, definitions, and postulates that created or led to the math concept of interest. The math concept you want to understand has to be, as close as possible, explained and defined using set concepts, because almost all math concept, no matter how misleading are the terms that reference them can be explained by using set theory.</p><p>Digression: problem with education is that student has more difficulty to associate a reference word to referenced concept instead of focusing on referenced concept, how it is derived, what are the major postulates and how it is instantiated from its axiomatic system it belongs to. <br /></p><p>As for other fields, other than math, the approach is similar. The term may need to be masked or replaced with your own reference, while the surrounding system needs to be defined from theorems, definitions, and postulates. Also, what is theorem in one system can be axiom or starting postulate in another. This is how you can creatively and effectively link systems from different fields like between math and biochemistry or biology and mechanical engineering. This method I developed for multidisciplinary teams I call Postulate Driven Implicative Truth Synthesis. More on this in my book, <a href="http://www.implicativetruthsynthesis.com" target="_blank">"Yes, You Can!"</a>.</p><p><br /></p><p><br /></p><p><br /></p>Aaron Powellhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-18467638756290968022022-03-13T08:29:00.007-07:002022-03-18T07:32:50.700-07:00The Role of Mind's Reference System in Our Understanding of Various Concepts<p>Felix Hausdorff used Old Greek words and letters for set theoretical concepts in his book, "Set Theory". But Greeks did not have set theory, so how Felix chose a word from Greek language for an nonexistent concept? Moreover, how a Greek or Latin word can be used for referring to a technology or mathematical concept, one that did not exist in that civilization, and is it a better way to do that rather than using a word from the language you know? It become evident how important is our inherent ability to create references, which are arbitrary in nature, but we want some kind of correlation with the meaning we know with the properties of a concept we are referring to. This, then, tells us that we can create truly arbitrary references, associations, relations, correlations, links between concepts, namely a reference name for an underlying concept. </p><p>If we can use a Greek or Latin word to refer to a concept (mathematical, engineering, or any other) that did not exist in their time and still can benefit from these references (we can manipulate them) then we can chose any word from any language to reference that concept. And this brings us to the point that we can conclude that the <b>referenced concept</b> is way <b>more important</b> than the reference word we are using to refer to it.</p><p>Our goal is to somehow manipulate concepts and their relationships as a whole using some kind of reference method, using references. Before that we need to understand as completely as we can the underlying inter-correlated concepts - concepts and their relationships, I will call it underlying system. What is the best way to do it? The answer is that they have to be understood in terms of the underlying axioms - we need to be aware of axioms and chosen postulates the underlying system is constructed from. In other words, we expect that underlying system needs to be axiomatized. We can give a best shot to axiomatize it and in many cases this can be a tough task, however, we need to have the best axioms we can come up with. Once the axioms are defined or discovered, postulates that define the underlying system components are discovered and identified we can move on to forming references to it. Then a more abstract step is required. We need to identify the fundamental concepts from which all the system elements are derived or constructed from. For instance, in math it is sets and operations on sets, in set theory it is a set membership relation (element x belongs to set A), in computer engineering is a structure stack and manipulation of it. </p><p>When acquiring new knowledge (in mathematics, engineering, or from most other fields) we often have to deal with reference words (references) put out by the authors of the book we are reading. In order to effectively acquire new knowledge, these references need to be <b>masked</b>! We have to cover them with a "cloth" so we can <b>uncover the underlying system</b>, with its axioms and postulates, referenced with now masked word. Then, we replace the masked word with a reference word of our choice. Why the masking is necessary? Because we are inclined to seek the link between the meaning of the reference word we already know, from its everyday use, and somehow use that to describe underlying mathematical, engineering (or other scientific) concept. This in many cases leads to confusion. The reference word can be misguided and misleading to us in our effort to understand the underlying referenced concept. It is misleading because we try to match the everyday meaning of the word with concepts, or their properties, within axiomatized system which is very specific and in many cases disconnected from ordinary word usage. That's why it is more beneficial to use an Old Greek or Latin word for a reference rather than a non-productive metaphor from ordinary word usage. As a matter of fact you can refer to any mathematical concept with randomly chosen strings of letters, like "aadvark12" to describe any mathematical concept, because this reference will not misguide you to seek match between its ordinary use (which it does not exist in this case) and underlying mathematical concept (which, anyway, can be described with set theoretical terms, which in turn can be reduced to the statements 'x belongs to set A', i.e. membership relations.) If the word 'aadvark12' does not exist in a dictionary that does not reduce in any way our understanding of the underlying concept; moreover it prevents us to use everyday word to refer to it and then be misguided. On the other hand, if a word is in the dictionary, it does not mean it is an effective reference to a mathematical concept.</p><p style="text-align: justify;">More on this and how to link different axiomatized systems from different fields using Implicative Truth Synthesis and how to recognize and utilize Thought Initiator<span style="background-color: white;"><b>©</b></span>, Thought Pusher<span face="HelveticaNeue-Light, -apple-system, AppleSDGothicNeo-Regular, "lucida grande", tahoma, verdana, arial, sans-serif, "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji", NotoColorEmoji, EmojiSymbols, Symbola, Noto, "Android Emoji", AndroidEmoji, "Arial Unicode MS", "Zapf Dingbats", AppleColorEmoji, "Apple Color Emoji"" style="background-color: white; font-size: 16px; font-weight: 700; text-align: justify;">©</span>, Thought Visitor<span face="HelveticaNeue-Light, -apple-system, AppleSDGothicNeo-Regular, "lucida grande", tahoma, verdana, arial, sans-serif, "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji", NotoColorEmoji, EmojiSymbols, Symbola, Noto, "Android Emoji", AndroidEmoji, "Arial Unicode MS", "Zapf Dingbats", AppleColorEmoji, "Apple Color Emoji"" style="background-color: white; font-size: 16px; font-weight: 700; text-align: justify;">©</span> mind faculties, in my book, <a href="https://www.amazon.ca/YES-YOU-CAN-Aaron-Powell/dp/B08BW46B4W" target="_blank">"Yes, You Can!".</a></p><p><br /></p><p><br /></p><p><br /></p><p><br /></p>Aaron Powellhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-36446131647318624042022-03-12T18:28:00.003-08:002022-03-17T09:58:20.769-07:00How to Determine if a Set is Open Set (Topology Examples)<p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhItDFr4-e-moGPAv_ZvAKGOQpfWSkSherPwMFDQ7CsqT4c07Wx7s_0-VgO_FQUIhkci0yLhSyYXp9_FLO3sSmNTdhQHLurDAxdvir-NFin40jKWLPjPj1j5SEEsphmd4VxbGRBd2DZJ0RuDleVpz3vrHZRStGdkKn8yEeX5EsgB2FaNdo-lhA4JQ=s413" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="271" data-original-width="413" height="210" src="https://blogger.googleusercontent.com/img/a/AVvXsEhItDFr4-e-moGPAv_ZvAKGOQpfWSkSherPwMFDQ7CsqT4c07Wx7s_0-VgO_FQUIhkci0yLhSyYXp9_FLO3sSmNTdhQHLurDAxdvir-NFin40jKWLPjPj1j5SEEsphmd4VxbGRBd2DZJ0RuDleVpz3vrHZRStGdkKn8yEeX5EsgB2FaNdo-lhA4JQ=s320" width="320" /></a></div><br /><div class="separator" style="clear: both; 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border: 0px solid black; box-sizing: border-box; color: #0f1419; display: inline; font-family: TwitterChirp, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif; font-size: 23px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px; min-width: 0px; overflow-wrap: break-word; padding: 0px; white-space: pre-wrap;">How to learn math. Reviewing several books with same topic opens different views on the same subject. Within a book you use definitions and previously proved theorems to prove new theorems. Postulating sets is essential. </span><span class="r-18u37iz" style="-webkit-box-direction: normal; -webkit-box-orient: horizontal; background-color: white; color: #0f1419; flex-direction: row; font-family: TwitterChirp, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif; font-size: 23px; white-space: pre-wrap;"><a class="css-4rbku5 css-18t94o4 css-901oao css-16my406 r-1cvl2hr r-1loqt21 r-poiln3 r-bcqeeo r-qvutc0" dir="ltr" href="https://twitter.com/hashtag/education?src=hashtag_click" role="link" style="background-color: rgba(0, 0, 0, 0); border: 0px solid black; box-sizing: border-box; color: #1d9bf0; cursor: pointer; display: inline; font: inherit; list-style: none; margin: 0px; min-width: 0px; overflow-wrap: break-word; padding: 0px; text-align: inherit; text-decoration-line: none; white-space: inherit;">#education</a></span><span class="css-901oao css-16my406 r-poiln3 r-bcqeeo r-qvutc0" style="background-color: white; border: 0px solid black; box-sizing: border-box; color: #0f1419; display: inline; font-family: TwitterChirp, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif; font-size: 23px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px; min-width: 0px; overflow-wrap: break-word; padding: 0px; white-space: pre-wrap;"> </span><span class="r-18u37iz" style="-webkit-box-direction: normal; -webkit-box-orient: horizontal; background-color: white; color: #0f1419; flex-direction: row; font-family: TwitterChirp, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif; font-size: 23px; white-space: pre-wrap;"><a class="css-4rbku5 css-18t94o4 css-901oao css-16my406 r-1cvl2hr r-1loqt21 r-poiln3 r-bcqeeo r-qvutc0" dir="ltr" href="https://twitter.com/hashtag/math?src=hashtag_click" role="link" style="background-color: rgba(0, 0, 0, 0); border: 0px solid black; box-sizing: border-box; color: #1d9bf0; cursor: pointer; display: inline; font: inherit; list-style: none; margin: 0px; min-width: 0px; overflow-wrap: break-word; padding: 0px; text-align: inherit; text-decoration-line: none; white-space: inherit;">#math</a></span><span class="css-901oao css-16my406 r-poiln3 r-bcqeeo r-qvutc0" style="background-color: white; border: 0px solid black; box-sizing: border-box; color: #0f1419; display: inline; font-family: TwitterChirp, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif; font-size: 23px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px; min-width: 0px; overflow-wrap: break-word; padding: 0px; white-space: pre-wrap;"> </span><span class="r-18u37iz" style="-webkit-box-direction: normal; -webkit-box-orient: horizontal; background-color: white; color: #0f1419; flex-direction: row; font-family: TwitterChirp, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif; font-size: 23px; white-space: pre-wrap;"><a class="css-4rbku5 css-18t94o4 css-901oao css-16my406 r-1cvl2hr r-1loqt21 r-poiln3 r-bcqeeo r-qvutc0" dir="ltr" href="https://twitter.com/hashtag/engineering?src=hashtag_click" role="link" style="background-color: rgba(0, 0, 0, 0); border: 0px solid black; box-sizing: border-box; color: #1d9bf0; cursor: pointer; display: inline; font: inherit; list-style: none; margin: 0px; min-width: 0px; overflow-wrap: break-word; padding: 0px; text-align: inherit; text-decoration-line: none; white-space: inherit;">#engineering</a></span></p><p><br /></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgjD7sMPwyULpfZjfbsljNkPKlY0ku5XZ9FxuLuA23fDjxp8lOZV9Q-eRaZ4pF3zXmf8f_mbWKFNO8MpX6uaoBZsc_zQzTeC8jkFNcPwPTQhee5cPHXAzWiCndwdw_qQp_HfwRg2bCCjGH9ov85A48G8gjNbLm44p6eNbGIyRbOLZOV5uhkFpq6ew=s1013" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1013" data-original-width="723" height="459" src="https://blogger.googleusercontent.com/img/a/AVvXsEgjD7sMPwyULpfZjfbsljNkPKlY0ku5XZ9FxuLuA23fDjxp8lOZV9Q-eRaZ4pF3zXmf8f_mbWKFNO8MpX6uaoBZsc_zQzTeC8jkFNcPwPTQhee5cPHXAzWiCndwdw_qQp_HfwRg2bCCjGH9ov85A48G8gjNbLm44p6eNbGIyRbOLZOV5uhkFpq6ew=w327-h459" width="327" /></a></div><br /><p><br /></p>Aaron Powellhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-9975361706997002332021-12-25T17:54:00.002-08:002021-12-25T17:54:17.881-08:00How to measure the pupils' distance when ordering glasses online<p> You will need a transparent (plastic) ruler and a phone with cameras front and back.</p><p>1. <span> </span>Turn on the phone camera and use the front one so you can see your face.</p><p>2. <span> </span>Position ruler on the root of your nose so that the numbers are just above the eye.</p><p>3. <span> Now comes the most important step. Try to position the ruler in the straight line, parallel with your eye level so the ruler is at equal distance from your pupils, iris, then take a photo. This can be difficult to achieve, however there is a trick. Move the ruler successively just a fraction of degree up and down around the imagined straight, parallel line and take a picture for each position. These are small changes in ruler's elevation. Make at least 10 pictures.</span></p><p><span>5. Send pictures from your phone to your email and open them in a picture viewer. I use Irfan View. If you downloaded picture all in one folder you will be able to view them in Irfan View one picture after another and you will clearly see which ruler's position and elevation is the most accurate. After you select that picture simply read the numbers (length) on the ruler between the centers of your pupils. Accuracy can be as good as 0.5mm although you will be ok with 1mm precision.</span></p><p>I used this method to order glasses from EyeBuyDirect, but any company that offers frames and prescription glasses can use your precision measurement of the distance between your pupils.</p><p>.</p>Aaron Powellhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-6764994576645058972020-09-01T14:35:00.000-07:002020-09-01T14:35:14.083-07:00Mathematics<div dir="ltr" style="text-align: left;" trbidi="on">
You can download all the important posts as <a href="http://implicativetruthsynthesis.com/engineering/Unlocking_the_Secrets_of_Quantitative_Reasoning.pdf" target="_blank">PDF book "Unlocking the Secrets on Quantitative Thinking"</a>.<br />
<br /></div>
Aaron Powellhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com1tag:blogger.com,1999:blog-2745689498448608006.post-88287708176594568862015-07-02T06:07:00.001-07:002020-09-01T14:43:27.145-07:00One Derivation of Euler's Equation for Complex Numbers<div dir="ltr" style="text-align: left;" trbidi="on">
You can download all the important posts as <a href="http://implicativetruthsynthesis.com/engineering/Unlocking_the_Secrets_of_Quantitative_Reasoning.pdf" target="_blank">PDF book "Unlocking the Secrets of Quantitative Thinking"</a>.<br />
<br />
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;">
</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>
Aaron Powellhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-59542933773453567892014-07-02T10:15:00.000-07:002020-09-01T14:43:45.544-07:00Why does zero factorial (0!) equal one, i.e. 0! = 1?<div dir="ltr" style="text-align: left;" trbidi="on">
You can download all the important posts as <a href="http://implicativetruthsynthesis.com/engineering/Unlocking_the_Secrets_of_Quantitative_Reasoning.pdf" target="_blank">PDF book "Unlocking the Secrets of Quantitative Thinking"</a>.<br />
<br />
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>
Aaron Powellhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com2tag:blogger.com,1999:blog-2745689498448608006.post-61598625156035080882014-06-11T10:09:00.000-07:002020-09-01T14:44:09.133-07:00What is Mathematics?<div dir="ltr" style="text-align: left;" trbidi="on">
You can download all the important posts as <a href="http://implicativetruthsynthesis.com/engineering/Unlocking_the_Secrets_of_Quantitative_Reasoning.pdf" target="_blank">PDF book "Unlocking the Secrets of Quantitative Thinking"</a>.<br />
<br />
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>
Aaron Powellhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-54975923612276296702013-02-28T13:24:00.003-08:002020-09-01T14:44:37.596-07:00From Basketball, Financial Math to Pure Math and Back<div dir="ltr" style="text-align: left;" trbidi="on">
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</xml><![endif]-->You can download all the important posts as <a href="http://implicativetruthsynthesis.com/engineering/Unlocking_the_Secrets_of_Quantitative_Reasoning.pdf" target="_blank">PDF book "Unlocking the Secrets of Quantitative Thinking"</a>.<br />
<br />
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<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>
Aaron Powellhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-35399848596772331292013-02-27T15:58:00.002-08:002020-09-01T14:44:53.377-07:00Mathematical Proof for Enthusiasts - What It Is And What It is Not<div dir="ltr" style="text-align: left;" trbidi="on">
You can download all the important posts as <a href="http://implicativetruthsynthesis.com/engineering/Unlocking_the_Secrets_of_Quantitative_Reasoning.pdf" target="_blank">PDF book "Unlocking the Secrets of Quantitative Thinking"</a>.<br />
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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. </div>
Aaron Powellhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-90084673844670663402013-02-27T13:55:00.003-08:002020-09-01T14:45:05.247-07:00More About the Concept of a Set and the Concept of a Number<div dir="ltr" style="text-align: left;" trbidi="on">
You can download all the important posts as <a href="http://implicativetruthsynthesis.com/engineering/Unlocking_the_Secrets_of_Quantitative_Reasoning.pdf" target="_blank">PDF book "Unlocking the Secrets of Quantitative Thinking"</a>.<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. </div>
Aaron Powellhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com1tag:blogger.com,1999:blog-2745689498448608006.post-36202239679320502822013-02-27T13:48:00.003-08:002020-09-01T14:45:18.317-07:00What is a number<div dir="ltr" style="text-align: left;" trbidi="on">
You can download all the important posts as <a href="http://implicativetruthsynthesis.com/engineering/Unlocking_the_Secrets_of_Quantitative_Reasoning.pdf" target="_blank">PDF book "Unlocking the Secrets of Quantitative Thinking"</a>.<br />
<br />
<div class="MsoNormal">
</div>
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I want to show methods how we can conceptualize a number in
order to actually understand what it is and how it is used in real life or in
pure mathematics. </div>
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<br /></div>
<div class="MsoNormal">
This introductory book should provide multiple benefits for
anyone interested in a deeper, more fundamental understanding of mathematics or
for a reader who wants to refresh or upgrade her knowledge in mathematics at
school or at workplace. If you ever have wondered where all those theorems come
from, can I create a theorem, how come math is so wide in scope, what are the
roots of mathematical thinking, this book is for you. The book will, hopefully,
put all your previous mathematical knowledge on firm and correct footing. It
can provide answers about what exactly is the subject of investigation and
research in mathematics and what mathematics is all about. The book should
provide firm, in depth intellectual tools for understanding a quantification
process that can be used virtually in any area of human activity such as
economics, finance, engineering, space sciences, physics, sales, planning, etc.</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
The explanation of a number concept, of a number definition can
be a useful and effective starting point for all those creative minds who ask
“why mathematics?” “what for we have to calculate all that?”, and “what is the
number actually?”.</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
So, let’s start. Let’s say that we can conceptually,
visually, in our minds, distinguish objects among themselves, and that we can,
and then that we want, to count those distinct objects. </div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
When we, for whatever reason, group objects into some
collections (they don’t have to be of the same kind nor similar at all), we can
be in a position to determine which collection has more objects, if we want to.
Don’t forget, we don’t have defined yet any concept of numbers, i.e. labels,
names for counts that reflect the size of a collection, its number of elements.
</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
How, then, we can determine which set, which collection has
more elements, objects? </div>
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<br /></div>
<div class="MsoNormal">
The essential method is to compare collections of objects by
pairing, matching, elements from one collection, with elements, objects from
another collection. Pairing has to be, obviously, one to one. <span style="mso-spacerun: yes;"> </span></div>
<span style="font-family: "times new roman"; font-size: 12.0pt;"><br clear="all" style="mso-special-character: line-break; page-break-before: always;" />
</span>
<br />
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Let’s say, we have a set of pens, set of apples, and a set
of bananas, as shown in the Picture 1.</div>
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</div>
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<br /></div>
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<br /></div>
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We can start first by determining which, if any, set has
more elements. </div>
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<br /></div>
<div class="MsoNormal">
In order to do that,<span style="mso-spacerun: yes;">
</span>we need to match, to pair, one object from one set with only one object<span style="mso-spacerun: yes;"> </span>from another set. If no objects are left
unmatched then two sets have the same number of elements. As we can see in the
Picture 1, by pairing apples and pens, and apples and bananas, no objects are
left unmatched, hence these three sets have the same number of elements. We do
not have the name yet for that count, for that number, but, the good thing is
we know what we are talking about! We are talking about certain number of
elements, <b style="mso-bidi-font-weight: normal;">defined by the exact match of these
three collections.</b> That property is what we are after! That “numberness” is
what we are looking for to capture and define.</div>
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<br /></div>
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<br /></div>
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<br /></div>
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Now, note one very interesting thing. If we replace bananas
with cars, and pair, match every car with every pen, we can see that the match
is again achieved! The pairs are again complete and no cars or pens or apples
are left unmatched. Hence, these two collections, two sets, have the same
number of elements.</div>
<div class="separator" style="clear: both; text-align: center;">
<a cars.jpg="" concept="" href="https://www.blogger.com/Number+concept+more+cars.jpg" more="" number="" width="320"></a></div>
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<br /></div>
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<br /></div>
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If we add a car then we can see that sets of cars has more
elements than the set of apples or set of pens. Or, if we add a pencil, we can
see that set of pens has more elements than the other two sets.</div>
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<br /></div>
<div class="separator" style="clear: both; text-align: center;">
</div>
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It is this property, this “numberness”, common in pairing
two or more sets, that we call a <i style="mso-bidi-font-style: normal;">number</i>.
You see, no matter what kind of objects are in two sets, if they match, that
property, is the actual <i style="mso-bidi-font-style: normal;">number</i>. That
is the concept of a number! It is, at this point, completely arbitrary how we
are going to label this numerical concept we have just discovered and defined.
Word, symbol, reference for it is completely arbitrary. We talk about number three
here. In English, it is called number <i style="mso-bidi-font-style: normal;">three</i>,
and the numeric symbol is 3. In
other languages it can have different name and nomenclature for it may be
different as well. Label for our new concept is really of very minor
significance at this point. It is the concept of “numberness” that adjective,
that property the sets have, the property sometimes used as a noun <i style="mso-bidi-font-style: normal;">number</i>, <span style="mso-spacerun: yes;"> </span>is way more important than the tag, than that curvy
trail of ink on the paper we use to reference it. What is important is what the
trail of ink on paper represents and not ink itself.</div>
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Let’s go back to our quantification adventure. </div>
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Of course, we could start with three object and obtain
number five, or seven objects, and obtain number seven, etc. Notice how we, now,
have this set property to work with. set property related to its number of
elements, the quantity of objects, the pure count of objects, that can be
abstracted (because that count is the same for all sets having that number of
elements). We have abstracted a property, quantity from any two sets of
objects, that we can call a number, or a count! That’s the actual concept of a
number. </div>
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Notice, also, how “number” in its essence, is not even a
noun, but more like an adjective, that describes “quantitative” aspect of two
sets, the number of paired elements of two collections.</div>
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Any time you have number in your mind first, and only then
you decide what you are going to count, you defined the number what it is.</div>
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As long as we know that these labels, symbols,
1,2,3,…represent that property of one to one pairing between the elements of two
sets (with the goal to determine if they have the same number of elements or
not) we are on a good path to work with numbers and quantification.</div>
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You may ask at this point “well, I don’t always see two sets
when I count objects of one set, I just count them without any pairing with the
elements of another set”. Good question! What you actually do, by, say,
counting CDs, or lemons, in your collection, is matching them with the set of
natural numbers in your mind, which is completely ok. But, note, you have
natural numbers at your disposal to use them for counting other objects, while
in our previous explanation we were actually just defining the numbers! We were
after very definition of a number. Once the numbers are defined, as we did for
number three, 3, you can use that number three and other numbers in counting
any kind of objects!</div>
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Look at that number five, say, the universal count of five,
for any kind of objects. One more interesting conclusion follows. You see, how
at this point, we can deal with counts only! We can deal with a count 3 and a
count 5, which we call a number 3 and a number 5, regardless which objects they
may represent count of! When we want to add them, it will always be 3 + 5 = 8,
no matter what we have counted! Apples, pears, cherries, lemons, their taste,
texture, color, cannot change the numerical result 3 + 5 = 8. That’s one of the
beautiful sides of mathematics. The search for truth about pure quantitative
relations. And this is exactly what pure math is about! Beauty of the applied
math, on the other hand, is in the challenge to find all kinds of relationships
between objects and concepts that need to be quantified.</div>
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From the application point of view, we, now, can use our generalized
knowledge that 3 + 5 = 8, and utilize it any real world situation. For
instance, if we have 3 cars and we buy 5 more cars, we will have 8 cars. </div>
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Note that “purity” of math is just related to the fact that
we do not care what we have just counted. We were only interested in adding,
subtracting, dividing pure counts, pure numbers we have abstracted from real
world objects counted. </div>
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Mathematicians have an exotic term for the number of
elements in a set, for the set’s size – it’s cardinality of a set.</div>
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You may ask “I can just start counting and continue counting
objects without putting them into any set. I can even stop counting at any
arbitrary time and still get a count, without specifying any set”. It is
actually completely true. You don’t have to have defined set first then count
elements within it to obtain a number. Technically, you are forming set “on the
go”, set whose elements can be defined as “anything I can see around me I can
put in set and count right then and there”. </div>
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This question is also interesting from another point of
view. In slightly different approach, you can start with number 5 and then
count any object you see around until you complete five counts. You see, at
this point, you used pure math in real world scenarios, perhaps even
unconsciously! You started with a pure number five, and it was up to you what
are you going to count. Of course in physics, engineering, economics, trading,
it matters what you count! That’s why we have to drag units beside pure numbers
to <b style="mso-bidi-font-weight: normal;">remind us</b> what we have counted
and what we may want to count when we go back from pure mathematical calculations
to the real world. More about that in a second.</div>
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If we want to mix pure math and real world scenarios and
objects we are counting, it’s easy, but, of course, it has to be carefully done!
What we need to do is to put a small letter beside the number, to keep track
what we have counted. Hence, in physics we have 3m + 5m = 8m, for distance,
length in meters. Then, also we can put 3 apples + 5 apples = 8 apples in
agriculture studies. We essentially do two steps here, during the additions of
real objects. When we want to add 3m + 5m, we actually separate pure numbers
from the meters counted, we enter with these two numbers the world of pure
math, where we do calculation of pure numbers only, 2 + 3 = 8. Then we go back
to real world of meters (because we have those small letters to remind us what
we have counted) with the result 8, and associate the name of the object, in
this case it’s a physical unit of length, or distance, which is meter, (m), to
the number 8. And, voila, we have just used pure math in the real world
application!</div>
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Aaron Powellhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-87333888245245077772013-02-24T15:02:00.001-08:002020-09-01T14:45:56.498-07:00Abstract Nature of Geomatrical Figures<div dir="ltr" style="text-align: left;" trbidi="on">
You can download all the important posts as <a href="http://implicativetruthsynthesis.com/engineering/Unlocking_the_Secrets_of_Quantitative_Reasoning.pdf" target="_blank">PDF book "Unlocking the Secrets of Quantitative Thinking"</a>.<br />
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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 />
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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 />
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Aaron Powellhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-83055083660868722312013-02-09T15:02:00.004-08:002020-09-01T14:46:13.406-07:00Function, map, pair in mathematics<div dir="ltr" style="text-align: left;" trbidi="on">
You can download all the important posts as <a href="http://implicativetruthsynthesis.com/engineering/Unlocking_the_Secrets_of_Quantitative_Reasoning.pdf" target="_blank">PDF book "Unlocking the Secrets of Quantitative Thinking"</a>.<br />
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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 />
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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 />
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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 />
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[ math, mathematics, mathematical function, function, map ] </div>
Aaron Powellhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-303514396917891552012-09-12T12:33:00.005-07:002020-09-01T14:46:29.680-07:00Flow of Quantification Results - Pure and Applied Math<div dir="ltr" style="text-align: left;" trbidi="on">
You can download all the important posts as <a href="http://implicativetruthsynthesis.com/engineering/Unlocking_the_Secrets_of_Quantitative_Reasoning.pdf" target="_blank">PDF book "Unlocking the Secrets of Quantitative Thinking"</a>.<br />
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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>
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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.</div>
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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>
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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 />
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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>
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Aaron Powellhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-91750769855272407262012-08-16T11:19:00.004-07:002020-09-01T14:46:46.531-07:00That famous Cauchy definition of the limit, and another view that explains the elusive concept.<div dir="ltr" style="text-align: left;" trbidi="on">
You can download all the important posts as <a href="http://implicativetruthsynthesis.com/engineering/Unlocking_the_Secrets_of_Quantitative_Reasoning.pdf" target="_blank">PDF book "Unlocking the Secrets of Quantitative Thinking"</a>.<br />
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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 />
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[ applied math, definition of limit, limit in mathematics, concept of a limit, limit, limiting value, integral, differential, ]</div>
Aaron Powellhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-80676578972908053582012-06-07T15:14:00.002-07:002020-09-01T14:47:47.167-07:00Set Theory, Units and Why We Can Multiply Apples and Oranges but We Cannot Add Them<div dir="ltr" style="text-align: left;" trbidi="on">
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You can download all the important posts as <a href="http://implicativetruthsynthesis.com/engineering/Unlocking_the_Secrets_of_Quantitative_Reasoning.pdf" target="_blank">PDF book "Unlocking the Secrets of Quantitative Thinking"</a>.<br />
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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Aaron Powellhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-24585106028167225432012-05-29T13:22:00.092-07:002020-09-01T14:48:03.963-07:00The Links Between Different Axiomatic Systems and Cross-Axiomatic Ideas Generation<div dir="ltr" style="text-align: left;" trbidi="on">
You can download all the important posts as <a href="http://implicativetruthsynthesis.com/engineering/Unlocking_the_Secrets_of_Quantitative_Reasoning.pdf" target="_blank">PDF book "Unlocking the Secrets of Quantitative Thinking"</a>.<br />
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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 />
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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 />
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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 />
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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 />
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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 />
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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 />
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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 />
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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 />
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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 />
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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 />
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<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="https://1.bp.blogspot.com/-aMioZ3JQWNk/T8Zuu7dZ-zI/AAAAAAAAAxY/wuW5pcxIRwE/s640/Mathematics+and+its+Axiomatic+Relations+With+Other+Fields.jpg" width="507" /></a></div>
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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 />
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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 />
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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 />
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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 />
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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 />
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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>
Aaron Powellhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-72799433753526075582012-05-07T14:42:00.003-07:002020-09-01T14:48:22.712-07:00Unlocked 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 />
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You can download all the important posts as <a href="http://implicativetruthsynthesis.com/engineering/Unlocking_the_Secrets_of_Quantitative_Reasoning.pdf" target="_blank">PDF book "Unlocking the Secrets of Quantitative Thinking"</a>.<br />
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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>
Aaron Powellhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-86613599129616390512012-05-03T21:08:00.009-07:002020-09-01T14:48:34.672-07:00Axioms and Theorems in Relation to the Mathematical Models of Real World Processes<div dir="ltr" style="text-align: left;" trbidi="on">
You can download all the important posts as <a href="http://implicativetruthsynthesis.com/engineering/Unlocking_the_Secrets_of_Quantitative_Reasoning.pdf" target="_blank">PDF book "Unlocking the Secrets of Quantitative Thinking"</a>.<br />
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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 />
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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 />
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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 />
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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 />
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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 />
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Aaron Powellhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0tag:blogger.com,1999:blog-2745689498448608006.post-7584356733109451702012-04-30T17:06:00.011-07:002020-09-01T14:48:50.525-07:00Mathematics, Intuition, Real World Mathematics Applications. Random Notes.<div dir="ltr" style="text-align: left;" trbidi="on">
You can download all the important posts as <a href="http://implicativetruthsynthesis.com/engineering/Unlocking_the_Secrets_of_Quantitative_Reasoning.pdf" target="_blank">PDF book "Unlocking the Secrets of Quantitative Thinking"</a>.<br />
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<b>Edison on the role of theory in his inventions</b><br />
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“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 />
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</b><br />
<b>Economics and mathematics</b><br />
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<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 />
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<b>About proof</b><br />
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The premise may be true because it gives true consequences. Let's prove it later.<br />
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Daydreaming is a comfortable investigation of a number of premises that can give true consequences without a need for premises' proofs.<br />
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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 />
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Even if pure math is clear if contradictions, it can not guarantee that non-axiomatized field of applied math will have meaningful results.<br />
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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>
Aaron Powellhttp://www.blogger.com/profile/06941521455903170510noreply@blogger.com0