slomobile

Thank you for the book reference. The topics do align with what I am studying. I agree with most of the observations, but the places I disagree are the core of my motivation for the study. "Most of what we take for granted is culturally motivated." I agree. "A culture... existed before he was born, and will continue to exist after him." Has been true throughout most of history. I question whether it is still true. The industrial revolution may have been the last "cultural period" that lasted an entire generation. World war changed culture everywhere. 50's rebuilding established a new culture, 60's and 70's established counterculture, 80's 90's globalization and cultural experimentation, information age culture was initially optional, no longer. Big data is becoming metaculture. Prepper/apocalyptic counterculture is rising. It is debatable whether these are proper cultures, but for my purposes, they also serve as a set of things taken for granted. "The principal mathematical element in the culture... will be chiefly possessed by the professional mathematician." Not in the age of the citizen scientist, makers, YouTube, and cheap global availability of information and scientific tools and materials. There would have been more, tying together notation, communication, teaching, and this new oddity culture, but I've run out of time and need to get on the road. Icy conditions, fingers crossed.

@Col Not Colin and @wtf thank you. That interaction was exactly what I was looking for. I appreciate the further external references. Non-Standard Analysis is still above my head, but I will be drinking it in over the next few weeks. That was part of the plan, to observe a mathematical conversation I do not understand. Then systematically research, and record all the foundational concepts I need to add to my repertoire to achieve some functional level of understanding. It is a step change in my understanding. In general, step changes are revelatory when examining complex dynamic systems, which is why I chose this strange method. In order to help me know when I have achieved "some functional level of understanding" I ask that you try to come up with some test for me. It can be just a single question and answer pair that I should be able to answer after understanding what you both just presented. Please send it to me via the message system on this forum with the word "TEST" in the subject line. I will not have it opened until after I have done what feels like enough research. At which point, I will have someone else open it and test me. What a wonderful bonus that the topic involves elements of what I am trying to do. I've also been fascinated by the foundational usefulness of set theory and been reading works about and by the Boole family. So this is front of mind. I have some further questions for @Col Not Colin about the circumstances surrounding receiving this book section "passed to me by another Mathematician." Did you ask for this specific information, or was it chosen by the other mathematician? I want to know who's judgement selected this particular book. If it was chosen by someone else, were you aware that it existed prior to receiving it? What actions on your part caused the mathematician to bring it to you. In what practical context was the information sought or delivered? Work project, idle speculation, academic study, other? The above answers may or may not prove revealing, but they are necessary for testing a particular theory of learning. Tell me whatever you can remember about the difference between what you expected to learn when you received this section of book, versus what you actually learned from it. I assume you were drawn to this "Notation Study" thread due to your own previous examination of notation, indicated by "It's when I started to appreciate that this might be all we need for mathematics." Correct me if I am wrong and feel free to expand. Or when you finish the dishes, teach me a better way to learn Calculus. Again, thank you for participating. Would either of you mind if I messaged you regarding NSA in case I get stuck? @wtfsorry if the thread seems a bit cryptic. As @Col Not Colinsaid, I am gathering information about how mathematicians communicate. If you don't mind, I'd like to ask you similar questions. How you came to this thread, was an interest in the mention of NSA, but what prompted you to open a thread titled "Notation Study"? It was obviously your choice, not someone else. Is your interest in this thread professional, idle speculation, academic, or other? Tell me about the difference between what you expected when you entered this thread, versus what you actually discovered here. Sorry it took me a while to respond to your excellent posts. I mistyped my password and locked myself out of the forum for a few days.

I definitely prefer "Distance is proportional to time" over s = ut, but introduce an additional concept like constant acceleration and one requires equations and units to relate to other concepts like average speed. Wouldn't it be nice if there were an intuitive ubiquitous system to quickly sketch free body diagrams with the limited information you have and the system generates all the standard equations and allows you to cherry pick the output information you require in terms of whatever variables you failed to specify? v^2 = u^2 + 2fs, s = (u + v)t/2 are filled in just in case you need them. DistanceAtoB(m, LinearPath) = Velocity@A(m/s) * DurationAtoB(s, LinearPath) + AccelerationAtoB(m/s^2, LinearPath) * DurationAtoB(s, LinearPath)^2 / 2 is how I would express your example in code I'm working on that lets you define a label such as DistanceAtoB in several explicit ways, while the system implicitly defines other relationships by recognizing the type of quantity and the points from the name and other info like units or path integration from passed in parameters. A specific path defined by several points with velocity at each point could be substituted for the LinearPath parameter. You can select undefined labels and the system will suggest candidate definitions like Intellisense for math. If the system does not have enough information to solve a system of equations it will tell you the minimum number of additional equations required and the variables that should be included in them. It allows aliases to be used for expressing one set of relations with relationships defined by another set. Example: Pressure, Restriction, and Flow substituted for Voltage, Resistance, and Current. It will keep track of all operations and units so they can be simplified or unwound on demand. Division in particular works much better when you just store the dividend and divisor for use in future calculations and only calculate the quotient and units when called upon to display an answer. It isn't a complete or consistent system by any measure. It just doesn't work. And the notation is way too long and complex. But it convinced me there is something there worth exploring further. "obscure and impenetrable" I fear I may have made that worse with this explanation. Whether I try to be verbose or concise, it seems to end up that way. I could blame the head injury. Ever since, I hear things extremely literal and tend to express things by sloppy analogy. After over a decade struggling to be understood, applying various edits to my habits, this is what I am capable of. I appreciate any honest critique of my writing style. It is how I improve. Honestly, I think my mental delay gives me a helpful perspective on some problems with an answer so obvious that it was selected before exploring other answers. When everyone else has heard the problem, processed immediately and moved on, I am still processing... and before reaching the obvious conclusion my mind drifts to something seemingly unrelated. When I snap back to what I was processing, the unrelated stuff gets mixed in and considered as well. It isn't very efficient, but it allows me to see connections I would have missed before the injury. Or maybe those aren't insights at all and just mental illness. Maybe. I choose to explore it anyway. I don't know what the result of this study will be. That is why it is being made. Yes, ML = Machine Learning. This is not associated in any way with Microsoft's Lean or work done at Imperial College, London. I was completely ignorant of that.work, but looks like I have some reading to do. The this link just pointed to specific content on the page explicitly linked further down the post. You didn't miss anything. It was such a long address I just didn't want it to interrupt the flow. If you are ever nervous about a link, just hover your mouse over it but don't click. Then look at the lower left of your browser window to see the URL it points too. Yours is a good general policy to avoid mischievous links. I'll keep that in mind for future posts. You can also hover over the link and invoke your context menu with a right click, or 2 finger tap on the touchpad. Select "Copy Link Address" to put it on your clipboard, then Ctrl V or paste into a text editor or browser address bar to see where it is they are trying to take you. Probably a wise choice to resist calling out examples of 'bad' forums. I retract that call to action. I'm glad you found value here! Take all the time you need to respond. I appreciate your engagement.

@Col Not Colin, good insights. The existing forums do provide a host of examples and I will use them. It just occurred to me that many forums 'sticky' guidance on how to ask good questions in order to get good responses. I would appreciate a metric that can be programmatically applied to the mass of forum posts to extract good posts worthy of attention. Perhaps the stickied advice can be turned into an algorithm. I like https://cr4.globalspec.com/ forum and the way it uses votes to rate posts as good and almost good. Throwing ML at it, treating the posts and votes as training data might reveal a model that could be applied here. I would need to cull populist responses like memes and one liners, but they probably carry a type of signature as well. What is a forum which you would consider the antithesis of this one, which also has a voting system? That is a whole different area of study, but as long as the posts remain, it can always be referenced as an addendum later on. A few good examples here from fully informed respondants can be used as prototypes to filter other sources of math communication.

Thank you, these answers are useful,. They are however the low hanging fruit, as it mirrors the history of teaching basic math concepts, which I already have access to. I would love some examples of professional math communication between adult peers (assumed similar level of education, but disparate experience). Those are data points which I do not have alternate access to, but which I assumed would be plentiful in this forum. The answers thus far have exposed that the more difficult math concepts to communicate will be those that lack personal usefulness or familiar analogy. Not everything can be made relatable. I am acutely interested in the difficult to communicate concepts, please share those even if you feel they will not generalize. For example, you were just hired as a very junior engineer to work with GPS satellites, but were a bit fuzzy on time dilation, what did a colleague say that helped you "get it". If I were pressed to identify ideal examples, I would choose the letters between Solvay Conference attendees if they had the benefit of all knowledge and tools up to now. Modern scientists are not the ideal choice because I believe we have lost a bit of the art of communication as a society since that epic era. I would very much like to be proven wrong on that point. Consider it a challenge.

That is fair given I am requesting anecdotal information. However, the "first principles' I am interested in are those required to COMMUNICATE mathematical information in practical, not academic situations. I am not interested in the math per se, but the form used to effectively communicate it. In order to determine the minimum elements("as simple as possible") required to effectively communicate math concepts, I must first establish the breadth of concepts that must be communicated so that I do not leave out essential but uncommon fundamental elements("but no simpler"). I fully expect to receive examples far above my head that will require me to do research and see exactly which prerequisite concepts were implied in the communication. This reveals a hierarchy of concepts which is a first principle. A hierarchy of math concepts. is certainly not a new idea. I have researched the history of notation and how math has been taught in the past as now.. It is part of what inspired the question. I don't need to reinvent the wheel, but I do intend to prepare alternate ways to describe wheel motion and compare various measures of efficiency, depth of understanding, and retention on various populations. I have access to archives.. I cannot possibly read it all nor understand most of it. I welcome pointers to specific illustrative historical examples because there is likely something Interesting I missed that you all will find. But it cannot provide everything I am after. Outside of internet resources, I do not have broad access to anecdotal examples of passing math concepts verbally, or handwritten on envelopes and Post Its, or email and text message as it is actually applied. I am interested in the incremental little bits you show someone that makes them say ":Oh, I get it now." How do you say this in a text message? Almost everything historical contains adopted conventions based on something that came before and was developed without the benefit of recent advances. I am interested in what a clean sheet approach might look like today with modern knowledge and modern tools. The nature of physics and underlying mathematical principles has not changed. How we use math has changed a great deal. What happens if we define integers as even multiples of PI and figure out object counting later? What would a base 60 numeric keypad look like? How can we use colored text, animated graphs, sound, vibration, and touch sliders as basic tools to imply math information given that the average user of advanced math is more likely to have a smartphone than paper and pencil on them at any given time. Matrix representations are now used far more than any time in history. It is a very useful notation for modern purposes. An advantage, it saves a lot of chalk by implying an awful lot . It is efficient, for those that understand the implied information. But chalk is mostly used for education and outdated even there. If you want to apply matrix math you use R, or Python, or Matlab, etc... not chalk or pencils. Maybe we should teach with those tools in second grade. I can imagine extensions that allow you to mouse over a matrix and get a tooltip showing the expanded linear equations it is generated from while a computer voice(for auditory learners) explains the operations to be performed on that matrix and its part within the overall Jupyter notebook. The remaining inspiration behind this thread was my own head injury and the mental adaptations I was required to make when my knowledge remained, but my processing ability diminished. I realized the notations of higher math had unpriced externalities. They are cheap(efficient) if you don't account for all the hidden prerequisites that allow those with ability and education to access them. Again using https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_University_Physics_(OpenStax)/Map%3A_University_Physics_I_-_Mechanics_Sound_Oscillations_and_Waves_(OpenStax)/11%3A__Angular_Momentum/11.02%3A_Rolling_Motion as an example, what if you don't know how to type Greek letters, or don't even know that the squiggly w is a Greek letter. Will you understand what to do with \vec{v}_{P} = -R \omega \hat{i} + v_{CM} \hat{i} \ldotp That is equally valid from some perspective. Are current notations making things harder than it needs to be. A person talking directly to another person will generally communicate more efficiently. I was hoping to gather some examples of that. This obviously didn't work. What is a better way to ask?

Yes, it is a weird question. Stealing fries is a useful communication of the subtraction concept. It is an example of what I am looking for. I am trying to get back to first principles in a field with a great deal of adopted convention. I'm trying to see if any of those conventions have become a crutch or shortcut for those in the know that slow or inhibit understanding by those unfamiliar or uncomfortable with the conventions. An example: Traditional single letter variables, even with sub- and super- scripts do not convey much information and require mental symbol substitution to process. Multi-character computer variables say what the variable is and leverage the highly practiced mental processes involved in reading and forming mental images from what is read. Symbol substitution is fast for some and slow to others. Perhaps this fact is unintentionally sorting people into scientific vs computer careers. If science were done with variables composed of whole words rather than single letters would the field have access to a different set of minds? Would existing scientists better grasp concepts because they access their emotional reading imagination, or are less burdened by cold symbol substitution. Of course reading is also symbol substitution, but many that see single letter variables first convert to the words the variables represent then do a second substitution from the word to the meaning represented by the word. The single letter variable convention has great space efficiency because it is unnecessary to demarcate the beginning and end of variables. Is that better than mental efficiency? I Single letters allow 2 letters together to imply multiplication, which must be explicit with multi-character variables. Do we need implicit multiplication? Is there a different way to imply coefficients with multichar variables? Can we settle on one representation of division? Are there better keyboard layouts for math that just work without LaTeX? PEMDAS applied blindly to endless practice problems (inconsistently between texts) confuses many and seems to delay understanding of how to properly form unambiguous equations. Order of operations is not fundamental and not necessary, just convenient. Is there value in experiencing the inconvenience? Tackling one notation convention at a time is doomed to failure. Tackling them all at once, is likely still doomed, but might produce some insight along the way to failure. I wish to compile several unconventional ways to do math, present them to test subjects, and see if any of those alternate methods are more efficient or intuitive than conventional notation. The results of that may inform development of a math software package.

This thread is an attempt to quantify the methods used to communicate a mathematical concept to another person for the purpose of applying the concept. Each response should ideally contain: A single math concept, as you would present it to an inexperienced colleague in your field, who asks for help. Any additions that you would include for the average person. Optional comments. Provide as many examples as you like, including historical examples, but please send each as its own post for ease of evaluating responses. There are no restrictions on field or complexity. The only restrictions on length and format are the practical limitations of this forum. You may provide links to external resources. Any document over 10 pages provided as a link or attachment should call out the most relevant page numbers in the body of the post. We specifically request examples that include unconventional, obsolete, historical, experimental, graphical, scanned handwritten, 3 dimensional, or otherwise odd notation provided it is useful to explain a math related practical concept. Computer code from any language is welcome. Please note if you would use computer code exclusively, or coupled with traditional notation. Please identify the notation type or language if it is not obvious to an English speaking C/C++ coder. We understand there is ambiguity in the question, such as "What is a single math concept?" Use your personal judgement. Disambiguation is another feature of the thread. Thank you. I hope the results are interesting.