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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.

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I demonstrated the concept of subtraction to my children by eating their French fries. It worked quite well, and served the parallel purpose of being an education in social interaction with people more powerful,than you... that life isn’t always fair. 

Sorry... yours is a weird OP

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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.

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34 minutes ago, slomobile said:

Yes, it is a weird question.

So it is difficult to know how or even if to answer.

So it good that your second post has improved my understanding over your first one.

Have you considered looking at history and the development of Maths teaching in schools ?

Much of what you describe already happens so is there to be studied.

Also you should review your conception of 'first principles'.

Maths has advanced to the point where many of the most basic principles are also the most difficult.

 

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  • 2 weeks later...
On 1/15/2021 at 11:53 AM, studiot said:

you should review your conception of 'first principles'.

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?

Edited by slomobile
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15 hours ago, slomobile said:

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.

 

OK so I will try to discuss communication of Mathematics, rather than principle of Mathematics.

I can't see where you have mentioned any basic Maths, computer code is hardly basic if it is indeed Maths at all.

However I beg to disagree with your outright  rejection of History.
Perhaps your experience of History at school was of the sort "History is a list of dates of battles, deaths and treaties to be learned by heart and regurgitated for the examiner".

History actually offers many lessons for those that care to peer into them.
Not the least being concerning computer code.
Coding languages have a very short lifetime; I have seen them come and go and stopped bothering to learn the new fashion decades ago now so I have little idea of the meaning of your example. The last serious program I wrote was PFortran TRIP (Trigonometric Intersection Program).

British schools went through a phase of demanding that every child learn 'programming'.
This mean resources were wasted on teaching first, different versions of BASIC, then PASCAL, then some early scripting.
None of which are current today.

History also tells us that the basic mathematical operation of counting is at least as old as writing, probably much older.
Now schools used to teach using the old fashioned balance scales. Good schools would actually get the pupils to set up pretend shops acting as customers and shoperkeeps.
They would weigh out amounts of materials, say potatoes or sand and also practice with pretend money.
This allowed a method of counting by the custemr presenting say a half crown coin and the shopkeeper saying That's one and fivepence and then making up the one and fivepence to half a crown with coins to provide the change.
Instant communication of arithmetic and fractions.
For those who were a dunce at school arithmetic there was the joke, you say you can't do maths but you can still instantly recognise that you need a treble eighteen, double top and single nineteen to finish in a darts match when you are on 113!

 

Would these be the sorts of examples you are looking for ?

 

 

 

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Having taught math successfully in Junior High and High school I though I would have a lot of great examples for you-- but on further reflection realized my best ones were not something that could be generalized.  I had the greatest success when I could connect the math lesson to the students' experiences.  For example, in my rural area the vast majority of the students have experience with guns and many also have reloaders in their families (people who make their own ammunition).  When I first tried to teach statistics I got blank looks from many students,  So that weekend I took my test equipment out to the rifle range and measured velocities of 10 rounds of ammo I had built.  On Monday I put the data up on the screen and asked the students if the load I had developed was consistent enough for hunting.  This lead to a successful lengthy discussion and the development of the idea of mean and standard deviation.

The lesson I learned and applied from then on is this:  The goal is not so much to make the students think differently, but rather to create a use for the math knowledge in a way that connects to their experiences.  I can think back to lots of examples of good teaching tricks, but realize they are were specific to a certain student or group of students.  Not much help to what you want.

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19 minutes ago, OldChemE said:

The goal is not so much to make the students think differently, but rather to create a use for the math knowledge in a way that connects to their experiences.

Yes, yes and yes.  +1

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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.

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Laboratory rats is something that comes to mind. 

You have a collection of forums here, people making and replying to comments from others all the time.  I cannot imagine how many times these people and posts have been analysed and even directly manipulated and interacted with for the purposes of someone's research.  That might very well be some experiment in Psychology, a study on how the internet is used, or something similar.  It's quite kind and ethically minded of you to let people know why you're asking for the thing you want.  I wouldn't recommend deliberately starting a false discussion or manipulating the forums but browsing through all those threads that already exist here seems like fair game.  This is a public forum and people should know their comments can be read by anyone.

Sadly, I don't think you'll find those letters between attendees of the fifth Solvay conference you were looking for.  Well, maybe with a lot of digging and after twenty years have passed so that you can see which people or ideas did become significant.  Public forums tend to attract people with time to spare, something to shout out about and then also tend to cultivate cliques and self-appointed experts.  Active scientists, going to a modern Conference, don't have the time for a forum like this.  I'm new here, with little experience of this forum and not pointing any fingers at anyone, that's just how it is in many other public forums.

Anyway, I'll have another read through all you've written and see if I can think of anything useful that fits your requirement.  The trouble is I'm new here and attaching, scanning and or even just seeing if I can drag-and-drop in formulae is something that will take me an hour.  To anyone reading this --->  Why isn't there a simple equation editor at the top of the toolbar?  If you are designing a new interface for this thing, Slomobile, please put one in.

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@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.

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On 1/15/2021 at 4:44 PM, slomobile said:

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.

I was going to say why do you not give us an example ?

but re - reading  I find you already did that but that your example is as obscure and impenetrable as the rest of your question.

I agree that is sounds good in principle to ane the variables more meaningfully, but consider this:

This was actually the way it was done years ago so Newton would have said,   "Distance is proportional to time"   (note they did not use equations in his day)

This sounds better than s = ut doesn't it  ?
And further this is what is taught junior school.

BUT

What about s = ut + ft2/2

How about putting that into words ?

Which would you rather learn ?

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3 hours ago, slomobile said:

Throwing ML at it, treating the posts and votes as training data might reveal a model that could be applied here.

ML = Machine Learning ?

Does this research have any connection with Microsoft's Lean?  Is that the ML abbreviation?  If not, you may have a passing interest anyway.  I'm cautious about giving out links but you sound competent enough to make your own searches for what is being done at Imperial College, London, UK - where  Lean is being used to try and develop a competent artificial mathematician.  For whatever it is worth, I was very relunctant to click the links you provided, especially the hyperlinked word  this you used in one earlier postI prefer to be in direct control of navigation and not pass extra information through the URL string sent to a server - but then I am old and don't own a mobile phone.

4 hours ago, slomobile said:

What is a forum which you would consider the antithesis of this one

  I wouldn't really want to comment or offend anyone or any organisation.  On the other hand, listing some positive points seems less problematic:   Studiot, the other person replying to you seemed very pleasant and I probably wouldn't have bothered coming back to this forum for a second day if there hadn't been a few like that on here.  Physics Stack Exchange is also another good site.

I promised you an example, I'm sorry, Real Life has got in the way and it took me half-an-hour to write this much text.  Another day please.

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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.

16 minutes ago, Col Not Colin said:

ML = Machine Learning ?

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.

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10 hours ago, slomobile said:

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.

 

Firstly this post seems to confirm that this is about computer programming so I don't know why you have placed it in Applied Maths, not Computer Science  ?

But thank you for the reply,  +1 , which is much more understandable and would have saved two weeks of floundering discussion if you had posted this in the first place.

As to the substance of it. It is interesting that what you are trying to do is an updated version of my (ironic) patented method of passing exams in kinematics from the 1960s.

In more modern times (the internet and more recently smartphones) many such 'calculators' have been offered for example Chemists can easily find pH calculators and Doctors/Pharmacists can find 'volume of distribution' calculatiors online very easily.
Even more advanced is the Wolfram Alpha and associated site which offers a sort of 'online mathematician' you talk about.

But these are just tools that should not be used blindly

The user should know and understand the subject to hand.

 

A second thought occurs to me, you have mentioned that some folks find plain English more understandable than a formula or bunch of formulae.

I agree.

But yet other folks, myself included, find pictures and diagrams even more informative.

I had (still have but sadly cannot now deploy it) a program callled DesignView.

This was an amazing program for its time, but serious hampered by the limitations of the computers of its day (Windows for Workgroups no less).

In essence the user is provided with an entry screen on which she can place directed line segments (vectors) graphically, thus describing say a structure or dynamic system.
The computer automatically turns this into an appropriate set of equations and solves them.

The user may then play about with the subject either by directly editing the equations or changing the line segments in the graphical model.
Non linear equataions such as spline curves may be generated and there was an animation section to the program for dynamical systems.

More recently other graphical system modelling programs such as Visio, CircuitMaker, Spice etc have been released.

Traditional drawing programs such as AutoCad have added some of this functionality.

There are also intermediate programs such as MathCad and Mathematica, ChemDraw and Chemsketch.

Many of these require considerable investment in time and effort for profficiency.

Alongside this Finite Element programs have become more user friendly with half ways decent GUIs.

Yet again the GUI is not the be-all and end-all, that many think.
I remember  in the 1970s visiting the Hydrographic Office of the Navy where they were starting the mass digitisation of Admiralty Charts.
They ahd developed a voice command system for editing the digital chart, so a new wreck, a new or shifted sandbank etc could be added at the speed of light in comparison to the old system (I don't know if you are familiar with the system of 'Notices to Mariners')

So it would be interesting to learn where your proposed AI fits into all this ?

 


 

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Hi.  Well, as promised, here's (a bit of) something that was passed to me by another Mathematician.

What is it?   An extract from a book, it was the actual physical book that was passed to me.

Formal reference:        H. Gerome Keisler,  Foundations of infinitessimal calculus,   Publisher: Prindle, Weber & Schmidt, incorporated,  1976.

An online version exists, although there's obviously been a serious re-juggling of pages and content since I had my hard copy (we called it a book).  The images in this forum post are screenshots from the appendix of that online copy.  You'd probably prefer to point your data collection tools at the electronic copy and these images are then pages 178~180 as described in that text  but pages  188~190 as described by my pdf reader (since some introduction and cover pages are counted).

Here's the URL at the time of writing:    https://www.math.wisc.edu/~keisler/foundations.pdf  

What does it mean?  Well, it's a theorem and proof showing how the hyperreal numbers relate to the ordinary Real numbers.  That's important for an area of Mathematics called Non-Standard Anaylysis but it's not what made this important to me.

Why was it important to me?  It's just a background story.  A bit like "the little boy who cried wolf" is a background story that seems to be an awfully wasteful way of explaining to children that they shouldn't tell lies.  What was important about this section of the book was that formal first order logic and set theory was used throughout the construction of the hyperreal numbers.  It's when I started to appreciate that this might be all we need for mathematics.  For example, sets might be the most fundamental structure in mathematics -  a bit like atoms for the Physicicists. 

   The actual symbols and notation used in mathematics are not important but that there is a language and some structure to that language may be the only important thing. 

Definition:  A Language is a set of symbols appropriate for the structure under consideration.

  A first order Language must possess the following symbols:   Connectives, Variables, Commas, Paranthesis.     

Connectives must include the following symbols    "Implies" ;   "if and only if" ;   "Not"   ;   "And"  ;   "Or

     (I'm skipping the rest of a definition for a Language, you can find a full description in the book by Keisler if you're interested.  What we have above is enough to get a flavour of the idea).

Typically we draw a little arrow   -> for the implies sign  and a double-headed arrow   <->   for "if and only if"     etc. etc.       BUT the exact choice of a picture for that symbol is entirely up to you, their Boolean logic values are all that matters and that is what makes it an  "Implies" symbol  (and they are, of course, exactly what you would expect as a computer scientist).

  With this idea of a language, what the  Elementary Extension Principle is saying is quite simple:

A  sentence of first order logic that is true in the Reals will translate to a sentence that is true in the Hyperreal Numbers.

I don't have any post-it notes or diagrams scribbled on the back of an envelope from that Mathematician but I don't thnk they would have helped much anyway.  I think I needed to read the full fairy-tale of Non-standard analysis to learn about Logic and Language and re-evaluate what this thing called "Mathematics" might actually be.

STOP here.  You said one thing per post is preferrable.  I've probably already covered several different things by mistake, sorry.

DATA, attachments etc. appear below:

Image, part 1 of 3.

image.png.66264f236c1239e447c22962443f5bea.png

Image, part 2 of 3:

image.thumb.png.251805da9cc189b40ab29ce2d153c8f9.png

Image, part 3 of 3:

image.png.6b9d68227ff51fb50d7e4030561333b4.png

 

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4 hours ago, Col Not Colin said:

A  sentence of first order logic that is true in the Reals will translate to a sentence that is true in the Hyperreal Numbers.

I haven't followed this thread so I don't know the context of your paste of an excerpt from Keisler's 1977 or so book on NSA. 

What you wrote here is perfectly true, since both the hyperreals and the standard reals are models of the first-order theory of the reals.

But in order to construct the hyperreals, you need a gadget called a non-principal ultrafilter on the natural numbers. Such a thing exists only in the presence of a weak form of the axiom of choice. So the logical principles needed to build the hyperreals exceed those needed to build the standard reals.

Secondly, the hyperreals do not satisfy the least upper bound property, because they are non-Archimedean. 

Third, Keisler's book is not about research, since the hyperreals were first constructed by Hewitt in 1948 and nonstandard analysis was developed by Robinson sometime afterward. Keisler's intent was to write an NSA-based textbook for freshman calculus. It's telling that in the 44 years since then, no other similar books have been written; and calculus is still overwhelmingly taught in the traditional manner based on limits.

There are occasional NSA-based calculus courses given, and studies show that by and large, students come away just as confused about NSA-based calculus as they do from traditional calculus. 

So we see that (1) NSA offers no pedagogical advantages (else more schools would have adopted it since 1977 and more texts would have been written); (2) NSA requires a strictly stronger logical foundation than the standard reals, namely a weak form of the axiom of choice; and (3) the hyperreals lack the fundamental defining property of the standard reals, namely the least upper bound property.

As I say I'm not sure what your point is in pasting this excerpt so I can't comment on that. I'm just mentioning some context for NSA that you should know about if you care about NSA or wish to make some point based on it.

Some light background reading of interest:

https://en.wikipedia.org/wiki/Transfer_principle

https://en.wikipedia.org/wiki/Ultrafilter

https://math.stackexchange.com/questions/1838272/why-do-we-need-ultrafilter-for-construction-of-hyperreal-numbers

https://en.wikipedia.org/wiki/Least-upper-bound_property

https://en.wikipedia.org/wiki/Archimedean_property

 

Edited by wtf
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3 minutes ago, wtf said:

I haven't followed this thread so I don't know the context of your paste of an excerpt from Keisler's 1977 or so book on NSA.

OK. 

1.  Rest assured, I wasn't telling the OP that this is the best way to learn calculus.  Although... at some later time, I feel I could make that argument but I've got dishes to wash first.  

2.  I'm quite delighted that someone else knows anything about NSA and I'm most impressed with your post.

3.  Yes the axiom of choice is an issue BUT it's a a set theory issue and that's great because set theory was also relevant.  You need a set theory just as much as you need a Language.

4.  I'm not sure you require the Ultrapower construction (or an Ultrafilter) just to construct an object that is like the hyperreals, especially if we soften the requirements on * slightly.  To be more specific, appendix 1E in my old copy of Keisler entitled "a simple construction of the Hyperreal Numbers" seems to construct a triple  (R, R*, *) such that these things A~D will hold:

    (A)  R is a complete ordered field.

    (B)  R* is a proper ordered field extension of R.

    (C)  * is a map from   { functions of n variables on R }    to   { functions of n variables on R* } .       * : f --> f*

     and, the field operations on R*  are the   image under * of the field operations on R.

    (D)  If two systems of formulae have the same real systems,  then  they have the same hyperreal solutions.   (mapping formulae and field operations by * as expected, but it will make the definition too confusing if I shove it all in here).

    That's a thing, a simplified thing, we could call the hyperreal numbers and I'm fairly sure that ultrafilters were not required.  You did still require the Axiom of Choice (or Zorn's lema).   The * transformation won't take all first order sentences across to R* but it's not too far off.

5.  The Hyperreals are non-Archimedian.  Yes, I agree but ... who said it was and does it matter?  It's interesting but not a problem.

 

37 minutes ago, wtf said:

Some light background reading of interest

  Thanks for that and I will have a look through.

 

 WHAT IS THIS THREAD ABOUT?

Well, the OP is gathering information from Mathematicians.  They were interested in things like how Mathematicians communicate.  Getting some concrete examples of this, like the post you have just sent as a reply to me etc.   You're probably in-line for inclusion in the study, so be warned but don't worry too much.  Any post can be viewed by anyone and data could be gathered, we have the advantage of knowing it is happening.   What are the OP's goals?  Various and you should probably read their posts instead of getting a corrupted opinion from me.  I would say they have an interest in designing software and determining if notation is important.

  Anyway, +1 for your post.  Anyone who knows a little NSA can't be all that bad and I'm very grateful for the discussion.

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5 hours ago, Col Not Colin said:

Definition:  A Language is a set of symbols appropriate for the structure under consideration.

  A first order Language must possess the following symbols:   Connectives, Variables, Commas, Paranthesis.     

Connectives must include the following symbols    "Implies" ;   "if and only if" ;   "Not"   ;   "And"  ;   "Or

If this is your interest, you'd be better served by a book on elementary mathematical logic, not a treatise on the hyperreals. 

 

17 minutes ago, Col Not Colin said:

The Hyperreals are non-Archimedian.  Yes, I agree but ... who said it was and does it matter?  It's interesting but not a problem.

The least upper bound property is the defining characteristic of the real numbers. It's important. The hyperreals are deficient in that respect.  

17 minutes ago, Col Not Colin said:

Anyway, +1 for your post.  Anyone who knows a little NSA can't be all that bad

 "Can't be all bad." I'll try not to blush with false modesty LOL.

I thought you were the OP but if not my apologies. Like I say I haven't followed this thread and should probably go back into my hidey hole. You drew me out mentioning the hyperreals, which I spent some time looking into a while back. 

I greatly recommend Terence Tao's brilliant blog post on nonprinciple ultrafilters as voting systems. That's the article that snapped all this into focus for me.

https://terrytao.wordpress.com/2007/06/25/ultrafilters-nonstandard-analysis-and-epsilon-management/

See also 

https://terrytao.wordpress.com/2012/04/02/a-cheap-version-of-nonstandard-analysis/

https://terrytao.wordpress.com/2010/11/27/nonstandard-analysis-as-a-completion-of-standard-analysis/

 

 

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@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.

Edited by slomobile
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1 hour ago, slomobile said:

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?

Hi.  Here are some answers:

1.  The entire book was handed to me, not just that one section.  However, that section is the most relevant part and copying the entire book to this post would probably violate copyright etc.  I certainly didn't read all of the book at the time and probably only about half of it to this day.

2.  No I didn't ask for this specific information.  I was interested in set theory and real analysis.

3.  I was not aware the book existed.

4.  What actions prompted the other mathematician?  This is speculation, I'm not the other mathematician.  I suspect it's that I showed an interest in the fundamental nature of mathematics and some desire to make sense out of what we (Mathematicians) have been doing for the last few thousand years.

5.  Practical context:  Final year project in an undergraduate degree.  The other mathematician was my supervisor.  I don't seperate my learning environment from a work environment too much because that is the area I went into shortly afterward as my work environment (I became a lecturer).

6.  I expected to learn about the minimal assumptions required to construct the Real numbers.  I ended up learning about logic, set theory and the ability for Mathematics to analyse itself.

7.  Yes, I have some interest in notation but in the broader context of identifying the fundamental nature of Mathematics.  Another thing that made me reply was that you had few replies.

8.   You can message me about NSA although I am NOT an expert in that area and these days I always have housework and real life stuff to do.  I'm only here at the moment because the covid situation is essentially cutting out all other avenues for discussion (or anything).  I would also advise caution about applying too much of your time to studying NSA.  As indicated by  wtf  It is widely regarded as an inefficient method for real analysis.  However, it's a great thing for pure mathematicians to have some familiarity with.  I'll try and figure out how the message system works and create a short "test".

 

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3 hours ago, slomobile said:

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.

Ok YOU are the OP. My apologies to everyone for that confusion on my part. NSA is a bit of a niche area of study, it won't do you much good in general. It's an alternate model of the real numbers. You'd be better off studying the standard model first; that is, the real numbers as taught in high school, and their formalization in the undergrad math curriculum as in a course on Real Analysis.

 

 

3 hours ago, slomobile said:

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.

I'll leave it under the rock two meters due north of the old oak tree in the park. Make sure you're not followed. 

There are many math resources on the Web, you should just consult some of them at whatever level of math you feel comfortable with.

If you want to see mathematicians discussing things, you should become a daily reader of https://math.stackexchange.com/. And if you want to see actual professional mathematicians talking among themselves, read https://mathoverflow.net/

 

 

3 hours ago, slomobile said:

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"?

I periodically check out this site to see if there are any new posts in the Math section. Your OP was new so I looked at it.

 

 

3 hours ago, slomobile said:

It was obviously your choice, not someone else.

Unless one is a determinist, in which case my looking at the thread was determined at the moment of the Big Bang. You can't discount that possibility and nobody knows for sure whether it's true.

 

 

3 hours ago, slomobile said:

Is your interest in this thread professional, idle speculation, academic, or other?

 

Math forum junkie going back to sci.math on Usenet.

 

 

3 hours ago, slomobile said:

Tell me about the difference between what you expected when you entered this thread, versus what you actually discovered here.

 

I haven't read the thread. I didn't understand your OP and didn't have an interest. Recently when the subject of NSA came up, I jumped in, because I have an interest in the subject and took the trouble to understand the basics some time ago.

 

3 hours ago, slomobile said:

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.

Could be worse, you might have  been one of those bitcoin holders hodlers who forgot their wallet password and lost millions.

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  • 2 weeks later...

@slomobile

I still find the information provided a tad conflicting. This thread is stated to be about notation. Later you say it is about communication. And teaching.

Then you introduce Non Standard Analysis, which is noted to be a minority sport.

So here is the beginning of one of the last chapters in a book from a well respected mid 20th century Mathematician which addresses all your topics and more that you probably haven't thought about.
I commend the book to you.

wilder1.thumb.jpg.4705e4dfa5cd8e55272c1382f713e7a6.jpgwilder2.thumb.jpg.562a5f4176266b7fb6c9ff560f4b9925.jpgwilder3.thumb.jpg.4514cb048aacf476c0c0a3f618bfea9b.jpg

 

It is also worth noting that there is a good introduction to NSA in chapter V of Thurstons book Differentiation and Integration.

Edited by studiot
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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.

 

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