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Holden

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Re-imaging Mathematics
« on: May 26, 2017, 04:10:49 am »
On the one hand I am greatly attracted towards maths & yet the way it is presented repulses me,that is the reason why first I want to dismantle mathematics and then re-build it from ground up:I was not the lion, but it fell to me to give the lion's roar.
Too often the outcome of mathematical philosophical inquires is to provide detailed answers to the "how "questions & barely any attention is given to the "why" questions.
Is it possible to lift the veil & demystify mathematics,to show that for all its wonders,it remains a set of human practices,grounded,like everything else,in the material world we inhabit.We know that Wittgenstein has argued for critical re-examination of traditional presuppositions about the certainty of mathematics.

The question is -is mathematics something which is beyond all social and historical factors?

To be continued..Lor' willin' an' th' crick don' rise.
La Tristesse Durera Toujours                                  (The Sadness Lasts Forever ...)
-van Gogh.

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Nation of One

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Re: Re-imaging Mathematics
« Reply #1 on: May 26, 2017, 10:46:37 am »
Quote from: Holden
On the one hand I am greatly attracted towards maths & yet the way it is presented repulses me,that is the reason why first I want to dismantle mathematics and then re-build it from ground up.

This is a rather lofty ambition.  To each his own.  I am going to continue to plug away at the more challenging exercises in the Dolciani series as planned, and then just peck away at undergraduate physics.   My life project is rather humble, I'll admit.   I intend to explore SymPy and some parts of SageMath along the way, tinkering with C++, Python, and even some old Basic and Pascal.

I do not intend to discourage you from re-imagining mathematics.  I'm just saying that I will be taking a more traditional, methodical approach which will consist mainly of going through a seemingly endless series of exercises, one textbook after another.  I understand why you have been repulsed by the way mathematics has been presented.

I want to be clear about why I have chosen the books in the Dolciani series.   She and her coauthors were mathematicians, not simply educators.  They present the material with set-builder notation.  Now, I suppose NJ Wildberger has some objections to the real numbers and likes to deal with rational numbers only.  He wants to present mathematics in a more pure manner without resorting to the transcendental functions or so-called infinite series. 

If I live long enough, I might give some attention to his work in this area.

I see the way life goes.  I witness the very limited amount of short term memory available to any of us.  My engagement with mathematics is more about learning how to work in a calm manner, developing patience and care, to not work in haste or give into despair.  Like I said, my devotion to rebuilding my own personal foundational understanding and skills is quite humble.

I do not have the energy or the confidence to dismantle mathematics and rebuild it from the ground up.  On the other hand, keeping Wildberger in mind, you are not alone in pointing out that the way it is presented may be more confusing than necessary.  Follow your bliss.

Maybe you will be taking a more philosophical approach to mathematics as a whole system, whereas my goal is simply to become a more skillful "mathematical technician" working within the existing system, mainly with the so-called real numbers, including the dreadful irrationals, radicals, and transcendentals despised by Wildberger.

I will learn to embrace the imperfections and obstacles introduced when we are working with digital computers and decimal representations.   There is a great deal of chaos and confusion.  I will learn not to despair and just peck away at understanding what is before me.

Just writing a program in C++ using <vectors> to solve systems of linear equations with matrices, Gaussian elimination, row-reductions, etc, can be remarkably satisfying and challenging. 

I will not discourage you from your dismantling project.   I'm just saying that I will continue to try to take the approach of rebuilding my own understanding of the existing mathematics (and programming).  So while you are re-imagining mathematics, I will be putting the precious little short-term memory to work to handle only as much as it is able to, figuring out ways to compensate for this short amount of such memory as we have been given.

« Last Edit: May 26, 2017, 10:49:08 am by Raskolnikov »
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Holden

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Is Traditional Mathematics Infallible?
« Reply #2 on: May 28, 2017, 02:54:37 am »
Quote
I will be taking a more traditional, methodical approach

Is it meant, by this, that "traditional,methodical" mathematics is quite exempt from any philosophical intellectual influence?
 Keynes writes :

The ideas of philosophers and economists, both when they are right and when they are wrong, are more powerful than is commonly understood. Indeed the world is ruled by little else. Practical men, who believe themselves to be quite exempt from any intellectual influence, are usually the slaves of some defunct economist or philosopher. Madmen in authority, who hear voices in the air, are distilling their frenzy from some academic scribbler of a few years back. I am sure that the power of vested interests is vastly exaggerated compared with the gradual encroachment of ideas. Not, indeed, immediately, but after a certain interval; for in the field of economic and  philosophy there are not many who are influenced by new theories after they are twenty-five or thirty years of age, so that the ideas which men apply to current events are not likely to be the newest. But, soon or late, it is ideas, not vested interests, which are dangerous for good or evil.


I'd say that the same is true with regard to mathematics as well. If we use the phrase" traditional mathematics" to mean mathematics without any philosophical baggage,then that is problematic,for the phrase has many,many implicit philosophical connotations.

We can spend our time working on mathematical proofs ,but surely, we cannot overlook the fact that the Dolciani book lays down what is intended to be secure base of absolute truth:the axioms of logic,the intuitively certain principles of mathematics, the self-evident axioms and rules.Each of these foundations is assumed without demonstration,leaving them open to challenge and doubt.This book uses deductive logic to demonstrate the truth of theorems of mathematics.Consequently,the book fails to establish the absolute validity of mathematical truth.For deductive logic can only transmit truth,not inject it,and the conclusion of a proof is no more certain than the weakest premise.

One can say that history or economics are uncertain.That one would rather just focus on algebra to stand on firm, certain ground. Well, that is highly problematic.
The quest for certainty in mathematics leads inevitably to a vicious cycle.Any mathematical system depends on a set of assumptions,and trying to establish their certainty by proving them leads to infinite regression. There is no way of discharging the assumptions.Without proof,the assumptions remain fallible beliefs,not necessary knowledge.All that can be done is to minimise them,to get a reduced set of axioms,which have to be accepted without proof.The only alternative is to replace one set of assumptions by another.But replacement merely starts off a further circuit of the vicious cycle.

One can direct this argument at the whole body of mathematical knowledge & is not framed for a single mathematical system.Many attempts to provide a foundation for a restricted part of mathematics manage to reduce assumptions in that formal system.What is done in such a case is to push some or all of the basic assumptions into the metalanguage,however,meta-mathematics does not stop the infinite regress in proofs which now reappear in the infinite hierarchy of ever meta-theories.
It follows that the assumptions have the status of beliefs ,not knowledge & must remain eternally open to challenge and doubt & are corrigible.

Any foundation of mathematics is in a certain fashion circular.That is,there always remain presuppositions which must be accepted on faith or intuition without themselves being founded.
Another central problem arises from the underlying logic on which mathematical proof rests.The establishment of mathematical theorems,that is,their deduction from axiom sets,requires further assumptions, namely, those of the axioms and rules of inference of logic itself.These are nontrivial & non-eliminable assumptions,and above argument about the ultimate irreducibility of assumptions on pain of a vicious cycle applies equally to logic.

Thus,mathematical truth as represented by theorems depends on essential logical as well as mathematical assumptions.This dependence increases the set of assumptions on which mathematical knowledge rests,and these cannot be neutralised by the "if-thenist" strategy.

To be continued..Lor' willin' an' th' crick don' rise.
« Last Edit: May 28, 2017, 03:02:55 am by Holden »
La Tristesse Durera Toujours                                  (The Sadness Lasts Forever ...)
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Holden

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Re-imaging Mathematics (Part II)
« Reply #3 on: May 28, 2017, 08:19:03 pm »
After the publication of mathematical proof many,many times ,after 20 ,30 40 or more years,the results are regularly found to be fundamentally flawed & rejected.The claim that all or most mathematical proofs prove their results absolutely,beyond any doubt,is simply false.

Rigor in mathematics is not a timeless ideal,because standards of rigor change historically,and it can be assumed that they will continue to evolve.Hence,the achievement of full rigorisation and hence total certainty could never be claimed for mathematical proofs.
There now exist humanly uncheckable informal proofs of mathematical results.These informal proofs are already too long for any mathematician to check.Translated into fully formal proofs they would be much longer.If they cannot possibly be surveyed by a mathematician ,they cannot be regarded as absolutely correct.IF such proofs were to be checked by computers,instead of humans,further problems would arise.No guarantees could be given  that the software and hardware were designed absolutely flawlessly,nor that the software ran correctly.Given the complexity of computers and softwares these can no more be checked by a single person than can a large proof.
Furthermore, such checks involve an irreducible empirical element.

Is this all nitpicking when mathematics is acknowledged to be the most certain knowledge known to mankind?

The history of mathematics exhibits unpredicted changes,reverses, ruptures just as complex as biological evolution and equally without teleology.
The context of the discovery of a proof cannot be overlooked,I mean the contingent circumstances of the human author & the historical forces are significant too.And as we well know,history is not a rational process and cannot be treated objectively and logically.

Traditional epistemology,from the British empiricists to the present ,take sense data that present themselves to the individual subject as the basis of empirical knowledge ,both in terms of genesis and justification.It is therefore inconsistent to absolutely exclude human and genetic considerations in epistemology as invalid,since these figure so significantly in its history up to the present day.

To be continued..Lor' willin' an' th' crick don' rise.

La Tristesse Durera Toujours                                  (The Sadness Lasts Forever ...)
-van Gogh.

Nation of One

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Re: Re-imaging Mathematics
« Reply #4 on: May 28, 2017, 08:35:24 pm »
Quote
I will be taking a more traditional, methodical approach

Is it meant, by this, that "traditional,methodical" mathematics is quite exempt from any philosophical intellectual influence?

No, not at all exempt from such influence.  It's just that I am not looking to examine any "utimate truths", but simply applying the axioms and the theorems, and doing some programming along the way, rolling my sleeves up and getting my hands dirty, so to speak.

The mathematics and programming is intellectually stimulating in its own right as it is.

Factoring polynomials and the rest of it.  I accept the definitions of the real numbers and I work with them in my head and on paper.

I'm revisiting algebra, geometry, trigonometry, etc. (putting calculus and physics on hold, knowing that what most attracted me to calculus and physics in the first place was all the ALGEBRA I had to use!) That is what I mean by a traditional approach.  I want to develop my skills in algebraic manipulations. 

Quote
we cannot overlook the fact that the Dolciani book(s) lay down what is intended to be secure base of absolute truth:the axioms of logic,the intuitively certain principles of mathematics, the self-evident axioms and rules.Each of these foundations is assumed without demonstration,leaving them open to challenge and doubt.

There is a considerable amount of demonstration, especially in working through the exercises.  I'm not really even trying to challenge or doubt any of this, not the field axioms or the real numbers in general.  I know that the real number line is just a mental representation that is infinitely divisible.

I can apply the theorems to solving the problems. 

Quote
This book uses deductive logic to demonstrate the truth of theorems of mathematics.Consequently,the book fails to establish the absolute validity of mathematical truth.For deductive logic can only transmit truth,not inject it,and the conclusion of a proof is no more certain than the weakest premise.

Which book are you refering to?  There are so many.  Are you speaking of the many books as "this book"?

"The book fails to establish the absolute validity of mathematical truth"   - ? -

Why do you demand absolute validity?

I think you may be making this more difficult than necessary.  If you don't want to study certain mathematics, then just don't study it.  Do you attack the foundations to relieve yourself of any compulsion to study it?

Is there anything wrong with studying things like "greatest common divisor" or
least common multiple" or how to program a computer to assist in finding such things just to see how it works without demanding that anything like integers or real numbers or rational numbers actually exist?

I'm not concerned with "absolute truth" or any kind of absolute validity.

I will accept the minor impurities and the universe of discourse, learning the rules as recipes as oposed to rules as laws. 

Hence, I just want to take some concepts for granted without getting bogged down in the crisis in the foundations of mathematics.

Quote from: Jose Ferreiros
... Mathematics and its modern methods are still surrounded by important philosophical problems. When a sizable amount of mathematical knowledge can be taken for granted, theorems can be established and problems can be solved with the certainty and clarity for which mathematics is celebrated.   But when it comes to laying out the bare beginnings, philosophical issues cannot be avoided.

I want to calculate, compute, factorize, solve, and, yes, sometimes even prove by deduction or induction.  I am not saying that what I am devoting myself to is exempt from philosophical doubt.  I just don't find it necessary.  The truth and falsity is relative to the axioms and theorems, the rules and patterns.  I am not interested in absolute truth, if there even is such a thing.

I just want to study some math.  Please don't think any less of me if I seem a little stubborn when it comes to this area of my life.  After all, sometimes it feels like all I have left in this world. 

Take care Holden. 

While I will butt out of this thread, for the most part, it would help a great deal if, when citing an author, you would leave a link to the source, such as Concluding Notes on the Social Philosophy Towards which the General Theory Might Lead when referencing John Maynard Keynes ( ... "madmen in authority, who hear voices in the air, ... )

This way I can follow the links at my leisure.

No hard feelings.  It is all worthy of our attention, no doubt.

Still, I am determined to focus on the nitty gritty details of my own humble study which includes a collection of obscure yet classical vintage textbooks, lots of notebooks and pencils ... and the technicalities of computer programming which demand I keep at least one foot planted firmly on the ground.
 ;D
« Last Edit: May 29, 2017, 09:39:05 am by Raskolnikov »
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Re: Re-imaging Mathematics
« Reply #5 on: July 08, 2017, 12:27:57 pm »
I was searching the board for a proper place for this link, and, if you don't mind I will place it here.

Upon further research into just what Zeilberger and Wildberger have in common besides their German?Jewish sounding names, I found that Perhaps the most prominent ultra-finitist working today is Doron Zeilberger, a mathematician at Rutgers. Zeilberger, like many ultra-finitists, believe there is a largest natural number. When asked the inevitable question about what happens when you add 1 to it, he replies that, in a very elegant circularity, you go back to 0.

Zeilberger is a Finitist who insists that today’s Mathematics (capital M) is a religion.  Its central dogma is thou should prove everything rigorously.

That page links to his Opinions page.

Mind you, I only stumbled upon his name in Wildberger's book and took notice because of the mention of RENE.  I have been curious to see a computer program in action proving elementary Euclidean Geometry proofs.  So ...

Also, a three dollar book I ordered from 1993 by Arthur Engel, Exploring Mathematics with your Computer, which interested me because of its uniqueness (It was all math code written in Pascal) - the wrong book arrived which shared the same "series ISBN" - New Mathematical Library.

Uncanny and strange that the WRONG BOOK has the title, Geometry Revisited by Coxeter and Greitzer, circa 1967.  Why do I use my most favorite word (uncanny) up there with the word weird?  Well, you see, the current volumes of my "mathematical diary" are called exactly that:  GEOMETRY REVISITED.

Besides that, I was born in 1967.

It's just a weird coincidence.  The full title of the art sketchbook I pencil in solutions in is called Geometry Revisited, Book Two, A Mathematical Diary by a disciple of The Weird. 

I have a great many of such math diaries which have replaced the scribblings I once called "A Philosophical Diary".

In the preface of the WRONG BOOK, they write, "Historically, it must be remembered that Euclid wrote for mature persons preparing for the study of philosophy.  Until our own century, one of the chief reasons of teaching geometry was that its axiomatic method was considered the best introduction to deductive reasoning."

Anyway, as you can see, even though my brain feels like it is ALL OVER THE PLACE, actually the chaos has a definite theme and pattern.

The one thing the professional mathematicians may not have is the leisure to think about what they may consider "high school and undergraduate" mathematics, and yet, for me, that material, along with the mathematical thinking of our ancestors, both ancient as well as prehistoric, require a great deal of leisure to explore and reflect upon.

Our brains have not changed physiologically throughout the eons.

What has changed is the unfathomable mass of information and knowledge passed down to us as well as the pace at which mass industrial society attempts to "inject" thousands upon thousands of years of thinking into our basically prehistoric brains!

No wonder we get headaches and suffer from insomnia.  No wonder we become distracted, frustrated, and overwhelmed.

I suggest we continue to play dumb, for our feelings of inadequacy must be a very common response.

What can we do but space out and meditate in between our feeble efforts to think?

What is the geometric space between our thoughts?

Holden, have you considered that you just might be a heretic of this religion?

And myself, I want at all costs NOT to be a crank or a phony elitist.  I apologize for responding to your "Re-Imagining Mathematics" as though I were a religious fanatic.

In reality, I most likely am some in some kind of chaotic, spontaneous, informal, unofficial cult of mathematically-oriented programming enthusiasts and hobbyists who, having found themselves marginalized and unable to keep pace with the more ambitious but less contemplative masses of the Industrial World, have latched on to LEISURE as a way of life.  There are many of us who choose books over automobiles,

Whereas in the ancient world, only the aristocrats could spend their lives practicing this religion of Mathematics, now a great many "disabled" or  "psychologically unemployable" would-be slaves, are taking to foraging for "information" as some kind of treasure.

You are a heretic, Holden, and I am an infidel.  Also, I am moving in an opposite direction from the PhDs and graduate students ... I find I am rejuvenated by this revisiting the college-prep high school curriculum from the vantage point of an escaped slave or time-traveler from prehistoric times.

The approach taken in the 1960's, what may have appeared too formal and rigorous (i.e., RELIGIOUS) back then (which is why there was such a strong resistance, that is, "the math wars"), is now very novel to me.  Yes, now that I have proven to be officially "resistant to traditional employment" and quite content living on government relief like our Ignatius Reilly from Toole's A Confederacy of Dunces, I can embrace the mathematics I only hated because we were so methodically marched through it during a period of our lives when we were experiencing emotional turbulence (adolescence).

Again, please do not let me discourage you from engaging with mathematics however the hell you want to.  Myself, I am leaning in the direction of riding the wave of alternating periods of enthusiasm and depression/disgust.

Where our minds meet, I think, is in our sense of inadequacy and inferiority - due to the premium we place on honesty, clarity, and self-deprecating humor.  We both may have violent resentments against obfuscation, so we are inclined to seek some balance by injecting a little humility and, if possible, even simplicity into our own personal engagement with what goes by the names of maths, math, or mathematics.


- Your Working Boy, paying homage to Ignatius J. Reilly.

PS: another excerpt from the book I did not ask for which has the same title as my (current) secret diaries, Geometry Revisited.

With a literature much vaster than those of algebra and arithmetic combined, and at least as extensive as that of analysis, geometry is a richer treasure house of more interesting and half-forgotten things, which a hurried generation has no leisure to enjoy, than any other division of mathematics.  ~ E.T. Bell

-------------------------------------------------------------------------
An aside:

A not so small but mostly overlooked tragedy of living in an age of information abundance is that there are many enthusiastic young scholars who may find themselves criminalized by the representatives of capitalism, commerce, and the corporate state for say, using "Student Editions" of mathematics software or distributing digitalized literature.

What kills me about so-called student editions of products such as Mathematica, Matlab, and Maple is that the Industry is claiming to define what a student is!

In other words, you are not considered a student unless you are in the system working toward a degree.  They are blatantly denying the far more prevalent practice of independent and exploratory "self-education."

My strong personal opinion is that such "Student Edition licenses" ought to be altered to include those who promise to use the products for educational purposes only, not commercial purposes.
« Last Edit: December 29, 2021, 11:54:55 am by Half-Crazy Nobody »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

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Re: Re-imagining Mathematics
« Reply #6 on: December 29, 2021, 11:43:24 am »
Quote from: Holden
On the one hand I am greatly attracted towards maths & yet the way it is presented repulses me, that is the reason why first I want to dismantle mathematics and then re-build it from ground up:I was not the lion, but it fell to me to give the lion's roar.

I finally get it, what you write above.  When I first replied to this in 2017, I must have been in a different state of mind.

Have we come around full circle?   Now your statement is crystal clear to me, although I can understand why one might proceed with caution ...

So, a little more than 4 years ago, I had this to say:

Quote from: Nobody
I want to be clear about why I have chosen the books in the Dolciani series.   She and her coauthors were mathematicians, not simply educators.  They present the material with set-builder notation.  Now, I suppose NJ Wildberger has some objections to the real numbers and likes to deal with rational numbers only.  He wants to present mathematics in a more pure manner without resorting to the transcendental functions or so-called infinite series.

I am glad to be drawn to these threads.  Sometimes I become panic-stricken, thinking that I might have lost interest completely.   

Back in 2017:
Quote from: Nobody
If I live long enough, I might give some attention to his work in this area.  [writing about Dr. Norman Wildberger]

I see the way life goes.  I witness the very limited amount of short term memory available to any of us.  My engagement with mathematics is more about learning how to work in a calm manner, developing patience and care, to not work in haste or give into despair.  Like I said, my devotion to rebuilding my own personal foundational understanding and skills is quite humble.

I do not have the energy or the confidence to dismantle mathematics and rebuild it from the ground up.  On the other hand, keeping Wildberger in mind, you are not alone in pointing out that the way it is presented may be more confusing than necessary.  Follow your bliss.

In a different thread, Ibra gives me some credit for, at the very least, pointing towards a trail ... In fact, today I am reading our old threads with renewed enthusiasm, as though I was beginning.  Maybe this is the nature of "being no one;" that is, we are consciousness confronting itself.  We circle around abstractions and try to find concrete/solid ground, or at least enough understanding that we might explain some idea or concept to ourselves while walking around aimlessly.

I do not think we take ourselves too seriously, and this might end up being our saving grace, as far as authenticity goes.  Any students of the future are bound to respect the blood and tears left on our notes.
« Last Edit: December 29, 2021, 12:26:15 pm by Half-Crazy Nobody »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

~ Tabak und Kaffee Süchtigen ~