Fundamentals, Consistency and Breakthroughs

Breakthroughs occur when deeper consistencies are found in the fundamentals of a subject. A breakthrough is always preceded by the discovery of some inconsistency. It then leads to the breakthrough and a deeper consistency.

For a long time, mathematics was based on numerals that did not include the zero. Look at the Roman numerals. There is no zero there. This is because a number was supposed to represent “something” only. For a long time it was inconceivable that the idea of “nothing” had any use in mathematics.

We know that the next number is arrived at by adding one to the number. There is no limit to the next number. It keeps on going for ever. We also know that the previous number is determined by subtracting one from the number. However, at one time the previous number stopped at one, because counting started from one. Nobody seemed to see any utility in seeking a number previous to one.

It was not until the 4th century BC that somebody took a hard look at the inconsistency of not going beyond one for the previous number. This new number was perceived as “void” or “nothing.” If a person spent all his money, he had no money left. This was consistent with the idea of previous number. But this was resisted simply because counting started from one, and “nothing” could not be counted. The moment that resistance was overcome, breakthroughs started to occur in the subject of mathematics. Zero could now be used as a reference point from which to measure. The biggest breakthrough came with the positional notation for the numbering system. Suddenly, it became much simpler to write numbers (compare current numbering system to the Roman system), and mathematics started to progress by leaps and bounds.

Inconsistencies at the fundamental level are hard to accept as inconsistencies because the popular consensus endorses it as the norm. Thus, it took several centuries before the possibility of numbers previous to zero could even be considered. The idea of negative numbers was resisted in Europe as late as the 17th century AD, even though they were known to facilitate the representation of debt. Finally, with the formal introduction of negative numbers later in 17th century, the “previous” numbers could also be extended without limit. A much deeper consistency was achieved at the fundamental level of mathematics. This led to incredible breakthroughs.

There are many such examples of overcoming mental resistance at the fundamental level for the sake of greater consistency. And each time it happened it was followed by wonderful breakthroughs. Here is one more example. We know that larger and larger magnitudes may be represented by the idea of a unit, such as, a planet, a galaxy, a cluster of galaxies, and a universe. Is there a limit in magnitude to which a unit is bound? The expanding universe seems to indicate that the answer is no. Now looking in the opposite direction of smaller and smaller magnitudes, is their another limit to which a unit is bound? The discovery of the irrational number seems to indicate that the answer is no again. So, the mathematical logic points to the possibility of discovering smaller and smaller particles endlessly. We don’t know for certain yet. We are simply looking at a deeper consistency.

In summary, breakthroughs occur when one seeks a deeper consistency in the fundamentals of a subject, even when this requires going against the general consensus.

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Comments

  • Chris Thompson  On November 22, 2011 at 12:05 AM

    This quite an interesting synopsis on the purpose and origin of zero. Also I would like to say thank you for your most recent answers to my questions on KHTK exercises. I found your replies clear and concise. Good luck with your writing on maths.

  • vinaire  On November 22, 2011 at 6:18 AM

    You are welcome, Chris, and thank you. 🙂

    .

  • Chris Thompson  On May 28, 2012 at 5:41 PM

    There is a developing story about a young man who is cutting new sign through physics solving old Newtonian problems with two recent papers on the dynamics of a projectile. Here is someone who may contribute mightily to the whole of us.

    http://math.stackexchange.com/questions/150242/teenager-solves-newton-dynamics-problem-where-is-the-paper

    http://www.reddit.com/r/worldnews/comments/u7551/teen_solves_newtons_300yearold_riddle_an/c4sxd91

  • IdealGoal  On October 7, 2012 at 3:01 PM

    Hi vinaire.

    Thanks for this writing showing the historical resistance to the changes in mathematical concepts.

    As I can see in the history of mathematics, only in the last centuries the theoretical aspects have been developped for themselves with a full application of the principle you develop here about the seak of a higher consistency by spotting inconsistencies first as aware intellectual action.

    What we should call “mathematics” didn’t exist at first 10.000 years ago, but “quantification” yes.

    When you speak on the seak of an improvement of the consistency of mathematics you do speak about “mathematics”, vinaire. But when Man was counting the number of his sheeps he didn’t make “mathematics”! He didn’t work on an “abstract”, an organised abstract of idealities that we could call like this!

    At first we have the accountant Babyloniens, for exemple, needing to handle inheritances, dividing of goods with few rules to apply, like “the oldest son will receives the half, the second a quarter, the third, a height, etc. If any girl, divide the remaing in equal parts. If only one son or one girl, of course he or she get the whole”.

    This example is not very complexe but were getting rather some times and same for the division of harvest for a village, especially a bit later in Egypte.

    So, first we had a “need”, to solve few problems about “quantification”: “how much the third daughter will receive in inheritance in a family of 5 sons and 3 girls?”

    Then, something very different happened: the accountants, to improve their efficiency and elarge the complexity of quantification problems they were asked to solve, they noticed few repetive patterns when they wrote their calculations. They decided to get more easy with, to use these patterns as such, so an abstraction, without to care for any quantity involved.

    They found then they were kinds of patterns and startted to play with seeing that they could get more and more easy with any problem they could meet in their work. And at this time, we were for the first time in this aera of the planet “making mathematics”, their first steps in algebra. (Babyloniens were not necessary the first to make this evolution, Chinees for example, developped their own way aside.)

    But when then at the end of the XIXth or beginning of the XXth, Cantor was working on the Sets Theory, he was no more connected with any concrete problem of quantification associated with daily life but purely in the world of abstraction and in reference to a whole set of very extented, organised and consistent patterns, essentially conceptual forms and objects.

    At this stage, yes, a true mathematician, someone who his aware of his seak for a better consistence, could exist.

    Many times in the history of mathematics, a new pattern or object was create by a practical guy and only after having mathematicians working on it to find how it could fit a consistence with the existant mathematical constructions.

    One of the main example of this, is when Faraday was working on electrostatics and needed to use a concept of a certain strength, attraction or répulsion, so oriented. He used arrows to have symbol for this concept, concept known as “vectors” today.

    But the point is that he didn’t try to see if the mathematical where inconsistent, he just spotted that he missed a conceptual tool and created it. Only after the mathematicians looked to see how it could fit.

    So, all this to say that, with my present viewpoints and reality, there was not a necessary aware seak for consistences and first inconsistences, but more in early age of our present civilisations a seak for intellectual tools to get more efficient in the concrete problems they had to face, and then only, they were guys who started to work only on these tools as concepts, and even an other stage, specific to the second part of the XXth century where few mathematicians started to seak for new patterns, new objects, for the seak of themselves, only with the self-obligation of self-consistence. Oddly enough, this last way releaved himself productive then for concrete uses, like in physics, economy, etc.

    Now, Dear vinaire, what you’ve spotted before shows, still from my own viewpoints, an effective and systematic heuristic method for any kind of researchs, and that’s looks to me a very useful point. 🙂

    What do you think?

  • vinaire  On October 7, 2012 at 4:21 PM

    Dear Didier,

    Thanks for taking time to write this wonderful response to my article. You are absolutely correct in pointing out that many mathematical concepts evolved out of practical necessity. Another example of this that comes to my mind is the formulation of calculus that came out of the researches into the phenomenon of motion that Newton delved into. I am sure there are still many more discoveries like that waiting out there for us.

    What led me to the above article was the amount of data that now seem to exist in every subject. In this Information Age the amount of data that one has to plow through is enormous and overwhelming, at least for me. I solved this problem for myself by diving into fundamentals of whatever subject I get interested in. Fundamentals are always very few. So, this nicely takes care of any overwhelm due to the volume of data. Knowledge of fundamentals then also helps me sort out the important bits from the volume of data that I can safely ignore.

    Now that I have the important bits, I can increase my understanding by looking at them more closely. I don’t have to worry about what is consistent, because that is easy to understand. It is the bit that is hard to understand, which pulls me forward.

    One day I was looking at the history of mathematics in Wikipedia, when the above article just hit me. So I wrote it down. I have also been looking closely at Scientology Axioms, and comparing them to my understanding of Hinduism and Buddhism. It was then that I found myself start becoming aware of this principle that I jotted down at the end of the above article.

    Thank you very much for your beautiful post on this blog. If you want to polish this up as an article, I would be happy to publish it as your contribution. This blog is about diving into knowledge and deriving enjoyment out of it. I do publish contributions from others that I find to be enjoyable.

    Again, thanks for appreciating this article. There are not many people sharing this passion who have visited my blog. I very much like having you here.

    .

    • IdealGoal  On October 7, 2012 at 5:55 PM

      Hi vinaire! You’re very welcome for my post and yes, you can publish it as my contribution but as I typed it from my smartphone could be I’ve missed some typing errors or even of grammar, spelling, etc., I couldn’t easily check this while typing it :/ If you can correct them, you may.

      For the “technical points” of your answer, yes, it’s my own knowledge too about Newton, differential calculus and integral calculus have a large agreement that he developed these tools while working on the description of motion, etc.

      Yes, it’s not so often to find someone who share this passion about Knowledge and I would say mostly “Understanding”.

      I make this difference because “knowledge” can be more about “the things we learn” while “understanding” shows a result of “workable knowledge”, as I could see it myself.

      Vinaire, I will be please to continue on these subjects, and I very share with you your way to go through the existent knowledge starting with fundamentals as I would say… “Stable Data”? 😉

      Indeed, I did like this with Scientology itself: studying first the Logics, Pre-Logics, Axioms and Factors.

      Having done that makes me feel the same way that you have shared in your general studies in sciences: nothing can confuse me and I have the ability now to spot immediately a false datum anywhere, it’s like it jumps to the eyes.

      An other advantage I have seen with the way we share, is that it gives the ability to create with the subject, especially to spot the “holes” that could exist in a theory, or to create an other for other uses or just for the fun! 😉

      But yes, your own contribution is again for me, very useful in this peculiar times of oceans of data, because it really gives a way for any student of researcher, to find their way to lead to possible new discoveries. This is, for me, what we do call “epistemology” 🙂 (Would be even more “épistémologie” in french, cause in for english culture, “epistemology” regroups two concepts which were separated first in the old continent: “épistémologie” itself, more about history, methodology, and validity of the reasoning, and “philosophie des sciences”, “philosophy of sciences”, which is more about “what would be interesting to look for as knowledge”, the pertinence of the knowledge itself, and the questing about “if” and “how” a being could “know”, named to for these last ones “théorie de la connaissance” – “theory of knowledge” -, but could be you know that it all already 🙂 )

      Ok, nice to read you again, but could be you’ll have to patient a few for a next post :/

      Regards.

      • vinaire  On October 7, 2012 at 8:57 PM

        Thank you DiDier.

        I plan to Look at Scientology Axioms more closely on my blog. What do you think of Scientology Axiom #1? It has been treated in great detail in Eastern philosophy.

        .

        • IdealGoal  On October 7, 2012 at 11:01 PM

          Great! Vinaire 🙂

          For the Scientology Axiom #1, I can only trust you for having been treated in great detail in Eastern philosophy. But looks to me that it was something like that Buddha has came too, but not very sure of this.

          All what I could say in this Axiom is that is the simplest truth it could be expressed but to say:

          – “We are the power of decision”, or more developed:

          – “we are fundamentally one and total cause, fully responsible, knowing everything”.

          But it’s just my opinion, my feeling, my intuition; even if based on the logic of the Factors too.

          But, for my present viewpoint, if I would have written it I would have written rather:

          ———————————————————————–

          Axiom 1 (Careful: ALTERED VERSION, NOT L. RON HUBBARD VERSION): Life is MEST ran basically by a static POWER OF CHOICE.

          DEFINITION: A static IS not a space, not a time, SO not an energy neither a mass, SO it has no motion, no wavelength; it has no location in space or in time, UNLESS HE DECIDE/CONSIDER/POSTULATE TO BE ANY OF THESE. It can decide to stay in communication/assume with its postulates, SO to “perceive” them, or missassume them, SO to not-is them, SO no more perceiving them.

          ————————————————————————-

          To compare with the none altered CoS Inc.’s version (I don’t have with me the old books :/ ):

          ————————————————————————-

          Axiom 1 Life is basically a static.

          DEFINITION: A life static has no mass, no motion, no wavelength, no location in space or in time. It has the ability to postulate and to perceive.

          ————————————————————————-

          Well, I wish you could have understood is not a change of the Tech of L. Ron Hubbard itself, because it is no more Ron’s Axiom, so no more “Scientology”, but this change would permit us to evaluate for ourself LRH writing by differentiations and similitudes.

          After all, it is precisely what he asked even in the old Basic Study Manual when he talked about “integrity”, aren’t you agree? Vinaire.

          So, do you think my changes pertinent?

          Regards.

        • vinaire  On October 8, 2012 at 3:16 AM

          OK, let’s discuss Scientology Axiom one. I have one question on what you have written.

          What pronoun can most suitably be applied to the static?

          .

        • IdealGoal  On October 8, 2012 at 1:22 PM

          I had the same question with myself! Vinaire. lol

          When I typed my modifications on the Axiom 1, I was to put “he” in a neutral way, but as I saw it was “it” in the original text I’ve written “it” too 😉

          Well, as “it” is often used for an object, or non-pet animals, we would say that “it” should not apply!

          But I’ve found in Wikipedia that:

          – “it …/…, singular English gender-neutral pronoun; …/… is used only as a dummy pronoun in various impersonal constructions and to refer to abstractions, places, inanimate objects or materials, and non-human life of low order or unknown gender. ”

          So looks to me it doesn’t let place finally for discussion here at it is used for “abstractions”, while “abstract” itself means (thefreedictionary.com):

          – “adj.
          1. Considered apart from concrete existence: an abstract concept.”

          And of course, if it is not a space, not a time, not an energy, so not a matter, for me, the “static” behind* a life, or living-organism, by definition is not part of the “concrete world”, while “concrete” has this definition (still thefreedictionary.com):

          – “adj.
          …/…
          2. Existing in reality or in real experience; perceptible by the senses; real: concrete objects such as trees.”

          Of course, as a Scientologists, this could raises concerns for us; right? Cause we speak about “reality” and thanks to LRH we know many things about “what is the reality” 😉

          To not substitute me the source of Scientology itself, I prefer to put few quotations form him, especially from his “Technical Dictionary” I like so much:

          “REALITY,

          1. is here on earth, agreement as to what is. …./… (Creation Of Human Abilities”, p. 249)

          2. that sequence which begins with postulates and ends with mass, which we originally defined as an agreed-upon thing. Reality is the agreed-upon apparency of existence. (Control and the Mechanics of Start Change Stop, p.11) ../…”

          So, we see there is here something recurrent here: the concept of “agreement”.

          But to “to be agree”, by definition, it supposes that we have several “terminals” (as Ron spoke) to be agree.

          So, that leads for me to the conclusion that no “reality” is possible unless several “entities” could “exist” having the “power of decision” of taking a agreement.

          So, looks to me that means that the “static” itself is “apart from” … “any agreement”! And would mean that if “a static” takes “agreements”, it will get involved subsequently with a concrete existence! lol

          Isn’t it an interesting conclusion? lol

          But well, “it” at least has no “gender”, by the english definition of “gender”, so we are sure “it” is “neutral”, and as the Wikipedia article explains very well, “he” would be subject to the gender-equitable controversies any way, and as the standard use of “it” includes “the abstract”, It’s ok for me to continue with “it” as pronoun.

          What do you think?

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