On 5/13/2022 12:48 PM, Timothy Golden wrote:
> On Thursday, May 12, 2022 at 3:55:34 PM UTC-4, sergio wrote:
>> On 5/12/2022 2:34 PM, Timothy Golden wrote:
>>> On Thursday, May 12, 2022 at 3:16:58 PM UTC-4, Timothy Golden wrote:
>>>> On Thursday, May 12, 2022 at 11:35:15 AM UTC-4, Timothy Golden wrote:
>>>>> On Wednesday, May 11, 2022 at 1:09:32 PM UTC-4, Timothy Golden wrote:
>>>>>> On Tuesday, May 10, 2022 at 9:13:35 AM UTC-4, Timothy Golden wrote:
>>>>>>> It seems uncontroversial at first.
>>>>>>> Discern unity on the continuum
>>>>>>> versus unity in discrete terms.
>>>>>>> The problem opens up considerably.
>>>>>>> That these two take the same representation '1' within our numerical representation is problematic.
>>>>>>>
>>>>>>> Set theory is supposed to address this, yet the natural values are formally a subset of the real values. This has been vetted by eons of mathematicians, right?
>>>>>>>
>>>>>>> Having gone through the long way around through the generalization of sign, which uncontroversially I have named polysign numbers, and early on in the past tense, we arrive at a treatment of number as sx, where s is sign and x is continuous magnitude. Sign is of course discrete in its quality; the real numbers being the two-signed numbers, and but for the introduction of a non-travelling identity sign (the zero sign) polysign are consistent with the real number in its present form. Of course three-signed numbers require attention, but if you focused long enough you would bump into them as the complex numbers in a new suit, and realize along the way that the real number is not fundamental. I don't mean to drive you into polysign, but it is this route of thought which leads me to the present interpretation. Having generalized the sign of the real number to what degree am I burdened dealing with the continuous magnitude of it?
>>>>>>>
>>>>>>> Along the way operator theory is encountered. Polysign come with sum and product algebraically defined in Pn. Geometry comes along for free through the balance of the signs. No Cartesian product is necessary. They are extremely close to the polynomial form, but already they possess their modulo sign character from their composition and so the ideal of abstract algebra, seemingly the curriculum where polysign are intersecting, that ideal is not necessary. That confusing load is gone, along with other confusing details such as the obfuscation of closure and the need to introduce real value coefficients. No. The real value is P2. P3 sits alongside P2 as a sibling; not as a child. Operator theory is directly falsified within the curriculum of abstract algebra, though possibly patchups are underway. Meanwhile their treatment of sum and product as fundamental I agree is sensible, though the term 'ring' is poor.
>>>>>>>
>>>>>>> Ultimately we see that mathematics has crossed up a fundamental distinction between operators and values and treats compositions of the two as if they are fundamental values. Instances of these include the rational values such as one fifth as well as the irrational values such as the square root of two. In hindsight the irrational value is foisted upon the student as a foil to the foibles of the rational value so quickly that there is no time to look back upon the problem. Firstly, division is not a fundamental operator. Secondly there is a lack of closure of the rational value. To what degree the rational value constructs the continuum versus happens to fit upon it can be taken as a matter of discussion. Clearly the camp that I have landed in is either deleted from current theory or has never even existed.
>>>>>>>
>>>>>>> The continuous and the discrete are distinct. The operator and the value too are so distinct that such blurry claims as modern mathematics makes deserve our scrutiny. Here I think we can lay a boundary where mathematics left philosophy.
>>>>>>>
>>>>>>> That we are near to discussing physical correspondence too at this early level of theory is good. This is as it should be; the three as one.
>>>>>> Unity in discrete terms can be represented without even uttering our usual sense of number. Any glyph will do to represent a concept such as a count of sheep in a flock; a practical instance of early need for accounting. A leather bag containing pebbles would suffice. On a clean piece of bark a series of blobs or tics made with a piece of carbon from the fire. These early marks are unital in nature. Their value as a transcribable record is complete. So long as no ellipses are used the mapping of a modulo ten value (though here some ambiguity creeps in) is possible, which is our usually presumed representation as say '14 sheep' being bbbbbbbbbbbbbb, the 'b' arbitrarily chosen.
>>>>>>
>>>>>> No geometrical significance is had in this sense of number. We do however witness that every practical instance that can be discussed and verified does occur in spacetime. In this regard the continuum is acting as a basis for the analysis and for the representation. The notion that we will somehow build off of this discrete form to recover the continuum cannot gain theoretical support under this awareness.
>>>>>>
>>>>>> As works in geometry progress and the appreciation of the line as defined by two positions in space (and here should we engage time?) we can eventually work up to this line as a concept of 'dimension' and with the use of the Cartesian product beget the three dimensional representation of the continuum. Frozen depictions on a piece of paper have sufficed and now we all do have the ability to animate a pixelated version on these displays. Still though our perception is not truly three dimensional. We do not see any galaxies beyond the tree. Not even a mountain. We have an occluded form of vision based on ray tracing. We should all be able to point to Hawaii and convey thanks to Tulsi and hope she does not get swallowed by the machine.
>>>>>>
>>>> https://www.youtube.com/watch?v=bhj8xTRjFA0
>>>>>> Meanwhile the orthogonal real valued approach leads to 4D spacetime, three dimensions of which are pulled out of a hat for the sake of physical correspondence and the other bidirectional in direct conflict with its properties as unidirectional. All the while one might ask how a zero dimensional point actually requires four dimensions to address... or was it three actually? This awareness leads to some concept of collapsing systems but also the care with which we could scrutinize existing theory as fictitious. In that modern mathematics has failed to yield an emergent spacetime candidate then its usage as a basis for physics is suspect. This is a fine position to land in, except for the fact that the openings are not well declared. We are engaged in a progression. Existing theory needn't be perfect, but as well it needn't be presented that way either. As we entrain ourselves on the works that came before; as we struggle to repeat their results; to what degree do we blind ourselves?
>>>>> If we accept the logic that rejects the rational value as fundamental based upon its embedded operator we land in the evaluated form of those values, for instance 3/5= 0.6, and we see that but for the decimal point the representation is fully back to a modulo ten natural value. The decimal place is indicating the unity position of the decimal value. In other words (confusion here: base 10 versus the secondary unital mark) the regard of the decimal number as a natural number is not within the ordinary interpretations yet mechanistically it holds. Computations such as sum and product will carry out on these values as natural values. Division as a reverse operator need not be burdensome.
>>>>>
>>>>> As we come to regard this format as the working format on the continuum it is because of its adjustable resolution that it is so. This is epsilon/delta theory playing out as digit chasing. This same awareness put the irrational values on the real line according to Dedekind et al. That this same applies to the rational values: here is a substantial change. Of course having just rejected the rational number what right do I have bringing it back in here? In its just form the instance above 3/5 becomes 3.0/5.0, and here of course the precision is quite limited, but the type of the numbers has brought closure. Natural values do not so easily creep into the continuum other than through the careful interpretation as I've laid it out here. Certainly with a pair of dividers and a straight edge one can develop the intervals needed with geometry on paper, but all will concede that their perfection is an assumption that does not hold true. I would think without too much effort accuracy to 0.01 might be possible but not to 0.001. Well is this mathematics or physics? Doesn't geometry actually lay between the two? The logic of the continuum as I am laying it out here is consistent with geometrical works on paper. 6.67430(15)×10−11: physicists would like to do better than this. Always this will be true.
>>>>>
>>>>> The invalid assumption of mathematicians is that three can exist as 3.000... in perfection. Why this value is invalid is because the natural value 3000... is invalid. No computation can be done on this form. This is because computation is actually done on natural values. Strange though it is for I who originally thought that I could grant the continuous magnitude as a pure concept without ever looking back; yes, polysign does work out this way on the side of sign... but as we go through the gyrations of mathematics from abstract algebra (rather high on the heap) back down into operator theory (which is actually a fairly direct course down to the bottom) the ambiguities that have accrued in mathematics do make themselves felt. The lack of regard for type sensitivity has been enforced on mathematicians under threat of failure. And this as they pay for the way of their teachers and institutions. Mimicry is one of the human's finest capabilities, but if ever there were a place that it ought to have been challenged it is in this subject. Of course there is a necessary tension: without the prior works we would have little to go upon and the gains to be had are quite slow in surfacing. As to how many gems we have still failed to uncover: I happily declare polysign numbers to be one of these gems and expose that polysign acts as a surrogate. I find it bizarre that those amazing minds who have come before did not find it and that I who really do not have so much ability did find them. That they could lead me all the way through to here back in the bowels of simple number theory; rather they empower me to treat this space as open. This notion of openness perhaps is the most threatening thing of all for it lays the professor and the student flat.
>>>>>
>>>>> This principle of openness is in some ways forgotten by me. I do remember back when I thought that the internet could yield it. The open system is not one ruled by a government in secrecy. Obviously the governments will have to practice openness. Whether it can be done in a way that the wrenches in the gears that corrupt capitalists will throw their way: yes, I suppose so. I hope so.
>>>
>>> https://www.youtube.com/watch?v=Kfe1Slac6W0
>>
>>
>> what happened after that ?
>
> You could try: https://www.youtube.com/watch?v=kfMwXsWbvuk
>
> When was the cold war with Russia reignited? 2016 begins an era.
>
> On the numerical front: Values such as 0.333... seem to be accepted by mathematicians while a value such as 333... is not accepted by mathematicians. As to why the addition of a decimal point makes one more acceptable than the other: here I believe my meanderings have come into a new crux. Of course the side effects are many, and this is but one vector of awareness. Still, computationally this arguments holds up very well. That modern mathematics is soaked in ambiguity I find provable.
>
> That epsilon delta theory has enough in it to handle the irrational value suggests that we simply apply it to all values unconditionally. This then exposes our representation of the continuum as gray in nature. The discrete and the continuous each take their representation through the natural value but their meanings, and the added complexity of a floating unit position, differentiates them. Sufficient versus perfection could be a loose form of the discussion, and of course the mathematician as more perfect than any other is a sort of aesthetic that draws certain types to the subject. They are purists. We like that, but still, when the perfection is not attainable some concession has to be made.
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