On Monday, 10 May 2021 at 08:58:35 UTC-4, timba...@gmail.com wrote:
> On Monday, May 10, 2021 at 8:49:47 AM UTC-4, Eram semper recta wrote:
> > On Monday, 10 May 2021 at 08:34:24 UTC-4, timba...@gmail.com wrote:
> > > On Sunday, May 9, 2021 at 5:41:15 PM UTC-4, Eram semper recta wrote:
> > > > On Sunday, 9 May 2021 at 12:13:52 UTC-4, timba...@gmail.com wrote:
> > > > > On Tuesday, May 4, 2021 at 11:45:59 AM UTC-4, Eram semper recta wrote:
> > >
> > > > > Beyond this, if it is correct that the following line finishes Dedekind's analysis:
> > > > > "For this we notice that if
> > > > > x > 0 , 2 - x ^ 2 = e > 0,
> > > > > then
> > > > > 2 - y ^ 2 <= e / 2
> > > > > for the y constructed above, this means that we have a sequence in A whose square can become arbitrarily close to 2, which finishes the proof."
> > > > > - https://en.wikipedia.org/wiki/Dedekind_cut#Construction_of_the_real_numbers
> > > > That is gibberish which in no way defines sqrt2 or any "irrational number".
> > > And so it is that you should refine your criticism to being against:
> > > https://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of_limit
> > Nonsense. The RATIONAL NUMBERS do not require any limit theory. In fact, limit theory is not even required in calculus.
> >
> > That we can calculate derivatives and definite integrals has nothing to do with limit theory. My New Calculus is proof of this.
> > > > >
> > > > > This same analysis can be done simply broaching an additional decimal place as well. There is no need of anything other than the ordinary computational form.
> > > > Nonsense. The need is for UNDERSTANDING. That a computer can be programmed to perform all these computations does not mean the computer understands the same.
> > > This sounds quite a lot like Dedekind's own words.
> > Only Dedekind's honest admission that he had no clue why calculus works (not in so many words). Other than this, Dedekind remained clueless and unfortunately his hilarious essay formed the basis of formal systems and logic.
> > >
> > > To resolve the problem through epsilon/delta is to admit that every number has a precision.
> > Rubbish. A number is a very well defined concept:
> > A number is a name given to a measure that describes a magnitude or size.
> > The only occasion when you can talk about precision is wrt measure of incommensurable magnitudes such as sqrt2 or pi, but neither of these are numbers, only symbols denoting magnitudes that have no measure.
> > > We can push the precision up, and this is the resolution. This is the best that we can do. That the value
> > > 2.3
> > > actually is a segment on the line: this is roughly what we are admitting to.
> > No, no, no. That is FALSE. You should first understand what it means to reify a number line. No mainstream academic I have ever met had a clue:
> >
> > https://drive.google.com/file/d/0B-mOEooW03iLMHVYcE8xcmRZRnc
> >
> > Moreover, the derivation of the RATIONAL number line requires the prior establishment of number and not the other way round.
> > > No different than a pencil has some thickness as we trace along a straight-edge so that the legible trace does not actually come perfect, then we do have actual correspondence within this understanding.
> > Clueless as can be you are. Geometry has nothing to do with pencil, ruler or compass which are mere tools or manipulatives used to convey the perfect concepts therein.
> > > Tick your point off along your imperfect first trace; it has some thickness.
> > Not even wrong!
> >
> > <I stop here. You need some serious education! >
> > > That we can push the number much farther than we can do the trace on the paper, and that you may think that the ideal model in your head is pure: it is not, for if you can see the trace then it is too thick. That your mind will not halt is as problematic as it is perfectionistic. We could have halted at say the 0.5mm trace that a mechanical pencil can deliver; scaled up to a legal size piece of paper; putting naught and unity at roughly ten millimeters from diagonal corners; matching straight edges to prove the one we will use to do the trace with. Then tick, tick, and you can be off plotting reciprocal values for the rest of your life through iterated means. No doubt somebody did plot 1/101 as an exercise; no matter how badly it came out. The trick of getting 1/102 on the same graph is really the problem, no? On the one hand it is merely a matter of counting, yet on the other upon fixing unity the dread of the paper form is well felt. That your compass barely knows the difference between the two; these problems are absolved by the pure mathematics. It seems as though there is some sense of absolution going on in the human mind which practices mathematics in the supposedly pure form. That you claim your mind to be more pure than everyone else's fits this paradigm, and I don't mean any harm or insult in saying it. We all are trying our best here as I understand it. We who attack the status quo system with our skepticisim are willing to break with our social animal mimicry responses, and this is yet another layer of the human problem, yet without any such we would not have anything to discuss here. None can claim to have arrived here on their own terms out of thin air. We can try, but proof is not going to be forthcoming. We are all dealing in a system of tremendous accumulation. Some ride the status quo like a railroad. If they stray even just a little their minds will falter. We on the other hand have no trouble slipping off the tracks, heading down the bank to have a look at the stream, and return dancing along a rail for another hour or so. Not too long, though. Not for me.
> https://archive.org/details/cu31924001586282/page/n37/mode/2up?ref=ol&view=theater
> "Here at the close we ought to explain the connec- tion between the preceding investigations and certain fundamental theorems of infinitesimal analysis.
> We say that a variable magnitude x which passes through successive definite numerical values ap- proaches a fixed limiting value a when in the course of the process x lies finally between two numbers be- tween which a itself lies, or, what amounts to the same, when the difference x^o. taken absolutely be- comes finally less than any given value different from zero.
> One of the most important theorems may be stated in the following manner : " If a magnitude ;*: grows continually but not beyond all limits it approaches a limiting value."
> I prove it in the following way. By hypothesis there exists one and hence there exist infinitely many numbers 02 such that x remains continually <a2; I designate by 2t2 the system of all these numbers ai, by iti the system 6f all other numbers o.\ ; each of the latter possesses the property that in the course of the process x becomes finally ^ai, hence every number ai is less than every number 02 and consequently there exists a number a which is either the greatest in 2ti or the least in 2(2 (V, iv). The former cannot be the . case since x never ceases to grow, hence a is the least number in 2t2 Whatever number 01 be taken we shall have finally ai< a; <o, i. &., x approaches the limiting value o.
> This theorem is equivalent to the principle of con- tinuity, i. e. , it loses its validity as soon as we assume a single real number not to be contained in the do- main 2J ; or otherwise expressed : if this theorem is correct, then is also theorem iv. in V. correct.
> Another theorem of infinitesimal analysis, likewise equivalent to this, which is still oftener employed, maybe stated as follows : "If in the variation of a magnitude a: we can for every given positive magni- tude S assign a corresponding position from and after which X changes by less than 8 then x approaches a limiting value." "

Oh dear. What the fuck are you even going on about here? I told you that all this is irrelevant.

You can't draw a line with any thickness, much less a "number". I've never heard anything so stupid in my life.

The concept of NUMBER has NOTHING to do with limits. As I've repeatedly told you:

A number is a NAME given to a MEASURE that describes a magnitude or size.

Not all magnitudes can be measured. For these it is insufficient to talk about a limit of any sort. We know these are constants, but as we cannot measure the same precisely, we cannot talk about giving them a NAME in terms of MEASURE, only in terms of MAGNITUDE.

Pi, sqrt2 are symbols for magnitudes. They are not symbols or NAMES for numbers. Pi is a constant ratio which has zero to do with limits.

pi = circumference : diameter

pi =/= circumference / diameter <---- because then this would be a NAME given to a number, but circumference / diameter CANNOT be a number because circumference and diameter have NO COMMON MEASURE. What is the matter with you?!