You are missing my point. Larson uses the sense of polarity, but not in the sense you are want to use it. Although, I admit it could be used to indicate above and below unit speeds (in fact, I use it that way at times), but it's important to understand that unit speed is the datum which is referenced by him, not the "directions" of the deltas, as you are presenting them.Horace:
The signs are needed in the canonical scalar speed notation in order to avoid the "directional" ambiguity. How else would you distinguish inward from outward scalar speeds? This is manifested in that passage above, written by Larson, where he is analyzing only the attributes of scalar speed at unity - not any speed deviated from unity. He was the one to use the words "negative" and "positive" in reference to scalar speed that he was expressing in the canonical form of 1Δs/1Δt , as evidenced by his words "each unit of motion consists of one unit of space in association with one unit of time...".
Each unit of motion that he refers to is an instance in a scalar time ordered, or a scalar space ordered sequence. This is easily understood with the help of the PAs. I cannot understand your reinterpretation of polarity in relation to a previous unit of motion only, To me, it's just bizarre. Sorry.
I don't think that's true, Horace. Here's the complete quote:Horace:
Not according to Larson. In that passage above he was considering unit speed only (not any deviated speed). Exchanging the numerator with the denominator does not affect the unit speed, thus your distinction between (s/t) and (t/s) does not make sense for the unity speed that Larson was considering.
I understand from this that the negative speed is less-than unit speed, caused by the sequence of reversals of space, while the positive speed is greater-than unit speed, caused by the sequence of reversals of time. We can express the greater-than unit s/t speeds, as greater-than-one rational numbers, but they are the equivalents of the less-than-one rational numbers, when we invert the space/time terms to time/space terms.Since motion exists only in units, according to the postulates that define a universe of motion, and each unit of motion consists of one unit of space in association with one unit of time, all motion takes place at unit speed, from the standpoint of the individual units. This speed may, however, be either positive or negative, and by a sequence of reversals of the progression of either time or space, while the other component continues progressing unidirectionally, an effective scalar speed of 1/n, or n/1, is produced.
I've just shown that he wasn't just referring to unity speed in that passage, but I don't quote Larson as an "authority," only to help clarify my own position, which differs from his at times.Horace:
That might be true in general when using the operational notation of a fraction, but Larson was not using that notation and he was writing only about unity speed in that passage above.
With reference to the image of the timeline diagram you posted, it's important to realize that Larson didn't understand Hastenes' concept of an operational number, as most people don't. At least he never indicated he did. So, distinguishing the difference between the quantitative interpretation of number and the operational interpretation is crucial to the development of the LRC's RSt.
For instance, In the quantitative sense, the rational number 1/3 is one-third of the whole number 1, but the in the operational sense, 1/3 is -2, as you have shown. Yet, the question of how the 1/3 speed can arise from the unit progression, given that it's not an even number, was answered by Larson in a way that strains credulity and is really hard to follow. Namely, that the reversals occur in triplets - In-out-in, in-out-in, in-out-in, etc.
Nevertheless, given the less-than-unit, "unit," we can see that this bizarre conclusion is not necessary, because the value of the unit operational number 1/2, can be combined multiple times to produce any other operational number in the sequence of your diagram; That is to say,
1/2+1/2 = 2/4 = -2,
1/2+1/2+1/2 = 3/6 = -3,
1/2+1/2+1/2+1/2 = 4/8 = -4...
So, in this way, we can eliminate the confusion that dogged even Larson. Just sayin.