My compliments to Agentredlum for so quickly spotting and pointing out the failure of my simplistic description of Plank points – and in such a concise and to the point way (pun intended). Concise and to the point is something I have trouble doing at times.
Your comments about my description of Plank points are correct. I used the simile of spheres to get the thought across in a less complicated and too idealized way. I tried to avoid use of too many words – but as a result I lost accuracy. So, rather than perfect or even “nearly like” spheres is not the best description because of the problems with spheres completely filling space which Agentredlum notes. I’ll no doubt still be too inaccurate, but perhaps a better description of Plank points is as a mushy configuration of space that may be more or less accurately visualized as tiny roundish marshmallows rather than spheres. These marshmallow like points have dimension of a sort and are compacted together to fill all of space - and they compose all of space. One can only say that their average diameter is the Plank length….quantized space with an average volume of a Plank volume. If a set of them could be arranged in a line, they would constitute a finite number line without partial pieces representing places between points – that is without irrational components. The assertion is based on the idea of quantum fluctuations which are tiny, messy creations of energy/matter on a similar scale. Your edits seem to acknowledge a similar path.
I can’t say whether the ideas supporting quantization are dead or not, but if they are, I hope they are revived. My posts here support quantization by way of suggesting the idea that the math used to calculate quanta of various sorts is based on a philosophy of number theory which includes infinites as part of a counting continuum in the form of the idealized standard number line. Infinities as reflected by the presence of irrational numbers confound calculations that are supposed to reflect the physical world and lead to results that either are not interpretable or lend themselves to ridiculous interpretations. So, you are also correct to note that extending the description of the hydrogen spectrum to other elements has met with difficulty. My answer is I believe the reason may be due to using math (including pi) based on a continuous numbering system.
…and I really like your edits… they strike me as thoughts that would yield results if they could be formally adopted – not to replace traditional math as a philosophy, but to replace certain mathematical applications as descriptions of the real world. The problem would be in how to actually formalize things. Particularly good is to use the DeBroglie wavelength and uncertainty principle as an approximation mechanism. It would certainly be a candidate. I suspect that is very nearly in line with what I am advocating. The alternate use of approximations by rounding the irrationals will eliminate the irrationals by substituting the closest rational number at the cutoff point. Pretty much the same as quantizing space to the Plank length. I have often wondered what a study of the 31st to maybe the 39th decimal place of the irrationals would yield – down to and just past the theoretical Plank length - would it be true randomness or would patterns be detected?
More to your edit 2, I’ll mention that the difference in “what’s the difference” is partial or in-between points that when included in calculations yield the kind of results I mentioned earlier. Too often none, confusing or ridiculous.
Lastly, your mention of the square root of 2 is interesting, but in a different way. I would propose that by rounding the root, one would have to either squeeze the triangle very slightly to make the end points connect, yielding a triangle whose internal angles add to less than 180 degrees… or stretch it slightly, yielding a triangle whose internal angles are more than 180 degrees. Neither of these are necessarily outside the realm of reality. My speculation is that quantized space reflects a closed universe by necessity, one in which the internal angles of triangles are less than 180 degrees – and not as an approximation, but as the reality of the nature of quantized space. I understand the physicists are trying to do experiments that will decide the question, but I’ve not heard any results. Hopefully the results will not be interpreted in a ridiculous way.
So, Agentredlum, I cannot say you are incorrect with your comments, but rather to say there is plenty of room for debate and exploration. I think I understand your point of view and I do appreciate the response. I believe you can see I am seeking to plant seeds rather than cut down the forest, but I sense that I’ve beaten this particular subject near to death and do not wish to wear out my welcome by overdoing it. For others who’ve seen and thought about where the irrationals are and witnessed my words…Thanks!
