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This is a description of a little project I played with in high school. The proper subcategory for it under "Science" would be "Nuclear Chemistry", but these days even ordinary Chemisty is a subdivision of Physics, so...
Before you jump to conclusions, I should say that this was basically just a
mathematical exercise; no explosions were involved (or even likely).
OK, some appropriate background information is now necessary. In Chemistry class (hopefully most of you reading this had at least a LITTLE exposure to the subject), they tell you about something called "Atomic Weight" or "Atomic Mass". Like ordinary masses such as kilograms or slugs, a particular atom is used as a "standard mass" against which all other masses are compared. (The standard kilogram is locked away in Paris somewhere; I don't know where they keep the standard slug, or if there even is one, for the "English" system of measuring units.)
If I recall right, due to the way chemists made their measurements in the early days of Chemistry-as-a-Science, it happened they could measure oxygen more accurately than other atoms. Having discovered that other elements mostly had weights that were close to whole-number values if oxygen was set equal to 16 units, they decided to Officially Define the weight of oxygen as having a mass of 16.0000 "atomic mass units" (or "amu").
A lot of data was then gathered, using that Standard.
Then came the discovery of "isotopes", which messed everything up. Oxygen has three stable isotopes, with approximate masses of 16, 17, and 18 amu, and it is the Natural MIXTURE that was Officially Defined as equalling 16 amu! (the heavier ones are relatively rare). The physicists who made this discovery didn't like that situation at all! They wanted one specific atomic isotope to be set as the Standard, in whole units. However, if they were to recalibrate the Standard, such that oxygen-16=16.0000 amu, then all the other atomic weights became seriously different from the data that the chemists had been painstakingly gathering for years.
A compromise was reached. It was decided to set Carbon-12=12.00000 amu, because the atomic weights of natural mixtures just happened to come out to be pretty much the same as before, when the old Standard gave the mixture of oxygen isotopes the weight of 16.0000 amu (now the oxygen mixture is 15.9994). Ever since, Carbon-12 has been the Standard for both Chemistry and Physics.
However, Science marches on, and keeps gathering data. The field of nuclear physics discovered something called "the curve of binding energy", that sort-of explains why different atoms, even though constructed from whole numbers of protons, neutrons, and electrons, had atomic masses that varied quite a bit from whole-number values. For example, while carbon-12=12.00000 amu, hydrogen-1=1.007825 amu, and tin-118=117.901606 amu. See the linked table.
Personally, I didn't like the idea that a huge percentage of the atoms had "accurate" atomic masses that were LESS than the whole number associated with the isotopes. It seemed to me that it would be more "aesthetic" if some sort of Standard was used, that let all the atomic masses be AT LEAST that of the sum of protons and neutrons in all the atoms' nuclei.
According to the curve of binding energy, the "lowest" point in the curve is iron-56. On the carbon-12 scale, it has a mass of 55.934942. What if we re-define that as 56.00000? How would all the other istopes compare to that "Standard"?
Dividing 56 by 55.934942 gave me a result of 1.0011631, which I then used as a multiplier for all the other atomic/isotopic masses. This was done on paper with an early calculator, long before I could use a computer and do such things as post it here. (I only recently decided to post this Idea, so please give me some time to make the computation results available.)
Anyway, after obtaining the list of values converted to the iron-56 "standard", it was indeed true that every single atomic/isotopic mass was at least equal, and usually slightly more, than the number of nuclear protons and neutrons. Hydrogen-1 was 1-and-a-fraction; carbon-12 was 12-and-a-fraction, tin-118 was 118-and-a-fraction, and so on.
It occurred to me that it might be interesting to create a graph of the data. On the X-axis of the graph, I put the whole-number values of the istopes. On the Y-axis of the graph, I put the fractions of each mass, that was greater than the whole number. The units on the Y-axis were something like 1/1000 the units on the X-axis, so that the differences could become obvious. In the near future I will scan the table and link it.
The cool thing is, without doing any conversion of the sort normally associated with E=mc-squared, this "Mass Fraction Graph" closely portrays the curve of binding energy, of the various nuclides.
Isotope Masses table
http://physics.nist...ions/stand_alone.pl
As mentioned in the main text [Vernon, Feb 15 2009]
Curve of binding energy
http://hyperphysics.../nucene/nucbin.html
As mentioned in the main text [Vernon, Feb 15 2009]
Stephen Hawking Loses a Bet
http://physicsworld.../article/news/19926
If Space and Time are continua, then Information cannot be Conserved; it would be lost forever in a black hole. However, the evidence we have suggests that information IS conserved.... [Vernon, Feb 17 2009]
The Stubbed T.O.E.
http://www.nemitz.net/vernon/STUBBED2.pdf
More on binding energies of different types (including "negative") than you probably want to know. [Vernon, Feb 17 2009]
Here's the Graph
http://www.nemitz.n...n/massfracgraph.GIF
Because of its size, not all the istopes are listed. But the most important parts are here, so it should be good enough. Also, keep in mind two factors that affect the shape of this curve, compared to the curve of binding energy. This curve is per-nucleus; the other is per-nucleon. And this curve is exaggerated because the Y-axis is in units of 1/10,000amu. [Vernon, Feb 18 2009, last modified Feb 19 2009]
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Well, yes, seems fairly sensible in a way. But it's still
arbitrary, in the sense that your iron 56 presumably has a
different mass from that of the sum of its components - ie,
it's the most stable isotope, but doesn't have zero binding
energy. Aside from the coolth of showing *relative* binding
energies directly, what would be the advantage of this
compared to the current system? |
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I think a more useful thing would be to redefine the signs for
electricity, now that we know the electron is negative. |
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I really think it should be callibrated for Hydrogen - 1. Very sepcifically, unbound monatomic Hydrogen, with (or without) one electron (if it turns out this exists). Or, perhaps, the actual mass of a free proton, if, too, that is possible to measure with accuracy. In theory (or in my theory, anyway) - that would make everyting else relative to that, and any mass ratio changes would directly correlate with the binding energy present. In my mind this is a little less arbitrary than just choosing a stable element. |
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It'd be interesting to get the masses of free protons and neutrons, as well as free electrons, and use that to make a graph of atomic masses VS ideal atomic masses. To me, that'd be the best way to show the E=mc^2 relationship that binding (and other...let's not forget the stability of the electron binding, especially in terms of ionic and covalent bonding should affect [italics] molecular [/i] mass as well) energy has on atomic mass. |
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[MaxwellBuchanan], yes, I misspoke a bit. Iron-56 has the most amount of mass-per-nucleon converted into binding energy, which means its measured mass, the sum of those nucleons, is minimized. The normal curve of binding energy shows binding-energy-per-nucleon, and therefore is maximized at iron-56. The graph is therefore sort of an inversion of the standard curve, yet just as precise, in its own way, as the curve. |
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Next, yes, of course the atomic weight of the natural mixture of iron isotopes would be fractionally greater than a whole number. It's not important; no chemist currently minds that all current atomic weights are fractionally different from whole-number values. What they WOULD mind is having all the existing data need to be converted to a new Standard, should this for some oddball reason ever be considered as a Standard. It's not really necessary. |
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For this single purpose, though, the iron-56 standard makes perfect sense. If there is any advantage in this graph over the usual curve of binding energy, it lies in not needing to do mass-energy conversions to compute that standard curve. This graph is conceptually and technically simpler. |
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[Custardguts], while certainly a fundamental unit of nuclei, an Atomic Mass Unit standard based on hydrogen-1 would have the same aesthetic problem as carbon-12; a great many masses of nuclei would have precise values that were less than the "simple" values; oxygen-16 would have a precise mass of 15-and-a-big-fraction, and so on. Not to mention, it could be argued that choosing the neutron would be just as valid as choosing the proton, as the Standard (and the aesthetic problem would exist for that, too). Choosing iron-56 as the standard is the ONLY way ALL precise nuclear masses will be at least equal to their "simple" values. That makes it unique, just as hydrogen-1 is unique in a different way. |
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// The cool thing is, without doing any conversion of the sort normally associated with E=mc-squared, this "Mass Fraction Graph" exactly portrays the curve of binding energy, of the various nuclides. // |
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If you graphed the differnces to whole numbers before your conversion, i am pretty sure you would (qualitatively) get the same picture. Multiplying everything with a constant will not alter the shape of things... |
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[loonquawl], no it doesn't work that way. Tin-118 has the lowest point on the graph if the carbon-12 scale is used (that's why I listed it in the main text). That number "118" makes a difference. |
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The difference is something I should have mentioned earlier. This graph is not a per-nucleon graph. One would have to divide each fraction by the associated whole number, to get the per-nucleon values. So, actually, the curve of this graph does differ in that respect from the usual curve of binding energy. It is a curve of per-nuclide values. Nevertheless, it is similar enough to the ordinary curve that the resemblance is obvious. |
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Arbitrary units of measurement are just that. Arbitrary. I used to be ambivalent about their use, but now, I am not so sure! |
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The same problem occurs with the chosen mathematical unit 1. There is certainly no point we could call "one" on the number line without an arbitrarily chosen point that equals one. Mathematically there is no point that is excactly one unit positive or one unit negative from zero. Unless we assign that unit. So far all we have is e^(i(pi)) = -1. Although this comprehesively defines a unit against zero it uses itself in the proof. |
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... could I get the Reader's Digest version of "binding energy" ? so a proton/neutron temporarily gives up some mass to energy to bond with others ? |
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In that case why not use the raw mass of 1 proton or neutron as "1" ? |
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[4whom], while mathematics can deal with continua, the physical world, so far as we can determine, is quantized into units, likely even including Space and Time. Even if continua were common in the physical realm, there are still situations where "1" is completely distinct. Remember "yes" and "no"? How about the famous phrase, "a little bit pregnant"? If something either exists, or does not exist, the possibilities are limited to exactly 2 discrete conditions/states, typically represented by 0 and 1. |
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While it will be difficult, it would be ultimately beneficial for Science to discover all the REALLY fundamental units of Nature, and to then base all our measurement systems on those units. Such a system would be completely independent of human egotism and human preferences; it would work across the Observable Universe. In one respect, actually implementing such a system is not so difficult; did you know that the "English" system of units used in the United States is actually based on the Metric System? The "inch" is exactly specified to be 2.54 centimeters, and so on. Such things can be done again.... |
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[FlyingToaster], any chemical or nuclear reaction that gives off energy involves the conversion of mass into energy. The reaction products, such as carbon dioxide and water if we burned gasoline, or helium if we fused hydrogen, are generally very stable substances and difficult to break apart. (While CO2 was unknown, water for thousands of years was thought to be a fundamental substance.) Obviously, we can say that the components of those reaction products are strongly bound together (keep in mind that nuclear particles are usually MUCH more strongly bound than chemical particles). It takes the same amount of energy to break them apart as was released when they were formed. THAT is "binding energy". |
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The raw mass of 1 proton or 1 neutron are values that are not associated with any binding energy. My only objection to using one or the other as a measurement standard is that the results are not aesthetic. Look again at the main text where it says under the carbon-12 scale, the atomic mass of tin-118 is 117.901606. We KNOW that that atomic nucleus has 50 protons and 68 neutrons in it (118 particles); why do we want to say its exact mass is less than 118? |
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Meanwhile, in iron-56, its 26 protons and 30 neutrons have fused to produce a nucleus that has the maximum possible amount of binding energy PER PARTICLE. That means one way of thinking about those particles (not widely accepted in Physics), in this particular nucleus, is that the individual protons and neutrons have lost mass; their masses are as low as they can go. NOW, BEING MINIMAL UNITS, they thereby become qualified as a Measurement Standard. |
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--It has suddenly occurred to me that one aspect of that last paragraph means I may need to reconstruct my graph! Alas! Nevertheless, thank you! (It won't be so hard to do; I'm writing a program to process that linked Isotopes table, so merely need to tweak the program...and this oddball thought might be erroneous, anyway. To be determined.) |
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Here are the atomic masses on the Fe-56 scale, rounded to the 4th decimal place. This is the data I'll use to reconstruct the graph. I should reconstruct it on the general principles that (1) the linked table, from which this data was generated, is more up-to-date than the table I used 30-odd years ago, and (2) I might have made a typo or calc error back then, which is avoided by getting a computer to do the conversions: |
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neutron: 1.0099
H-1: 1.0090; H-2: 2.0164; H-3: 3.0196
He-3: 3.0195; He-4: 4.0073
Li-6: 6.0221; Li-7: 7.0242
Be-9: 9.0227
B-10: 10.0246; B-11: 11.0221
C-12: 12.0140; C-13: 13.0185; C-14: 14.0195
N-14: 14.0194; N-15: 15.0176
O-16: 16.0135; O-17: 17.0189; O-18: 18.0201
F-19: 19.0205
Ne-20: 20.0157; Ne-21: 21.0183; Ne-22: 22.0170
Na-23: 23.0165
Mg-24: 24.0129; Mg-25: 25.0149; Mg-26: 26.0128
Al-27: 27.0129
Si-28: 28.0095; Si-29: 29.0102; Si-30: 30.0086
P-31: 31.0098
S-32: 32.0093; S-33: 33.0098; S-34: 34.0074; S-36: 36.0089
Cl-35: 35.0095; Cl-37: 37.0089
Ar-36: 36.0094; Ar-38: 38.0069; Ar-40: 40.0089
K-39: 39.0090; K-40: 40.0105; K-41: 41.0095
Ca-40: 40.0091; Ca-42: 42.0074; Ca-43: 43.0087; Ca-44: 44.0066
Ca-46: 46.0071; Ca-48: 48.0083
Sc-45: 45.0082
Ti-46: 46.0061; Ti-47: 47.0064; Ti-48: 48.0037; Ti-49: 49.0048
Ti-50: 50.0029
V-50: 50.0053; V-51: 51.0032
Cr-50: 50.0041; Cr-52: 52.0009; Cr-53: 53.0022; Cr-54: 54.0016
Mn-55: 55.0019
Fe-54: 54.0024; Fe-56: 56.0000; Fe-57: 57.0016; Fe-58: 58.0007
Co-59: 59.0017
Ni-58: 58.0027; Ni-60: 60.0005; Ni-61: 61.0019; Ni-62: 62.0004
Ni-64: 64.0023
Cu-63: 63.0028; Cu-65: 65.0033
Zn-64: 64.0035; Zn-66: 66.0027; Zn-67: 67.0050; Zn-68: 68.0039
Zn-70: 70.0067
Ga-69: 69.0057; Ga-71: 71.0072
Ge-70: 70.0056; Ge-72: 72.0057; Ge-73: 73.0083; Ge-74: 74.0072
Ge-76: 76.0097
As-75: 75.0087
Se-74: 74.0085; Se-76: 76.0075; Se-77: 77.0094; Se-78: 78.0079
Se-80: 80.0095; Se-82: 82.0120
Br-79: 79.0101; Br-81: 81.0104
Kr-78: 78.0110; Kr-80: 80.0093; Kr-82: 82.0088; Kr-83: 83.0106
Kr-84: 84.0091; Kr-86: 86.0105
Rb-85: 85.0106; Rb-87: 87.0103
Sr-84: 84.0110; Sr-86: 86.0092; Sr-87: 87.0100; Sr-88: 88.0079
Y-89: 89.0093
Zr-90: 90.0093; Zr-91: 91.0114; Zr-92: 92.0119; Zr-94: 94.0155
Zr-96: 96.0198
Nb-93: 93.0144
Mo-92: 92.0137; Mo-94: 94.0143; Mo-95: 95.0162;
Mo-96: 96.0162
Mo-97: 97.0187; Mo-98: 98.0193; Mo-100: 100.0237
Tc-97: 97.0191; Tc-98: 98.0211; Tc-99: 99.0213
Ru-96: 96.0191; Ru-98: 98.0192; Ru-99: 99.0210; Ru-100: 100.0204
Ru-101: 101.0229; Ru-102: 102.0229; Ru-104: 104.0263
Rh-103: 103.0252
Pd-102: 102.0241; Pd-104: 104.0249; Pd-105: 105.0271; Pd-106: 106.0267
Pd-108: 108.0294; Pd-110: 110.0330
Ag-107: 107.0294; Ag-109: 109.0314
Cd-106: 106.0296; Cd-108: 108.0297; Cd-110: 110.0308; Cd-111: 111.0332
Cd-112: 112.0329; Cd-113: 113.0357; Cd-114: 114.0358; Cd-116: 116.0396
In-113: 113.0354; In-115: 115.0375
Sn-112: 112.0350; Sn-114: 114.0353; Sn-115: 115.0370; Sn-116: 116.0365
Sn-117: 117.0389; Sn-118: 118.0387; Sn-119: 119.0416; Sn-120: 120.0417
Sn-122: 122.0452; Sn-124: 124.0494
Sb-121: 121.0444; Sb-123: 123.0472
Te-120: 120.0435; Te-122: 122.0448; Te-123: 123.0472; Te-124: 124.0469
Te-125: 125.0497; Te-126: 126.0497; Te-128: 128.0532; Te-130: 130.0573
I-127: 127.0521
Xe-124: 124.0500; Xe-126: 126.0507; Xe-128: 128.0523; Xe-129: 129.0547
Xe-130: 130.0546; Xe-131: 131.0573; Xe-132: 132.0576; Xe-134: 134.0611
Xe-136: 136.0653
Cs-133: 133.0600
Ba-130: 130.0574; Ba-132: 132.0585; Ba-134: 134.0602; Ba-135: 135.0626
Ba-136: 136.0626; Ba-137: 137.0651; Ba-138: 138.0656
La-138: 138.0675; La-139: 139.0679
Ce-136: 136.0652; Ce-138: 138.0664; Ce-140: 140.0682; Ce-142: 142.0743
Pr-141: 141.0715
Nd-142: 142.0728; Nd-143: 143.0760; Nd-144: 144.0775; Nd-145: 145.0811
Nd-146: 146.0828; Nd-148: 148.0889; Nd-150: 150.0953
Pm-145: 145.0813; Pm-147: 147.0860
Sm-144: 144.0794; Sm-147: 147.0858; Sm-148: 148.0869; Sm-149: 149.0904
Sm-150: 150.0916; Sm-152: 152.0964; Sm-154: 154.1012
Eu-151: 151.0954; Eu-153: 153.0991
Gd-152: 152.0965; Gd-154: 154.0999; Gd-155: 155.1028; Gd-156: 156.1035
Gd-157: 157.1065; Gd-158: 158.1078; Gd-160: 160.1131
Tb-159: 159.1102
Dy-156: 156.1056; Dy-158: 158.1081; Dy-160: 160.1112; Dy-161: 161.1141
Dy-162: 162.1151; Dy-163: 163.1182; Dy-164: 164.1198
Ho-165: 165.1221
Er-162: 162.1171; Er-164: 164.1199; Er-166: 166.1233; Er-167: 167.1262
Er-168: 168.1277; Er-170: 170.1331
Tm-169: 169.1307
Yb-168: 168.1292; Yb-170: 170.1324; Yb-171: 171.1351; Yb-172: 172.1364
Yb-173: 173.1394; Yb-174: 174.1412; Yb-176: 176.1472
Lu-175: 175.1442; Lu-176: 176.1473
Hf-174: 174.1424; Hf-176: 176.1460; Hf-177: 177.1490; Hf-178: 178.1507
Hf-179: 179.1539; Hf-180: 180.1558
Ta-180: 180.1568; Ta-181: 181.1585
W-180: 180.1560; W-182: 182.1598; W-183: 183.1630; W-184: 184.1649
W-186: 186.1706
Re-185: 185.1681; Re-187: 187.1732
Os-184: 184.1664; Os-186: 186.1701; Os-187: 187.1732; Os-188: 188.1744
Os-189: 189.1779; Os-190: 190.1794; Os-192: 192.1847
Ir-191: 191.1827; Ir-193: 193.1874
Pt-190: 190.1809; Pt-192: 192.1843; Pt-194: 194.1883; Pt-195: 195.1915
Pt-196: 196.1929; Pt-198: 198.1981
Au-197: 197.1956
Hg-196: 196.1937; Hg-198: 198.1970; Hg-199: 199.1997; Hg-200: 200.2009
Hg-201: 201.2040; Hg-202: 202.2055; Hg-204: 204.2107
Tl-203: 203.2084; Tl-205: 205.2128
Pb-204: 204.2103; Pb-206: 206.2140; Pb-207: 207.2166; Pb-208: 208.2185
Bi-209: 209.2234
Po-209: 209.2255; Po-210: 210.2271
At-210: 210.2314; At-211: 211.2329
Rn-211: 211.2360; Rn-220: 220.2673; Rn-222: 222.2758
Fr-223: 223.2791
Ra-223: 223.2779; Ra-224: 224.2808; Ra-226: 226.2883; Ra-228: 228.2963
Ac-227: 227.2918
Th-230: 230.3007; Th-232: 232.3079
Pa-231: 231.3046
U-233: 233.3107; U-234: 234.3132; U-235: 235.3173; U-236: 236.3201
U-238: 238.3277
Np-237: 237.3239; Np-239: 239.3310
Pu-238: 238.3264; Pu-239: 239.3302; Pu-240: 240.3330; Pu-241: 241.3372
Pu-242: 242.3403; Pu-244: 244.3481 |
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I never liked the term "binding energy" anyhoo - it makes it sound like it's stored energy, in which case nuclei with large binding energies should have more mass, not less. The reality is that the stable nuclei lose mass, because so much energy is given up dropping them down into such a stable state. "Binding stability" would be better, but loses it's reference to energy and mass. Hell, "UNbinding energy" is actually a more logical term. |
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I hate (umm love) to speculate as to what mass is but figure that the different masses correspond to left over (weak/strong) nuclear force. Which ties with the fact that so far there is no subatomic nature that carries mass. |
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Sort of like putting a shade over a bulb and getting a lamp... |
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The possibility that guage variance/invariance can apply to all scales invalidates all quantum arguments. However, quantum theory is pertinant and predictive. Something you cannot ignore, at least not without another explanation under you belt, or up your sleeve. All likely scenarios, contain an element of *unit*. Even SuSy, and p-branes/ M-branes. Even guage variance/invariance has a cut-off point including units. |
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The standard response to the "half a hole"/ " a little pregnant" argument is that "pregnant" and "hole" have *already* been defined. It really is not given, at least philosphically (and hence the resurgance of philosophy), that we have accurately described the units we describe with. |
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But the unit has never, and I mean never, been adequately described. I am of the opinion that it can never be. There are the Zeno arguments, there are the Zeno argument debunkers, and there are the Godel-o-philes. |
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If space or time or both proved to be "granular" we may have some space to accomodate this [Vernon]'s argument. However, if one or the other, proved a continuum, all is lost. And further, accepting S. Hawking and R. Penrose arguments of Anthropic principles. It might well be the what we observe (granular) and the actual nature of things (continuum) are the same observationally to us. |
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[4whom], you neglected to mention the Euclidean approach, which is simply to postulate the concept of "unit". Like several other Euclidean postulates, it's something most people will understand, even without a specific ultimate definition. |
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I am with Vernon on the subject of units --- I can define exactly 1 apple with units apples. |
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In the interest of better understanding Vernon may come along and decide that he needs to plot a variance of appleness with respect to mass. By doing this he is not saying anything more about apples but rather about mass. Given that we have not defined the carrier of mass it is reasonable not to be able to define the unit of mass. So we only have apples --- but that has been enough so far to work out the properties of mass... |
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With regards to quantum physics it seems to me that smashing stuff up and sorting through the remains has been fairly fruitless. And we are still left with a discrepency regarding mass... |
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Anyway, loving to beat the same drum over and over... Since the major property of mass is gravitation (discounting inertia) it seems reasonable that this attraction is all that mass is. So I like Vernons description of atomic level bond energies because it leaves unanswered the question of what happens to the left overs... |
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With regards to the links. I am puzzled by Hawking and his radiation --- it is in direct contradiction to his big bang theory. I would have expected him to propose that information is stored in a black hole and that at some point the body will become unstable and (well) explode (releasing its store of information). |
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//[4whom], you neglected to mention the Euclidean approach, which is simply to postulate the concept of "unit". // |
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I most definitely said, arbitrary units are arbitrary. If you want to measure everything according to an apple, then go for it! What I did say is we have not found, or articulated, a *fundamental* unit, not even mathematically. I also said it is not likely that we ever could. |
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Ratio and proportion are the most accurate we could ever get, as there is no arbitrary unit. The "units" cancel out. |
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There is good news, though. The lack of proof of a fundamental unit lets you, safely, choose your own. And that is why I approve of this idea. However, take it from whence it comes. |
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[4whom], perhaps I misunderstood you. You seemed to be saying that it was the concept of "unit" itself that had no provable definition. I'm well aware that in the natural world there are lots of things that could be used as the basis for measuring other things, and no good reason to choose most any of them over the rest. But that is an issue distinct from the concept of "unit" itself. Would you be more clear about this, please? Thanks in advance! |
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[Vernon], as you have said already, there is a gap between a continuum, and a corpuscular or granular view. All I said was that maybe we may not be able to tell the difference, observationally. |
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Maybe *I* am not being clear. I love the units (plural intended) and all that they have given us in mathematics. But because they do not refer to anything more specific than themselves, they have a safe home. They can be used to represent themselves, fractions of themselves, irrationals, transcendentals, and even "imaginaries". But because they, "units", mean themselves, they just become nice toys. VERY nice toys. |
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When we take the giant step from units that mean themselves to units that mean something physical (or physical within a field (eg space/time)), we have some problems. Sure one apple is fine. And there is nothing manifestly wrong with an infinity of apples. It is when we try and describe an infintesimally small (or even finitely (above a limit) small) part of the apple that we encounter problems. Observationally (at these limits) we observe lattice guage theories as having confinement (read arbitrary unit of choice,) (guage theories themselves have a kind of "stretching" unit, which I am partial towards ). This confinement allows us a "cut-off", physically. Mathematically it remains a huge problem. But not from *just* the mathematics side (these kinds of cut-offs can be dealt with). But rather from the merger between the two philosophies, of continuum and corpuscular (quantum). |
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Taken from the apple point of view. One can prove that given the apple as a unit of measurement, there is no way to construct a rational number in apples, without the apple. It sounds trivial but it is not so. I can also construct several *ratios*/*roots* that are independant of the notion apple, but that can be expressed in apples. But that only depends on the apple being infinitely divisible. Accepting that apples cannot be divided infinitely (ie they are corpuscular/ physical) means we cannot describe those *ratios*/*roots* in apples without encountering a point where we just have to say "Ahhh, wtf!" and assign an arbitrary apple unit. It is easy to accept that we can have a string of apples infinitely long, it is harder to accept we can have an infinitely long string of apples between any two apples. And harder still to accept that there is an infinitely long string of apples between any two rational partitions like "apple/x(apple)" and "apple/(x+apple)apple". |
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Of course, any mathematical formulation of the Yang-Mills mass gap will leave a lot of egg on my face... |
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And I have one! But it is a few fries short of a Big MacMeal. (Anyone here considered a one dimension Penrose "tiling"?) |
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4Whom --- as you say given the apple mathematics cannot determine half an apple. But then that is not what maths is for... |
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[4Whom], as soon as it became convenient to count things, words representing quanties were created: "one, two, three, many". Later, the numerical quantities were ABSTRACTED from the original purpose of counting actual objects with them, and we could say that is how Math was born. I think it is the neglecting-to-remember this connection, between the origin of counting units and their abstraction, that is causing your philosophical dilemma. Sure, numerical units may not be definable AS abstract things, but it doesn't matter so long as we remember they came into existence via a non-abstract definition. |
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[up_on_cloud_nine], the Atomic Number is the number of protons in a nucleus. The Mass Number is the sum of protons and neutrons in a nucleus. The Atomic Mass is a relative value, of a given atom compared to a declared-to-be-"standard" atom. The typical standard is Carbon-12, arbitrarily declared equal to 12.00000 "atomic mass units". This causes many relative Atomic Masses to become less than their Mass Numbers, which I consider to be un-aesthetic. If a different standard, Iron-56, is chosen, arbitrarily declared equal to 56.00000 amu, then all the other Atomic Masses, relative to this, become fractionally greater than their Mass Numbers. A nifty meaningful graph was just a side-benefit of increasing the aesthetics of the situation. |
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