What Does Differ Mean In Math latest 2023

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Electron-Ness: Why Are All Electrons Identical?

Go to your local store and buy several items of the same product – say a package of three golf balls. Though the golf balls appear identical, closer examination will reveal ever so slight differences. One ball maybe fractionally larger; another ever so slightly less spherical; perhaps the third is ever so slightly lighter. The generality that extends from this is that any two seemingly identical products will have nevertheless slight variations in their properties.

Now buy a packet of three electrons (or their antimatter equivalent, the positron). Each electron, or positron, will be identical in size, mass and electric charge to as many decimal places as you care to measure. All electrons (and positrons) are 100% absolutely identical clones.

Take one electron and one positron and bring them together. They annihilate releasing a fixed amount of energy. Take another electron and another positron and repeat the scenario. The pair will annihilate releasing an identical amount of energy in the process. The amount of energy released in each electron-positron annihilation case is the same, to as many decimal places as you can measure. That’s quite unlike taking a match from a box of matches, striking same and releasing its stored chemical energy as heat energy. Another match from the same box wouldn’t release, to as many decimal places as you care to measure, the absolutely identical amount of heat energy.

How come identical golf balls aren’t but identical electrons (or positrons) are?

Electrons (or positrons), having mass, can be created from energy (just like mass can be converted to energy as in the case of the electron-positron annihilation process). You (human intelligence) can’t create identical golf balls, but a seemingly non-intelligent natural process (Mother Nature by any other name) can create or produce copies of a fundamental particle, like an electron (or positron), that are clones of one another down to the nittiest-grittiest detail.

Even with quantum mechanics in force, you’d think energy could create or be converted into an electron with twice the standard electron mass or twice the electric charge, or thrice even. But no. You see one electron you’ve seen them all – every electron that is, was or will be, anywhere, everywhere, anytime, every time in our Universe. Electrons, like Black Holes, have no hair. That means they have no individual personality. In fact Black Holes can be said to have some fuzz because they can and do differ in terms of size, mass and electric charge. Electrons have the exact same size, mass and electric charge, so absolutely no hair! Relative to Black Holes, electrons (and positrons) are absolutely bald!

Invoking all things quantum is still a bit of a cop-out in that while quantum means things are this or that, one unit or two, one energy level or two energy levels, there’s no explanation as to why it’s only one or two, not one & a half. It just is, but why remains a mystery.

Why are all electrons (and positrons) identical?

1) Of course THE cop-out answer is that that’s just the way God wanted it and no correspondence will be entered into regarding the matter.

Unfortunately, there is no real evidence for the existence of any deity past and/or present that stands up to any detailed scrutiny.

2) One could resort to an explanation via string theory merged with quantum physics. String theory just replaces elementary particles as little billiard balls for elementary little bits of string (albeit not string as we know it). Now maybe, as in all things quantum, these strings can be one unit in length, or two units, or three units, or four units, etc. Any positive whole number multiple of one string length is okay. Now say that a two length unit of string is an electron. A two unit length of anti-string is therefore a positron.

Or, one can suggest that strings vibrate and can only vibrate at specific frequencies as any musician playing a stringed instrument knows. So, a string vibrating at one allowed frequency is an electron; if it vibrates at another allowable frequency maybe that’s a proton or a neutron. Again, a vibrating anti-matter string would produce manifestations of the antimatter particles, a positron being dependent upon one of the allowable vibrating frequencies.

Of the two possibilities, it’s the vibration rate theory that’s preferred. All strings are of the same fundamental length – their rate of vibration can differ, but at precise intervals. What causes strings to vibrate at the rate they do, and how they can change rates of vibration (morph from one kind of particle into others) are questions better left for another time.

Unfortunately, string theory has no credibility in terms of any actual experimental evidence, and, to add insult to injury, it requires the postulation of ten to eleven dimensions in order to fit the pieces together. If string theory gets some experimental runs on the board then, and only then, will it be time to take strings seriously.

3) Well, one other possible explanation is that all electrons are absolutely identical because there is only one electron in actual existence. If you see the same object twice, thrice of a zillion times over, then it’s the same object and the fact that it is consistently identical is not a great mystery. But how can the Universe contain only one electron? That seems to be the least obvious statement anyone could ever make – the statement of a total wacko.

Well, one explanation goes something like this. Our one electron has zipped back and forth between the Alpha and Omega points again, and again, and again. Now multiply ‘again’ by zillions upon zillions upon zillions of times. When you take a cross section at any ‘now’ point in time between the Alpha and the Omega, there will be zillions upon zillions upon zillions of electrons visible ‘now’. Simple, isn’t it?

Unfortunately, while there is no violation of physical laws at the micro level in travelling through time (apart from going forward at a rate of one second per second which we do whether we like it or not), no exact causality mechanism has been proposed to explain how and why an elementary particle shifts gear into time reverse (or forward again).

Back to the original question, why are all electrons identical? Or not, as the case may be.

4) Perhaps in other parallel universes, ones that have different physics, all electrons (if they have electrons at all) might not be identical. That possibility is akin to asking about numbers of angels dancing on pinheads. There’s just no way of ever knowing since parallel universes are beyond the reach of science as we know it.

But say each member of the particle zoo weren’t identical to every other member in kind. Say electrons came in a thousand variations of mass and electric charge; ditto the other elementary particles. You’d have a particle jungle. If that were the case, presumably it would prove to be very difficult to create identical atoms of the elements and identical molecular compounds and ultimately it would prove difficult to build up the structure of our Universe as we know it, including us. An analogy might be that it’s far easier to assemble a ten piece jigsaw puzzle and one with a billion pieces. Our particle zoo seems to be a Goldilocks zoo – not too many particles and variations thereof; not to few either (I mean a universe composed of just identical electrons is equally as bad for life as we know it). Of course if that – the Goldilocks particle zoo – weren’t so, we wouldn’t be here to ponder the issue.

Moving on up the chain, assuming all members of the particle zoo are identical then atoms of any particular element must be identical – if you’ve seen one gold atom, you’ve seen them all (though owning them all is quite a different matter). If elements come in different isotopes, then all the specific isotope atoms of that element are identical.

Further moving on up the chain, if identical atoms combine with other different identical atoms, then presumably the resulting molecules will be identical. While that’s true, it’s only true up to a point, because eventually you can get molecules that while seemingly identical, have handedness. That is, your hands, while identical, aren’t identical because one has a left-handed orientation; the other has a right-handed orientation. That’s the point things start to fall apart or break down.

That apart, macro objects, like golf balls, are composed of millions of atoms and/or molecules. If a golf ball has one more, or one less molecule than another, well the two aren’t identical.

5) Introducing the maths connection: Here, there and everywhere, on a flat surface, the shortest distance between two points is a straight line; triangles have a sum total 180 degrees; 2 + 2 = 4. In each case, it is so to as many decimal places as you care to calculate. Every 7 is identical to every other 7 – no more and no less. That’s true whether or not one is dealing with base ten, or in binary (base two).

So what’s the connection? All computer generated simulations, in whatever context, for whatever purpose, are ultimately software programs, which in turn are just mathematical constructions. All you see are ultimately expressions of maths, of binary bits, of 0’s and 1’s, something on or off. So if you simulate some object using binary software programming, and you create another object using the exact same binary software coding, then those two virtual objects are identical. Now, call what you have simulated, ‘electrons’. So if all electrons are identical, maybe it’s because they are mathematical constructions – the end products of computer software/programming.

In simulations, virtual objects can interact with other virtual objects (more mathematical wizardry). Change happens. Well, that’s what we observe in our reality too. The question is, is our reality really real reality, or simulated reality? Are our electrons identical because each is the product of an identical piece of binary software programming? That may not ultimately be the answer, but it’s an answer. Electrons are the same since they are all constructed from the same mathematical whole cloth of binary bits – of 0’s and 1’s.

Discussion: One may argue that there are indeed differences between electrons (and positrons), we just haven’t measured to enough decimal places yet. While that might be true, I personally wouldn’t want to bet on it.

Conclusion: I started out with the question of why all electrons are identical. The answer is, I don’t know and neither, I suspect does anyone else. However, the foundation of physics (itself the foundation for the other sciences) is grounded in maths, and maths, as noted above, has no problem with the concept. All identical equations yield identical results; the ‘equals’ sign itself demands identicalness. Perhaps maths has more fundamental ‘reality’ than anyone has given it credit for. That’s certainly the case if we should happen to be inhabiting a software generated, simulated Universe

Postscript: The concept of identicalness can bring us into some weird scientific and philosophical territory. Two people examining the same object will not agree to the Nth degree that the object under consideration is the exact same object, an identical object, when compared from each person’s perspective. Perception is ultimately a function of brain chemistry and no two people have the exact same brain chemistry due to various factors like genetics, age, physiology, disease, fatigue, and/or intakes of various solid, liquid and gaseous elements and compounds that directly affect brain chemistry. The differences may be really tiny and nitpicky but nevertheless present. To take another case, if three court stenographers all record and transcribe a days worth of testimony, no doubt there will be (ever so) slight differences in the final three versions.

Even the same person experiencing the same object or event a second, third, etc. time – say watching a film again or listening to a CD track again, won’t have identical experiences, again due to the internal brain chemistry being slightly different on each occasion. That’s apart from the fact that external influences like temperature, humidity, pressure, and general wear and tear (entropy) all affect that object or event and the environment between that object/event and the person experiencing the object/event. Those external factors also change from moment to moment.

People though tend to agree (brain chemistry not withstanding) on what an independent umpire says about an object or event – the independent umpire being an instrument or measuring device. Instruments are of course also subject to external influences, but aren’t affected by brain chemistry – they have no brains!

Measurements tend to be numerical, and numbers are pretty straight forward. However, all measurements are subject to some uncertainty or error margins, especially analogue devices like a ruler – is it 1.510 cm or 1.511 cm or 1.509 cm? Or a thermometer – is it reading 31.37 degrees or 31.38 degrees or 31.36 degrees? Or take a standard watch or clock – is it 12:00:00 or 12:00:01 or 11:59:59?

Digital instruments however have readouts that have a finite number of places in which to display the result, so they don’t tend to give you a plus or minus uncertainty error bar. A digital instrument will readout that the length IS 1.510 cm; the temperature IS 31.37 degrees; the time IS 12:00:00, and everyone looking at the readout will agree.

In the case of an electron, the independent umpire gives the same numerical results for each electron it measures. Of course there are still error bars, but with each further decimal place reached, identicalness holds and the error bars get less and less.

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