For what it's worth -- You're absolutely right in sticking to your guns, Alice the Curious -- I can hardly believe the Monty Hall problem of 'changing odds' can be so misunderstood by (apparently) many or seen as anything but an artifact from how the problem is explained. I think that's the ONLY real problem here -- in that the problem isn't accurately explained. As you pointed out, the game-player's odds of selecting the prize ALWAYS ends up being 50 percent. Monty's showmanship is part of the equation, always eliminating a no-prize choice -- thus always collapsing the probability odds to one in two.<br><br>Perhaps the truth here is counter-counter intuitive. What the 'change to improve odds' folks ignore is that Monty's collapsing statistical odds from 2/3 to 1/2 of the two-door non-selection 'choice' is a given -- Monty will ALWAYS reveal that one of the two-doors is empty -- so the initial choice will ALWAYS result in two possible outcomes for Monty -- both of his doors are either empty, or only one of them is. But the rules of the game are that Monty ALWAYS shows us at least one empty door. So, the game-show odds at the very beginning are constained by this rule -- the player is ALWAYS betting on only one of two outcomes -- the door you select is either empty or has a prize: the odds are 1:2. With the same rules, ie., Monty ALWAYS shows you an empty door and lets you switch to another until there are only two doors -- it doesn't matter if there are 100 doors. Let's say -- You choose Door #76 out of 100 -- But each time Monty asks you to change, you stick -- Until the last choice, two doors remain -- What are your chances the prize is behind door number 76 -- 1:100? Of course not! That's because the choices are limited to only two doors. The argument '1:100' would be nothing but an artifact, an argumentative anomoly distorted by confusing initial conditions with final conditions. One doesn't improve one's odds by changing at that last choice opportunity. The counter-intuitive false-logic is but a fluke of how the choice is described.<br><br>But of course, you saw thru the problem and knew that. Perhaps the 'real' point to the professor making the 'increase your odds to 2:3 by switching' argument is to bamboozle the class, to distort the class 'logic' by exercising his academic authority or see whether anyone would refute and challenge him.<br><br>The dichotomy (and other) argument of Zeno's paradox is fascinating in its way because of what it says about the inherant property of fuzziness, ie. indeterminacy, of space and time at or approaching irreducable units. The 'short answer' solution to resolving Zeno's paradox is, obviously, recognizing that since Einstein we know that mass and energy are convertible, and so are space and time (ie., influenced by energy densities). But a closer examination of what Zeno's paradox focrces us to consider leads us to: What does it mean to 'halve' the distance of some incredibly small measure, ie. an atomic radiius? And: Is 'time' fluid and continuous, or does it, can it. must it occur in quantized 'jumps'? <br><br>(For those who can't follow this or are bored to tears by meta-philosophical/cosmic/quantum discussion, you might wanna scroll down past the rest of my comment -- S)<br><br>Beyond the distance of 'space' occupied by an atom's outer electron shell, reality and the universe get 'wierd' and inexplicable; The properties of these 'units' of space merge with the properties of time when the time units considered --by successive 'halving' operations-- approach some fantstically diminutive, elusive value -- such as the Plank time, at 1X10^-43 second -- which astute hobbyist cosmologists will recognize is equivalent to that given age of the universe (according to our theory of the Big Bang) at which 'time', our understanding of physics completely breaks down -- it is a kind of backward 'looking' into the distant history spacetime foam of the universe's blackhole navel. (Actually, at t = 10^-32 s, we can only guess at what the universe 'looked' like -- our understanding gets more and more bizarre beyond that, when the base forces (weak, strong, electromagnetic, gravity(?)) first 'crystallized' or metamorphised out of the incredible dense maelstrom energy of nuclei particle soup when the entire cosmos existed in a pointsource state essentially impossible to understand.)<br><br>The radius of a proton (mean measure as this measurement depends on the vagaries of light properties as affecting measurement at these illimitable distances) is about 2X10^-16 m, or rounded to a 10^-15 m. Well before the meaning of this absolute distance 'unit', measuring space becomes problematic -- extremely problematic, as we're talking about the nuclear forces of how the 'stuff' of the universe interact with each other, the bits and pieces of the 'real' world of atoms and their various bonding properties -- the 'meaning' of an object's 'boundaries' and its relation to the space it occupies gets 'fuzzy' and indeterminate. Most of the stuff we 'touch' in the world we ordinarily 'know' and exist in, we never get much closer to than 'about' 10^-15 m -- or, one-tenth billion of a meter, ie., 1/10,000,000,000 meter -- about the range at which the electromagnetic properties of objects' and people's atom's electrons determine how the world's bits and pieces interact and which we can meaure (and describe). This is an extension of the wierdness of stuff at atomic distances, where there is more 'voidness' than substance in a macro-scale object, like tables and chairs and rocks and people.<br><br>This gets back to the 'nature' of the world and the relative meaningless of describing it in terms of these kinds of quantumized intervals and distances, measures of space and time -- in which the universe is essentially opaque to our understanding and hence, we have no real way to describe it -- that 'edge' where 1X10^-43 s 'meets' 2X10^-16m.<br><br>So, getting back to the hare and rabbit 'race' -- Our common-sense logic tells us, of COURSE the fleetfoot hare catches-up-to and passes the slow-but-steady turtle -- the 'paradox' is that we can't reconcile it with our explanation of why an infinitude of separation-distance halving operations doesn't imply the necessity of an 'always one more half-distance' before the hare catches up to the turtle -- at least, unless we recognize that our description and measurement of the world fails at these incredibly small time-intervals and space-measurements. Which makes me wonder if Zeno and the Classical thinkers intimated as much?<br><br>This gets to the idea of a quantized measure of time, ie., an irreducable unit, which for all practical purposes, and esp. for purposes of deconstructing the argument of Zeno's paradox, ought to be something real, ie, tangible and practical and minimally discrete, such as the time light, C, takes to traverse the smallest practical (measurable, ie. knowable --with a small measure of error) a proton's diameter (roughly 10x^-15 m). <br>Dividing C (3 x 10^8 m/s) by a proton diameter = 3 x 10^23 hertz ? Frequency?<br>(Plank energy = 4.14 x 10^-15 eV)<br>Alternatively: How LONG would it take light, C, to cross the distance of a proton diameter?<br>(2 x (10^-16) m) / c = 6.6712819 × 10^-25 seconds<br>Hmmm .... Is it more practical then to consider 6 x 10^-25 s to be the smallest practical time interval, rather than the figure I first used, 10^-43 s? This is like, 1/10,000,000,000,000,000,000,000,000, or one one-ten-billion-trillionth of a second (!!!!!)<br>In a similiar vein re: the smallest meaningful measures of time and space, one web post I found claims to be able to quantize a universal constant mass from quantum states of time and space, as per:<br><br>The physically smallest distance – elementary length, r0 – and the physically smallest time interval – elementary time, t0 – are closely connected to Planck's constant h, the total energy/matter-mass M0 of the Universe and the velocity of light c0 in what is called vacuum. <br><br>For anyone wanting to check-out this intriguing concept, the link is:<br><!--EZCODE AUTOLINK START--><a href="http://www.rostra.dk/louis/quant_15.html">www.rostra.dk/louis/quant_15.html</a><!--EZCODE AUTOLINK END--><br>The author suggests: 'The smallest physical energy/mass quanta in the Universe are identical to the ’particles’ which I have given the name unitons. Some consequences of the existence of unitons, – the true atoms of the Universe – you can study at':<br><!--EZCODE AUTOLINK START--><a href="http://www.rostra.dk/louis/">www.rostra.dk/louis/</a><!--EZCODE AUTOLINK END--><br>-- A VERY intriguing site featuring treatises re: a new theory of the Universe by Louis Nielson, who wrote: Holistic Quantum Cosmology with Decreasing Gravity.<br><br>Perhaps the essence of this system derived via deduction is this: 'My discovery of a connection between the total mass of the Universe, elementary length, elementary time, Planck's Constant and the light constant gives a holistic context of our Universe. There is a connection between microcosmos and macrocosmos. As the smallest physical distance is determined by the total mass of the Universe, it can be expressed: "The Greatest is contained in the Smallest!"<br><br>From thence:<br>In my Holistic Quantum Cosmology I have shown which connection there is between the actual extent of the Universe, its age (the cosmic time parameter), elementary length and elementary time. Likewise I have shown the connection between the total mass of the Universe and the smallest possible mass – elementary mass.<br><br>The Universe has developed from one quantum – the cosmic embryoton. This cosmic embryoton had an extension equal to elementary length, and the Universe – space, time, and thereby the nature laws – were formed within the first cosmic quantum interval. The evolution of the Universe is guided by a cosmic evolution quantum number "ticking" up through the natural numbers. The Universe was "born" when this number was one (unity)! "Today" this cosmic quantum number is extremely high, 7.2 · 10127.<br><br>The evolution of the Universe is characterized by the original cosmic embryoton being split up in more and more elementary quanta. The actual number of these elementary quanta is equal to the cosmic evolution quantum number.<br><br>The division and spreading of the energy originally concentrated in the cosmic embryoton is in accordance with the entropy law, "demanding" that a system evolves from a less probable and dense state to a more probable and less dense state.<br><br>As the Universe develops from one single quantum mass to more and more quantum masses, spreading in the expanding space, a consequence will be that the cosmic unit for mass will vary as the Universe evolves! This cosmic variation of the physical basic unit for mass has very interesting consequences for the physical quantities we have defined in order to describe the Universe, of which we are ourselves a part. As it will be shown in the following, the evolution of the Universe can be characterized by a development up through the natural numbers! Given in cosmological units, many of the physical quantities we presume constant will vary . This is f.i. the case with Planck's "constant".<br><br>Physical Quantities and their Units<br>In order to describe the phenomena in the Universe, of which we are part ourselves, we have defined physical quantities. A physical quantity is characterized by a mathematical number, a unit and maybe a direction.<br><br>Our apparent feeling, that we live in a Universe controlled by certain laws, is a.o. caused by the discovery that between certain physical quantities there exists a mathematical connection, often even a simple one.<br><br>Experience has shown that all physical quantities can be defined by a few fundamental physical quantities. Even if we have more possibilities to choose physical fundamental quantities, it looks like the Universe "itself" has decided which basic quantities shall be used when deducting other physical quantities. These fundamental quantities are:<br>Distance in space between two "points",<br>Time interval between two events, and<br>Mass of a matter/energy containing system.<br><br>In order to be able to indicate definite distances, time intervals and masses it is necessary – also for communication of these quantities to others – to define units for distance, time interval and mass. We have innummerable possibilities for choice of definitions of these units, but the interesting question is: Does there exist a universal and absolute system of units, independent of living creatures and specific local systems? My reply to this question is: Yes! A yes, because I believe to have found these absolute cosmological fundamental units!<br><br>We shall revert to these cosmological units, but first we shall look at some remarks to the measuring units we normally use in our physical description of the world. Almost all countries have agreed to use the same physical units, which are: the distance meter, the time second and the mass unit kilogram. Previously there was chaos regarding basic units, different units were used in different countries, sometimes even different units were used within the same country.<br><br>The basic units used today, meter, kilogram and second, are far from being cosmologic, on the contrary they have an origin tightly connected to the human dimensions and rhythms. One meter is approximately equal to the length of a leg of an adult person. One second is approximately equal to the interval between two pulse beats, and a kilogram is approximately equal to one third of the weight of a newly born baby.<br><br>Gradually it was realized that it was practical to define the physical basic units independent of human relations. As an example, the unit of length, one meter, was introduced during the French revolution, and in 1791 it was defined as one ten millionth of the distance between the North Pole and Equator, measured along the longitude circle through Paris. Based on measurements of degrees, a standard was manufactured of platin, the so-called archive meter. This should then be a secondary normal meter.<br><br>Later measurements have shown that the archive meter is 2/10 mm shorter than 1/107 of the length of the earth quadrant. The archive meter was then chosen as primary normal meter, and the meter was defined as the distance between the two plane ends of the platine stick at 0° Celcius. Thus there is no more connection between the meter and the natural object Earth. Since then the meter was adjusted and defined several times. At the 11th General Conference for Measure and Weight in Paris 1960 a new definition was given. <br><br>This definition is determined by a definite emission of photons from the krypton-86 atom. It was decided: One meter is 1650763.73 times the wavelength in vacuum of the emission by a jump from level 2 p10 to 5 d5 in the atom.<br>Stoney's and Planck's "Units"<br><br>At a meeting in Belfast in 1874, the Irish physicist George Johnstone Stoney (1826-1911) ['On the Physical Units of Nature', Philosophical Magazine, vol. 11, p. 381 (1881)] forwarded a proposal to some "natural" units for length, time and mass, independent of human existence. By suitable combinations of Newton's gravitation constant, the velocity of light and an electric unit for charge, which Stoney had himself proposed and calculated, he could construct a length, a time and a mass.<br>Stoney found the following values:<br>Length = 10^-37 meter, time = 0.3 × 10^-45 second and mass = 10^-7 gram.<br><br>It should be noted that the electron was only discovered in 1897 and that 'electron' was introduced by Stoney as the name of the smallest electrolytic unit of charge. Likewise it should be noted that Stoney's units are introduced purely ad hoc without connection to any theory.<br><br>In 1899 Max Planck (1858-1947) ['Über irreversible Strahlungsvorgänge'. Sitzungsberichte der Preußischen Akademie der Wissenschaften, vol. 5, p. 479 (1899)] gives a mathematical description of the electromagnetic radiation from an 'absolutely black body'. In order to be able to explain the experimental facts, Planck had to break with the classical continuous physics, and had to presume that the radiation was emitted in discrete energy portions – called quanta.<br>After the introduction of his quantum constant, Max Planck showed, by a pure consideration of dimensions, that it was possible to construct a length, a time and a mass, by means of Newton's gravitation constant, the velocity of light and the new Planck constant.<br><br>The energy of one radiation quantum was proportional to the frequency of a specific radiation. This proportionality constant, connecting the energy of a quantum and the frequency, we today call Planck's Constant. By the quantization of the radiation energy and the introduction of Planck's Constant, quantum physical considerations were introduced in the description of nature.<br>The equations for calculation of what we today call Planck length lp, Planck time tp and Planck mass mp are given by: (all gif file images -- see url)<br>....<br>Today several cosmologists use these Planck quantities for conditions in the early Universe, and this in spite of the fact that the quantities can not be motivated as being fundamental for any physical theory. The motivation for their use is their small size, but nobody has foreseen the small elementary quantities from my theories.<br>--unquote--<br>***<br>From these initial propositions, Nielson concludes:<br>"All physical quantities, such as velocity, acceleration, force, work, energy etc. are defined as usual, but we must take in account the quantization of space, time and mass. This fundamental "atomization" causes that all movement is discontinuous – all movements are "jumps". Likewise all physical processes in a definite system will be characterized by discontinuous changes of certain physical quantities."<br>Basically, Nielson is making the same assumptions that I have, although we come from different starting premises. As I hope to have shown, a consideration of Zeno's paradox(s) leads one to such a premise -- that there MUST be fundamental, absolute units of space and time, these must be quanta, ie. irreducable measures, a characteristic of our spacetime universe. The challenge is to accurately define just WHAT these values are, how they relate to other properties of mass and energy forces, and then to see what further discoveries via relationships of mass and energy they enable us to make about the universe.<br><br>A few further tantalyzing links on the journey of deducing the inevitability of space and time quanta implied by consideration of Zeno's paradox:<br><br>ZENO'S PARADOX: A RESPONSE TO MR. LYNDS (by Eric Engle)In a world that (generally) presumed that matter and energy were very different quanta, Zeno's paradox forced one to ask what is meant by motion and time ...<br>www.lexnet.bravepages.com/ZENO.html<br><br>spaceThe text proposes that space is a quantum system, as proven so by Zeno's paradox. The quanta of space are essentially photon holes, the absence of the ...<br>www.johnkharms.com/space.htm <br><br>PHOTON EMISSIONA photon is the addition of energy to a vacuum energy quanta. ... of space and time as it relates to the thought experiment known as Zeno's paradox. ...<br>www.johnkharms.com/photon.htm <br><br>Snapping and Zeno's Paradox, by Rev. Chuan Zhi ShakyaSnapping and Zeno's Paradox. by Chuan Zhi Shakya, OHY ... initiated instantaneously by light quanta that convert the sun's energy into chlorophyll. ...<br>www.hsuyun.org/Dharma/zbohy/ Literature/essays/czs/snapping.html <br><br>Student Challenges Basic Ideas of Time | MetaFilterPretty soon, Achillles - or rather, the quanta which Achilles is composed of - can zap across Zeno's Paradox boundary in no time at all! ...<br>www.metafilter.com/mefi/27402 <br><br>QI Talk Forum | View topic - Beast NumberZeno's Paradox is only one weapon in the armoury of the commited Diabolicalist. ... Interesting about infinite division and quanta - I know our current ...<br>www.qi.com/talk/viewtopic.php?t=668& start=2&sid=49085f2d75f51663f9ad44ea82a849ed <br><br>EBTX - The Nature of Collisions in TVF's MM ModelTom has erred in stating that the problem is similar to those in Zeno's paradox. Zeno's paradox consists of an infinite number of finites summing to a ...<br>www.ebtx.com/ntx/colidtvf.htm <br><br>scamper labs / advice / may 2000... a biology textbook, and so on smaller, until I saw quanta spinning wildly. ... Every chain of thought play-acted Zeno's paradox, as I went halfway to a ...<br>scamper.org/advice/adv_000529.html <br><br>The essence of life is self-reference-- Light quanta inside IQS are trapped like matter is trapped by a black hole in space. ... This also shows that the Zeno paradox (the movement ...<br>www.mdpi.org/fis2005/F.30.paper.pdf <br><br>DIAMOND In this volume, paradoxes by Russell, Cantor, Berry and Zeno are all resolved. ... Voter's Paradox; Metamathematics: Gödelian Quanta; Meta-Logic ...<br>www.worldscibooks.com/mathematics/3271.html <br>****<br>An important observation, essential to understanding the 'value' of paradoxes:<br>'... paradoxes exist not to be solved but rather to teach problem solving! It is axiomatic that a paradox presents a "red herring" - that it present a problem other than the real problem that it presents - in order to force the student to discover a solution by questioning their ordinarily unquestioned assumptions.' -- Eric Engle,
www.lexnet.bravepages.com/ZENO.html<br>Ennyway -- Sorry to range so far afield here -- I imagine most folks will get glazed eyes and scroll to the next comments, but mebbe a few folks will be intrigued to reflect on these and related implications that extend from considering what Zeno's paradox can teach is.<br><br>Thanks for the opportunity for tantalyzing digression, AlicetheCurious!<br>Starman<br> <p></p><i></i>