Fred56

How do you guys get a series of (potentially) infinite terms to converge? Or are mathematicians who do this deluding themselves, maybe? Alternatively, are you claiming that an indefinite integral is an illusion of some kind?

Sure we are. But lots of people want to get rid of the pesky critters who come onto their property looking for food (elephants, wolves, bears, anything that is a potential 'danger', birds, reptiles, ...). So our concerns aren't really mirrored by our actions in general. There simply isn't enough room left for many wild animals, who are losing ground to us on a daily basis.

If you sit somewhere quiet and try to concentrate on the area between your eyes, you start seeing stuff (like flashing or coruscating light), it's pretty cool. After a while you start to hear stuff (not thoughts) inside your head. These phenomena are maybe from the pineal gland, but I don't know that is the case (I don't think anyone else does either). Even just focusing on the sound of your own breathing can be kind of trance-inducing, or whatever you might call it. These are not imagined or hallucinated, because they make you feel relaxed -they are in fact quite centering (not distracting) and help your focus. The more often you meditate like this, the more stuff happens, but you have to give it a good try, as it were.

Fredrik: I have considered that the learning process (acquisition of information) which requires energy, must necessarily be equal or proportional to the energy expended. The energy expended in a brain is not the only requirement (an observer must stay alive, and find food and so on). In this sense, all expenditure of energy (including that made to find and ingest food), therefore represents the entropy of the acquired learning (knowledge or information). In other words all observers are the result of their entire lifetime of 'observation': is eating something 'measuring' it, or is there any expenditure that is not part of a measuring or observing 'process'? A brain cannot exist in isolation, it requires a body, a body requires food, and so on. Can you see a problem with the conclusion that knowledge is the sum of all expenditure, for any organism (observer)? Which means that every lifeform is its own knowledge or information store (the entire organism), not just its 'brain' or equivalent? P.S. I guess that means I agree with Mr Zurek's statement: there is no way to separate the information from the observer, because the information is the observer. Not sure I can say I'm a Quantum Darwinist yet though. Life is able to acquire information and use it to extend itself. Inanimate objects cannot do this -they don't convert part of themselves into energy, then use it to move around, or observe, and etc..

Math does 'let' you do this. You should take a closer look at the posts before yours. There are an infinite series of real values between 1 and 2. There are also an infinite series (of real values) between 2 and 3, (with all values real numbers). So what's the difference between them (is it zero or something else)? Although they are bounded differently, the intervals are the same size...

Actually it depends how you 'define' a particular infinity (there are lots of them -an infinite variety). Math lets you subtract infinities if they are 'undefined'. Two undefined infinities can be equal (or just assumed to be). Likewise with bounds for an infinite set, as above. I think your definition might run into difficulties with the idea of limits.

People who get into yoga (especially Westerners), often start to believe in things like that, things that can't be illustrated except by "going there". I wouldn't get too into what your instructor is saying though, it sounds a lot like a Westerner's ideas of what it's all about (so it's her own explanation -to herself). Read some good yoga books by practitioners, the earlier ones are good. There are supposed to be different levels you can get to with meditating, but then yoga is actually quite a broad subject (not just exercises and breathing).

Have you tried the hp users site (try googling hp calc)...?

If you're talking about the impending climate situation, I would say humans will almost definitely survive (we're fairly resourceful). But we can't predict how long H. sapiens will persist before we get replaced by another species. This has been the pattern for a lot longer than we have been around (according to the paleaontologists). Or maybe we'll evolve into the subsequent species (but this doesn't always happen, usually a species or an entire genus goes extinct eventually).

Here's a discontinous function: [math] f(x) = \frac 1 {|x-1|} [/math] The domain (input) of [math]f [/math] is: [math] x-1[/math] if [math]x-1 \ge 0 [/math] and: [math] -(x-1) [/math] if [math] x-1 < 0 [/math]; and excludes (is closed to) the set {1} (from the real numbers). So [math]f [/math] is discontinous, about the value 1, at which it's an asymptote (x can be any value less than or greater than 1, but not equal). You can define the domain of any function to itself be discontinous, or "patch" things so the range (output) isn't...

Suck it in dude... Wooa looks like he sucked in a bit too much. He´s gone then, it´s safe to come out... I personally was offended by his aberrant use of ´the Kyngge´s Englysshe´... Growing and gathering store The idea that our knowledge is a store, like some record which has been “carved in the sides of great mountains”, say, for all time is only whimsical at best, as any 'knowledgeable' man should realise. In spite of our hope, all knowledge --all records and symbols and argument-- is all completely useless, meaningless or absurd, without a mind to understand it. The store, ultimately, is in our heads. Knowledge (of mathematics) is considered both in a static “knowable” sense, and in a growing or evolving “unknown” sense. All knowledge has this static/changing nature. It is up to us to observe and record, but the record, 'external and internal', is necessarily a work in progress, and one we will never complete: there is, for an observer with limited, or “defined and bounded”, connection to an external world, no such thing as complete knowledge (of any thing). The only “complete knowledge” we have available, in that sense, is that information will never be a complete thing, there will always be uncertainty in our measurement. This “observer” uncertainty -an inherent characteristic, is mirrored in an inherent uncertainty in the fine structure of the world -something we have not known about for “very long”. This discovery has (and is) changing the way we look at things. We check the store's inventory constantly, and evaluate the “storability” of new information, we sort and gather, strew and scatter, connecting and severing the links that become a lasting record on the “wallpaper of the mind”. But if the mind were paper we would not need to cut down so many trees, perhaps. The external record, which we place greater faith in, is more a convenient way to compare individual knowledge, and represents the application of a group of observers who can compare individual knowledge (observation), more efficiently. Thus the group mind has more status, generally as the selector agent of persistent records. Today the external symbolic record is considered large enough and stable enough to provide (static, learned) knowledge for a very large group, but none of it makes any sense at all to, say a whale, or an amoeba, or any of the many other kinds of observers. We are one of the few species to extend their learning this way (and the most successful at doing it). But the external record is transient (it has its own entropy). ¨She tells Max to stay, when the class has gone away, so he waits behind. Writing 50 times, ´I must not be so´, oh oh oh. But as she turns her back on the boy, he creeps up from behind...¨ A comment about that mass stuff: Mass is defined several different ways in Physics, there is potential and kinetic energy (and chemical, nuclear, and thermodynamic, ...). Einstein concluded that energy, a scalar quantity which is always conserved, is photons, or photons (light waves or particles) are energy, and have zero (rest) mass. Rest mass is an extrapolation (introduced in the early 20th century by deBroglie), which can never be determined empirically (directly), because nothing is ever at rest. There is relativistic mass, intrinsic mass, rest mass, inertial mass; there is only, however one kind of radiation energy. Photons have a mass equivalent. Cosmologists refer to energy, rather than mass (maybe they do this because they are 'the same thing'). This is my position too, sorry if anyone has got all confused about it, but I thought the "issue" had been settled, like 90 odd years ago... If someone can illustrate what's important about saying "photons have zero mass", I'll move it off the page with all the other things it "doesn't have", (which can't be all that useful a way of describing anything). Dogs don't have wings, and cows don't jump over houses, the Moon (apparently) is not made of green cheese, this is all useful information, for some reason or other...

Isn't 'important', except that it maybe doesn't exist, like mass and change in amplitude are non-existent, perhaps.. What I want to nail down is how the momentum is "transferred", if both vectors are varying about a zero point, or does the "packet" get delivered during an oscillation (of both components), regardless of what the instantaneous values are for either (so the delivery occurs over time, or over a wavelength, or something)?

Right. But later on I change that particular sweeping statement to "it can be zero", which is true if the subtraction/division is done before any limit is taken (I've done this, and you have too, with a series of infinite terms that cancel so the series converges...). Yea, but what does size mean? they have the same cardinality, but different bounds. If you define the two to have the same bounds, the difference will be zero, won't it?

Here's what Wikipedia says about photons and momentum: k is the directional wave(number) vector =[math] \frac 1 {\lambda} [/math]. Doesn't look like there's any room left for volume. The Wikipedia entry for photons goes on to describe how the momentum is the result of the two moments, or components, that have an angular frequency (just the e and m components). The spin angular momentum is independent of the frequency -it is related to the way a photon propagates (as a wave of EM energy, and the fact it has no mass, but has a mass equivalent like Einstein says). Wikipedia again: So the momentum is due solely to the frequency, which means that the area (of the 2-d "wavefront") is constant --at least it must be when the photon collapses, or if it varies with time, then somehow it "becomes" its actual value...? P.S. About the zero-mass of photons: the photon is without intrinsic mass (it doesn't have any at all, how sad). Since it doesn't have mass, isn't saying "a photon's mass doesn't exist" a bit pointless? Especially since energy (which is photons) is equivalent to mass? I think it kind of is, myself...

Observation: We classify information as the symbols we use for ´messages´, and measure this content, or certainty, that such external symbolic information has (as books, electrical signals or magnetic regions on a spinning disc, or photons from a screen). But it has absolutely zero content unless it is processed, ultimately by some observer, who must expend energy to ´interpret´ the message and its content. So where is the information? If it is the content of the message and this reduces the internal entropy of the observer, what corresponding increase in entropy is needed for this reduction in uncertainty (by the observer´s processing of the message)?

I thought that could mean that volume is the same thing as ´mass´ for a photon. I could however be incorrect (along with a few other guys, like Mr. Einstein, say). Hang about... If a photon moves through a vacuum and has 2-DOF along a traveling surface, and the two components are like changing vectors, the momentum at any point in time is some area on this 2-d surface. It must be a constant, rather than cyclic area, because it represents the photon´s momentum -the sum of both moments: the electric and magnetic moments. Otherwise how is a photon´s momentum transferred, or conserved? The area described by both moments or vectors must be constant, and depend on frequency -could that make the momentum look like a volume after all... there´s the ´area´ described by the two perpendicular wave fronts, and the frequency could look like a ´depth´, so a shorter cylindrical ´volume´ would be a more energetic photon -the height of the cylinder is inversely proportional to the momentum...? At least when the wave collapses or whatever.

There´s solar towers. They built one somewhere in the Middle East I think. There are plans to build a big prototype in Australia. But they are only about 3% overall efficient and to get a significant output you need a pretty big area to collect solar energy. And an efficient turbine to harness the airflow...

Maybe you should have a read of the other thread about mass and photons (information). I started this one off and it´s a tough game but I think the ref could award a try here and there (maybe even a penalty). He hasn´t blown his bloody whistle much but.

It does mean that yes, there are an infinite number of intervals (bounds) that you can define with an infinite number of real values in them (on any interval of the real numbers). Any infinite set also has an infinite number of possible subsets. That´s a general rule, I think. Maybe someone who knows could say if that is the case (I´m not too good at number or set theory).

If you are really interested about what makes your ´self´ tick, then you should prepare your expectations accordingly, but if you´re just after something better than boredom, it probably won´t be what you expect (either way). This could explain why back in the day the strong stuff was under the purveyance of a scary guy, who only administered same at special (i.e. initiative, or expiative, or supplicative -of some god or spirit, maybe) occasions. The kids got the picture that it was special stuff and they weren´t supposed to mess with his ju-ju. Or any of his other stuff...

You can imagine a photon as a traveling wave, or as having a 2d surface that projects at right angles to the direction of travel or movement. There are 2 degrees of freedom for a photon, along this traveling 2d surface (let´s say it looks like a little circle), and there are 2 components, an electrical and a magnetic component, which ´resolve´ into a momentum. The wave is said to collapse when this happens. But this is a model, an idea of what a photon being absorbed or emitted ´looks like´. A 2-d surface can´t have volume so it´s meaningless in such a space, but area isn´t. The radius for photons in this (mathematical) space is never more than the same constant value (but the two components vary sinusoidally about a zero point, so that the area is also cycled this way, from zero to a constant value, and the cycle time, or frequency, determines the energy of a particular photon), this (maximum constant amplitude) appears to be related to its apparent velocity, somehow. The energy is not related to the distance traveled, unless the photon interacts with another photon (or an electrical or magnetic field, or collides with an electron or other charged bit of matter). In other words you could say that the energy in a photon is bounded by (integrable over) a single period of its cycle (or something similar), like a packetised bit of energy, rather than the integral of all the periods it has cycled through on its journey.

This is true in the sense of a mind being able to evaluate, especially to evaluate the same ability in another mind. Without group knowledge there would be only individual transient (unstable) knowledge. This extends to organisms that share any sort of resource, say bacteria that swap bits of genetic material -the group observes the (external) ´knowledge´ and selects some of it -this becomes group knowledge because it persists, because of this selection. This models what humans do (or bees, or ants, or probably every organism that requires knowledge --of itself and its orientation to the world and other organisms, and what that necessarily requires). (Even if I say this myself -the mind with the supposedly critical evaluation function)...

All life is changing, or evolving. This appears to be a teleological kind of process, in which all forms of life compete with each other and with other lifeforms, here on this planet. The past 400 or 500 million years of our planet´s history is known by humans (us), with some detail. We know that the continents were all much closer together back then, and that they have split up and dispersed, spread around the globe by forces that act inexorably, but with great energy, and build new ocean floor constantly, as the continents are pulled and pushed around like a bunch of bumper cars, except very slowly moving ones. Also, during the Cretaceous, there were no masses of frozen water at either pole, and the newly-separating continents (mostly two large masses, a northern, and a southern continent) were flooded, or had shallow inland seas. There was generally more tectonic and volcanic activity, because of all the plate movement and the new island chains, and hot spots -that are believed to be responsible for the welling up of heat beneath continental cratons, and then forming rift valleys that become new seas, and then oceans -this is how the Atlantic started. The climate was much warmer. The Holocene era saw the dispersal of the continents reach an apogee, and the process should soon (within the next 15-20 million yrs) start to reverse and the oceans will start to close up again, and new rifts form (perhaps under Yellowstone´s hotspot, or Siberia). This process is cyclic, and has presumably seen several episodes of the breakup and re-aggregation of all the continental, less dense shields, or cratons (formed in the Permian era, and so referred to as Permian shields), that float on a plastic layer of basaltic rock, which subducts great swathes of the edge of an oceanic plate, or plates crash into and compress another plate, as in the case of India, which has raised a great table (the Tibetan Plateau) of mountains, and Africa itself, which continues to rotate anticlockwise and push north into Europe, so that the Mediterranean will eventually pinch off into a small inland sea, or series of inland saltwater bodies. These changes have seen big climatic change as well, and the planet is now locked, it seems into a cycle of freezing and thawing that has an interval, or a frequency, of hundreds (about 1.3 per glaciation) of thousands of years. The frequency correlates with subtle changes in the earth´s orbit and inclination: Milankovitch cycles. Life also changes its environment necessarily, in order to itself change and keep changing. The first major global change that life made to the planet was arguably the growth of oxygen-producing organisms (cyanobacteria and blue-green algae), in the earth´s new oceans. These changed the atmosphere and altered the oxygen content to the extent that life could now learn to adapt (was obliged), to the new ´resource´ that oxygen, actually a very toxic substance, had to offer (because it readily accepts electrons it ´oxidises´ things -one of the first things that happened to it was the oxidation reaction with all the iron lying around, and in the oceans). So change is ongoing and a necessary thing. And it is a balance, an ongoing cycle of exploitation, depletion, and learning -to find new resources or exploit existing ones with more efficiency. Can an organism that changes its environment do so too efficiently, so that its own ability (to exploit and change the environment) places it in danger (from that change), so to speak?

blue_cristal: Have you seen the research (a single review of IQ results) that indicates there is a balance of highly intelligent people with an equal ´representation´ of lowly intelligent (i.e. morons)? And this is largely a male trait in humans (but could be true of primates or other social animals -I don´t know about any research in these areas)... Just thought I´d chuck it in there...

The same, uncountable ´number´ of discrete values is between any two such arbitrary bounds, and has the same cardinality (in fact is identical -except for the limits or bounds -the excluded values)... Is this a quiz? These are all sets which are bounded in value but have unbounded cardinality. There´s something of a symmetry, in that you can exclude any cardinal number (by bounding it between any two, or between any two arbitray real values, but each discrete element has a value which means it also has an order , or selectability, within the set. Since by definition all members of such a set are discrete, counting them to determine this trivial fact is, ..well, trivial, or actually pointless (pun intended).