Anyone here into math?

[Guest [Kp...]]

ok, i confess i am addicted to benzobuddies. i cannot lurk or not post -- this post is testimony.

 

randomness

 

(an interesting article on it.)

 

Gamblers, by definition, are optimists. They can win only by beating the odds, and--over the long term--the odds cannot be beaten.

But what about the short term? Many gamblers believe winning streaks, known as "hot hands", are real, and that if they are in such a streak it makes sense to keep on betting.

 

Conversely, many also believe bad luck is sure to reverse itself not merely by reverting to the mean, as a statistician would predict, but to the extent that the gambler will recoup his losses. This is known as the gambler's fallacy.

 

Non-gamblers might reasonably assume both approaches to be equally fantastical. But research by Juemin Xu and Nigel Harvey at University College, London, just published in Cognition, has shown that in some areas of gambling hot hands do actually exist.

Conversely, and just as oddly, they found that in these same areas the gambler's fallacy is yet more fallacious than a statistician would predict. Punters who continue to punt after losing do not even manage to revert to the mean.

 

Using the power of the internet to round up a huge sample, the two researchers examined 565,915 bets made by 776 people on sports such as horse-racing and football. Because these were online bets their timing could be established precisely. Ms Xu and Dr Harvey looked at winning and losing streaks up to six bets long.

 

The probability of a first bet winning was 48% and that of a follow-up winning again was 49%. After that, the streak took off. The third bet won 57% of the time. The fourth, if the third had won, won 67% of the time, the fifth, 72% of it and the sixth 75%. As for the losers, after ploughing their first bets, their success with their second slipped to 47% and thence held at 45%.

 

The explanation of the puzzle, Ms Xu and Dr Harvey found, was not that Lady Luck actually does smile on winners and frown on losers. Rather, as winners' winning streaks increased in length they started choosing safer and safer odds, which led them to win more often, though less profitably. In contrast, those who had experienced a losing streak went for ever riskier bets, making it more likely the streak would continue.

 

Hot hands, then, are real. But they are created by a gambler's behaviour rather than by fortune's wheel. The gambler's fallacy, though, is made worse by his behaviour. The moral is to believe in maths, not luck, and probably not to bother betting in the first place.

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i do not doubt probability theory, randomness or the existence of it in nature (quantum mechanics). but somehow i am not convinced by the explanation given in the above article for why lucky or unlucky streaks happen. i have myself experienced repeated "lucky/unlucky streaks" in an online game called yatzy that involves dice and is played against real opponents. i have also noticed that i seem to have a permanent unlucky streak with very few lucky breaks in between (observed over 5 years) -- resulting in a net outgo of $$ to keep buying game currency. what's weird is that it hasn't averaged out (and i have tried all tactics like playing more when i am winning and taking a break when i am losing). what do you do if it doesn't average out in a lifetime? wait for another lifetime? of course, my sample size of 1 is too small to generalize. 

 

one more case -- http://www.dailymail.co.uk/news/article-2214409/The-7m-streak-luck-Mayfair-casino-thinks-good-true.html

 

[Guest [ba...]]

Horse racing and football odds are set by gamblers.  They often have little to do with reality.  The line in football is set so that half of the people will bet each side which assures the casino of maximum winnings.  Some people bet pretty fanatically on their team, while others actually study the match-ups and look for a line that doesn't quite match up with their predictions.  There's definitely a lot of luck in picking football winners, but there is some degree of skill.

 

Horse racing is the same.  The betters set the odds, and betters can be really stupid.  If you 'know' something about a horse or its owner, you might have an edge over the rest of the betting field.

[Guest [Kp...]]

Horse racing is the same.  The betters set the odds, and betters can be really stupid.  If you 'know' something about a horse or its owner, you might have an edge over the rest of the betting field.

 

this is true. here are 2 examples (link):

 

1. Edward Thorp:

 

Thorp is the father of card counting. Not only was he successful using it in real-world situations, he was the one who invented the original system. A mathematics professor who possessed a master’s degree in physics and a doctorate degree in mathematics, he clearly had above-average intelligence.As an adult in the early 60s, Thorp knew next to nothing about casino games and the world of gambling. But when a friend, Claude Shannon, brought him and his wife to Las Vegas he became interested in blackjack, and after playing the game a number of times became convinced that there was a mathematical way in which the player could gain an advantage.He studied the game in a systematic method and exhaustively examined every facet of the game. Using a computer owned by the university he taught at, he simulated billions of blackjack hands to delve even further into the mathematics of the game. This computer was so massive it filled an entire room, yet it was less powerful than today’s laptops. Through his calculations and observations he created his system which “accounted for the variations in those (cards) that remained after certain hands were dealt”. Basically, he realized that smaller cards were more advantageous for the dealer and when they left the deck, advantage shifted in the player’s favor so they should bet more. Concurrently, larger cards were more advantageous to the player and when they left the deck the advantage shifted to the house, or dealer, so less money should be bet. Using this method Thorp calculated that the player could own a 1% to 5% advantage over the house.Thorp and Shannon hit the casinos and would return with their pockets filled with cash. One typical weekend would net $70,000 in today’s money. The gambling industry was no match for Thorp’s flawless execution of his card counting method. After all, card counting did not yet exist to them so they had no idea what was happening. Thorp drew the attention of casino bosses when he began winning unusually high sums and most were convinced he was cheating. They intensely watched him play and studied videotapes of his play, but they could see nothing nefarious occurring. Before long, some casinos asked Thorp to leave because he was simply winning too much, yet they still did not know how he was doing so.In 1962 Thorp wrote his book “Beat the Dealer” which detailed his card counting “ten count system” and became an instant hit as well as a modern-day classic. With its sales he amassed a pretty handsome fortune. In 1966 he wrote a second edition which expanded on the intricacies of the system. It’s interesting to note that the sudden explosion in card counting worked in favor of the casinos since many people attempting to do it just couldn’t pull it off as effectively as Thorp did. However, his findings and methods have been the basis for every card counting system following, including that of the aforementioned MIT team’s.Following his gambling exploits, Thorp applied his mathematical genius to the stock market and made a huge fortune in securities and hedge funds. Due to his dominance over the casinos and revolutionary thinking, Thorp was one of the first seven inducted into the Blackjack Hall of Fame.

 

2. Gonzalo Garcia-Pelayo

 

Garcia-Pelayo of Spain was initially a record producer but proved to not be very successful in that endeavor. As a result, he decided to devote all his energies to his passion, roulette, and became the first person to successfully exploit wheel bias in the 90s.Others had suggested the idea of wheel bias but had never taken advantage of it in a casino situation. Wheel bias is the belief that not all roulette wheels are perfectly random, and that each individual wheel is unique in that certain numbers are more likely to drop than others. This aberration was a result of wheels being ever so slightly off level or because of other minute inaccurate measurements such as tiny differences in pocket sizes, or the way the wheel’s gears worked.He began in casinos in Spain by tediously staring at a specific wheel for thousands of spins, recording his results, then analyzing them with a computer. He also recruited his 5 children to help record results. Before he bet a cent, he did many observation sessions on the same wheel. When he felt it was time to bet on the wheel’s “hot numbers” he swung a 5% house edge to a 15% player’s edge and raked in the cash. When he began to feel heat he found a new casino and did the process over again. When every casino in Spain knew who he was he took his method to the United States and Las Vegas where he continued to profit. When he became just too well-known by casinos around the world he retired with an estimated $1.5 million in the bank. One casino sued to recover their losses but Spain’s Supreme Court ruled in Garcia-Pelayo’s favor saying that all he did was use “ingenuity and computer techniques. That’s all”.

 

-----------------------

 

i suppose there are no truly random casinos, games or races (zero odds). lucky/unlucky streaks depend on odds -- knowingly or unknowingly.

[Guest [Kp...]]

Energy, matter, and information equivalence

 

Shannon's efforts to find a way to quantify the information contained in, for example, a telegraph message, led him unexpectedly to a formula with the same form as Boltzmann's. In an article in the August 2003 issue of Scientific American titled "Information in the Holographic Universe", Bekenstein summarizes that "Thermodynamic entropy and Shannon entropy are conceptually equivalent: the number of arrangements that are counted by Boltzmann entropy reflects the amount of Shannon information one would need to implement any particular arrangement..." of matter and energy. The only salient difference between the thermodynamic entropy of physics and Shannon's entropy of information is in the units of measure; the former is expressed in units of energy divided by temperature, the latter in essentially dimensionless "bits" of information.

 

The holographic principle states that the entropy of ordinary mass (not just black holes) is also proportional to surface area and not volume; that volume itself is illusory and the universe is really a hologram which is isomorphic to the information "inscribed" on the surface of its boundary.

[[...]]

I really impressed with these numbers.

[[be...]]

Casinos, by average, earn $1 million a day. They would not be in business unless there are plenty of losers. Another reason why drinks are free. To get people looped so they will take chances they cannot win.

[[Be...]]

I remember being in the Trump Casino in Atlantic City many years ago and had lots of trouble getting my free drinks.  I guess they could see that I wasn't a high roller and wouldn't be spending alot of money there.  The buffet dinner in their restaurant was the best food I had ever eaten.  Their green beans were perfectly made.  I guess not related to math much; going off topic. 

[Guest [ou...]]

On this day, a child was born who would change the world before the age of 33; Newton.

Happy holidays! By the way, Jesus wasn't born on Christmas. He wasn't even born on 0.

He was born somewhen in the spring of 5BC.

[[Be...]]

outis, I believe I read the same thing somewhere about when Jesus was actually born.

[Guest [Kp...]]

By the way, Jesus wasn't born on Christmas. He wasn't even born on 0. He was born somewhen in the spring of 5BC.

 

i didn't know this. let me google.

 

On this day, a child was born who would change the world before they turn 33; Newton.

Happy holidays!

 

outis, most of these geniuses, in art and science, whip out their masterpieces before they hit 30 (including jesus). are they living life backwards?

 

i was reading about dirac's beliefs the other day and was amused to see how his opinion about god changed with the passage of time and physics:

 

dirac as a young man:

If we are honest—and scientists have to be—we must admit that religion is a jumble of false assertions, with no basis in reality ... If we are honest—and scientists have to be—we must admit that religion is a jumble of false assertions, with no basis in reality ... If religion is still being taught, it is by no means because its ideas still convince us, but simply because some of us want to keep the lower classes quiet. Quiet people are much easier to govern than clamorous and dissatisfied ones. They are also much easier to exploit. Religion is a kind of opium that allows a nation to lull itself into wishful dreams and so forget the injustices that are being perpetrated against the people. Hence the close alliance between those two great political forces, the State and the Church. Both need the illusion that a kindly God rewards—in heaven if not on earth—all those who have not risen up against injustice, who have done their duty quietly and uncomplainingly. That is precisely why the honest assertion that God is a mere product of the human imagination is branded as the worst of all mortal sins.

 

dirac near the end of his career:

It could be that it is extremely difficult to start life. It might be that it is so difficult to start life that it has happened only once among all the planets... Let us consider, just as a conjecture, that the chance life starting when we have got suitable physical conditions is 10−100. I don't have any logical reason for proposing this figure, I just want you to consider it as a possibility. Under those conditions ... it is almost certain that life would not have started. And I feel that under those conditions it will be necessary to assume the existence of a god to start off life. I would like, therefore, to set up this connexion between the existence of a god and the physical laws: if physical laws are such that to start off life involves an excessively small chance, so that it will not be reasonable to suppose that life would have started just by blind chance, then there must be a god, and such a god would probably be showing his influence in the quantum jumps which are taking place later on. On the other hand, if life can start very easily and does not need any divine influence, then I will say that there is no god.

 

i am not pointing out the above to validate the existence of god but pointing out that nature is so unpredictable and deeply mysterious that no matter how well founded your belief might be, you can trust nature to, in time, a) reverse your belief, b) leave it unchanged, c) make your belief ambiguous... etc. as hawking said -- and i hope he is right -- the mystery will never end and there will always be a job for mathematicians and scientists (and for people like me who try to understand what they mean or could mean).

 

[Guest [Kp...]]

on randomness

 

important caveat -- my understanding has been formed by the conversations in this thread, inputs from two members in this thread and wikipedia. i do not profess to be painting an accurate or truthful picture of things. if you think i am wrong, feel free to point out -- i promise not to get belligerent and i promise to relearn the subject starting with the premise that there is a 99% chance that i am wrong.

 

randomness is ingrained in logic and analytical thought. it is the property of any logical system that is sufficiently complex (multiplication in arithmetic or self-recursion in logic). we are familiar with only that randomness that is computable like the decimal expansions of algebraic irrationals and pi. so, in a sense, there is nothing random about randomness (or random sequences in arithmetic). if we move from this point in the informal definition of randomness to the other end, i.e. uncomputable or pure randomness, then everything is random about those random sequences, but, they are absolutely impossible to generate¹ (to the extent that one can wonder if the existence of such randomness is at all a valid premise... constructivists think not). these contradictory properties that define the two boundaries of randomness are united by the property that all these random sequences are normal (equal distribution of all digits and digit sequences in the larger random sequence), but, it is not possible to prove they are normal (except in retrospect on n digits of the number sequence). randomness, thus described, is a strait jacket in which our minds are trapped because this randomness is a product of the way we reason (which is also the only way in which we can reason for we have not known any other method).

 

thus to determine whether we learn randomness from nature or we impose randomness on nature (when reconstructing nature, through our senses, in our minds) is not possible. it is an epistemological question like the chicken-egg or thought-action paradox or, can-we-prove-we-have-free-will paradox. in other words, the question is irrelevant to science (because it lies outside logic or it can be investigated only by meta logic). it is interesting to note that humans are very poor at "creating" random sequences (this might be logical if human thought was itself based on "randomness" of the computable sort).

 

1. despite there being a definition (algorithm) to generate it (chaitin's constant) -- chaitin

 

by g. chaitin --

 

Now, instead of looking at individual instances of Turing's famous halting problem, you just put all possible computer programs into a bag, shake it well, pick out a program, and ask: "what is the probability that it will eventually halt?". This probability is the number Omega.

 

An example will make this clearer: suppose that in the whole wide world there are only two programs that eventually halt, and that these programs, when translated into bit strings, are 11001 and 101. Picking one of these at random is the same as randomly generating these two bit strings. You can do this by tossing a coin and writing down a 1 if heads comes up, and a 0 if tails comes up, so the probability of getting a particular bit is 1/2. This means that the probability of getting 11001 is $1/2 \times 1/2 \times 1/2 \times 1/2 \times 1/2 = 1/2^5.$ So the probability of randomly choosing one of these two programs is $1/2^3+ 1/2^5 = 0.15625.

 

Of course, in reality there are a lot more programs that halt, and Omega is the sum of lots of terms of the form $1/2^ N.$ Also, when defining Omega, you have to make certain restrictions on which types of programs are valid, to avoid counting things twice, and to make sure that Omega does not become infinitely large.

 

Anyway, once you do things properly you can define a halting probability Omega between zero and one. Omega is a perfectly decent number, defined in a mathematically rigorous way. The particular value of Omega that you get depends on your choice of computer programming language, but its surprising properties don't depend on that choice.

[[be...]]

On this day, a child was born who would change the world before the age of 33; Newton.

Happy holidays! By the way, Jesus wasn't born on Christmas. He wasn't even born on 0.

He was born somewhen in the spring of 5BC.

 

Here ya go outis. [credit: wiki]

 

The date of birth of Jesus is not stated in the gospels or in any secular text, but most scholars assume a date of birth between 6 BC and 4 BC.[1] The historical evidence is too incomplete to allow a definitive dating,[2] but the date is estimated through two different approaches—one by analyzing references to known historical events mentioned in the nativity accounts in the Gospels of Luke and Matthew, and the second by working backwards from the estimation of the start of the ministry of Jesus.[3][4]

[[Be...]]

I also read that they based Jesus' true birth on astrological alignments according to the wise men who found him and brought him gifts and knew when he'd be born.  The Magi wise men studied the placement of the stars and planets.  Sounded interesting to me. 

[[al...]]

on randomness

 

important caveat -- my understanding has been formed by the conversations in this thread, inputs from two members in this thread and wikipedia. i do not profess to be painting an accurate or truthful picture of things. if you think i am wrong, feel free to point out -- i promise not to get belligerent and i promise to relearn the subject starting with the premise that there is a 99% chance that i am wrong.

 

randomness is ingrained in logic and analytical thought. it is the property of any logical system that is sufficiently complex (multiplication in arithmetic or self-recursion in logic). we are familiar with only that randomness that is computable like the decimal expansions of algebraic irrationals and pi. so, in a sense, there is nothing random about randomness (or random sequences in arithmetic). if we move from this point in the informal definition of randomness to the other end, i.e. uncomputable or pure randomness, then everything is random about those random sequences, but, they are absolutely impossible to generate¹ (to the extent that one can wonder if the existence of such randomness is at all a valid premise... constructivists think not). these contradictory properties that define the two boundaries of randomness are united by the property that all these random sequences are normal (equal distribution of all digits and digit sequences in the larger random sequence), but, it is not possible to prove they are normal (except in retrospect on n digits of the number sequence). randomness, thus described, is a strait jacket in which our minds are trapped because this randomness is a product of the way we reason (which is also the only way in which we can reason for we have not known any other method).

 

thus to determine whether we learn randomness from nature or we impose randomness on nature (when reconstructing nature, through our senses, in our minds) is not possible. it is an epistemological question like the chicken-egg or thought-action paradox or, can-we-prove-we-have-free-will paradox. in other words, the question is irrelevant to science (because it lies outside logic or it can be investigated only by meta logic). it is interesting to note that humans are very poor at "creating" random sequences (this might be logical if human thought was itself based on "randomness" of the computable sort).

 

1. despite there being a definition (algorithm) to generate it (chaitin's constant) -- chaitin

 

by g. chaitin --

 

Now, instead of looking at individual instances of Turing's famous halting problem, you just put all possible computer programs into a bag, shake it well, pick out a program, and ask: "what is the probability that it will eventually halt?". This probability is the number Omega.

 

An example will make this clearer: suppose that in the whole wide world there are only two programs that eventually halt, and that these programs, when translated into bit strings, are 11001 and 101. Picking one of these at random is the same as randomly generating these two bit strings. You can do this by tossing a coin and writing down a 1 if heads comes up, and a 0 if tails comes up, so the probability of getting a particular bit is 1/2. This means that the probability of getting 11001 is $1/2 \times 1/2 \times 1/2 \times 1/2 \times 1/2 = 1/2^5.$ So the probability of randomly choosing one of these two programs is $1/2^3+ 1/2^5 = 0.15625.

 

Of course, in reality there are a lot more programs that halt, and Omega is the sum of lots of terms of the form $1/2^ N.$ Also, when defining Omega, you have to make certain restrictions on which types of programs are valid, to avoid counting things twice, and to make sure that Omega does not become infinitely large.

 

Anyway, once you do things properly you can define a halting probability Omega between zero and one. Omega is a perfectly decent number, defined in a mathematically rigorous way. The particular value of Omega that you get depends on your choice of computer programming language, but its surprising properties don't depend on that choice.

 

I am not ashamed to admit that I don't understand half of what you wrote as I am about as far from being a mathematician as I person can get, but it seems to me that flipping a coin an infinite number of times will not produce a rough 50/50 split between heads or tails if true randomness exists because such a state would be free from the constraints of probability (not to mention external variables that might come into play during each toss).

[Guest [ch...]]

[Guest [Kp...]]

 

:laugh: :laugh: omegawd!

 

 

[Guest [Kp...]]

I am not ashamed to admit that I don't understand half of what you wrote as I am about as far from being a mathematician as I person can get, but it seems to me that flipping a coin an infinite number of times will not produce a rough 50/50 split between heads or tails if true randomness exists because such a state would be free from the constraints of probability (not to mention external variables that might come into play during each toss).

 

Randomness is a difficult concept to explain (which is why I did not make a stab at it earlier). Let me see if I can explain it. If I can, I will consider it an achievement. And, by the bye, don't underestimate yourself: even though you might not be inclined towards math, you brain is performing the most complex mathematics known to the universe, every second, just to create a mesmerizing and seamless picture of the world in your mind, wherein you can feel unfettered and unaware (of the constant computations) to pursue whatever you like.

 

What you feel about randomness is correct. Intuitively I too feel that way. If we try to define it, we would say that randomness is a fair coin, in a fair space, behaving in a way that is not subject to any physical law that could bias it to one side; certainly not subject to a 50/50 law or any other probability law. Then if we tried to make the definition concise, we could say, "Randomness is a fair coin operating in a fair space and not constrained by any law." It is now that we can appreciate the enormous difficulty of making this definition operative. This is because every random sequence will have to be studied and compared with all the extant laws known to man before we can announce that the sequence "does not obey any law." (If you can come up with another method to detect its randomness, do share). The enormity of this task is such that our definition becomes unworkable -- no computer in the world can make this comparison in finite time.

 

If you read the definition of randomness formulated by Chaitin, you will see that he has defined it exactly as we have and has also stated that an implication of our definition is that "pure randomness" is practically impossible to generate.

 

Another problem with our definition is that what looks random to man today, might not look random to the man of tomorrow, or yesterday. So our definition is tied to the moment when this definition is exercised. To the man of yesterday, the shape created by mountains, snowflakes and jungles looked random. But to the man of today they are predictable to a large degree because they follow an equation called fractals. We can, however, make observations that some sequence is random, using our definition, provided we confine ourselves to a time in history, a database of our knowledge and a finite length of the sequence (or shape). That would be a workable definition in finite space. Our definition, on the other hand, is of a model in infinite space.

 

(I did not know that there is a proof that such a sequence should be normal or 50/50 and that the proof follows from the definition above. I have thus deleted my intuitive reasons for why it should be normal.) 

 

As chessplayer said above, the universe is finite while our definition is abstract or infinite. So pure randomness probably does not exist. But we will look for only those patterns, in our finite life and finite universe, that approach our definition closest, before we declare them random.

[[al...]]

Yes, I suppose that if it where possible to fully understand the interrelationships of all things and of all actions we will see that everything is indeed a predictable reaction and that true randomness does not exist, but since that is not possible, randomness will remain only a concept that can never be proven to exist or not exist.

 

However, I just realized that if you consider the universe to be truly infinite (as I do), then it is perhaps possible to have an infinite number of different outcomes, each being repeated an infinite number of times, allowing for randomness to simultaneously exist and not exist.

[Guest [Kp...]]

Yes, I suppose that if it where possible to fully understand the interrelationships of all things and of all actions we will see that everything is indeed a predictable reaction and that true randomness does not exist, but since that is not possible, randomness will remain only a concept that can never be proven to exist or not exist.

 

Yes, it can never be proven. I believe (and this is strictly my imagination) that randomness is the equation to ourselves. Since it is logically impossible for us to discover our own equation, we will not be able to discover all the laws of the universe and brain. We will be able to discover their "statistical distributions" but not be able to predict how they will behave in the future. Life, then, will remain a mystery (as it probably should).

 

I also believe that the brain is a computer, as is everything else in the universe. Our premise that only we have free will and consciousness is unfounded. If we are endowed with consciousness and free will, then this privilege should extend to the entire universe (at least relatively). It is entirely possible that if we plot the evolution of the universe, from big bang to the big rip, on a logarithmic scale, the universe might resemble (but not exactly be like) a living cell and the solar system might resemble a nitrogen atom. A mountain may walk around and an ocean might slither away. Everything in the universe, then, would seem to have free will except us (or relative to us). We might simply appear as vacuum, or degrees of it, that fills whatever space the rest of the universe creates.

 

No matter how hard we try, we cannot but have an anthropomorphic vision of things everywhere. It seems we cannot do anything but study ourselves relative to that which is not us. We are helpless because this is a boundary of reason. And even when confronted, steeped and saturated with reason, our whole life, we cannot understand it. This might be because it is from pure reason, as Chaitin suggests, that pure randomness appears. And then this randomness goes on to pervade every aspect of reason.

 

For those interested, here is a paper that disagrees with the philosophical implications of randomness raised by Chaitin:  Franzen

 

However, I just realized that if you consider the universe to be truly infinite (as I do), then it is perhaps possible to have an infinite number of different outcomes, each being repeated an infinite number of times, allowing for randomness to simultaneously exist and not exist.

 

Wow! And it was you who said that you do not have an inclination for math? The words you have used are exactly how the digits of pi are described. And we have computed pi to millions of decimal places. We notice this pattern in it very strongly but that still helps us predict nothing about the rest of the digits of pi and it is not a coincidence that we probably can never prove that pi is a "normal" number. (Normal = 50/50.)

 

Six nines in pi.

[[al...]]

Our entire universe may very well be like an expanding nitrogen atom, but I wish to emphasis "our". Upon an infinite backdrop of nothingness there may very well exist an infinite number of companion nitrogen atoms as well as other atoms, all coming together to form molecules and structures, expanding forever outward while the process is simultaneously repeated inward on a progressively smaller and smaller scale. That is the playing field that can allow for an infinite number of outcomes all repeating an infinite number of times. I like to call this "infinity squared". In this process who is to say what is possible and what is not. A mountain may walk, but it will not walk alone.

[Guest [Kp...]]

Our entire universe may very well be like an expanding nitrogen atom, but I wish to emphasis "our". Upon an infinite backdrop of nothingness there may very well exist an infinite number of companion nitrogen atoms as well as other atoms, all coming together to form molecules and structures, expanding forever outward while the process is simultaneously repeated inward on a progressively smaller and smaller scale. That is the playing field that can allow for an infinite number of outcomes all repeating an infinite number of times. I like to call this "infinity squared". In this process who is to say what is possible and what is not. A mountain may walk, but it will not walk alone.

 

I think your imagination is more beautiful than mine. I could not have expanded my mythical idea any better! I especially love the last line: In this process who is to say what is possible and what is not. A mountain may walk, but it will not walk alone..

[[pe...]]

Easy problem

 

You put a 1kg rock inside a bucket and fill it with 99kg of water.

You leave the bucket in the sun for the water to evaporate,

until it contains 98% water. How much does the bucket weigh now?

 

1kg rock + 99kg water = 100kg

 

after evaporation, bucket contains  98% water

 

assume bucket contains x kg after evaporation

 

so 0.98 x = water_after_ev

so x = water_after_ev / 0.98

we also know that water_after_ev = x - rock = x-1

so x = (x-1)/0.98

so x = 50

 

ans. 50 kg  (good one!)

 

Problem solved!

PS - Algebra >= Intuition

Actually there is no solution to the question as to the weight of the bucket. It could be any weight and does not change based on the contents.

[[al...]]

Our entire universe may very well be like an expanding nitrogen atom, but I wish to emphasis "our". Upon an infinite backdrop of nothingness there may very well exist an infinite number of companion nitrogen atoms as well as other atoms, all coming together to form molecules and structures, expanding forever outward while the process is simultaneously repeated inward on a progressively smaller and smaller scale. That is the playing field that can allow for an infinite number of outcomes all repeating an infinite number of times. I like to call this "infinity squared". In this process who is to say what is possible and what is not. A mountain may walk, but it will not walk alone.

 

I think your imagination is more beautiful than mine. I could not have expanded my mythical idea any better! I especially love the last line: In this process who is to say what is possible and what is not. A mountain may walk, but it will not walk alone..

 

Kpin99,

Thank you. I very much enjoyed our little exchange. Your mind is sharp.

[Guest [Kp...]]

Chaitin and Randomness once again, and Art

 

I believe fiction (and poetry) is a study of patterns, like mathematics, in a way that is so abstract that it transcends the highest of mathematics. I believe it transcends because in order to study, store and recall the images, or patterns, that we create in these arts, our brain requires unimaginable mathematical computation -- one of a scale that mathematics isn't yet sure of how to perform. This is just my personal opinion.

 

What Chaitin says is that were it possible for us to distance ourselves from mathematics, and observe the big picture painted by the whole of mathematics, we would see that the picture is neither linear nor homogeneous (Chaitin does this using a cantor set and a prefix free universal Turing machine). The painting would appear to be totally chaotic with no logic whatsoever. This is surprising, for, of all sciences, we had expected this the least from mathematics. So mathematics isn't as logical as we think. This painting would be rubbish actually, were it not for the fact that at the bottom of the picture is scribbled the name of the painter -- Man. What was man thinking, it makes you wonder, when he painted this -- and suddenly the painting becomes invaluable.

 

So the connection between the author and his art is not always obvious (to say nothing of the connection between mathematics and art). It is not so easy to determine these boundaries. But, this is just a construct, for even though this picture ought to be thought of as existing (as his proof shows), it remains that we cannot recede and see the big picture mathematics creates.

 

The connection between the artist and his art, thus remains an open question. We can have prejudices about how the two are connected, but never an objective reason.

[[al...]]

Chaitin and Randomness once again, and Art

 

I believe fiction (and poetry) is a study of patterns, like mathematics, in a way that is so abstract that it transcends the highest of mathematics. I believe it transcends because in order to study, store and recall the images, or patterns, that we create in these arts, our brain requires unimaginable mathematical computation -- one of a scale that mathematics isn't yet sure of how to perform. This is just my personal opinion.

 

What Chaitin says is that were it possible for us to distance ourselves from mathematics, and observe the big picture painted by the whole of mathematics, we would see that the picture is neither linear nor homogeneous (Chaitin does this using a cantor set and a prefix free universal Turing machine). The painting would appear to be totally chaotic with no logic whatsoever. This is surprising, for, of all sciences, we had expected this the least from mathematics. So mathematics isn't as logical as we think. This painting would be rubbish actually, were it not for the fact that at the bottom of the picture is scribbled the name of the painter -- Man. What was man thinking, it makes you wonder, when he painted this -- and suddenly the painting becomes invaluable.

 

So the connection between the author and his art is not always obvious (to say nothing of the connection between mathematics and art). It is not so easy to determine these boundaries. But, this is just a construct, for even though this picture ought to be thought of as existing (as his proof shows), it remains that we cannot recede and see the big picture mathematics creates.

 

The connection between the artist and his art, thus remains an open question. We can have prejudices about how the two are connected, but never an objective reason.

 

If reality can be said to be relative to the observing consciousness (and I believe that it can because a garden slug lives in the same, yet totally different world than the gardener who plucks it from a plant), then the expression of imagination that we call art is relative to artist who creates it as well as the consumer of the art. It cannot exist on its own. One also has to wonder to what extent does mathematics really exist without a consciousness seeking to bring order to a universe that might very well not need it.