The great-grandmother of airports, Heathrow was closed yesterday. There will be tremendous chaos, but not of the type that characterizes war, or a natural disaster such as a large earthquake. Somewhat disturbingly, some part of me is actually enjoying seeing how the part of the world that I detest grinds to a halt.
I am of course speaking about the make-believe world where well-heeled executives jet around busily, shiny briefcase in hand, converging at precise locations and timings to discuss how they're going to screw the rest of us over again. I enjoy seeing how a few inches of snow manages to stop our great, big capitalist machine in its tracks, and how we will all actually survive this catastrophe. Even better is the irony that the man stranded at the airport might, just a few days ago, have been trying to convince everyone of his superior abilities to foresee the future.
Meanwhile, business is as usual in the rest of the universe, proving that 1000 sweatshop workers and 100 paper-pushers didn't really have to pay for their CEO's Christmas vacation to London. Of course, when the snow melts, we shall all scurry back to our designated positions, work doubly hard to clear the backlog, just in time to ensure that all the cogs in the machine will be turning nicely for my flight from Gatwick on New Year's Day.
Sunday, December 19, 2010
Tuesday, December 14, 2010
Hey mum, erm... I'm studying to become a physicist, not a physician
Apart from the usual incredible memories I have from my trip to Iceland, which you can find pictures of on some other people's blog/Facebook page, I have also brought back two other things. The first is URTI, which I think is medical jargon for "the unknown infection that's stuffing up your nose". A few panadols later, I am up-and-running, ready to serve out some Icelandic humour (that's the second thing I brought back).
Unfortunately for my mum, I am not training to become a physician. I trust that she has figured that out over the years. As I have myself made a host of English language gaffes, I am not entitled to berate her for the unintentional mix up with "physicist". However, and this is no by-product of mathematical snobbery, the following accidental rediscovery of the trapezium rule cannot be forgiven:
I actually found this via some other blog, which had scathing comments about plagarizing integration from Newton. Actually, the blogger was probably being kind. Newton's invention of the calculus certainly required quite a profound insight. Independent discovery of integration, even if it took place centuries after Newton's, might arguably require some cognition. The definite integral roughly refers to the area under a curve, and through the Fundamental Theorem of Calculus, one can relate this geometrical quantity to the slope of the curve, i.e, differentiation That is at least Newton's insight. As for the trapezium rule (or in modern terminology, Tai's Model), the insight is that the area under a curve lies between the following two quantities: (1) the area of the trapeziums that you can squeeze under the curve, but almost couldn't, and (2) the area of the trapeziums that you can't but almost could. Most people will probably have played with one of those toys with holes of different shapes and corresponding pegs for those holes. A pleasant lesson from this toy is that you cannot squeeze a square peg into a round hole, and then you develop the commonsense intuition of area (which, incidently (or incidentally?), may not be quite right since it leads to the Banach-Tarski paradox). Equipped with this, you might discover, as Dr. Tai did, that you can approximate an area by breaking it up into little rectangles and triangles, and then summing up their individual areas. It might be advisable to check if your surgeon is familiar with Tai's Model before your operation.
At the other end of the spectrum, some people in the medical profession do have a wonderful sense of humour:
Unfortunately for my mum, I am not training to become a physician. I trust that she has figured that out over the years. As I have myself made a host of English language gaffes, I am not entitled to berate her for the unintentional mix up with "physicist". However, and this is no by-product of mathematical snobbery, the following accidental rediscovery of the trapezium rule cannot be forgiven:
For the record, Dr Tai's mathematical discovery was published in a top medical journal, peer-reviewed by a number of world-leading experts, and, gasp..., cited over 100 times by other researchers. If you have bothered to sift through the abstract, you might have suspected that the confidently-named "Tai's Model" is really the "trapezium rule", which is something we have all learnt in secondary school when the teachers were preparing us for calculus.A mathematical model for the determination of total area under glucose tolerance and other metabolic curves.
Abstract
OBJECTIVE--To develop a mathematical model for the determination of total areas under curves from various metabolic studies. RESEARCH DESIGN AND METHODS--In Tai's Model, the total area under a curve is computed by dividing the area under the curve between two designated values on the X-axis (abscissas) into small segments (rectangles and triangles) whose areas can be accurately calculated from their respective geometrical formulas. The total sum of these individual areas thus represents the total area under the curve. Validity of the model is established by comparing total areas obtained from this model to these same areas obtained from graphic method (less than +/- 0.4%). Other formulas widely applied by researchers under- or overestimated total area under a metabolic curve by a great margin. RESULTS--Tai's model proves to be able to 1) determine total area under a curve with precision; 2) calculate area with varied shapes that may or may not intercept on one or both X/Y axes; 3) estimate total area under a curve plotted against varied time intervals (abscissas), whereas other formulas only allow the same time interval; and 4) compare total areas of metabolic curves produced by different studies. CONCLUSIONS--The Tai model allows flexibility in experimental conditions, which means, in the case of the glucose-response curve, samples can be taken with differing time intervals and total area under the curve can still be determined with precision.
I actually found this via some other blog, which had scathing comments about plagarizing integration from Newton. Actually, the blogger was probably being kind. Newton's invention of the calculus certainly required quite a profound insight. Independent discovery of integration, even if it took place centuries after Newton's, might arguably require some cognition. The definite integral roughly refers to the area under a curve, and through the Fundamental Theorem of Calculus, one can relate this geometrical quantity to the slope of the curve, i.e, differentiation That is at least Newton's insight. As for the trapezium rule (or in modern terminology, Tai's Model), the insight is that the area under a curve lies between the following two quantities: (1) the area of the trapeziums that you can squeeze under the curve, but almost couldn't, and (2) the area of the trapeziums that you can't but almost could. Most people will probably have played with one of those toys with holes of different shapes and corresponding pegs for those holes. A pleasant lesson from this toy is that you cannot squeeze a square peg into a round hole, and then you develop the commonsense intuition of area (which, incidently (or incidentally?), may not be quite right since it leads to the Banach-Tarski paradox). Equipped with this, you might discover, as Dr. Tai did, that you can approximate an area by breaking it up into little rectangles and triangles, and then summing up their individual areas. It might be advisable to check if your surgeon is familiar with Tai's Model before your operation.
At the other end of the spectrum, some people in the medical profession do have a wonderful sense of humour:
Friday, December 3, 2010
Inappropriateness of the concept of a price
Over a coffee today, I made a bold statement about economics, or more specifically, the fatally-flawed concept of pricing.
I claimed that using numbers (i.e. dollars and cents) to "measure" the value of goods is fundamentally wrong. I have always believed this statement to be obvious, although I have had a hard time convincing people. Often those with the economics education that I lack will introduce complicated words like "utility", "fluidity", "non-linearity", etc. But let's not get bogged down by these distractions. I have a simple situation: In front of me is a cup of coffee, a cup of tea, and a cake. I prefer the coffee to the tea, the tea to the cake, and the cake to the coffee. Well, then, there is no consistent way of pricing these three objects with numbers! Apart, of course, from giving all of them the same price.
A common way out of this conundrum is to say that the notion of value is subjective. Fine. But I think there is an objective way to value things. Just not with numbers. Here's what I think is wrong with using numbers in pricing objects. Suppose you claim that the cup of coffee is worth 2 dollars. Do you know what you have just done? Based on the premise that there is a meaningful numerical price to any valuable object, you have just simultaneously compared the value of the cup of coffee to everything of value in the universe! You would have made, implicitly, the claim that the cup of coffee is more valuable than anything, anywhere in the world, that you will possibly place a $1 price tag on. Now, is that cup of coffee still worth $2?
The problem, I feel, is not that numbers are inadequate for use in pricing, but that numbers are overly adequate for that purpose. For numbers are not as innocent as they seem. They are extremely well-behaved. They have the property that if Number A < Number B, and Number B < Number C, then necessarily Number A < Number C. For all choices of numbers. Now, at some point in our lives, we must all have faced the paradoxical situation in which you prefer product A to B, B to C, and C to A. For instance, suppose I have a choice of going out on a date with Amy, Betty, and Cindy. I prefer Betty's smile to Amy's. But I like Cindy's legs even more. Yet Amy has this mysterious aura about her that draws my attention. How can I attach numbers/price tags on the value of these three dates? Well, the point is, you simply can't! Unless you have a preference that goes like A<B<C, there is just no way to use numbers consistently in a A<B<C<A situation.
Furthermore, it is completely natural not to have a "most-preferred date". But this is precisely what you have to commit to when you put price tags on, for among any three numbers, there is surely a biggest number! In this sense, the ordering that humans attribute to valuable objects, simply does not have the requisite properties of numbers! Why then, insist on using dollars and cents to label objects? Incidently, in the more primitive system of barter trade, such inconsistencies never arise, because one only ever compared the relative value of goods, and never made the leap of faith of operating on a dollar-and-cents system.
Now, the economists-armed-with-calculus (I'm trademarking this phrase) will come at my throat with all sorts of jargon to explain away this inconsistency. In retaliation, I could present some nasty algebra to expose the inherent divergence in the properties of valuation, and the properties of numbers. I find it a burden to try to look for the correct "price" for everything, and have it all work out consistently. Why not just concede that numbers are not the appropriate mathematical objects to use?
This is not to say that pricing is not useful. It certainly saves us some trouble, especially when we never had any preferences to begin with. It also allows us to feel a sense of satisfaction when we have more of these dollars than our neighbours. But to take the bold move of developing an entire theory based on dollars-and-sense might just be misguided.
*At least two possibilities can arise from my little rant. Quite likely, I am naively commenting on a field that I have little expertise in. Or, maybe, I have made a truly profound observation, for which I stake a proprietary claim to. In any case, this is just my 2 cents worth.
Wednesday, November 24, 2010
Denmark vs Germany
"The Talmudic philosopher doesn't give a hoot for 'reality'." - Einstein on Bohr
"Stop telling God what to do" - Bohr, after Einstein's proclamation that God does not play dice.
I have recently been reading more closely about the famous debate between Neils Bohr and Einstein regarding quantum mechanics, which actually plays out like a drama. I only wished I could comprehend their thoughts and the German. It really is remarkable that 80 or so years later, the same sort of arguments reappear in various guises. But what is a physicist doing, speaking about philosophy with all its associated "-isms"?
John Wheeler is on (video) record, saying that "philosophy is too important to be left to the philosophers". Stephen Hawking claims in the first line of his latest book that "Philosophy is dead". It might be that modern physics is modern philosophy, much like physicists used to be considered "natural philosophers". In any case, anyone who looks closely at the foundations of any modern physical theory (in particular, quantum mechanics) is bound to hit some fundamental questions about the ontological or epistemological aspects of reality.
The fascinating thing about Einstein is not so much in his genius as evident from his work when he was alive, as it is in the remnants of his genius left over long after his death. For instance, it seemed for the longest time that Bohr had famously defeated Einstein, so that orthodox quantum mechanics dominated up till the 1980s. This prevailing interpretation of quantum mechanics takes the name "Copenhagen interpretation", to rub salt into the great German's wounds. Yet, in that duration of time, no completely satisfactory account of the meaning of quantum mechanics had been settled on. Today, Einstein's EPR paradox is the symbol of all that feels wrong with quantum mechanics, namely, the high-handed way it deals with objective reality. I must also state that no completely conclusive experiment has ruled out Einsteins's doctrine of Trennungsprinzip, which is the idea that separability constitutes objectivity. The notion of entanglement denies this sort of local realism, and is often regarded as the characteristic feature of quantum theory. Today, it seems that quantum mechanics is in good shape, with all its spectacular successes. In truth, with each success, the problem of what it all means only gets deeper, but of course, no quantum physicist would openly admit that the enterprise from which he makes his living is built on shaky grounds.
The other example of Einstein coming back from the dead is the revival of his proposed cosmological constant - once the "biggest blunder in his life". It almost seems, befittingly, that he had figured out the intricacies of time-travel, and is regularly making visits from his grave to tease the rest of us. So how was it that he managed to think so deeply? I stole the excerpt below:
Well, you can't blame physicists for being weird.
"Stop telling God what to do" - Bohr, after Einstein's proclamation that God does not play dice.
I have recently been reading more closely about the famous debate between Neils Bohr and Einstein regarding quantum mechanics, which actually plays out like a drama. I only wished I could comprehend their thoughts and the German. It really is remarkable that 80 or so years later, the same sort of arguments reappear in various guises. But what is a physicist doing, speaking about philosophy with all its associated "-isms"?
John Wheeler is on (video) record, saying that "philosophy is too important to be left to the philosophers". Stephen Hawking claims in the first line of his latest book that "Philosophy is dead". It might be that modern physics is modern philosophy, much like physicists used to be considered "natural philosophers". In any case, anyone who looks closely at the foundations of any modern physical theory (in particular, quantum mechanics) is bound to hit some fundamental questions about the ontological or epistemological aspects of reality.
The fascinating thing about Einstein is not so much in his genius as evident from his work when he was alive, as it is in the remnants of his genius left over long after his death. For instance, it seemed for the longest time that Bohr had famously defeated Einstein, so that orthodox quantum mechanics dominated up till the 1980s. This prevailing interpretation of quantum mechanics takes the name "Copenhagen interpretation", to rub salt into the great German's wounds. Yet, in that duration of time, no completely satisfactory account of the meaning of quantum mechanics had been settled on. Today, Einstein's EPR paradox is the symbol of all that feels wrong with quantum mechanics, namely, the high-handed way it deals with objective reality. I must also state that no completely conclusive experiment has ruled out Einsteins's doctrine of Trennungsprinzip, which is the idea that separability constitutes objectivity. The notion of entanglement denies this sort of local realism, and is often regarded as the characteristic feature of quantum theory. Today, it seems that quantum mechanics is in good shape, with all its spectacular successes. In truth, with each success, the problem of what it all means only gets deeper, but of course, no quantum physicist would openly admit that the enterprise from which he makes his living is built on shaky grounds.
The other example of Einstein coming back from the dead is the revival of his proposed cosmological constant - once the "biggest blunder in his life". It almost seems, befittingly, that he had figured out the intricacies of time-travel, and is regularly making visits from his grave to tease the rest of us. So how was it that he managed to think so deeply? I stole the excerpt below:
It seems to have been an emotional need for Einstein to detach himself from his fellow humans in order to devote himself to the study of the Cosmos. For example, Einstein’s former associate Adriaan Fokker wrote in his highly perceptive obituary of Einstein: ‘His true passion was to penetrate the riddle of the immeasurable cosmos, which stood high above the muddle and the confusion of personal interests, feelings and low impulses of men. Such thought comforted him when he had seen through the hypocrisy of the common ideals of decency. The consideration of this external reality lured him as a liberation from an earthly prison.’. Einstein made a similar point himself: ‘I mercifully belong to those people who are granted as well as able to dedicate their best efforts to the consideration and the research of objective, time-independent matters. How fortunate I am that this mercy, which makes one quite independent of personal fate and of the behaviour of one’s fellow humans, has befallen me.’
Well, you can't blame physicists for being weird.
Monday, November 15, 2010
A generous Aussie
What follows shall be a personal post. I am playing squash for my college, and today I was up against an Aussie from some other college. We engaged in the usual pre-match conversation, where I learned that he was a PhD student in law. After revealing that I was studying mathematics, he lit up and informed me that he had met the Fields medallist Cedric Villani just yesterday. It turned out that I had skipped his rare lecture to turn up in time for the game with him. Of course, I didn't tell him that. I also found out that he had done a mathematics major during his undergraduate days, so I was leaning towards categorizing him on the "cool" side.
We proceeded to the best-of-five-sets match. I would be no match for him in terms of power, so I played a patient retrieving game. The first set was very close, and I thought I had it, at 8-something up, but I made a few mistakes to lose the set. Well, it was still early, and I didn't commit too much during the set, so I was ready for at least three more sets. I took a comfortable lead in the second set to reach game point, but allowed him to claw back to 6-8. Then an incident occurred, which I believe led to his post-match comments, which I shall reveal later. I retrieved an impossible looking shot, which he reciprocated with a return which I suspected was a double-bounce. I didn't play the return, and held up two fingers, which is the gesture to indicate a double bounce. I believe he nodded and then proceeded to serve. This surprised me, because I had the impression that he agreed that his shot wasn't good, so I asked him for the ball. A discussion ensued, and not wanting to create a scene, I offered to replay the point. Apparently, he thought I had held up two fingers to indicate that my retrieval shot was a double-bounce. Admittedly, I was not completely clear with my gesture, but I was also certain that my shot was good. Ordinarily, the point would be replayed, but I let him have the point anyway, which brought him to 7-8. I kept my cool, and won the remaining points to take the second set.
During the break, I clarified the matter with him, but I could sense his unhappiness. Which was strange, since I had given him the point in question. Perhaps it was just in-match behavior. Anyway, we went to the third set, which was another very close set that I took. So I was up 2-1 in sets, and only required one more for the win. My plan for the fourth set was to go for winners to hopefully build an early lead. This went horribly wrong, and I was behind very quickly. I realized that there was no chance of claiming this set, so I conserved energy and tried to make him run as much as possible in the remaining points. I lost the fourth set 9-1. Ouch.
Therefore, it was down to the fifth set, which I think he was pretty confident of taking, having destroyed me in the earlier set. But this was precisely what I wanted. Well-rested after my break in the fourth set, I came out with all cylinders firing. One of my first shots was a lucky flick from the back corner off the side of my racket. He claimed that it was a bad shot because he heard a weird sound (which actually came from the the ball hitting side of my racket, rather than the tin at the front of the court). Not wanting to get into another debate, I conceded the point, bewildered. I later hit a very nice shot just above the tin, which he also claimed to be a bad shot. I gave him the point. In the subsequent points, I played very well, and retrieved many shots that he believed were winners, which must have been frustrating. Eventually, I took the final set and hence the match. The mandatory handshake was firm and felt friendly.
Five-setters are draining, and outside the court, I thanked him for the great match. His reply was something like "Yeah,..., I think I was kind of generous, you know...".
WHAT???
Then he continued, "I think you have a warped (not sure if this was the exact word) view of your double bounces,...some of your shots I thought were double-bounce, but you didn't call it".
I replied, "Is that the case? I really thought my shots were good. If I thought it was bad, I would have called it. I wouldn't want to win on dubious points." (Note: as with all recollections of conversations, the exact words were probably a little bit different.)
"... I don't know. Maybe I'm wrong. But I just have a different philosophy when playing squash. It's just a college match. I prefer to err on the side of my opponent. Some of your shots, I was sure they were double-bounce, but you didn't call it..."
Now, I was actually quite enraged at the accusation, but I understood that he might not have been feeling so good after losing a 5-setter. (The last time I played a 5-setter, I lost, and that was painful. It still haunts me a bit today. But that was IHG, which is a far more emotional affair.) I let him cool down and pack his stuff, while I sat down to try to recall the basis of his claims. Perhaps I was really poor in my judgements. So I offered him an apology, "Sorry if I had made any mistakes just now. I wouldn't play on intentionally if I knew I hit a bad shot....So we're cool?"
He didn't reply and left.
That was really sucky. I felt completely maligned, yet unable to make sense of his accusations. I sat down for a good ten minutes to try to recall which points had been dubious, and it seemed that in almost all of them, I had given him the benefit of the doubt. Then, again, maybe I overestimated myself? But I had never before been accused of playing unfairly. Besides, I have always played a retriving style, which can frustrate opponents who think that they've got a point won. But he doesn't look like the type who would accuse his opponents of poor sportsmanship. I even resorted to "all's fair in the game" to try to feel better in case I was really playing unsportingly, which I am certain I wasn't. Maybe it's quite annoying to lose to a skinny Asian? Finally, a realization dawned on me.
He's a lawyer.
We proceeded to the best-of-five-sets match. I would be no match for him in terms of power, so I played a patient retrieving game. The first set was very close, and I thought I had it, at 8-something up, but I made a few mistakes to lose the set. Well, it was still early, and I didn't commit too much during the set, so I was ready for at least three more sets. I took a comfortable lead in the second set to reach game point, but allowed him to claw back to 6-8. Then an incident occurred, which I believe led to his post-match comments, which I shall reveal later. I retrieved an impossible looking shot, which he reciprocated with a return which I suspected was a double-bounce. I didn't play the return, and held up two fingers, which is the gesture to indicate a double bounce. I believe he nodded and then proceeded to serve. This surprised me, because I had the impression that he agreed that his shot wasn't good, so I asked him for the ball. A discussion ensued, and not wanting to create a scene, I offered to replay the point. Apparently, he thought I had held up two fingers to indicate that my retrieval shot was a double-bounce. Admittedly, I was not completely clear with my gesture, but I was also certain that my shot was good. Ordinarily, the point would be replayed, but I let him have the point anyway, which brought him to 7-8. I kept my cool, and won the remaining points to take the second set.
During the break, I clarified the matter with him, but I could sense his unhappiness. Which was strange, since I had given him the point in question. Perhaps it was just in-match behavior. Anyway, we went to the third set, which was another very close set that I took. So I was up 2-1 in sets, and only required one more for the win. My plan for the fourth set was to go for winners to hopefully build an early lead. This went horribly wrong, and I was behind very quickly. I realized that there was no chance of claiming this set, so I conserved energy and tried to make him run as much as possible in the remaining points. I lost the fourth set 9-1. Ouch.
Therefore, it was down to the fifth set, which I think he was pretty confident of taking, having destroyed me in the earlier set. But this was precisely what I wanted. Well-rested after my break in the fourth set, I came out with all cylinders firing. One of my first shots was a lucky flick from the back corner off the side of my racket. He claimed that it was a bad shot because he heard a weird sound (which actually came from the the ball hitting side of my racket, rather than the tin at the front of the court). Not wanting to get into another debate, I conceded the point, bewildered. I later hit a very nice shot just above the tin, which he also claimed to be a bad shot. I gave him the point. In the subsequent points, I played very well, and retrieved many shots that he believed were winners, which must have been frustrating. Eventually, I took the final set and hence the match. The mandatory handshake was firm and felt friendly.
Five-setters are draining, and outside the court, I thanked him for the great match. His reply was something like "Yeah,..., I think I was kind of generous, you know...".
WHAT???
Then he continued, "I think you have a warped (not sure if this was the exact word) view of your double bounces,...some of your shots I thought were double-bounce, but you didn't call it".
I replied, "Is that the case? I really thought my shots were good. If I thought it was bad, I would have called it. I wouldn't want to win on dubious points." (Note: as with all recollections of conversations, the exact words were probably a little bit different.)
"... I don't know. Maybe I'm wrong. But I just have a different philosophy when playing squash. It's just a college match. I prefer to err on the side of my opponent. Some of your shots, I was sure they were double-bounce, but you didn't call it..."
Now, I was actually quite enraged at the accusation, but I understood that he might not have been feeling so good after losing a 5-setter. (The last time I played a 5-setter, I lost, and that was painful. It still haunts me a bit today. But that was IHG, which is a far more emotional affair.) I let him cool down and pack his stuff, while I sat down to try to recall the basis of his claims. Perhaps I was really poor in my judgements. So I offered him an apology, "Sorry if I had made any mistakes just now. I wouldn't play on intentionally if I knew I hit a bad shot....So we're cool?"
He didn't reply and left.
That was really sucky. I felt completely maligned, yet unable to make sense of his accusations. I sat down for a good ten minutes to try to recall which points had been dubious, and it seemed that in almost all of them, I had given him the benefit of the doubt. Then, again, maybe I overestimated myself? But I had never before been accused of playing unfairly. Besides, I have always played a retriving style, which can frustrate opponents who think that they've got a point won. But he doesn't look like the type who would accuse his opponents of poor sportsmanship. I even resorted to "all's fair in the game" to try to feel better in case I was really playing unsportingly, which I am certain I wasn't. Maybe it's quite annoying to lose to a skinny Asian? Finally, a realization dawned on me.
He's a lawyer.
Tuesday, October 26, 2010
I am alive, 1 is not equal to 0 (sometimes), and the continuum hypothesis
Here is evidence that I have not been abducted by some aliens. While taking an afternoon nap just now, I had the most ridiculous dream about UFO's in the shape of skulls made of some beige-coloured material. These were really tiny UFO's. They could fit into your palm! And so I was swatting one particular UFO with my rusty squash strokes, and caught the cracked object with my hand. Then the skull spoke to me before it took it's last breath.... "take a look at what we are made of...". And I saw that it was made of silicone.
Haha. "You meant silicon right?"
Well, I don't know because I woke up immediately after. It's a little bit bothersome that I cannot rule out the possibility that I had been beamed up by these little critters made of the material that enters your favourite breast implants, with the intention that I spread the message that the future of civilization is in silicone-based lifeforms. Utterly bewildering.
That possibility aside, I should be in the version of the universe where I am freezing my butt off in my poorly-heated room, trying to revive a blog that has neither direction nor pictures of my house (those will come eventually, maybe in the summer). Well, suffice to say, I have been trying my darndest to wrap my head around these algebraic topology concepts that are truly quite intimidating. That should be enough evidence that all is well with me.
On to the mandatory geeky stuff. I often hear people describe mathematicians as people who spend their time "proving that 1 is equal to 0". This is completely false. Firstly, it is ludicrously simple to "prove that 1 is equal to 0". All you have to do is to define a set of objects containing "1" and "0" and declare them to be one and the same thing! Or, if you are not satisfied with this trivial manner of construction, you can also define, abstractly, a set endowed with certain properties that ultimately require "1=0" to hold. The simplest example would be the "zero ring".
The point is, in most algebraic structures of any interest to anyone operating in the real world, "1=0" does not hold. There is no point in trying to prove it, because it is false, although there is some romantic notion in proving absurdity. This is not simply because a vertical line looks different from an oval. The question is not whether the symbols "1" and "0" are equal, but whether the abstract object they represent are really one and the same thing (what the hell is "same-ness" anyway??). The arabic symbol "1", the chinese character "--" and the roman character "I" are all just things you write on the blackboard to refer to "unity", which is this object that stubbornly refuses to do anything to every other fellow object even if you leave them in a room overnight and force them to multiply. "Unity" does not belong to you or me, or even on the planet earth. It's obstinateness is much further-reaching. Skull-shaped UFO's share the same "unity" as us, for example, although they may have curly fingers which led them to represent "unity" with "S" instead.
Now, this is getting scary. It seems that the concept of "1" is so compelling as to permeate the entire existential universe. Just to be clear, the "1" and "0" that I am referring to are "integers", the kind of stuff we were all force-fed with when we were in primary school, and not things in the "zero-ring" (where they are really Jekyll and Hyde). Is that reasonable? Leopold Kronecker famously said:
"God made the integers, all else is the work of man."
Maybe you believe that 1,2,3.... are God-given. Fair enough, Kronecker agrees. Well, he probably would have said the same thing about numbers and fractions. How about the real numbers? Now, this is the bomb. Everyone who has ever used any calculus has implicitly declared that these creatures exist. But, wait. What is the problem? Can't you just invent anything in mathematics? Just like you pretended that integers existed, you can also pretend that real numbers exist!
Here's the problem. With integers, there was already the pesky question of "what is the largest integer?", which already leads us headlong into the realm of infinity. This sort of infinity is not so bad, since we can count our way there. But the real numbers are terrible. There are so many of them, that mathematicians had to invent a new sort of infinity to create them. A kind of monster that led us to also create calculus, Fourier analysis, and all sorts of associated appendages. All sorts of important mathematical tricks require us to be able to handle such humongous infinities, and you will have grounds to question whether we can handle that. In typical narcissistic style, we invoke the Axiom of Choice to endow us with that power. (So God gave us the integers, and free will to create and play with real numbers...).
It requires, quite literally, a much bigger leap of faith to embrace the real numbers, even though we use it routinely in our everyday earth-bound lives. The problem does not even end there. It turns out that we are not clever enough to create a set with more objects than the integers, but fewer than the real numbers. Which is a little silly don't you think? David Hilbert regarded this anomaly, commonly called the "continuum hypothesis", as such an important question that it is the first of his famous "Twenty-three problems". This problem has already been solved, in some sense. The answer is that, we can never know whether or not we are clever enough to create such an intermediate set; even invoking the Axiom of Choice does not help at all. Pretty smart eh?
Unfortunately, the alien died before I could ask him whether he used calculus. It appears that even in my wildest dreams, the answer to the above questions continues to elude me.
Haha. "You meant silicon right?"
Well, I don't know because I woke up immediately after. It's a little bit bothersome that I cannot rule out the possibility that I had been beamed up by these little critters made of the material that enters your favourite breast implants, with the intention that I spread the message that the future of civilization is in silicone-based lifeforms. Utterly bewildering.
That possibility aside, I should be in the version of the universe where I am freezing my butt off in my poorly-heated room, trying to revive a blog that has neither direction nor pictures of my house (those will come eventually, maybe in the summer). Well, suffice to say, I have been trying my darndest to wrap my head around these algebraic topology concepts that are truly quite intimidating. That should be enough evidence that all is well with me.
On to the mandatory geeky stuff. I often hear people describe mathematicians as people who spend their time "proving that 1 is equal to 0". This is completely false. Firstly, it is ludicrously simple to "prove that 1 is equal to 0". All you have to do is to define a set of objects containing "1" and "0" and declare them to be one and the same thing! Or, if you are not satisfied with this trivial manner of construction, you can also define, abstractly, a set endowed with certain properties that ultimately require "1=0" to hold. The simplest example would be the "zero ring".
The point is, in most algebraic structures of any interest to anyone operating in the real world, "1=0" does not hold. There is no point in trying to prove it, because it is false, although there is some romantic notion in proving absurdity. This is not simply because a vertical line looks different from an oval. The question is not whether the symbols "1" and "0" are equal, but whether the abstract object they represent are really one and the same thing (what the hell is "same-ness" anyway??). The arabic symbol "1", the chinese character "--" and the roman character "I" are all just things you write on the blackboard to refer to "unity", which is this object that stubbornly refuses to do anything to every other fellow object even if you leave them in a room overnight and force them to multiply. "Unity" does not belong to you or me, or even on the planet earth. It's obstinateness is much further-reaching. Skull-shaped UFO's share the same "unity" as us, for example, although they may have curly fingers which led them to represent "unity" with "S" instead.
Now, this is getting scary. It seems that the concept of "1" is so compelling as to permeate the entire existential universe. Just to be clear, the "1" and "0" that I am referring to are "integers", the kind of stuff we were all force-fed with when we were in primary school, and not things in the "zero-ring" (where they are really Jekyll and Hyde). Is that reasonable? Leopold Kronecker famously said:
"God made the integers, all else is the work of man."
Maybe you believe that 1,2,3.... are God-given. Fair enough, Kronecker agrees. Well, he probably would have said the same thing about numbers and fractions. How about the real numbers? Now, this is the bomb. Everyone who has ever used any calculus has implicitly declared that these creatures exist. But, wait. What is the problem? Can't you just invent anything in mathematics? Just like you pretended that integers existed, you can also pretend that real numbers exist!
Here's the problem. With integers, there was already the pesky question of "what is the largest integer?", which already leads us headlong into the realm of infinity. This sort of infinity is not so bad, since we can count our way there. But the real numbers are terrible. There are so many of them, that mathematicians had to invent a new sort of infinity to create them. A kind of monster that led us to also create calculus, Fourier analysis, and all sorts of associated appendages. All sorts of important mathematical tricks require us to be able to handle such humongous infinities, and you will have grounds to question whether we can handle that. In typical narcissistic style, we invoke the Axiom of Choice to endow us with that power. (So God gave us the integers, and free will to create and play with real numbers...).
It requires, quite literally, a much bigger leap of faith to embrace the real numbers, even though we use it routinely in our everyday earth-bound lives. The problem does not even end there. It turns out that we are not clever enough to create a set with more objects than the integers, but fewer than the real numbers. Which is a little silly don't you think? David Hilbert regarded this anomaly, commonly called the "continuum hypothesis", as such an important question that it is the first of his famous "Twenty-three problems". This problem has already been solved, in some sense. The answer is that, we can never know whether or not we are clever enough to create such an intermediate set; even invoking the Axiom of Choice does not help at all. Pretty smart eh?
Unfortunately, the alien died before I could ask him whether he used calculus. It appears that even in my wildest dreams, the answer to the above questions continues to elude me.
Tuesday, August 31, 2010
Physics envy
"Take a little bad psychology, add a dash of bad philosophy and ethics, and liberal quantities of bad logic, and any economist can prove that the demand curve for a commodity is negatively inclined." --Andrew Lo/Mark Meuller
I promised Thaddeus to write something about physics envy. The problem is, a simple Google search will lead to numerous good articles that are much more eloquent than any I could write, and probably better thought out as well. I myself am sitting on the fence with regards to this. While it is easy to bash pretentious economics papers replete in arcane mathematics, part of me suspects that the "mathematization" of the soft sciences may actually be fruitful eventually. We just need to recognize when not to take our mathematics too seriously.
Furthermore, I am also wondering why "physics envy" is not called "maths envy" instead. Perhaps physicists hide their intellectual insecurities rather better? But more pertinently, why does mathematics work so darned well in physical theories but not in economics or biology or sociology? Is the physicist who gorges on mathematical texts akin to the beefcake in the gym -- merely engaging in muscle-building of another kind?
Here's an excerpt from the famous paper "The Unreasonable Effectiveness of Mathematics in the Natural Sciences", by physicist/mathematician Eugene Wigner, which I highly recommend.
I promised Thaddeus to write something about physics envy. The problem is, a simple Google search will lead to numerous good articles that are much more eloquent than any I could write, and probably better thought out as well. I myself am sitting on the fence with regards to this. While it is easy to bash pretentious economics papers replete in arcane mathematics, part of me suspects that the "mathematization" of the soft sciences may actually be fruitful eventually. We just need to recognize when not to take our mathematics too seriously.
Furthermore, I am also wondering why "physics envy" is not called "maths envy" instead. Perhaps physicists hide their intellectual insecurities rather better? But more pertinently, why does mathematics work so darned well in physical theories but not in economics or biology or sociology? Is the physicist who gorges on mathematical texts akin to the beefcake in the gym -- merely engaging in muscle-building of another kind?
Here's an excerpt from the famous paper "The Unreasonable Effectiveness of Mathematics in the Natural Sciences", by physicist/mathematician Eugene Wigner, which I highly recommend.
- There is a story about two friends, who were classmates in high school, talking about their jobs. One of them became a statistician and was working on population trends. He showed a reprint to his former classmate. The reprint started, as usual, with the Gaussian distribution and the statistician explained to his former classmate the meaning of the symbols for the actual population, for the average population, and so on. His classmate was a bit incredulous and was not quite sure whether the statistician was pulling his leg. "How can you know that?" was his query. "And what is this symbol here?" "Oh," said the statistician, "this is pi." "What is that?" "The ratio of the circumference of the circle to its diameter." "Well, now you are pushing your joke too far," said the classmate, "surely the population has nothing to do with the circumference of the circle."
I also found this amusing drawing:
I have a mild mental block now, which suddenly brings me to a stark realization:
I have literary envy.
Friday, August 27, 2010
The tautology problem
A recent discussion about Ohm's Law led me to think a bit more deeply about the character of physical law. My mech. eng. friends may recall a few occasions when I pointed out that Newton's second law, F=ma, is merely a definition of what force is; likewise, I am inclined to point out to at least one electrical engineer who reads this blog that resistance is merely defined via Ohm's law. Before anyone starts accusing me of undermining the foundations of their profession, let me first state that I am a physicist, and these issues trouble me greatly -- if the Second Law is tautologically true by definition, then have we really been talking in circles for the last 300 years? But I am already digressing, because I actually want to discuss... "survival of the fittest".
First, the wikipedia article explaining the tautology problem:
- "Survival of the fittest" is sometimes claimed to be a tautology. The reasoning is that if one takes the term "fit" to mean "endowed with phenotypic characteristics which improve chances of survival and reproduction" (which is roughly how Spencer understood it), then "survival of the fittest" can simply be rewritten as "survival of those who are better equipped for surviving". Furthermore, the expression does become a tautology if one uses the most widely accepted definition of "fitness" in modern biology, namely reproductive success itself (rather than any set of characters conducive to this reproductive success). This reasoning is sometimes used to claim that Darwin's entire theory of evolution by natural selection is fundamentally tautological, and therefore devoid of any explanatory power.
Now, the evolutionists will go up in arms at this blasphemy. "How dare anyone contest the views of the great prophet Darwin? He must be from the Creationist side!" There are a lot of heated online discussions on this issue, and they are mostly quite amusing. The defence of Darwin usually goes something like this: "...survival of the fittest is not tautological because it is falsifiable... there are instances where the fittest do not survive...", or, "fittest is not merely defined as being better equipped for survival and reproduction...there are criteria for fitness that are independent of survival..." Now, these arguments look to me like they are desperately trying to salvage the phrase in question from tautological status. There are even arguments like "...every scientific law can be phrased as a tautology: Newton's law of gravitation can be rephrased as 'the ball is falling, therefore it falls'. Relativity can be phrased as 'light bends in a gravitational field, therefore, if I shine light into a gravitational field, it bends'..." There is of course, creative name-calling and insult hurling between intellectuals as well...
One type of tautology is a statement that has the form "if A, then A". For instance, "light bends in a gravitational field, therefore, if I shine light into a gravitational field, it bends" is such a tautology. However, it certainly does not follow that "A", the physical law under scrutiny, is tautological. One the other hand, if you define "fittest" as "best survivability/reproductivity", then "survival of the fittest" takes on the form "A=>A(<=>B)" which is tautological and also the content of the phrase itself. A possible escape route is to do a sneaky redefinition of "fittest". Under this new definition, one can accomodate "fit" creatures that wind up dead (these were not considered fit under the original definition), providing the desired counterexample. But this approach compromises the universal validity of "survival of the fittest", and besides, should one be satisfied with this sleight-of-hand? Putting it in another way, does the omniscient God Himself not know the set of factors that determine the likelihood of survival of any particular species? Does such an exhaustive list not exist in principle, albeit hidden from our finite-sized brains? Is the proposed redefinition, then, an admission of our ignorance rather than a logical necessity?
But, one protests, "Bull-crap! How can the concept of 'survival of the fittest' be tautological when it has so evidently taught us so much?" Contrary to what a defensive Darwinist might think, the tautological status of "survival of the fittest" does not render the entire theory of evolution meaningless. Let me elaborate on my point of view. Use of the word "fittest" implicitly suggests that one can compare, quantitatively, the survivability of a species versus another. Mathematically speaking, this means that there exists a map from the set of parameters describing the species and the environment in question to an ordered (or even partially ordered) set. That is, we can, provided we think hard enough, attribute (say) a number to each species, compare the numbers for the species under consideration, and then decide which of these is the fittest. In my opinion, this is the essence of Darwin's evolution -- that there actually exists a well-defined concept of fitness/survivability/whatever-you-wish-to-call-it that allows us, mere mortals, to meaningfully predict what we observe to be natural selection. Combined with the postulate of heredity, one then has a mechanism to explain the phenomenon of evolution. The existence of such a function is not a priori clear, which is why we have a theory of evolution. Indeed, the theory is falsifiable; there simply may not be such a function, for example, if there really is a Creator. The job of the scientist is then to search for the functional dependence of fitness/survivability on the parameters at hand, or at least, to suggest a plausible simplified approximation for such a function. And then we can start talking... :)
So, do not worry! The theory of evolution is still a legitimate and rich one, despite "survival of the fittest" being a tautology. Therefore, we should not hope to get "free" explanatory power from this popularized catch-phrase (e.g. the dinosaurs died out because the mammals were fitter than them / humans dominate the planet today because we are the fittest, etc.), but should instead seek a comprehensive characterization of fitness, and look actively for possible improvements in the details of the theory. Tautologies are commonplace in a good theory (a proof can be argued to be a convoluted tautology), but the real danger is using a tautological argument to explain something, or worse still, to justify certain positions (think social Darwinism).
By the way, I also think that F=ma suffers from the same problem, in that it is quite unsatisfactory to say that "the ball is accelerating because there is a force acting on it". But that is another story for another time, and suffice to say, classical mechanics does not reductio ad vacuo (bad grammar) even if you treat F=ma as a definition.
*I have no formal training in the theory of evolution beyond a secondary school level, and only a basic understanding of formal logic. But I understand that professional biologists and logicians do not necessarily speak in the same language.
Wednesday, August 25, 2010
The plunge
So I have opened a can of worms/Pandora's box/a blog. This was extremely difficult, mainly because of the burden of having to invent a clever blog title. Having recently proved that 0=0, I am not particularly confident of my inventiveness; furthermore, it is not clear what exactly this blog will be about, or what grand purpose it will serve. Maybe one day when I have finally attained some wisdom, this will evolve into an academic blog of some worth. Meanwhile, I will probably pen down trivial stuff, like my thoughts, new music that I discovered, my latest state of emotional distress, pictures of my lunch (of course not!), etc...
Now, what DO mathematicians do? Well, apart from sipping coffee, scratching their heads, fretting about cosmologically urgent problems nobody cares about, and joking about economists armed with calculus, they actually write blogs. See terrytao.wordpress.com for example. I'm not so sure about facebook, although Terence Tao does have a fan club there. So indeed, it is possible to be a Field's medallist, handsome, blog-writer, and have a wife and son. So far, I have taken the first plunge and started a blog, so let's see how deep this rabbit hole goes.
Now, what DO mathematicians do? Well, apart from sipping coffee, scratching their heads, fretting about cosmologically urgent problems nobody cares about, and joking about economists armed with calculus, they actually write blogs. See terrytao.wordpress.com for example. I'm not so sure about facebook, although Terence Tao does have a fan club there. So indeed, it is possible to be a Field's medallist, handsome, blog-writer, and have a wife and son. So far, I have taken the first plunge and started a blog, so let's see how deep this rabbit hole goes.
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