There remains a lot to be learned… 😉
There remains a lot to be learned… 😉
Besides physics and philosophy I invest quite a lot of time and energy in sports. Several years ago a friend of mine asked me to join him to go climbing in a gym. When he saw me wavering he spoke the magic words “perhaps it’s not such a good idea. Your disability will make it very difficult for you to get up there”. That’s when I was sure that I was going to go climbing. And so it started. Now, almost four and a half years later, I go climbing every week and I have made some nice rock climbing trips abroad. Continue reading
In sci-fi movies there is often talk of “going to another dimension” as if there is some kind of barrier in between dimensions that can be crossed only if the circumstances are very special – usually the filmmakers are wise enough not to specify what these very special circumstances are. I will show in this blogpost that “Travelling to another dimension” is not only physically impossible; the phrase makes no sense from a logical point of view either. Continue reading
I want to show in this post that the natural logarithm is not natural – it is not a characteristic of objective nature.
The natural logarithm pops up everywhere in science: biology, sociology, economics… and the list goes on. Every student of physics knows that many of the calculations in physics (for example in electrodynamics and quantum mechanics) would be undoable if it weren’t for the natural logarithm. The universal applicability of the natural logarithm suggests that it is something that exists in the world in which we live; that it is a characteristic of the physical world. We will see, however, that the naturalness of the natural logarithm (having base ) is a characteristic of the definition of derivative – and of any logarithm to which this definition can be applied.
The natural logarithm is a logarithm with as its base the mathematical constant . is a number such that the function for every has a value that is equal to its rate of growth . The existence of a number with that characteristic follows logically from our definition of derivative.
The derivative of a function is defined as follows:
In mathematics all exponential functions are of the form Therefore, if is an exponential function then the following holds:
We can move outside the limit so that the limit no longer depends on the variable (and it is therefore a constant in ):
Let’s take a closer look at the latter limit. In it there are the two functions and . We may calculate the value of the limit with the help of l’Hôpital’s rule: by determining the derivatives of and and computing . Determining is easy; . The value of is less straightforward, because it depends on the value of . If in has the value then (in the figure you can see that for and have both the same value and the same slope; and also have the same value). Returning with our findings about this limit to the equation for we see that the only exponential function for which is the exponential with base ; only if .
Consider again the general form of the exponential function, . There are two ways in which we can transform any function of the form into a natural exponential function.
Suppose for example that a biologist, let’s call her Fleur, studies a colony of bacteria in a Petri dish. Fleur finds that the number of bacteria in the Petri dish grows exponentially with a linear increase in time. Say she counts the time, , in seconds and formulates the function to describe the number of bacteria. If then n(t) is cumbersome to work with mathematically. If Fleur could rewrite in terms of a natural logarithm then any differential equation with in it would become much easier to solve.
Suppose Fleur indeed wants to rewrite in terms of a natural logarithm. She has measured the number of bacteria at and found it to be . It would be a bad idea for Fleur to use the second method (2) of naturalising because that would change and then would conflict with her initial measument. Fleur, being the clever biologist that she is, realises that and therefore opts for method (1). Suppose that she sees that doubles every 2.5 seconds. Then she might try substituting the variable for a variable for which each unit-step is not 1 second but 2.5 seconds (). The function can now be rewritten into a function which has a natural exponential form: . This will make Fleur’s life much easier.
Now we may ask the question whether the natural exponential growth-rate is a characteristic of the system that is studied or a characteristic of the way we describe that system. The facts that is a characteristic of the definition of derivative and that any exponential function can be written as a natural exponential might suggest that is a characteristic of our number system and not of nature. Such a conclusion, however, disregards a distinguishing feature of logarithmic functions.
The distinguishing feature of logarithmic functions is that their rate of growth at a certain moment scales with the value of the function at that moment. A natural logarithmic function is a special case of this in that the scaling is very simple: the rate of growth is at every moment exactly equal to the value of the function at that moment.
Any dataset that can be described in terms of exponentials can also be described in terms of natural exponentials (any exponential function can be transformed such that its base is ), but we have stumbled upon an objective fact about nature (as it comes to us in our observations) whenever we discover that some phenomenon allows a description in terms of exponentials at all.
The fact that we can describe nature in terms of natural logarithms is a consequence of the way we describe nature, but the fact that we can use logarithms at all tells us something about nature’s dynamics.
The part of Kant’s philosophy that I’ll be discussing in this post is Kant’s view on human knowledge. The central idea of Kant is rather straightforward: the world must be such that knowledge is possible. The world, according to Kant, must be structured in such a way that our knowledge of that world is possible. And since we have knowledge (otherwise we couldn’t live our lives) the world must have certain structural features that make this knowledge possible.
It sounds like Kant is stating the obvious here. We have knowledge of everyday objects; for example, we know that heavy objects (such as stones) fall. In Kant’s view it follows from this that it must be possible to think about heavy objects and about ‘falling’. Ok, so there must be the possibility of knowledge in order to have knowledge – sounds pretty simple, doesn’t it?
Things become complicated when Kant tries to specify what it means to be able to think about objects and processes. Thinking about a falling stone is impossible, Kant believed, without thinking about this process in terms of causal laws. Kant’s argument therefore boils down to the following: causal laws are inherent in knowledge; we have knowledge, so causal laws must exist.
When Kant wrote about these matters (in the 18th century) he had in mind Newton’s physics as a model for human knowledge. The possibility of knowledge therefore went hand in hand with the possibility of Newton’s physics. Causal laws are essential elements in Newton’s physics (manifesting themselves as the conservation of momentum) so it shouldn’t surprise us that Kant identified causal laws as essential elements in knowledge.
Modern science presents numerous challenges to Kant’s and Newton’s view on causal laws. In the theory of special relativity the concept of time loses its absolute character and it therefore becomes impossible to say which events precede which events. What does it mean to say that some event causes another if there is no ‘earlier-than’ relation? Causes in quantum theory are even more problematic: in quantum theory there are events which have no cause in the Newtonian sense at all.
These scientific developments suggest that Kant was mistaken when he said that causal laws necessarily exist. Modern science shows that knowledge (in the form of scientific theories) is still possible without causal laws. And yet Kant’s central idea – that the world must be such that knowledge is possible – is still a truism. It’s just that we’ve come to realise that our knowledge does not coincide with knowledge in Newton’s physics. How should we make sense of Kant’s idea in the face of modern physics? One option is to replace Newton’s causal relations between events with relations of probability. That works as follows.
Suppose a scientist repeats a measurement a number of times. For example, she repeatedly measures the length of a stick. No matter how accurate the measurements are, they never yield exactly the same results because there are always infinitely many possible disturbing influences (the wind, a varying temperature of the stick, ambient air temperature, quaking due to seismic activity; you name it). If the results of the measurements were put in a graph then we would not find a straight line (representing equal measurement-results), but a normal distribution (a Gaussian, or bell-shaped, curve; see graph). Someone with a strict belief that everything in nature follows the same causal laws may believe that different measurement-outcomes are due to measurement disturbances, but she may as well believe that the different measurement-outcomes refer to differently sized sticks (compare a sociologist who is not sure whether the variations in her data are due to variations pertaining to one individual or to variations within a population of individuals).
But wait! Such a normal distribution is actually an approximation of what is measured. The real results form a dotted line which (if the scientist has done a proper job) only roughly follows a smooth, normal distribution. Before we can use experimental results to construct a mathematical model of the experiment we must assume that the experimental results can be extrapolated to a smooth curve. That’s the first assumption we need to make. But that is not enough. Suppose our scientist is asked by a fellow scientist what is the length of the stick. It would not be satisfactory if our scientist could only point at the normal distribution and say “the actual length of the stick is somewhere in that graph”. No, what the scientist must also assume is that the average value of the normal distribution corresponds to the length of the actual physical stick. To be able to construct useful mathematical models using experimental results we must make at least these two assumptions: 1) experimental data can be extrapolated to a smooth, normal distribution, and 2) the average value of this distribution corresponds to something real.
Back to Kant and to his possibility of knowledge. After our analysis of the role of probability in science, we might now determine how assumptions about probability theory make knowledge possible. The philosopher/physicist Hans Reichenbach (early 20th century) diagnosed a deficit in Kant’s philosophy: Kant’s idea is relevant not for knowledge per se, but for scientific knowledge. The above considerations show that the possibility of scientific knowledge requires that the scientist makes certain assumptions about the theory of probability. Probability theory serves to flesh out the import of Kant’s philosophy for modern science.
The probability that a dart will hit any specific point of a dartboard is zero because there are infinitely many points on the board. And yet if you throw a dart at a dartboard you’ll always hit some point (assuming you hit the dartboard). Hitting a specific point at a dartboard is highly improbable, but not impossible.
The responses to a similar statement in a previous post were mixed. Some people were delighted by its counterintuitiveness, whereas others were skeptical – what if we assume that the tip of the dart has a size that is not a mathematical point? What if it were, say, 0.1 square mm? That wouldn’t change the probability of hitting any one point, because there would still be an infinite number of points on the board. True, the probability of hitting any one of those points would increase by a certain factor, but no matter how large this factor is, the probability of is still – zero-probability.
What we can do is divide the area of the dartboard into a (finite) number of smaller areas and assume that if two of such areas are equal in size, then the probability of hitting them is also equal. There are two fundamental problems with such an approach:
To grasp the second of these problems it might be helpful to realise that this problem occurs whenever we try to subdivide an infinite set. Just think of the set of positive integers (1, 2, 3, 4, 5…etc.) and try to subdivide it into equal subsets. These equal subsets may have any size we choose. For example, we can choose subsets each of which contains two elements, like this: (1, 2), (3, 4), (5, 6), (7, 8)…(etc.). But we might also opt for the smaller subsets (1), (2), (3), (4)…(etc.) or the larger subsets (1, 2, 3), (4, 5, 6), (7, 8, 9), (10, 11, 12)…(etc.). There are infinitely many choices, because we’ll never run out of integers! The same argument goes for areas (just assign an integer to every point) or any other infinite set.
Both of the questions facing the probability theorist can only be answered by adopting a suitable convention. The scientist must assume that equal areas have equal probability of being hit and she must assume that there is a preferred way to divide up a continuous area that is shared by other scientists. This shows that statements of probability do not have universal validity. The probabilistic problem of uniquely dividing up infinite sets has become known as Bertrand’s paradox.
If probability is in its core subjective, as we’ve seen, then why does science seem to possess an objective quality? One may suspect, perhaps, that the crux lies in finding the right convention. But that is not the case, since different conventions are logically equivalent; none of them is objectively better than the others. All we can do is what scientists have done ever since the dawn of science: find out by trial and error what are the most useful conventions in different situations.
For a thrower of darts the situation is simple because the conventions have already been decided upon. All players know beforehand that equal areas have equal probabilities (without being aware that this is a convention). The other convention, about the subdivision of the surface of the dartboard, has already been decided upon by the manufacturer of the dartboard; all players tacitly agree upon this conventional subdivision of the surface of the dartboard.
For the scientist – perhaps working in the field of particle physics or in cosmology – it is not obvious which conventions are useful: perhaps there is a particle for which equal size does not imply equal probability. Or perhaps two cosmologists from opposite points on the earth’s surface study some galaxy without knowing the angle from which they observe the galaxy. These cosmologists will disagree on which areas on the distant galaxy are equal (and hence they might disagree on the exact source or intensity of incoming radiation).
Science is, in the end, a capricious affair.
When I had exchanged email addresses with the Japanese girl and we had said goodbye to each other, our wheelchairs were pushed off by one of the airport’s employees. With great dexterity he steered both wheelchairs at the same time to our next stop: customs. The customs check is one of the most unpredictable parts of a journey in a wheelchair. Sometimes the check is very thorough: the douaniers check every inch of the wheelchair; frisk me while I remain seated and even swab the wheelchair’s tires for explosive residue. At other times merely seeing the wheelchair is enough to just let me pass and wishing me a pleasant journey. The only constancy that I can detect is that never once have they checked the tubes in the frame of my wheelchair – I wonder what could fit in there.
When we had passed customs the wheelchair pusher dropped me off before the gate from which my plane was to leave. I was lucky, I thought, because the gate was straightly opposite from a coffee bar, so I wouldn’t have to walk very far for my ‘daily worship of the black gold’. Neither were the toilets very far from my gate. I sat down and made myself comfortable. Out of my bag I took a sandwich and the book I wanted to read. To get into the spirit of the conference I had chosen a German book on Kantianism. At the conference I was going to give a talk on something I’ve been working on the past few years. I’ve been working on the role of Kant’s philosophy in modern philosophy of physics. Many physicists see little value in philosophical systems, no matter how well thought-out, of over two centuries old. At the other extreme there are those who believe that modern physics, and particularly quantum mechanics, present philosophers and physicists alike with problems that can only be resolved within a Kantian approach.
Looking up from my book, rather sleepily, I noticed on the view screen that the regular boarding was to be preceded by what they call ‘priority boarding’. People with babies or other disabilities or people who are willing to pay for priority boarding are allowed to board the airplane before the horde of regular passengers. Since I fall in the category of people with disabilities I’m allowed to make use of priority boarding.
By the time the actual boarding of the airplane begins I’m not in a wheelchair anymore, and as long as I’m not walking I don’t really look disabled so I always try to make sure that the people behind the boarding-counter see me walk up to them so that they’ll allow me to ‘board with priority’.
When I walked up to the counter to tell the lady behind it that I wanted to make use of priority boarding she had been very busy with a conversation up until that moment and hence had not seen me walking up to her desk. So when I asked her whether it be possible to make use of priority boarding I could hear her starting a sentence “but why do you need…” As I was quickly trying to think of a way to convince her of my disability (should I show her the scar on the back of my head, which was due to the latest brain surgery I’d had?) I almost fell over backwards. When I had regained my balance the lady behind the counter was a lot more willing to accept that I belong in the disability priority class.
The stewardess behind the counter, made anxious by her experience, now wanted me to board the plane with extra priority – even over the other priority passengers (she was probably afraid that I would fall). Once in the plane I could relax: “if anything goes wrong now it’s not my fault” I thought. I always like flying because on a flight you can read or work without being disturbed. But not only that. Not only without being disturbed but also without the possibility of distracting yourself with Google or Facebook or what-have-you: you can sort of force yourself to do the work that you have taken with you. For most people this strategy will not work because it will only make them stare out of the window of the airplane. For me the situation is somewhat different because I have so much double vision (because of the spasticity of the muscles moving my left eye) that staring out the window while actually seeing things requires a lot more effort than reading. For me the strategy works perfectly: often I look forward to a flight for weeks because I have already decided upon what to read.
They sometimes say that if you try something very often then you will succeed. Say you play a game of dice and in every round there is a non-zero probability that you’ll win. As the number of rounds you play becomes larger and larger then you must, eventually, win – right?
Wrong. What you can say with certainty is that the probability that you will keep losing as you play many rounds becomes very small. That means that the probability that you will win at least once becomes very large. In the limit of the number of rounds going to infinity the probability of winning actually equals one. But that does not mean that winning is a certainty. It is possible, even after many rounds, that you’ll never win. It may be infinitely improbable, but it is not impossible.
Saying that something has a zero-probability of occurring is not the same as saying that it is impossible. Just think of throwing a dart at a dartboard. The probability of the dart hitting any specific point is zero (as there are infinitely many points on the board), and yet if you throw a dart at the dartboard you’ll always hit some point. Hitting a specific point at a dartboard is highly improbable, but not impossible.
What happens if you repeat a process that can have different outcomes (such as throwing a die) very often is described by Bernoulli’s Law of Large Numbers (LLN). There are actually two different laws that bear the name LNN: the strong and the weak LLN. In a future post I will explain the difference between the strong and weak versions of the LLN and also show how these laws are often misinterpreted in modern physics.
At international conferences the university or institute hosting the conference often provides a list of hotels which are a relatively small distance from the conference site. The hotels on this list are usually neither the cheapest nor the cosiest, but rather well-suited for grave and grey professors. Since I am neither grave nor grey I often search for a (youth-)hostel myself. I can’t always find hostels that are close to the conference, but the nice thing about hostels is that they force you to have some social interaction. When I had found a suitable hostel near to the University of Vienna (which was the place the conference was held) it was time to book a flight.
In daily life, as I told you earlier, I make use of a tricycle. I even enter supermarkets riding on it. The obvious alternative, a wheelchair, is impractical for me because due to my spasticity I can’t use my left arm to spin the wheels of a wheelchair. Travelling by plane pretty much rules out the tricycle as possible mode of transport when I’m abroad, because it costs a small fortune to take the trike with me. A wheelchair it is then. Because I have ‘special needs’ – such as the wheelchair – I often opt for one of the larger airlines to fly with, because their service towards disabled travellers is better than the service provided by smaller airlines (matters are even worse with the so-called low-cost airlines). For my trip to Vienna I chose to fly with KLM, the Royal Dutch Airline. To travellers in a wheelchair KLM provides ‘wheelchair assistance’, which you can simply ask for during the check-in at the counter (officially you need to book this in advance, but I never do that).
The lady who checked my passport and printed my boarding pass told me to report to the special ‘disability counter’ and there await being picked up for a wheelchair-push to the gate from which my flight was to depart. At the disability counter they told me to take a seat in their waiting room and wait for one of their colleagues. I looked around and I saw four rows of chairs most of which were empty, save for a somberly gazing elder couple and a Japanese girl with a rather blank stare. I sat down in, or rather stumbled into, one of the chairs in the row closest to me. The Japanese girl sat in the same row, with several empty chairs in between. I was about to start reading the book I had brought with me, when I noticed that the girl’s blank stare hadn’t changed direction since I first saw her. She was blind. While I was busy trying to imagine what it must be like to be blind I suddenly realized that I was staring at her. People often stare at me too. Especially when I ride my tricycle. I can understand why people stare. At my tricycle in the supermarket, naturally, but also at my tricycle in general, out on the street. I think people stare for the simple reason that they do not often see tricycles – let alone tricycles with adults on it. This curiosity for the outlandish, as I see it, has a significant advantage. When I meet people I don’t know they are often favourably inclined to having a conversation; they want to find out why I’m on a tricycle.
Blindness probably also induces the curiosity of strangers, but for blind people it is more difficult than for me to exploit this curiosity (if they even want to) because they can’t see the people with whom they could potentially have a nice conversation – life is not fair. I decided then and there that I would not let this particular girl be the victim of such ethical asymmetry. I decided to start a conversation with her. It turned out that she wasn’t as helpless as she seemed: she had been in the Netherlands to represent the Japanese Society for Blind Athletes. After learning about each other’s jobs and impairments and exchanging other pleasantries we decided to go for a coffee, as we had to wait another 30 minutes before one of the airport’s crewmembers would come to pick us up. I was a bit alarmed because I didn’t know how far we needed to walk to the nearest coffee bar. I had told Tomoe (I don’t know whether that was her first or last name; the other name was Takada) I could only walk very short distances, but perhaps she had misunderstood? Her English was not perfect, to put it mildly, so perhaps she didn’t even know I was disabled? I became even more alarmed when she asked me whether it was ok to hold my arm because she didn’t know the route to the bar. I told her that I would do my best and started lumbering out of the waiting room with the Japanese girl clasping my arm. I must admit that I felt a little proud at the situation; I didn’t feel like the handicapped one for a change. So we had coffee and our ways parted; hers back to Tokyo, and mine to Vienna. For a very brief period of time the half-spastic man was king in the land of the blind.