Paraclimbing World championship – Paris, 2016

UPDATE: I ended fourth in my category. That’s less than I had hoped for, but a lot more than I had expected (I was the only Dutch finalist). The next worldchampionship (Innsbruck, 2018) I’ll do better! Continue reading

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Does Science Describe Reality?

In the philosophy of science there is a debate about whether scientific theories tell us what the world is really like, or whether scientific theories are nothing more than ‘tools’ or ‘instruments’ – useful for making predictions, but not for telling us what the world is really like. This debate is called the realism-debate. Most working scientists are realists: they believe that scientific theories tell us what the world is really like. They argue that realism is the only philosophy of science that can explain the success (in terms of the accuracy of predictions) of science. For example, they argue that “the theory of atoms allows us to predict that a gas expands when it is heated; wouldn’t that be a mystery if atoms did’t exist?” Continue reading

This Is What Climbing With A Hemiparesis Looks Like

There remains a lot to be learned… 😉

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Olympic paraclimbing

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

What Is A Dimension?

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

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Conference Vienna – lunchtime!

[disability as a networking skill] Continue reading

How Natural Is The Natural Logarithm?

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 $e$) is a characteristic of the definition of derivative – and of any logarithm to which this definition can be applied.

Wherever there’s derivation, there’s $\bold{e}$

The natural logarithm is a logarithm with as its base the mathematical constant $e$. $e$ is a number such that the function $e^t$ for every $t$ has a value that is equal to its rate of growth $[e^t=\frac{de^t}{dt}]$. The existence of a number with that characteristic follows logically from our definition of derivative.

The derivative of a function $f(t)$ is defined as follows:

$\frac{df(t)}{dt}=\lim_{h \to0}\frac{f(t+h)-f(t)}{h}$.

In mathematics all exponential functions are of the form $f(t)=ca^t$  Therefore, if $f(t)$ is an exponential function then the following holds:

$\frac{df(t)}{dt}=\lim_{h\to0}\frac{ca^{t+h}-ca^t}{h}$

We can move $ca^t$ outside the limit so that the limit no longer depends on the variable $t$ (and it is therefore a constant in $f(t)$):

$\frac{dca^t}{dt}=\lim_{h\to0}\frac{ca^{t+h}-ca^t}{h}=ca^t[\lim_{h\to0}\frac{a^{h}-1}{h}]$.

Let’s take a closer look at the latter limit. In it there are the two functions $f_1(h)=a^h-1$ and $f_2(h)=h$. We may calculate the value of the limit with the help of l’Hôpital’s rule:  by determining the derivatives of $f_1$ and $f_2$ and computing $\frac{f_1'}{f_2'}(0)$. Determining $f_2'$ is easy; $f_2'=\frac{dh}{dh}=1$. The value of $f_1'$ is less straightforward, because it depends on the value of $a$. If $a$ in $f_1$ has the value $e$ then $\frac{f_1'}{f_2'}(0)=\frac{e^h}{1}(0)=1$ (in the figure you can see that for $h=0$ $f_1$ and $f_2$ have both the same value and the same slope; $f_1'$ and $f_2'$ also have the same value). Returning with our findings about this limit to the equation for $\frac{da^t}{dt}$ we see that the only exponential function for which $\frac{df(t)}{dt}=f(t)$ is the exponential with base $e$; $\frac{dca^t}{dt}=ca^t$ only if $a=e$.

Every exponential can be expressed as a natural exponential

Consider again the general form of the exponential function, $f(x)=ca^x$. There are two ways in which we can transform any function of the form $ca^x$ into a natural exponential function.

1. Firstly, we can choose a suitable variable transformation ($t \rightarrow x$, where $x=\log_a e^t$) so that we may write $f(x)=ca^x$ as $f(t)=ce^t$.
2. Alternatively, if we don’t want to adjust the variable, we may rewrite $ca^x$ as $ce^{c_1x}$ (for $a=e^{c_1}$) and then let $e^{c_1}$ merge with $c$ into some other constant $c_2$ so that we end up with $f(x)=c_2e^x$

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, $t$, in seconds and formulates the function $n(t)=ca^t$ to describe the number of bacteria. If $a\neq e$ then n(t) is cumbersome to work with mathematically. If Fleur could rewrite $n(t)$ in terms of a natural logarithm then any differential equation with $n(t)$ in it would become much easier to solve.

Suppose Fleur indeed wants to rewrite $n(t)$ in terms of a natural logarithm. She has measured the number of bacteria at $t=0$ and found it to be $n(0)=c$. It would be a bad idea for Fleur to use the second method (2) of naturalising $n(t)$ because that would change $n(0)$ and then $n(t)$ 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 $n(t)$ doubles every 2.5 seconds. Then she might try substituting the variable $t$ for a variable $x$ for which each unit-step is not 1 second but 2.5 seconds ($x=\frac{1}{2.5}t$). The function $n(t)$ can now be rewritten into a function $n(x)$ which has a natural exponential form: $n(x)=ce^x$. 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 $e$ is a characteristic of the definition of derivative and that any exponential function can be written as a natural exponential might suggest that $e$ is a characteristic of our number system and not of nature. Such a conclusion, however, disregards a distinguishing feature of logarithmic functions.

But not every phenomenon can be described by an exponential

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 $e$), 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.

Kant & Modern Physics

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?

Immanuel Kant (1724-1804)

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).

A normal distribution, which is symmetrical around some average value$\mu$.

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.

Hans Reichenbach (1891-1953)

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.

Posted in Philosophy of Physics, Probability | | 1 Comment

Probability “0” Is Not Impossibility

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 $\frac{1}{\infty}$ is still $\frac{1}{\infty}$ – 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:

1. How do we know that equal areas have equal probabilities?
2. There are infinitely many ways to subdivide an area. How do we know which one is the correct one?

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.

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Conference Vienna, part II – customs, Kant and quantum mechanics

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.