## Paraclimbing plans for 2017

2017 is going to be a great year for me. I’m getting married to the woman I love; in a month or two I hope to finish my PhD and, on top of that, there’s going to be a lot of climbing. Last September’s Worldchampionship in Paris was the first international climbing-competition that I participated in. I did better then I had expected two months before, so the competition really whetted my appetite. The next Worldchampionship will be in 2018 in Innsbruck (AUT). But I’m not going to wait that long!

## Worldcup

In 2017 the IFSC (International Federation for Sportsclimbing) organises a paraclimbing Worldcup: a series of climbing events at different locations. At the time of writing the IFSC  has yet to announce the locations and dates of the worldcup-events, but there are rumours about three of the locations. There will probably be paraclimbing events in Edinburgh (UK) and Sheffield (UK) sometime around September 2017 and another will be organised in Imst (AUT), but the precise date of the event in Imst hasn’t yet been made public. I plan to take part in all three events. Last year – before my joining the Dutch paraclimbing team – there was a paraclimbing event organised in Campitello di Fassa (IT). I’d really like to compete in Campitello di Fassa, so I hope that they’ll organise another event this year!

## Boston?

Another trip which I’m considering would take me to Boston (MA). On June 23rd the Brooklyn Boulders climbing community in Somerville will host the USA Climbing National championships. I would very much like to visit the city of the MIT and the Boston Tea Party. It would also be a great opportunity for me to get a feeling for different styles (formats) of competition: at the Worldchampionship in Paris I had to climb ‘on-sight’; in Boston they’ll be climbing ‘red-point’ (I’ll explain the difference in a later blogpost).

Because I don’t know the precise dates at which I’ll have to get the best out of myself (or even what counts as ‘the best’; because that depends on the competition-format) it’s difficult to make plans about training. It’s difficult to set short-term goals if you don’t yet know your long-term goals. That is why I’ll write about my goals and training-methods in a later blogpost. Don’t forget to subscribe to the updates on this blog! 😉

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## Bestaat Lege Ruimte?

Wat zou er gebeuren als uit de ruimte plotseling alle objecten zouden verdwijnen? Blijft er dan lege ruimte over, of is er helemaal niets meer? Met andere woorden: is de lege ruimte zelf ook een soort ‘object’? Continue reading

## 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

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## 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

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## 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

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## 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.

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## 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.

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