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    Lehrstuhl für Mathematik VIII - Statistik

    Philosophical Questions & Answers

    In the following, I answer some questions that I have come across in my life - they either came up in discussions with students, other discussion partners, or I simply had to ask them myself. I answer these questions to the best of my knowledge and conscience, and with the - for such a Q & A appropriate - brevity. I should point out that I am neither particularly knowledgeable in philosophy, nor in physics, apart from some university courses in physics and personal reading. My answers are based on my knowledge, common sense, logic, and - maybe most of all - naivety. I make an effort to not contradict present science, but beliefs and assumptions are not avoided. If you find logical or scientific flaws, or if you simply want to help me improve my answers, send an e-mail to tom.fischer@uni-wuerzburg.de. Finally, note that this is an unfinished work in progress and that many (or all?) of the answers given below might have been given elsewhere by someone else before. This web page should by no means be mistaken for a scientific article (I also only refer to widely known scientific results whose sources are easy to find), but if new ideas should be contained in any of my writings below, then I herewith claim priority for them. Tom Fischer

    Contents:
    On probability

    On probability

    It is my impression that the philosophical meaning of probabilities sometimes is a source of confusion even for mathematically trained academics. Whether the following could help to ease the confusion, I cannot be sure about, but it helps me.

    Q: What is a probability?

    A: Probabilities are a concept in mathematical measure theory. They describe the content (weight, mass, volume, or area) of a mathematical set relative to a larger one in which it is contained.

    Q: Are probabilities real?

    A: Probabilities are as real or unreal as any mathematical object. For instance, is the number Pi = 3.14159265... real? (In the numeric sense, probabilities are of course real numbers.)

    Q: Do probabilities exist in physical reality?

    A: They exist in mathematical reality. However, physical reality and its mathematical description (= physics), that might utilize probabilities, are two different things. If Quantum Mechanics was an extremely good description of physical reality, then probabilities could have a serious claim to be considered as essentially real in the sense that they helped to supply a very well-functioning non-deterministic mathematical model of the world. However, for all we know, the world could be deterministic. No one can disprove this physically or logically. Perhaps an agnostic stance is the safest.

    Q: Do you believe in a deterministic world?

    A: Not at all, but that is something for another question below.

    Q: What about the probabilities to win the lottery, or to get six eyes when casting a die? They seem to work, they seem to be correct!

    A: In reality, one can never directly observe a probability, but only a relative frequency, meaning the percentage of the occurrance of a certain outcome or event when an experiment or game is repeated a finite number of times. In physics, probabilities are used to mathematically model occurances for which a seemingly stable relative frequency is observed in a series of experiments that are executed as independently of one another as possible. Lotteries, coin tosses or casting dice are examples of such experiments. The next two answers will explain in more detail how and why these models work.

    Q: What is the connection between relative frequencies and probabilities?

    A: Kolmogorov's Strong Law of Large Numbers. This theorem of mathematical probability theory (= stochastics) states that in a mathematical model of an experiment in which a certain event has a certain probability, the relative frequency of this event will always mathematically converge to its probability when the consecutively considered experiments are assumed to be independent. Note that always, relative frequency, and independent have a precise mathematical meaning not further explained here.

    Q: What is the connection between theoretical model and real world?

    A: The connection with the real world is that relative frequencies can be observed and that mathematical independence can be replaced by physical independence in the sense that any influence of one particular instance of the experiment on any other instance in the series must be avoided as much as possible (they are isolated). So, some properties of the mathematical model and of the real experiment seemingly match. Under these conditions, it therefore seems to be justified to model an event whose relative frequency seemingly converges to a limit by means of a probability (with the hypothetical limit being taken as its probability). Anyone who has ever thrown a fair die for, say, 10,000 times and recorded the occurrance of the event 6 eyes might therefore agree with assigning this event a probability of 1/6 = 0.16666... However, while the observed frequencies are real, the assumed probability is just that: a mathematical model assumption. I am happy to be called an agnostic frequentist for that answer, and - at the risk of contradicting Glenn Shafer - no, I do not think that frequentism necessarily is naive.

    Q: So, relative frequencies and probabilities are not the same?

    A: Correct. As explained earlier, probabilities are essentially used to describe the weight or size of a mathematical set relative to a larger one in which it is contained. As such, they are also used to mathematically model relative frequencies which are observed in the real world. Colloquially, in every day life, the two often get confused, as one might say that the "probability" of getting a "head" in a coin toss is "50%" (every now and then I might be guilty of that, too). However, this is not a mathematical fact, and instead based on the popular knowledge that the observed relative frequencies for a head are approximately 50% for possibly most circulated coins.

    Q: What are the limitations of probabilistic models?

    A: Just take the notion of physical independence in the sense of total isolation. The execution of independent identical experiments is essentially impossible in reality as various forces (such as gravitation, electromagnetism etc.) and their aftereffects will spoil any potential setups. Probabilistic models of the real world can only be understood as idealizations, as is the case with essentially all physical theories.

    Q: What is the most important theorem in probability theory?

    A: It must be Kolmogorov's Strong Law of Large Numbers for the earlier explained reasons.

    Q: What is the most remarkable theorem in probability theory?

    A: It must be the Central Limit Theorem since, to say it with Götz Kersting's paraphrased words, "the normal (Gauss) distribution suddenly appears out of nowhere!"

    ...

    Expansion pending!

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