Can Mathematics Solve Philosophical Problems?

This post is not an attempt to answer the question of the subject, but merely to discuss related topics. The idea dates back to the time when I first learned what is the likelihood function in probability theory.

When one encounters a philosophical problem, it is often the case that the problem is stated clearly enough to cook up a paradox in your mind, but vaguely enough not to see any reasonable way to approach the solution. A famous example is the free will problem: whether or not people have an actual, true free will. The problem is (or seems to be) clearly stated: we all (think that we) know what is meant by free will and so it makes sense to ask whether it exists; also there are two opposite intuitions about the issue: on the other hand we do experience (something that we interpret as) free will, but on the other hand aren’t our brains just a highly deterministic physical system? But this question (stated this way) gives no clue on what is the correct way (if any) to approach the solution.

I often think that many such philosophical questions might be solved by practicing the specific science which deals with the concepts in question. For instance the aspect of free will probably belongs to cognitive science and psychology. (Some people think that it belongs to religion; fine with me.) Thinking that way, I believe that many (but not all) philosophical problems concerning mathematics, can be solved mathematically. Gödel’s theorems provide a good example: they have certainly deeply influenced the philosophy of mathematics, not to mention many other branches of philosophy.

I would like to discuss the so called Hume’s argument given by David Hume. As the free will problem, it is also a philosophical problem stated such that the paradox is evident. How do I know that if I now jump out of the window, then I will fall down? Maybe I will fly up to meet the gods? How do we know that the Sun will not turn into an elephant tomorrow? Or rhinoceros? How do we know that all the rhinoceroses in the world will not turn into elephants tomorrow? We could refer to our scientific theories, but how do we know that they (will) hold (tomorrow)?

I’ll give two reasons why the Hume’s argument is not worth taking seriously (in some sense) and one of the arguments is mathematical and other is philosophical (I guess). The “in some sense” above takes place because of course, Hume’s right. We cannot know for sure. But since we do not know for sure, we should attach probabilities to the various possible events, do not we? By Hume, no! Because the probability that the Sun turns into an elephant any second might be 0.999! It might be the case that we have just been unbelievably lucky in that the Sun is still the Sun! It is like flipping a coin: even if you get 10000 heads in a row, the probability of getting a tail on the next move is still one half, provided the coin is fair…wait!…how did we know that the coin is fair? Did we flip it before to measure that?

Fortunately, there is the concept of likelihood. Suppose you have a coin and you do not know whether it is fair or not, and you do not know what is the probability of getting tails or heads. After you get 10000 heads in a row without getting any tails, you can ask: what is the likelihood that the probability of tails is 0.5? Whatever it is, the likelihood must be less than the same for 0.25 and so on.

Mathematically, writing X for the set of observed data and Θ for the set of parameter values, the expression P(X | Θ), the probability of X given Θ, can be interpreted as the expression L(Θ | X), the likelihood of Θ given X. The interpretation of L(Θ | X) as a function of Θ is especially obvious when X is fixed and Θ is allowed to vary. (I copypasted this paragraph from wikipedia, forgive me.)

When I learned about this, I immediately saw a solution to Hume’s argument. Thus, as we observe the Sun rising every day, the likelihood for the “parameters” (the real and possibly absurd laws of nature) to be such that the probability of the Sun rising is 1 is approaching 1. Similarly the likelihood for it to turn into an elephant with probability 1/2 approaches zero.

The difference of this to the response “the probability of sun rising is approaching 1” is that we cannot know the “real” probability, but we can calculate the likelihoods.

For the sake of completeness I would also like to give the obvious philosophical (pragmatic) response to the Hume’s argument. Suppose the world is a totally arbitrary place and tomorrow something very weird could happen. Thus there are two options: either everything will be as we normally predict, or not. So we would like to prepare ourselves to both events. Next we notice that we can be prepared for the “normal tomorrow”, but we cannot be prepared for the “absurd tomorrow”, simply because we have no idea what it could be. So the only reasonable thing to do is to prepare ourselves for the “normal tomorrow” and hope that it will happen. Thus, believe we in the laws of physics (or whatever) or not, we should act as we believed them to be true. Sounds like a pragmatists argument, that is why I put “pragmatic” into brackets above. But I claim further that it is not only a pragmatic argument (that’s why brackets!). Namely if you see yourself acting according to certain believes, shouldn’t you consider believing them already?

Probably the following argument is fallacial, but you can see what I mean. If you are say a platonist and do not want to base your believes on “what is the most rational way to act” -paradigm, you can use Occam’s razor to infer that “if we know anything to be true at all, then we know the laws of physics are not a matter of chance”, whatever “true” means for you. Then apply Occam’s razor and infer “We do know anything at all” and then apply modus ponens to infer that the laws of physics are not a matter of chance…

I still prefer the mathematical argument, although it says the same, I think.

About Vadim Kulikov

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2 Responses to Can Mathematics Solve Philosophical Problems?

  1. Kalle Timperi says:

    Hello :)

    I felt like commenting on your probabilistic attack against Hume’s argument.

    The likelihood approach seems very intuitive and ingenious at first sight, but it has an inherent methodological problem, which is: how to specify the probability distribution, based on which we perform our likelihood calculations? In order to calculate any kind of likelihoods you obviously need to have the probabilistic parameter that you want to calculate the likelihood for. This parameter pertains to a particular probability distribution or, in practice, to a certain random variable, which obeys that distribution.

    So, it makes a difference whether we think that ever since the birth of our Sun the gods of chance have flipped a coin every time planet Earth rotates once around its axis (which would correspond obviously to your coin tossing analogy) to decide if the transformation of Sun should take place, or whether the probability of the transformation follows for example some continuous probability distribution, in which case it might be possible that the density function of this probability still has a value significantly different from zero, even though an enormous amount of time has elapsed since the birth of our solar system. This could then translate into a reasonably “big” probability of something funky happening during the following year, for example.

    It is true that in any model that reasonably depicts the reality as we see it, the likelihood for “big” probabilities of something funky happening does approach zero, but without any a priori assumptions about the distribution we cannot really judge how small the probability for an unexpected event during, let’s say the next 24 hours, actually is.

  2. zenBen says:

    Without meaning to derail the conversation on probabilities, it is incumbent on me to go back to “…the aspect of free will probably belongs to cognitive science and psychology.”

    Philosophical questions dealing with ourselves rely a great deal on introspection. The brain, introspectively cognisizing itself. The issue is that nosce te ipsum has only very rarely delivered accurate results, by the current measures of what we know of the workings of the bio-mind. Plato effectively diverted a great deal of the subsequenct millenia of philosophy, IMO, from an approach that would render some truth.

    The possibility of free will in general is maybe a question of physics – do quantum behaviours translate to the macro scale? Lately results seem to suggest that they can. The possibility of free will for people is a question tightly bound to how we actually ARE in the world, the how of thinking, emoting, reacting etc. From an accurate model of that, I could see the tools of mathematics begin to be applied to work on the free will question. And of course the tools will help in building that model. And of course you don’t have to start from a complete model…what is a complete model anyway? :)
    BUT, start from the wrong model, and where will you end up?
    As a computer scientist might say, GIGO :D

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