Probability, Mathematics and Philosophy
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Does God exist? Is my Penis smaller than average? Does Science have rigorous foundations? Is the universe deterministic? Let’s talk about the foundations of probabilistic reasoning. Oh, and some Wittgenstein.
Motivation
Why even think about this? Why not just sit back and let the boffins do what they do?
Because, even the boffins don’t know what they’re doing.
Here’s a few examples to whet your appetite.
- Fine-tuning argument for God.
This argument says that physical constants are so well aligned that God is the most reasonable cause.
This statement is probabilistic. Let’s dissect this. It says, were God not to exist then the probability of us existing would be small. Well, for starters we run into the problem that maybe the statement ‘were God not to exist’ or ‘were God to exist’ is completely meaningless. After all, if all the universe does depend on some omnipresent all-powerful deity, then the counterfactual where God doesn’t exist doesn’t make much sense. On the flip side, if the concept of all-powerful etc are just words we’ve attached some vague nonsensical meaning to, I may as well say ‘the universe is finely tuned, so if there isn’t a floogidoop with magical powers to make 1=2 then it would be unlikely’. So it’s a nonstarter. In contrast, there’s something, say, about a dice roll, where supposing there hadn’t been a 3 rolled seems very plausible.
2. What is the probability I have a smaller-than-average Penis?
This question supposes the counterfactual that, were you to measure every man’s penis then you could compare the sum divided by the number and finally compare it to your own size.
But now we are in the same position as the God question. After all, the average penis size exists (although this is an interesting bit of abstract reasoning — the number just exists out there, before being measured??). So surely the probability is 1 or 0?
But in another sense this question isn’t meaningless. After all, if I had to place a bet on the answer, I might have a pretty good idea what it was (okay — it’s 0.56mm shorter, if you must know), but wouldn’t be certain.
3. Is Science ‘correct’?
Imagine a world where you live in a room with an apple tree. You make the inference that apples fall to the ground. In fact, you go further and say that, in general, things fall to the ground. (Or you say that they probably do, or do with near certainty)
But one day I turn on the anti-gravity device hidden in the floorboards. Suddenly your theories are entirely wrong. While the correct theory was:
- I live in a room with anti gravity floor boards
- things fall to the ground provided the anti gravity floorboards aren’t activated
I could never have made a reasonable theory on this basis. I made a theory to cope with the relevant information, but one day it failed me, potentially with disastrous consequences (maybe I bumped my head?)
4. Is the universe deterministic?
Suppose we have a probabilistic world, where different things are possible. But we only observe one outcome. Thus we do not know we are not in a deterministic world.
Counterfactuals and imagination
Now, most philosophy arguments on this subject are split along different lines to what I am about to argue.
I am going to argue that probability is a meaningless idea. We simply aren’t conceptually able to peek very deep into what reality is.
For something to be assigned a probability, we need to be able to conceive of the alternative. Whether or not that alternative is genuinely ‘possible’ is a fruitless quest. This isn’t only because, say, a dice roll considered random 100 years ago can now be predicted. More fundamentally, to define ‘true randomness’, people appeal to concepts like ‘other possibilities’ (when we observe one outcome), while determinism says only one thing is possible (when I can imagine other things and but only experience one thing). Do you see how circular this is?
Our brains have a model of the world. From this, probabilistic statements are actually a coping mechanism, or mathematically a function or decision rule, which turns limited information into a decision. Given our lack of information, it is our conceiving of other possibilities which helps us avoid getting stuck in a bad spot. And this, generally speaking, has worked enough. Even if it isn’t rigorous. (Because it can’t be rigorous)
Thus, the child can conceive of the chance ‘Santa’ brings them more presents (in the parent’s eye: impossible), the parent thinks of alternative ways a dice can roll (in a classical physicist’s eye: impossible), and quantum physicists get very confused as to what the results of their experiments mean (currently they seem to think that quantum physics means that systems must be everywhere deterministic or nowhere deterministic, although no one is really sure what either term means).
Other worlds branching off in alternative dimensions, my house having anti-gravity floorboards, ambiguity over what God means and whether he exists, these are all possibilities I will have to live with … although perhaps ambiguity in some areas is preferable.
Oh, and Wittgenstein. (Philosophical Health Warning!)
P.S. If you were wondering, this links into why philosophers are basically all skeptics who aren’t sure if their own toes exist. Because we can’t define counterfactuals! Wittgenstein argued that many things Philosophers said were valid thought experiments were actually meaningless. But he too got in a muddle. Maybe me theorizing about myself being tormented by an evil demon is meaningless. But what about that fly whose wings are being torn off by a sadistic child? Maybe it cannot express a though logically, but that doesn’t stop the truth of it being tortured by some sadistic child.
