Is Occam’s razor about preferring the simpler theory or the theory with fewer unexplained assertions?

Question

Yes, I know that simplicity and complexity are terms that have different meanings to different folks, but so is basically anything in philosophy. Most people will consider God creating the world as more simple than God 1 creating God 2 who then creates the world for example.

Moving that aside, if we have two theories that supposedly explain the same thing, is it ever justified to pick the one that has more unexplained assumptions? And if so, is this the same as the theory being more complex?

There are of course theories that have been tested for its truth that seem more complex than other theories that were simpler. However, arguably, they were accepted since there were evidence for those theories, not because they were more complex.

Once you have evidence for a theory, no matter how seemingly complex, it is not an unneeded assumption or postulate anymore.

Answer 1

The key point is that you need to take into account the plausibility of the assumptions, not just the number of them. A theory with a dozen plausible assumptions can be preferable to a theory with a single incredible assumption. You should be able to see why this is the case if you consider probability theory. The probability of n assumptions all being true is the product of their individual probabilities, so every time you add a new assumption the overall probability goes down. In that sense, you can increase the probability if you can eliminate extraneous assumptions. However, the overall probability depends on the probabilities of the individual assumptions as well as the number of them, so a theory with fewer assumptions is not necessarily better than a theory with more.

Answer 2

The principle named “Occam’s razor” is about two theories which ceteris paribus (= other things equal) differ only that one introduces less additional concepts than the other. That’s all.

Note that Occam’s razor is not an elaborated theory, but just a heuristic principle propagated by Occam and by other philosophers before and after him. One of the later philosopher coined the formulation:

Non sunt multiplicanda entia sine necessitate (Entities are not to be multiplied without necessity).

Answer 3

Is Occam’s razor about preferring the simpler theory or the theory with less unexplained assertions?

There is more to the strength of an explanation than the number of ingredients.

Situationally it can come into play, the properties and characteristics of the proposed ingredients.

1000 philosophers witness 100 consecutive demonstrations by 100 magicians, each in turn sawing a lady in half, then re-assembling her all with no harm.

Asked to make suggestions as to how this feat may be being accomplished. The philosophers begin philosophizing...

• "Magic" says one.

1 ingredient. Simple as simple can be.

But there is no quality in the explanation. It is just a make-believe word and associated claim that "word" did it.

No demonstrations of the same act will prove "magic" to be an invalid explanation. Only if the magicians reveal their secret will there be knowledge.

Yet... "Magic" fails, completely on the "explains anything" scale.

• "Fake automated legs" suggests another group of philosophers.

A higher quality suggestion. More detail. Yet simple, plausible. Possible.

• "Mirrors" says another group. "Magicians love their mirrors".

Meh, say most others, that mirrors don't seem compelling. Still... more substance to the claim than "magic" which is simple, but lacking in substance, and therefore lacking in plausibility. Unless we can come up with a drawing as to how "mirrors" explains it, actually... "mirrors" seems weak, compared to "automated legs".

• "Two women" says a little kid in the back row.

All the philosophers turn to look at one another... Turns out, one of them couldn't get a babysitter. But hey, maybe the kid is right??

Bottom line, observation, simplicity, confirmation by prediction and repetition of a phenomenon... they can only take you so far.

Ockham's razor is often helpful... but... must be judiciously applied.

Simple is good. Simple and right is way better.

"The Devil is in the details" is one way of putting it that I have heard voiced often.

Answer 4

It prefers simpler theories that explains the exactly same phenomenon.

If you have evidences for one theory, that usually means other theories failed to predict the phenomenon in the exactly same situation. In this case, they can't be compared using Occam's razor, because they don't explain the exactly same phenomenon. The failed theories should be immediately rejected because they are simply wrong.

A quick patch is to make exceptions. That is, to add the unexplained phenomenons into the failed theories and say exactly how they are different. But the complete description of the failed theories with the new additions would quickly grow very big, surpassing the working theories even if they seem complicated. In reality, there could be unexplained phenomenons in every theory, say the exact values of the inherent randomness in the physical world. But they are not explained by any theory, so it doesn't change what should be prefered. Starting from here, people may imagine how a theory could be fixed in better ways and evaluate it based on more assumptions. So we feel they didn't directly reject everything with small flaws. But technically Occam's razor could only be applied after the fix.

You could say the complexity of a part of the theory is "explained" by some complicated phenomenon. But by what I said, you should be compressing the description of the phenomenon using the theory, instead of compressing the description of the theory using the phenomenon. Actually it doesn't matter much, as they are both increasing I(X;Y) if we consider H(X) and H(Y) their Kolmogorov complexity. But compressing the description of the phenomenon is easier to think, if we consider technically we should add the complete description of all randomness into the theory to evaluate it, but we don't, and we compress a part of the randomness in different situations as approximation. Doing the opposite would either create duplicated work, to describe the theory in too many different ways, and make it difficult to say which ones are the same theory to allow us to compare between theories, or leave irregularities that the relations between the theory and reality are in different directions for the first set of evidences and other evidences.

Another answer says: "You need to take into account the plausibility of the assumptions." This is wrong. Occam's razor means you could measure the plausibility of assumptions, if not contradicting with reality, exclusively by the length of the description of the assumptions. If you have considered other means of plausibility, you might be using Occam's razor together with something else. It could be better or worse, but that part isn't from Occam's razor itself.

So, "there is a god" is indeed a simpler expression than the physical theories by Occam's razor, even if you consider it implausible. But it doesn't say what the god is like and what does the god do to the world, and you still have to add the complete description of a physical theory to make it able to explain things. "There is a god" plus the physical theory together would be more complex than the physical theory alone.

Answer 5

...if we have two theories that supposedly explain the same thing, is it ever justified to pick the one that has more unexplained assumptions?

I want to point out first that the answer to the question is "yes." Occam's razor is not the only aspect of justification. It's just one of many tools. But you are clearly focusing on cases where Occam's razor applies, so I will focus there as well.

I think one very valid case where one might accept a more complex solution with more unexplained elements is flexibility. In an "ideal" world, theories cost nothing to produce or discard. We can always run with one solution and immediately switch over to another when it suits us better. In practice, human beings are more rigid than that. It can be very difficult to toss away an old theory, even if the new theory is simpler.

In the presence of the unexplained, we recognize that there may be more evidence out there than we have collected so far. We may prefer a more complicated theory due to a belief that it is more flexible to accommodate new information.

I think String Theory may be an interesting place to investigate this pattern. String theory introduces an astonishing array of initial states which makes it very flexible (something like 10^500 states). Some consider this to be a virtue -- flexibility as we learn more about our world.

Of course, Occam's razor is focused on "all else being equal." One may argue this is an unequal thing. Occam's razor is, after all, nothing more than a principle. It is indeed easy to argue that there are no two theories in most topics that are perfectly equal in every way except complexity.