I am not historian enough to pronounce upon the relation of what is standardly called Occam's Razor to the writings of the 14th century William of Ockham. The different spellings of his name will serve as a reminder to be careful about reading contemporary concerns into the works of philosophers long dead. Setting aside historical concerns, Occam's Razor is standardly taken to be a principle of theoretical economy or parsimony that states:
OR. Do not multiply entities beyond necessity.
It is sometimes formulated in Latin: Entia non sunt multiplicanda praeter necessitatem. The principle is presumably to be interpreted qualitatively rather than quantitatively, thus:
OR*. Do not multiply TYPES of entity beyond necessity.
Thus it is not individual entities that are not to be multiplied, but types or kinds or categories of entity. To illustrate. Some criticized David Lewis' extreme modal realism on the ground that it proliferates concreta: there are not only all the actual concreta, on his view, there are all those merely possible ones as well. He responded quite plausibly to the proliferation charge by pointing out that the Razor applies to categories of entity, not individual entities, and that category-wise his ontology is sparse indeed.
'Multiply' is a picturesque way of saying posit. (Obviously, there are as many categories of entity as there are, and one cannot cause them to 'multiply.') And let's not forget the crucial qualification: beyond necessity. That means: beyond what is needed for purposes of adequate explanation of the data that are to be explained. Hence:
OR** Do not posit types of entity in excess of what is needed for purposes of explanation.
So the principle enjoins us to refrain from positing more types of entity than we need to explain the phenomena that need to be explained. It is obvious that (OR**) does not tell us to prefer theory T1 over theory T2 if T1 posits fewer types of entity than T2. What it tells us is to prefer T1 over T2 if T1 posits fewer types of entity AND accounts adequately for all the data. So there is a trade-off between positing and accounting.
It seems to me that the Razor as I have just described it ought to be in every philosopher's tool box. But how useful is it? Not very. For it tells us not to posit more than we need, but it does not tell us what we need. For example, do we need mathematical sets? Given Manny, Moe, and Jack, do we need to add to the ontological inventory the set {Manny, Moe, Jack}? It is not obvious that we do. But it is also not obvious that we don't. There are arguments on both sides which I won't go into now.
Here's the punch line: simply brandishing the Razor has no tendency to show that there are not such abstract objects as sets. That would be an abuse of the Razor. It would be the mistake of thinking that T1 is to be preferred to T2 solely on the ground that T1 posits fewer types of entity.
Note that I presupposed above that philosophy is an explanatory enterprise. Is that obvious? As Hilary Putnam says somewhere, "It ain't obvious what's obvious."
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