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Larks comments on Open Thread: May 2010, Part 2 - Less Wrong
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Around the turn of the last century, the logicists, like Frege and Russell, attempted to reduce all of mathematics to logic; to prove that all mathematical truths were logical truths. However, the systems they used (provably) failed, because they were inconsistent.
Furthermore, it seems likely that any attempt at logicism must fail. Firstly, any system of standard mathematics requires the existence of an infinite number of numbers, but modern logic generally has very weak ontological commitments: they only require the existence of a single object. For mathematics to be purely logical, it must be tautological - true in every possible world*, and yet any system of arithmetic will be false in a world with a finite number of elements.
Secondly, both attempts to treat numbers as objects (Frege) or concepts/classes (Russell) have problems. Frege’s awful arguments for numbers being objects notwithstanding, he has trouble with the Julius Caesar Objection; he can’t show that the number four isn’t Julius Caesar, because what this (abstract) object is is quite under-defined. Using classes for numbers might be worse; on both their systems, classes form a strict hierarchy, with a nth level classes falling under (n+1)th classes, and no other. Numbers are defined as being the concept which has all those concept’s whose elements are equinumerous; the class of all pairs, the class of all triples, etc. But because of the stratification, the class of all pairs of objects is different from the class of all pairs of first level classes, which is different from the class of all pairs of second level classes, and so on. As such, you have an infinite number of ‘2’s, with no mathematical relations between them. Worse, you can’t count a set like {blue chair, red chair, truth, justice}, because it contains objects and concepts.
What seems more likely to me is that there are an infinite variety of mathematical structures, purely syntax without any semantic relevance to the physical world, and without ‘existence’ in any real sense, as a matter of induction we’ve realised that some can be interpreted in manners relevant to the external world. As evidence, consider the fact that different, mathematics are applicable in areas: probability theory here, complex integration here, addition here, geometry here...
*strictly speaking, true in every structure.
No, that's not right. Russell and Whitehead's Principia Mathematica is the fullest statement of logicism, and its system was never proved inconsistent.
Here I'm less certain, but I'm pretty sure that that's not right either. You would have relations among two such 2s, but those relations would be of a higher type than either 2. But, again, I'm definitely vaguer on how that would work.
Yes, sorry, I meant that Frege failed his system was inconsistent (though possibly not if you replace Basic Law 5 with Hume's Law). Russell, on the other hand, simply runs into Incompleteness; you can't prove all of mathematics from logic because you can't prove it full stop.
You'd have '2's of all cardinalities, so to have a relation between them, you would need to move into the uncountables - but then there are new pairs to be formed here... Essentially, you can reconstruct Russell's original paradox, comparing the cardinality of the set with the cardinality of certain things that fall under it.
You could mitigate this but cutting short the recursion, and simply allowing the relation to hold between the first n levels of concepts or so., on pain of arbitrariness.
I'm curious as to the downvotes; was I off-topic, too long, or simply wrong? Edit: And (if it's acceptable to ask about other people's downvotes) why was zero call downvoted?
That's not a problem for logicism per se. Logicism isn't really a claim about what it takes to prove mathematical claims. So it doesn't fail if you can't prove some mathematics by a certain means. Rather, logicism is a claim about what mathematical assertions mean. According to logicism, mathematical claims ultimately boil down to assertions about whether certain abstract relationships among predicates entail other abstract relationships among predicates, where this entailment holds completely regardless of the meaning of the predicates. That is, mathematical claims boil down to claims of pure logical entailment.
So, if you discover that your particular mathematical system is incomplete, then what you've really done is discover that you had missed some principles of logic. It's as though you'd known that P ∧ P entails P, but you just hadn't noticed that P ∨ P entails P as well.
(But you were right about why logicism ultimately failed to convince everyone: Mathematics seems to have ontological commitments, where pure logic does not.)
I didn't downvote either comment. Your comment was probably downvoted because some readers considered its arguments to be wrong or unclear. zero call's comment was probably downvoted because it smacks of the mind projection fallacy, especially here:
The organization of facts into axioms, rules of inference, proofs, and theorems doesn't seem to be an ontologically fundamental one. We superimpose this structure when we form mental models of things. That is, the logical structure of things exists in the map, not the territory.
I wish you would have made this last comment on the post directly, so that I could reply to that there. Anyways, the point I was offering was that the logical structure does exist in the territory, not just the map. Our maps are merely reflecting this property of the territory. The fundamental signature of this is the observation that physical systems, when viewed in a map which exists only as a re-representation or translation (as opposed to an interpretation) amenable to logical analysis, are shown to prohibit logical contradiction. (For example, the two statements (if A, then B) and (if A, then not B) cannot both be true, where A and B are statements in some re-representation of the physical system.)
I'll move that part of my comment there, with my apologies.
That's quite alright -- thank you for your discussion.
I appreciate your comments but I'm having trouble seeing your point with regards to the idea. To reiterate, with regards to your last paragraph,
I'm proposing that these interpretations work because the internal physical systems (the territory) obeys the same properties as consistent mathematical systems -- see my comment to TM below.
There is a great deal of difference between it operating, in certain regards, on the same sort of rules (rules isomorphic to) mathematics, and mathematics being applicable because physics isn't logically inconsistent. It's not a logical contradiction to say that two points have the same position, nor to say that 2+2=1 (for the latter, consider arithmetic modulo 3). Nor can maths be deduced purely from logic; partly because logic doesn't require the existence of more than one object.
Russell did try to deduce maths from logic plus some axioms about how the world worked - that there were an infinite number of things, etc., but the applicability of the maths is always going to be an empirical question.