Preamble In my last post, I discussed some basic logical connectives. Before moving on to more advanced topics, I want to discuss formal syntax. I think many people will find this post dry and boring, but I think it's neccessary to cover at some poin Up till this point, I've...
Preamble In my last post I gave the philosophical prerequisites needed to understand my approach to logic and mathematics. Before getting into the subject at hand, here's a clarifying remark inspired by some comments on my last post. When I say that mathematics is a social activity, I mean it's...
Preamble I've wanted to write a series of posts here on logic and the foundations of mathematics for a while now. There's been some recent discussion about the ontology of numbers and the existence of mathematical entities, so this seems as good a time as any to start. Many of...
In my last post, I discussed some basic logical connectives. Before moving on to more advanced topics, I want to discuss formal syntax. I think many people will find this post dry and boring, but I think it's neccessary to cover at some poin
Up till this point, I've been assuming an understanding which bears dwelling on. I've previously stated that a judgment is a series of mental tokens which one may declare. While I've technically adhered to that, it's obvious that I'm assuming more structure. Furthermore, I've made usage of variable binding and substitution, which have not been properly accounted for.
This short post is intended to specify the nature of syntax as... (read 3517 more words →)
In my last post I gave the philosophical prerequisites needed to understand my approach to logic and mathematics. Before getting into the subject at hand, here's a clarifying remark inspired by some comments on my last post. When I say that mathematics is a social activity, I mean it's an activity that is social. Politics is also a social activity. This means that saying, for example, that mathematics is incomplete or inconsistent makes as much sense as saying that politics is incomplete or inconsistent (in the same way a formal logic might be). It's very common for someone to make mention of mathematics having some property (e.g. Gödelian incompleteness) when, in fact,... (read 5330 more words →)
I've wanted to write a series of posts here on logic and the foundations of mathematics for a while now. There's been some recent discussion about the ontology of numbers and the existence of mathematical entities, so this seems as good a time as any to start.
Many of the discussed philosophical problems, as far as I can tell, stem from the assumption of formalism. That is, many people seem to think that mathematics is, at some level, a formal logic, or at least that the activity of mathematics has to be founded on some formal logic, especially a classical one. Beyond that being an untenable position since Gödel's Incompleteness Theorems, it also... (read 4891 more words →)
My goal is mostly to collect my understanding of mathematics into some coherent collection that people can read. Mostly, I just want to expose people to an alternative way of thinking, as I've seen people from this community say things that rely on a narrow view of foundations. My plan for the series is not to be especially philosophical. The first post consists only of things which I think are obvious. As far as I can tell, the main reason people thought it might have been controversial is that they didn't understand what I was getting at in the first place.
The content of this post is actually the meat of my series.... (read more)
"1 What is "formalism"? "
"I think it isn't, because positions like F1 and F2 involve such assumptions and aren't made untenable by the incompleteness theorems "
I think it's ironic that you're arguing with me over the meaning of a word, considering the content of my essay. I stated at the begining of my essay what I meant by "formalism". If you don't think that word should be used that way, that's fine, but I'm not interested in arguing about the meaning of a word. By pretending that I'm arguing against any and all forms of what may be called formalism, you are replacing what I actually said with something else. That's not... (read 388 more words →)
No. I'm not advocating for some sort of finitism, nor was Brouwer. In fact, I didn't actually mention computability, that's just something gjm brought up. It's irrelevant to my point. Mathematics is a social activity in the same way politics is a social activity. As in, it's an activity which is social, or at least predicated on some sort of society. Saying that mathematics is incomplete is as meaningful as saying that politics is incomplete in the same way a formal logic might be. It just doesn't make sense.
Note that the intuitions which justify the usage of a particular axiom is not part of an axiom system, but those intuitions would still be part of mathematics. That's largely the intuitionist critique of "old" formalism. It was also used as a critique of logical positivism by Gödel.
"If what the universe does is computable then there is no way the whole community of mathematicianscan do mathematics that avoids both inconsistency and completeness. "
I don't think you understand what I'm getting at. It's not that completeness shouldn't be worried about, it's that it doesn't make sense if you aren't already assuming that mathematics is a formal logic. If you worry about formal logic then you worry about completeness. If you don't assume that mathematics is a formal logic, then worrying about mathematics does not lead one to consider completeness of mathematics in the first place. I'm saying that it does not make sense to talk about mathematics being complete or... (read 358 more words →)
See this SEP section.
"if what our brains can do is computable then there is no way we can do mathematics that avoids both inconsistency and incompleteness."
This sentence illustrates the formalist essentialism that I'm criticizing. If we consider mathematics as a social activity, as Brouwer did, then the notion of completeness doesn't come up in the first place, and it's useless to worry about such a thing. This perspective, in part, influenced Gödel to make his discoveries in the first place.
Much of the point of Hilbert's program (and the wider goal of formalism/logicism) was to prove mathematics in entirety consistent by providing a formal logic which could be considered mathematics itself. Without that,... (read more)