# Need guidance on Proof theory and Model theory

1. Sep 18, 2014

### unko

Hi, I'm planning to start my Master programme this december. I am very interested in philosophy of logic. And i am inclined to read more proof theory rather than model theory. And i really wanted to engage in proof theory as my first reading materials towards mathematical logic area.

Unfortunately none of my professors in my university are in mathematical logic's sphere. But one of my professor agreed supervising me and his current interest is on Group theory. His suggestion was to make use Proof theory as a basis to study Algebra for my dissertation. I never went further into Model theory but i'm guessing Model theory is closer to Algebra than Proof theory.

So, i'm wondering if i can just continue my reading on Proof theory, hoping that it can somehow help me with my dissertation later on Logic in Algebra. Because i really wanted to finish my reading on Proof Theory (Takeuti) and Handbook of Proof theory (Samuel R. Buss) before this December

If Model theory is really indispensable(is it?) in writing thesis for logic in algebra, and since Proof theory and Model theory shared many basic theorems, then can i just read Proof theory, and when i finished reading it, i can easily understand Model theory later for my thesis in Algebra. Can i ?

2. Sep 19, 2014

### verty

I think model theory will be indispensable. Logic, for example, didn't change much since Aristotle and Chryssipus, until Frege with his strange symbols (see Begriffsschrift). Then Russell invented his symbols $\forall, \exists$ and all that. And 30 years later you had GĂ¶del's theorems. So I think the general means of progress in these areas is to create an algebraic system of symbols and then go much further with the analysis.

You can also look at set theory as a semantics of logic. A set is just the extension of a predicate, etc, etc. This led to a total reworking of mathematics and no one can say that it hasn't been better since. Even Descartes with his cartesian plane, equations as names for curves, led to incredible progress, surely the whole of analysis for example.

Calculus is similar, Leibniz's symbols are still with us and I think no one can say they haven't made rates of change problems very easy to handle.

Perhaps symbols and algebra are not the same thing but they go together, they are surely indispensable. This reminds me of something, I recall reading that after Russell and Whitehead wrote the Principia Mathematica, there was a tendency, till around the 40's or perhaps 50's, to reduce everything to symbols, for example genetics. Supposedly that trend died away but that is the mindset you want to have, I think. Otherwise, you'll be adding to the verbiage that no one is ever going to read a few years later.

3. Sep 19, 2014

### unko

My interest is solely based in the area of philosophy of logic or metamathematics or the history of them. And i would rather go to a philosophical department but for example in japanese universities almost all of the logicians are in the computering departments, and only around 2 professors on logic are in mathematical departments, and none of the logicians here are residing in any philosophical departments. Most of them here are in the area on ethics or metaphysics.

And since i really do interested in the rigorousity of mathematics in logic, i really do try my best to stick on studying mathematics in logic and not to engage completely in the study of logic in mathematics per se. So I thought starting my master and phd programme in studying logic in mathematical department would be nice. So in future, when i finished my phd, that might help me as a ticket to go further to postdoc schorlarship in philosophical logic in other countries.

As from what i know that model theory is the study of logic in mathematics instead of studying mathematics in logic, and proof theory is more likely applicable in philosophy.

so if you said model theory is really indespensable for algebra, does studying further in proof theory (atleast for this 3 months before i start master programme) can help me understand model theory easily when i start my master programme? or it doesnt really help and, reading rigoroualy on proof theory for 3 months is a waste of time?

ps: Im not sure if my choice of word is correct, but what i meant by "mathematics in logic" is like studying logic using mathematics and vice versa.

Last edited: Sep 19, 2014
4. Sep 21, 2014

### verty

What people like Aristotle did was to take thought and argumentation and try to write it down. They wanted a language to describe convincing arguments. When does an argument convince and when does it fail to commute? What they in fact did was to create an algebra. At least, it seems that way to me. Logic is all about algebras, is it not?

You could argue that this is a philosophical position, that whether it is or isn't about algebras can't be solved because it is just a point of view. But this is not really to the point: knowledge advanced when symbols and algebras were used. This is reason enough, you must use them.

I know that this isn't all that it is about, there is also how to interpret things. I'll use the example of plural quantification: "there are elephants that live in Asia". One can interpret this as a plural quantifier, not to mean there is an elephant but that there are myriad elephants. But see, I think this has to be turned into an algebra and symbols, and how could one do that? The idea that you can lose precision and somehow learn something is wrong a priori, is it not?

Or take possible world semantics, you can say that this is a way to interpret logical statements. Ok, but give a possible world a symbol, develop a model theory for it, this is the right approach. Whatever you can say in words, give it an ontology of symbols, this is something I very strongly believe in. To not do this is to repeat the mistakes of the past.

I think there is no option for you to but to do this mathematical approach. I've seen a few times before that people ask questions here, they want to study in the USA and have a masters in something like phenomenology. It is a warning sign because you just know there wasn't a lot of math in it. In fact, I'll give this quote:

Some ideas are just plain bad.

Do you know the story of the blind men and the elephant? They felt the elephant; one thought it was a snake, one thought it was a tree, etc. It was none of those things. This is what philosophy is about (if it is about anything), realising that it isn't a snake or a tree, realizing what it is and then being very precise to define it, the more precise the better because that is the whole point, to remove the uncertainty.

Now proof theory is like the snake theory, it professes to be about proofs but this is just one aspect of logic. Proofs work the way they do because of the algebraic structure of the logic. So that is what you want to study, not what is going to be superficial.

I want to make a point here that hopefully will be valuable to you. You can see this as choosing a foundationalist theory over a coherentist theory, coherentism being the idea that as long as a theory is coherent, it doesn't need to be founded in any sort of common sense. Given what I said about philosophy being about removing uncertainty, it is very clear than coherentism is wrong. And this right here is the point I want to make, you must choose a side. Looking at all sides is an overrated notion, it takes too much time and is not how you want to be thinking.

Your question was, is it a waste of time? Yes, it will waste your time to study rigorous proof theory. That said, everyone has some time to waste so if you are interested, have a look at it.